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Page 1: SPOR Monograph Recent Advances in Improved Oil Recovery ...s-skj/CoNing/SPOR-monograph/spor-mono.pdf · SPOR Monograph Recent Advances in Improved Oil Recovery Methods for North Sea

SPOR Monograph

Recent Advances

in Improved Oil Recovery Methods

for North Sea Sandstone Reservoirs

Svein M. SkjævelandEditor

Professor of Reservoir Engineering atRogaland University Center.

Jon KleppeEditor

Professor of Reservoir Engineering atNorwegian Institute of Technology, University of Trondheim

First PrintingNorwegian Petroleum Directorate

Stavanger1992

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ISBN 827257340-7YA 753

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Authors

Idar Akervoll is manager ofthe reservoir technology sectionat IKU and was previously agroup leader and scientist atIKU and SINTEF for researchon gas- and chemical ooding.He holds an MS degree in chem-istry from Trondheim U.

Børre Antonsen is a seniorengineer with Norsk Hydroand previously worked forIFE. He holds an MS de-gree in mathematics fromOslo U.

Tor Austad is associate pro-fessor of petroleum engineeringat Rogaland U and consul-tant to Rogaland Research.He holds a PhD degree inphysical-organic chemistryfrom Bergen U. Austad is therecipient of the SPOR research

prize.

Tor Bjørnstad is head of thetracer technology departmentat IFE. Previously he was aresearch fellow at CERN inGeneva, assistant professor innuclear chemistry at Oslo Uand associate professor in radio-chemistry at Bergen U. He holds

a PhD degree in nuclear chemistry from Oslo U.

Per Arne Bjørkum is sectionmanager at the Geological Lab-oratories at Statoil. He was se-nior geologist at Esso and re-search supervisor in geology atRogaland Research. His maininterest is sandstone diagene-sis. He holds an MS degree

in petroleum geology from Bergen U. and a PhD-candidate in petroleum geology from Oslo U.

Vilgeir Dalen is a specialist inreservoir modelling with Statoil,and adjunct professor of reser-voir engineering at NTH. Heholds MS and PhD degrees incivil engineering from NTH. Hehas previously worked in reser-voir simulation research at IKU

and SINTEF.

Magnar Dale is assistant pro-fessor of mathematics at Roga-land U and consultant to Ro-galand Research. He holds adegree in music from BergenConservatory and a PhD de-gree in pure mathematics fromBergen U.

Anna Inger Eide is princi-pal engineer at NPD and wasprogram manager for SPOR inthe period 198591. She workedon reservoir engineering and im-proved recovery methods. Sheholds an MS degree in organicchemistry from Bergen U.

Steinar Ekrann is head ofresearch at Rogaland Researchand previously worked in ap-plied numerical modelling atIFE. He holds an MS degreein physics and a PhD degreein numerical mathematics fromNTH. His main interest is nu-

merical and analytical reservoir modelling.

Eli Eikje is a reservoirengineer with Saga whereshe works on IOR methods.She was a research engineerat Rogaland Research andholds BS and MS degrees inpetroleum engineering fromRogaland U.

iii

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Jón-Steinar Gudmundssonis a professor of petroleum en-gineering at NTH. His researchinterests are within productionengineering, reservoir engineer-ing, and natural gas engineer-ing. He was with the Geother-mal Division at the National

Energy Authority of Iceland and the Petroleum En-gineering Department at Stanford U. He has degreesin chemical engineering from Heriot-Watt U (BS) andBirmingham U (PhD).

Øistein Glasø is senior re-search engineer with IKU andcurrently engaged in research ongas miscible ooding. He holdsa BS degree in chemical engi-neering from Gothenburg U anda BS degree in economics fromTrondheim U.

Bjørn Gulbrandsen is a se-nior reservoir engineer withSaga. Previously he was withIFE performing mathematicalmodelling of industrial pro-cesses. He holds an MS degreein applied mathematics fromOslo U.

Jan Erik Hanssen is a co-ordinator of IOR research atRogalnd Research. He hasworked there and at RogalandU. mainly with foams and gas-based improved recovery meth-ods. He holds an MS degree inpetroleum chemistry from Oslo

U. and is a PhD candidate in petroleum engineeringunder the Aalbord U./Rogaland U. doctoral program.

Adolfo Henriquez is sectionmanager of the reservoir tech-nology section at Statoil. Hehas worked in development,maintenance and user supportof reservoir simulation tools,and in several eld develop-ment projects. His professional

interests are reservoir description, supercomputing,evaluation of uncertainty in production prognosesand improvements in reservoir modelling technology.He holds MS and PhD degrees in physics fromOslo U and a PhD in petroleum engineering fromRogaland U.

Leif Hinderaker is principalengineer at NPD, and was headof the reservoir technologysection. He holds an MS degreein mathematics from Bergen U.His work in NPD spans fromwaterooding of chalk reser-voirs to new eld developments

and IOR methods for sandstone reservoirs.

Odd Hjelmeland is managingdirector of ResLab and was asection manager in reservoirengineering at IKU. He holdsan MS degree in petroleumengineering from Leoben Uand a PhD degree in reservoirengineering from NTH. His

main interests are reservoir fundamentals and exper-imental techniques.

Lars Holden is research direc-tor at Norwegian ComputingCenter where he works withreservoir evaluation and inparticular the problem of scal-ing. He holds an MS degree innumerical analysis and a PhDdegree in mathematics from

Oslo U.

Arne Hole is manager ofpetroleum technology at Sta-toil. He holds an MS degreein mathematics and physicsfrom Oslo U and has 15 yearsof experience in reservoirengineering, drilling and pro-duction.

Torleif Holt is group managerat IKU's reservoir lab. Heholds a degree in chemicalengineering and a PhD degreein physical chemistry, bothfrom NTH.

iv

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Odd Steve Hustad is leaderof the reservoir simulationgroup at IKU. His main eldof activity is related to the de-velopment and use of reservoirsimulation tools. He holds a BSdegree in applied mathematicsfrom Waterloo U, an MS degree

in physics from Trondheim U and a PhD degree inpetroleum engineering from Trondheim U.

Stein Rune Jakobsen issenior consultant with Petec,working on reservoir evaluationand technology development.Previously he was with Ro-galand Research simulatingprocesses related to water-based enhanced oil recovery, in

addition to working on production chemistry andtechnology. He holds an MS degree in petroleumengineering from Rogaland U.

Torgrim Jacobsen is aresearch scientist in sedimen-tology at IKU. He holds an MSdegree in sedimentology fromOslo U. He worked as a researchassistant in petroleum geologyat NTH. His main interest isreservoir characterization, espe-

cially the integration of quantitative sedimentologicaldata in reservoir modelling.

Tom A. Jelmert is associateprofessor of reservoir engineer-ing at NTH. He holds a BSdegree in electrical engineeringfrom Purdue U and MS andPhD degrees in petroleum engi-neering from NTH. Previouslyhe was a research engineer with

SINTEF and taught mathematics and physics at theAcademy of the Royal Norwegian Air Force.

Thormod Johansen hassince 1981 been a researchscientist and a section leaderin reservoir modelling at IFEand is now a principal engineerat Hydro. He holds an MSdegree in mathematics fromOslo U.

Knut Jorde is geological advi-sor at Saga, working on reser-voir related topics. He has beenstate geologist with the Nor-wegian Geological Survey. Heholds a BS degree in chemistry,mathematics, and physics andan MS degree in geology from

Oslo U.

Jon Kleppe is a professor ofreservoir engineering at NTHand Dean of the School of EarthScience and Metallurgy. Hehas BS and MS degrees in me-chanical engineering from SouthDakota U and a PhD degreein petroleum engineering from

Texas A&M U. He is an advisory board member ofSPOR.

Jostein Kolnes is assistantprofessor of reservoir technol-ogy at Rogaland U and seniorresearcher with Rogaland Re-search. He holds an MS de-gree in organic chemistry fromBergen U. During the period1985-91 he was supervisor of

SPOR-WATER.

Torgeir Kydland is sectionmanager in the reservoir depart-ment at Hydro and previouslyworked with Statoil. He holdsan MS in degree in mathematicsand physics from Oslo U. He hascoauthored several publicationsregarding the use of horizontal

wells for thin oil zone reservoirs.

Arild Lohne is a research engi-neer at the EOR group at Roga-land Research. His main inter-ests are surfactant phase behav-ior and ooding, experimentsand simulation. He holds a BSdegree in environmental engi-neering from Rogaland U.

v

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Eva Ljosland is production en-gineer supervisor with Statoil.She has worked with reservoirevaluation and oshore data ac-quisition. Her interests are en-hanced oil recovery, tracer tech-nology and optimization of wellinow performance. She holds

an MS degree in physical chemistry from Bergen U.

Torgeir Lund is senior staengineer at the Statoil pro-duction laboratory where heworks on chemicals for IOR andpolymer/polymer-gel technol-ogy. He was a senior researchscientist at the Phillips U onnuclear reactions with semirela-

tivistic heavy ions. He holds an MS degree in nuclearchemistry from Oslo U.

Rune Mjøs is senior geologistat Statoil's exploration depart-ment and was a geologist atHydro and senior scientist atRogaland Research. He holdsan MS degree in petroleumgeology and sedimentologyfrom Bergen U.

Kari Grete Nordli Børve isassistant professor of chemistryat Bergen U. She holds a PhDdegree in physical chemistryfrom Bergen U. Her interestsare within colloid and surfacechemistry and include mi-croemulsions, phase behavior,

and the properties of thin lms containing macro-molecules.

Johan Petter Nystuen is ge-ological advisor with Saga. Heholds an MS degree in geologyand a PhD degree from Oslo U.He was assistant professor ingeology at the Agricultural Uand Oslo U and performedresearch in regional geology,

structural geology, sedimentology and reservoirgeology.

Bjørn V. Nystrand is areservoir engineer with theNorwegian Petroleum Direc-torate (NPD) with interestsin improved oil recovery. Heholds BS and MS degrees inpetroleum engineering fromRogaland U.

vi

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Edvard Prestholm is seniorscientist in geology at RogalandResearch. He holds an MSdegree in sedimentology fromBergen U. His main interestsare delta front sediments,early deformation of deltaicsequences, and thermal conduc-

tivity of sedimentary rocks.

Per Kristian Munkerud isresearch engineer with IKU.He holds an MS degree inengineering from NTH and isa PhD candidate in reservoirengineering.

Gudmund Olsen is a seniorengineer with Hydro. He holdsan MS degree from Stan-ford U.

Henning Omre is a statis-tician at the NorwegianComputing Center. He washead of the SAND-Group andholds a degree in statistics fromNTH and a PhD degree ingeostatistics from Stanford U.His main interests are spatial

statistics and reservoir characterization.

Sjur A. Rogde is section man-ager of the reservoir simulationsection in Statoil. He workedwith reservoir performanceevaluation, eld developmentplanning, prospect evaluation,and improved oil recovery. Hisinterests include improvements

in reservoir evaluation technology and reservoir dataintegration. He holds MS and PhD degrees fromBergen U.

Arne Skauge is head of re-covery processes department atNorsk Hydro Research Centre.He is an adjunct professor ofreservoir physics at Bergen U,where he received a degreein physical chemistry. He isan advisory board member of

SPOR.

Svein M. Skjæveland is aprofessor of petroleum engineer-ing at Rogaland U. He holdsPhD degrees in physics fromNTH, and in petroleum engi-neering from Texas A&M U. Hewas elected 1990 - Oil Man ofthe Year by the Stavanger SPE

Section and president of Rogaland U for 1992-94. Heis an advisory board member of SPOR.

Johan Sjøblom is a profes-sor of physical chemistry atBergen U. He holds a PhDdegree from Åbo Akademi andis docent at Helsinki U. His in-terests cover surface and colloidchemistry involving micelles,emulsions, microemulsions,

lyotropic liquid crystals, Langmuir-Blodgett lmsfrom a fundamental as well as applied point of view,and material science.

Arne Stavland is seniorresearch engineer at RogalandResearch. He has worked withenhanced oil recovery, mainlypolymer ooding and gel tech-nology. He holds an MS degreein physics from NTH.

Ron Steel is professor of reser-voir geology and sedimentologyat Bergen U and was chiefgeologist with Hydro. He isan advisory board member ofSPOR and a scientic advisorto IKU. He holds a PhD fromGlasgow U, and his current

research activity is focused on reservoir descriptionand on sequence stratigraphy.

vii

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Ole Torsæter is professor ofreservoir engineering at NTH.He holds MS and PhD degreesfrom NTH and has been aresearch engineer with SIN-TEF and Phillips, Oklahoma.During 1991-92 he is visitingprofessor at New Mexico Tech.

His research interests are experimental studies ofow in porous media.

Jann-Rune Ursin is assistantprofessor at Rogaland U wherehis elds of interests are inreservoir technology. He holdsa PhD degree from Bergen U.

Olav Vikane holds a PhDdegree in chemistry fromBergen U. He is advisor inchemical ooding at the Statoilproduction laboratory. He iscurrently engaged in projectson surfactant ooding and theapplication of gel technology

for water prole modication.

Kirsti Veggeland was re-search coordinator of IOR atRogaland Research. She holdsa degree in physical chemistryfrom Bergen U and has recentlystarted on a doctorate projectin surface chemistry studyingsurfactant and polymer interac-

tions.

Jan Ole Aasen is alead engineer at Sta-toil. He holds a de-gree in engineeringphysics and a PhDdegree in numeri-cal mathematics fromNTH.

viii

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Contents

I Preliminaries iii

Authors v

Preface xiiiThe SPOR Program, by A.I. Eide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiThe Monograph, by S.M. Skjæveland and J. Kleppe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Acknowledgements xv

1 Introduction by S.M. Skjæveland and J. Kleppe 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

II Fundamentals 3

III Fundamentals 5

2 Phenomena 72.1 Surface and Interfacial Tension, by J. Sjøblom and K.G. Nordli Børve . . . . . . . . . . . . . . . 7

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Adsorption at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Contact Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.5 Experimental Techniques for Measuring Tensions. . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Miscibility, by Ø. Glasø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Adsorption, by T. Austad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.3 Phase Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.4 Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Dispersion, by T. Holt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.2 Microscopic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Field Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Molecular Diusion, by J.S. Gudmundsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.3 Diusion Coecient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.5 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5.6 Diusion in Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Frontal Instabilities, by J.R. Ursin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.3 Hele-Shaw Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

ix

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2.6.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6.5 Theoretical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Equations 373.1 Conservation Laws, by T. Johansen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.3 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.4 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.5 Well-Posedness of the Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.1 Phase Behavior, by O.S. Hustad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.2 Rheology, by A. Stavland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Rock Properties, by O. Torsæter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.1 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.2 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Fluid-Rock Interaction, by O. Torsæter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4.1 Capillary pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4.2 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4.3 Three-Phase Models, by S.R. Jakobsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4.4 Hysteresis, by S.R. Jakobsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 Transport Equations, by O.S. Hustad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.5.1 Simplied Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5.2 Black Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5.3 Compositional Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.4 Chemical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Solutions 754.1 Shocks and Simple Waves, by T. Johansen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1.2 Riemann Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1.3 The Buckley-Leverett Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.1.4 Multicomponent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1.5 Polymer Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.6 Solvent Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.1.7 Surfactant Flooding. Type II(-) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.1.8 Thermal Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1.9 Three-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.1.10 Extensions of Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Cross-Sectional Displacement, by S. Ekrann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2.1 Displacement with Negligible Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2.2 Gravity-Dominated Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2.3 Field Examples, by G. Olsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

IV Reservoir Description 113

5 Geological Description 1155.1 Geological Models in Reservoir Description, by R. Steel . . . . . . . . . . . . . . . . . . . . . . . 115

5.1.1 Information Databases and Outcrop Analogues . . . . . . . . . . . . . . . . . . . . . . . . 1155.1.2 Depositional Environments and Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 1165.1.3 Base-Level Changes and Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Macroscopic Heterogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.2.1 Depositional Heterogeneity, by R. Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.2.2 Geometry and Heterogeneity in Deltaic Reservoirs, by R. Mjøs and E. Prestholm . . . . . 1195.2.3 Structural Heterogeneity, by R. Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2.4 Diagenetic Barriers, by P.A. Bjørkum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3 Microscopic Heterogeneties, by P.A. Bjørkum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.3.2 Kaolinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.3.3 Illite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

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5.3.4 Chlorite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.3.5 Nonclay Cements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.3.6 Secondary Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.3.7 Permeability and Petrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.3.8 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.3.9 Retention of Chemicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.4 Minipermeameter Use, by T. Jacobsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.4.1 The Electronic Field and Laboratory Permeameter . . . . . . . . . . . . . . . . . . . . . . 1355.4.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.4.3 Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6 Heterogenity Models 1436.1 Stochastic Models for Reservoir Characterization by H. Omre . . . . . . . . . . . . . . . . . . . . 143

6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1.2 Facies Architecture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.1.3 Petrophysical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.1.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.2 Field Examples, by K. Jorde, B. Gulbrandsen, and J.P. Nystuen . . . . . . . . . . . . . . . . . . 1516.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.2.2 Large-Scale Heterogeneity Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.2.3 Modelling of Heterogeneity Distribution Within Single Sandbodies . . . . . . . . . . . . . 1546.2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7 Tracer Testing 1597.1 Tracer Types, by T. Bjørnstad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.1.2 Tracers for Injected Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.1.3 Tracers for Injected Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.1.4 Future Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.2 Reservoir Characteristics, by B. Antonsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.2.2 Reservoir Heterogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.2.3 Residual Oil Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.3 Field ExamplesRadioactive Tracers, by S. Rogde and E. Ljosland . . . . . . . . . . . . . . . . . 1667.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.3.2 Monitoring the Gullfaks Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.3.3 Veslefrikk Tracer Implementation Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.3.4 Ekosk Field Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.3.5 Murchison Field application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8 Reservoir Modelling 1738.1 Review of Numerical Models, by V. Dalen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.1.2 Black-Oil Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.1.3 Grid Eects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1748.1.4 Other Volumetric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.1.5 Compositional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1758.1.6 Chemical Flood Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.2 Geological Structures, by A. Henriquez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.2.1 Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1788.2.2 Assignation of Petrophysical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.2.3 Gridblock Size and Representation of Physical Flow Mechanisms . . . . . . . . . . . . . . 1798.2.4 Comunication Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.2.5 Heterogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.2.6 Number of Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.2.7 Quality Control: Three-Dimensional View of the Reservoir . . . . . . . . . . . . . . . . . 180

8.3 Eective Properties, by S. Ekrann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.3.1 Single Phase, by L. Holden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.3.1 Two Phases, by M. Dale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.4 Dynamic Pseudofunctions, by S. Ekrann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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8.5 Uncertainty in Forecasting, by J.O. Aasen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.5.2 General Techniques of Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.5.3 Parametric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928.5.4 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.5.5 Use of Geostatistical Methods in Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . 194

8.6 Field Examples, by A. Henriquez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1968.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1968.6.2 Troll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1968.6.3 Troll Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1968.6.4 Results of the Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1978.6.5 Troll Oil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1988.6.6 Statfjord Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

V Methods 207

9 Gasooding 2099.1 Displacement Mechanisms, by I. Akervoll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2099.1.2 Opportunities and Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2099.1.3 Factors Aecting Local Displacement Eciency . . . . . . . . . . . . . . . . . . . . . . . . 2119.1.4 Miscible Processes, by Ø. Glasø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2149.1.5 Immiscible Gas Injection, by O. Hjelmeland and T.A. Jelmert . . . . . . . . . . . . . . . . 216

9.2 Flow of Gas Condensate, by P.K. Munkerud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2189.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2189.2.2 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2189.2.3 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

9.3 Tertiary Gasooding, by T. Holt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.3.2 Some Characteristics of Tertiary Oil Recovery . . . . . . . . . . . . . . . . . . . . . . . . . 2219.3.3 Three-Phase Relative Permeability of Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . 2239.3.4 Laboratory Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2249.3.5 Field Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2249.3.6 Tertiary Gas Injection in North Sea Oil Reservoirs . . . . . . . . . . . . . . . . . . . . . . 225

9.4 Gravity-Stable Displacement, by A. Skauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2269.5 Water-Alternate-Gas Injection (WAG), by T. Holt . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

9.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289.5.2 Alternating Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2299.5.3 Gravity Segregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2299.5.4 High Water Saturations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2309.5.5 Three-Phase Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2309.5.6 WAG Injection in the North Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

9.6 Field Examples, by A. Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

10 Surfactant Flooding 24110.1 Displacement Mechanisms, by J. Kolnes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

10.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24110.1.2 Process Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24110.1.3 Surfactant Flooding in North Sea Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . 245

10.2 Relative Permeability, by E. Eikje and S.R. Jakobsen . . . . . . . . . . . . . . . . . . . . . . . . 24710.2.1 Two-Phase Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24710.2.2 Three-Phase Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24710.2.3 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24810.2.4 Desaturation Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

10.3 Phase Behavior, by K. Veggeland and A. Lohne . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.3.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.3.3 Modelling and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25310.3.4 Representation of Simple Surfactant Systems by Ternary Diagrams . . . . . . . . . . . . . 25410.3.5 Multicomponent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

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10.3.6 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25610.4 Loss of Chemicals, by T. Austad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25910.4.2 Methods to Measure the Loss of Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . 25910.4.3 Type of Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26010.4.4 Chromatograpy Eects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26110.4.5 Nonequilibrium Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26210.4.6 Sacricial Chemicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

10.5 Temperature Eects, by S.R. Jakobsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.5.1 Standard Waterooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.5.2 Polymer Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26610.5.3 Surfactant Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26610.5.4 Polymer Gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26610.5.5 Single-Well Surfactant Tracer Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26610.5.6 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

10.6 Field Examples, by O. Vikane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26710.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26710.6.2 Oshore Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26710.6.3 Surfactant Injection Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26710.6.4 Data Needed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26710.6.5 Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26810.6.6 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

11 Sweep Improvements 27911.1 Foams for Gasooding, by J.E. Hanssen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

11.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27911.1.2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27911.1.3 Bubble Generation and Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28011.1.4 Foam Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28111.1.5 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28411.1.6 Field Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28411.1.7 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

11.2 Polymer Flooding, by A. Stavland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.2.2 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28611.2.3 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28611.2.4 North Sea Reservoir Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28611.2.5 Laboratory Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

11.3 Polymer Gels, by J. Kolnes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28911.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28911.3.2 Gelation in Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28911.3.3 Gelation in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29311.3.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

11.4 Field Examples, by T. Lund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29511.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29511.4.2 North Sea Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29511.4.3 The Vorhop-Knesebeck Polymer Flood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

12 Special Topics 30712.1 Thin Oil Zones, by S. Ekrann and S.M. Skjæveland . . . . . . . . . . . . . . . . . . . . . . . . . 307

12.1.1 Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30712.1.2 Production of Free Gas and Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30712.1.3 Vertical Well Coning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30812.1.4 Production Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30912.1.5 Horizontal Well Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

12.2 Field Examples, by T. Kydland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

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VI Appendices 319

A The SPOR Program, by A.I. Eide 321A.1 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

A.1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321A.2 Statistics and Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

A.2.1 Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322A.2.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322A.2.3 SPOR Board and Advisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322A.2.4 Subprogram Management and Secretariat . . . . . . . . . . . . . . . . . . . . . . . . . . . 322A.2.5 Institutes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323A.2.6 Seminars and Meetings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

A.3 SPOR Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323A.3.1 Technical Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323A.3.2 International Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324A.3.3 Database of SPOR Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

A.4 New R & D Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

B Key Parameters for Norwegian Sandstone Reservoirs, by L. Hinderaker, S.M. Skjæveland,

and B.V. Nystrand 325B.5 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325B.6 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

B.6.1 Properties Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325B.6.2 Volume Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326B.6.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

B.7 Formations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

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Preface

The SPOR Program

STATE R & D

PROGRAM FOR IMPROVED

OIL RECOVERY AND

RESERVOIR TECHNOLOGYSPOR

The Norwegian State R&D Program for Improved OilRecovery and Reservoir Technology (SPOR) was ini-tiated in 1985. The program was founded at the re-quest of governmental authorities and thus signalledtheir future active attitude towards using new or non-conventional technology for Improved Oil Recovery(IOR) from Norwegian reservoirs. Bearing in mindthat the rst eld to be developed on the NorwegianShelf, Ekosk, was discovered in 1969, this early focuson IOR-methods was a consequence of realizing that

optimal recovery strategy had to be considered and implemented at an early stage of eld development.The framework and purpose of the SPOR Program were stated in a Proposal to the Storting and in the

General Guidelines given by the Ministry of Petroleum and Energy (MPE), which says:

The main objective of the program is to develop an independent, national R&D environment on apar with the best available internationally and with rst-hand knowledge of possibilities and limitationsregarding methods for improved recovery from oil reservoirs on the Norwegian Shelf.

The program is assumed to be designed and implemented in such a manner that the NorwegianPetroleum Directorate (NPD) can make use of the developed expertise in carrying out its administrativetasks. An important objective for the program will therefore be to contribute towards establishing refer-ence research environments which the Directorate can draw upon when evaluating various developmentschemes on the Norwegian Shelf.

The organization of SPOR, with NPD in a central management position, reected this purpose of the program.SPOR's Board, holding the nancial and professional responsibility for the program, has been chaired by NPD'sDirector of Resource Management Division, and the location of its Secretariat within the oces of NPD ensuredthe authoritiesÕ direct inuence and benet. From the start in 1985 three established Norwegian researchinstitutes were appointed to perform the R&D within SPOR and thus become leading in designated areas ofimproved oil recovery technology in Norway.SPOR's objective has been to promote research on IOR-methods, which are relevant to the Norwegian elds.

The program therefore consisted of three major subprograms: (1) reservoir characterization, (2) water injectionincluding the use of chemicals, and (3) gas injection techniques. The scope was initially described in the SPORProgram Outline, and the program strategy can be summarized in the following key words: Development ofNorwegian expertise, focus on quality and applicability, national coordination, technical guidance from industryexperts, international collaboration, and making results publicly available.It has been important for SPOR to relate to other national and international R&D activities and keep close

contact with oil companies to ensure a suitable prole of the Norwegian R&D groups. The decision to divideSPOR into subprograms with responsibility given to three dierent research institutes was based upon the needfor focusing on the basis of existing expertise at the start of the program in 1985.The SPOR projects have continuously been evaluated by an Advisory Group of experts from oil companies and

universities in Norway. Based upon their industry experience and valuable dialogues with research environments,the experts have advised the SPOR Board on their priorities for the program. International experts have alsobeen consulted on a yearly basis to ensure the right overall prole of the SPOR Program.The intention of SPOR has been to use the expertise developed through the program in reservoir related

education, and make eorts to disseminate the results to the oil producing industry. A yearly SPOR Seminarhas been arranged to present and discuss the results openly. The research has generated a large number ofinternational publications, in addition to project reports prepared during the program period.

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SPOR was completed by end of 1991, after having achieved its overall goals. Starting at a rather low level ofknowledge in 1985, the IOR-expertise and interest have grown very rapidly within Norwegian R&D environmentsand oil companies during the last years. The rst surfactant and WAG pilots being performed by lisencees inNorwegian North Sea reservoirs in 1991 and 1992 are important milestones, and these eld implementations,together with the preparation and publishing of the monograph, give a worthy nal conclusion to SPOR'sseven-year period.Additional information on the SPOR Program is given in Appendix A.

The Monograph

The monograph has ve parts:

• Preliminariesincluding an introductory chapter, a list of authors, table of content with attributed authors,preface, and acknowledgement.

• Fundamentalsscientic/technical basis for the consecutive two parts, and a state-of-the-art exposition of(1) phenomena, (2) equations, and (3) analytical solutions, that are important for the understanding andapplication of modern methods of improved oil recovery.

The important physiochemical phenomena are rst described and explained, then formulated by equationsor correlations, and nally transformed into transport equations. Some of these are solvable in modiedform, and solutions are presented.

• Reservoir Descriptionmodern methods for transforming reservoir heterogeneity of sandstone reservoirsinto quantied, geological models that in digitized form are used in reservoir simulation models. Theessential problem is how to catch and quantify the geological features dominating uid ow, and hence anyIOR-method.

• Methodsnew methods for improvement of oil recovery; four chapters are included: Gas Flooding, Surfac-tant Flooding, Sweep Improvements, and Special Topics.

• Appendicescontain an exposition of the SPOR Program, a summary of key parameters of North Seasandstone reservoirs, and subject index.

Each chapter is a self-contained entity, with separate Nomenclature and References. However, cross referencesto other chapters are included.We have tried closely to follow the Society of Petroleum Engineers recommendations regarding style, units,

symbols, and references. It is a formidable task to achieve consistency, and sometimes compromises have beenmade to increase readebility. Also, the monograph contains highly theoretical chapters and practical, engineeringexamples, and is a conglomerate of science and engineering disciplines from elds like physics, mathematics,statistics, chemistry, geology, etc. Even though we have reneged on complete consistency, much eort has beenspent to make the presentation clear and precise.Many of the authors have participated in the SPOR Program and several engineers and scientists from Nor-

wegian oil companies have written contributions to exemplify applications of improved recovery methods on aeld scale.A short biography and photo of each author is included at the start of this part of the monograph. The

authorship is attributed in the table of content.

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Acknowledgements

This monograph is the result of a sustained eort from many persons in addition to the authors. We gratefullyacknowledge their assistance.The Norwegian authorities represented by the MPE and the NPD are acknowledged for their initiative and

funding of the SPOR Program which led to the preparation of this monograph.The SPOR Steering Committee and Advisory Board were generous in their encouragement and support of the

work. We also acknowledge the Norwegian oil companies and research institutes for their interest and support.This monograph consists of contributions from a large number of individuals. We are indepted to all the

contributors, and thank them for their great eorts in writing the rst drafts, and their patience during thereview process.A separate subproject of the monograph was to prepare the appendix on Key Parameters for Norwegian

Sandstone Reservoirs. We thank NPD and the oil companies for preparing the reservoir data.The SPOR Secretariat has unfailingly assisted and inspired the editors to carry the project through. We

especially thank Anna Inger Eide for her support, and Marta Eliassen for the follow-up and coordination of thecontributions from all the authors. She also typed several chapters and assisted in the nal editing.Ola Ketil Siqveland of Rogaland University Center did the tremendous job of technically assembling the

contributions from the authors, including corrections, adjustments, and compilation of the chapters with list ofreferences and nomenclature. Many of the gures had to be scanned and retraced in order to get it all togetherin a typeset, printer-ready Latex-Postscript le. We thank him for his dedicated and expert eort.

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Chapter 1

Introduction

1.1 Background

One of the most productive hydrocarbon provincesin the world is found in the North Sea, with someof the largest oshore oil and gas elds that are onproduction today. Typically, an oil eld in the NorthSea consists of deep sandstone reservoirs of high pres-sures and temperatures, containing light oil. Pro-ductive oil zones are very thick and the rock highlyproductive, resulting in high well rates and low welldensities. Natural waterdrive is normally not of su-cient strength to maintain pressure, and many eldsare therefore supported by water injection. Relativelyhigh oil recoveries are obtained through stable waterdisplacement, even though well spacings are relativelylarge. A few elds are produced through miscible dis-placement by hydrocarbon gas, thus obtaining evenhigher oil recoveries than would normally be expectedfrom waterooding.In such an environment, and also considering the

high cost of drilling and production in the North Sea,the potential for additional recovery of oil by mosttertiary methods would appear to be small comparedwith less productive regions of the world. For in-stance, thermal methods are not considered promis-ing under these conditions. However, it was rec-ognized that some secondary and tertiary recoverymethods could succeed. Specically, both improvedwaterooding through polymer addition and reduc-tion of residual oil saturation behind a waterood bysurfactant addition were recognized as technically fea-sible methods of some potential regarding extra oil re-covery, and thus warranted further studies for NorthSea conditions.On the gas injection side, both secondary and

tertiary gas injection, including WAG and miscibleooding could be suitable for these conditions. Addi-tionally, it is obvious that any improvements in reser-voir description at an early stage for these reservoirsof large well spacings would directly improve the re-covery of oil through better planning of the produc-tion.The SPOR Program set out in 1985 to investi-

gate the conditions and potential methods describedabove. A research program was initiated which putstrong emphasis on development of expertise in theareas of reservoir technology and improved oil recov-ery methods in some key Norwegian research insti-

tutes. A series of research projects were started fo-cussing on research related to the methods above, lim-ited to the processes in the reservoirs. Thus, the mainemphasis of the program has been on reservoir tech-nology, including microscopic and macroscopic dis-placement mechanisms. The three research institutesthat were selected, were assigned the responsibilitiesfor Water injection projects, Gas injection projects,and Reservoir description projects, respectively.

Although the reservoir description part of the pro-gram naturally would overlap with the geosciences,actual research in geology and geophysics was not in-cluded in the program. The reasoning behind thisexclusion was that much knowledge in these areaswas already available but not properly utilized by thereservoir engineers. Therefore, advances in reservoirdescription for the purpose of improving the dynamicdescription would be still be possible by integration ofexisting geological knowledge in reservoir technologyresearch projects for improving oil recovery.

Only research pertaining to sandstone reservoirswas included in the research program. Although im-portant elds in the North Sea, such as the chalkreservoirs in the Ekosk area, thus were excludedfrom the program, this limitation of the SPOR Pro-gram was made partly because of the abundance ofsandstone reservoirs in the North Sea, and partlybecause research programs on chalk reservoirs haveexisted in Norway for several years, sponsored byoil companies having interests in the Ekosk area.Furthermore, it was considered that a research pro-gram limited to sandstone reservoirs could be morefocussed and thus more eciently managed than abroader program.

During the course of the research program thatwas started in 1985 and ended in 1991, many ad-vancements in the area of improved oil recovery meth-ods have taken place, not only in projects sponsoredby SPOR, but through research and developmentprojects carried out by oil companies, research in-stitutes and universities throughout the world. Itseemed therefore appropriate to document these de-velopments in a single volume, for the purpose of ad-vancing the knowledge of these new methods to theindustry and the research communities. The initia-tive to this monograph was therefore taken by theAdvisory Board of the SPOR Program in 1990.

1

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2 CHAPTER 1. INTRODUCTION

1.2 Scope and Objectives

The purpose of a monograph is to provide an au-thoritative, up-to-date treatment of the fundamentalprinciples and state of the art in a selected eld (oftechnology).The scope of the present monograph is advances

in IOR methods for North Sea sandstone reservoirs.Classical IOR methods such as primary water- andgas-ooding are appropriately covered in many othertextbooks and monographs and are therefore nottreated in detail here. The emphasis is on advancesthat have been made during the last 5-6 years. Thefundamental principles of the methods are covered indetail, and in order to provide sucient basis, a three-chapter part on fundamentals has been included, cov-ering phenomena, equations, and solutions.The monograph is thus not intended as a textbook

on IOR methods in general. Rather, it presents se-lected IOR methods particularly suited for North Seasandstone reservoirs. The presentation is at an ad-vanced level and monograph may serve as a referencebook for the industry and research community, andas a graduate-level university textbook on this specialtopic.Four main objectives were formulated for the

monograph. One is to organize and to document inone publication the information currently availablefrom laboratory research, engineering studies, andeld testing to provide a more comprehensive futuresource document on the subject.Another objective is to teach those principles and

basic phenomena that are general to recent advancesin IOR methods for North Sea sandstone reservoirs,regardless of the specic methods developed.A third objective is to give an up-to-date engineer-

ing assistance in design and performance projectionsof the modern IOR methods. These will be demon-strated through sample calculations, preferably on ac-tual eld data.A nal objective is to give an up-to-date assess-

ment of how the various modern IOR methods haveperformed in eld trials, through actual eld exam-ples.The scope of the SPOR Program was initially

described in the Program Outline, formulated in1984/85. Since that time, the scope has been cul-tivated and adjusted through continual discussions inmeetings of the Advisory Group and the Board ofSPOR, through discussions with oil companies andresearch organizations, and at seminars arranged bySPOR. It is therefore believed that the nal assort-ment of research areas, the philosophy and objectivesof the SPOR Program, reect a consensus of priori-ties for the industry, government, and research insti-tutions.The topics covered in the monograph therefore are

close to those of the SPOR Program. However, themonograph is not a summary of the SPOR Programresults. Rather, each topic is presented at the state-of-the-art level, regardless of the SPOR results, en-

compassing the available literature on the subject.The monograph is aimed at the international com-

munity and each author has written the contributionswith this in mind. SPOR results are referred to onlyif they are relevant. However, reference is permittedto unpublished SPOR reports, when the quality ofthe results warrants such reference. Many condensedresults from the reports are in the process of beingpublished in international journals.

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Part I

Fundamentals

3

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Chapter 2

Phenomena

2.1 Surface and InterfacialTension

2.1.1 Introduction

The explotation of hydrocarbons oshore is a com-plex process of controlling interactions in systems in-volving crude oil, water, gas and solid formation. Insuch complicated systems, one must master the sur-face properties of oil/rock, water/rock and, in com-bination, the interface oil/water. The most centralproperty, when giving an overall picture of the inter-facial conditions, is the surface or interfacial tension.This property is very sensitive to changes at the in-terface. In this section we will briey illustrate howthe tension may be used in determining surface excessconcentrations due to adsorption, and how the ten-sion is governing wetting properties. Due to the greatsignicance of the surface/interfacial tension, severalexperimental methods have been developed in orderto measure this physical property. This section re-views some of the most commonly used techniques.

2.1.2 Denitions14

An interface is known as the boundary region betweentwo adjacent bulk phases. The equilibrium bulkphases can be: liquid-vapor (LV), liquid-liquid (LL),liquid-solid (LS) and solid-vapor (SV). The bound-aries towards vapor (LV and SV) are commonly de-noted surfaces whereas the LL and LS boundaries arecalled interfaces. In the following we will mainly beconcerned with the systems of LL and LV.Any surface is in the state of lateral tension; this

leads to the concept of surface tension. For plane sur-faces, the tension is a force acting in the plane of thesurface with equal magnitude in all directions. Forcurved interfaces, the denition is similar but slightlymore complex. The surface tension, denoted σ, can berelated to the work required to increase the area of asurface. A molecule at a surface is in a state of higherenergy than a bulk molecule, due to anisotropic, in-termolecular interactions. This means that energy isrequired to move a molecule from the interior to thesurface of a phase, i.e., to increase the surface area ofthe system.Keeping the temperature, pressure and amount of

material in the system constant, the following expres-

sion for surface tension may be written:

σ =

(∂G

∂A

)T,p,ni

(2.1)

The unit of surface tension is that of energy per area,i.e., J/m2, or more commonly N/m.An analogous analysis may be undertaken for two

pure, immiscible liquids in equilibrium. The free en-ergy required to form a fresh interface is referred to asthe excess interfacial energy, and hence σ is termedinterfacial tension. The denition is in accordancewith Eq. 2.1.In principle, this analysis has so far been limited

to at interfaces with no pressure gradients. With apressure dierence across the interface between twophases, the interface will show a net curvature withthe larger pressure on the concave side. The rela-tionship between the pressure dierence ∆p and thecurvature is given by Laplace's equation

∆p = σ

(1

r1+

1

r2

), (2.2)

where r1 and r2 are the principal radii of curva-ture, and σ is the tension. For a spherical droplet(r1 = r2 = r), ∆p = 2σ/r. Across a planar inter-face/surface r1 = r2 →∞ and ∆p = 0.The dierence in pressure across a curved surface

gives rise to a phenomenon as capillary rise. Eq. 2.2is of special importance for enhanced oil recovery.

2.1.3 Adsorption at interfaces4

When molecules are accumulated (adsorbed) at aninterface, σ will change. The Gibbs adsorption equa-tion is a thermodynamic expression which relates thesurface excess concentration of a species to both σand the bulk activity of the adsorbate. In systemswhere σ is easily measured (LL and LV), the Gibbsadsorption equation thus may be used to determinethe surface excess concentration. In other systems,where the surface tension can not be measured di-rectly (SV and LS), the Gibbs adsorption equationmay be used to calculate the change in σ.Consider adsorption from a two-component liquid

mixture at the liquid-vapor interface. The Gibbs ad-sorption equation then becomes:

−dσ = Cs1dµ1 + Cs2dµ2, (2.3)

5

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6 CHAPTER 2. PHENOMENA

where the subscripts 1 and 2 refer to the componentsof the mixture; Csi is the surface concentration andµi is the chemical potential of component i.By dening the surface concentration of the sol-

vent (component 1) equal to zero, and utilizing thefact that there is a chemical equilibrium between bulkand surface phase, Eq. 2.3 may be rewritten as

(Cs)2 = − 1

RT

d(ln a2), (2.4)

where a2 is the activity of component 2. For verydilute solutions concentration may replace activity inEq. 2.4.If the surface tension of a solution decreases with

an increasing concentration of solute, so that Cs2 ispositive, there is an excess of solute at the interface.This is the usual situation with surface active agents(surfactants); an accumulation at the interface leadsto a lowering of the surface tension.Eq. 2.4 is commonly used to determine the sur-

face excess concentration of solutes. From a practicalpoint of view this is found from the slope of a graphof σ as a function of the logarithm of the solute con-centration.

2.1.4 Contact Angles

When a liquid is brought into contact with a solidsurface, the liquid can either expand over the wholesurface or form small drops on the surface. In the rstcase the liquid will wet the solid completely, whereasin the latter case a contact angle Θc > 0 will developbetween the surface and the drop (Fig. 2.1). At equi-librium,

σLV cos Θc = σSV − σLS , (2.5)

where σ is the surface/interfacial tension between thephases in contact. Eq. 2.5 is generally known as

liquid

solid

vapor

σLS σSV

σLV

Θc

Figure 2.1: A drop of a liquid on a solid surface.

Young's equation. From this relation it is clear thatthe tension between the dierent phases involved willcompletely dictate what kind of contact two (or sev-eral) phases may have with regard to wetting andspreading.

2.1.5 Experimental Techniques forMeasuring Tensions.

There exists a broad variety of experimental tech-niques for accurate determination of surface and in-terfacial tension.24 Basically, many of the methods

can be traced back to the fundamental equations inthe previous section, i.e., Laplace's and Young's equa-tions.The capillary rise method is based on the behav-

ior of a liquid in a thin capillary as predicted byLaplace's equation. The tension can be calculatedby means of the liquid column height and the dimen-sions of the capillary. In addition, the knowledge ofthe density dierence between the equilibrium phasesand the wetting properties of the liquid is required.The drop weight method is based on the idea of an

accurate weighing of the drop falling from the tip of acapillary of radius r. Since not only a single drop willbreak but also a thin neck of the subsequent drop,one must introduce a correction factor depending onrV −1/3, where V is the volume of the drop. Themass of the drop, the correction factor, the gravityand the geometry of the tip is then used to calculatethe surface tension.Du Noüy tensiometer measures the force required

to pull a platina wire ring through a surface. An ele-mentary equation is constructed in an analogous wayas for the drop weight method described above. Acorrection factor is needed due to the fact that someliquid clings to the ring, and also to the descriptionof the dimension of the eective contact circle of thering. The principle of the Wilhelmy plate method issimilar to that of the Du Noüy ring technique, buthere a rectangular plate (e.g. of rubbed platina) isused. The main diculty with this method is estab-lishment of a required zero contact angle between theliquid and the plate.A common method to evaluate the surface tension

is to accurately determine the contours of a pendant(or a sessile) drop. Gravity, surface tension and pres-sure gradients will be responsible for the drop shape.Practically, this method is based on the use of exist-ing tables of numerical values which may be used inorder to t the drop shape.The techniques described above are usually applied

to systems in which the surface tension exceeds a cou-ple of mN/m. A complementary technique for verylow tensions is the spinning drop technique. A drop ofone liquid is placed in another of higher density in along, rapidly rotating tube. During the rotation, thedrop elongates into a cylindrical shape determined bythe magnitude of the interfacial tension, the densitydierence between the liquids and the speed of rota-tion. With this technique, tensions of 10−3 to 10−4

mN/m can be measured.Table 2.1 lists some values for surface and inter-

facial tensions between pure liquids and their vapor,and between two liquids where one of them is water.Pure liquids against their vapor usually have surfacetensions from 10 to 80 mN/m. One exception is liq-uid metals with much higher surface tensions. Thestronger the intermolecular attractions in the liquid,the greater is the work needed to bring bulk moleculesto the surface, i.e., the greater is the value of σ.Table 2.2 gives the surface tension at the critical

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2.2. MISCIBILITY 7

micelle concentration (CMC) for some common sur-factants in water.

Table 2.1: The surface tension (σLV ) and the inter-facial tension towards water (σLW ) for some liquids,T = 293 K, from Aveyard and Haydon

Liquid σLV (mN/m) σLW (mN/m)Water 72.8 n-octane 21.7 51.7n-dodecane 25.4 52.9n-hexadecane 27.5 53.8Dichloromethane 28.6 27.7Benzene 28.9 35.0Mercury 476.0 375.0

Table 2.2: The surface tension for aqueous solutionsof surfactants at the critical micelle concentration(CMC), from Vold and Vold2

Compound σ (mN/m)C12H25OSO3Na 39.3C12H25C6H5NBr 42.8C6H4C12H25OSO3Na 44.8C12H25(OC2H4)6OH 31.8

2.2 Miscibility

2.2.1 Criteria

Criteria for developing miscibility are suggested byseveral investigators. Such criteria are the disappear-ance of two simultaneously owing phases (when onephase results from a mixture of two uids) and/orachieving a maximum oil recovery of 90 to 95% afterinjection of 1.2 PV of the displacing agent. How-ever, earlier work58 has shown that miscibility is notalways related to high microscopic displacement ef-ciency of oil. When the transition zone needed toobtain miscibility is long compared to the length ofthe ow path (as with nitrogen), the recovery of oilwill be relatively low (60 to 80%). If miscibility isobtained, then it is characterized by a plateau on thepressure vs. recovery curve. This plateau, togetherwith visual observation of the euent during the dis-placement test, should be used as a criterion for mis-cibility.

2.2.2 Methods

Experiments

Flow experiments oer the most reliable method todetermine the minimum miscibility pressure (MMP)for injection of CO2, N2, or hydrocarbon gas. The

slim-tube method, where the eect of viscous nger-ing is reduced to a minimum, has been most widelyused. The permeable medium consists of glass beadsor unconsolidated sand packed in long coil tubes ofvery thin cross section. The combination of tubelength and diameter, particle size of porous mate-rial, and the displacing velocity are important factorsin the design of a slim-tube apparatus for miscibledisplacement.9

The limitation of the slim-tube test and the prob-lems associated with miscible displacement in porousmedia have been described by several authors.1014 Itis important to note that the packed tube displace-ment apparatus is a device for bringing about multi-ple contact between concurrently owing uids. It isnot intended to simulate reservoir conditions. Hence,slim-tube tests will not in general be indicative ofultimate recovery, macroscopic sweep eciency, tran-sition zone length, etc.High-pressure, slim-tube displacement tests are

performed at specic temperatures and pressures.The slim tube is saturated with oil at the requiredexperimental condition and gas is injected at a con-stant rate by means of a high-pressure displacementpump. As an example, oil recovery at 1.2 PV injectedis plotted in Fig. 2.2 versus a series of displacementpressures, thus determining the MMP at prevailingtemperature.

100

90

80

70

70 75 80 85 90 95

Miscibleimmiscible

CO2 MMP

Displacement tests pressure, bar%re

cove

ryat

1.2

PVof

CO

2in

ject

ed

Figure 2.2: Displacement tests of reservoir uid withCO2.

Equation of State

Construction of pseudoternary diagrams at dierentpressures and constant temperature can be made byan equation of state, and the graphical representationof the phase envelopes in a ternary diagram will per-mit the estimation of MMP. This limit is determinedby selecting the pressure at which the limiting tie lineof the associated phase envelope has the reservoir oilcomposition just to its right.There are, however, several limitation to this ap-

proach, as described by Benham et al.15 Jensen andMichelsen16 have shown that the generalization of theternary-diagram approach to a multicomponent sys-tem may give unexpected results. They found no ob-

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8 CHAPTER 2. PHENOMENA

Table 2.3: Comparison of miscibility pressure ob-tained from slim-tube experiments with simulationmodel results.16

Gas MMP (atm) Dev.Gas Drive Exp. Sim. %NG Vap. 474 445 -6NG Cond. 125 121 -3NG Vap. 396 449 14NG Vap. 409 498 21C1 Vap. 430 474 10

vious relation between an MMP determined by a slim-tube experiment and the result of an MMP calcula-tion. In addition, the ability of an equation of statemodel to predict phase properties near the criticalpoint is not always good. Typical deviations betweenexperimental determined MMP and those simulatedusing the container method of Jensen and Michelsenare shown in Table 2.3. Fig. 2.3 shows the phase en-velopes of a synthetic oil-nitrogen system at three dif-ferent pressures based on gas-liquid equilibrium cal-culations. The ternary diagram indicates that mis-cibility is achieved for all pressures investigated andthat the MMP lies just below 300 bar.

+

++ +

+

+

XX

XX X

X

X

X

X

X

O

OO O

O

O

300 bar350 bar400 bar

N2

C1 - C3C10

Reservoir fluid

Figure 2.3: Predicted MMP by simulation for a syn-thetic oil/N2-system.

Correlations

Correlations for predicting MMP have been proposedby several investigators10,15,1721 and may be used,for an actual reservoir, to estimate the of potential ofmiscible gas ooding.Few of the CO2 MMP correlations10,1821 can be

used with condence for nal project design. Theyare useful, however, for screening and preliminarywork, but they disagree regarding the eect of oil type(e.g., C7+ properties of the oil) on MMP.Alston et al.21 have shown that the volatile and

intermediate fractions of reservoir oil signicantly can

aect the MMP when their ratios depart from unity.Compared with CO2 miscible ooding, very little

has been published on high-pressure, hydrocarbon-gas miscible ooding.2224

In 1960, Benham et al.15 presented empiricalcurves for estimation of miscibility conditions forreservoir oils that are displaced by rich gas withina pressure range of 1 500 to 3 000 psia (10.34 to 20.68MPa). They assumed a limiting tie line (at the criti-cal composition on a ternary diagram) parallel to theC1-C7+ axis and estimated mole% methane in the in-jection gas from calculated critical points with pres-sure, temperature, molecular weights of C2 throughC4 in the gas, and the C5+ molecular weight of theoil as variables.Based on the data of Benham et al., Glasø25 pro-

posed correlations for predicting MMP. These equa-tions are results of curve tting (error <1%) and anextension of Benhams et al.'s correlation.The amount of published data on N2 miscible dis-

placement and MMP correlations is very limited.2426

An MMP correlation for N2 has recently been devel-oped by Glasø.8 Input parameters are the molecularweight of C7+ in the stock tank oil, temperature, andmole% methane and intermediates (C2 to C6) in thereservoir uid. The eect of the input parameters onMMP with N2 gas is related to the API gravity of theoil.

Hydrocarbon gas. The following equations areproposed for predicting MMP,25

(pmm)x=34 =

6 329.0− 25.410y − (46.745− 0.185y)z

+ [1.127× 10−12y5.258 exp(319.8zy−1.703)]T,

(2.6)

(pmm)x=44 =

5 503.0− 19.238y − (80.913− 0.273y)z

+ [1.700× 10−9y3.730 exp(13.567zy−1.058)]T,

(2.7)

(pmm)x=54 =

7 437.0− 25.703y − (73.515− 0.214y)z

+ [4.920× 10−14y5.520 exp(21.706zy−1.109)]T,

(2.8)

where

y =

(2.622

γ−0.846o,C7+

)6.588

is the corrected molecular weight of C7+ in the stock-tank oil; x is the molecular weight of C2 through C6

in injection gas; pmm is minimum miscibility pressurein psig; z is the mole% methane in injection gas; T isthe temperature in F; and γo,C7+ is specic gravityof C7+.

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2.3. ADSORPTION 9

The mole% methane represented in Eqs. 2.6through 2.8 is related to the type of miscible displace-ment considered.For vaporizing gas drive, the minimum allowed

mole% of C2 through C6 in the injection gas for misci-ble displacement is assumed equal to that existing atthe critical composition. This corresponds to a limit-ing tie line parallel to the C1-C7+ line in the ternarydiagram (Fig. 2.4) that intersects the reservoir uidcomposition. Mole% methane required for miscibilityequals 100% minus estimated mole% C2 through C6.

100% C1

100% C2 - C6100% C7+

Estimatedmole-% methanin injection gasfor miscibilityLimiting

tie line

Reservoir fluid

Injectiongas used

Figure 2.4: Estimation of mole% methane in injectiongas; vaporizing gas drive miscibility.

For condensing gas drive, the mole% methane inthe injection gas is equal to the actual compositionof the injection gas used. The accuracy of predictedMMP from Eqs. 2.6 through 2.8 is related to the ac-curacy of the input data, such as mole% methane inthe gas and the molecular weight of C7+ in the stock-tank oil.For volatile oil, the eect of temperature on MMP

is less pronounced then for black oil, probably causedby the attening of the pressure-temperature diagramnear the critical point. By changing the temperatureterm to T 0.94 in Egs. 2.6 through 2.8, it is possible toestimate MMP for volatile oils with the same accu-racy as for black oils.

Carbondioxide. The following equation is pro-posed for predicting MMP for CO2-ooding of liveoil:21

pmm =

8.78× 10−4 (T )1.06(MC5+)1.78

× (xvol/xint)0.136 (2.9)

The minimum pressure, pmm (psig) required for CO2

miscibility with live oil is represented as a function ofreservoir temperature in degrees, T (F); oil pentaneand heavier molecular weight (MC5+

); and the ratio ofvolatile to intermediate mole fractions in the reservoiroil. The oil volatile fraction (xvol) is considered toconsist of methane and nitrogen (C1 and N2); and

the oil intermediate fraction (xint) is considered toconsist of ethane through butane, carbon dioxide, andhydrogen sulde (C2-C4, CO2 and H2S).

Nitrogen. The following equations are proposedfor predicting MMP when nitrogen is injected:8

Oil with gravity < 38 API.

(pmm)API<38 =

− 1.01 + 22.42 p∗mm + 0.63 (p∗mm)2,

(2.10)

where

p∗mm =MC0.88

7+T 0.11

C0.502−6 C0.30

1

. (2.11)

Oil with gravity > 38 API.

(pmm)API<38 =

− 307.10 + 1 578.11 p∗mm − 591.31 (p∗mm)2,

(2.12)

where

p∗mm =MC0.48

7+

T 0.28 C0.112−6 C0.39

1

. (2.13)

pmm is here the minimum miscibility pressure in bar,MC7+

is the molecular weight of C7+ in the stock tankoil, T is the reservoir temperature in C and C2−6

and C1 is the mole% intermediates and methane inthe reservoir uid.

2.3 Adsorption

2.3.1 Introduction

The feasibility of using chemicals to improve the oilrecovery from the reservoir is very often linked tothe retention of the injected material by the reser-voir rock. Consequently, in order to design and opti-mize a chemical ooding process, the mass transportof injected chemicals in porous media should be un-derstood. Chemical ooding processes may involveinjected slugs of dierent types of chemicals such assurfactants, alcohols, polymers, caustics, carbon diox-ide etc. The scope of the present presentation is re-stricted to surfactants, and to some extent alcohols.Various mechanisms by which surfactants are re-

tained by the reservoir rock, have been identied asprecipitation, phase trapping and adsorption. Withreference to the large amount of research in this areaduring the last 20 years, it is today possible techni-cally to prevent loss of surfactants due to precipita-tion and phase trapping, by using salt tolerant surfac-tants, and being condent that changes in parameters(salt, hardness, pressure, temperature, cosurfactant,etc.) responsible for the phase behavior only takeplace within acceptable limits. In many cases it ispossible to design a surfactant system tting such re-quirements. However, adsorption of surfactants atthe solid-liquid interface will always take place, and

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10 CHAPTER 2. PHENOMENA

factors inuencing the extent of adsorption are impor-tant to understand. Thus, very brief comments willbe made on precipitation and phase trapping, whileadsorption will be handled in more details.

2.3.2 Precipitation

Somasundaran's group at the Colombia Universityhas carried out a number of studies in order to evalu-ate the mechanisms responsible for abstraction of sur-factants in porous media that can be related to theprecipitation of negatively charged surfactants andmultivalent cations:2830

2R-SO−3 + M2+ (R-SO3)2 M(s). (2.14)

In general, an increase in temperature will increasethe solubility of the surfactant-metal complex. Inter-estingly, at moderate salinities the addition of mono-valent ions results in increased solubilization of theprecipitate.29,31 However, the above equilibrium ismostly governed by the physiochemical properties ofthe surfactant solution, being able to form micellesand large molecular aggregates. Fig. 2.5 illustratesthe relative concentration of surfactant in solutionwhen adding a negatively charged surfactant to a so-lution containing multivalent cations. The dierentregions of the diagram are29

Region 1. The surfactant is dissolved as ions, R-SO−3 , or ion-pairs, R-SO3M+.

Region 2. Equilibrium exists between R-SO−3 , R-SO3M+ and ( R-SO3)2 M(s).

Region 3. Micelles are formed which have the po-tential to dissolve the precipitate. Equilib-rium exists between R-SO−3 , R-SO3M+, (R-SO3)2M(s) and micelles.

Region 4. All the precipitated material has dis-solved, and an equilibrium between R-SO−3 , R-SO3M+, micelles and perhaps (R-SO3)3M− is es-tablished.

Region 5. Reorganization of the surfactant aggre-gates, forming liquid crystals may happen, caus-ing a reprecipitation of the surfactant.

Alumina is present in many clay minerals, i.e.,kaolinite and illite, which are part of the reservoirrock. The activity or concentration of Al3+ in solu-tion is related to the pH of the brine. Assuming aconcentration of R-SO−3 close to the critical micelleconcentration, CMC, precipitation of (R-SO3)3Al canbe expected at pH < 5.30

The precipitation and resolubilization of complexesbetween anionic surfactants and multivalent cationscan in many cases explain the maxima and minimaobserved in adsorption isotherms.28

At moderate salinities and temperatures the toler-ance of petroleum sulfonates, R-SO−3 , towards mul-tivalent cations can be enhanced by adding ethoxy-lated nonionic surfactants, R-(O-CH2-CH2)x-OH.32

1.0

01 2 3 4 5

Total Surfactant Concentration

CS

CSi

CMC

Figure 2.5: Change in dissolved anionic surfactantversus the amount of surfactant in the presence ofmultivalent cations.

Increased tolerance towards hardness can also beobtained by using ethoxylated sulfates, R-(O-CH2-CH2)x-OSO−3 , and sulfonates, R-(O-CH2-CH2)x-SO−3 .

3335 The former is unstable and it will hy-drolyze at elevated temperatures.36

2.3.3 Phase Trapping

Experiments have proved that the optimal recoveryof oil is obtained if the surfactant system is optimizedwhen such a way that equal volumes of oil and waterare solubilized by the surfactant into a microemul-sion phase called the III state. Various reservoir pa-rameters like salinity, temperature and pressure aresensitive to the phase behavior, especially when us-ing a one-component surfactant without alcoholic co-surfactants.37,38 Petroleum sulfonates can enhancethe cationic exchange between the rock and the brineby increasing the amount of divalent cations in thesolution. The surfactant system may then form awater-in-oil microemulsion, i.e., the II(+) state. Theinterfacial tension, IFT, increases, and the surfactantwill be trapped in the nearly immobilized oil phase.These diculties can probably be avoided by usingethoxylated sulfonates having an ethoxylation degreegreater than three. From chromatographic studiesthese compounds are found to show lower electro-static interaction with cations than nonethoxylatedsulfonates.39

Anionic surfactant systems, showing normal phasebehavior, will usually move towards the II() stateupon increasing the temperature.40 Thus, a positivetemperature gradient from the injection well to theproduction well will normally not cause phase trap-ping of the surfactant due to temperature eects.Pressure does eect the phase behavior of multi-

phase microemulsion systems. Drastic changes in thesolubilization parameters can be observed within apressure interval of 100 bar for live crude oil sys-tems.38 Normally, decrease in the pressure willmove the microemulsion system towards the II(+)state.37,38,40 Thus, large pressure drops close to theproduction well may cause a phase transition of thesurfactants into the oil phase.

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2.3. ADSORPTION 11

2.3.4 Adsorption

Surfactant systems at the hydrophilic/lipophilic bal-ance, i.e., being at the optimum for oil recovery, arepoorly soluble in both of the excess phases. In anactual oil reservoir the surfactants are most likelyat the oil/water interface and at the solid/liquid in-terface. Very often adsorption of surfactants at thesolid/liquid interface is initiated by electrostatic in-teraction between the solid (adsorbent) and the sur-factant (adsorbate). At the natural pH value of thebrine, most of the reservoir minerals (quartz, kaolin-ite, etc.) show a net negative charge, i.e., the Zeta-potential is negative. However, some of the clayminerals have positively charged edges. In order tolower the adsorption, negatively charged surfactantsare usually considered as the main surfactant speciesin the slug.Adsorption onto the mineral surfaces is a complex

process during which adsorption of the surfactantmay occur successively by ion exchange, ion pairing,and hydrophobic bonding mechanisms. Some knowl-edge about the phenomena is obtained by studyingthe prole of the adsorption isotherm. A typical ad-sorption isotherm for the adsorption of a negativelycharged surfactant onto an adsorbent, having posi-tively charged sites, is S-shaped, Fig. 2.6. The dier-ent regions reect the distinct modes of adsorption:

Region 1. The surfactant adsorbs mainly by anionexchange, showing a linear relation between ad-sorbed material and the equilibrium concentra-tion.

Region 2. A marked increase in adsorption, result-ing from the interaction between the hydropho-bic chains between the oncoming surfactants andthe surfactants already adsorbed, takes placeabove the CMC of the surface.

Region 3. The degree of adsorption decreases be-cause adsorption has to overcome the electro-static repulsion between the surfactant and thesimilarly charged solid.

Region 4. A plateau adsorption is obtained at thecritical micelle concentration, CMC. The concen-tration of monomers is constant above the CMC.The micelles do not adsorb onto the solid.

Above the CMC of the surface (the line betweenRegion 1 and 2) surfactant aggregates are formed.The structure of the adsorbed surfactant aggregates,which may vary along the isotherm, has been debatedin the literature. Two model structures have beenproposed, i.e., hemimicelles and a bilayered structuretermed admicelles, Fig. 2.7. Yeskie and Harwell41

have performed model calculations, and they con-cluded that high counterion bindings, high surfacecharge densities, and high dielectric constants of thesolvent all favor the formation of an admicelle beforethe formation of a hemimicelle. The opposite con-ditions favor formation of a hemimicelle at a lower

CMC

1 2 3 4

Equilibrium Surfactant Concentration

Ads

orpt

ion

Figure 2.6: Schematic S-shaped adsorption isothermfor an anionic surfactant and a substrate having pos-itively charged sites.

surfactant concentration than necessary for an admi-celle to form.

Admicelles Hemimicelles

Figure 2.7: Schematic of proposed structure of sur-factant aggregates adsorbed to the surface.

It has to be noted that the shape of the adsorptionisotherm may vary considerably for dierent systems.Maxima and minima in the adsorption are very oftenobserved, and dierent explanations have been putforward. It appears to be documented in the litera-ture that these phenomena can be explained by a)precipitation/resolubilization of complexes betweenanionic surfactants and multivalent cations, b) us-ing surfactant mixtures, c) using dierent solid/liquidratios, and d) micellar exclusion from the negativelycharged surfaces of the porous medium.28,42

The interfacial tension between the water and oilphase decreases until the concentration of the sur-factant reaches the CMC. Thus, in chemical ood-ing situations where the key problem is to obtainas low IFT as possible, the surfactant concentrationshould be above the CMC. Some factors inuencingthe plateau adsorption will be discussed. The discus-sion is, however, limited to pure anionic and nonionicsurfactants. More complicated adsorption behaviormay be observed using commercial mixed surfactantsystems.

Surfactant Structure. An increase in the adsorp-tion results from an increase in the alkyl chain lengthof n-alkylbenzenesulfonates. Branching of the alkylchain for similar surfactant systems has been found todecrease the ability of the surfactant to adsorb. Alky-larylsulfonates appear to adsorb more strongly thanalkylsulfonates of comparable chain length.43 De-

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12 CHAPTER 2. PHENOMENA

crease in the number of the ethoxy-groups in ethoxy-lated alkylarylsulfonates, R-Ph-(O-CH2-CH2)x-SO−3 ,will increase the adsorption.34,44

Salinity. Generally, it is observed that the plateauadsorption of anionic surfactants increases by increas-ing the salinity. The eects are mainly explained byneutralizing the electrostatic interaction between thehead-groups of the surfactants by forming an electri-cal double layer.45 Monovalent water structure break-ing cations like K+ and NH+

4 are found to cause anincrease in adsorption relative to a structure makingcation like Na+.45 In the presence of divalent cationscomplexes of the type RSO3M+ may be formed in so-lution, which can adsorb onto negatively charged siteson the rock surface.45,46 For nonionic surfactants in-crease in the salinity may cause phase separation, andthe adsorption increases drastically.

pH-value. The Zeta-potential of the mineral ox-ide surface is sensitive to the pH of the solution. Ingeneral, decreasing the pH will increase the numberof positively charged sites on the surface. Thus, in-creased adsorption of negatively charged surfactantsis found.44,45

Temperature. In general, molar enthalpies of ad-sorption of anionic surfactants on mineral oxides arenegative, which suggest that adsorption decreasesas the temperature increases. However, in somecases increased temperature can result in increaseddissolution of the minerals causing precipitation ofsurfactant-metal complexes, which are experimen-tally registered as adsorption. The adsorption of non-ionic surfactants very often increases as the temper-ature increases.47

Wettability. The distribution of uids in pores isstrongly aected by the wettability of the porousmedium. Due to the inhomogeneous nature of a reser-voir rock it is dicult to perform a single analysiswhich clearly tells you whether the reservoir is water-wet, oil-wet, or neutral. Conicting eects of wetta-bility on surfactant adsorption have been reported.However, later studies appear to show that adsorp-tion of an anionic surfactant on an oil-wet surface isgreater than adsorption on a waterwet surface underno salt conditions. The presence of salt in solution re-sults in higher adsorption on the water-wet surface.48

Kinetics. The equilibrium adsorption constant,Keq = k1/k2, where k1 and k2 are the adsorbing anddesorbing rate, respectively. The rate constants in-crease, but Keq decreases upon increasing the tem-perature, which is explained by a more acceleratingincrease in k2 relative to k1.47 The kinetics of adsorp-tion/desorption of alkylsulfonates on reservoir clayminerals have indicate long-term eects at elevated

temperatures. However, possible precipitation of alu-mina species released slowly from the clay is exper-imentally dicult to discern from adsorption. Re-cently it has been veried that salt tolerant ethoxy-lated sulfonates require several weeks in order to ob-tain adsorption equilibrium in dynamic adsorptionexperiments in reservoir cores containing fairly largeamounts of detrital clay minerals. The rate and ex-tent of adsorption is a function of the properties ofthe micropores of the clay minerals and the oil satu-ration.35,49

Surfactant mixtures. Commercially availablesurfactants consist of mixtures of unknown compo-sition rather than well-characterized mixtures. In or-der to describe the adsorption behavior of such a sys-tem, the mixed hemimicelles/admicelles and mixedmicelles in solution have to be taken into account.Due to the dierent geometrical structure of the twogroups of surfactant aggregates, it may exist dierenttype of synergism between the individual surfactantsin the aggregates. Mixtures of anionic and nonionicsurfactants very often show increased adsorption rel-ative to the pure anionic surfactant, while mixture oftwo anionic surfactants can show a decrease in ad-sorption. Adsorption models for the two cases havebeen proposed which describe the adsorption behav-ior quite well.50,51

Preventing adsorption. Various precautionarymeasures for the dierent mechanisms have beenstudied in order to reduce the retention of surfactants,and the literature has been reviewed by Surkalo andPouska.52 In most cases, a preush using dierentchemicals is suggested, i.e., injection of (1) sodiumchloride solution in order to reduce hardness; (2) al-kaline additives sodium hydroxide, carbonates, phos-phates, and silicates, to both decrease hardness andto render the rock more negatively charged;53 (3) sac-ricial chemicals, i.e., lignosulfonates,54,55 polybasiccarboxylic acids,56 and polyethylene oxide,57 that willadsorb and block the active sites of the rock.Coinjection of low concentration surfactant and a

biopolymer, termed Low Tension Polymer Flood,LTPF, has been found to lower the adsorption of thesurfactant drastically. The mechanism by which thesystem works is not yet well understood.58

Adsorption Models

Very often the static equilibrium adsorption of surfac-tants from a solution is found to t the Langmuir59

adsorption isotherm,

Γ =Γm CsCs + a

, (2.15)

where Γ is surface excess of surfactant, Γm is surfaceexcess of surfactant at monolayer adsorption, Cs isthe equilibrium surfactant concentration and a is aconstant.

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2.4. DISPERSION 13

The Eq. 2.15 may be transformed into the linearform

CsΓ

=CsΓm

+a

Γm. (2.16)

The constants Γm and a can be determined by plot-ting Cs/Γ against Cs.Theoretically it is proposed that this equation is

valid only under the following conditions (1) the ad-sorbent (solid) is homogeneous, (2) the solvent andthe surfactant have equal molar surface areas, (3)both surface and bulk phases exhibit ideal behavior,(4) the adsorption is monomolecular.The fact that so many experimental adsorption

data t the Langmuir equation does not mean thatthe assumptions are fullled. Mutual compensa-tion of dierent factors that aect the shape of theisotherm may be an explanation.The continuity equation for a solute (surfactant)

owing through a one-dimensional porous mediumwith dead-end pore volume, containing adsorptiononto the pore surface, is according to Bidner andVampa60 given by

−v ∂Cs∂x

+K∂2Cs∂x2

=

f

(∂Cs∂t

+Awrφ

∂Γ

∂t

)+ k(Cs − C∗s ), (2.17)

where

k(Cs − C∗s ) =

(1− f)

(∂C∗s∂t

+Awrφ

∂Γ∗

∂t

). (2.18)

Here, v is average constant velocity, Cs is the surfac-tant concentration, x is distance, K is the longitudi-nal dispersion coecient, f is fraction of pore spaceoccupied by mobile uid, Awr is rock area per unitvolume of rock, φ is porosity, k is mass transfer coef-cient, and Γ and Γ∗ are surfactant adsorption frommobile and stagnant uid, respectively.The following assumptions have been made: (1) the

ow is isothermal, single phase, one dimensional andbicomponent; (2) the porous media is homogeneousand uniform with constant cross section; (3) uid androck are incompressible; (4) no chemical reactions; (5)only longitudinal dispersion.The adsorption model is based on reversible ad-

sorption of the surfactant,

S + Θk1k2

SΘ (2.19)

where S is surfactant of concentration, Cs; Θ is unad-sorbed amounts per unit surface area, Qa−Γ, (Qa istotal adsorbent capacity); SΘ is surfactant adsorbedper unit area; and k1 and k2 are rate constants.From Eq. 2.19, the following second-order rate

equation can be written,

∂Γ

∂t= k1Cs(Qa − Γ)− k2Γ. (2.20)

For very long times, ∂Γ/∂t = 0, and Eq. 2.20 re-duces to the Langmuir equilibrium adsorption model,Eq. 2.15,

Γ =k1QaCsk2 + k1Cs

=QaCs

Cs + k2/k1, (2.21)

which is equal to Eq. 2.15 with the substitutionsQa =Γm and a = k2/k1.An identical adsorption mechanism can be assumed

in the stagnant uid pore volume,

∂Γ∗

∂t= k1C

∗s (Qa − Γ∗)− k2Γ∗. (2.22)

This general model, which considers several mecha-nisms simultaneously, is a useful tool to simulate dif-ferent types of ow of miscible uids through porousmedia. However, in a homogeneous reservoir sand-stone containing detrital clay minerals, the stagnantvolumes are associated with the micropores of theclay. This volume is nearly negligible compared to thetotal pore volume, i.e., f ≈ 1, but a large surface areais related to these micropores.35,49 The nonequilib-rium adsorption process taking place in such systemsis discussed in details in a later section.

2.4 Dispersion

2.4.1 Introduction

The combined eect of the two physical phenom-ena, diusion and convection induced mixing is com-monly referred to as dispersion. Dispersion actsagainst maintaining concentration dierences, or inother words, the absolute values of all concentrationgradients will be reduced due to dispersion as theuid is transported through the system (e.g. a porousmedium).Dispersion will reduce variation in concentrations

both along and transverse to the direction of ow.These two types of dispersion are thus called longi-tudinal and transversal dispersion, respectively. Fora cylindrical porous medium, the notations axial andradial dispersion are also commonly used.Both longitudinal and transversal dispersion are

modelled by dispersion coecients in the continuityequation. For axial ow in a cylindrical core thisequation may be formulated by61

1

V

∂ni∂t

+∂(q Ci)

∂V−Kl

∂2Ci∂x2

−Kt1

r

∂r(r∂Ci∂r

) = R,

(2.23)where V is the volume of a dierential annular ring, nithe moles of component i, Ci concentration of com-ponent i, q volume ow, t time, r radius, Kl lon-gitudinal, and Kt transversal dispersion coecient.R represents transfer of component i from a sourceor to a sink, e.g. mass transfer to another phase, orgeneration/consumption in a chemical reactor.Dispersion may occur in various scales. Mixing

at the pore scale is often referred to as microscopic

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14 CHAPTER 2. PHENOMENA

dispersion, at core scale as macroscopic, and onreservoir scale as macroscopic or megascopic disper-sion. This classication somewhat reects the dif-ferent mixing processes involved. Whereas molecu-lar diusion and ow in a single pore or in a fewadjacent pores constitute the mixing mechanisms atmicroscopic scale, large-scale heterogeneities such asstratication, shales, blocks of variable permeabilityetc. will dominate the mixing at the megascopic scale.At a laboratory core scale, all types of mixing mayoccur, but for homogeneous media the main mecha-nisms are likely to belong to the microscopic class. Toclarify, dispersion will in the following be divided intotwo categories. Dispersion at a core scale or less willbe called microscopic. Large scale dispersion (macro-scopic, megascopic) will be referred to as dispersionat a reservoir scale.

2.4.2 Microscopic

Fig. 2.8 illustrates a transport zone along which theconcentration of component i changes. To the left of

Ci1Ci2

A

_x

_x

B

Figure 2.8: Einstein's diusion model.

reference plane AB (unit area), the concentration ofcomponent i is Ci1, and similarly Ci2 on the righthand side. During the characteristic time, τ , all par-ticles will move a characteristic distance, x, due tothermal motions in the system (average displacementdue to Brownian movement during τ). As the proba-bility of a molecule to move in either direction is thesame, the amount of component i, ni, crossing ABfrom the left during the time τ is

~n(Ci1) =1

2Ci1x, (2.24)

and a corresponding amount of component i movingfrom the right, crossing AB is

~n(Ci2) =1

2Ci2x. (2.25)

The net ux, Ji, across AB then becomes

Ji =~n(Ci1) + ~n(Ci2)

τ=Ci1 − Ci2

x

x2

2τ= − x

2

dCidx

.

(2.26)This may be compared to Fick's law for uncoupleddiusion,62

Ji = −DoidCidx

, (2.27)

where Doi is the bulk phase diusion coecient ofcomponent i. This indicates that a diusion coe-cient may be regarded as the ratio of the square ofthe characteristic length of a diusive event dividedby two times the corresponding time, or

Doi =x2

2τ. (2.28)

Eq. 2.28 is often called Einstein's relationship.Fig. 2.9a illustrates a part of an idealized porous

medium which may be viewed as an array of voidsinto which uid ows at a high velocity from small-area ports. As a result of acceleration in the ports andretardation upon entering the voids, mixing occurs.In the limiting case, if perfect mixing occurs in each

void, the porous medium may be regarded as a se-ries of perfectly mixed vessels interconnected by portsconsisting of closely packed regions.In terms of Eq. 2.28, the distance x over which

the uid travels before void-cell mixing is, on the av-erage, dp, and the corresponding time τ is equal todp/v, where v is the interstitial velocity. Insertinginto Eq. 2.28 yields

K ′l = dpv/2, (2.29)

where K ′l has replaced Do to denote that mixing inthe direction of ow now is a result of mechanicalmixing, rather than pure molecular diusion.A porous medium will generally not be regularly

packed as shown in Fig. 2.9a. As a stream of uidows through the porous medium, it can be imaginedthat at some time it strikes a piece of packing and issplit in two by the collision, Fig. 2.9b. On the average,one half of the stream moves laterally to the right, theother to the left. This event occurs repeatedly, withthe result that the original single stream is laterallydispersed, or fans out towards the wall of the system.In terms of the Einstein relation it can be said that

upon splitting, the stream moves a diusive distancex equal to one half the particle diameter. The timefor a jump, or a laterally dispersing split to occur,is of the order of the time required for the uid totransverse one axial layer of packing, or τ ' dp/v, 61

and by Eq. 2.28,

K ′t = dpv/8. (2.30)

The total microscopic dispersion in a porous mediummay be due both to molecular diusion and convec-tive mixing by mechanisms as already shown. Otherpossible mixing processes will be discussed below. As-suming that the eects of molecular and mechanicalmixing are additive, the total dispersion may be ex-pressed as

Kl = Do/τ + 0.5 dpv (2.31)

Kt = Do/τ + 0.125 dpv, (2.32)

where the bulk diusion coecient, Do, is modiedby the tortuosity factor, τ . This factor is a measureof how much longer a particle must move in a porous

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2.4. DISPERSION 15

dp

(a) Idealization of longitudinal mechanical mixing.

dp

(b) Idealization of transversal dispersion by stream splitting.

Figure 2.9: Mixing in porous media.

medium compared with a bulk phase or a straighttube.Fig. 2.10 shows a plot of Peclet numbers for longi-

tudinal and transversal dispersion, both for gas andliquid, as functions of the Reynolds number, NRe.The Peclet number in the gure is dened as dpv/Kk

(k = l or t). With the denition of the Reynolds num-ber used, the region between laminar and turbulentow occurs at NRe ' 40. 61

For ow of liquid it is seen that convective owdominates the total dispersion at relatively low NRe

numbers compared to gaseous systems.At high NRe numbers, the Peclet number for lon-

gitudinal dispersion varies between 0.3 and 1, thoughat even higher NRe it appears to approach the value2 found for gas dispersion. The Peclet number fortransversal dispersion in liquid is dominated by con-vective mixing at NRe > 10−2, and approaches avalue of 10 at large NRe.A similar, but less complicated behavior is observed

for gaseous systems. When convective mixing startsto dominate at NRe numbers of approximately 1 and10 for longitudinal and transversal dispersion, respec-tively, the corresponding Peclet numbers approach 2and 10. By rearrangement of Eqs. 2.29 and 2.30,Peclet numbers of 2 and 8 are obtained.Although mixing-cell and stream-splitting theory

104

102

100

10-2

10-5 10-3 10-1 101 103

NPe

=d p

/Kk

LiquidGas

Moleculardiffusion

Moleculardiffusion

Transversaldispersion

Longitudinaldispersion

NRe = ρ dp/µ

Figure 2.10: Wilhelm's display of longitudinal andtransversal dispersion.61

often yield good agreement with experimental resultsfrom packed beads, other mixing mechanisms mayalso appear during ow in porous media.Fig. 2.11 illustrates dispersion in a capillary tube.

B A

JD

JD

Figure 2.11: Longitudinal dispersion in a capillarytube.

Imagine that a uid B is injected into the tube orig-inally lled with uid A. The two uids may haveidentical ow properties. Due to the velocity varia-tions across the tube, v = 0 at the wall of the tubefor laminar ow, liquid B will penetrate into liquid A.This penetration is partly counteracted by moleculardiusion, JD.The net result will be a mixing zone growing at a

more rapid rate than would be obtained from diu-sion alone. Theoretical considerations show that ifdiusion nearly can dampen out radial concentrationvariations, a symmetrical, longitudinal mixing zonewill be established. The mixed zone would travel withthe mean speed of the injected uid, and will be dis-persed as if there were a dispersion coecient givenby63

Kl = Do +v2a2

48Do, (2.33)

where a is the radius of the capillary.Convective mixing may also be explained as a result

of other microscopic phenomena such as incompleteconnectivity of the porous medium, obstructions, andrecirculation caused by local regions of reduced pres-sure.4 The level of dispersion by these mechanisms issuggested proportional to the ow velocity,65

K ′l ∝ v. (2.34)

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16 CHAPTER 2. PHENOMENA

(a) Incomplete connectivity in a porous medium.

(b) Mixing caused by obstructions.

(c) Recirculation of uid.

Figure 2.12: Mechanisms for convective mixing.

These mechanisms are illustrated in Fig. 2.12.Apparently, dispersive behavior may also be due to

dead-end pores and adsorption. Coats and Smith66

have shown that the eect of dead-end pores is bet-ter described by a dierential capacitance modelrather than a pure dispersion model. The appar-ently anomalous behavior of longitudinal dispersionin Fig. 2.10 is likely to be due to capacitive eects,generating lower Peclet numbers.61 Dispersion byadsorption is again an unsteady-state phenomenon.Just as with dead-end pores, a concentration frontwill deposit or remove material, and therefore con-tributes to attening of the concentration proles inthe interstitial uid.

Dispersion in Laboratory Cores

The currently best description of the mixing proper-ties of porous media is an equation of the form14,67

Kk

Do=

1

τ+Ak

(vdpσ

Do

)mk, (2.35)

where Kk is the eective dispersion coecient (k = lor t). The tortuosity factor is related to the electri-cal resistivity factor (formation factor), F , and theporosity, φ, by τ = Fφ. Ak and mk are constantsand σ is called a packing, or inhomogeneity factor.For longitudinal dispersion, Al is equal to 0.5, re-

ecting its origin in the mixing-cell theory, Eq. 2.29.The values of ml in the range 1.0 to 1.6 are frequentlyseen, and often ml is reported to be close to 1.2.12 A

value of ml in the range between 1.0 and 2.0 can beexpected if all the mixing processes discussed con-tribute to the total dispersion, Eqs. 2.29, 2.33, 2.34.Perkins and Johnston63 that a value of ml = 1.2 iscorrect only for measurements at high owrates, orhigh Peclet numbers, NPe = vdp/Do > 50. Thisis explained by the mixing-cell theory, and presum-ably shows that diusion is not entirely equalizingthe concentration within each cell due to low resi-dence time. This explanation is, however, somewhatin conict with the results in Fig. 2.10. Here, thebehavior expected from mixing-cell theory occurs athigh owrates, close to and behind the region of tur-bulent ow, as expected, since turbulent mixing isbelieved to be important in order to equalize the con-centrations in each cell.The value of ml also sensitive to the range of

owrates used in the experiments. If the data used tot Eq. 2.35 is measured in a region where both termsof the equation are of signicance, an exponent ml

larger than 1 is to be expected.By comparing the second terms of Eqs. 2.33 and

2.36, the relative importance of capillary dispersionto the total convective dispersion may be determined.The inhomogeneity factor, σ, is experimentally deter-mined in the range 1 to 10, where 1 is the theoreticalminimum for regular bed packs.63

By assuming that dp and a are approximatelyequal, and by using appropriate diusion coecientand owrates, it is seen that capillary dispersion maybe a signicant part of the total dispersion in liquidsystem, but insignicant in gaseous systems, whereDo may be several orders of magnitude larger.For transversal dispersion, a unit value of mt and

At = 0.0157 is commonly used.63 Sometimes Eq. 2.35is applied without the inhomogeneity factor, and thenAt = 0.055. The transition between these two valuesis due to an average value of 3.5 for random packs ofsmall particles.A value At = 0.055 is approximately one half of

what is expected from the theory of stream splitting,Eq. 2.30, mt = 1. As seen from Fig. 2.10 for liquidsystems, the Peclet number approaches 10 only closeto the transition to the turbulent ow regime. Atlower Reynolds numbers, lower values of At are to beexpected.

Measurements of Dispersion

Dispersion coecients may be determined by experi-mental methods and analyses as described by severalauthors61,63,6769 for longitudinal dispersion, and fortransversal dispersion.61,63,70 By Eq. 2.35, τ , dpσ,and mk are determined using data from displace-ment experiments or tracer injections into a carrierstream. These three quantities are all presumed toreect properties of the porous medium in question.A requirement is that the displacement experimentsare performed with uids of equal density and mobil-ity.If these conditions are not fullled, the measured

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2.4. DISPERSION 17

dispersion is also inuenced by uid properties, e.g.densities and viscosities. This is discussed by Perkinsand Johnston,63 where some other factors, such asparticle to column-diameter ratio, particle-size distri-bution and particle form, also are discussed.An example of a laboratory setup for measure-

ments of longitudinal dispersion coecients is shownin Fig. 2.13. Here a pulse of Argon is introduced intothe carrier stream of nitrogen, and the variance ofthe input signal is recorded at the exit of the system.This is done both with and without a core in the core-holder, and the dierence in variance of the outputsignals in the two cases is directly related to the dis-persion coecient, the ow velocity, and the lengthof the porous medium.67,69 This experimental setupis fast and accurate due to a high level of automa-tion. The measurements can easily be performed inthe presence of an immobile phase (e.g. at Siw).

Eect of Dispersion on Laboratory Experi-ments

Dispersion is a measure of the degree of mixing in aporous medium. Returning to the mixing-cell theory,it can be shown that a porous medium of length Lbehaves like n mixing chambers in series, where n isgiven by61

n =Lv

2Kl+ 1. (2.36)

Two extremes thus exist. If Kl approaches zero, themixing in the system is negligible. The ow in the sys-tem will be perfect plug ow, maintaining sharp con-centration proles, and a slug of material may keep itsintegrity throughout the system. On the other hand,if Kl approaches innity, dispersion dominates, andthe system becomes equivalent to one mixing cell. Aslug process whose eectiveness depends on the con-centration of an active component is unlikely to beeective in such a system.Possible eects of dispersive mixing in laboratory

experiments will be illustrated by some examples.Solution of the continuity equation for a one-

dimensional, nonreactive system, and equal viscositymiscible uids, predicts that the mixing-zone lengthwill be a function of the square root of the distancetravelled,71

(∆L)10−90

L= 3.625

(Kl

vL

)0.5

, (2.37)

where (∆L)10−90 is the distance between the 10% and90% concentration point of the injected uid.Insertion of Eq. 2.35 (forml = 1) into Eq. 2.37, and

remembering from Fig. 2.10 that convective mixing islikely to dominate the total longitudinal dispersion inliquid systems, Eq. 2.37 becomes

(∆L)10−90

L= 2.563

(σdpL

)0.5

. (2.38)

Thus, for a given porous medium, the relative lengthof the mixing zone can only be reduced by increasingthe length of the core.

During micellar ooding, the eectiveness of theprocess may depend on the concentration of surfac-tant. If for example, the process is eective only aslong as the surfactant concentration is larger than60% of injected value, a 10 cm long core may requirea slug size of 25% PV in order to be eective through-out the system. If the core length can be increasedto 100 cm, a slug of only 8% PV needs to be injectedin order to maintain the eectiveness.71

For multicontact, miscible displacement of oil bygas, or any other multicontact miscible process, thereexists strong evidence that the total recovery e-ciency depends on the size of the transition zone. Therelative size of the transition zone increases with de-creasing core length. A high level of dispersion, ex-pressed by Kl/vL, thus decreases oil recovery, bothfor multicontact miscible N2 injection72 and rst-contact miscible or multicontact miscible displace-ment by CO2.73 For rst contact miscible CO2 dis-placement, an increased level of dispersion reducedoil recovery by 5% of OOIP. For a multicontact misci-ble displacement, oil recovery was reduced by 12% ofOOIP, going from low to high dispersion. The eectof dispersion on recovery was explained by dierentprocess paths in the compositional space, illustratedby ternary phase diagrams. The result of a high levelof dispersion may be that miscibility is not reachedfor systems being multicontact miscible in systemswith lower dispersion.73

Because of high mobility, injected gas has a ten-dency to nger through the oil phase. The result isa reduced oil recovery as demonstrated by Gardnerand Ypma74 for both rst contact and multicontactmiscible CO2 coreoods. Viscous ngers are formedfor both types of processes, but for rst-contact mis-cibility, the oil mixed into the ngers is miscible withthe gas. First contact miscible displacement there-fore gave the best recovery. By reducing the owrateof injected gas in the rst-contact miscible system,recovery was increased because viscous ngering wassuppressed by transversal dispersion.This phenomenon is characterized by a transversal

dimensionless time, Θt, as the ratio between convec-tive and transverse dispersion time, given by

Θt =KtL

vd2, (2.39)

where d is the diameter of the core. A low value ofΘt gives an unstable displacement; a high value givesa stable displacement.The diameter of the core is usually large compared

to the critical width of viscous ngers, λc, see below,and viscous ngers are thus formed. Therefore, at lowvalues of Θt, the process is characterized by unstabledisplacement. At high values for Θt, however, disper-sion will solubilize oil into the ngers which growtogether to one large nger covering the whole crosssection of the core. The result is a displacement e-ciency characterized by stable displacement. The twoextreme situations are connected by a transition zonewith intermediate behavior.74

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18 CHAPTER 2. PHENOMENA

Timer

Ar

H2O

N2 N2

3-way valve

Air

Solenoid valve

∆p

Core

Filters

2XRi-detector

Recorder/computer

Flow-meter

Figure 2.13: Apparatus for measurement of longitudinal dispersion.

For multicontact miscible displacement processes,transversal dispersion will inuence the growth of vis-cous ngers as described by the equation9

λc = 25/2 πµo + µgµo − µg

Kt

v, (2.40)

where λc is the critical nger width. Viscous ngerscannot grow if the diameter of the porous mediumis less than λc. A typical value for λc during slim-tube experiments is 0.1 cm for a owrate of 0.2 cm/s.The diameter of a slim-tube is usually of the order0.5 cm. The oil recovery is therefore dependent onthe injection rate through the formation of viscousngers unless the injection rate is reduced, giving aλc greater than the diameter of the slim tube. Longi-tudinal dispersion may also aect the oil recovery inslim-tubes, as demonstrated by displacement experi-ments and simulations.75

2.4.3 Field Scale

Dispersion on a microscopic level is usually modelledby a second-order derivative with respect to concen-tration in the continuity equation, Eq. 2.23. As al-ready discussed, the total dispersion coecient con-sists of two terms, Eq. 2.35. The rst, 1/τ , representsmolecular diusion which in the simplest case in de-scribed by Fick's law. With basis in Einstein's rela-tion, Eq. 2.28, mixing-cell and stream-splitting the-ory, it is shown that dispersion on a microscopic levelmay be viewed as a process of the same nature asmolecular diusion. The addition of the two termsof Eq. 2.35 and modelling of dispersion as diusiontherefore seem to be justied.On the reservoir scale, the conditions may be dif-

ferent. Here mixing will generally be caused by largescale heterogeneities such as stratication, shales,blocks of variable permeability, etc. If a reservoir isdivided into blocks of variable size and permeability,and a miscible displacement with unit mobility ra-tio is simulated, the response at the system exit willresemble that of a dispersive process, even thoughthe simulation model contains no dispersive terms.

By analyzing the system response with a dispersionmodel, i.e., Eq. 2.37, an eective dispersion coecientmay be determined. This coecient will be consid-erably larger than the dispersion coecients for theseparate homogeneous blocks, and are caused by mix-ing due to permeability variations between the blocks,and not microscopic dispersion phenomena.Once the cause of mixing on a reservoir scale is

determined, the question arises if it can be considereddispersive, i.e., a process modelled by a second-orderdierential equation. Based on various studies andanalyses, this question has been answered with bothyes76 and no.77

The possible mixing processes in a reservoir may bedivided into three categories:78 (1) dispersive mix-ing as already described, i.e., mixing that can bedescribed by a Fick's law type equation; (2) capac-itive mixing due to mass transfer between uids ow-ing with dierent velocities, also stationary phases;(3) ux-induced mixing due to permeability varia-tions, stratications, shales, etc. If a one-dimensionalinjection process is modelled as dispersive, only areal dispersive process should yield a constant dis-persion coecient. Processes in the two other cate-gories will give variable dispersion coecient as func-tion of amount of uid injected. Correspondingly, thelength of the mixing zone will develop dierently forthe three mixing processes.78

Analyses of an displacement process in a porousmedium divided into blocks of variable and more-or-less correlated permeabilities, showed that a low de-gree of heterogeneity, and a low ratio between lengthand width of the reservoir, favored a dispersive natureof the mixing process.78

Permeability variations and other inhomogeneitiesare therefore the controlling factors for large-scale dis-persion. This is illustrated in Fig. 2.14, where a cor-relation between the extent of the porous mediumand the dispersivity, Kl/v, is shown. The gure isbased on both eld and laboratory measurements ofdispersion.78 Microscopic mixing phenomena deter-mines the total dispersivity in core samples. Theeld-scale dispersivity is often several orders of mag-

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2.4. DISPERSION 19

103

102

101

100

10-1

10-2

10-3

10-1 100 101 102 103 104

Distance, m

Kl/

,m

Field dataLab dataAll dataField data

Figure 2.14: Field and laboratory dispersivity asfunction of the extent of the porous media.78

nitude larger.Microscopic mixing phenomena may also be impor-

tant on a reservoir scale. Fig. 2.15 shows how a slug ofsurfactant may be diluted due to the combined eectsof longitudinal and transversal dispersion for dier-ent transversal dimensionless times Θt (= KtL/vd

2).Suppose that the length of the reservoir is 1000 mand the interstitial velocity 1.75 ·10−6 m/s (0.5 ft/d).With a value of Kt = 5 · 10−10 m2/s for diusion ofsurfactant in liquid, values for Θt of 0.28 and 0.0028are obtained for layer thicknesses of 1 m and 10 m,respectively. With a larger Kt, the importance oftransverse dispersion may increase considerably, e.g.in gas/gas systems.Returning to Fig. 2.15, most of an injected slug may

enter the high permeability layer. In the low perme-ability layer, the slug may become so small that itis destroyed by longitudinal dispersion. If a chemicalslug is injected, adsorption of surfactant will dilutethe injected slugs further, and the result may be thatthe adsorption capacity is only satised in the high-permeability regions. For such systems, transversaldispersion may become an important mechanism forthe process, and large transversal dispersion will bebenecial. Eects of transversal dispersion on pro-cesses in stratied reservoirs are further discussed byLake and Hirasaki.80

Asgarpour et al.81 discuss the importance of mix-ing processes on miscible gas injection. It is shownhow the size of a miscible hydrocarbon slug was de-termined from the dispersivity of the reservoir. Basedon local heterogeneities and transport conditions, amoderate (6% HCPV) slug size was rst calculatedto be sucient. As a result of a re-evaluation, mixing

θt ≈ 0.0

θt ≈ 0.05

θt ≈ 0.2

θt ≈ 0.7

θt > 1.0

Figure 2.15: Eect of transversal dispersion on thedistribution of a surfactant slug in a two-layered sys-tem of dierent permeability.79

eects on a larger scale was taken into consideration,resulting in a doubling of the slug. The process wasrst-contact miscible. A multicontact miscible pro-cess would have required a larger slug, but speciedcalculations was not made.

As a nal example of the eect of dispersion onreservoir scale, gas injection into a gas condensatereservoir will be considered.

Gas condensate reservoirs are often produced byinjection of dry gas to maintain reservoir pressureabove the dewpoint pressure. Nitrogen gas may re-place dry gas as injection uid. Investigations haveshown that the dewpoint pressure increases by addi-tion of nitrogen. The results of slim-tube experimentsnevertheless indicate that more than 98% of the gascondensate may be recovered by use of nitrogen.82

In a standard slim-tube experiment, the dispersivity,Kl/v, is in the order of 3 · 10−5 to 10−4 m, resultingin almost no mixing between injected nitrogen andcondensate.

On a reservoir scale, displacement mixing of theuids by dispersion may reduce the recovery of con-densate signicantly compared with the recovery inslim-tube experiments. By Fig. 2.15, a dispersivity of100 m for a 1000 m long reservoir can be expected.This reduces the recovery of condensate by approxi-mately 30%. 83

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20 CHAPTER 2. PHENOMENA

2.5 Molecular Diusion

2.5.1 Introduction

The transport of molecules in the absence of bulk owis referred to as molecular diusion. In reservoir en-gineering, molecular diusion is of particular impor-tance in miscible displacement, where it expresses thetransport of molecules from one location to anotherdue to a gradient in their concentration.When two miscible uids are brought in direct con-

tact, for example, they will initially be separated bya sharp interface. With time, the two uids will dif-fuse into one another such that the sharp interfacegradually changes to a diuse mixing zone. Molecu-lar diusion arises because of the random motion ofthe molecules in each uid.Molecular diusion can occur in gases, liquids and

dense phases. The diusion of molecules in gases iswell known and can be estimated for high- and low-pressure conditions.84 The diusion of molecules inliquids is less well known, particularly at reservoirpressure conditions. The diusion of molecules indense phases (above the critical point) is the leastwell know condition where diusion occurs. In reser-voir engineering, when estimating the diusion ofmolecules in dense phases, molecular diusion in liq-uids can generally be used as a basis.The mathematics of diusion are presented by

Crank85 and the diusion of gases in porous mediaby Cunningham and Williams.86

The diusion of molecules in uids is expressed interms of a diusion coecient Do, also called diu-sivity. The diusion coecient of molecules in liq-uids is much smaller that the diusion coecient ofmolecules in low-pressure gases. However, diusionrates in liquids need not necessarily be low, since theconcentration gradients can be large.84

Molecular diusion is generally presented as one ofthe phenomena contributing to dispersion, Stalkup.9

The phenomenon of dispersion in porous media ex-presses the longitudinal and transverse mixing thattakes place due to molecular diusion and bulk ow.In the limiting case, where bulk ow approaches zero,molecular diusion is the only phenomenon contribut-ing to dispersion.A review of molecular diusion in hydrocarbon u-

ids and porous media has been presented by Perkinsand Johnston,63 who also discussed dispersion. Theprediction of molecular diusion in gases and liquidsat reservoir conditions has been presented by Sig-mund.87,88 Recent papers on molecular diusion anddispersion in hydrocarbons are those of Grogan etal.89 and Renner.90 Other relevant papers includethose of Arya et al.78 and Correa et al.91

2.5.2 Equations

Consider a cross section perpendicular to a concentra-tion gradient in the mixing zone between two miscibleuids. In the mixing zone, there will be two opposite

concentration gradients for each pure uid relative tothe other. The rate of transfer of a diusing substancethrough a unit cross sectional area is proportional tothe concentration gradient measured normal to thesection.For steady-state, one-dimensional diusion the

mass ux of each miscible uid component is ex-pressed by Fick's rst law,

J = −DodC

dx, (2.41)

where J is diusion ux, Do diusion coecient,C concentration and x distance. The subscript odenotes that the diusion occurs in the absence ofporous media. Fick's rst law can also be expressedas

dn

dt= −DoA

dC

dx, (2.42)

where n is amount of chemical component (mole-cules), t is time, and A cross-sectional area. Normalderivatives (d's) are used in the above equations be-cause the molecular diusion process is stead-stateand the diusivity coecient assumed constant.Unsteady-state, one-dimensional diusion, how-

ever, is described by Fick's second law,

∂C

∂t= Do

∂2C

∂x2. (2.43)

This is the well know diusion equation.85 Moleculardiusion at an initially sharp interface, such as be-tween two miscible uids, is illustrated in Fig. 2.16.Fick's second law describes the concentration of each

Solvent

Solution

Time = 0 Time = 0 Time = t Time = tc c c

(a) (b) (c) (d)

Figure 2.16: Diusion at an initially sharp boundaryin a cell of uniform cross section.93

of the miscible uid components with distance x andtime t. The general solution of the diusion equationfor uids at rest can be represented as

C =1

2

[1± erf

(x

2√Dot

)], (2.44)

in terms of the error function. For miscible dis-placement situations, the initial and boundary con-ditions/values need to be specied. The details of us-ing more involved solutions of the diusion equationto study molecular diusion are presented by Killie,92

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2.5. MOLECULAR DIFFUSION 21

for example. The solution of the diusion equationindicates that the spread of the mixing zone in mis-cible displacement will be proportional to the squareroot of the diusion coecient.12 This fact is gener-ally used in laboratory experiments to determine themolecular diusion coecient.The above general solution of the diusion equation

is for two miscible uids at rest. The distance x canbe thought of as the distance away from the initialsharp interface in either direction. However, if thetwo uids are moving at some uid velocity u, thedistance x can be replaced by the eective distance

x = x− u t. (2.45)

If the interstitial velocity v = q/(φA) is used insteadof the uid velocity u = q/A for moving uids, theeective diusivity coecient D should be used inthe above general solution of the diusivity equation.The term q is volumetric owrate and φ formationporosity.The steady-state and unsteady-state diusion

equations, Fick's rst and second law, respectively,can be used in the interpretation of laboratory mea-surements of the diusion coecient.12,90,92 The the-oretical concentration proles at ve dierent times(unsteady-state) in a 13 mm long diusion cell areshown in Fig. 2.17.92 The diusion coecient used

t1

t2t3

t4

t5

Cell length, m

Con

cent

ratio

n

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-0.20.000 0.005 0.010

Figure 2.17: Concentration proles in a diusioncell.92

was 0.775×10−9 m2/s and the timesteps t1 to t5 cor-respond to 0, 2, 5, 20, and 120 minutes, respectively.

2.5.3 Diusion Coecient

The diusion coecient (diusivity) is generally notconstant. It depends on temperature and also variessomewhat with concentration and pressure. Thevalue of the diusion coecient also depends uponwhat other kinds of molecules are present. For exam-ple, the diusivity coecient is more likely constantin nonpolar hydrocarbons than highly polar aqueoussolutions. In the majority of reservoir engineering ap-plications, the diusion coecient can be treated asa constant.The actual driving force of diusion is the gradient

of chemical potential, and not the concentration gra-dient; that is, chemical activity instead of chemical

concentration should ideally be used to characterizemolecular diusion. The chemical potential is denedas the rate of change in Gibbs free energy of a systemwith number of moles of a chemical component whenthe temperature, pressure, and the sum of moles ofall components are held constant.93 However, in di-lute solutions the chemical activity can be replacedby the chemical concentration, especially when themolecules are nonpolar.Diusion in porous media can been described by

the general diusion equation with the introductionof an eective diusion coecient D, that dependsupton the texture of the porous medium.94 Further-more, it can be argued that molecular diusion is sim-ilar to the conduction of electricity; that is, molecularconcentration, mass ux, and diusion coecient areanalogous to electrical potential, intensity and con-ductivity, respectively.Experiments to determine the electrical conductiv-

ity of porous media saturated by an electrolyte givevalues for the formation electrical resistivity factor F .It is dened as the ratio of the electrical conductivityof the uid phase alone, to the electrical conductivityof the saturated medium. The formation electricalresistivity factor can be correlated by the expression

D

Do=

1

Fφ. (2.46)

Published data for the eective diusivity coe-cient in porous media presented by Perkins and John-ston63 show that the value of Fφ typically rangesfrom about 1.25 to 1.65, depending on the texture ofthe medium, see also Ref. 12 for details. The electri-cal resistivity factor-porosity term Fφ is sometimesreplaced by the tortuosity factor τ . It is reportedby Ruthven95 that experimental tortuosity factors forporous media generally fall within the range 2 to 6.

2.5.4 Measurements

Comparatively few studies have been concerned withthe measurement of molecular diusion coecients atreservoir conditions.89,92 What follows are examplesof reported measurements.Brow et al.96 measured the diusivity of methane

in crude oil at 18 MPa and 70 C. They report adiusion coecient of 3.34×10−9 m2/s. This value issimilar in magnitude to those for methane in normalparan solvents measured at atmospheric pressureand temperature from 0 to 50 C reported by Haydukand Buckley.97

Sigmund87 presented measurements of the diusiv-ity of methane in propane and butane for pressures inthe range of 1.5 to 20 MPa and temperatures from 35to 105 C. The reported diusion coecients are inthe range of 16× 10−9 to 76× 10−9 m2/s. These arehigher than the diusivities for methane in normalparans measured at atmospheric pressure by Browet al.96

It has been showed by Hayduk and Cheng98 thatdiusivity in hydrocarbons increases with decreasing

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22 CHAPTER 2. PHENOMENA

solvent viscosity. Grogan et al.89 suggested that thedierences in the diusivity measurements Brow etal.96 and Sigmund87,88 are caused by dierences insolvent viscosity.The diusivity (diusion coecient) of nitrogen

in octane and decane with temperature has recentlybeen reported by Killie.92 The results show that thediusivity increases markedly with temperature, indi-cating that the diusivity decreases with solvent vis-cosity, as expected. However, the results also indicatethat diusivity decreases with pressure. The diusiv-ity of nitrogen in octane at 15 MPa and 100 C wasabout 12× 10−9 m2/s. The diusivity of nitrogen indecane at 100 C was also about 12 × 10−9 m2/s at7.5 MPa and slightly lower, at 10×10−9 m2/s, at thepressure of 15 MPa.Additional diusivity measurements in hydrocar-

bon systems at reservoir conditions (high pressureand temperature) are also reported.99102

2.5.5 Correlations

The prediction of diusion coecients in liquids isdiscussed by Reid et al.84 They point out that liquidstate theories for estimating diusion coecients arequite idealized and none are satisfactory. However,the form of the Stokes-Einstein equation has provideda framework for several useful prediction methods.For example, Grogan et al.89 used

Do = 5.72× 10−12T

µ(2.47)

to estimate the diusion coecient of carbon dioxidein liquid water and

Do = 1.41× 1

µ0.47(2.48)

to estimate the diusion coecient of carbon diox-ide in liquid hydrocarbons. The viscosity term in theabove equations is that of the liquid phase; the sol-vent.In hydrocarbon reservoirs, it is important to know

the eects of both temperature and pressure on thediusion coecient. The eect of temperature canbe considerable as expressed in the above Stokes-Einstein type equations, both directly and indirectly(through the liquid viscosity). The eect of pressureon the diusion coecient in liquid hydrocarbon isreported much less than that of temperature. Con-icting reports on whether pressure aects the diu-sion coecient have recently appeared in the litera-ture.89,90

Grogan et al.89 state that the evidence availablesuggests that diusivities for methane at reservoirconditions are not greatly dierent from those mea-sured at atmospheric conditions, provided that sol-vent viscosities are similar. Furthermore, they con-cluded that correlations for the diusivity of car-bon dioxide in hydrocarbon solvents in terms of sol-vent viscosity, developed from measurements at at-

mospheric pressure, provide realistic estimates for thediusion coecient at reservoir conditions.Renner,90 however, states that high-pressure diu-

sion coecients are signicantly dierent from thosemeasured at atmospheric pressure. To represent bothhigh- and low-pressure diusion coecients in liq-uid hydrocarbons with one empirical Stokes-Einsteintype equation, Renner90 correlated 141 experimentaldiusion coecient values (including the data pre-sented by Grogan et al.89) for the following parame-ters: temperature 32 to 140 F (0 to 60 C), pressure14.7 to 2560 psia (0.1 to 17.7 MPa), liquid hydrocar-bon viscosity 0.2 to 134 cp, gas viscosity 0.0088 to0.019 cp, molecular weight of gas 16 to 44 kg/kmol,molar volume of gas 0.145 to 26.5 m3/kmol. The gasviscosity was not found to be a signicant parameterin the correlation work.A least-squares t of the 141 experimental diusion

coecients as a function of liquid viscosity µL, gasmolecular weight Mg, gas molar volume VMg, pres-sure p and temperature T , led Renner90 to the follow-ing relationship for the diusion coecient in liquidhydrocarbons

Do = 10−9 × T 4.524

µ0.4562L M0.6898

g V 1.706Mg p1.831

, (2.49)

where the diusion coecient has the unit m2/s, andthe other parameters the eld units given above. The141 experimental points and the correlation are plot-ted in Fig. 2.18. The Renner-correlation is based on

20

10

1

0.10.1 1 10

Dif

fusi

onco

effi

cien

t,10

-9m

2 /s

(µj-0.4562 Mi

-0.6898 Vi-1.706 p-1.831 T 4.524)

Figure 2.18: Diusion coecient in liquid hydrocar-bons, is Eq. 2.49, from Renner.90

the diusion coecient of carbon dioxide, methane,ethane and propane in hydrocarbon mixtures; it isrecommended for diusion coecients at reservoirconditions.Renner90 also correlated experimental data for the

diusion coecient of carbon dioxide in liquid waterand brine. It was found that water/brine (liquid)viscosity and gas viscosity both will correlate with thediusion coecient. Molecular weight of gas, molarvolume of gas, pressure and temperature were not

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2.5. MOLECULAR DIFFUSION 23

found to be signicant. A least-squares t of bothhigh- and low-pressure data led to the expression

Do = 6.391× 103µ6.911g

µ0.1584L

, (2.50)

where the unit of viscosity is cp. The correlation andexperimental data are plotted in Fig. 2.19.

10

10.5 1 10

Dif

fusi

onco

effi

cien

t,10

-9m

2 /s

10-12 (µj-0.1584 µj

6.911)

Figure 2.19: Diusion coecient of carbon dioxidein liquid water and brine: 4 high pressure, 4 atmo-spheric pressure, atmospheric pressure, and isEq. 2.50, from Renner.90

An empirical correlation has been presented bySigmund87,88 for molecular diusion in hydrocarbongases and multicomponent mixtures at reservoir con-ditions. The correlation is formulated as a density-diusion coecient product expressed in terms of apolynomial,

ρMDo

ρoMDoo

= a1 + a2ρMr − a3ρ2Mr + a4ρ

3Mr, (2.51)

where ρMr is the reduced molar density

ρMr =ρMρMc

. (2.52)

Note that ρM and ρMc are the molar density and crit-ical molar density, respectively, not the mass density.The superscript o in the polynomial equation refersto the original physical state. That is, if the diu-sion coecient for a particular hydrocarbon mixtureis known at some temperature and pressure (originalstate), the diusion coecient of the same mixturecan be estimated at any other physical state (othertemperature and pressure). It should be noted thatthe subscript o on the diusion coecient in Fick'slaws refers to molecular diusion in uids (gases, liq-uids, dense phase) in the absence of porous media.The polynomial coecients in the Sigmund87,88

correlation have the values a1 = 0.99589, a2 =0.096016, a3 = 0.22035, and a4 = 0.032874. The cor-relation has recently been discussed by da Silva andBelery103 who recommended the following equationswhen ρMr > 3

ρMDo

ρoMDoo

= a5 exp(3− ρMr) (2.53)

where the constant a5 = 0.18839. They pointed outthat the Sigmund87,88 correlation is convenient inreservoir engineering applications, primarily becauseit requires only the critical properties of the chemi-cal components of a hydrocarbon mixture and othercommon parameters used in any equations of state.They include an overview of molecular diusion inhydrocarbon reservoirs.

2.5.6 Diusion in Reservoirs

The molecular diusion coecient Do of hydrocar-bons in mixtures (gases, liquids and dense phase) atreservoir conditions is reported to be in the range 1-10×10−9 m2/s. The corresponding eective moleculardiusion coecient D in porous media is reportedto be signicantly lower, perhaps 50% lower, if thetortuosity factor τ=2. Nevertheless, molecular dif-fusion is an important phenomenon in miscible dis-placement and other mass transfer processes in hy-drocarbon reservoirs.The mass ux of chemical components (kg/m2·s)

across a reservoir interface (m2) depends on theirconcentration gradients dC/dx and diusivity coef-cients D (m2/s). The total mass transfer of thechemical components (kg), however, depends also onthe extent (size) of the eective interface; that is,the larger the interface, the larger the exchange ofmass. It follows that molecular diusion in hydrocar-bon reservoirs is sensitive to the details of uid owin porous and fractured media; the details of complexinteractions between uids and formations.Two contrasting examples serve to illustrate the

importance of molecular diusion in reservoir masstransfer processes. Consider a piston-like (mobilityratio of unity) miscible displacement process in anideal homogeneous reservoir formation. In this sit-uation, the interface between the two uids will berelatively sharp such that the mixing zone developsslowly with time, Fig. 2.16. The amount of chemicalcomponents diusing between the two uids will belimited.Consider a miscible displacement process between

uids of greatly diering mobilities taking place ina highly heterogeneous reservoir, resulting in chan-nelling and ngering. In this situation, the interfacebetween the two uids will be highly diuse and tendsto be spread over a large volume of the reservoir. Theamount of chemical components diusing between thetwo uids will consequently be large, primarily be-cause of the large interface.The relationship between reservoir heterogeneity

and dispersion (including diusion) has, for examplebeen studied by Arya et al.78 The eect of molecu-lar diusion in naturally fractured reservoirs has beenstudied by da Silva and Belery.103 In this situation,the interfacial area between the miscible displacementuid and liquid saturated matrix blocks is so largethat molecular diusion markedly aects the recov-ery of liquid hydrocarbons. It was concluded by daSilva and Belery103 that molecular diusion in natu-

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24 CHAPTER 2. PHENOMENA

rally fractured reservoirs is a rapid phenomenon andmay override other more common hydrocarbon dis-placement mechanisms.

2.6 Frontal Instabilities

2.6.1 Introduction

The phenomenon of frontal instability appears onmany scales but is most frequently studied in the lab-oratory.Instabilities originate on the interface between e.g.

oil and a displacing uid that may be part of a sec-ondary oil recovery process. They take place both inmiscible and immiscible displacement processes andare often characterized by patterns of penetrating n-gers of displacing uid, so-called viscous ngers.

2.6.2 Basics

We will restrict our investigation to the study of hy-drodynamically initiated instabilities,104 and there-fore only to Newtonian uids, for which Darcy's lawis applicable. We start with the case of idealized im-miscible ow where dispersion is neglected, i.e., thereexists a sharp front between the two uids and no cap-illary forces are active. To distinguish the two uids,we call them oil and water.With gravity included, we may write

∇p = −µk~u+ ρ~g. (2.54)

In Fig. 2.20 is shown the linear displacement of oil bywater where the average front has reached position x.Innitesimal perturbations on the front have causedpoint A− to slow behind and point B+ to be some-what ahead of x. The pressure drop δp between thetwo extreme positions x+ and x− is, from Eq. 2.54,

δpA =

−µokou− ρog

δx, (2.55a)

δpB =

−µwkw

u− ρwgδx, (2.55b)

where δx = x+ − x−, and ~u and ~g are parallel andopposite (Fig. 2.20).The pressure dierence, ∆p = δpB − δpA, now de-

termines further development. If ∆p > 0, the per-turbation will grow and a water nger will developinto the oil, and an oil nger in the opposite direc-tion (with the average front position as reference).If ∆p < 0, the perturbations are dampened and thedisplacement process is stable.Using Eq. 2.55b, we may express the pressure dif-

ference as

∆p =

(µoko− µwkw

)u− (ρw − ρo)g

δx. (2.56)

Using the mobility λ = k/µ and the mobility ratioM = λw/λo, Eq. 2.56 can be written

∆p =

(M − 1)

µwkw

u−∆ρwog

δx, (2.57)

A+ B+

A- B-

x

x+

x-

→g

water

oil

Figure 2.20: Innitesimal front perturbation, velocityand gravity are parallel and opposite.

where ∆ρwo = ρw − ρo is the density dierence.If ρw > ρo, gravity is a stabilizing force. When

M > 1, for ow pattern as in Fig. 2.20, with velocityand gravity in opposite directions, the pressure dif-ference may be positive (∆p > 0) if M is large, i.e.,unstable displacement.If M > 1, the front may be conditionally stable,

dependent on the magnitude of the Darcy velocity.When ∆p = 0 in Eq. 2.57, the critical velocity, uc, isfound by

uc =∆ρwog

(M − 1)µw/kw. (2.58)

Eq. 2.58 is a direct result of the more general hy-drodynamic stability analysis given by Saman andTaylor.105

The heuristic and approximative analysis per-formed related to Fig. 2.20 should be considered moreillustrative than complete.105

Evaluating the instability analyses presented in theliterature, one should be aware of the limitations ofthe pertubative approach. Since the perturbationsare assumed to be innitesimal, no predictions fromthis method can be made for the growth of ngers andtheir inuence on breakthrough times and recovery.Several important factors are not taken into ac-

count in the analysis:

1. In the immiscible case, surface tension and wet-tability should be included,

2. Dispersion phenomena, describing the micro-scopic mixing of uids on the pore level, are im-portant. In miscible ow, diusion processes alsohave to be considered,

3. The shape of the porous medium and the bound-ary condition have to be considered in all ood-ing experiments.

2.6.3 Hele-Shaw Cells

Improved understanding of the mechanisms of frontalinstabilities has been gained through various ow

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2.6. FRONTAL INSTABILITIES 25

studies using Hele-Shaw cells,106,107 both experimen-tally and through numerical investigations. The owtakes place between two parallel plates with interplatedistance of capillary dimensions (small compared toall other length scales).The similarity with ow through porous media is

based on the fact that hydrodynamic ow in Hele-Shaw cells can be described by Darcy's law (restrictedto single-phase ow for low Reynolds numbers).108,109

The permeability is given as a function of the widthbetween the plates, k = b2/12.The analogy between ow in Hele-Shaw cells and

in porous media is far from perfect. For miscible owthrough a porous medium, mixing on pore level leadsto dispersion both parallel and normal to the ow di-rection, characterized by longitudinal and transversedispersion coecients Kl and Kt. In Hele-Shaw cells,only longitudinal dispersion occurs. Also, multiphaseand multicontact ow takes place for immiscible owin porous media, with menisci between oil and watermaking a transition region. This is not observed inHele-Shaw cells.The popularity of Hele-shaw cells is therefore

hardly their resemblance to porous media, but morelikely that they readily can be inspected, with directoptical access and ease of operations.Surface tension and dispersion are not believed to

alter the stability criteria, Eq. 2.56, but rather tomodify the nonstabilized ow. Therefore, Hele-Shawcells have been quite popular in eorts to gain moreknowledge about ow behavior in porous media.

2.6.4 Experiments

For immiscible ow, it is practical to quantify theeect of surface tension by the capillary numberNca = µu/σ, the ratio between viscous and capillaryforces.For unstable ow, the surface tension both pro-

motes and dampens the growth of viscous ngers.This slightly contradictory statement is to be under-stood by the following arguments: The general eectof surface tension is to suppress any enlargement ofthe surface that incipient ngers represent. In thecase of an already well developed nger, surface ten-sion will prevent small perturbations to develop onthe nger surface, forcing all uid entering the ngerto promote its growth. For a given set of parameters,there is therefore a preferred size (width and length)of ngers that will develop in a porous medium.When Nca increases, the viscous forces will domi-

nate and instabilities appear on many scales. WhenNca is large, the dynamics of ngering is very complexand the system becomes chaotic.At low Nca-numbers, a few or a single nger may

characterize the ow, and the evolution of those n-gers can be described through a process of shielding,spreading and splitting:104

shielding: A nger ahead runs faster than ngersbehind due to the instability condition.

spreading: Surface tension acts to spread the dom-inant nger to a preferable width.

splitting: Fingers that grows broader than the op-timal width, will become unstable, leading to asplitting at the nger tip.

New ngers created by tip splitting are stable andagain subjected to shielding and spreading which en-sure a cyclic behavior. To promote this type of n-gering, the surface tension should be suciently highto allow the tip to become unstable and low enoughto cause spreading (maintaining the nger form).For miscible ow, no capillary forces are active,

σ = 0, and dispersion plays the same role as sur-face tension for immiscible ow. In analogy, thecompetition between viscous ow and dispersion isgiven by the Peclet number, NPe, see Christiansenand Fanchi110 and Sec. 2.3.4.The net eect of dispersion is to reduce instabilities

by dampening the growth of incipient ngers. In prin-ciple, all miscible displacement processes are stable ifdispersion is given enough time to act. Therefore,the importance of the Peclet number is observed tobe similar to that of the capillary number for immis-cible ow. The miscible ow processes of spreadingand splitting are active as described before, while theprocess of shielding does not occur at the same rateas observed in immiscible ow.104

Nearly all experimental investigations of instabilityand viscous ngering, with a few exceptions,111 havebeen carried out using unconsolidated porous mediaor models in two and three dimensions. Most porousmedia studied therefore tend to have higher eectivepermeability than normally found in reservoirs. Also,a more homogeneous pore-throat distribution is ob-served in these media, leading to a rather at cap-illary pressure curve.If viscous-nger development may be scaled to

pore-size level, results obtained through laboratoryexperiments may not accurately describe what hap-pens in consolidated porous media, where dispersionand surface tension would both depend on the char-acteristics mentioned.The formation of viscous ngers is to a large ex-

tent determined by properties of the uids owing.The quantitative aspects of surface tension, whetherσ > 0 or σ ∼ 0, are not decisive for instabilities to oc-cur. However, surface tension does inuence the formand evolution of viscous ngers, when they develop,together with other parameters like σ(Nca), K(NPe),and Swi.In a pioneering work, Chuoke112 found surface ten-

sion to consolidate the form and promote the enlarge-ment of the ngers. When the capillary number in-creases, viscous forces dominate and many small n-gers are formed. It has also been observed that inthis case narrow ngers are dampened.113

In the opposite case, when Nca is small, large n-gers are allowed to grow while small instabilities aredampened. For a given set of parameters leading

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26 CHAPTER 2. PHENOMENA

to unstable ow, instabilities on separate scales aredampened dierently, leading to an optimum, pre-ferred number of ngers formed with a specic widthto length ratio.114

It is also observed that viscous ngering commencesat lower capillary number than needed to mobilizeresidual oil.115 The oil saturation behind the viscousfront may therefore be strongly dominated by the pro-cess of viscous ngering.In the miscible case, important parameters like vis-

cosity and diusion are functions of concentration andif it varies smoothly between the two uids, stabilityis improved.In Hele-Shaw experiments, both dispersion and dif-

fusion are important, while in porous media diusionis less important. Dispersion is found to be activeboth longitudinally and transversally to the directionof ow, and increasing values of Kl and Kt reducethe eect of unstable ow by adding to the smooth-ing process. The eect of dispersion in miscible owis analogous to the eect of surface tension in im-miscible ow. Transversal dispersion Kt is found tosmear out the ngers more eciently than Kl, whileKt may damp the nger growth in laboratory exper-iments more than in the reservoir.116

Existence of connate water may promote ngeringby enhancing dispersion112 and larger ngers may de-velop.117 In waterwet porous media, or in cases whereconnate water is present, water ngers are observedto grow much wider than oil ngers.118 The increasedinstability caused by these eects also lead to moreirregular ngers and frequent tip splitting.104

Analogous to diusion, dispersion is considered astatistical process and the presence of connate wa-ter tend to increase instability, but at the same timeto make the ow pattern more predictable.119 Thengers grow faster but also more regular.Instability and unstable ow patterns may de-

pend on the dimensions of the actual porousmedium.113,120 The global dimensions of the porousmedium is normally expressed by the dimensionlessaspect ratio AR = h/b, height divided by the widthof the porous medium. In Fig. 2.21 are shown threeporous media of dierent aspect ratios, illustratingthe range from 2D vertical models in the laboratory,to horizontal reservoirs of limited range.

A R = 1 AR= 0.05

Lab Reservoir

A R =5

Figure 2.21: Shape of three porous media for dierentaspect ratios.

In laboratory experiments, aspect ratios aroundunity is frequent. For reservoirs, aspect ratios vary,

and are often much less than one, AR 1. It istherefore somewhat disturbing when it is observedthat ow becomes potentially more unstable whenthe aspect ratio decreases.114 Displacement processesthat are stable or conditionally stable in the labora-tory, will for the same choice of parameters becomeunstable in the reservoir. Enhanced instability willlead to an increased number of ngers, but partly dueto merging, relatively fewer ngers may be observedwhen AR decreases.Stability also depends on the nature of hetero-

geneities in the porous medium.113 Heterogeneitiesembedded in the medium and characterized bymacroscopic dimensions have a tendency to force n-gers to join and thereby reduce their number. In con-tradiction, ngers are also known to be initiated byand to feed on heterogeneities.Abrupt permeability variations near the entrance

of the porous medium will normally generate morengers than if the same permeability variation hadoccurred farther downstream.121 It seems that initi-ation of viscous ngering is more sensitive to hetero-geneities than the development of individual ngers.It has been observed in simulation of unstable, mis-

cible displacement processes that generation of ngersdoes not happen unless instabilities of some kind areintroduced. When concentration gradients or perme-ability variations are introduced in these simulationmodels, then also ngers arise. This could lead to theconclusion that onset of instability is initiated by hy-drodynamic perturbations, perhaps on a microscopicscale, and that the onset mechanism is statistical bynature. In a porous medium, both permeability vari-ations (heterogeneities on a local scale) and hydrody-namic perturbations could be considered the sourceof instability.The growth of a stable nger under unstable ow

conditions is known to be proportional to time whileits shape depends on the rate of ow.111 The n-ger width is observed by several researchers122124 tovary with the square root of time. The number ofngers created is thought to be independent of owrate,111 but dependent on the mobility ratio and thepermeability variation.125

2.6.5 Theoretical Development

Introduction

Numerous theoretical attempts have been made todescribe frontal instability. They may be grouped inthree classes:

• Theories based on percolation and Diusion-Limited Aggregation (DLA) have been used tosimulate ngering.126 Other researchers havetried statistical models like random walk127 ormodels based on a probabilistic approach likeMonte Carlo simulations.128,129 Common tothese approaches is the lack of generality in thesolutionsone simulation is the answer only fora particular set of parameters.

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2.6. FRONTAL INSTABILITIES 27

• A second class is based on the analy-sis of a Buckley-Leverett type of displace-ment,113,130133 where the solution of nonlineardierential equations is an integrated part of theproblem. This is normally done using numeri-cal methods, and numerical dispersion and in-stability become an additional complicating fac-tor. These methods rely on relative permeabilityrelationships and capillary pressure data derivedfrom experiments that are interpreted as if theywere stable displacements.

• A third class of researchers use perturbationtheory as a tool in the investigation of nger-ing.105,112,118,123 The basic idea has been pre-sented above.

Description of unstable ow behavior was intro-duced through perturbation analysis by Taylor in theearly 1950's. Later, Saman and Taylor105 intro-duced a formal instability theory. Chuoke112 modi-ed the theory for the immiscible case to include sur-face tension. Continued interest in this eld seemsto be declining until Gaupta123 in 1974 included sys-tem permeabilities and wettabilities. In 1981, Pe-ters and Flock118 presented a theory where a dimen-sionless group of parameters (an instability number)was used to delineate the stable ow regime. Inthese studies, only incipient ngers were consideredand the importance of the physical dimensions of theporous medium were not properly recognized. Later,Bentsen134 introduced a force potential, making itpossible to study growing ngers by the solution ofa moving boundary problem. In this theory, severalnew conditions were imposed and the physical dimen-sions are included.In presenting a more complete theoretical descrip-

tion of viscous ngering, it is interesting to note thatthe theory has been veried through experiments,such as in Hele-Shaw cells,135 in a rectangular un-consolidated porous medium,136 and in long tubularporous media.137

Pertubation Analysis

The ngers are treated as perturbations on a moving,piston-like surface, where a force potential, Φ(σ, µ, u),is evaluated across the interfacial surface; Φ shouldhere be associated with the liquid pressure p inEq. 2.54. The surface separating the two immisci-ble uids is found by solving the Laplace equation bythe method of separation of variables. The solutiondescribing the interface is given in terms of a charac-teristic wave function where the frequency is associ-ated with the number of ngers occurring and theirrespective widths. When the force potential acrossthe interface becomes larger than zero (∆Φ > 0), vis-cous ngers develop.When proper boundary conditions (e.g. maximum

nger width is equal to tube radius) and the contin-uum principles are considered, then ∆Φ = 0 givesthe critical wavelength which is to be associated with

the physical situation when the viscous forces exactlybalance the gravity and capillary forces. The stabil-ity analysis, including all parameters of importance,produces a dimensionless number,118 Isr, which fortubular porous media is given by137

Isr =µwu(M − 1−NG)

krwσe

(M2/3 + 1

2M2/3

)K2

≤ 13.65, (2.59)

where

M =krwµw

µokro

, NG =∆ρgkrw cosα

µwu.

Here, krw and kro are the endpoint relative perme-abilities of water and oil, respectively; NG the dimen-sionless gravity number, relating gravity to viscousforces; σe the eective interfacial tension, incorporat-ing the thickness of the transition zone between thetwo uids.When Isr ≤ 13.65, the ow is stable. As seen from

Eq. 2.59, the criterion for stability depends on a groupof parameters and not just the mobility-gravity rela-tion as in Eq. 2.58.The relationship between frontal stability and

groups of parameters in Eq. 2.59 has been veried ex-perimentally. Data from several investigators,118,138

indicate that a pseudostable region starts at Isr =450. Fig. 2.22 shows a comparison between theoryand experiments,137 where the recovery at break-through is constant both in the stable and pseu-dostable region.

10-1 100 101 102 103 104

1.0

0.8

0.6

0.4

0.2

0.0

Instability Number, Isc

Bre

akth

roug

hR

ecov

ery,

(fra

ctio

nof

OO

IP)

Stableregion

Transitionregion

Pseudostableregion

Figure 2.22: Recovery at breakthrough as function ofthe instability number.137

Nomenclature

A = area, m2

= constantAR = aspect ratio, dimensionless

Awr = rock area per unit volume of rock, m2/m3

a = activity= parameter for Langmuir adsorption model

Eq 2.15, kg/m3

= radius of capillary tube, m= coecients/constants, a1 · · · a5

b = width, mCs = excess surface concentration, mol/m2

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28 CHAPTER 2. PHENOMENA

= surfactant concentration, kg/m3 ormol/m3

f = fractional pore space occupied by mobileuid

D = (eective) diusion coecient, m2/sDo = (bulk) diusion coecient, m2/sd = thickness or diameter, mF = formation resistivity factorG = Gibb's free energy, Jg = gravitational constant, m/s2

h = height, mIsr = instability number, dimensionlessJ = ux, mol/m2sK = (longitudinal) dispersion coecient, m2/sK ′ = coecient for convective mixing, m2/sKeq = adsorption constant (= k1/k2)k = mass transfer coecient, s−1

= permeability, m2

k1 = kinetic rate constant for adsorption,m3/kg s

k2 = kinetic rate constant for desorption, s−1

L = length of porous medium, m∆L = mixing zone length, mM = molecular weight

= mobility ratio, dimensionlessm = exponent (Eq. 2.35)

Nca = capillary numberNG = gravity number

NPe† = Peclet number, udp/Do or udp/Kk (k = l

or t)NRe

‡ = Reynolds number (here ρudp/µ)n = amount, mol= number of mixing chambers

p = pressure, Pa or psi or bar or atmpmm = MMP, bar or psigp∗mm = correlating number of calculating MMPQa = adsorbent capacity, kg/m2

q = volumetric owrate, m3/sR = gas constant, 8.314 J/mol K= source/sink, mol/m3 s

r = radius, mS = surfactant

SΘ = adsorbed surfactantT = temperature, K or C or Ft = time, su = Darcy velocity, m/sV = volume, m3

VM = molar volume, m3/molv = interstitial velocity, m/sx = molecular weight of C2 through C6 in in-

jection gas= mole fraction= distance, m

x = characteristic distance of diusive event,m

y = corrected molecular weight of C7+ in

†note that Peclet numbers are related either to moleculardiusion or dispersion coecients.‡dp is used as characteristic length rather than hydraulic

diameter. This gives turbulent ow at lower NRe than usuallyseen.

stock-tank oilz = mole% methane in injection gasα = tilt angle, radians

Γ = surface excess of surfactant in oodedpores, kg/m2

Γ∗ = surface excess of surfactant in stagnantvolumes, kg/m2

Γm = surface excess of surfactant at monolayeradsorption, kg/m2

γ = specic gravityΘc = contact angle

= rock surfaceΘt = dimensionless timeλ = width of viscous nger, m= mobility

µ = chemical potential, J/mol= viscosity, Pa·s

ρ = density, kg/m3 or mol/m3

σ = surface/interfacial tension, N/m= packing or inhomogeneity factor

τ = tortuosity factorτ = characteristic time of diusive event, sΦ = force potential, Paφ = porosity

Subscripts

c = criticalD = diusionale = eectiveg = gasi = component

int = intermediatej = phasek = k ∈ l, tL = liquidl = longitudinal

M = molarm = minimum

= miscibilityo = oilp = particler = reducedS = solids = surface or surfactantt = transversalV = vaporvol = volatileW = waterw = water

Superscripts

∗ = stagnant volume = endpointo = original or initial

Operators

∆ = dierenceδ = innitesimal dierence

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REFERENCES 29

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30 CHAPTER 2. PHENOMENA

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32 CHAPTER 2. PHENOMENA

[88] Sigmund, P.M.: Prediction of Molecular Diu-sion at Reservoir Conditions. Part II - Estimat-ing the Eects of Molecular Diusion and Con-vective Mixing in Multicomponent Systems,JCPT (July-Sept. 1976) 532.

[89] Grogan, A.T., Pinczewski, V.W., Ruskau,G.J. and Orr, F.M. Jr.: Diusion of CO2 atReservoir Conditions: Models and Measure-ments, SPERE (Feb. 1988) 93102.

[90] Renner, T.A.: Measurement and Correlationof Diusion Coecients for CO2 and Rich-GasApplications,SPERE (May 1988) 51723.

[91] Correa, A.C., Pande, K.K., Ramey Jr., H.J.,and Brigham, W.E.: Computation and In-terpretation of Miscible Displacement Per-formance in Heterogeneous Porous Media,SPERE (Feb. 1990) 6978.

[92] Killie, S.: Interferometric Technique for Deter-mination of Diusion at Elevated Temperaturesand Pressures, Dr. Ing. Thesis, Norwegian In-stitute of Technology, Trondheim (1989).

[93] Daniels, F. and Alberty, R.A.: Physical Chem-istry, 3rd ed., John Wiley & Sons, New Yory(1967).

[94] Fried, J.J. and Comparnous, M.A.: Dispersionin Porous Media, Advances in Hydroscience,V.T. Chow (ed.), Academic Press (1971) 7, 169-282.

[95] Ruthven, D.M.: Principles of Adsorption andAdsorption Processes, John Wiley & Sons, NewYory (1984).

[96] Brow G.T., Cupps, C.Q., and Fry, J.: Methodfor Testing Rate of Gas Diusion in CrudeOil by Periodic Measurements of ConcentrationProles, Report RI-7359, U.S. Bureau of Mines(March 1970).

[97] Hayduk, W. and Buckley, W.D.: Eect ofMolecule Size and Shape on Diusivity in Di-lute Liquid Solutions, Chem. Eng. Sci. (1972)27, 19972003.

[98] Hayduk, W. and Cheng, S.C.: Review of Rela-tion Between Diusivity and Solvent Viscosityin Dilute Liquid Solutions, Chem. Eng. Sci.(1971) 26, 63546.

[99] Reamer, H.H., Opfell, J.B., and Sage,B.H.: Diusion Coecients in HydrocarbonSystems. Methane-Decane-Methane in LiquidPhase, Ind. Eng. Chem. (1956) 48, 27582.

[100] Gavalas, G.R., Reamer, H.H., and Sage, B.H.:Diusion Coecients in Hydrocarbon Sys-tems. Homogeneous Phases at Elevated Pres-sures, Ind. Eng. Chem. Fundamentals (1968)17, 30612.

[101] Atwood J.G. and Goldstein, J.: Measurementsof Diusion Coecients in Liquids at Atmo-spheric and Elevated Pressure by the Chro-matographic Broadening Technique, J. Phys.Chem (1984) 88, 197585.

[102] Erkey, C. and Akgerman, A.: Translational-Rotational Coupling Parameters for Diusion

in n-Octane, Trans., AIChE J. (1989) 35, 443-8.

[103] Da Silva, F.V. and Belery, P.: Molecular Diu-sion in Naturally Fractured Reservoirs: A De-cisive Recovery Mechanism, paper SPE 19672presented at the 1989 SPE Annual TechnicalConference and Exhibition, San Antonio, Oct.811.

[104] Homsy, G.M.: Viscous Fingering in PorousMedia, Ann. Rev. Fluid Mech. (1987) 19, 271311.

[105] Saman, P.G. and Taylor, G.I.: The Penetra-tion of Fluids into a Porous Medium or Hele-Shaw Cell Containing a More Viscous Liquid,Proc., Soc. London Ser. A245 (1959) 312329.

[106] Howison, S.D.: Fingering in Hele-Shaw Cells,J. Fluid Mech. (1986) 167, 43953.

[107] Park, C.W. Gorell, S., and Homsy, G.M.: Two-Phase Displacement in Hele-Shaw Cells: Ex-periments on Viscously Driven Instabilities, J.Fluid Mech. (1984) 141, 25787.

[108] Saman, P.G.: Viscous Fingering in Hele-Shaw Cells, J. Fluid Mech., (1986) 173, 7394.

[109] Saman, P.G.: Exact Solutions for the Growthof Fingers From a Flat Interface Between TwoFluids in Porous Medium or Hele-Shaw Cell,Quart. Journ. and Applied Math. (1959) XII,Pt.2.

[110] Christiansen, R.L. and Fanchi, J.R.: Initi-ation and Propagation Mechanisms of Misci-ble Viscous Fingers, paper presented at the1990 AIChE Spring National Meeting, Orlando,March 1822.

[111] Pavone, D.: Observations and Correlationsfor Immiscible Viscous Fingering Experiments,paper SPE 19670 presented at the 1989 SPEAnnual Technical Conference and Exhibition,San Antonio, Oct. 811.

[112] Chuoke, R.L., van Meurs, P., and van der Poel,C.: The Instability of Slow, Immiscible, Vis-cous Liquid Liquid Displacement in Perme-able Media, Trans., AIME (1959) 216, 18894.

[113] Jerauld, G.R. Davis, H.T., and Scriven, L.E.:Stability Fronts of Permanent Form in Im-miscible Displacement, paper SPE 13164 pre-sented at the 1984 SPE Annuall Technical Con-ference and Exhibition, Houston, Sept. 1619.

[114] Aleman, M.A., and Slattery, J.C.: A Lin-ear Stability Analysis for Immiscible Displace-ments, Transport in Porous Media (1988) 3,45572.

[115] Wardlaw, N.C.: Eects of Capillary Numberand Its Component Variables on Waterood Ef-ciency and Oil Mobilisation, AOSTRA Jour-nal of Research, 4, 35, 1988.

[116] Lee, S.-T. Gary Li, K.-M., and Culham, W.E.:Stability Analysis of Miscible DisplacementProcesses, paper SPE/DOE 12631 presentedat the 1984 SPE/DOE Symposium on EOR,

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REFERENCES 33

Tulsa, April 1518.[117] Huang, A.B. Chikhliwala, E.D., and Yortsos,

Y.C.: Linear Stability Analysis of ImmiscibleDisplacement Including Continuously Chang-ing Mobility and Capillary Eects: Part II -General Basic Flow Proles, paper SPE 13163presented at the 1984 SPE Annual TechnicalConference and Exhibition, Houston, Sept. 1619.

[118] Peters, E.J. and Flock, D.L.: The Onset ofInstability During Two-Phase Immiscible Dis-placement in Porous Media, SPEJ (April 1981)24958.

[119] Peters, E.J. and Hardham, W.D.: A Compar-ison of Unstable Miscible and Immiscible Dis-placements, paper SPE 19640 presented at the1989 SPE Annual Technical Conference and Ex-hibition, San Antonio, Oct. 811.

[120] Chang, S.-H. and Slattery, J.C.: Stability ofVertical Miscible Displacements With Develop-ing Density and Viscosity Gradients, Trans-port in Porous Media (1988) 3, 27797.

[121] Moissis, D.E. Miller, C.A., and Wheeler, M.F.:A Parametric Study of Viscous Fingering inMiscible Displacement by Numerical Simula-tion, Rice University, Houston, (1987).

[122] Stoke, J.P. et al.: Interfacial Stability and Im-miscible Displacement in Porous Media, Phys.Rev. Lett., (Oct. 1986) 57, 14, 171821.

[123] Gaupta, S.P. and Greenkorn, R.A.: An Experi-mental Study of Immiscible Displacement WithUnfavourable Mobility Ratio in Porous Media,Water Resours. Res. (April 1974) 10, 2, 3714.

[124] Sigmund, P.M. Sarma, H.K. Sheldon, P., andAziz, K.: Rate-Dependence of Unstable Water-oods, paper SPE 14368 presented at the 1985SPE Annual Technical Conference and Exhibi-tion, Las Vegas 2225.

[125] Moissis, D.E. Miller, C.A., and Wheeler, M.F.:Simulation of Miscible Viscous Fingering Us-ing a Modied Method of Characteristics: Ef-fects of Gravity and Heterogeneity, paper SPE18440 presented at the 1989 SPE Symposiumon Reservoir Simulation, Houston, Feb. 68.

[126] King, M.J. and Scher, H.: Probabilistic Stabil-ity Analysis of Multiphase Flow in Porous Me-dia, paper SPE 14366 presented at the 1985SPE Annual Technical Conference and Exhibi-tion, Las Vegas, Sept. 2225.

[127] Araktingi, U.G. and Orr, F.M. Jr.: ViscousFingering in Heterogeneous Porous Media, pa-per SPE 18095 presented at the 1988 SPEAnnual Technical Conference and Exhibition,Houston Oct. 25.

[128] King, M.J.: Viscous ngering and probabilis-tic simulation, Standard oil Research and De-velopment, Cleveland (1988).

[129] Fanchi, J.R. Shank, G.D., and Christiansen,R.L.: Chaos: A Source of Miscible ViscousFingering Instablities, Petroleum Society of

CIM/Society of Petroleum Engineers, paperNo. CIM/SPE 9099 (1990).

[130] Hagoort, J.: Displacement Stability of WaterDrives in Water-Wet Connate-Water-BearingReservoirs, Trans., AIME (1974) 257.

[131] Yortsos, Y.C. and Huang, A.B.: Linear Stabil-ity Analysis of Immiscible Displacement Includ-ing Continuously Changing Mobility and Cap-illary Eects: Part 1 - Simple Basic Flow Pro-les, paper SPE/DOE 12692 presented at the1984 SPE/DOE Symposium on EOR, Tulsa,April 1518.

[132] Skaugen, E.: Analytical Model of Viscous Fin-gering Including Buckley-Leverett Type Dis-placement, paper presented at the 1985 Ero-pean Meeting on IOR, Roma April 1618.

[133] Hughes, D.S. and Murphy, P.: Account-ing for Unstable Immiscible Flow Within aConventional Reservoir Simulator by Use ofPseudo Relative Permeabilities, Proc., 1987Fourth European Symposium on EOR, Ham-burg, (Oct. 2729).

[134] Bentsen, R.G. and Saeedi, J.: Liqiud Liq-uid Immiscible Displacement in UnconsolidatedPorous Media, JCPT (Jan-March 1981) 93103.

[135] Coskuner, G. and Bentsen, R.G.: On the De-velopment of a Functional Form for the Surfaceof An Immiscible Viscous Finger and the Use ofThis Surface in Stability Theory, Chem. Eng.Res. Des, 65 (Jan. 1987).

[136] Sarma, H.K. and Bentsen, R.G.: An Exper-imental Verication of a Modied InstabilityTheory for Immiscible Displacements in PorousMedia, JCPT (July-Aug. 1987).

[137] Chakrabarty, C. and Bentsen, R.G.: Insta-bility Theory for Immiscible Displacements inTubular Systems, Journal of Petroleum Sci-ence and Engineering, (1991) 6 1535.

[138] Demetre, G.P. Bentsen, R.G., and Flock, D.L.:A Multi-Dimensional Approach to Scaled Im-miscible Fluid Displacement, JCPT (July-Aug. 1982).

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34 CHAPTER 2. PHENOMENA

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Chapter 3

Equations

3.1 Conservation Laws

3.1.1 Introduction

Classical theory of uid dynamics is based uponthe fundamental physical conservation principles, i.e.,conservation of mass, energy and momentum (theNavier-Stokes equations). Hence, the formulation ofa mathematical model for a uid ow problem startswith a precise formulation of conservation of a setof physical quantities. Such a conservation law for aphysical quantity α states that the rate of change ofthe amount of α contained in an arbitrary but xedvolume V is equal to the total ux of α into V plusthe total source of α in V . If ωα denotes the amountof α per bulk volume and ~Fα is the ux density, i.e.,the amount of α transported per unit area and time,this statement may be more precisely expressed as

∂t

∫ ∫V

∫ωαdV = −

∫S(V )

∫~Fα · ~n dS +

∫ ∫V

∫qα dV,

(3.1)where the source term qα is measured in units of α perbulk volume and time. In Eq. 3.1, S(V ) is the surfaceof V and ~n is the outer unit normal to S(V ). Assum-ing that ωα and Fα are suciently regular functionsof position and time, the divergence theorem yields∫ ∫

V

∫ [∂ωα∂t

+∇ · ~Fα − qα]dV = 0. (3.2)

Since V is arbitrary, we obtain the dierential formof the conservation law for α:

∂ωα∂t

+∇ · ~Fα = qα. (3.3)

Consider a uid system composed of a number ncof independent components owing through a porousmedium subject to a given set of boundary condi-tions. In principle, the properties of the uid systemin thermodynamical equilibrium are uniquely deter-mined by the overall composition, pressure and tem-perature. The determination of the phase behavior ofthe uid system is discussed elsewhere in this mono-graph. Here, we simply assume that the number npof phases formed (1 ≤ np ≤ nc), the phase composi-tions, and properties are unique functions of overallcomposition, pressure and temperature. Likewise, we

also assume that the interactions between the owingcomponents and the medium are uniquely determinedby these state variables. In the following three sec-tions we give a compact description of the basic con-servation laws most commonly used in macroscopicmodelling of ow processes in porous media. A moredetailed treatment can be found in Lake.1 In addi-tion, we derive simplied forms of these conservationlaws on which the treatment of analytical methodsfor dissipation-free problems in Sec. 4 is based. Fi-nally, in Sec. 3.1.5 we give a brief discussion of well-posedness of the ow models.

3.1.2 Conservation of Momentum

It is beyond what is practically attainable to solveNavier-Stokes type problems for ow in individualpore channels on a scale larger than a few pores. Con-sequently, volume averaging techniques must be in-voked in order to generate macroscopic ow models.Such averaging methods constitute an area of activeresearch and a comprehensive treatment of this canfor example be found in Quintard and Whitaker.2 Itcan be shown that conservation of momentum for asingle phase in the interior of a ow domain is prop-erly modelled on a macroscopic scale by Darcy's law,which we also adopt as a correct model for multiphaseow:

~uj = −λj(∇pj − ρj~g). (3.4)

Here, u is volumetric ux density (Darcy velocity),λj = kj/µj , g is the acceleration of gravity and sub-script j refers to eective properties of phase j.

3.1.3 Conservation of Mass

Let ωij be the mass fraction of component i in phasej; i = 1, . . . , nc; j = 1, . . . , np. The volume of phase jper bulk volume is φSj . Thus, the mass of i per bulkvolume entering Eq. 3.1 is

ωi = φ

np∑j=1

ρj Sj ωij , (3.5)

and the mass ux density in Eq. 3.1 is

~Fi =

np∑j=1

ρj [ωij ~uj −Dij Sj ∇ωij ], (3.6)

35

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36 CHAPTER 3. EQUATIONS

where uj is given by Eq. 3.4 and the last term rep-resents transport caused by molecular diusion andphysical dispersion,3 see also Sec. 2.3.4. Thus, Dij isthe total diusion coecient of i in phase j. For sim-plicity, we assume production/injection of uids tobe incorporated in time-independent boundary con-ditions. Furthermore, we assume that chemical reac-tions between owing components do not occur. Thesource/sink term qi in Eq. 3.1 then represents rate ofretention of component i, i.e.,

qi = − ∂

∂t[(1− φ)ρs ωis] , (3.7)

where ωis is stationary mass of i per unit mass ofrock. As mentioned in the introduction, we assumethat a model for ωis is given. The conservation lawfor mass of component i is then given by Eqs. 3.1,3.4, 3.5, 3.6 and 3.7. For analytical studies, this formis in general too complicated. In order to prepare foranalytical treatment, we therefore rst introduce thefollowing assumptions:

i) Flow is one dimensional,

ii) Rock and uids are incompressible,

iii) Molecular diusion is negligible,

iv) Volume change upon mixing of uid componentsis negligible.

In particular, this allows a formulation of the conser-vation laws in terms of volume fractions cij of compo-nent i in phase j. For ow problems on a sucientlylarge scale, (iii) is an adequate assumption.1 This im-plies that Dij = Dj , where Dj measures dispersionin phase j. Dividing all terms in the conservation lawfor i by the density ρi of the pure component i andusing the above equations, we obtain

φ∂

∂t

np∑j=1

Sj cij + ai

+∂

∂x

np∑j=1

cij uj

=∂

∂x

np∑j=1

Sj Dj∂cij∂x

,(3.8)

where we have represented retention as volume of sta-tionary component i per unit pore volume,

ai =1− φφ

ρsρiωis. (3.9)

By denition,

np∑j=1

Sj = 1;

nc∑i=1

cij = 1. (3.10)

Using Eq. 3.10, summation of Eq. 3.8 yields

∂uT∂x

=∂

∂x

np∑j=1

uj

= 0 (3.11)

or uT = constant. In Eq. 3.11 we have neglected therate of change of total volume of stationary compo-nents per pore volume. Dening capillary pressurePkj as

Pkj = pk − pj , (3.12)

a straightforward algebraic manipulation shows thatuj can be written

uj = fj uT +λjλT

np∑k 6=j

λk∂Pkj∂x

, (3.13)

where the fractional ow fj of phase j is dened by

fj =λjλT

1 +1

uT

np∑k 6=j

λk∆ρjk g sinϑ

. (3.14)

Here, ϑ is the angle of inclination between the hori-zontal plane and the ow direction (with an appropri-ate sign convention), λT =

∑λk and ∆ρjk = ρj−ρk.

Using the above expressions, the conservation law forcomponent i becomes

φ∂ci∂t

+ uT∂Fi∂x

=

∂x

np∑j=1

(SjDj

∂cij∂x− cij

λjλT

) np∑k 6=j

λk∂Pkj∂x

,(3.15)

where the overall volume fraction of i is

ci =

np∑j=1

Sj cij + ai (3.16)

and the overall fractional ux of i is

Fi =

np∑j=1

fj cij . (3.17)

3.1.4 Conservation of Energy

If Uj denotes internal energy per unit mass of phasej, the internal energy of j per unit bulk volume isφSjρjUj . In most porous ow problems, it is ade-quate to neglect kinetic energy. Therefore, the termωE for energy in Eq. 3.1 may be written

ωE = φ

np∑j=1

ρj Sj Uj , (3.18)

and the corresponding energy ux becomes

~FE =

np∑j=1

ρj Uj ~uj − λ∇T, (3.19)

where the last term in Eq. 3.19 is Fourier's law forheat conduction, λ being the overall thermal con-ductivity of the saturated medium. We recall thatinjection/production terms are incorporated in the

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3.1. CONSERVATION LAWS 37

boundary conditions. The source term qE in Eq. 3.1is therefore composed of heat transferred from themedium to the uid system and external work doneon the uid system. Let Us denote specic internalenergy of the rock. The rate of heat transfer per unitbulk volume is therefore

qH = − ∂

∂t[(1− φ)ρs Us] . (3.20)

In all nonisothermal ow problems arising in oil re-covery, this term plays an important role. The signif-icance of source terms representing mechanical workdone on the uid system must be evaluated for theparticular process under consideration. It can beshown that the source term representing volume com-pression/expansion of uids is given by

qpdV = −np∑j=1

pj∇ · ~uj , (3.21)

and the source term corresponding to work done bybody forces is

qb = −np∑j=1

~uj · ∇pj . (3.22)

For simplicity, work done by gravity has been ne-glected. The term qE = qH + qpdV + qb in Eq. 3.1can then be written

qE = − ∂

∂t[(1− φ)ρs Us]−

np∑j=1

∇· (pj ~uj). (3.23)

Eqs. 3.1, 3.4, 3.18, 3.19, and 3.23 constitute the en-ergy balance equation. ( It is convenient to combinethe last term of Eq. 3.23 with the term ∇ · ~FE togive ∇ · [

∑ρjHj~uj ], where Hj is the specic phase

enthalpy dened by Hj = Uj + pj/ρj .)We next invoke the simplifying assumptions stated

in the previous section in order to make analyticaltreatment possible even in nonisothermal ow. Inaddition, we will assume that specic heat capacitiesare independent of temperature. Thus, ρjUj = κjT ,ρsUs = κsT , where the κ's are specic volumetricheat capacities. Using Eqs. 3.13 and 3.14 and ne-glecting work terms, we arrive at the following formof the energy balance equation:

φ∂

∂t

np∑j=1

κj Sj T + κs T

+ uT∂

∂x

np∑j=1

κj fj T

=

∂x

λ∂T∂x

+

np∑j=1

λj κj TλT

np∑k 6=j

λk∂Pkj∂x

,(3.24)

where κs = (1− φ)/φκs.

3.1.5 Well-Posedness of the Models

A mathematical model is well-posed if the solution ofthe initial-value problem for the model depends con-tinuously (in some precise sense) of the initial data.

As described in the previous paragraphs, the formu-lation of ow models in porous media always involvessome kind of simplifying assumptions enforced by thecomplexity of such problems. Therefore, when model-ing a stable ow phenomenon, i.e., one which is stableto small perturbations, we must rst of all ascertainthat our model behaves accordingly. In porous ow,unstable phenomena may occur when one uid is dis-placing another at an adverse mobility ratio. In suchcases we must ascertain that the model is capable ofcapturing the physical instabilities correctly. What-ever the nature of a ow phenomenon is, stabilityconsiderations become essential.In the ow models discussed in the previous sec-

tions, dierent choices of primary unknowns in themodel formulation are possible. Let the choicebe denoted u1(x, t), . . . , uN (x, t) and let ~u(x, t) =(u1, . . . , uN )(x, t). Assuming that the capillary pres-sures Pkj occuring in Eqs 3.15 and 3.24 can be ex-pressed in terms of ~u, we can write

∂Pkj∂x

=

N∑n=1

∂Pkj∂un

∂un∂x

.

A ow model based on these conservation laws maytherefore be written on the general form of a 2. orderPDE:

∂~g(~u)

∂t+∂ ~f(~u)

∂x=

∂x

[D(~u)

∂~u

∂x

], (3.25)

where ~g(~u) = (g1(~u), . . ., gN (~u)), ~f(~u) = (f1(~u), . . .,fN (~u)) are N vectors and where D(~u) is an N × Nmatrix. Let dg and df denote the Jacobians of g andf respectively, and assume that the eigenvalues ofdg are real and positive (which seems to be satisedfor most real ow problems). Suppose we freeze thecoecient matrices dg, df and D of Eq. 3.25 at a xedstate ~u0. Let λf (~u0) and λD(~u0) be eigenvalues ofdf(~u0) and D(~u0) respectively. A necessary conditionfor Eq. 3.25 to be well-posed at ~u0 is that either

i) Re[λD(~u0)] > 0, or

ii) Re[λD(~u0)] = 0 and Im[λf (~u0)] = 0.

For details, we refer to Smoller.4 If (i) is satised,the model Eq. 3.25 is parabolic at ~u0. If (ii) is sat-ised, the model is hyperbolic at ~u0. If the condi-tion is violated, the model is elliptic. In other words,time-dependent elliptic problems in this sense are ill-posed. For linear problems, the condition is both nec-essary and sucient for well-posedness. We also re-mark that even if a model violates the condition atsome points ~u0, the model might nevertheless behavewell if the solution tends to avoid staying inside ellip-tic regions in composition space. Examples of suchbehavior seem to occur in certain models for threephase ow with zero capillary pressure, see Bell etal.,5 Holden et al.6 However, this depends on the as-sumptions made about the model parameters and itremains an unresolved problem whether or not this

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38 CHAPTER 3. EQUATIONS

reects real physical behavior. If not, this is an ex-ample where inadequacy of the model parameters canbe disclosed through stability considerations.The assumption of zero capillary pressure is rea-

sonable from a physical point of view on eld scaleproblems and at ow rates typical for oil recovery pro-cesses. However, this assumption is more problematicmathematically. For example, in a completely immis-cible process, where cij = 0 for all i 6= j and cii = 1in Eq. 3.8, neglecting the Pkj-terms reduces the owmodel to a 1. order system, which in the notation ofEq. 3.25 may be written

∂~u

∂t+∂ ~f(~u)

∂x= 0. (3.26)

Here ~u represents phase saturations. As we have seen,well-posedness of Eq. 3.26 as dened above requireshyperbolicity, but this in general is not a sucientcondition.It is a well-known fact (consider for instance the

scalar Buckley-Leverett initial value problem for wa-terooding) that (smooth) solutions of hyperbolicsystems may cease to exist in nite time and thatshock solutions then must be constructed from dis-continuous versions of the conservation laws (jumpconditions). However, such jump conditions do notin general produce unique solutions and additionalconstraints (entropy conditions) must be imposed inorder to pick the physically relevant solution.The discontinuous nature of hyperbolic problems

makes the determination of unique solutions of initial-value problems for such models a highly nontrivialmatter. For strictly hyperbolic systems with Rie-mann data, see Sec. 4, where the initial and injectedconditions are suciently close, Lax7 determined anentropy condition subject to which these problemsalways have unique solutions. For a more generalclass of data, Glimm8 proved existence of solutions,although the uniqueness problem remains unresolved.Beyond this, very little is known about existence anduniqueness for hyperbolic systems in general.Since it is the consensus of opinion of workers

in the area, Smoller,4 that parabolic systems haveunique smooth solutions, a possible strategy to ob-tain uniqueness for hyperbolic problems is to regardsolutions of Eq. 3.26 as limits to solutions of the cor-responding model expressed by Eq. 3.25 as the r.h.s.tends to zero. This strategy has been carried out rig-orously for the general (hyperbolic) Buckley-Leverettinitial-value problem (N = 1), Oleinik,9 and physi-cally motivated by Aavatsmark10 using capillary pres-sure as model for diusion. Similar results for gen-eral systems (N > 1) have yet to be derived. In fact,recent results indicate that ill-posed hyperbolic sys-tem for porous ow processes exist, see Azevedo andMarchesin,11 and Johansen and Winther.12

Nevertheless, such models play an important rolein the understanding of dierent ow mechanisms,Sec. 4.

3.2 Fluid Properties

3.2.1 Phase Behavior

The objective of any phase behavior model is todescribe the relative phase volumes based on inde-pendent variables such as pressure, temperature andoverall composition. In addition to the phase relativevolumes (saturations), the properties of the phases(densities and viscosities), and related properties (in-terfacial tensions) are also determined. These proper-ties are then applied in models describing the move-ment of phases through a porous medium, the reser-voir rock.Thermodynamic theory that describes phase be-

havior is summarized below, followed by an illustra-tion of the nature of phase behavior. An outline oflaboratory experiments describing processes of reser-voir uids and uid models used in reservoir simula-tors are then presented. Although models represent-ing phase behavior are simple, the theory from whichthey are derived is more complex. A more extensiveoutline of thermodynamic theory is therefore givento illustrate the concepts from which phase behaviormodels are generated

Theory

The building blocks of thermodynamics are based onthree laws. These laws have been deduced from a setof postulates. The postulates and the terminologywhich is common to the subject of phase equilibriumare presented below. The denition of extensive andintensive parameters and their relation to the equa-tions used for describing phase equilibrium are thenpresented. Euler, Gibbs-Duhem and Maxwell rela-tions are introduced to describe how the various pa-rameters are related and their importance to variousenergy functions. Lastly, the equation of state, thelever rule, the principle of corresponding states andGibbs phase rule are discussed to complete a basic re-view of thermodynamic theory necessary to do phaseequilibrium predictions.The principles of thermodynamics have been de-

veloped for simple systems that are macroscopicallyhomogeneous, isotropic, uncharged and chemically in-ert. The rst postulate states:13

There exist particular equilibrium statesof simple systems which, macroscopically,are completely characterized by the spec-ication of the internal energy, U , and aset of extensive parameters, volume V andmole numbers N1, N2,. . . , Nn of the chemi-cal components.

The extensive parameters play an important rolethroughout thermodynamic theory. They have theproperty that their value in a composite systemequals the sum of its values in each subsystem. Molenumbers, as an example, are extensive parameterswhose sum are always the same if no molecules are

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3.2. FLUID PROPERTIES 39

added or removed from a system. The law of conser-vation of energy is therefore referred to as the rstlaw of thermodynamics.The postulates to follow are referred to as the en-

tropy maximum postulates. The principle of maxi-mum entropy is the fundamental principle of thermo-dynamics. The second postulate states:

There exists a function, entropy S, of theextensive parameters of any composite sys-tem, dened for all equilibrium states andhaving the following property. The valuesassumed by the extensive parameters in theabsence of an internal constraint are thosethat maximize the entropy over the manifoldof constrained equilibrium states.

Continuing, the third postulate states:

The entropy of a composite system isadditive over the constituents subsystems.The entropy is continuous and dierentiableand is a monotonously increasing functionof the energy.

The entropy is a homogeneous rst-order functionof the extensive parameters for a simple system, i.e.,if λ is a constant then

S(λU, λV, λN1, . . . , λNn) =

λS(U, V,N1, . . . , Nn), (3.27)

and the entropy may be interpreted as a measure ofdisorder. Postulate three implies that(

∂S

∂U

)V,N1,...,Nn

> 0, (3.28)

and that the entropy can be inverted with respect tothe energy, i.e.,

S = S(U, V,N1, . . . , Nn), (3.29)

orU = U(S, V,N1, . . . , Nn). (3.30)

The second law of thermodynamics states that for allprocesses, the total entropy change is positive anda limiting value of zero is approached for processeswhich approach reversibility.The fourth postulate is an extension of the so-called

Nerst postulate (third law of thermodynamics) whichis due to Planck and states:

The entropy of any system vanishes inthe state for which(

∂S

∂U

)V,N1,...,Nn

= 0

at absolute zero temperature.

Eq. 3.30 is referred to as the fundamental equa-tion. In phase equilibrium the interest is often con-cerned with the changes in U with respect to the ex-tensive parameters. In dierential form the funda-mental equation is

dU =

(∂U

∂S

)V,N1,...,Nn

dS

+

(∂U

∂V

)S,N1,...,Nn

dV

+

n∑i=1

(∂U

∂Ni

)S,V,Nj 6=Ni

dNi. (3.31)

The partial derivatives are then referred to as inten-sive parameters and they are dened as the temper-ature,

T =

(∂U

∂S

)V,N1,...,Nn

, (3.32)

the pressure,

p = −(∂U

∂V

)S,N1,...,Nn

, (3.33)

and the electrochemical potential of component i

µi =

(∂U

∂Ni

)S,V,Nj 6=Ni

. (3.34)

Eq. 3.31 may also be expressed as

dU = TdS − pdV +

n∑i=1

µidNi. (3.35)

By using the denition of the homogeneous rst-orderproperty and Eq. 3.35, the Euler relation

U = TS − pV +

n∑i=1

µiNi (3.36)

may be derived.The Gibbs-Duhem relation is obtained from Euler's

relation by dierentiating with respect to the inten-sive parameters,

SdT − V dp+

n∑i=1

Nidµi = 0. (3.37)

In order to make use of the Gibbs-Duhem relation,one has to know the intensive parameters in terms ofthe extensive parameters. This is usually describedby an equation of state,

p = p(S, V,N1, . . . , Nn). (3.38)

However, for practical purposes, it is more convenientto use intensive parameters as the independent vari-ables rather than extensive parameters. Also, it ismore convenient to use temperature as a variable formatching experimental data instead of entropy.

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40 CHAPTER 3. EQUATIONS

By the use of Legendre transformation the internalenergy, U, may be transformed to the Helmholtz freeenergy, A, by replacing the entropy with temperature

U = U(S, V,N1, . . . , Nn)⇒A = A(T, V,N1, . . . , Nn), (3.39)

or the Gibbs free energy, referred to as Gibbs func-tion,

U = U(S, V,N1, . . . , Nn)⇒G = G(T, p,N1, . . . , Nn), (3.40)

where the entropy and volume are replaced by tem-perature and pressure, respectively.The Helmholtz free energy and the Gibbs function

are used to determine the state of a system. TheHelmholtz potential minimum principle states:

The equilibrium value of any uncon-strained internal parameter in a system indiathermal contact with a heat reservoirminimizes the Helmholtz potential at con-stant temperature, that is equal to the heatreservoir.

Similarly the Gibbs function minimum principlestates:

The equilibrium value of any uncon-strained internal parameter in a system incontact with a temperature and a pressurereservoir minimizes the Gibbs function atconstant temperature and pressure, that isequal to those of the respective reservoirs.

These energy functions are represented in dierentialform as

dA = −SdT − pdV +

n∑i=1

µidNi, (3.41)

and

dG = −SdT + V dp+

n∑i=1

µidNi. (3.42)

In integrated form the equations are

A = −pV +

n∑i=1

µiNi, (3.43)

G =

n∑i=1

µiNi. (3.44)

The partial derivatives of the energy functions gener-ate a set of relationships referred to as the Maxwellrelations. These relationships are rather numerousand the reader may refer to a textbook on thermody-namics.1315 These relationships may be applied tosimplify the set of equations in a practical manner forobtaining the best solution method.For the problem of nding the equilibrium state of

a uid consisting of a set of components at a constant

temperature and pressure, one has to locate the min-imum of Gibbs function where the independent vari-ables often are represented by an equation of state.16

The oil industry applies various equations of statefor various purposes. Of the most popular equationsof state are the volume cubic equations of state ofthe form of van der Waal's equation.17 The volumecubic equation of state may be written in the generalform of a pressure function in terms of molar volume,temperature, and composition.18

p =RT

V − β− Θ(V − η)

(V − β)(V 2 + δV + ε), (3.45)

where the greek letters in principle are functions oftemperature and composition. Eq. 3.45 is written inthe form of a repulsive term and an attractive termto account for the relative incompressibility of liquids.However, the simple repulsive term has been criticizeddue to its poor ability to represent the contributionof the repulsive forces.19

From this form, the two most popular volume cubicequations of state applied in the oil industry may beformed, namely the Redlich-Kwong2022 and Peng-Robinson23 equations of state. More recently, theSchmidt-Wenzel24 equation of state has gained pop-ularity.The cubic form is simple to use and it requires the

lever rule to describe how the phases should split.This is due to the fact that the volume of a two-phasesystem is mathematically discontinuous.Fig. 3.1 shows a representation of the lever rule,

where the equation of state describes the relationshipbetween pressure and volume at constant tempera-ture.

A C

B

ΙΙ

Volume

Pres

sure

D

Figure 3.1: The lever rule

The lever rule states that equilibrium area I shouldbe equal to area II, where area I is represented byABC and area II is represented by CDE. This is arather simplied picture and for multicomponent sys-tems the picture becomes more complex. As an ex-ample, a multicomponent system requires certain sta-bility criteria to be satised.It is appropriate here to introduce the theorem of

corresponding states. The compressibility factor or

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3.2. FLUID PROPERTIES 41

Z-factor, Z, for all pure uids may according to thistheorem be represented by a universal function of thereduced temperature, Tr = T/Tc, and reduced pres-sure, pr = p/pc,

Z = Z(Tr, pr), (3.46)

where the subscript c represents the critical statewhich will be discussed below. Eq. 3.46 gives an ex-cellent representation for a class of simple uids (Ar,Kr and Xe) but a systematic deviation is noted formany other substances.15

The relationship between pressure, volume andtemperature may therefore be represented by allow-ing a compressibility factor to be inserted in the idealgas law to compensate for any dierences a real gasmay inherit. The relation is written as

pV = ZNRT, (3.47)

where R is the gas constant and N is the number ofmoles of uid mixture.One important thermodynamic rule is the Gibbs

phase rule, dened by

f = n− P + 2, (3.48)

where f is the number of thermodynamic degrees offreedom, n is the number of components and P is thenumber of coexisting phases.An example of the usefulness of Gibbs phase rule

is as follows. For a single component system theremay exist three phases when there is zero degree offreedom. This implies that the triple point is uniquelydened for a single component. Similarly for a twocomponent system, four phases coexist at a unique setof values for pressure, temperature and composition.A hydrocarbon uid system consisting of many

components, say n, may have n+ 2 coexisting phasesaccording to Gibbs phase rule. In reservoir engineer-ing practice, however, one operates with two-phasesystems, namely vapor and liquid. In some casesthree phases are necessary, where for example, sig-nicant amounts of carbon dioxide is present to forma second hydrocarbon liquid phase.The conditions at the critical point for a single com-

ponent system are(∂p

∂V

)T

=

(∂2p

∂V 2

)T

= 0. (3.49)

This condition yields a unique critical compressibil-ity factor for a pure component. Pure componentsvary in their values of critical compressibility,25 andfor hydrocarbons the trend is, the greater the car-bon number the smaller is the critical compressibilityfactor. For multicomponent systems, the critical con-ditions are more complex.26

To determine a phase equilibrium solution at xedpressure and temperature, the minimum of Gibbs freeenergy with respect to composition must be solved.This requires that three restrictions must be satised:

(1) the material balance must be preserved; (2) nodriving force must exist to cause a net movement ofany component from one phase to another; (3) thechemical potentials for each component must be equalin all phases.The equality of the chemical potentials is neces-

sary but not a sucient condition for minimizing theGibbs free energy. Therefore, identication of falsesolutions may in many cases be of great importance,especially in the near phase transition region.The mathematical description of the phase equilib-

rium solution may be obtained by locating a tangentplane to the Gibbs free energy surface for a givenpressure, temperature and overall composition. Thepoints of tangency must then satisfy the three restric-tions mentioned above.To check for the solution's validity, a stability anal-

ysis may be performed in order to determine if thetangent plane lies below all possible equilibrium com-positions. As a rule of thumb, the more phasespresent, the lower the Gibbs free energy. More pre-cisely, if a solution exists having P phases then theGibbs free energy is less than the Gibbs free energy ofa solution having P − 1 phases.27 Also, at the pointof tangency, the Gibbs free energy surface must beconcave upward. An illustration of this condition isgiven in Fig. 3.2.

x z y

Gmin

Mole fraction

Gib

bsen

ergy

Figure 3.2: Gibbs energy curve.

In principle, the stability technique is a matterof nding the roots where the Gibbs energy surfacecrosses the tangent plane. Therefore, if the tangentplane lies above the Gibbs free energy surface at anypoint, the predicted equilibrium solution is false. Theconverse is also true, if the tangent plane lies entirelybelow or tangent to the Gibbs free energy surface, thesolution describes the equilibrium state.A constrained minimization formulation of a vapor-

liquid equilibrium problem of the Gibbs free energyhas been presented by Trangenstein.28 The formula-tion makes use of the rst-order Kuhn-Tucker nec-essary condition which is simplied by the Gibbs-Duhem relation. It also applies the second order nec-essary and sucient conditions for the minimum. Theset of equations in the problem makes use of a trust

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42 CHAPTER 3. EQUATIONS

region to compute both the step direction and lengthto minimize the Gibbs free energy. This is a combina-tion of the methods of steepest descent and Newtondirection.

The Nature of Phase Behavior

The visualization of phase equilibrium states is usefulto understand its behavior. This section describes thesingle and binary component systems, and ternary,quaternary, and multicomponent systems. A systemcontaining more than four components may not com-positionally be visualized completely for a xed pres-sure and temperature, but, some partial illustrationsmay be useful.

Single and Binary Systems. For many compo-nents there exists a critical point; hydrocarbon com-ponents such as methane through normal decane, n-decane, have critical points. However, for compo-nents with a carbon number greater than n-decane,this is not necessarily the case. For the sake of argu-ment, the illustrations to follow apply to componentshaving a critical point.Two single component systems with their phase

boundaries are shown in Fig. 3.3. The two lines end-ing at points A and B, are referred to as the criticalpoints for the rst and second component, respec-

Temperature

Pres

sure

A

C

D

B

Figure 3.3: One and two component phase envelopes.

tively. A pure liquid state exists above the phase linefor the single components, and a pure vapor state be-low. However, for temperatures and pressures abovethe critical point, there is a single phase state with-out any clear denition of whether it is a liquid or avapor. That means that if a pressure and tempera-ture path around the critical point is followed fromliquid to vapor, the pure component uid will go frompure liquid to pure vapor without any phase transi-tion, meaning the uid is always in single phase. Ifthe phase line is crossed, a two-phase intermediatestate will occur.A dotted line is drawn between critical points A

and B to mark the critical locus of mixtures of the

two components. Any uid consisting of a combi-nation of these two components will have its criticalpoint on the critical locus. Two-phase envelopes arealso shown in Fig. 3.3 with critical points C and D.The uid with critical point C is richest in compo-nent one while the uid with critical point D is richestin component two. For a given temperature, a two-component uid has a pressure interval where twophases exist simultaneously. This is not so for a sin-gle component uid, where only one single pressurevalue will contain two phases for a given temperature.To highlight the phase envelopes in Fig. 3.3,

Fig. 3.4 shows a pressure-composition plot of the twotwo-component uid systems with overall composi-

Mole fraction

C

D

x yT1

T2

T2 > T1

Pres

sure

Figure 3.4: Two two-component phase split en-velopes.

tions dened by the critical points C and D, respec-tively. At a particular pressure and a composition de-ned by point D, the uid will form two phases, oneconsisting of x amount of component one in the liquidphase and y amount of component one in the vaporphase. The line connecting x and y is referred to asa tie line. Associated to each tie line are K-valuesdened as K = y/x for each component, where x andy usually are in units of mole (or weight) fractions.At the critical point, the K-values become unity andin the single phase region they are either undenedor zero.In Fig. 3.4 the uid system dened by point C with

temperature T1 may have its temperature increasedto T2 with the resulting eect that the envelope willoverlap the envelope for the uid system with com-position D at temperature T2. No equilibrium statemay exist within the phase envelope. All overall com-position dened within the phase envelope will splitinto phases dened by the tie line K-values runningthrough the overall composition.

Ternary Systems. In a three-component uid sys-tem, the components may be classied by their molec-ular weights. Typically, a ternary system may be rep-

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3.2. FLUID PROPERTIES 43

resented by a light (zL), an intermediate (zI) and aheavy (zH) component, where zj is component j'smole (or weight) fraction.29

For a xed pressure and temperature, the composi-tional space may be illustrated by a ternary diagram.Fig. 3.5 depicts two-phase envelopes each with onecritical point. The two-phase envelopes represent thesame three components at given temperature but at

ZL

ZIZH

Criticallocus

x

y

z

A

pi

pi+1

B

Figure 3.5: Ternary phase envelopes.

two dierent pressures. For any overall compositionz = (zL, zI , zH) within a phase envelope, the uidwill split into two phases with compositions x andy. These compositions are related by the materialbalance relation

z = Lx+ V y, (3.50)

where L and V represent the liquid and vapor phasefractions, respectively. Eq. 3.50 represents one tieline and it is unique in the sense that no other tie linecrosses it.Fig. 3.6 illustrates the eect of pressure on the

phase envelope; here pi is greater than pi+1 and thetemperature is constant. The critical locus at con-stant temperature and varying pressures and compo-sition is represented by the dashed line AB.The two-phase envelopes having the same overall

composition do not necessarily have tie lines that areparallel. Therefore, a change in pressure for a xedoverall composition will result in a change in the tieline and the phase compositions. All overall composi-tions along a particular tie line will split into phaseshaving the same compositions. The change in over-all composition along a tie line results in a changein the phase fractions only. As the pressure changeswhile the temperature is kept constant, the angle ofthe tie lines changes. A prism is drawn to illustratethis point in Fig. 3.6. Note that for pressures abovep1, only a single phase exists.Figs. 3.5 and 3.6 are simple illustrations of phase

envelopes. It may well be the case that two-phaseenvelopes exist at a particular pressure and tem-perature as shown in Fig. 3.7. Each envelope hasits own critical point. The various combinations ofphase envelopes are often dependent on their system

0.8 0.6 0.4 0.2

Mole fraction light

ZL

p1

p2

p3

p4

p5

p6

B

Criticallocus

Pres

sure

ZL

0.2

0.4

0.6

0.8

0.8

0.6

0.4

0.2

ZH

Mol

efra

ctio

nin

term

edia

teMole

fractionheavy

Bubbl

e-

poin

t

ZI

A

Dew-point

Figure 3.6: Ternary phase envelope with varying pres-sure.

components.30 As the pressure is reduced, assumingconstant temperature for the example illustrated inFig. 3.7, the two-phase envelopes will merge into onephase envelope, causing the critical points to mergeand vanish.

Quaternary Systems. A four-component uidsystem phase envelope may be illustrated in a tetra-hedron as shown in Fig. 3.8. The sides of the tetrahe-dron represent four combinations of ternary systems.This type of representation of a four component sys-tem is unable to show the complete compositionalspace as well as the pressure and temperature range.Therefore, for example, if the pressure domain is ofinterest, only part of the compositional space may beillustrated.Fig. 3.6 may represent such an illustration. The

requirements is then that one of the apexes, say the

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44 CHAPTER 3. EQUATIONS

CP

ZH ZI

ZL

Figure 3.7: Alternative ternary phase envelope.

x

z

y

ZL

ZIHZH

ZIL

Criticallocus

Tieline

Figure 3.8: Quaternary phase envelope.

intermediate (zI), must represent the sum of two com-ponents, i.e., a pseudocomponent. The pseudocom-ponent may be dened via the use of an equationof state31 or transformed from data generated by aprocess spanned by two extreme states.32 These twodenitions of pseudocomponents may result in twodierent representations of the phase envelope.The critical point in a ternary system is a point

on a critical locus in a quaternary system as seen inFig. 3.8. As the number of components is increased,the dimensionality of the critical region may also in-crease. For multicomponent systems, the critical re-gion can therefore not always be dened.

Multicomponent Systems. A multicomponentsystem may contain anywhere from ve to severalmillions of components. These systems are usuallyrepresented in a manner showing their dependenceon pressure and temperature having their composi-tion xed, see Fig. 3.9. The gure illustrates a bubblepoint line, a critical point, a retrograde dewpoint lineand a dewpoint line. The bubble point line representspressures and temperatures at which all the gas inequilibrium with the oil condenses to oil. That is theboundary between a single oil phase and two phases,oil and gas. Likewise for the dewpoint line, it repre-sents the boundary between a single gas phase andtwo phases, oil and gas. The cricondenbar is denedas the maximum pressure at which two phases may

Temperature

Pres

sure

Dewpoint

Criticalpoint

Retrograde

Dew-point

Bubbl

epoi

nt

Figure 3.9: Pressure-Temperature phase envelope.

coexist, and similarly the cricondentherm is denedas the maximum temperature at which two phasesmay coexist.In the retrograde region, oil or gas may come out

of solution for then to return to solution as either thetemperature or pressure is altered.A third dimension may be added to Fig. 3.9, show-

ing how the phase envelope may vary with composi-tion, as shown in Fig. 3.10. The variation in composi-tion may be representative of a particular process orit may represent the relative change in concentration

Temperature

Pres

sure

Criticallocus

Compo

sition

Figure 3.10: Pressure-temperature phase envelopeswith compositional variation.

of a particular component within the multicomponentsystem.The variation of the phase envelope with pressure

and temperature can also be represented by the in-dividual component K-values, as shown in Fig. 3.11.This is a more common way to represent composi-tional variations for multicomponent systems. Aswith the phase envelope, the pressure axis in Fig. 3.11may also be replaced by a particular component orcomposition that relates to a process.Although only a few representations of the nature

of phase behavior have been illustrated here, they arevery useful when trying to understand how particularproperties of a uid may change. In conclusion itmay be added that for every composition at a xedpressure and temperature, an oil or gas phase hasunique density and viscosity values that represent itsproperty. If the two phases coexists in equilibrium,these phases have an interfacial tension that is uniquefor their particular tie line.33

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3.2. FLUID PROPERTIES 45

Pressure

ZL

ZI

ZH

10

1.0

10-1

10-2

K-

valu

e

Figure 3.11: K-values variation with pressure.

Experiments for Obtaining pVT Properties

The pVT experiments most commonly performed onlive hydrocarbon systems are34 (1) constant massexpansion (CME), (2) constant volume depletion(CVD), (3) dierential liberation (DL), (4) swellingtest (ST), (5) multistage separation (MS), and (6)multicell contact (MC).The experiments are designed for studies of the vol-

umetric behavior of reservoir uid through processesthat are representative for those occuring in the reser-voir during its production period. The process foreach of the experiments is summarized below.

Constant Mass Expansion. This experiment de-scribes the uid's behavior when the pressure is de-creased from pressures above the saturation pressureto pressures well below the saturation pressure keep-ing the overall composition unchanged. The exper-iment consists of loading a cell (a container able towithstand high pressures and temperatures) with auid of a particular overall composition representa-tive of the reservoir uid under study. The cell'svolume is altered at constant temperature in variousstages allowing the uid to come to equilibrium foreach stage. The experiment determines the uid'ssaturation pressure and the relative volumes of oiland gas at the various pressure stages.A ternary diagram with the various pressure stages

showing the process development is illustrated inFig. 3.12. A complementary diagram of the pressure-temperature space is shown in Fig. 3.13. In bothgures, the saturation pressure (bubblepoint) is rep-resented by pressure ps. For each pressure stage, theequilibrium compositions denoted in Fig. 3.12 willhave associated to it a volume, density and viscosityas mentioned earlier. These properties are process de-pendent and it is common to represent, say, the liquidvolume, V , relative to its saturation volume, Vs, asshown in Fig. 3.14.

Constant Volume Depletion. Constant volumedepletion is similar to constant mass expansion inthat the process goes through a series of decreas-ing pressure stages. This experiment however, starts

0.8 0.6 0.4 0.2

Mole fraction light

ZL

p1

p2

p3

p4

p5

p6

B

Criticallocus

Pres

sure

ZL

0.2

0.4

0.6

0.8

0.8

0.6

0.4

0.2

ZH

Mol

efra

ctio

nin

term

edia

teMole

fractionheavy

Bubbl

e-

poin

t

ZI

A

PS

Z

Dew-point

Figure 3.12: Pressure-composition phase envelopesshowing the process for a CME experiment.

at the uid's saturation pressure having a cell vol-ume, Vs. For each pressure stage below the satura-tion pressure two phases exist. An amount of equi-librium gas is removed from the cell for each of thepressure stages so to regain the cell's saturation vol-ume, Vs. The overall composition in this experimenttherefore changes at each pressure stage. The processpath in the compositional space is therefore dierentfrom a constant mass expansion. A benet of thisexperiment is that it allows for the analysis of theequilibrium gas at each pressure stage. This impliesthat the phase compressibilities, Z, see Eq. 3.47, andcomposition may be derived for both phases throughmaterial balance.

Dierential Liberation. Like the two previousexperiments, dierential liberation is performed onoils at constant temperature condition, starting at the

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46 CHAPTER 3. EQUATIONS

Temperature

Pres

sure

p1Criticalpoint

p2

pS

p3

p4

p6

p5

Figure 3.13: Pressure-temperature phase envelopeshowing the process for a CME experiment.

Pressure

Rel

ativ

evo

lum

e

pS

1.0

Figure 3.14: Liquid relative volume for a CME exper-iment.

bubblepoint pressure and decreasing the pressure in aseries of stages. At each pressure stage, all the equi-librium gas is removed from the cell before decreas-ing the pressure to the next pressure stage. The nalpressure is often atmospheric pressure. Thereafter, anal step reducing the temperature to standard con-ditions (15 C or 60 F) is performed. The relativeoil volumes dened as formation volume factors arethe ratio of liquid volumes at a particular pressurestage to liquid volume at standard condition. Theformation volume factor is a major parameter usedin black-oil models as will be discussed below.The process path in a ternary compositional-

pressure space is shown in Fig. 3.15. The last fewstages at low pressures are omitted here. One maycompare this process to the process of constant massexpansion, Fig. 3.12, and observe the dierence inliquid phase composition paths. This dierence in liq-uid compositions will result in dierent liquid volumesnot depicted in these gures. Pressure-temperaturephase envelopes may also be used to illustrate the dif-ferential process, keeping in mind that for each pres-sure stage a new phase envelope exists.

Swelling Test. Swelling tests are usually per-formed on undersaturated oils for the purpose of in-vestigating their volumetric behavior when in contactwith another uid such as carbon dioxide, CO2, or ni-

0.8 0.6 0.4 0.2

Mole fraction light

ZL

p1

p2

p3

p4

p5

p6

B

Criticallocus

Pres

sure

ZL

0.2

0.4

0.6

0.8

0.8

0.6

0.4

0.2

ZH

Mol

efra

ctio

nin

term

edia

teMole

fractionheavy

Bubbl

e-

poin

t

ZI

A

Dew-point

Figure 3.15: Pressure-Compositional phase envelopeshowing the process for a DL experiment.

trogen, N2. These gases as well as hydrocarbon gasesand mixtures thereof, will swell undersaturated oilsdierently depending on the uid system's composi-tion. That is, when a gas gets dissolved into an oilleaving no free gas, the volume expansion is depen-dent on the resulting composition. The pressure andtemperature are kept constant in order to measurethe increase in volume. In addition, in order to mea-sure the increase in volume, a measurement of theuids saturation pressure is also made. This gives anindication of how the saturation pressure is aectedby amount of gas dissolved.

Multistage Separation. During the production ofoil and gas in the eld the uids from the wells gothrough a process of decreasing pressures and temper-atures before it is stored at standard conditions. Theprocess consists of a sequence of separator tanks sep-

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3.2. FLUID PROPERTIES 47

arating the oil from the gas as illustrated in Fig. 3.16.Each separator tank has a dierent pressure and tem-perature. The oil that goes to storage is termed stock

Gas Gas Gas

p1

T1

p2

T2

p3

T3Fluid

stream

Stocktankoil

Gas

Oil Oil Oil

p1 > p2 > p3

T1 > T2 > T3

Figure 3.16: Multistage separation

tank oil. The stock tank oil properties are dependenton its composition which is again dependent on theprocess path and initial reservoir composition. Thevolume ratio between the gas and oil at standard con-ditions is termed the gas/oil ratio, GOR, and is de-noted by Rs. The oil/gas ratio from the reservoir gasis denoted by rs. The rs represents the liquid contentin the gas.

Multicell Contact. This experiment consists of aseries of cells coupled to each other as illustrated inFig. 3.17.32,35,36 The experiment reveals how com-ponent exchange inuences the phase volumes. The

Wetgas

Drygas

Figure 3.17: Multicell contact experiment.

experiment may be performed with any number ofcells, but most commonly, one or two cells are used.One cell may be used to study how oil vaporizes or gascondenses as it comes into contact with an injectionuid (usually gas) of constant composition at con-stant pressure and temperature. Two cells are usedto study how the injection uid changes property asit contacts fresh oil or gas at constant pressure andtemperature. For a single cell the experiment may besummed up in the following steps.1. Given the overall composition of a uid and the

process temperature, it is necessary to determine thesaturation pressure, ps, and its saturated volume, Vs.2. The saturated uid which is contained in the

cell has its pressure reduced to an operating pressurecausing two phases to form. Part of the equilibriumphase (usually gas) is then removed to regain the sat-uration volume, Vs, for the initial conditions for theuid system.3. Maintaining constant operating pressure, a spec-

ied amount of injection uid is injected into the cell,causing its volume to increase.

4. After equilibrium is attained at the operatingpressure, the equilibrium phase (gas) is removed untilthe saturation volume, Vs, is regained.5. Steps 3 and 4 are repeated until all specied

amounts of injection uid have been injected into thecell.At each equilibrium state (step 4), the phase prop-

erties such as density, compositions, phase fractionsetc. are determined. In addition, the amount of gasremoved from the cell must also be determined formaterial balance.The experiment with multiple number of cells is

similar to the single cell, but coupled in series. Thatis, the rst cell undergoes a vaporization process asexplained above. The second cell undergoes the sameprocess except that the injection uid now is the re-moved equilibrium phase from the rst cell. Similarly,the third cell has removed equilibrium phase from thesecond cell as its injection uid. This transport ofvarying equilibrium phase compositions continues forall cells.

Models Describing Phase Behavior

Three common model types applied in reservoir engi-neering to represent phase behavior of uids are de-scribed below. The models are termed the (1) black-oil Model, (2) the compositional model and (3) thechemical model.

The Black-Oil Model. The phase behavior treat-ment in conventional black-oil simulators is based onpressure dependent formation volume factors, Bg, Bo,and Bw, and the solution gas/oil ratio, Rs.37 Theseterms relate the oil's volumetric phase behavior atreservoir conditions to those at surface conditions.This model's phase behavior equations may be sum-marized through its description of the phase densitiesas

ρj =ρsjBj

, j = g and w , (3.51)

andρo =

1

Bo(ρso +Rsρ

sg), (3.52)

where superscript s denotes surface conditions. Thesurface densities are constants, and the stock tank oiland surface gas are termed the oil and gas compo-nents, respectively.The conventional black-oil simulator has received

wide acceptance for its application to depletion pro-cesses and simple cases of injection. In recent times,the formulation has been improved to model gas con-densate reservoirs,38 referred to as the symmetricalblack-oil formulation.The symmetrical formulation requires the gas den-

sity to be described by

ρg =1

Bg(ρsg + rsρ

so) (3.53)

where the rs represents the liquid/gas ratio from thereservoir gas.

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48 CHAPTER 3. EQUATIONS

The parameters Bg, Bo, Rs and rs are usually ex-pressed as functions of reservoir pressure and satura-tion pressure. Extensions of these parameters havealso been proposed.39,40 The extensions make theseparameters dependent on composition or the amountof injection gas that has contacted a given reservoirvolume. The variation of the parameters caused byan injection gas, for example, may be analyzed in amulticell contact experiment. The extension intro-duces an extra free variable. This variable accountsfor the variation in uid phase properties that are de-pendent on the amount of injection gas contacted aswell as on the pressure. The viscosity is also a phaseproperty that is dependent on pressure and amountof gas contacted.There are certain restrictions phase behavior pa-

rameters must satisfy in order to ensure nonnegativetotal uid compressibility

cp = − 1

V

∂V

∂p> 0 (3.54)

where V is the total volume of all phases. For thesymmetrical black-oil model where the uid parame-ters are pressure dependent only, this implies

cp =SgBg

(−dBgdp

+drsdp

(Bo −RsBg)(1−Rsrs)

)+SoBo

(−dBodp

+dRsdp

(Bg − rsBo)1−Rsrs)

),(3.55)

where it is assumed that

mso

ρso=

mo

ρoBo+mgrsρgBg

, (3.56)

andmsg

ρsg=

mg

ρgBg+moRsρoBo

. (3.57)

The parameters Bg, Bo, Rs and rs are usually ob-tained from pVT laboratory experiments such asthose described above. However, the laboratory ex-periments in some instances may give too little infor-mation. The above parameters are especially dicultto obtain for gas condensates, and even more so whenthese terms are dependent on a multicontact process.Therefore, estimation of these parameters are oftenmade by pVT simulators that are based on volumecubic equations of state.34,41,42

The Compositional Model. The densities in acompositional model are in addition to being depen-dent on the pressure as in the black-oil model, alsodependent on the concentration of each componentand temperature. The compositional model appliesthe thermodynamic theory reviewed at the beginningof this section in order to determine the densities,given the pressure, temperature and overall composi-tion.This model requires that the properties of the indi-

vidual components must be known such as the molec-ular weight, critical pressure, critical temperature,

critical compressibility factor and acentric factor.43

The model also incorporates binary interaction pa-rameters44 and volume correction parameters45 toimprove its modeling capability.46

All these component properties are incorporatedinto mixing rules that generate parameters in the vol-ume cubic equation of state20,21,23,24 that relate thesystem's pressure, temperature and volume.43

To solve the equilibrium problem, there are threemain solution techniques that are in common use,namely successive substitution,47,48 minimum vari-able Newton-Raphson49 and minimization of Gibbsfree energy,28 with a stability analysis.50 These solu-tion techniques all have in common to solve the ma-terial balance equations, Eq. 3.50. Once solved, thephase compositions with their component propertiesmay be incorporated into the mixing rules and theroots of the volume cubic equation of state is solvedfor the phase volumes. The phase masses are the com-ponent sum of the product of molecular weight andmole fraction for each component. Having the aboveinformation, the phase densities can be determined.The phase viscosities are determined from correla-

tions that incorporate the compositions and the re-duced phase densities.43,51 Likewise the interfacialtension may be determined, often through the para-chor formulation.33,52

The condition of a positive total uid compressibil-ity, Eq. 3.54, must also be satised. The derivation ofcp must therefore include terms containing the deriva-tives with respect to composition. Even for simplemodels this may lead to extensive expressions.32

Experience shows that using this type of model isnot without problems. In many cases there seem to bea gap between theory and experiment.46,53 This maybe understandable since the constants in the equationof state have been determined from simple systemscontaining few components and the equation of stateis applied to complex systems containing many morecomponents than used in the model. It is also arguedthat the forms of the volume cubic equations of stateare too simple, at least for the repulsive term.19

The Chemical Model. Chemical models are usedto model the interaction between liquid phases suchas formation water, oil and microemulsion. The as-sumption is made that these liquid phases are in-compressible, and no gas phase is present. Thismakes properties such as phase densities independentof pressure, i.e., constant. Chemical phase behaviormodels describe how the liquid phases change in rel-ative volumes with respect to the relative amount ofthe components. They also describe the property ofthe phases such as viscosities and interfacial tensions.There are four main components in this model

type, namely brine (salt water), oil, surfactant andalcohol. Polymer is also an important componentmodeled by this model type. These componentsmay partition into three possible phases, namely anaqueous, an oleic and a microemulsion phase.

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3.2. FLUID PROPERTIES 49

The phase behavior is modeled by solving thebinodal curves, phase boundaries, and distributioncurves, tie line, for a pseudoternary system.1,54 Themodel also applies partition coecients, which areanalogous to K-values, to the alcohol componentin order to dene pseudocomponents for a pseu-doternary diagram, Fig. 3.18. The prex pseudo takeson a dierent meaning here compared to its previ-

Water

Surfactant

Oil

Alcohol

Pseudoternarydiagram

Overallconcentration

Distributioncurve

Binodalcurve

Figure 3.18: Typical type III phase envelope.

ous usage, since pseudocomponents may be depen-dent on overall concentration. Although concentra-tion dependent partition coecients are not thermo-dynamically rigorous, they are often expressed as con-stants.55

The model is termed compositional because it alsomodels ion exchange between chloride anions (Cl−)and, sodium (Na+) and calcium (Ca2+) cations.These ions expresses the ionic strength in the solu-tion or the eective salinity. The eective salinitythen is an expression of the type of phase behavior,classied as either type II(), type III or type II(+).Fig. 3.18 illustrates the compositional space wherethe alcohol partitions to form three pseudocompo-nents lying in a plane having type III phase enve-lope.Adsorption is also important for this model type.

The loss of surfactant and the ion exchange betweenthe uids and reservoir rock inuence the phase be-havior of the system. Adsorption is history depen-dent and requires an additional component, namelythe reservoir rock.

3.2.2 Rheology

Introduction

Rheology is the science of deformation and ow ofmaterials in response to stress. The basic equation ofdeformation is given by

τ = ηγ, (3.58)

where τ is the shear stress, γ is the shear rate denedas ∂vx/∂y, and η is the viscosity. Viscosity is denedas the friction of the uid to ow. A shear force mustbe applied to the upper plate, Fig. 3.19, in order tokeep it in motion relative to the lower plate with ve-

x

y

vx = 0

vx = v

vx(y)

Figure 3.19: Steady-state velocity prole of a uidentrained between two at surfaces.56

locity v in the x-direction. By uid viscosity, the forceis transmitted through the uid to the lower plate insuch a way that the x-component of the uid velocitylinearly depends on the distance from the lower plate.It is implicitly assumed, and experimentally veried,that the uid does not slip at the plate surface.Fluids can be grouped into two main groups. New-

tonian uids, such as water, have shear-independentviscosity and the shear stress is proportional tothe shear rate. Non-Newtonian uids have shear-dependent viscosity. Examples are polymers, poly-mer melts, gels, and drilling mud, and they are cate-gorized as pseudoplastic, Bingham plastic, tixotropicetc. In this monograph we will limit the expositionto pseudoplastic uids.EOR polymers are pseudoplastics; the viscosity

is decreasing by increasing shear rate. Some u-ids have an increasing viscosity with increasing shearrate. They are called dilatant and their behavioris shown in Fig. 3.20. Some uids have a time-dependent viscosity; tixotropic if the viscosity is de-

Newtonian

Bingham plastic

DilatantPseudoplastic

Shear rate.γ

Shea

rst

ress

τ

τ1

τ0

.γ1

τ1.γ

η =

Figure 3.20: Relationship between shear stress andshear rate for dierent materials.56

creasing with increasing time at constant shear rate;rheopectic if the viscosity is increasing by increasingtime, Fig. 3.21. Some polymer gels are thixotropic.

Eect of Concentration

Einstein57 found that by adding spherical particles toa solvent, the viscosity will increase according to

ηsol = ηsolv(1 + 2.5ϕ). (3.59)

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50 CHAPTER 3. EQUATIONS

Thixotropic

Time

τ Rheopectic

Figure 3.21: Relationship between shear stress andtime.56

Here ηsol is the viscosity of the solution, ηsolv is theviscosity of the solvent, ϕ is the volume fraction ofspheres, and ϕ 1. For nonspherical particles thisequation was later generalized to

ηsol = ηsolv(1 + νϕ), (3.60)

where ν is a shape factor that depends on the particle.Eq. 3.60 is known as the Einstein-Simhas equation.58

When the particle concentration is increased, higherorders of ϕ must be accounted for,

ηsol = ηsolv(1 + νϕ+ κϕ2 + ξϕ3 + . . .). (3.61)

Simha58 found that the shape factor for an ellipsoidalparticle can be written

ν ∝(ab

)1.8

, (3.62)

where a and b are the axes, and a/b 1. For poly-mers, it is more convenient to use other dimensions,and we dene the relative viscosity ηr by

ηr =ηsol

ηsolv, (3.63)

and the specic viscosity ηsp as

ηsp =ηsol − ηsolv

ηsolv. (3.64)

Let the volume fraction ϕ be expressed by concentra-tion c,

ϕ = vhc, (3.65)

where vh is the hydrodynamic volume. SubstitutingEqs. 3.64 and 3.65 into 3.61, we get

ηsp = νvhc+ κv2hc

2 + . . . . (3.66)

The intrinsic viscosity [η] is dened by

[η] = limc→0

ηsp

c. (3.67)

Thenηsp = [η]c+ k′[η]2c2 +O(c3), (3.68)

orηr = 1 + [η]c+ k′[η]2c2 +O(c3). (3.69)

The intrinsic viscosity gives information about themolecular size and shape, and can be calculated byplotting the reduced specic viscosity ηsp/c versusconcentration. The slope of the curves gives the valueof Huggins constant k′. This constant is a measure ofthe intramolecular interaction.59 Theoretical valuesfor nonaggregating molecules are in the range 0.35to 0.45.60 Both intrinsic viscosity and Huggins con-stant are important factors in polymer descriptionand quality control. Large values of k′ indicate ag-gregation.Another useful method to determine intrinsic vis-

cosity is given by Flory.60 Taking the logarithm ofEq. 3.69 and performing series expansion for low con-centration give

ln ηr = [η]c+ k′′[η]2c2 + . . . , (3.70)

where k′′ = k′ − 1/2.Since k′′ is smaller than k′ (and k′ is normally close

to 0.5), the slope of the curve is smaller and the ex-trapolation of (ln ηr)/c is preferred over extrapolationof ηsp/c.The intrinsic viscosity can also be calculated as sug-

gested by Staudinger,61

[η] = K ′Maw, (3.71)

where K ′ and a are constants and Mw is the molec-ular weight which can be measured by light scatter-ing. The relation is experimental. For simple polymerstructures, we can nd theoretical values for the ex-ponent a; exible chain molecules in the range 0.5 to0.8 and for rodlike structures around 1.8.

Eect of Shear Rate

In Sec. 3.1.1, a non-Newtonian uid is described as auid where the viscosity is a function of shear rate.Limiting the discussion to pseudoplastic uids, thesimplest and most common relation between stressand shear rate is a power law of the form

τ = mγn, (3.72)

where n is the ow index and m is the consistencyparameter. The viscosity is found from Eq. 3.58:

η = mγn−1. (3.73)

For Newtonian uids n = 1, and for n < 1, the vis-cosity will decrease with increasing shear rate. Thisphenomenon is called shear thinning and is a char-acteristic property of pseudoplastic uids. Eq. 3.72is called the power law model or Ostwald de Waelemodel.62

Many non-Newtonian uids have a Newtonianregime for low shear rates that the power law modelcannot handle. Models with more than two parame-ters have been proposed, such as the Carreau model63

η(γ)− η(γ =∞)

η(γ = 0)− η(γ =∞)= [1 + (λγ)2]−α/2, (3.74)

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3.3. ROCK PROPERTIES 51

where η(γ =∞) is the viscosity at innite shear rate,normally the viscosity of the solvent; η(γ = 0) is theviscosity at zero shear rate; and λ is a characteristictime constant. For γ 1/λ the viscosity η(γ) isconstant. For γ 1/λ the equation will be reducedto a power-law model for ηsp, where n = 1−α. Later,the Carreau model has been generalized to

η(γ)− η(γ =∞)

η(γ = 0)− η(γ =∞)= [1 + (λγ)a]−α/a, (3.75)

and called the Carreau-Yasuda model.64 A similarexpression is Meter's equation,65

η(γ)− η(γ =∞)

η(γ = 0)− η(γ =∞)= [1 +

γ

γ1/2]1−α, (3.76)

where γ1/2 is the shear rate fulllling

η(γ1/2) =1

2η(γ = 0). (3.77)

For a simplied polymer model, called rigid dumb-bells, the time constant λ is given by65

λ =(ηsol − ηsolv)Mw

cRT, (3.78)

where Mw is the molecular weight, R is the gas con-stant, and T is the temperature. By using Eq. 3.67,λ0 is dened as the time constant at zero concentra-tion,

λ0 =[η]ηsolvMw

RT. (3.79)

The relation between the time constant, the intrinsicviscosity, and Huggins constant then is

λ = λ01 + [η] k′c. (3.80)

Mixing of Two (or More) Solutions

For a mixture of two dierent solutions the viscos-ity can be estimated by an experimental fourth rootmixing rule

1

η1/4mix

=

[ϕ1

η1/41

+ϕ2

η1/42

]. (3.81)

Here ηmix is the viscosity of the mixture, and ϕ1 andϕ2 are the volume fractions. This mixing rule is ex-tensively used in petrochemical calculations and canbe extended to more than two solutions. Eq. 3.81 hasbeen used by Koval66 to estimate the eective viscos-ity ηe in viscous ngers,

ηe =

[0.78 + 0.22

(η2

η1

)1/4]4

. (3.82)

Another mixing rule is given by Harris67 forpolystyrene solutions. The relation works best whenthe dierence in molecular weight is not too great,

log(ηmix) = w1 log(η1) + w2 log(η2), (3.83)

and w1 and w2 are the weight fraction of the mixture.

3.3 Rock Properties

3.3.1 Porosity

Porosity is the ratio of void or pore volume to macro-scopic or bulk volume and for most naturally occur-ring media the porosity is between 0.10 and 0.40. Theporosity can be divided into an interconnected or ef-fective porosity available for uid ow and a discon-nected porosity unavailable for uid ow. The latterporosity is of no interest for the ow processes dis-cussed here, so in the rest of the text the term poros-ity means eective porosity. However, some IOR-processes exhibit behavior whereby some of the ef-fective porosity is shielded from the displacing agent.These pore volumes are denoted dead end (local het-erogeneity in the permeable medium) or inaccessiblepore volume (IPV). The most common explanation ofinaccessible pore volume is that the smaller portionsof the pore space will not allow large molecules likepolymers to enter. In extreme cases, IPV can be 30%of the total pore space.1

Porosity is a function of many factors as shown inTable 3.1. The most important of these are grain

Table 3.1: Factors controlling rock pore structure.

Original DiageneticGrain sorting Grain dissolutionGrain packing Pressure solutionGrain shape Grain overgrowthsIntergranular matrix CementsClay laminae Authigenic clays

size and grain-size distribution (including clay typeand content), shape of grains, packing and cemen-tation. Most porosity data provide little, if any, in-formation regarding the eect of these rock parame-ters on porosity. However, the objective of numerousstudies has been to relate porosity other rock prop-erties. These correlations are subject to severe lim-itations and are seldom used. In applied reservoirengineering, it is important to study each reservoirrock carefully, and occasionally it is possible to corre-late porosity with authigenic clay cement content orestablish cross plots between porosity and permeabil-ity. In Fig. 3.22 is shown an example from the TrollField with relationship between porosity, permeabil-ity and lithofacies classes.The porosity of a reservoir rock may be determined

by core analysis or an appropriate logging technique.From a statistical viewpoint, both logging and coreanalysis provide an inadequate sampling of porositydata. However, the measuring techniques and statis-tical analysis are outside the scope of this monograph.When the reservoir pressure declines, slight reduc-

tions in volume of the rock grains due to rock graincompressibility, and in porosity due to pore volumecompressibility, are obtained. For most reservoirs,the change in rock grain volume is much less than

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52 CHAPTER 3. EQUATIONS

10000

1000

100

10

1.0

0.10 10 20 30 40

Porosity %

Perm

eabi

lity

(md)

CS1 coarse sand,poorly sorted

CS2 coarse sand,well sorted

MS1 heterogenous,medium sand

MS2 medium sand,well sorted

FS1 bioturbated,fine sand(micaceous)

FS2 laminatedfine sand(micaceous)

VS1 bioturbated,very fine sand/silts (micaceous)

CS1

CS2

MS

1

MS2

VS

1

FS1

FS2

Figure 3.22: Relationship between porosity, permeability and lithofacies in the Troll eld.68

the change in the porosity, and

cf =

(1

Vb

)(dVpdp

), (3.84)

where cf is formation compressibility, Vb is reservoirbulk volume, Vp is reservoir pore volume and p is porepressure.The formation compressibility may either be mea-

sured in the laboratory or determined from correla-tions. One well known correlation is that of Hall:69

cf = 1.94× 10−9φ−0.438. (3.85)

where φ is porosity in percent.The formation compressibility for limestones and

sandstones are of the order 0.3 × 10−9 to 3.6 × 10−9

Pa−1. Therefore, any study of oil recovery from un-dersaturated reservoirs requires that the formationcompressibility is known.

3.3.2 Permeability

Permeability is a property of the porous medium andis a measure of the capacity of the medium to trans-mit uids. Permeability is a tensor that in general isa function of pressure. Usually, the pressure depen-dence is neglected in reservoir calculations, but the

variation with position can be pronounced. Very of-ten the permeability varies spatially by several magni-tudes, and such heterogeneity will of course inuenceany IOR process.Preceeded by Hagen's (in 1839) and Poiseuille's (in

1846) work on the laws aecting the ow of waterthrough capillary tubes, Darcy (in 1856) performed aseries of experiments on the relationship aecting thedownward ow of water through sands. A compre-hensive discussion are given in these classical publica-tions.3,70,71 The generalized equation called Darcy'slaw may be written in the form:

~u = −

~~kµ

(∇p+ ρ~g), (3.86)

where ~u is supercial velocity, ~~k is permeability ten-sor, µ is uid viscosity, ∇p is pressure gradient, ρ isuid density and ~g is gravitational vector.

Permeability Model

Formation permeability may be determined or esti-mated on the basis of core analysis, well tests, pro-duction data, well log interpretations, or correlationsbased on rock parameters. In the evaluation of an

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3.4. ROCK PROPERTIES 53

IOR process, pore models are very often used, for in-stance the Kozeny-Carman model.71 The volumetricow rate q in a horizontal capillary of radius R andlength Lt is given by Hagen-Poiseuille equation

q = πR4

∆p

Lt, (3.87)

and the average velocity in the tube is

v =q

π R2=R2

∆p

Lt. (3.88)

We have to transform this equation to the scale of arepresentative elementary volume (REV). The REVis dened as a volume below which local uctuationsin permeability is large.3 If we make the travel timein the capillary tube equal to that in a REV, then(

Ltv

)t

=

(L

v

)REV

. (3.89)

The relation between interstitial and supercial ve-locity is v = u/φ. Darcy's law can be used to elim-inate v in Eq. 3.89 and we obtain the permeabilitycomponent k

k =R2φ

8τ, (3.90)

where τ = (Lt/L)2 is the tortuosity. This is an im-portant permeable media property and is usually esti-mated from electrical resistivity measurements. Thetortuosity is in the range of 2 to 5 for most reservoirrocks.The capillary radius R in Eq. 3.90 is dicult to de-

ne for a porous medium, but may be approximatedby the hydraulic radius Rh that expresses the ratiobetween volume open to ow and the wetted surfacearea,

Rh =φ

a(1− φ), (3.91)

where a is internal surface area per volume, an impor-tant intrinsic property of permeable media. By sub-stituting this expression for R in Eq. 3.90 and solvingthe equation for an assemblage of uniform sphereswhere a = 6/D we get the Kozeny-Carman equation:

k =1

72τ

φ3D2

(1− φ)2. (3.92)

Here D is sphere or particle diameter, and we seethat permeability is a strong function of particle sizeand packing through φ. The Kozeny-Carman equa-tion is often used to make order-of-magnitude esti-mates of pore size from the knowledge of permeabil-ity. However, the capillary tube model is of limitedvalue since it does not provide alternate pathways foruid ow within each REV. The consequence is thatwe cannot predict relative permeabilities or trappedphase saturations, parameters of major importancein IOR processes. The two-phase parameters will bediscussed later and specic applications of the per-meability models in IOR processes are discussed byLake.1

Eect of Clay

Clays are generally located on the pore grain surfaces.They have a large specic surface area, and are chem-ically reactive. The particle size is small (< 50nm)so the permeability of clay is generally very low. Seg-regated clays or shales are therefore not included asa part of productive reservoir thickness. Shales arebarriers to uid ow, particularly vertical ow, andwill hinder gravity segregation. Dispersed clay aredistributed among the pores of sedimentary rocks andmay aect the permeability in several ways dependingon the type of clay. Swelling clays, such as smectiteor degraded illite, tend to reduce rock permeabilitywhen contacted by fresh water. Kaolinite very of-ten moves within the rock pore system and lodges insmall passages, thereby reducing permeability. Illitecreates more tortuous paths through the pore systemby bridging gaps between grains.

3.4 Fluid-Rock Interaction

3.4.1 Capillary pressure

The capillary forces are very often the strongest forceswithin the REV in multiphase ow at typical ow ve-locities. When more than one uid phase is presentin a porous rock, there are at least three sets of ac-tive forces aecting capillary forces, namely the forcesthat are active at the interface between two immisci-ble uid phases and between each uid and the solid.The combination of all the active surface forces de-termines the capillary pressure of a porous rock. Webegin with a brief discussion of surface tension, wetta-bility and the capillary tube concept, before proceed-ing to the capillary phenomena actually observed inmultiphase ow.

Surface tension

A direct evidence of the existence of tension in aninterface is the tendency of a soap bubble to decreaseits size. The stress causing this decrease is calledsurface or interfacial tension.Fig. 3.23 shows a spherical cap which is subjected

to a surface tension σ around the base of the cap

σ σ

p2

p1

Figure 3.23: Capillary equilibrium of a spherical cap.

and to normal pressures p1 and p2 at each point onthe surface. The eect of the surface tension σ is to

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54 CHAPTER 3. EQUATIONS

reduce the size of the sphere unless it is opposed by asuciently great dierence between pressures p1 andp2. The surface tension has the dimension of forceper unit length.The Laplace equation for the mechanical equilib-

rium of an arbitrary surface is71

p2 − p1 = σ

(1

r1+

1

r2

), (3.93)

where r1 and r2 are the principal radii of curvature.Introducing the mean radius of curvature rm denedby

1

rm=

1

2

(1

r1+

1

r2

), (3.94)

the Laplace equation becomes

p2 − p1 =2σ

rm. (3.95)

Note that the phase on the concave side of the surfacemust have a pressure p2 which is greater than thepressure p1, on the convex side.

Wettability

When two immiscible uids contact a solid surface,one of them will tend to spread or adhere to it moreso than the other. This is the result of the surfacetension of the uids and solid-uid interfacial tension.A measure of wettability is the contact angle Θ whichis related to the surface energies by the Young-Dupreequation. For example, for a water-oil-solid system,we have

σos − σws = σow cos Θ, (3.96)

where σos is interfacial tension energy between theoil and solid, σws the water and solid, σow betweenthe oil and water, and Θ is contact angle measuredthrough the water phase.Neither the oil-solid nor the water-solid interfa-

cial energies can be measured directly. However, theequivalent terms, the oil-water interfacial tension andthe contact angle, can be determined independentlyin the laboratory. The contact angle is the mostuniversal measure of the wettability of surfaces andFig. 3.24 shows idealized examples of contact anglesand spreading.For much of the past 50 years, a large body of

reservoir engineering practice has been based on theassumption that most reservoirs are strongly water-wet, i.e., the reservoir rock surface always maintains astrong anity for water in the presence of oil. But re-search the last years has revealed evidence of crude oilwetting behavior, and it is now widely accepted thatmost reservoirs are at wettability conditions otherthan very strongly water-wet.7274 This conclusionhas led to increased interest in improving the proce-dures for measuring reservoir wettability. Especiallythe eect of wettability on oil recovery in variousenhanced recovery processes has been studied inten-sively by methods involving core samples. The two

OIL

WATER

OIL OIL

θ = 0oSOLID

OIL

WATER

θ = 110o

SOLID

θ = 180o

θ = 40o

Figure 3.24: Idealized examples of contact angles andspreading.72

most used methods of quantifying wettability basedon rock/brine/oil displacement behavior are the mod-ied Amott test and the USBM test.75,76 The readersinterested in advanced core analysis for wettabilityshould refer to extensive bibliographies provided byAnderson.7782

Capillary Pressure

Fig. 3.25 shows a capillary with a nonwetting phaseon the left and a wetting phase on the right. If the

Nonwettingphase

Wettingphase

p p2 1

Interface

θ

R

Figure 3.25: Interface between two phases in a tube.

phases and interface are stagnant, a static force bal-ance across the interface in the direction parallel tothe tube axis gives the following expression

p2 − p1 =2σ cos Θ

R= Pc. (3.97)

This equation relates the capillary pressure across aninterface to the capillary radius R, the interfacial ten-sion σ, and the contact angle Θ.For more complicated geometries, the capillary

pressure is inversely proportional to a generalized in-terfacial curvature, which is usually dominated by thesmallest local curvature of the interface. The entryof a nonwetting phase into a single pore of toroidalgeometry bounded by a sphere assemblage is shownin Fig. 3.26. Fig 3.27 shows the capillary pressure

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3.4. ROCK PROPERTIES 55

corresponding to the various entry positions of theinterface.

AB C D

Nonwetting

Figure 3.26: Schematic of interface entrance into atoroidal pore.1

AB

C

D

E

F

GNonwetting phasenot constrainedby walls

Nonwettingphaseconstrainedby walls

Isolated globuleof nonwettingphase

S nw

Pc

Figure 3.27: Resulting capillary pressure for nonwet-ting uid entry, adapted from Lake.1

The behavior of the capillary pressure curve whenuid is moving from points A, B, C and D is di-rectly interpreted from the interface curvature. Be-yond point D, the interface will leave the pore, andneither the capillary pressure nor the saturation willchange. Another realistic situation is that the porebody is much larger than the pore entrance. Then theinterface will collapse, creating a disconnected glob-ule of the nonwetting phase within the pore body.The globule conforms to the pore body to minimizeits energy, and the curvature again increases, caus-ing an abrupt decrease in the capillary pressure frompoints D to E. If the nonwetting phase saturation isagain increased, the globule is forced farther into therock-rock contacts causing increased capillary pres-sure (E,F,G).A reservoir rock has continuous pore size distribu-

tion and the capillary pressure curve will be continu-ous as in Fig. 3.28. The curve exhibits many of thesame features as the discontinuous one in Fig. 3.27,such as entry pressure at low nonwetting phase sat-uration and sharp increase in capillary pressure atlarge saturations.Fig. 3.29a shows an idealized medium where a non-

wetting phase is forced into the system. This is calleda drainage process, nonwetting saturation increases.When the nonwetting phase saturation decreases we

b d

a

c

Pc

0 1S

wS

wi

Figure 3.28: A primary drainage capillary pressurecurve (a) with drainage-imbibition hysteresis loops(b,c,d).

have an imbibition process, Fig. 3.29b. The dierence

OIL

WATER

b

a

Figure 3.29: Schematic of drainage (a) and imbibition(b) process.

in drainage and imbibition capillary pressure curve inFig. 3.28 is called capillary hysteresis. It is caused bydierences in advancing and receding contact angles(drag hysteresis) and dierence in capillary retention(trapping hysteresis). Lake1 and Stegemeier83 treatthe physics of trapping hysteresis. Its numerical for-mulation is discussed by Aziz37 and Killough.84

Leverett85 demonstrated that a single, dimension-less capillary pressure function could represent a ge-ological formation, independent of variations in poresize distribution. In Eq. 3.97, the tube radius R is re-placed by a function (R/J), where J is a dimension-less function of the nonwetting phase saturation Snw.If R is replaced by

√k/φ, as indicated in Eq. 3.90,

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56 CHAPTER 3. EQUATIONS

we arrive at the Leverett J-function

J(Snw) =Pc

√kφ

σ cos θ. (3.98)

The numerical constant tortuosity is included in theJ-function. Alternate capillary pressure correlationsand application of such correlations are discussed byseveral authors.8688

The pore size distribution has great impact on therecovery eciency of a displacement process. Thedrainage capillary pressure curve is often used to cal-culate pore size distribution. From Eq. 3.90, withσ cos θ known, we can calculate the radius R of thesmallest pore being entered at that nonwetting satu-ration. This information can be converted into a fre-quency of pores at a given R. A typical pore size dis-tribution curve for a North Sea sandstone89 is givenin Fig. 3.30.

0.01 0.1 1 10 100Mean pore throat diameter, (µm)D

0.4

0.3

0.2

0.1

0.0

d /

d (l

og

)V

D

Figure 3.30: Pore size distribution curve.89

3.4.2 Relative Permeability

Relative permeability is the ratio of the eective per-meability of a given uid at a xed saturation to thepermeability at 100% saturation. Eective perme-ability is the ability of the porous material to conducta uid when its saturation is less than 100% of thepore space,

krj =kjk, kr ∈ [0, 1], (3.99)

where krj and kj are the relative and eective perme-abilities of phase j, and k is the absolute permeability.The controlling factors are believed to be pore geom-etry, wettability, uid distribution, saturations andsaturation history.Oil and water relative permeabilities in an oil-water

system are usually plotted as functions of water sat-uration as typically shown in Fig. 3.31. The relativepermeability of a phase decreases as the saturationof that phase decreases. The saturation Sor is calledthe residual oil saturation and the reduction of thistrapped oil saturation is an important objective of im-proved oil recovery projects.

Drainage

Imbibition

S w

k rS or

1.0

0.01.00.0 S wi

Figure 3.31: Typical relative permeabilities; oil-watersystem.

Eects of Wettability

Wettability of the rock has an important eect on therelative permeability and has been subject to com-prehensive literature reviews.81,87 The eect of wet-tability on the relative permeability endpoints is stillsubject for discussion. The rules of thumb presentedin several books on the subject87,88,90 are:

Waterwet OilwetIrreducible watersaturation 2025% 15%

Saturation at whichoil and water relative 50% 50%permeabilities are equal

Relative permeabilityof water at residual 30% 50%oil saturation

These rules are questioned by several researchers,especially for nonuniform or mixed wettability sys-tems.7274

Hysteresis

Since the saturation history aects the uid distribu-tion and causes a hysteresis in the capillary pressurecurve, similar hysteresis eect is to be expected in thedrainage and imbibition relative permeability curves,Fig. 3.31. Hysteresis in the wetting phase relativepermeability is usually considered to be negligible, asshown in Fig. 3.31. In improved recovery projectswe may often encounter a change from one processto another. Secondary drainage is obtained after dis-placing all mobile oil as shown in Fig. 3.32.

In practice, the ow reversal may occur at any in-termediate saturation and the relative permeabilityfollows an intermediate path.

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3.4. ROCK PROPERTIES 57

Secondary drainage

Intermediate path

0.0 1.0

k ro

S w

0.0

1.0

Figure 3.32: Eect of ow reversal on relative perme-ability.

Empirical Functions

The general shape of the relative permeability curvesmay often be approximated by the following exponen-tial forms,87

kro = kero

(So − Sor

1− Sor − Swi

)n, (3.100)

krw = kerw

(1− So − Swi1− Sor − Swi

)m, (3.101)

where kero and kerw are endpoint relative permeabil-ities, Swi and Sor are irreducible water saturationand residual oil saturation respectively and n and mare constants. These equations t most experimentaldata and separate explicitly the relative permeabilitycurvatures and the endpoints.

Measuring Methods

The relative permeability can be measured at roomor reservoir conditions and by either steady-state orunsteady-state methods. At reservoir conditions it isobvious that reservoir core samples and uids are usedand that the maintenance or restoration of these con-ditions are performed carefully.In the steady-state method, two uids are injected

simultaneously at a xed ratio. The pressure dif-ferential during ow is measured and from this therelative permeabilities are determined. The primaryconcern in these experiments is to reduce the satura-tion gradients at the inlet and outlet, and the variousunsteady-state methods only dier in the practicaldetails. A comprehensive review of the methods isgiven by Honarpour et al.87 Some investigators preferthe steady-state methods to unsteady-state methodsespecially for rocks of intermediate wettability. How-ever, the unsteady-state method is by far the mostcommonly used and convenient method of measuringrelative permeability.In 1952, Welge92 presented the unsteady-state

technique for measuring relative permeability. Onephase is displaced from a core by injecting the other

phase, and the relative permeability ratio is calcu-lated from the produced uid ratios. The mathemat-ical basis for interpretation of the data is an extensionof the Buckley-Leverett93 frontal advance theory. Bycombining Darcy's law and capillary pressure in dif-ferential form we obtain

fw2 =

1 +koµou

(dPcdx− g∆ρ sinϑ

)1 +

kokw

µwµo

, (3.102)

where fw2 is the fraction of water in the outlet stream,u is the supercial velocity of total uid leaving thecore, ϑ is the angle between direction of ow andhorizontal, and ∆ρ is the density dierence betweendisplacing and displaced phases. Welge92 developedthe relationship

Sw − Sw2 = Qwfo2, (3.103)

where Sw is average water saturation, Sw2 is the wa-ter saturation at the exit end of the test sample, andQw is pore volumes of cumulative injected uid.In practice, Sw, Qw, and fo2 can be determined at

any time from the production history of a ow exper-iment. Then the saturation at the outow face canbe calculated from Eq. 3.103, and since the viscosi-ties are known, the relative permeability ratio can bedetermined from

fo2 =1

1 +µo/kroµw/krw

. (3.104)

A limitation of this method at the time it was intro-duced was that it gave only the relative permeabilityratio, not the individual relative permeabilities. How-ever, in 1959 Johnson et al.94 showed how individualrelative permeabilities could be obtained from the re-sults of an external drive experiment. Later severalalternative techniques have been reported.95,96

Advantages of the unsteady-state method includespeed and simplicity. The main limitation is the re-quirement of large pressure gradient to minimize cap-illary end eects. However, today test runs at reser-voir rates are frequently modelled with numerical sim-ulators, and the results are applied in a correctionprocedure for end eects. As mentioned earlier, reser-voir rocks of intermediate or mixed wettability arequite common and there is growing evidence that endeects for wateroods at reservoir rates in such sys-tems are quite small.72 In addition, the new methodsfor in-situ measurements of saturation to check forend eects and variations in uid distribution haveprovided increased condence in the unsteady-stateprocedures and results.Several other laboratory based methods for deter-

mining relative permeability are available, but mostof them have had limited utility. In comparison withthe steady and unsteady-state technique, the onlymethod with some additional features is the cen-trifuge method. It involves monitoring liquids pro-duced from core plugs during the rotation. The

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58 CHAPTER 3. EQUATIONS

test requires an automated centrifuge with imagingand data collecting systems. Description of suchequipment is given by O'Meara and Lease97 andMunkvold and Torsæter.98 The calculations are ei-ther based on an analytical solution99 or numeri-cal simulators.100,101 Measurement of relative per-meability with the centrifuge is substantially fasterthan other techniques and are not subject to the vis-cous ngering problems in the unsteady-state mea-surements. On the other hand, the centrifuge methodis subject to capillary end eect problems and mobil-ity ratio problems due to the requirement of negli-gible oil/gas mobility ratio in the analytical model.At present, the centrifuge method does not provide ameans for determining relative permeability of the in-vading phase. However, in gravity drainage processesthe centrifuge method is recognized to give reliableoil relative permeabilities.98100

3.4.3 Three-Phase Models

A number of three-phase relative permeability mod-els has been published over the years. None of thesemodels have been accepted as denite, a model thatwould predict the correct three-phase relative perme-ability for any combination of rock type and uid sat-uration history. The obvious reason is that the ow ofthree phases in porous media is controlled by a num-ber of physical and chemical parameters that mayco-exist in a vast number of combinations. In addi-tion, for practical purposes it is impossible to takemore than a few of these combinations into consid-eration during laboratory relative permeability mea-surements. With this in mind, it is not very sur-prising that three-phase relative permeability modelsthat predict correct values for some sets of data failfor others, i.e., the model is not necessarily as univer-sal as the rst evaluations might have indicated.The number of published relative permeability data

is very limited compared to the number of model pa-rameters. In addition, many of the published data aresubject to discussion whether saturation history andend eects should be taken into consideration whenevaluating the measured data. Also, many of the pub-lished relative permeability data are scattered, andthe tracing of isoperms is therefore subject to inter-pretation.102 It is still an open question whetherthree-phase relative permeabilities are functions oftheir respective phase saturations only.The most important rock parameters that controls

the relative permeability are pore throat size dis-tribution and the pore throat to pore body ratio.Important uid-rock parameters are interfacial ten-sion, wettability and residual uid saturation. Resid-ual uid saturation is normally related to capillarynumber through the so-called capillary desaturationcurve. Important liquid parameters are uid-uid in-terfacial tension and viscosity. Unfavorable viscos-ity ratio may lead to viscous ngering. In addition,both relative permeability and residual saturation arenormally functions of the saturation history. Finally,

relative permeability may be a function of more thanone saturation.It is, for practical reasons, not possible to establish

three-phase relative permeability models that take allthese variables into consideration explicitly. The so-lution has been to group the variables into dimension-less numbers like the capillary number, the instabil-ity number or the wettability number, then establisha relationship between these numbers and capillarypressure or relative permeability and nally calculatethe three-phase relative permeability using two-phasedata measured at, or approximated to, the actualthree-phase conditions.Some three-phase relative permeability models will

be presented based on pore network theory and somebased on measured two-phase relative permeability.For simplicity, only the relative permeability of theintermediate-wetting phase will be presented since itis generally accepted, but necessarily true, that therelative permeability of the wetting and nonwettingphase are functions of their own saturation only. Theintermediate-wetting uid is assumed to be oil.

Models Based on Network Theory

A number of three-phase relative permeability mod-els based on various capillary tubing/pore networkconcepts have been published, though none of theseseem to have gained the same degree of popularity asthose based on two-phase relative permeability. Thereason is probably that these models are not veryexible with respect to tting observed data.

Introduction. Burdine103 developed a two-phasedrainage relative permeability model based on owin a bundle of capillary tubes. Relative permeabilityis proportional to the mean hydraulic area of eachphase. It was necessary to introduce a tortuosity fac-tor to make the model t measured relative perme-ability data. Burdine showed that the average tortu-osity factor for each phase was linearly proportionalto the phase saturation. His expression for the non-wetting phase is

krnw,dr = (1− S?w)2

∫ 1

S?w

1

P 2c

dS?w∫ 1

0

1

P 2c

dS?w

, (3.105)

where the capillary pressure is related to the radiusof the capillary tubes through the expression

Pc =2σ

r.

Using the Corey drainage capillary pressure vs. wet-ting phase saturation relationship

S?w = (PcPe

)λ, (3.106)

Eq. 3.105 can be written in the form

krnw,dr = (1− S?w)2[1− (S?w)2+λλ ], (3.107)

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3.4. ROCK PROPERTIES 59

where λ is a pore size distribution index that de-creases as the pore-size distribution broadens.

Three-Phase Drainage Model. The theory out-lined for two-phase relative permeability can be ex-tended to three-phase behavior. The expression S?win Eq. 3.107 must be replaced by S?l , which is the to-tal liquid saturation. It is assumed that the capillarypressure vs. total liquid saturation obtained when gasdisplaces oil in the presence of water, will be the sameas when gas displaces water with no oil present.104

The relative permeability of oil in an oil-water-gassystem, can then be expressed as

kro,dr = (S?o )2

∫ S?l

S?w

1

P 2c

dS?l∫ 1

0

1

P 2c

dS?w

, (3.108)

assuming oil is the intermediate-wetting phase. If thecapillary pressure term in Eq. 3.108 is replaced byEq. 3.106, it can be written in the form

kro = (So

1− Slr)×[(

So + Sw − Slr1− Slr

) 2+λλ

−(Sw − Slr1− Slr

) 2+λλ

].

(3.109)

Using λ = 2, which is a typically value for consoli-dated sandstone, Eq. 3.109 can be written

kro =(Sl − Sw)3

(1− Slr)4(Sw − Sl − 2Slr). (3.110)

The parameters Slr and λ in Eq. 3.109 can be used tot either two-phase relative permeability or capillarypressure data. The constants are then be valid forthree-phase relative permeability calculations. Themodel can only t the endpoint on either the gas-oil or the water-oil two-phase relative permeabilitycurve. If these are very dierent the model may notbe suciently exible.

Three-Phase Imbibition Models. The relativepermeability of the nonwetting phase is normallylower during imbibition than during drainage. Thisis explained by capillary trapping of the nonwet-ting phase as the wetting phase invades the porousmedium. Because of the trapping of nonwettingphase, the concept of capillary tube bundles can notbe used directly to establish an imbibition model.Naar et al.105,106 used the concept of capillary tubesrandomly interconnected to build their imbibitionmodel. It was then possible to model the phenomenawhere the nonwetting phase is bypassed and trappedby the invading wetting phase. Also, Naar assumesthat relative permeability is proportional to the hy-draulic area:

kro,im =

(1− Sfw1− Swc

)3

×

S∗∗of

∫ S∗of

0

(S∗∗of − S∗∗)P 2c

dS∗∗∫ 1

0

1− S∗

P 2c

dS∗∗. (3.111)

Using the Corey type of capillary pressure vs. wettingphase saturation relationship with λ = 2, Eq. 3.111can be written

kro,im = (S∗of )3(S∗of + 3S∗fw). (3.112)

This equation has no free parameters to t measuredrelative permeability, except for the irreducible watersaturation. But Eq. 3.112 can be derived for othervalues of λ, giving better t measured data.A slightly dierent model was proposed by

Parker,107

kro,im = (S?t − S?w)0.5

∫ 1−SgS?w

1PcdS∫ 1

01PcdS

2

. (3.113)

Unlike Burdine, Parker assumes that the relative per-meability is proportional to the mean hydraulic radiusof each phase. The tortuosity factor is also dierent.Instead of Eq. 3.106, Parker uses another relation tomodel the capillary pressure curve:

S? = [1 + (υPc)n]−m, m =

1

n. (3.114)

Including Eq. 3.114 in Eq. 3.113, it can be written

kro,im = (S?t − S?w)0.5×[(1− (S?w)

1m )m − (1− (S?t )

1m )m]2,(3.115)

where m can be used to t either measured capillarypressure or measured two-phase relative permeabilitycurves. According to Baker,108 Eq. 3.115 could betted to most of the two-phase relative permeabilitydata he used in his evaluation study with m = 1.Eq. 3.115 then becomes

kro = (S?t − S?w)2.5, (3.116)

which indicates that the three-phase relative perme-ability of oil is a function of its own saturation only.

Models Based on Two-phase Relative Perme-ability

The most popular three-phase relative permeabilitycorrelations are undoubtedly those based on mea-sured two-phase relative permeability. One advan-tages is that they will approach the measured two-phase relative permeability at the two-phase limits,and the models can therefore be used continuouslybetween the two- and three-phase regions. The dif-ference between the models is how well they pre-dict relative permeability in the three-phase region.For simplicity, the oil phase is assumed to be theintermediate-wetting phase.

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60 CHAPTER 3. EQUATIONS

Stones Models. Two of the most popular three-phase models are those proposed by Stone.Stone's rst model for the relative permeability to

oil in a three-phase system is109

kro = S∗o (krow

1− S∗w)(

krog1− S∗g

), (3.117)

where krow and krog are the measured two-phase oilrelative permeabilities; krow is measured as a func-tion of water saturation, while krog is measured asa function of gas saturation. The minimum residualoil saturation, Som, is included. It must be deter-mined by interpolation between two-phase data, or itmay be used as a tting parameter if some measuredthree-phase relative permeability data are available.Both Stone's models require that the relative perme-abilities of the wetting and nonwetting phases arefunctions of their respective saturations only. Theargument is that water, being the wetting phase, al-ways occupies the smallest pores, while the gas phasealways occupies the largest pores, and the oil phasetherefore spatially separates them.Oil relative permeability in the second model of

Stone is given by155

kro = (krow+krw)(krog+krg)− (krw+krg). (3.118)

Improvement of Stone's Models. Aziz andSettari37 suggested a normalized form of Stones mod-els,

kro =S∗okrocw

krow1− S∗w

krog1− S∗g

, (3.119)

and

kro = krocw×

[(krowkrocw

+ krw)(krogkrocw

+ krg)− (krw + krg)].

(3.120)

The minimum residual oil saturation, Som, is com-monly used as a tting parameter. Fayers andMatthews112 found that the prediction of relative per-meability by Stone rst model could be improved byintroducing a linear expression for Som,

Som = ωSorw + (1− ω)Sorg, (3.121)

whereω = 1− Sg

1− Swc − Sorg. (3.122)

This expression is, however, not valid if gas istrapped. Morse and Holmgren113 found that both theresidual oil saturation and the oil relative permeabil-ity decreased as the trapped gas saturation increased.This observation may be explained by the assumptionthat the trapped gas phase occupies large pores whichthe owing and trapped oil phase normally wouldhave occupied. Based on published results, Fayersand Matthews proposed an empirical correlation be-tween residual oil saturation and residual gas satura-tion to be used instead of Eq. 3.121, when trapped

gas is present,

Som = Sorw − 0.5Sgr. (3.123)

Later, Fayers114 extended Eqs. 3.121 and 3.122 to

Som = Sorw − [Sorw − Sorg

1− Swc − Sorg]Sg

−ε[(1− Sw − Sorg)Sg − S2g ], (3.124)

where ε is a free parameter. Here Som will approachesSorw when the gas saturation approach zero, and Sorgwhen the water saturation approaches Swc.

Other Models. Several other models have beenproposed for oil relative permeability in a three-phasesystem. In the following, three of them are briey dis-cussed.The Linear Interpolation Model.108 The

model interpolates linearly between the two-phasevalues krow and krog. The oil isoperms drawn in aternary diagram will therefore be straight lines. Thethree-phase relative permeability at a given oil satu-ration is found by iteration, and the model thereforerequires more computer time than other models. Itcan also be used to calculate three-phase water andgas relative permeability if the required two-phase rel-ative permeability data is available. The motivationfor proposing the model in the rst place is not clear,but this model, unlike many others, will denitely notgenerate any unexpected curvatures of the isopermsin the three-phase region and is therefore probablythe safest model to use if no measured data is avail-able to which the free parameter of the other modelsmay be tted.The Saturation Weighted Interpolation

Model.108 The model calculates the saturation av-erage relative permeability of krow and krog, wherekrow is weighted by the eective water saturation,and krog is weighted by the eective gas saturation,

kro =(Sw − Swc)krow + (Sg − Sgr)krog

(Sw − Swc) + (Sg − Sgr). (3.125)

The model forces the oil relative permeability to thecorrect two-phase limits,

kro → krow when Sg → Sgr,

kro → krog when Sw → Swc.

The model will therefore produce isoperms withstrong curvatures if Sorw is much dierent from Sorgand the oil saturation in the three-phase region closeto the smallest of these two. When Sorg > Sorw,the oil isoperm will be concave towards the 100% oilapex, and convex against the 100% oil apex whenSorg < Sorw. Also, this model can be used to calcu-late three-phase water and gas relative permeabilityif the required two-phase relative permeability dataare available.

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3.4. ROCK PROPERTIES 61

Pope's Model. The model was presented byDelshad and Pope,115

kro = k0row[aSα0 (1−Sw)β+(1−a)Sγo (1−Sδg ]. (3.126)

The free parameters α, β, γ, δ can be determined ei-ther by tting the equation to measured two-phaseoil relative permeabilities krow and krog, or by ttingboth two-phase and three-phase data. Though themodel includes many free parameters, it may be ap-plied if other more preferred models fail to predictacceptable three-phase values.

Discussion

The ability of the models to predict correct three-phase relative permeability has been discussed byFayers and Matthew,112 Delshad and Pope,115 andBaker.108 The majority of the models predict reason-ably well for high relative permeability values. Butthree-phase ow in the reservoirs is maybe most oftenpresent during tertiary processes at low oil saturation,and attention should therefore be given to the behav-ior of the models in the lower range of oil saturation.

Fayers and Matthew112 tested the normalizedStones models, Eqs. 3.119 and 3.120, for a number ofmeasured three-phase relative permeability datasets.They concluded that Stones rst model is superior ifEqs. 3.121 or 3.122 is used to calculate Som, and thatwater and gas relative permeability are functions oftheir respective saturation only.

Delshad and Pope115 evaluated six dierentmodels using several sets of three-phase relative per-meability data. In addition to the Stone's mod-els, they tested the saturation-weighted interpola-tion model, the Pope model, Parkers model, and -nally an model by Lake presented in Sec. 10.2 onpage 247. They concluded that Pope's model andthe saturation-weighted interpolation model gave thebest t to the measured data. Pope's model was t-ted to both two-phase data and a combination of two-and three-phase data. The latter technique gave bet-ter t, as expected. Both versions gave, however,better t than the Stone's models.

Baker108 evaluated twelve dierent three-phaserelative permeability models. Five of these were porenetwork type models and another ve were variationsof Stone's models. The two last models were the lin-ear interpolation models and the saturation-weightedinterpolation model. The models were tested for eightsets of published three-phase data. Baker concludedthat most of the models predicted the measured datawell for high values of relative permeability. The lin-ear interpolation model and the saturation-weightedinterpolation model gave the best t based on an av-erage of the eight datasets. These two models wereamong the three best for all eight datasets. The

models based on pore network theory gave gener-ally a poor t. The exception was Parker's modelwhich behaved almost as a linear interpolation model.The Stone models predict some dataset superbly andother poorly. On an average of all the datasets,Stones rst model, Eq. 3.119, in combination withthe expression for Som as suggested by Fayers andMatthews, gave the best overall t.

3.4.4 Hysteresis

There are several situations during hydrocarbon pro-duction where three phases may ow simultaneouslyand changes in the displacement sequence occur.Among these are gas injection after waterooding,continuous water-alternating-gas (WAG) injection,surfactant injection after waterooding, and water in-jection in a reservoir produced below the bubblepoint.None of the above models handle hysteresis in rel-

ative permeability caused by a change in phase dis-placement sequence. The pore network models usea single two-phase drainage capillary pressure curveas input. The drainage models are able to han-dle monotonous decrease of wetting phase saturationwhile the imbibition models can handle monotonousincrease of the wetting phase saturation.Killough84 suggested that hysteresis in two-phase

relative permeability should be measured and that anappropriate hysteresis scheme could be used to modelthe two-phase hysteresis. He further suggested thisscheme to be included in Stone's models to predicthysteretic three-phase relative permeability.

Land

Land116,117 formulated a relationship between themaximum and the residual saturation of the nonwet-ting phase,

1

S∗rnw− 1

S∗rnw,M= C. (3.127)

Although gas was the nonwetting phase in Land'swork, Eq. 3.127 can in principle be used for any non-wetting phase. Land also developed an equation forcalculating the mobile fraction of nonwetting phaseduring imbibition,

S∗nw,f = 0.5[(S∗nw − S∗rnw)

+

√(S∗nw − S∗rnw)2 +

4

C(S∗nw − S∗rnw) ].

(3.128)

These equations give important input to the hystere-sis models in this section.

Killough

Killough84 published a scheme for modelling two-phase relative permeability hysteresis, both for wet-ting and nonwetting phase. The input parametersare the primary drainage and imbibition curves. Thedrainage curve should be measured up to maximum

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62 CHAPTER 3. EQUATIONS

nonwetting phase saturation and the imbibition curveshould measured from this saturation and down to the(maximum) nonwetting phase residual saturation.From the imbibition relative permeability curve,

Land's constant C, from Eq. 3.127, is calculated andconsidered independent of the process. Eq. 3.127 isused by Killough to calculate the residual nonwet-ting phase saturation at any historical maximum non-wetting phase saturation. Historical maximum sat-uration is the maximum saturation the nonwettingphase has reached during the process. Killough as-sumes that drainage following imbibition does notexhibit hysteresis. He uses imbibition relative perme-ability at saturations below the historical maximumnonwetting phase saturation for both the imbibitionand drainage process. Killough normalizes the ex-perimental imbibition curve to t the relative perme-ability endpoints at the nonwetting phase historicalmaximum and residual saturations. The model hasone disadvantage. If the relative permeability dur-ing a drainage process following an imbibition processshows strong hysteresis, Killough's model will fail.

Lohne

A new scheme developed by Lohne118 allows forhysteresis loops between extreme saturations. Thismodel also uses the measured primary drainage andimbibition curves, the latter starting at maximumnonwetting phase saturation. The residual saturationof the nonwetting phase is calculated from Eq. 3.127.When the nonwetting phase saturation is higher thanthe historical maximum saturation, the relative per-meability is calculated using the primary drainagecurve. For simplicity, the nonwetting phase will beused to illustrate the concept of the model. When adrainage process switch to imbibition, the model cre-ates a rectangle into which the measured imbibitionrelative permeability is normalized. The upper right-hand corner of the rectangle is dened by the satu-ration and relative permeability at the turning point.The lower left-hand corner is given by the residualsaturation and zero relative permeability. The rela-tive permeability curve between the upper right-handand lower left-hand corner is the normalized primaryimbibition curve.This concept is valid not only when the process re-

verses from the primary drainage curve, at the maxi-mum historical nonwetting phase saturation, but alsowhen the process reverses at any possible saturationafter a number of imbibition-drainage cycles.When the process reverses from imbibition to

drainage, the model also generates a rectangle. Butnow the upper right-hand corner is given by the max-imum nonwetting phase saturation and the drainagerelative permeability at this point. The lower left-hand corner is given by the saturation and relativepermeability at the turning point. The relative per-meability curve between the upper right and lower leftcorners is the normalized primary drainage curve.The primary drainage and imbibition relative per-

meability curves cannot be normalized directly if thecurvature of the relative permeability curve is a func-tion of saturation. If the two-phase relative perme-ability curve is represented by

krnw = k0rnw(Sηnw)ξ(S

ηnw), (3.129)

the exponent is given by

ξ(Sηnw) =

log((Sηnw)ξ(S

ηnw)−(Sη

nw,L)ξ(S

ηnw,L

)

(Sηnw,U

)ξ(S

ηnw,U

)−(Sη

nw,L)ξ(S

ηnw,L

)

log(Sηηnw),

(3.130)where L denotes the lower left-hand corner and U theupper right-hand corner of the rectangle. The relativepermeability is given by

krnw = krnw,L + (krnw,U + krnw,L)(Sηηnw)ξ(Sηnw).(3.131)

The procedure for the wetting phase is similar, exceptthat the endpoint relative permeability at the residualnonwetting phase saturation is found by linear inter-polation between the endpoint relative permeabilityvalues of the primary drainage and imbibition curve.

Carlson

Carlson49 proposed a simple hysteresis model basedon the work of Land, and the assumption that thewetting phase relative permeability is not subject tohysteresis. Carlson uses Land's Eq. 3.127 and 3.128to calculate the residual nonwetting phase saturationand the mobile fraction of the total nonwetting phasesaturation. Then he assumes that the relative perme-ability of the mobile fraction of the nonwetting phaseduring the imbibition process is equal to the drainagerelative permeability, implying that the whole non-wetting phase is mobile during drainage process whenthe saturation is higher or equal to the maximum non-wetting phase saturation. This can be expressed as

krnw,im(Snw) = krnw,dr(Snw,f ). (3.132)

In addition to the fact that the model does not takeinto consideration hysteresis of the wetting phase rel-ative permeability or the imbibition relative perme-ability of the nonwetting phase, Carlson assumes thatthe shape of the imbibition and the drainage relativepermeability are equal.

3.5 Transport Equations

The recovery of hydrocarbon uids from NorthSea reservoirs is usually described by three dier-ent model types. They are presented below withthe derivations of their transport equations andare termed here, the black-oil model,3739,120,121

the compositional model,122125 and the chemicalmodel.1,124

The black-oil model is most commonly appliedwhen studying North Sea reservoirs. A majority of

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3.5. TRANSPORT EQUATIONS 63

all simulation studies performed on reservoirs in theNorwegian sector applies this model type. Othermodels are available for more special studies. Theprocesses studied are usually depletion and water in-jection, where the change in reservoir pressure hasthe main inuence on the change in uid properties.Variation in temperature, due to seawater injection,may also cause changes in uid properties. However,black-oil models do not usually model temperature-dependent uid properties.For gas cycling of gas condensate reservoirs,41 com-

positional simulators are used to model varying uidproperties due to mass exchange between uid phases.These models incorporate volume cubic equations ofstate for phase behavior and uid property descrip-tion.The chemical model,1,124 also applied in more spe-

cial studies, diers from the other models in that itdescribes the nature of surface active agents and slugpropagation. These agents, often injected as slugsfollowed by water, reduce the interfacial tension be-tween the oil and water phases and may in someinstances develop a third microemulsion intermedi-ate phase. Assumptions are that only liquid phasesare present in the reservoir and that the liquids andthe reservoir rock are incompressible. The transportequations are constructed from conservation laws, seeEq. 3.3, which in dierential form for substance i maybe expressed as

∂tWi = −∇ · ~Ni − qi, (3.133)

whereWi is the mass per bulk volume, ~Ni is the massux, and qi is the source term. Eq. 3.133 is regardedas the basic equation from which the following threemodel's transport equations are derived.

3.5.1 Simplied Equations

The principle on which all reservoir models are basedis the conservation of mass, expressed through one ormore continuity equations, depending on the numberof uid (and solid) phases and the number of uid (orsolid) components present. To the conservation equa-tions must be added constitutive relationships givinguxes as functions of pressure gradients, concentra-tion gradients etc., equation of state, and equationsfor phase equilibria. If heat ow is important, anequation for energy conservation must be includedwith the mass conservation equations.The transport equations take their simplest form

in models of ow of one or more uids, each contain-ing a single component and with no mass transportbetween the uids, or ow of a single uid phase con-taining one or more components. In the rst case thetransport equations have the well known form

∂t(φρjSj) = −∇ · (ρj~uj)− qj , (3.134)

where the component mass per bulk volume is φρjSj ,and the component mass ux is ρj~uj . The left-hand

side of Eq. 3.134 is the mass accumulation term whereφ is the porosity, the bulk volume fraction of intercon-necting pores, ρj is the mass density of phase j, andSj is phase saturation or volume fraction occupied byphase j; ~uj is the phase ltration velocity (Darcy's ve-locity) or ux. The q is the sink term dened as massper unit volume per unit time.The ltration velocity is denoted by

~uj = −K krjµj

(∇pj − ρjg∇D), (3.135)

whereK is the absolute permeability tensor, krj is therelative permeability to phase j, µj and pj are theviscosity and pressure of phase j. The accelerationdue to gravity is denoted by g, and the depth by D.These equations are coupled with equations of state

for each phase,ρj = ρj(pj), (3.136)

where the temperature is assumed constant; capillarypressure relationships,

Pc = pn − pw, (3.137)

where subscripts n and w denote the nonwetting andwetting phases, respectively; and normalization con-ditions ∑

j

Sj = 1. (3.138)

For ow of a single phase, e.g., a rst contact miscibledisplacement, the equations take the form

∂t(φρ ωi) = −∇ · ~Ni − qi, (3.139)

where ωi is the mass fraction of the i-th component inthe uid and ρ is the density of the uid. The vector~Ni, denoting the mass ux of the i-th component, hasa convective and a dispersive component. The con-vective component is the Darcy velocity of the uid,and the dispersive component is usually modeled byFick's law which relates it to the gradient of the com-ponent fraction, ωi:

~Ni = ρ ωi~u− ρ ~vi, (3.140)

~vi = T · ∇ωi, (3.141)

T = φ DI + |~u| (αl~e · ~e+ αt(I − ~e · ~e)) , (3.142)

D is a molecular diusion coecient, I is the unit ten-sor, ~e is a unit vector parallel to ~u, and αl and αt arelongitudinal and transversal dispersion coecients.

3.5.2 Black Oil

Black-oil models3739,120,121 are constructed on theapproximation that there are three independent com-ponents, namely dry gas, stock tank oil and water, alldened with constant properties at standard condi-tions. These components may distribute themselvesamong three phases, gas, oil and water, at reservoirconditions. The water component is assumed to exist

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64 CHAPTER 3. EQUATIONS

in the water phase only, which in turn is immisciblewith the two hydrocarbon phases. Mass transfer ofcomponents is therefore assumed only to occur be-tween the gas and oil phases.The transport equation for the water component

assumes that the density at surface conditions is con-stant. If mass and the surface volumes are conserved,the water mass density is

ρw =ρswBw

, (3.143)

where superscript s denotes surface conditions. Bwis the formation volume factor, i.e., reservoir/surfacevolume ratio. The mass of water per bulk volume isthen

Ww = φρwSw = φρswBw

Sw, (3.144)

and the water component's mass ux is

~Nw = ρw~uw =ρsw~uwBw

. (3.145)

Substituting Eqs. 3.144 and 3.145 into Eq. 3.133 anddividing through by the surface density, the transportequation for water becomes

∂t

(φSwBw

)= −∇ ·

(~uwBw

)− qw. (3.146)

The production term, qw, represents the volume ofwater produced at standard conditions per unit reser-voir volume per unit time.The transport equation for the oil component

(stock tank oil) may be derived in a similar man-ner, noting that the oil and gas densities at reservoirpressure are, see Sec. 3.2.1,

ρo =1

Bo

(ρso +Rs ρ

sg

)(3.147)

andρg =

1

Bg

(ρsg + rs ρ

so

). (3.148)

These two-phase densities both contain the oil com-ponent. Therefore, the mass of the oil component perbulk volume consists of the sum of two terms, (1) thesurface volume of oil from the oil phase, V soo, and (2)the surface volume of oil from the gas phase, V sog,

Wo = φ(V soo + V sog

)ρso = φ

(SoBo

ρso +SgBg

rs ρso

).

(3.149)Similarly, the oil component mass ux is

~No =~uoBo

ρso +~ugBg

rs ρso (3.150)

Substituting Eqs. 3.149 and 3.150 into Eq. 3.133 anddividing through by the constant stock tank oil den-sity, the transport equation for the oil component be-comes

∂t

(SoBo

+Sg rsBg

)]=

−∇ ·(~uoBo

+rs ~ugBg

)− qo. (3.151)

In a similar manner, the transport equation for thegas component, dry gas, may be derived:

∂t

(SgBg

+So RsBo

)]=

−∇ ·(~ugBg

+Rs ~uoBo

)− qg. (3.152)

Eqs. 3.146, 3.151 and 3.152 are the transport equa-tions for a black-oil model.38 These equations mustsatisfy the conditions∑

j

Sj = 1, (3.153)

Pcw = po − pw, (3.154)

andPcg = pg − po, (3.155)

where subscripts g, o and w refer to the gas, oil andwater phases, respectively. The water is assumed tobe the wetting phase, the gas the nonwetting phase,and the oil the intermediate wetting phase. Theporosity term, φ, is pressure dependent along withthe phase behavior terms Bg, Bo, Bw, rs and Rs.The black-oil model, although accepted for its ap-

plication to depletion and water injection processes,has also been extended to model more complex gasinjection processes.39 The extended model modiesthe phase behavior parameters (Bg, Bo, Bw, Rs andrs) to be functions of the amount of injection gascontacted, in addition to being functions of reservoirpressure and saturation pressure (gas/oil ratio).

3.5.3 Compositional Model

The compositional model122124 is constructed on thesame principles as the black-oil model. Instead ofmodelling the hydrocarbon uids with two compo-nents as in the black oil model, the compositionalmodel describes the hydrocarbon uids, oil and gas,by their constituent hydrocarbon components. Theassumption is also made here, that these hydrocar-bon components are immiscible with the water phaseas in the black-oil model. This is not completely truesince the lighter components such as methane andethane do dissolve into water. Components such ascarbon dioxide and nitrogen are also soluble in waterand may for certain processes have a signicant eecton the uid properties. However, most compositionalmodels approximate the transport equation for thewater component by Eq. 3.146. For the other com-ponents, the mass of component i per bulk volume isexpressed as

Wi = φ∑j

ωij ρj Sj , (3.156)

where the summation is over all phases, j. The cor-responding mass ux is

~Ni =∑j

ωij ρj ~uj . (3.157)

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3.5. TRANSPORT EQUATIONS 65

Inserting these terms into Eq. 3.133, the transportequation for each component becomes

∂t

φ ∑j

ωij ρj Sj

=

−∇ ·

∑j

ωij ρj ~uj

− qi, i = 1 . . . nc,

(3.158)

where nc denotes the total number components, ex-cluding the water component.Compositional models, however, conserve moles

rather than mass to be consistent with units used forthe equilibrium equations. Mass fractions, ωij , arerelated to mole fractions, xij , by

ωij = xij Mi/Mj , (3.159)

whereMi is the molecular weight of component i, andthe average molecular weight of phase j is

Mj =∑i

xijMi. (3.160)

The mass density, ρ, is related to molar density, ξ, by

ρj = ξj Mj . (3.161)

Eqs. 3.159 and 3.161 may be substituted intoEq. 3.158 and divided through by the molecularweight of component i, Mi, to obtain

∂t

φ∑j

xij ξj Sj

=

−∇ ·

∑j

xij ξj ~uj

− qi, i = 1 . . . nc.

(3.162)

The source term qi is the component sum of the oiland gas rates that applies to the well stream.The constraints for black-oil models also apply

here. In addition, the constraint∑i

xij = 1 (3.163)

must also be satised. Generally, there may be asmany phases as permitted by Gibbs phase rule. How-ever, two phases, gas and oil, are sucient for mostpractical applications.The phase molar densities, ξj , and phase mole

fractions, xij , are derived from the assumption thatwithin any volume element there is local equilibrium.The governing equations for phase equilibrium aremodel dependent.The component diusive and dispersive terms are

important in certain applications,125 such as slugpropagation. They model component mixing due

to molecular diusion and mechanical mixing causedby dierent ow paths within the porous medium.However, most compositional models are based onEqs. 3.146 and 3.162 and omit the diusive and dis-persive terms. It may be argued that the equilibriumassumption is in itself a diusive property.

3.5.4 Chemical Model

The chemical model1,124 denes its componentssomewhat dierently from the two previous models.The uids often consist of an oil, a surfactant, a wa-ter and an alcohol component. The oil component,or pseudocomponent, represents all its hydrocarbonconstituents. The components are thus dened to de-scribe phase behavior for systems slug injection thatmay contain a variety of substances. Three importantprocess phenomena accounted for the chemical modelare the surface properties between uid phases, theuid-rock interaction, and the mixing of uids due todispersion.The model is applied to processes containing liquids

only, namely oil, microemulsion, and water. Incom-pressibility is therefore assumed for both uids androck.The overall concentration consists of two parts, (1)

that of the uid which is permitted to ow, and (2)that which is adsorbed to the rock,1

Wi = φ∑j

ωij ρj Sj + (1− φ)ρs ωis. (3.164)

Eq. 3.164 assumes that the adsorption to the rockphase has little inuence on the porosity and mainlyaects the nature of the phase behavior. The massux,

~Ni =∑j

(ωij ρj ~uj − φ ρj SjTij · ∇ωij) , (3.165)

also consists of two parts, (1) the convective compo-nent (Darcy's velocity) and (2) the dispersive compo-nent.The dispersion tensor, Tij , may be dened for

porous media that are homogeneous and isotropic bytwo components,1

(Tij)l =Dij

τ

+(αlj − αtj)φ Sj |~uj |

(ul)2j +

αtj |~uj |φ Sj

, (3.166a)

(Tij)l =(αlj − αtj)φ Sj |~uj |

∣∣∣(ul)j (ut)j

∣∣∣ , (3.166b)

where subscript l refers to the longitudinal (parallel)direction to the ow, and subscript t refers to thetransverse (perpendicular) direction to the ow. Thedispersion tensor becomes important when describingslug propagation, which has an increasing inuencewith increasing velocity.

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66 CHAPTER 3. EQUATIONS

Eqs. 3.164 and 3.165 may be inserted into Eq. 3.133to give the transport equations. Since incompressibil-ity is assumed, the transport equations may be ex-pressed either in terms of volume fractions1 or molesper unit volume124 as

∂Ci∂t

= −∇ ·

∑j

(Cij ~uj − Tj · ∇Cij)

− qi,i = 1 . . . nc.

Ci represents the total number of moles of compo-nent i per unit bulk reservoir volume, and Cij de-notes moles of component i per unit volume of phasej; qi is the source term. In contrast to the black oiland compositional models, the chemical model alsoapplies Eq. 3.5.4 to the water component.

Nomenclature

A = Helmholtz free energy, Ja = retention per unit pore volume, m3/m3

= exponent= internal surface area per volume, m−1

= empirical constant= free parameter

a, b = axes of ellipsoid, mBj = formation volume factor of phase j,

Rm3/Sm3

C = trapping constantCi = total (overall) moles of component i per

unit bulk reservoir volume, mol/m3

Cij = moles of component i per unit volume ofphase j, mol/m3

c = phase volume fraction= concentration, g/cm3

= compressibility, Pa−1

D = diusion/dispersion coecient, m2/s= diameter, m= depth, m= (arbitrary) matrix

Dij = eective binary diusion coecient ofcomponent i in phase j, m2/s

~e = unit vectorF = fractional uxFα = ux density of quantity α, α/m2sf = fractional ow= thermodynamic degrees of freedom= fugacity= fraction of given phase in the produced

stream= (arbitrary) function

G = Gibbs free energy, Jg = acceleration of gravity, m/s2

= (arbitrary) functionH = specic enthalpy, J/kgI = unit tensorJ = Leverett J-functionK = absolute permeability tensor, m2

Ki = equilibrium constant (K-value) of compo-nent i

K ′ = empirical constantk = permeability, m2

~~k = permeability tensor, m2

krocw = oil relative permeability at irreducible wa-ter saturation and zero gas saturation

krog = relative permeability to oil in presence ofgas

krow = relative permeability to oil in presence ofwater

k′ = Huggins constantL = liquid phase fraction= length, m

M = molecular weight, kg or Daltonm = mass, kg

= consistency parameter= exponent

N = number of moles~N = mass ux of components, kg/m2·sn = number or exponentP = number of phasesPc = capillary pressure, N/m2

p = pressure, PaQ = cumulative injected uid in pore volumesq = volumetric ow rate, m3/s= source term, J/m3·s or kg/m3·s

q = source or sink term, Sm3/Rm3·sq = source or sink term, mol/Rm3·sR = Molar gas constant, 8.314 J/mol K= capillary radius, m

Rs = solution gas/oil ratio, Sm3/Sm3

r = radius, mrs = vapor oil/gas ratio, Sm3/Sm3

S = saturation= entropy, J/deg

Sfw = Sw + SobSj = saturation (volume fraction) of phase jSob = oil blocked by invading waterSorg = residual oil saturation in presence of gasSorw = residual oil saturation in presence of waterSt = Sw + SoT = temperature, K= transmissibility, Sm3/Pa·s= dispersion tensor, m2/s

t = time, sU = specic internal energy, J/kgu = supercial velocity, m/s= primary unknown

V = volume, m3

= vapor phase fractionv = velocity, m/svh = hydrodynamic volume, cm3/gWi = mass of component i per bulk volume,

kg/m3

w = weight fractionx = liquid composition or length, m

xij = mole fraction of component i in phase jy = vapor compositionZ = compressibility factor (Z-factor)z = overall compositionα = dispersivity coecient, m

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3.5. TRANSPORT EQUATIONS 67

= physical quantity= free parameter

α, β, γ, δ = free parametersβ = equation of state parameterγ = shear rate, s−1

δ = equation of state parameterε = free parameterε = equations of state parameterη = equation of state parameter= viscosity, Pa·s

[η] = intrinsic viscosity, cm3/gΘ = equations of state parameter= contact angle, radians

ϑ = dip angle, radiansκ = volumetric heat capacity, J/m3 K

κ, ξ = higher order constantsλ = mobility, m2/Pa s= thermal conductivity, W/m2 K= eigenvalue= time constant, s= pore size distribution index= free parameter

µ = viscosity, Pa·s= electrochemical potential, J/kg·mole

ν = shape factorξ = molar density, mol/m3

ρ = mass density, kg/m3

σ = interfacial tension, N/mτ = shear stress, N/m2

= tortuosityυ = free parameterφ = porosity, fractionϕ = volume fractionω = weighting parameterωi = overall mass fraction of component iωij = mass fraction of component i in phase jωα = amount of quantity α per bulk volume,

[α]/m3

Subscripts

b = bulkc = critical or capillary or connate or compo-

nentdr = drainageE = energye = eective or entry

exp = experimentalf = formation or owingg = gasH = heavy or heath = hydraulicI = intermediatei = component label or irreducible

im = imbibitionj = component or phase labelk = phase labelL = light or lower left corner or liquidl = longitudinal

M = maximumm = mean or minimum or microemulsion

n = number of componentsnw = nonwettingo = oilof = oil owingp = pore or phase

pdV = volume work, Jr = reduced or residual or relatives = stationary or solid or saturation

sol = solutionsolv = solventsp = specicT = totalt = tortuous or transversal or tubeU = upper right corner

x, y = cartesian coordinatesw = water or wetting

Superscripts

e = endpointn = free parameter

n,m = exponentsm = free parameter0 = endpointη = normalizedηη = normalized∗ = normalized∗∗ = normalized? = normalized¯ = normalized or averages = surface or standard conditions′ = modied

Operators

df = jacobian of f

Normalized Saturations

S∗of = Sof1−Sfw

S∗∗of = Sof1−Swi

S∗ = S−Swi1−Swi

S∗∗ = S−Sfw1−Sfw

S?0 = So1−Swi

S?w = Sw−Swi1−Swi

S?t = St−Swi1−Swi

S∗w = Sw−Swc1−Swc−Som

S∗o = So−Som1−Swc−Som

S∗g = Sg1−Swc−Som

So = 1−Sw−Sg−Sor1−Swr−Sor−Sgr

Sw = Sw−Swr1−Swr−Sor−Sgr

Sg = Sg−Sgr1−Swr−Sor−Sgr

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68 CHAPTER 3. EQUATIONS

Sηηnw = Snw−Snw,LSnw,U−Snw,L

Sηnw = Snw−Srnw1−Srnw−Srw

References

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70 CHAPTER 3. EQUATIONS

tute of Technology, Cambridge (1979).[65] Bird, R.B., Armstrong, R.C., and Hassager, O.:

Dynamics of Polymeric Liquids, John Wiley &Sons (1987) 1.

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[67] Harris, E.K. Jr.: J. Appl. Polymer. Sci., (1973)17, 167992.

[68] Osborne P. and Evans S.: The Troll Field:Reservoir Geology and Field DevelopmentPlanning, North Sea Oil and Gas Reservoirs,Kleppe et al. (eds.), Graham & Trotman, Lon-don (1987).

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[71] Dullien, F.A.L.: Fluid Transport and PoreStructure, Academic Press, New York (1979).

[72] Morrow, N.R.: Wettability and Its Eect onOil Recovery, JPT (Dec. 1990) 147684.

[73] Cuiec, L.E.: Evaluation of Reservoir Wetta-bility and Its Eect on Oil Recovery, Interfa-cial Phenomena in Oil Recovery, N.R. Morrow(ed.), Marcell Dekker, New York (1990) 31975.

[74] Torsæter, O.: A Comparative Study of Wetta-bility Test Methods Based on Experimental Re-sults from North Sea Reservoir Rocks, paperSPE 18281 presented at the 1988 SPE AnnualTechnical Conference and Exhibition, Houston,Oct. 25.

[75] Amott, E.: Observations Relating to the Wet-tability of Porous Rock, Trans., AIME (1959)216, 15662.

[76] Donaldson,E.C., Thomas R.D., and Lorenz,P.B.: Wettability Determination and Its Eecton Recovery Eciency, SPEJ (March 1969)1320.

[77] Anderson, W.G.: Wettability LiteratureSurvey-Part 1: Rock/Oil/Brine Interactionsand the Eects of Core Handling on Wettabil-ity, JPT (Oct. 1986) 112544.

[78] Anderson, W.G.: Wettability LiteratureSurvey-Part 2: Wettability Measurement, JPT(Oct. 1986) 124662.

[79] Anderson, W.G.: Wettability LiteratureSurvey-Part 3: Eects of Wettability on theElectrial Properties of Porous Media, JPT(Dec. 1986 ) 137178.

[80] Anderson, W.G.: Wettability LiteratureSurvey-Part 4: Eects of Wettability on Capil-lary Pressure, JPT (Oct. 1987) 12831300.

[81] Anderson, W.G.: Wettability LiteratureSurvey-Part 5: Eects of Wettability on Rel-ative Permeability, JPT (Nov. 1987) 145368.

[82] Anderson, W.G.: Wettability LiteratureSurvey-Part 6: Eects of Wettability on Wa-terooding, JPT (Dec. 1987) 160522.

[83] Stegemeier, G.L.: Relationship of TrappedOil Saturation to Petrophysical Properties ofPorous Media, paper SPE 4745 presented atthe 1974 SPE Symposium on IOR, Tulsa.

[84] Killough, J.E.: Reservoir Simulation with His-tory-Dependent Saturat ion Functions, Trans.,AIME (1976) 261, 3748.

[85] Leverett, M.C.: Capillary Behavior in PorousSolids, Trans., AIME (1941) 142, 15972.

[86] Monicard, R.P.: Properties of Reservoir Rocks:Core Analysis, Editions Technip, Paris (1980).

[87] Honarpour, M., Koederitz, L., and Har-vey, A.H.: Relative Permeability of PetroleumReservoirs, CRC Press Inc., Boca Raton(1987).

[88] Koederitz, L.F., Harvey, A.H. and Honarpour,M.: Introduction to Petroleum Reservoir Anal-ysis, Gulf Publishing, Houston (1989).

[89] Pallatt, N. and Palmer, T.: The Role ofPore Geometry in the Interpretation of ShaleySands, paper presented at the Society of CoreAnalysis European Core Analysis Symposium,London (May 2022, 1991).

[90] Craig, F.F.: The Reservoir Engineering As-pects of Waterooding,Monograph Series, SPE,Richardson, Texas (1971) 3.

[91] Chilingarian, G.V.: Relative PermeabilityConcepts, Journal of Petroleum Science andEngineering (1989) 2, 245.

[92] Welge, H.J.: A Simplied Method for Comput-ing Recovery by Gas or Water Drive, Trans.,AIME (1952) 195, 91.

[93] Buckley, S.E. and Leverett, M.C.: Mechanismof Fluid Displacement in Sands, Trans., AIME(1942) 146, 107.

[94] Johnson, E.F., Bossler, D.P., and Naumann,V.O.: Calculation of Relative Permeabil-ity from Displacement Experiments. Trans.,AIME (1959) 216, 370.

[95] Saraf, D.N. and McCaery, F.G.: Two- andThree-Phase Relative Permeabilities: a Re-view, Petroleum Recovery Institute ReportNo. 81-8, Calgary, Alberta (1982).

[96] Jones, S.C. and Roszelle, W.O.: GraphicalTechniques for Determining Relative Perme-ability from Displacement Experiments, JPT(May 1978) 80717.

[97] O'Meara, D.J., Jr. and Lease, W.O.: Mul-tiphase Relative Permeability MeasurementsUsing an Automated Centrifuge, paper SPE12128 presented at the 1983 SPE Annual Tech-nical Conference and Exhibition, San Francisco,Oct. 58.

[98] Munkvold, F.R. and Torsæter, O.: Rela-tive Permeability from Centrifuge and Un-steady State Experiments, paper SPE 21103presented at the 1990 SPE Latin AmericanPetroleum Engineering Conference, Rio deJaneiro, Oct. 1419.

[99] Hagoort, J.: Oil Recovery by Gravity

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REFERENCES 71

Drainage, SPEJ (June 1980) 20, 13950.[100] O'Meara, D.J., Jr. and Crump, J.G.: Measur-

ing Capillary Pressure and Relative Permeabil-ity in a Single Centrifuge Experiment, paperSPE 14419 presented at the 1985 SPE AnnualTechnical Conference and Exhibition, Las Ve-gas, Sept. 2225.

[101] Guo, Y.: Centrifuge Experiment and Rela-tive Permeabilities, Dr.ing. dissertation, TheNorwegian Institute of Technology, Trondheim(1988).

[102] Parmeswar, R and Maerefat, N.L.: A Com-parison of Methods for the Representation ofThree-Phase Relative Permeability Data, pa-per SPE 15061 presented at the 1986 SPE Cal-ifornia Regional Meeting, Oakland, April 24.

[103] Burdine, N.T.: Relative Permeability Calcula-tions from Pore Size Distribution Data, Trans.,AIME (1953) 198, 718.

[104] Standing, M.B.: Notes on Relative Permeabil-ity Relationships , The Norwegian Institute ofTechnology (Aug. 1974).

[105] Naar, J. andWygal, R.J.: Three-Phase Imbibi-tion Relative Permeability , SPEJ (Dec. 1961)2548.

[106] Naar, J. and Henderson, J.H.: An ImbibitionModel- Its Application to Flow Behaviour andthe Prediction of Oil Recovery, SPEJ (June1961) 629.

[107] Parker, J.C., Lenhard, R.J., and Kuppusamy,T.: A Parametric Model for Constitutive Prop-erties Governing Multiphase Flow in PorousMedia, Water Resources Research (April 1987)23, 4, 61824.

[108] Baker, L.E.: Three-Phase Relative Permeabil-ity Correlations , paper SPE/DOE 17369 pre-sented at the 1988 SPE/DOE Symposium onEOR, Tulsa, April 1720.

[109] Stone, H.L.: Probability Model for EstimatingThree-Phase Relative Permeability, JPT (Feb.1970) 2148.

[110] Stone, H.L.: Estimation of Three-Phase Rel-ative Permeability and Residual Oil Data, J.Cdn. Pet. Tech. (Oct. - Dec. 1973) 5361.

[111] Dietrich, J.K. and Bondor, P.L.: Three-PhaseOil Relative Permeability Models , paper SPE6044 presented at the 1976 SPE Annual Tech-nical Conference and Exhibition, New Orleans,Oct. 36.

[112] Fayers, F.J. and Matthews, J.D.: Evalua-tion of Normalized Stone's Method for Esti-mating Three-Phase Relative Permeabilities,SPEJ (April 1984) 22432.

[113] Holmgren, C.R. and Morse, R.A.: Eect ofFree Gas Saturation on Oil Recovery by WaterFlooding, Trans., AIME (1951) 192, 13540.

[114] Fayers, F.J.: Extension of Stone's MethodI and Conditions for Real Characteristics inThree-Phase Flow, paper SPE 16965 presentedat the 1987 SPE Annual Technical Conference

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[116] Land, C.S.: Calculation of Imbibition RelativePermeability for Two- and Three Phase Flowfrom Rock Properties, SPEJ (June 1986) 14956.

[117] Land, C.S.: Comparision of Calculated withExperimental Imbibition Relative Permeabil-ity, SPEJ (Dec. 1971) 419425.

[118] Eikje, E., Jakobsen, S.R., Lohne, A. andSkjæveland, S.M.: Relative Permeability Hys-teresis in Micellar Flooding, presented at theEuropean Symposium on IOR, Stavanger, May1991.

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[121] Allen, M.B. III, Behie, G.A., and Trangenstein,J.A.: Multiphase Flow in Porous Media, Lec-ture Notes in Engineering, 34, Springer-VerlagNew York Inc. (1988).

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[123] Ewing, R.E.: The Mathematics of ReservoirSimulation, SIAM, Philadelphia (1983).

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72 CHAPTER 3. EQUATIONS

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Chapter 4

Solutions

4.1 Shocks and Simple Waves

4.1.1 Introduction

In Sec. 3.1, the most commonly used macroscopicmodels for ow in porous media were derived andtheir basic properties briey discussed. In particular,the mathematical problems arising when neglectingdiusion terms in the models were mentioned. Theabsence of these terms reduces the models from 2.order to 1. order partial dierential equations. Inthis section, we will mostly disregard the problemsassociated with this degeneracy and we will focusupon the construction of analytical solutions of one-dimensional, diusion-free problems for incompress-ible ow. In addition, we will neglect volume changesupon mixing of uid components. For a general ap-proach including the latter eect, see Dumore et al.1

Solutions of such simplied problems must be re-garded as visualizations of ow phenomena and assuch they are qualitative in nature. Nevertheless,they do provide considerable insight into the owmechanisms governing dierent enhanced oil recov-ery processes: The global eects of a ow processare consequences of local behavior of the individ-ual uid components through their mutual interac-tions and their interactions with the medium throughwhich they ow. The resulting chromatographic sep-aration of individual uid components may there-fore as a rst approximation be regarded as a one-dimensional problem, which can be investigated on apurely mathematical basis if diusion and eects ofvolume changes are neglected.Throughout the section, we will use dimensionless

variables xD, tD dened by

xD =x

L; tD =

uT t

φL,

where L is the length of the reservoir. The subscriptD will be left out for an easier notation. Subject tothe assumptions made above, the conservation law,Eq. 3.15, for mass of component i becomes

∂Ci∂t

+∂Fi∂x

= 0, (4.1)

where the overall volume fraction of i in the uid

system is given by

Ci =

np∑j=1

Sj cij + ai, (4.2)

and overall fractional ux of i is

Fi =

np∑j=1

fj cij . (4.3)

For nonisothermal ow problems, we will also neglectheat conduction. The conservation law for energy,Eq. 3.24, then reads

∂t

np∑j=1

κj Sj T + κs T

+∂

∂x

np∑j=1

κj fj T = 0. (4.4)

The conservation laws described by Eqs. 4.1 and 4.4constitute the general mathematical model for any ofthe processes considered in this section.Excellent descriptions of analytical solutions of ow

problems arising in enhanced oil recovery can befound in Pope2 and Lake.3 Here, we will concen-trate on the structure of dierent ow models and wewill demonstrate the close relationship that exists be-tween most diusion-free, incompressible models fortwo-phase ow in enhanced oil recovery. Throughthis relationship, increased insight into the nature ofsuch ow processes will be obtained and the gener-ality of the solution procedure for a class of initialvalue problems (Riemann problems) for these modelswill be revealed. This will be demonstrated throughsome selected examples. Brief discussions on possi-ble extensions of the theory and also on three-phaseow models are given at the end of the Sec. 4. Gen-eral mathematical aspects of hyperbolic conservationlaws can be found in Courant and Friedrich,4 Courantand Hilbert,5 Lax,6 and Smoller.7

4.1.2 Riemann Problems.

All ow problems for models of the type Eqs. 4.1 and4.4 considered in this section will be formulated inan innitely long, one-dimensional and homogeneousmedium, −∞ < x < +∞. The uid system in thereservoir (0 ≤ x ≤ L) will initially be in a givenconstant state (the initial condition), which will be

73

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74 CHAPTER 4. SOLUTIONS

labelled by a superscript R. At time t = 0, ow inthe medium is initiated at a constant total volumetricuid rate uT > 0. The uid system thus injected intothe reservoir is also assumed to be in a given constantstate (the injected condition), which will be labelledL. If the unknowns of the uid ow problem are de-noted ~u(x, t) = [u1, . . . , uN ](x, t), we are consideringan initial value problem with initial data of the form

~u(x, 0) =

~uL if x < 0~uR if x > 0.

(4.5)

Such problems are called Riemann problems.All models considered in this section can be written

in the following form

∂~u

∂t+A(~u)

∂~u

∂x= 0, (4.6)

where A(~u) is an N ×N -matrix* with real eigenval-ues. In the terminology of Sec. 3.1.5, this means thatthe models considered will be hyperbolic. If in addi-tion, the eigenvalues λi of A are distinct everywherein composition space, say

λ1(~u) < λ2(~u) < . . . < λN (~u), (4.7)

the model is called strictly hyperbolic. Most modelsarising in enhanced oil recovery are nonstrictly hyper-bolic, which strongly complicates wave behavior andthus the construction of solutions of Riemann prob-lems in general.

4.1.3 The Buckley-Leverett Problem

For immiscible and isothermal displacement of oil bywater, the model Eq. 4.1 for conservation of water(and oil) reads

∂S

∂t+∂f(S)

∂x= 0. (4.8)

Here, S is water saturation and f is fractional ow ofwater given by Eq. 3.14, i.e.,

f =λwλT

(1 +

λouT

(ρo − ρw)g sinϑ

). (4.9)

In this section we will briey discuss the initial valueproblem for Eq. 4.8. In particular, we will investigatethe properties of the Riemann problem, which willbe discussed in some detail, because it also forms thebasis for corresponding analysis for multicomponentEOR-processes.Let γ = [t, x(t)] denote any smooth curve in (t, x)-

space, and let S0(x) be a given water distribution(−∞ < x < +∞). Suppose S(x, t) is a solution of thecorresponding initial value problem. The directionalderivative of S along γ is

dS

dγ=∂S

∂t+dx(t)

dt

∂S

∂x. (4.10)

*e.g., in the diusion-free model Eq. 3.25, A = dg−1df , seealso Sec. 4.1.4

Since S is also a solution of Eq. 4.8, we have

∂S

∂t+df(S)

dS

∂S

∂x= 0. (4.11)

It follows from Eqs. 4.10 and 4.11 that if γ satises

dx(t)

dt=df(S)

dS(4.12)

then dS/dγ = 0, i.e., S is constant on γ, which againfrom Eq. 4.12 implies that γ is a straight line calleda characteristic** for Eq. 4.8. It next follows thatfor any given x′, the solution S (provided it exists) isconstant and equal to S0(x) on the straight line

x = x′ +df

dS(S0(x′))t. (4.13)

For S(x, t) to exist and be given implicitly byEq. 4.13, the characteristics must not intersect, sinceeach characteristic is carrying a certain saturationvalue S0(x′). Hence, at points of intersection, mul-tiple values of S may occur, which is nonphysical. Asis also well known, for a physical solution of the con-servation problem to exist even after the time whensuch an ambiguity rst occurs, the introduction ofdiscontinuous (or shock) versions of the conservationlaws (jump conditions) is required. This can be donerigorously through the notion of weak solutions, seeSmoller,7 or as in Lax.6 It can also be derived bya simple argument considering a propagating shock,that a necessary condition for mass conservation is

σ(S+ − S−) = f(S+)− f(S−), (4.14)

where σ is the shock velocity and where S+, S− aresaturation values on the right and left side of theshock, respectively. Consider the Riemann problemfor Eq. 4.8. If S(x, t) is a solution of such a prob-lem, substitution into Eq. 4.8 shows that S(αx, αt) isalso a solution of the same problem for any α > 0.If S is unique, this means that S must be constanton any straight line through (0,0) in (t, x)-space, sayξ = x/t. Thus, S is a function of the single variableξ (self similarity). Substituting S(ξ) into Eq. 4.8 wend

− xt2dS(ξ)

dξ+

1

t

df(S(ξ))

dS

dS(ξ)

dξ= 0, (4.15)

ordf(S(ξ))

dS= ξ, (4.16)

which represents the speed of propagation of the satu-ration value S(ξ), in accordance with Eq. 4.12. Obvi-ously, the velocity of Eq. 4.16 must increase from theinjected towards the initial condition. The type of so-lution dened by Eq. 4.16 is called a spreading wave.Clearly, any constant state also satises Eq. 4.8. Alto-gether, the building blocks for the construction of so-lutions of Riemann problems are therefore the shock

**For a general denition of characteristics see for exampleSmoller.7

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4.1. SHOCKS AND SIMPLE WAVES 75

waves satisfying Eq. 4.14, the spreading waves satis-fying Eq. 4.16, and the constant states.Unfortunately, Eq. 4.14 does not resolve the prob-

lem of discontinuous solutions uniquely. To see this,consider a Riemann problem for Eq. 4.8 with injectedcondition and initial condition satisfying SR < SL,and suppose f is as shown in Fig. 4.1. Fig. 4.2 shows

f

SSLS*S0SR

σ1

σ2

Figure 4.1: Waterooding. Construction of possiblesolutions.

two possible solutions S1 and S2 of the Riemann prob-lem at a given t. The solution S1 satises Eq. 4.16

S

SL

S*

SR

x

S2

S1

σ1σ2

Figure 4.2: Waterooding. Possible solutions.

for S∗ ≤ S1 ≤ SL, and Eq. 4.14 with S− = S∗ andS+ = SR. The solution S2 satises Eq. 4.14 withS− = SL and S+ = SR. Thus, both S1 and S2 sat-isfy conservation of mass. In fact, any solution of theform S1 with S∗ ≤ S− ≤ SL does. In order to distin-guish the physically relevant solution, an additionalcondition is required. In the mathematical literature,such conditions are referred to as entropy conditions,reecting the physical fact that information is lostacross shocks. Rigorous derivations of such condi-tions are generally dicult and tedious. In this sec-tion we therefore take a simple heuristic approach.The relation Eq. 4.14 does not say anything aboutstability of shocks. From a physical point of view, itis natural to require the shock to be self-sharpening,otherwise a shock would not have formed in the rstplace, or it would disintegrate immediately. In otherwords, for any pair of saturation values S′, S′′ satis-fying S− ≥ S′ ≥ S′′ ≥ S+, the speed of propagation

for S′ and S′′ given by Eq. 4.16 should satisfy

df(S′)

dS≥ df(S′′)

dS. (4.17)

Inspection of S2 above shows that S′ = SL andS′′ = S0 violate this condition. (When SL < SR,the inequalities are reversed). With the additionalrequirement that overall wave velocity must increasefrom SL towards SR, we nd that the only possiblesolution of the problem is S1. In fact, this is the so-lution given by Buckley and Leverett8 and extendedby Welge76 to a simple graphical procedure for de-termination of average water saturation behind thefront.Subject to Eq. 4.17, the existence and uniqueness

of solution of the initial value problem Eq. 4.8 witharbitrary water distribution and arbitrary f has beenproved rigorously, Oleinik.10,11 In particular, the so-lution is shown to be the limit of vanishing diusionsolutions. Aavatsmark12 shows that this is also trueusing capillary pressure terms to model diusion. Theanalysis of well-posedness of the waterooding prob-lem in porous media is therefore complete.

4.1.4 Multicomponent Problems

In Sec. 3.1.5, the general conservation form of a mul-ticomponent system was written

∂~g(~u)

∂t+∂ ~f(~u)

∂x= 0. (4.18)

By the chain rule,

∂fi∂x

=

N∑k=1

∂fi∂uk

∂uk∂x

; i = 1, . . . , N, (4.19)

and similarly for g. Letting df = (∂fi/∂uj)ij and dgdenote the Jacobians of f and g respectively, we cantherefore write Eq. 4.18 as

∂~u

∂t+A(~u)

∂~u

∂x= 0, (4.20)

where A = dg−1df . We rst determine the smoothsolutions of the Riemann problem for Eq. 4.18. Ex-actly as in Sec. 4.1.3, we nd that such a solution isa function of the single variable ξ = x/t only. There-fore, as in Eq. 4.15, the solution must satisfy

(A(~u)− ξI)d~u(ξ)

dξ= 0. (4.21)

Here, I is the N × N -identity matrix. This is aneigenvalue problem for A. If λ1, . . . , λN denote the(real) eigenvalues of A, Eq. 4.21 implies that

λi(~u(ξ)) = ξ, (4.22)

and that d~u(ξ)/dξ is an eigenvector. Eq. 4.22 is theanalogue of Eq. 4.16. The geometric interpretation ofthis is that each eigenvalue of A generates a charac-teristic family of curves in composition space. Along

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76 CHAPTER 4. SOLUTIONS

each curve, ~u(ξ) satises Eq. 4.21. These N familiesof curves therefore dene all possible smooth solu-tions (simple spreading waves) of the Riemann prob-lem. However, the relation Eq. 4.22 requires that theeigenvalue associated with such a curve is increas-ing from injected towards initial condition; see alsoEq. 4.16. The spreading wave curves are thereforeoriented and cannot be reversed to yield a solution ofthe Riemann problem with initial and injected condi-tions interchanged. In order to resolve this, the intro-duction of shock waves is again necessary. The jumpcondition Eq. 4.14 for multicomponent problems isderived component-wise and reads

σ(~g(~u+)− ~g(~u−)) = ~f(~u+)− ~f(~u−). (4.23)

If ~u− is considered xed, Eq. 4.23 represents N equa-tions with N+1 unknowns u+

1 , . . . , u+N , σ. Since there

are N characteristic directions (corresponding to thespreading wave curves) through ~u−, Eq. 4.23 will (un-der appropriate assumptions) also dene N curvesthrough ~u−, each of which is tangent to a spread-ing wave curve at ~u−.* These are the shock wavecurves. In general, they do not coincide with the cor-responding spreading wave curves globally. In fact,it can be shown that such curves coincide globally ifand only if they are straight lines or the waves areindierent.**

The shock wave curves dened by Eq. 4.23 are alsooriented: The stability of a shock requires a self-sharpening behavior as discussed in Sec. 4.1.3. Forstrictly hyperbolic systems without linear degenera-cies, Lax6 showed that the admissibility of a shockassociated with the i-th eigenvalue λi requires thatthis self-sharpening persists in λi only:

λi(~u−) > σ > λi(~u

+), (4.24)

while for k 6= i, λk must violate Eq. 4.24. This isthe analogue to Eq. 4.17 for N > 1. This imposes anorientation on the shock curves. Together with thetrivial constant states, the simple spreading waves de-termined by Eq. 4.21 and the shock waves determinedby Eqs. 4.23 and 4.24 constitute the building blocksfor construction of solutions of Riemann problems.†

The criterion needed for gluing together these sim-ple waves was stated in Sec. 4.1.3: The overall wavevelocity must increase from injected towards initialcondition. For strictly hyperbolic problems, Lax6

showed that if the two states of the Riemann prob-lem are suciently close, the solution of the Riemannproblem can always be uniquely constructed in thismanner, subject to the condition Eq. 4.24. However,for arbitrary initial and injected conditions, or fornonstrictly hyperbolic problems, little is known ingeneral.‡ As we shall see in the rest of this section

*To see this let ~u+ → ~u− in Eq. 4.23**A wave is indierent if the characteristic velocity is con-

stant along the wave curve.†In somewhat rewritten form,13 Eq. 4.21 and 4.23 are

known as the coherence condition introduced by Helerich.14‡See, however, Liu15 for some result for N = 2.

the solutions of specic ow problems can neverthe-less be constructed.

4.1.5 Polymer Flooding

Polymers are used as additives to injection water inorder to improve sweep eciencies and displacementeciency since polymer solutions exhibit strong in-crease in viscosity even at low polymer concentra-tions. Therefore, polymer ooding is primarily usedwhen displacing high viscosity oils. Since sweep e-ciency is related to multidimensional problems, onlythe eect on displacement will be considered here.In this section, we will consider a model for poly-mer ooding in the context of the theory described inthe previous section. As will be shown in subsequentsections, this is a prototype model for all two-phasemodels to be discussed. We will therefore focus uponthe general structure of the simple waves, since thisrelates to the other models as well. However, thedetermination of Riemann solutions will be limitedto examples with most relevance to practical situ-ations. For a complete description of the solutionswith arbitrary Riemann data we refer to Isaachson16

for the case of linear adsorption and to Johansen andWinther71 in the general case. Mathematically, theRiemann problem for polymer ooding represents thesimplest extension of the Buckley-Leverett problemto systems of conservation laws arising in enhancedoil recovery. At least this statement is valid if we as-sume that the polymer solution is a uid composedof two pure components (water and polymer) with nopartitioning to the oil phase, and that the oil phaseis also a pure (nonpartitioning) component. Makingthe approximation that the volume fraction of purewater in the aqueous phase is identically equal to 1,the model becomes

∂S

∂t+∂f(S,C)

∂x= 0, (4.25a)

∂t(SC + a(C)) +

∂x(Cf) = 0, (4.25b)

where C is polymer concentration and a(C) is agiven local equilibrium model for the stationary poly-mer component, commonly taken to be the Lang-muir isotherm for adsorption. However, any mono-tonically increasing function of C passing through(0,0) can in principle be used, even multilayer ad-sorption models.18 The polymer concentration entersf primarily through the aqueous phase viscosity inλw = kw/µw in Eq. 4.9. According to the assump-tions made, the gravity factor multiplying λw/λT inEq. 4.9 is independent of C. Thus, for updip dis-placement (sinϑ ≥ 0) with ρ0 ≤ ρw, a typical shapeof f is shown in Fig. 4.3, where f(S0, C) = 0 for allvalues of C. For any xed S between Swc and S0, fis monotonically increasing with increasing values ofC, while for S between S0 and Swor, f is decreasing.

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4.1. SHOCKS AND SIMPLE WAVES 77

SSworS2S0

S1

Swc-h(C)

λ1

λ2

λ1=λ2

λ1=λ2

f(S,C1)

f(S,C2)

C1>C2

f

Figure 4.3: Polymer ooding. Fractional ow andeigenvalues.

Expanding the derivatives in Eq. 4.25 and rearrang-ing, we obtain

∂S

∂t+∂f

∂S

∂S

∂x+∂f

∂C

∂C

∂x= 0, (4.26a)

(S + h(C))∂C

∂t+ f

∂C

∂x= 0. (4.26b)

Here we have dened h(C) = da(C)/dC, which by as-sumption is positive. Letting u = (S,C), the matrixA(~u) in Eq. 4.20 is

A(S,C) =

∂f

∂S

∂f

∂C

0 fS+h(C)

. (4.27)

Clearly, A has the eigenvalues λ1 = ∂f/∂S and λ2 =f/(S+h(C)), which can be interpreted geometricallyas the slopes indicated in Fig. 4.3. We also observethat for S = S1; S = S2 in Fig. 4.3, λ1 = λ2. Hence,the model is not strictly hyperbolic.Obviously, if polymer concentration is xed, say

C = C0, the model of Eq. 4.25 reduces to the Buckley-Leverett model of Eq. 4.8 with f(S) = f(S,C0). Thespreading waves and shocks appearing in the Buckley-Leverett problems therefore also enter Riemann prob-lem solutions for Eq. 4.25. These waves correspondto the λ1-eigenvalue, as can be seen by substitutionof λ1 for ξ in Eq. 4.20. Similarly, substitution of λ2

in Eq. 4.20 yields

(λ1 − λ2)dS(ξ)

dξ+∂f

∂C

dC(ξ)

dξ= 0,

or equivalently, if C = C(S),

(λ1 − λ2) +∂f

∂CC ′(S) = 0, (4.28)

where C ′(S) = dC(S)/dS. The spreading λ2-wavescan then be determined by integration of Eq. 4.28which can be done in closed form when h(C) is con-stant (linear adsorption). Then, C(S) given implic-itly by the relation f(S,C(S))/(S + h) = Constantsatises Eq. 4.28.

We observe from Eq. 4.28 that at points whereλ1 = λ2 we must have C ′(S) = 0. Furthermore, since∂f/∂C = 0 at S0, see Fig. 4.3, the curves C(S) can-not intersect S = S0. Using Eq. 4.28 and the signs ofλ1−λ2, ∂f/∂C, one can infer the shape of the C(S)-curves shown in Fig. 4.4. We next determine the ori-

C

SSwc S0 Swor

λ1 = λ2

λ2 - spreading

λ2 - shock

λ1 - waves

Figure 4.4: Polymer ooding. Simple wave curves.

entation of these curves. Let ∇ = [∂/∂S, ∂/∂C]. Thedirectional derivative of λ2 along C(S) is:

∇λ2 · [1, C ′(S)] =

1

(S + h)2

[(S + h)

∂f

∂S− f, (S + h)

∂f

∂C− h′(C)f

]·[1, C ′(S)] =

1

S + h

(∂f

∂S− f

S + h+∂f

∂CC ′(S)

)− h′(C)f

(S + h)2C ′(S) = − h′(C)f

(S + h)2C ′(S),

since C(S) satises Eq. 4.25. If we assume h′(C) < 0(as for a Langmuir isotherm), we nd that for S > S0

the spreading λ2-waves must be directed towards in-creasing values of C, while for S < S0 the direction isopposite, see Fig. 4.4. We also observe that for linearadsorption (h′ = 0) the λ2-waves are indierent.It remains to determine the λ2-shock waves. The

jump condition given by Eq. 4.23 for the system ofEq. 4.25 reads

σ(S+ − S−) = f+ − f−, (4.29a)

σ(S+C+ + a+ − S−C− − a−) = f+C+ − f−C−.(4.29b)

Solving Eq. 4.29a for σ, substituting into Eq. 4.29band rearranging, we obtain

σ =f+

S+ +D=

f−

S− +D, (4.30)

where D = (a+−a−)/(C+−C−). This is interpretedgeometrically in Fig. 4.5. Since the entropy conditionof Eq. 4.24 was derived for strictly hyperbolic sys-tems, it cannot be applied in the present situationwithout additional considerations. However, we willdisregard these subtilities here and simply observe byinspection of Fig. 4.5 that if C+ < C−; S > S0 the

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78 CHAPTER 4. SOLUTIONS

f

S

L(-)R(+)

S0

LL-h(C+) -D -h(C-)

λ 2+ < σ < λ 2

-

λ1+

,λ1- > σ

R R

a(C)

CC -C+

h (C-)h(C+)

D

Figure 4.5: Polymer ooding. Shock orientation. Ad-sorption isotherm.

relation Eq. 4.24 holds, while if C+ > C− it is vio-lated. For S < S0 the opposite is true. Thus, we con-clude that λ2-shocks are oriented opposite to the cor-responding spreading wave curves, but not coincidingglobally unless h is constant. It can also be seen fromEq. 4.30 that a λ2-shock cannot cross S = S0.The only waves which can connect the two regions

S < S0, S > S0 are therefore the Buckley-Leverettwaves along C is constant. Since such a connectingwave is faster than any wave inside each region, theleading wave in a Riemann solution where the initialand injected conditions are located in separate regionsmust always be a Buckley-Leverett wave.We conclude this section with some examples uti-

lizing the above results. The solutions of the rstexample were derived by Patton et al.19

Example 1. Consider a polymer solution being in-jected into a reservoir initially saturated with oil atconnate water saturation, i.e., CL = Cmax, CR = 0,SL = Swor and SR = Swc. The fractional ow func-tions associated with the two constant states of thisRiemann problem is shown in Fig. 4.6. Accordingto the above, the change in C-values will take placethrough a λ2-shock, since CL > CR. Together withthe λ1-waves (Buckley-Leverett waves), it is easy tosee that Fig. 4.7 represents a solution of the problemsatisfying all requirements discussed above. In partic-ular, we observe that a bank of pure water is formingahead of the polymer displacement front. However, ifthe polymer is injected after a waterood, SR = S, anoil bank will form ahead of the polymer displacementfront.

Example 2. We next consider behavior at the trail-ing edge of a polymer slug. If all mobile oil is dis-placed by the slug, SL = SR = Swor and CL = 0,

LR

σ2

σ1

σ1

f(. ,CR)

f

-D R S0 S1 S2 S

f(. ,CL)

Figure 4.6: Polymer ooding. Construction of solu-tion.

σ2 σ1

σ1

Swor

S2

S1

Swc

_S

Polymerfront

Figure 4.7: Polymer ooding. Saturation proles.

CR = Cmax. Thus, C will go through a spread-ing λ2-wave, where the largest velocity is λmax =1/(1 + h(Cmax)). From the sharpening behavior ofthe leading polymer shock λmax = λ− > σ, and con-sequently the trailing and leading edges of the slugmay interfere during the ood causing a breakdown ofthe slug. It is therefore important that a sucientlylarge slug is injected in order to avoid this situation.A rst estimate of the slug size so required can bemade by these considerations. See also Walsh andLake13 for application of the same method to solventooding with chase uids and for the determinationof WAG-ratios.In the polymer model considered this far, certain

important eects are neglected. For example, if afraction of the pore volume is inaccessible to the largepolymer molecules, an increase in the characteristicvelocity of the polymer will result. For simplied sit-uations, this can also be analyzed by simple wavetheory, see Lake.3 Another important eect is per-meability reduction caused by large molecules block-ing narrow pore channels. Since permeability doesnot enter one-dimensional problems, this eect can-not be studied by such theories. By the same token,the eect of shear thinning is also excluded from suchanalysis.In Johansen et al.,20 a model for multicomponent

polymer ooding is introduced, and the eects of dif-

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4.1. SHOCKS AND SIMPLE WAVES 79

ferent polymer adsorption models and variation inmolecular weight of polymer components are studiedby simple wave theory. It is shown there that for acontinuous distribution of polymer components, thepolymer displacement front is not a shock as depictedin Fig. 4.7, but rather has the character of a spreadingwave. Also, it is shown that when polymer compo-nents are competing for the vacant adsorption sites,polymer components tend to accumulate at the front,which can enhance the displacement compared to asituation where components adsorb independently.

4.1.6 Solvent Flooding

The main achievement in polymer ooding is the en-hancement of displacement eciency; the theoreti-cal limit for ultimate oil recovery is the same as forpure waterooding. In solvent methods (injection ofCO2, N2, hydrocarbon gas, . . .) the main eect ismobilization of trapped hydrocarbons, through themechanism of dierent uid components partition-ing between the phases, tending to form a miscibleor partly miscible uid system. In particular, theseinteractions cause chromatographic separation of dif-ferent uid components. In this section we investi-gate such behavior through the simple wave theory.The model for solvent ooding to be discussed canalso be applied directly to alcohol ooding, Taber etal.,21 Wachman,22 and to carbonated waterooding,Claridge and Bondor,23 DeNevers.24 For simplicityof the description, we will assume that residual phasesaturations are zero. Inclusion of nonzero values is asimple extension.We rst consider a three-component uid system,

which according to Gibbs' phase rule can form nomore than three phases at a given pressure and tem-perature. However, in this section we will assume amaximum of two phases, the three-phase problemsbeing discussed in Sec. 4.1.9. We will assume a xedphase behavior represented by a ternary diagram asshown in Fig. 4.8, where Ci (i = 1, 2, 3) are the overall

Q C3 C1

C2

LC

λ1=λ2 λ1=λ2Cγ

Tie-line

Phas

e2

Phase

1

Ω1

Ω2

B

P

spreading (λ2)

shock (λ2)

Figure 4.8: Solvent ooding. Ternary diagram. Sim-ple wave curves.

volume fractions of the solvent and two hydrocarboncomponents (

∑Ci = 1). We rst choose C1, C2 as

state variables for the problem.We assume the existence of a unique Plait point P ,

dened as the overall composition for which the giventemperature and pressure is the critical point of theuid system. The binodal curve B separates the two-phase region Ω2 from the one-phase region Ω1, andwe label the phases as shown in the gure. The lineLc is the tangent to the binodal curve at P . Thetie lines are a family of nonintersecting straight linescovering Ω2. Along a tie line, phase compositions areconstant given by the endpoints of the tie line in Ω2.If S denotes saturation of phase 1, see Eq. 4.2, then

Ci = Sci1 + (1− S)ci2; i = 1, 2, (4.31)

and if f is fractional ow of phase 1, which we assumeis a given function of S along each tie line,

Fi = fci1 + (1− f)ci2; i = 1, 2, (4.32)

see Eq. 4.3. With these denitions, the model underconsideration here is Eq. 4.1; i = 1, 2. A tie line canbe represented in (C1,C2)-space by

C2 = ηC1 + a(η), (4.33)

where η is the slope of the tie line. The function a(η)describes the geometry of the tie lines. Since tie linesdo not intersect inside Ω2,

∂C2

∂η= C1 +

da(η)

dη> 0 in Ω2. (4.34)

For example, if all tie line extensions are passingthrough a common point Q(−C∗, 0) in Fig. 4.8,a(η) = ηC∗ and C1 +da/dη = C1 +C∗, which clearlyis positive in Ω2.Using Eq. 4.32 and Eq. 4.33, we may write

F2 = f(ηc11 + a(η)) + (1− f)(ηc12 + a(η)),

i.e.,F2 = ηF1 + a(η) in Ω2. (4.35)

Since C2 and S are uniquely determined by η and C1,we may write the model as

∂C

∂t+∂F (C, η)

∂x= 0, (4.36a)

∂η

∂t+F (C, η) + h(η)

C + h(η)

∂η

∂x= 0, (4.36b)

where we have expanded derivatives using Eq. 4.35,rearranged and put C = C1, F = F1. Furthermore,we have dened h(η) = da(η)/dη. According to whatwas obtained above, C + h(η) > 0 in Ω2. Therefore,Eq. 4.36 is well dened in Ω2, but we remark herethat this representation is not valid in general in Ω1.In the notation of Eq. 4.20,

A(C, η) =

∂F

∂C

∂F

∂η

0 F+hC+h

(4.37)

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80 CHAPTER 4. SOLUTIONS

F

(-h,-h)

C12 C11

λ1=λ2

λ1=λ2

λ1

λ2

C γ :F=C

C

F(.,η)

Figure 4.9: Solvent ooding. Fractional ux andeigenvalues.

and we observe the similarity with the model for poly-mer ooding; C, F , η in Eq. 4.36 correspond to S, f ,C, respectively, in Eq. 4.32. A typical shape of thefunction F (C, η) is shown in Fig. 4.9, where we havealso depicted the eigenvalues

λ1 =∂F

∂Cand λ2 = (F + h)/(C + h)

of Eq. 4.37. Considering the similarity with the modelfor polymer ooding, Fig. 4.3, the characteristic anal-yses in the two situations are entirely analogous: TheBuckley-Leverett type of waves are along η=constant(tie-line waves). If we assume dh(η)/dη > 0, thesimple wave curves corresponding to λ2 (non-tie-linewaves) are as depicted in Fig. 4.8. As in Sec. 4.1.5, itcan be shown that λ2-waves cannot intersect the bin-odal curve nor the curve Cγ dened by F (Cγ(η), η) =Cγ(η). If dh(η)/dη = 0, the spreading λ2-waves areindierent. This corresponds to the case when alltie-line extensions are passing through the point Qin Fig. 4.8. If dh(η)/dη < 0, the orientations of λ2-shocks and spreading waves in Fig. 4.8 are reversed.This has also been qualitatively observed by Cere andZanotti.25 Finally, the regions on each side of Cγ areconnected by fast tie-line waves.So far only Riemann problems with initial and in-

jected states inside Ω2 have been considered. In orderto extend the wave analysis to the entire compositionspace, we need to consider the jump condition for themodel, which reads

σ(C+i − C

−i ) = F+

i − F−i ; i = 1, 2. (4.38)

Clearly, this is satised with σ = 1 (unit velocitywave) and when ~u− and ~u+ are both located in Ω1,or on the curves B, Cγ . A physically motivated en-tropy condition for the present model has yet to berigorously derived. However, if ~uL, ~uR are locatedin Ω1 (including B) inside the region of tie-line ex-tensions, a multiple contact argument indicates thatsuch states cannot be connected by a unit velocity

wave. If at least one of the states lies outside the re-gion of tie-line extensions (condensing or vaporizinggas drives, Dumore et al.1) the same argument in-dicates the existence of a unit velocity wave joiningthe two states (developed miscibility). This denesa method for determination of Minimum MiscibilityPressure (MMP) in gas drives: The MMP is the pres-sure for which the initial condition lies on the criticaltie line Lc, Defrenne et al.26 (The denition is notstrictly correct if tie-line extensions can intersect Lcin composition space.) If one of the states is locatedin Ω1 and the other state in Ω2, they cannot be joinedby a simple spreading wave. Suppose they are con-nected by a shock and assume ~u+ is in Ω2. Then byEq. 4.33 and Eq. 4.34 we have

C+2 = η+C+

1 + a(η+); F+2 = η+F+

1 + a(η+), (4.39)

where η+ is the slope of the tie line passing through~u+. Since ~u− is in Ω1: F−i = C−i i = 1, 2. Substitut-ing this and Eq. 4.39 into Eq. 4.38 we nd

σ(C+1 − C

−1 ) = F+

1 − C−1 , (4.40)

σ(η+C+1 + a+ − C−2 ) = η+F+

1 + a+ − C−2 . (4.41)

Solving Eq. 4.40 for F+1 , substituting in Eq. 4.41 and

rearranging, we nd

(σ − 1)C−2 = (σ − 1)(η+C−1 + a+). (4.42)

Hence, for nonunity velocity shocks (σ 6= 1)

C−2 = η+C−1 + a+, (4.43)

which expresses that the state in Ω1 must lie on theextension of the tie line η+. Therefore, the only wavesthat can join the two regions Ω1 and Ω2 are shocksalong tie-line extensions.In order to demonstrate lack of uniqueness in so-

lutions of Riemann problems for this model, considerthe example in Fig. 4.10, where the initial conditionis in Ω2 on C2 = 0 and the injected condition is thepoint in Ω1 where Lc intersects C2 = 0. Clearly, bothcomposition paths depicted in Fig. 4.10 satisfy the re-quirements discussed above. Path I represents a unitvelocity wave, while Path II is a (composite) tie-linewave. The corresponding solutions for C1 are shownin Fig. 4.11. A further discussion on this can be foundin Ref. 27, providing clear evidence that if it is physi-cally possible for tie-line extensions to intersect in Ω1,the model is not well posed.

Example 1. As an example on construction of so-lutions for immiscible solvent ooding, consider pureCO2 displacing a two-component oil as shown in thecomposition space in Fig. 4.12. We assume dh/dη >0, in which case the tie-line extensions cannot inter-sect inside the composition space. The orientationof λ2-shocks is also shown in Fig. 4.12. The con-struction of the solution which is performed on frac-tional uxes for CO2 is shown in Fig. 4.13, wherehLR = (a(ηL) − a(ηR))/(ηL − ηR) = a(ηR)/ηR. The

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4.1. SHOCKS AND SIMPLE WAVES 81

C3 C1

C2

_u Rrσ1

P

Path I

Path II

σ=1

σ=1

B

_u

r

σ1

PathI

R

LC1

PathII

F(C,0)F

L

_u RL Shock σ1 Spreading wave r

Figure 4.10: Solvent ooding. Nonuniqueness. Con-struction of possible solutions.

solution is shown in Fig. 4.14, where we observe thatthe light hydrocarbon component is extracted by theCO2-rich phase forming a bank of lighter oil betweenthe displacing and displaced phases.

Example 2. The results described above can alsobe applied if we replace one of the hydrocarbon com-ponents with a pure water component, in which casethe two-phase region Ω2 covers the entire composi-tion space, Fig. 4.15. For example, this is useful forevaluating the performance of WAG's and chase u-ids in solvent ooding, see Walsh and Lake.13 Wethen regard the water(w) and oil(o) as nonpartition-

C1

r1

σ=1

σ1

x / t

Figure 4.11: Solvent ooding. Possible solutions.

CO2heavy

light

u1 σ1

σ2

σ3

u3

u2

R

PηR

Cγ L

Figure 4.12: Immiscible CO2-displacement. Compo-sition path.

F

C1R

(-hLZ,-hLR)

L

u1

u2u3

F(. ,0)

F(. ,ηR)

σ3

σ2

σ1

Figure 4.13: Immiscible CO2-displacement. Con-struction of solution.

ing components residing entirely in the aqueous(w)and oleic(o) phase respectively, while the solvent(s) isassumed to exhibit local equilibrium partitioning be-tween the two phases: cso = Kcsw. We assumeK < 1is constant. However, an extension to include com-position dependent K-values is straightforward. (IfK > 1, simply use oil as the reference phase. K = 1is trivial). The governing model becomes

∂Cw∂t

+∂Fw(Cw, Cs)

∂x= 0, (4.44a)

∂Cs∂t

+∂Fs(Cw, Cs)

∂x= 0, (4.44b)

where Cw = Scww; Fw = fcww; Cs = Scsw + (1 −S)Kcsw; Fs = fcsw + (1− f)Kcsw. By inspection ofFig. 4.15, a tie line for this problem may be writtenas

Cs =(1−K)csw

1− cswCw +Kcsw. (4.45)

Putting η = (1 − K)csw/(1 − csw) we nd Kcsw =ηK/(η + 1 − K), yielding Cs = ηCw + a(η) and

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82 CHAPTER 4. SOLUTIONS

heavy

light

CO2

σ3

σ2

σ1

C

x / t

Figure 4.14: Immiscible CO2-displacement. Solution.

Cs

Cw

csw

cso (Cs,Cw)=Kcsw aqueous

olei

c

cww

Figure 4.15: Composition space for WAG process.

Fs = ηFw + a(η), where a(η) = ηK/(η + 1 − K).We nd that dh/dη = d2a/dη2 < 0 and the Riemannproblem for Eq. 4.44 falls completely into the contextdescribed previously.In North Sea reservoirs, most oils are rich in light

components. In recovery of such oils, the high contentof methane will play an important role, in particularfor miscible or miscible-tending displacements. Forsuch uid systems, the method of using ternary orpseudoternary diagrams in estimation of MMP canhardly be used. It would therefore be a major achieve-ment if a fourth component could be included, and asimilar method developed for quaternary diagrams.Such an approach is taken by Monroe et al.28 forCO2 displacement of live oils. Although the compo-sition space is considerably more complex comparedto the three-component case, the wave analyses arefairly similar. The importance of such analyses forNorth Sea applications is clearly demonstrated whenreviewing the main conclusions of Ref. 28. In CO2-displacement of live oils, the recovery can be higheven in the two-phase immiscible regime. A methanebank will form, and if the displacing pressure is abovethe MMP for the CO2/dead oil system left behind themethane bank, the recovery will be ecient. In gen-eral, the eciency of such a process can be estimated

by one-dimensional wave analysis using compositionroutes in the quaternary diagrams. An extension ofthe analysis by Monroe et al.28 to systems with morethan four components seems within reach.

4.1.7 Surfactant Flooding. Type II(-)

Residual oil trapped by capillary forces in the porousmedium can be mobilized by reduction of interfacialtensions. This is the main mechanism in surfactantooding, since surfactants in contact with oil/watersystems tend to form microemulsions with low inter-facial tensions to the oleic and aqueous phases.Sec. 4.1.5 emphasized the fact that the prototype

model for polymer ooding considered there had aknown solution for arbitrary initial and injected con-ditions, and with arbitrary retention and ux func-tions. In Sec. 4.1.6 it was demonstrated how thiscan be transferred to solvent ooding maintaining thesame generality. From a mathematical point of view,the phase diagrams and model parameters arising inType II(+) and II(-) surfactant ooding are the sameas in solvent ooding, and consequently no additionaleort is required in order to solve the Riemann prob-lems for such systems. The most signicant dier-ence is in the phase diagrams, as the residual satu-rations in surfactant ooding can be zero at overallcompositions dierent from the Plait point. However,for construction of Riemann solutions this dierencedoes not represent an obstacle. Of course, the aboveremarks apply only to the simplest possible approachin modelling of surfactant ooding. Nevertheless, thisapproach has been used by several investigators2,2934

because it provides good qualitative information andinsight. In reality, the process is very complex, in-volving such eects as three-phase ow (Type III sys-tems), salinity dependence, and complicated reten-tion mechanisms, which all together act as a coupledsystem.As an example on construction of solutions for sur-

factant ooding, we consider an immiscible Type II(-)(constant salinity) displacement of oil by a low con-centration surfactant solution in brine, Fig. 4.16. LetC1 and C2 be overall concentrations of brine and sur-factant respectively, and let subscript 1 refer to theaqueous phase, which we also assume is the wettingphase. We then assume that adsorption of surfactantis a function A of concentration of surfactant in theaqueous phase only; A = A(C21). For any overallcomposition inside the II(-)-region, C21 is uniquelydetermined by the tie line passing through that com-position, say C21 = α(η). The governing model maythen by Eq. 4.33 and Eq. 4.36 be written (puttingC = C1, F = F1):

∂C

∂t+∂F (C, η)

∂x= 0, (4.46a)

∂t(ηC + b(η)) +

∂x(ηF + a(η)) = 0, (4.46b)

whereb(η) = a(η) +A(α(η)).

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4.1. SHOCKS AND SIMPLE WAVES 83

Oil Brine (C1)R

P

L

Surf. (C2)

λ2-shock

CγηL

σ1

σ2

σ3

Figure 4.16: II(-) surfactant ooding. Compositionpath.

Then,db

dη=da

dη+dA

dη> 0

and

d2b

dη2=d2a

dη2+d2A

dα2

(dα

)2

+d2α

dη2

dA

dα,

whered2A/dα2 < 0

for Langmuir type of adsorption isotherms. By in-spection of Fig. 4.16, d2α/dη2 < 0. For the sakeof simplicity (although not necessary), we assumethat d2a/dη2 < 0, which implies that d2b/dη2 < 0.The orientation of the λ2-shocks are therefore as in-dicated in Fig. 4.16. The construction of the solutionin Fig. 4.17 is shown in Fig. 4.18, where

h1 =

a(ηL +A(α(ηL)− a(ηR)−A(α(ηR))

ηL − ηR=

a(ηL) +A(α(ηL))

ηL

andh2 = a(ηL)/ηL.

We observe that an oil bank forms between the initialand injected states. The leading edge of the bank is aBuckley-Leverett type of shock along the ηR-tie line,followed by a surfactant front (λ2-shock), a spread-ing Buckley-Leverett wave and nally a solubilizationfront, which is a Buckley-Leverett type of shock alongthe ηL-tie line connecting the one- and two-phase re-gions. This example is also discussed in Pope.2

4.1.8 Thermal Methods

The most important mechanism in all thermal meth-ods is the reduction in oil viscosity, although a vari-ety of other mechanisms may contribute to increaserecovery. Thermal methods are primarily applicable

surf.

oil

sol. front surf. front x/t

C

σ1σ2

σ3

Figure 4.17: II(-) surfactant ooding. Solution.

LR

σ1

σ2

σ3

F1(.,ηL)

F1(. ,ηR)

F1

C1

-h2

-h1

Figure 4.18: II(-) surfactant ooding. Construction.

in heavy oil reservoirs, but their use even in recoveryof light oils can be economical. For North Sea con-ditions, the inverse" problem is also important, i.e.,cold uids being injected into hot reservoirs.In general, thermal methods is probably the class

of EOR-processes for which the simple wave theoryis least applicable. For in-situ combustion, no suchattempt seems to have been made, since chemical re-actions, change in volumes and heat loss eects mustbe taken into account, and because multiphase owoccurs.Some applications of simple wave theory to steam

ooding exist. An interesting approach is made inShutler and Boberg,35 dividing the uid ow intodierent isothermal zones separated by temperatureshocks. The condensation of steam caused by heatloss to adjacent strata is incorporated through aniterative technique utilizing the Marx-Langenheim36

theory and the simple Buckley-Leverett theory. How-ever, when considering dierent ow zones more orless independently, the injected state occurring inthe Riemann problem associated with the leadingsteam front is dicult to determine. In order to re-solve this, additional assumptions must be imposed.A discussion on this can be found in Lake,3 also pro-

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84 CHAPTER 4. SOLUTIONS

viding an estimate of steam front velocity as a func-tion of steam quality and heat of vaporization.A more general approach to steam ooding us-

ing simple wave theory is given by Wingard andOrr,37 where the eects of three-phase ow, densityvariations caused by temperature gradients, volumechanges upon mixing and condensation are taken intoaccount.For hot (or cold) waterooding, the assumption of

incompressibility is adequate. If also heat loss to ad-jacent strata and thermal expansion can be neglected,the governing model becomes:

∂S

∂t+∂f(S, T )

∂x= 0 (4.47a)

∂t(κwST + κo(1− S)T

+κsT ) +∂

∂x(κwfT + κo(1− f)T ) = 0,

(4.47b)

see Eq. 4.4. The temperature T enters the fractionalow function through the oil viscosity which is de-creasing with increasing temperature. Similarly toprevious derivations, it can be shown that the systemmatrix is

A(S, T ) =

λ1∂f

∂T

0 λ2

, (4.48)

where the eigenvalues are

λ1 =∂f

∂S,

λ2 = (f + h1)/(S + h2),

h1 = κo/(κw − κo),

h2 = (κo + κs)/(κw − κo).

The geometrical interpretation of λ1 and λ2 is simi-lar to Fig. 4.17, the λ2-ray is drawn from the point(−h2,−h1). Since normally h1 h2, propagation oftemperature fronts (λ2-shocks) is very slow comparedto saturation fronts (λ1-shocks). The analogy to themodels considered previously is obvious.The general Riemann problem for Eq. 4.47 was rig-

orously analyzed by Barkve.38

4.1.9 Three-Phase Flow

The characteristic feature of the models consideredabove has been the upper triangular form of the ma-trix A(~u) in Eq. 4.6. Consequently, the models havebeen hyperbolic for all choices of model parameters.For processes involving simultaneous ow of three im-miscible phases, this simplicity no longer persists. Avariety of three-phase relative permeability modelshave been suggested in the literature. For a com-prehensive treatment of their dierent physical prop-erties, see Delshad and Pope.39 The most commonly

used model in black oil reservoir simulation is theStone model which is based on two-phase relative per-meabilities for oil-water and gas-oil systems.40 Thesimplest model (Corey type) assumes that relativepermeability for each phase is a function of satura-tion of that phase only. In all cases, the conservationlaws Eq. 4.1 for three-phase immiscible ow can bewritten

∂Sw∂t

+∂fw(Sw, Sn)

∂x= 0, (4.49a)

∂Sn∂t

+∂fn(Sw, Sn)

∂x= 0, (4.49b)

together with the constitutive relations

Sw + So + Sn = 1; fw + fo + fn = 1,

where w, o, n refer to wetting, intermediate, andnonwetting phases respectively, and where fractionalows are given by Sec. 3.1. The matrix A(~u) forEq. 4.49 is

A(Sw, Sn) =

∂fw∂Sw

∂fw∂Sn

∂fn∂Sw

∂fn∂Sn

. (4.50)

The discriminant of the quadratic equation determin-ing the eigenvalues of A is

D =(∂fw∂Sw

+∂fn∂Sn

)2

− 4

(∂fw∂Sn

∂fn∂Sw

− ∂fw∂Sw

∂fn∂Sn

).

(4.51)

As can be seen from inspection of the Stone andCorey models, there is no immediate physical reasonfor D to be positive. Consequently, elliptic behav-ior, see Sec. 3.1, is possible. In fact, Bell et al.41

demonstrated through numerical experiments withthe Stone model in the absence of gravity, that suchbehavior actually occurred. Furthermore, it was indi-cated that states inside elliptic regions entering Rie-mann solutions tend to leave this region. For mod-els of mixed type (elliptic and hyperbolic), Holden etal.42 prove that if the states of the Riemann problemare both located outside the elliptic regions, the Rie-mann solutions will not penetrate these regions. Fur-thermore, if both states are inside an elliptic region,the Riemann solution will stay outside except for sin-gle shock waves between the given states and states inthe hyperbolic part of composition space. Therefore,it is not at all obvious that elliptic regions are non-physical. However, an indication of ill-posedness ofmodels of mixed type is given by Azevedo and March-esin.43 They present an example of a Riemann prob-lem for a three-phase model of mixed type with twodistinct solutions, which are both limits of vanishingdiusion solutions, and thus equally acceptable froma physical point of view.

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4.1. SHOCKS AND SIMPLE WAVES 85

Fayers and Matthews44 conjectured that gravityterms would counteract elliptic behavior. However,later numerical experiments indicate that this is notnecessarily correct. For the Corey type models,Marchesin and Medeiros45 proved that Eq. 4.49 isalways hyperbolic. This result was extended byTrangenstein,46 proving that the Corey type modelis the only one guaranteed to be hyperbolic forall choices of model parameters (including gravity).For any other model, elliptic behavior may occur.Trangenstein also makes the point that there seems tobe a conict between choices of three-phase relativepermeability models: The Corey type models, whichare the only models guaranteed to be hyperbolic forall choices of model parameters, are not capable ofreproducing laboratory measured two-phase data forboth oil-gas and water-oil systems.The investigation of three-phase relative permeabil-

ity models is an area of active research and the prob-lem has caught a considerable interest in the math-ematical community. For further details on this, werefer to Shearer47 and the references given there.In black-oil systems, the simultaneous ow of all

three phases seldom takes place. However, in EOR-processes like steam ooding and surfactant oodingsuch ow is common. In fact, in surfactant ood-ing, three-phase behavior is desirable since interfa-cial tensions between the middle-phase microemul-sion and the oleic and the aqueous phases can be verylow if the surfactant slug is carefully designed withrespect to the environment it will encounter. Theanalysis of the Riemann problem for Type III surfac-tant ooding is rendered by possible elliptic behav-ior in the three-phase region, unless Corey-type rela-tive permeability models are employed. Such modelshave been reported appropriate for surfactant ood-ing, by Delshad et al.48 Assuming a Corey-typemodel, Aanonsen49 considered the Riemann prob-lem for Type III surfactant ooding, where the ini-tial condition was located inside the three-phase re-gion (at zero surfactant concentration) and the in-jected condition was located in the one-phase region(at zero oil concentration) but outside the II(+) tie-line extensions. In Aanonsen,49 the unique solutionof this problem was constructed (subject to a self-sharpening" type of entropy condition). The solutionis similar to the one constructed in the example inSec. 4.1.7 for Type II(-) systems: An oil bank formsbetween a leading Buckley-Leverett type of shock andthe surfactant/solubilization fronts.

4.1.10 Extensions of Theory

Several other applications and extensions of the sim-ple wave theory beyond what has been discussed herecan be found in the literature. We limit the discus-sion of this by referring to some key papers related tothe specic subjects.One obvious extension of simple wave theory to in-

clude two space dimensions is to formulate the modelas one-dimensional problems in individual stream

tubes by introducing the dimensionless variables

xD =

∫ x0w(ξ)dξ∫ L

0w(ξ)dξ

; tD =tqT

φ∫ L

0w(ξ)dξ

, (4.52)

where qT is volumetric ow rate, x is distance alongstream line and w(x) is width of the stream tubetaken along equipotential lines perpendicular to bulkow. The stream-tube method is exact only whenmobility ratio is unity and when gravity eects arenegligible. Nevertheless, the method can be superiorto the use of conventional numerical methods even ifthese conditions are violated.A more general approach also utilizing analytical

solutions of local Riemann problems is the method offront tracking.50,51 The relations Eq. 4.52 can alsobe directly applied to extend the simple wave theoryto radial ow.A common approach in reservoir engineering is to

reduce inhomogeneous, two-dimensional ow prob-lems to homogeneous, one-dimensional equivalents"(for which simple wave theory applies) by introduc-ing average model parameters (pseudofunctions) asfunctions of average saturations and concentrationsover cross sections perpendicular to bulk ow. Thisapproach was rst used by Dykstra and Parsons52

for piston-like displacement of oil by water in layeredmedia without communication between layers. Thesame approach was also used by Koval53 for repre-sentation of viscous ngering in rst-contact miscibledisplacements. Numerous generalizations52,53 havebeen suggested and we refer to Orr et al.,54 Pande etal.,55 Hewett and Behrens56 and Lake3 for descrip-tions and discussions of the limitations of such meth-ods.In surfactant ooding, the salinity of the aqueous

phase may strongly inuence ood performance, sincethis is the main parameter determining the overallphase behavior at a given temperature and pressure.Therefore, the inclusion of sodium and calcium ascomponents in the Riemann problem for surfactantooding is desirable. Additional local equilibriummodels for cation exchange are then required. As-suming one phase ow, such a Riemann problem in-volving ion exchange with clay minerals was solved byPope et al.,57 and in the presence of surfactant mi-celles by Hirasaki.58 In both references, the inuenceof cation exchange on surfactant ood performanceis discussed. An extension to two-phase ow may bepossible by application of a theory described by sev-eral authors,18,20,59 where a general algorithm is pre-sented for the construction of Riemann solutions formulticomponent, two-phase ow problems from givensolutions of corresponding (simpler) one-phase prob-lems. In Dahl et al.,18 this algorithm was applied togeneralize a theory of chromatography developed byRhee et al.60 for single-phase ow with an arbitrarynumber of solutes adsorbing according to a competi-tive adsorption model.When water containing solutes is owing through

a porous medium containing soluble minerals, pre-

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86 CHAPTER 4. SOLUTIONS

cipitation and dissolution waves will occur. Assum-ing local equilibrium between owing and stationarycomponents and constrained by solubility products,Bryant et al.61 investigated such Riemann prob-lems. The analysis of hyperbolic systems with suchconstraints, and hyperbolic systems with source/sinkterms are research areas with a substantial poten-tial for attaining increased insight in dierent owphenomena. For example, by including source/sinkterms in the models, the eects of dead-end pores,heat losses in thermal methods, nonequilibrium ad-sorption/partitioning,62 ow in layered media withcommunication between layers63,64 and ow in natu-rally fractured media51 can be studied.

4.2 Cross-Sectional Displace-ment

The previous section discussed displacement pro-cesses in homogeneous 1D media. This geometricallysimple setting allowed complex physical processes tobe treated analytically. In the present section we al-low somewhat more realistic geometries, at the ex-pense of being able to include only quite simplisticphysical processes, for analytic treatment to be at allpossible.We consider layered reservoirs in 2D cross-sectional

geometries, as illustrated in Fig. 4.19. It is assumedthat all reservoir parameters are constant along thealong-dip coordinate, x. They are allowed to vary,in a piecewise constant manner, with the dip-normalcoordinate, y.

Water

Oil

y

x

Front

Figure 4.19: Layered reservoir.

A simple displacement process is assumed to takeplace in this geometry, with the displacing uid beinginjected at the left boundary and the displaced uidbeing produced at the right. For deniteness we shallhenceforth refer to the displacing uid as water andto the displaced uid as oil, although other uids areconceivable, of course. We shall also occasionally refer

to the boundaries as injection and production wells,respectively.The geometry outlined above does not capture all

important features of real situations. In particular,eects from ow convergence around wells are not in-cluded. Also, layered media are idealizations of realheterogeneous ones, of course. Still, a reasonable rep-resentation of reality can be obtained in many cases,at least for those parts of a reservoir which are su-ciently far from wells.We shall make the important assumption through-

out that the displacement is piston-like, in the sensethat a sharp interface (front), not necessarily recti-linear, exists between water and oil. On either sideof the front only one uid is mobile. Displacementfronts tend in reality to have a transition zone causedby capillary or dispersive eects. The assumption ofa sharp front is, therefore, an approximation which isnot always realistic. It has been suggested65,66 thatthe approximation is acceptable if the true transitionzone is less than some fraction of the reservoir thick-ness, 10% say.It will also be assumed that uids, as well as the

permeable medium itself, are incompressible. Thisshould be an acceptable representation of reality,since displacement processes tend to maintain reser-voir pressure, thus reducing the importance of com-pressibility eects.

4.2.1 Displacement with NegligibleGravity

With the assumptions made previously, and with theneglect of gravity, ow is governed exclusively by vis-cous forces. There are two fundamentally dierentapproaches in use when analyzing this situation. Inthe rst approach one neglects crossow between lay-ers. In the second approach one makes the oppositeapproximation of no resistance to crossow. This isthe vertical equilibrium (VE) approximation. In bothapproaches it is traditional to assume the front to bedip-normal in each layer.The choice of approach should be based on the

properties of the actual reservoir and the particulardisplacement process. Some quantitative informationon the relative merits of the two approximations hasbeen provided by Zapata and Lake.63

The choice may be important. It will be demon-strated that no crossow reservoirs and VE reservoirsrespond very dierently to variations in the water-oilmobility ratio, M .

The No Crossow (Dykstra-Parsons) Approx-imation

Imagine layer boundaries in Fig. 4.19 to be imper-meable strata, preventing crossow between layers.Then the layers are mathematically decoupled, ex-cept perhaps in wells.Such a no crossow situation has been analyzed by

a number of authors. The method of Stiles67 is consis-

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4.2. CROSS-SECTIONAL DISPLACEMENT 87

tent only for unit mobility ratio displacements. Dyk-stra and Parsons52 allowed a nonunity mobility ratio,and were able to determine recovery and producingwater-oil ratios at water breakthrough in each indi-vidual layer. El-Khatib68 generalized the Dykstra-Parsons results slightly to allow porosity and end-point saturations to vary between layers. Reznik etal.69 included variable mobility ratios, and also gavemethods for describing the process as a continuousfunction of time.The decoupling of layers implied by the no crossow

assumption is mathematically very convenient. It al-lows the problem to be treated essentially as a seriesof independent 1D problems. This suggests that itmay be possible to combine 1D techniques with theDykstra-Parsons method, in order to capture morecomplex displacement behavior in each layer. TheBuckley-Leverett8 solution is an obvious candidate.Several attempts to combine these two classical re-sults have been reported in the literature.7072

Returning to the piston-like displacement assump-tion, we reproduce the core of the Dykstra-Parsonsanalysis, which gives oil recovery at breakthrough ofwater. This result is needed for comparison with thecorresponding VE result.Assume without loss of generality that layers are

arranged such that the intrinsic front velocity, vf =λo/φ∆S, decreases upwards. Here λo, φ, and ∆S areoil mobility, porosity, and dierence between initialand residual oil saturations, respectively. (To obtainthe dimension of velocity, imagine vf to have beenmultiplied by a unit potential gradient.)Fig. 4.20 gives a snapshot of the displacement

process, and denes some geometrical quantities. Su-

Water

Oil2

N

xi i hi

1

L

Figure 4.20: Geometry of displacement in layeredreservoir.

perscript i refers to the layer number. Assume thatthe front starts out from the injection well, i.e., thereis no mobile water present initially. Let ER be therelative recovery at breakthrough of water, i.e., thevolume of produced oil as a fraction of total recover-able (mobile) oil initially present. ER is given by

ER =

∑Ni=1 h

ixiφi∆Si

L∑Ni=1 h

iφi∆Si, (4.53)

where x1 = L by assumption. With Darcy's law

uio = −λio ∂xΦio , (4.54)

uiw = −λiw ∂xΦiw , (4.55)

and the continuity equation

dxi

dt=

uiwφi ∆Si

=uio

φi ∆Si, (4.56)

one obtains

dxi

dt=

∆Φλiw λio

λiw (L− xi) + λio xi

1

φi ∆Si. (4.57)

Here ui and ∂xΦi are individual layer Darcy veloc-ities and potential gradients, respectively; ∆Φ(t) isthe total well-to-well potential (pressure) drop.Assume that the mobility ratio M is a constant,

i.e., such that λiw = M λio for all i. Division ofEq. 4.57 by dx1/dt then produces

dxi

dx1=M (L− x1) + x1

M (L− xi) + xiλio φ

1 ∆S1

λ1o φ

i ∆Si. (4.58)

Integration of Eq. 4.58 between x1 = 0 and x1 = Lyields the front positions at breakthrough in layer 1:

xi

L=M −

√M2 + (1−M2)F i

M − 1. (4.59)

Here

F i =λio ∆S1 φ1

λ1o ∆Si φi

(4.60)

is the ratio between intrinsic front velocities in layersi and 1.Insertion of Eq. 4.59 into Eq. 4.53 now produces

the answer sought.

The Vertical Equilibrium Approximation

The term vertical equilibrium was apparently rstcoined by Coats et al.73 The essentials of the VE ap-proximation had been employed previously, however,e.g., by Dietz74 in his study of gravity-dominated dis-placement. These two papers are representative ofthe two slightly dierent uses made of the VE ap-proximation, i.e., to generate VE pseudofunctions forreservoir simulation,66,73,75,76 or for analytic predic-tion of displacement processes, which is the presenttopic.The central point of the VE approximation is the

neglect of dip-normal potential gradients. This math-ematically transforms the 2D nature of the probleminto 1D, making it amenable to analysis.Zero dip-normal potential gradients can be ob-

tained by neglecting crossow. Neglect of crossowwould also generally imply nonconservation of mass,however, which is a distinct disadvantage. It is moresatisfactory to regard VE as the result of a limitingprocess in which one lets dip-normal permeabilitiesapproach innity.77 VE is then characterized by noresistance to crossow. With innite permeabilities

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88 CHAPTER 4. SOLUTIONS

and zero potential gradients, dip-normal velocitiesmust be deduced directly from the continuity equa-tion. This turns out to be straightforward.77 Theresulting dip-normal velocities become nite, exceptwhere the front is dip-normal. With gravity eects in-cluded, innite dip-normal velocities also result wherethe front slope is discontinuous.The VE approximation has been used in the neg-

ligible gravity situation by several authors. A paperby Zapata and Lake63 combines VE with Buckley-Leverett type analysis in each layer, in a two-layersystem. The resulting mathematical structure is oneof two coupled Buckley-Leverett problems.The simpler piston-like displacement problem of in-

terest here has been analyzed by e.g., Hiatt,78 War-ren and Cosgrove,79 Hearn,76 and El-Khatib.68 Formathematical convenience, all these authors make theassumption that layers are ordered according to theirintrinsic front velocity. In reality, of course, one canhardly expect reservoirs to be strictly ordered. Incontrast to the no crossow case, however, rearrange-ment of layers is now not permitted. To see this,consider the simple three-layer system in Fig. 4.21.It is easy to show80 that VE implies that the poten-

x*

q-3

q-2

q-1

q+

3

q+

2

q+

1

Figure 4.21: Ordered three-layer reservoir.

tial gradient ∂xΦ is constant with y and the same inboth phases. Assume for deniteness that the mo-bility ratio M is favorable, i.e., such that mobility islower to the left of the front. Then obviously ∂xΦ−must be larger in magnitude than ∂xΦ+ for mass con-servation to prevail, since it takes a larger pressuregradient to drive a low mobility uid to a given ve-locity. Subscripts − and + signify limiting values asx∗ is approached from the left and from the right, re-spectively. Individual layer ow rates qi, indicated inFig. 4.21 by bold arrows, experience discontinuities atx = x∗. The direction of the associated crossow isindicated by thin arrows. The crossow is seen to beconsistent with the postulated front, in the sense ofallowing the front topology to remain essentially un-changed. Front velocities vi can be computed from

v1 =q1+

φ1 ∆S1, (4.61)

v2 =q1− + q2

− − q1+

φ2 ∆S2, (4.62)

v3 =q − q1

− − q2−

φ3 ∆S3, (4.63)

and may be dierent in each layer. The totalowrate, q =

∑Ni=1 q

i, is constant with x. The abovetype of analysis has been carried out by several au-thors.68,76,78,79

Imagine, however, that the ordered reservoir hasbeen obtained by rearranging layers in the reservoirof real interest, which is assumed to have the highvelocity layer in the middle. The front congurationabove would then correspond to a front congurationin the real reservoir as illustrated in Fig. 4.22. The

x*

Figure 4.22: Unordered three-layer reservoir.

front now cannot retain its integrity. Oil must cross-ow out of the top layer at x∗, to pinch o the layer2 water tongue, thereby altering the front topology.We conclude from this example that rearrangementof layers inuences the displacement process in anessential manner. When computing e.g., productionproles, the true sequence of layers should be used.When M > 1, the common assumption of a front

which is dip-normal in each layer is not consistentwith other assumptions. It can be shown80 that theleading water tongue will develop a sharp-nosed tipin this case, as illustrated in Fig. 4.23.

j

x*

Figure 4.23: Sharp-nosed leading tip of displacementfront

It turns out that a complete analysis of displace-

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4.2. CROSS-SECTIONAL DISPLACEMENT 89

ment front behavior for a generally layered reservoiris a rather formidable task. Elements of such ananalysis have been given by Ekrann.80 Presently weshall consider only the relatively simple problem ofdetermining the velocity of the leading tip of the dis-placement front, which suces to estimate recoveryat breakthrough.Consider rst the M > 1 case, illustrated in

Fig. 4.23. At x = x∗, Darcy's law and continuityimplies

q = −∂xΦ∗N∑i=1

λio hi , (4.64)

andu∗w = −λjw ∂xΦ∗ . (4.65)

Here u∗w is the (along-dip) water Darcy velocity in thetip of the leading tongue, supposing the tongue tip tobe positioned in layer j.The front velocity v∗ of the leading tip becomes

v∗ =u∗w

φj ∆Sj=

q λjw

φj ∆Sj∑Ni=1 λ

io h

i. (4.66)

It appears reasonable that the tip will nd the po-sition where it obtains its maximum possible velocity,

vmax = maxj

[qM λjo

φj ∆Sj∑Ni=1 λ

io h

i

], (4.67)

where we again assume the mobility ratio M to beconstant.In the favorable mobility ratio case, the lead-

ing tongue will be blunt-nosed,80 as illustrated inFig. 4.24. The tongue may cover more than one layer.In this case, the potential gradient behind the front

x*

h2

h1

H

Figure 4.24: Blunt-nosed leading tip of displacementfront.

must be deduced from

q = −∂xΦ∗− Λ(h1, h2) , (4.68)

where

Λ(h1, h2)def=∫ h1

0

λo(y) dy +

∫ h2

h1

λw(y) dy +

∫ H

h2

λo(y) dy .

(4.69)

The front velocity of the leading tip becomes

v∗ =M∫ h2

h1λo(y) dy

Λ(h1, h2)· q∫ h2

h1φ(y) ∆S(y) dy

. (4.70)

If h1h2 spans more than a single layer, water cross-ow is implied by Eq. 4.70, such that the same frontvelocity is obtained in all layers in question.As in the unfavorable mobility ratio case, we as-

sume the leading tip to obtain its maximum possiblevelocity

vmax =

maxh1,h2

[M∫ h2

h1λo(y) dy

Λ(h1, h2)· q∫ h2

h1φ(y) ∆S(y) dy

].

(4.71)

It can be shown that h1 and h2 coincide with layerboundaries when the maximum has been obtained.The leading tip front velocity as given in Eqs. 4.71

or 4.67 is independent of the distance to wells. As-suming, as in the no crossow case, that there is nomobile water present initially, and assuming the lead-ing tip to instantly obtain its maximum velocity, weobtain

ER =vavr

vmax, (4.72)

where the average front velocity vavr is given by

vavr =q∑N

i=1 hi φi ∆Si

. (4.73)

Sensitivity to Mobility Ratio

We are now equipped to compare the consequences ofthe no crossow and the vertical equilibrium approx-imations. As an example, consider a ve-layer reser-voir with data in Table 4.1. Units are not given, sincethe absolute scale of layer thicknesses and mobilitiesis irrelevant. Porosities and endpoint saturations areassumed to be identical in all layers. Rearrangementof layers to conform to the numbering in Eq. 4.53 iseasily done. Layer 3 becomes the new layer 1.

Table 4.1: Data for example reservoir

Layer Thickn. Mobility5 10 14 10 23 5 102 10 11 10 5

Fig. 4.25 shows relative recovery at breakthroughversus mobility ratio for this example reservoir, ac-cording to the formulae given above. As can beseen, the VE approximation makes the process dra-matically more sensitive to the mobility ratio. Thisimportant fact has been observed by several au-thors.63,68,79,81 In the example reservoir, mobility

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90 CHAPTER 4. SOLUTIONS

1.0

0.8

0.6

0.4

0.2

0.010-1 100 101

ER VE

No

cross flow

M

Figure 4.25: Recovery at breakthrough in no cross-ow and VE approximations.

control would seem recommendable only if the VEapproximation is realistic. A reduction in mobilityratio from 1 to 0.4 approximately doubles the relativerecovery at breakthrough. If the no crossow approx-imation applies, this same reduction would increaserecovery at breakthrough by a factor of only about1.2. Keeping in mind that additives to increase vis-cosity may be costly, and also other drawbacks of mo-bility control (see Sec. 11.1 and 11.2), this would thenhardly seem an attractive proposition. A no crossowreservoir, on the other hand, would be an ideal candi-date for polymer gel treatment (see Sec. 11.3), wherethe purpose is to block o completely one layer. Ifcrossow between layers is absent, the needed selec-tive placement of gel is obtained by simply injectingonly into the layer to be treated. With crossow, thisstrategy is generally much more problematic.82

Mechanism of Viscous Crossow

The striking dierence between VE and no cross-ow reservoirs illustrated above has been explainedby many authors,63,68,79,81,83,84 who all observe thatviscous crossow is benecial ifM < 1, and detrimen-tal if M > 1. Thus, if crossow is prevented by im-permeable strata, sensitivity to variations in M mustdiminish.Fig. 4.26 illustrates these crossow eects. An ar-

bitrary displacement front (heavy curve) has beendrawn in a two-layer reservoir. Layers are equallythick, with the top layer having three times the per-meability of the bottom one. VE streamlines are dis-played, computed as indicated by Ekrann.77

In the favorable mobility ratio case, streamlines di-verge in the water tongue, thereby reducing the frontvelocity at the tip of the tongue. Water crossows outof, and oil crossows into the high permeability layer.Judging from the front velocities (not displayed), thefront will eventually become blunt-nosed (compareFig. 4.24) with the leading tongue occurring in thetop layer.The unfavorable mobility ratio case has the oppo-

site streamline pattern. Streamlines converge in thewater tongue to increase the tip front velocity. Water

(a) M = 0.5

(b) M = 2

Figure 4.26: VE crossow.

crossows into, and oil crossows out of the high per-meability layer. Judging again from the front veloc-ities (not displayed), the front tip would in this caseremain sharp-nosed, with a possible shift in its verti-cal position from that arbitrarily chosen in Fig. 4.26.Fig. 4.26 also illustrates some other features of the

viscous crossow mechanism. In the setting chosen,the driving force for crossow is the displacementfront itself. A one-phase case (or a unit mobility ratiodisplacement) would have no crossow. For a givenfront, crossow velocities increase with mobility ratiodeparture from unity. This eect is seen by com-paring Fig. 4.26 with Fig. 4.27, which has the same

Figure 4.27: VE crossow with M = 0.1

data, except that M = 0.1. Crossow is more dra-matic with the lower mobility ratio. Higher crossowvelocities manifest themselves as steeper streamlineslopes.Similarly, a steeper front implies larger crossow

velocities. This eect is visible in Fig. 4.26. Fora second illustration, compare Figs. 4.28 and 4.26.Fig. 4.28 has the same data as Fig. 4.26 (a), exceptthat the total front length has been halved. In the VE

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4.2. CROSS-SECTIONAL DISPLACEMENT 91

Figure 4.28: VE crossow with a short front.

approximation, in fact, for a front whose slopes areeverywhere zero or innite (as in Figs. 4.21 and 4.22),crossow would take place only at the exact positionswhere front slopes are innite, making crossow ve-locities innite there. This is an artifact, however.With real nite dip-normal permeabilities, crossowshould taper o from the steep parts of the front.Similarly, the exact parallel ow immediately outsidethe front regions is an artifact of the VE approxima-tion. In reality, one would expect a transition zone,with some crossow also outside the front regions.

4.2.2 Gravity-Dominated Displace-ment

In the remainder of the section we shall take into ac-count a positive density dierence, ∆ρ, between wa-ter and oil, and assume that the density dierence islarge enough, or the injection rate small enough, forthe displacement to be gravity-dominated in the senseof water monotonically underriding the oil, Fig. 4.29.Thus, the front height, h, will be supposed to be a

A

B

C

D

h(x)

H

dx

dh

_g

Figure 4.29: Gravity-dominated displacement front.

single-valued, decreasing function of the along-dip po-sition x.We shall refer to the displacement as gravity-stable

if it proceeds under circumstances allowing a sta-ble displacement front of nite length. With gravitytonguing, on the other hand, we shall mean a situa-tion where the front would continue to grow in lengthindenitely.

The vertical equilibrium approximation will beused throughout. The no crossow approximation isnot of much interest in this context, since it preventsthe otherwise essential gravity crossow between lay-ers. It should be noted, however, that even the nocrossow situation allows gravity to have some inu-ence: The hydrostatic pressure gradient will be dif-ferent in injection (water) and production (oil) wells,thus producing a well-to-well pressure drop whichvaries with depth. This variation is constant in time,and should be straightforward to include in the stan-dard Dykstra-Parsons analysis.

Dietz' Theory

Dietz' classical paper74 considered gravity-dominateddisplacement in homogeneous reservoirs. We shall re-peat the essentials of his analysis here.In Fig. 4.29, assume equality of oil and water pres-

sures at point A. Oil and water pressures at pointC can then be obtained by collecting the appropriateviscous and hydrostatic pressure losses in each phase.Equating pressures so obtained,

p(A) + gx ρo dx+ ∂x Φo dx+ gy ρo dh =

p(A) + gy ρw dh+ gx ρw dx+ ∂x Φw dx .

(4.74)

Note that dip-normal viscous pressure losses havebeen neglected. Thus Eq. 4.74 conforms to the VEapproximation. For a gravity-stable front, i.e., a frontmoving through the reservoir without changing itsshape, Dietz argued that (along-dip) uid velocitiesmust be equal in both phases. Thus, by Darcy's law

uw = −λw ∂xΦw = −λo ∂xΦo = uo . (4.75)

Eqs. 4.74 and 4.75 produce an expression for theslope, dh/dx, of a gravity-stable front,

dh

dx= − gx

gy+

u (1−M)

∆ρ gy λoM, (4.76)

where u is the uid velocity, u = uw = uo. Note thatthe slope is constant. Thus, a gravity-stable front ina homogeneous reservoir is rectilinear.A zero slope corresponds to an innitely long front.

A critical rate, qcrit = ucritH, for gravity stabilitycan then be dened as the rate which produces a zerofront slope, from Eq. 4.76

qcrit = HM λo gx ∆ρ

1−M. (4.77)

For downdip water injection (i.e., water injectedstructurally low), gx < 0, and qcrit becomes negativeif M < 1. Thus, a critical rate exists only for un-favorable mobility ratios M > 1. For M < 1, alldisplacements will be gravity stable. In fact, grav-ity is not necessary to obtain stability in this case atall, but inuences the slope of the stabilized front,making it nite for nite ow rates.

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92 CHAPTER 4. SOLUTIONS

Dietz also considered nonstable fronts, i.e., grav-ity tonguing, in which case the assumption of equalvelocities in the oil and water phases can no longerbe made. Retaining the approximation that individ-ual phase velocities do not vary with y, Dietz insteadassumed

uwuo

= M , (4.78)

i.e., he assumed equality of potential gradients. Letqw = huw be the water ow rate (ux), a function ofx. By mass conservation

∂xqw = −φ∆S ∂th = φ∆S v ∂xh , (4.79)

where v is the along-dip front velocity. Using Eq. 4.78it is easy to show that

qw =M hq

M h+H − h, (4.80)

and with Eq. 4.79

v = v(h) =q

φ∆S

M H

(M h+H − h)2. (4.81)

The density dierence ∆ρ does not enter this ex-pression. Thus, gravity appears to play no role inthe development of a gravity tongue. Note also thatthe front velocity v is a function of front height honly. Prediction of front evolution is straightforward,therefore. A given point on the front will move witha constant velocity determined by its dip-normal po-sition.Dietz actually recommended a somewhat dier-

ent deduction of Eq. 4.81: Substituting ∂xΦw =−qw/(hλw) and ∂xΦo = −(q − qw)/(H − h)λo intoEq. 4.74 and dierentiating with respect to x, oneobtains an equation containing qw and ∂xqw. Insert-ing Eqs. 4.80 and 4.79 into this equation, one recap-tures Eq. 4.81 except for an additional right-hand sideterm,

1

φ∆S

∆ρ gy λoM h (H − h)

M h+H − h∂2xh (∂xh)−1 . (4.82)

When the actual rate q is several times the criticalrate qcrit, Dietz argued, the front curvature ∂2

xh willbe small and the additional term can be neglected.

Ekrann's Extension to Stratied Reservoirs

Dietz' main results were

• Critical rate for gravity stability (Eq. 4.77)

• Shape of a gravity-stable front (Eq. 4.76)

• Front velocity of a nonstable front (Eq. 4.81)

and limited to homogeneous reservoirs.Using the VE approximation, Ekrann77 was able to

generalize Dietz' results also to cover stratied reser-voirs, i.e., reservoirs where parameters ∆S, φ, λw, λoare allowed to be arbitrary (not necessarily piecewiseconstant) functions of the dip-normal coordinate y.

Stratied reservoirs exhibit eects not encounteredin homogeneous ones, such as partial gravity stabilityand breakdown of gravity dominance. Results will bebriey outlined. Details can be found in Ekrann.77

Eq. 4.79 is still applicable in a stratied reservoir,but should be written

∂xqw = φ(h) ∆S(h) v ∂xh (4.83)

to emphasize that parameters should be evaluated aty = h. Eq. 4.80 for the water ux is too simplistic,however. Dening

Λo(h)def=

∫ H

h

λo(y) dy , (4.84)

Λw(h)def=

∫ h

0

λw(y) dy , (4.85)

Λ(h)def= Λw(h) + Λo(h) , (4.86)

and integrating Darcy's law dip-normally, we get

qw = −Λw(h) ∂xΦw , (4.87)

qo = −Λo(h) ∂xΦo , (4.88)

since individual potential gradients are constant withy in the VE approximation.77

The total ow rate q is constant with x by assump-tion, and

q = qw + qo . (4.89)

The four equations Eqs. 4.74, 4.87, 4.88, and 4.89have four unknowns: individual phase uxes and po-tential gradients. Solving for the water ux, one ob-tains

qw = Λw(h)q + Λo(h) ∆ρ (gx + gy ∂xh)

Λ(h). (4.90)

Eq. 4.90 gives the correct water ux in a stratiedreservoir, in the VE approximation, whether the frontis stable or not. Note that Eq. 4.80 is recaptured ifthe reservoir is homogeneous, and if ∆ρ is neglected.Eq. 4.75 is recaptured in a homogeneous reservoironly for a special front, namely that described byEq. 4.76.Let us now rst employ Eq. 4.90 to determine the

shape of a stable front, which has to move throughthe reservoir without changing its shape. The frontvelocity v must be constant with h. By mass conser-vation

v =q∫H

0φ∆S dy

. (4.91)

The front height h is by assumption a single-valuedfunction of x. This function can be inverted, i.e.,x = x(h). Integrating Eq. 4.83 between x = x(0) andx = x(h), and using Eqs.4.90 and 4.91, one obtainsafter rearrangement

dh

dx=

u

∆ρ gyF (h)− gx

gy, (4.92)

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4.2. CROSS-SECTIONAL DISPLACEMENT 93

where

F (h)def=

1

φ∆S·

[∫ h0φ∆S dy

Λw(h)−∫Hhφ∆S dy

Λo(h)

].

(4.93)An overbar indicates averaging over the full reservoirthickness, and u is the average Darcy velocity q/H.Eq. 4.92 is an equation for the slope of a gravity-

stable front. In a stratied reservoir, the slope willgenerally vary with h, giving a curvilinear front.For a homogeneous reservoir, the rectilinear front ofEq. 4.76 is recovered.As in the homogeneous case, for the stable front

to be of nite length, one requires the slope to beeverywhere nonzero (and negative). With downdipwater injection (gx < 0, ∆ρ > 0) and noting that gy <0, it can be seen from Eq. 4.92 that this requirementis always possible to satisfy, provided that the rateq is small enough. Thus, a critical rate qcrit can bededuced, below which gravity stability prevails,

qcrit = H ∆ρgx

minh F (h), (4.94)

supposing minF (h) < 0. Henceforth qcrit will be re-ferred to as the critical rate for onset of gravity tongu-ing.Eq. 4.77 is recovered for a homogeneous reservoir.

As in the homogeneous case, the displacement maysometimes be unconditionally stable. For downdipwater injection, this is the case if F (h) > 0 for all h.In contrast to the homogeneous case, such uncondi-tional stability may occur even with (slightly) unfa-vorable mobility ratios M , provided that the strati-cation is suciently favorable (permeabilities increas-ing upwards).If the displacement is unconditionally stable for

downdip injection, then even updip (gx > 0) waterinjection will be gravity-stable, provided that the in-jection rate exceeds a critical rate. This critical rate isalso given by the formula above. Note only that F (h)now is supposed to be positive throughout. Stabilitywith updip water injection was termed type II stabil-ity by Lake,3 who considered homogeneous reservoirs.Fig. 4.30 shows a gravity-stable front in the ex-

ample reservoir of Fig. 4.26 (M = 2), with a dip of10, and at 90% of the critical rate. Also shown arestreamlines. The vertical scale is exaggerated.

Figure 4.30: Gravity-stable front with streamlines.

Crossow takes place in the front region. This isin contrast to the homogeneous case, where it canbe shown that there is no crossow associated with agravity-stable front. The solution implied by Eq. 4.76is therefore exact. Generally, crossow takes placeand is governed by a combination of viscous and grav-itational forces. In the example, gravity eects arelarge enough to make water crossow out of the highpermeable layer, even though viscous forces in thiscase have the opposite eect.To complete the generalization of Dietz' results, we

need to develop a general expression for the frontvelocity v. This can be accomplished by insertingEq. 4.90 into Eq. 4.83. Carrying out the dierentia-tion, one obtains after rearrangement

v =1

φ∆S Λ2·

[ q (λw Λo + λo Λw)

+∆ρ gy Λw Λo Λ ∂2xh (∂xh)−1

+∆ρ (gx + gy ∂xh) (λw Λ2o − λo Λ2

w) ] ,

(4.95)

where it is understood that φ, ∆S, λw, and λo are tobe evaluated at y = h, and the mobility integrals arefunctions of h.The expression is rather complicated. All three

right-hand side terms will remain even for a homo-geneous reservoir. The rst term corresponds to theoriginal Dietz result, Eq. 4.81. The second term wasdeliberately neglected by Dietz (Eq. 4.82). The thirdterm does not appear in Dietz' development at all,because of his neglect of gravity when computing thewater ux (compare Eqs. 4.80 and 4.90).Van Daalen and van Domselaar85 derived Eq. 4.95

for a stratied reservoir, except that they neglectedthe dip-normal component gy of gravity. The resultthen simplies considerably, since v becomes a func-tion of front height h only, and not of front slope andfront curvature. Sheldon and Fayers86 and Fayers andMuggeridge87 have obtained the complete Eq. 4.95 fora homogeneous reservoir, through a somewhat dier-ent reasoning.For downdip water injection, if the critical rate

qcrit is exceeded, one expects gravity tonguing totake place. In a homogeneous reservoir, it is con-sistent to assume that the front slope ∂xh would thenasymptotically (i.e., as time approaches innity) ap-proach zero everywhere. To see this, note rst that∂2xh (∂xh)−1 = ∂h(∂xh). Thus, if ∂xh approaches zeroeverywhere, so does ∂2

xh (∂xh)−1. Neglecting termscontaining these quantities, v becomes a function ofh only. Then

x(h, t) = x(h, t0) + v(h) (t− t0) (4.96)

and

∂hx(t) = ∂hx(t0) + ∂hv (t− t0) (4.97)

provided derivatives exist. For homogeneous reser-voirs it can be shown that ∂hv is negative for all

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94 CHAPTER 4. SOLUTIONS

h (recall M > 1). By Eq. 4.97, ∂hx (always < 0)would then tend to innity with time. The frontslope ∂xh = (∂hx)−1 would then tend to zero, consis-tent with the original assumption of a negligible frontslope.In a stratied reservoir this reasoning does gener-

ally not apply. The neglect of ∂xh and ∂2xh (∂xh)−1

may well result in a velocity v(h) with ∂hv > 0in some interval in h, if the intrinsic front velocityincreases suciently rapidly with h, which in turnwould imply that the front slope would tend to in-crease with time. Thus, the front slope will not nec-essarily approach zero asymptotically, even if gravitytonguing takes place.It turns out77 that gravity tonguing in stratied

reservoirs may proceed with partial (local) gravitystability, in which case the asymptotic front consistsof one or more stable parts with nite front slopes.Each such interval of local stability moves with itsindividual velocity, but such that velocities decreaseupwards. Interspersed between the intervals of lo-cal stability, therefore, are parts of the front wherethe slope approaches zero. The total front length in-creases with time. Ekrann's theory77 allows determi-nation of the dip-normal position of each interval oflocal stability, as well as its velocity and front slope.Fig. 4.31 gives an example of such partial gravity-

stability, computed from the reservoir data given inTable 4.2. The vertical scale is strongly exaggerated.The gure displays stabilized front shapes in the in-

Figure 4.31: Intervals of local stability for examplereservoir.

tervals of local stability, and also indicates the frontvelocity of each interval. Imagine these intervals oflocal stability to be tied together through zero slopeparts of the front (making the full front innitelylong).The example is taken from Ekrann.88 Porosities

and endpoint saturations are constant. The oil vis-cosity is 1 cp. The reservoir is horizontal, withM = 2.

Fig. 4.32 refers to the same reservoir, except that ithas been tilted 10. The gure shows relative recov-ery at breakthrough ER vs. average Darcy velocity u.ER is computed from

ER =vavr

vtip, (4.98)

Table 4.2: Data for example reservoir

Layer Thickn. Oil(m) perm.(D)

10 10 1.9 10 5.8 10 1.7 10 2.6 10 5.5 10 5.4 10 0.53 10 3.2 10 10.1 10 1.

1.0

0.9

0.8

0.7

0.60.05 0.10 0.15 0.20 0.25

ER

_u (m/d)

Figure 4.32: Relative recovery at breakthrough vs.average Darcy velocity for example reservoir.

which essentially is a rewrite of Eq. 4.72; vtip is thefront velocity at h = 0, i.e., the asymptotic velocityof the leading tip of a gravity-dominated front. Asfor Eq. 4.72, Eq. 4.98 requires that no mobile wateris present initially, and that the asymptotic velocityis obtained instantly. Also, if the leading tip belongsto an interval of local stability, note that its actualshape is not taken into account. Eq. 4.98 is thereforestrictly correct only for innitely long reservoirs.Below a velocity of about 0.05 m/d, the Fig. 4.32

displacement is fully gravity stable, i.e., there is asingle interval of local stability which extends fromh = 0 to H. In the approximation chosen, this corre-sponds to a unit relative recovery at breakthrough.Above that velocity, gravity tonguing sets in, butstill such that one single interval of local stabilitycovers the lower nine layers. Thus, gravity tongu-ing implies merely that the top layer is badly swept,without large consequences for the recovery at break-through. Above about 0.16 m/d, the front starts tobreak into several intervals of local stability. Aboveabout 0.18 m/d, the lowermost interval encompassesonly the bottom two layers, and the front congura-tion is similar to the one depicted in Fig.4.31. Fromthis point on, ER starts to fall rapidly with the injec-tion rate.The development so far rests on vertical equilib-

rium, i.e., no resistance to crossow. In reality, thereis always some resistance. One could expect, there-fore, that the VE approximation will deteriorate as

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4.2. CROSS-SECTIONAL DISPLACEMENT 95

Figure 4.33: Onset of viscous tonguing.

dip-normal velocities increase relative to along-dip ve-locities, since this would increase the relative impor-tance of dip-normal potential gradients.Ekrann77 used such ideas to estimate when the

assumption of asymptotic gravity dominance wouldbreak down: For a (partly or fully) gravity-stablefront, determined in the VE approximation as de-scribed previously, he estimated dip-normal poten-tial gradients ∂yΦw and ∂yΦo by division of VE dip-normal velocities by the real (nite) dip-normal mo-bilities. Introducing these gradients into a generalizedversion of the pressure continuity equation, Eq. 4.74,and retaining the VE along-dip potential gradients,one can deduce an adjusted front slope ∂xh

′. If this

adjusted front tips over, i.e., if ∂xh′> 0 at some

point, then one would expect that gravity dominanceis no longer possible asymptotically. A viscoustongue would be expected to form internally in thereservoir, as illustrated in Fig. 4.33.For a front which is fully gravity-stable in the VE

approximation, the reasoning above gives the follow-ing inequality, to be satised for 0 ≤ h ≤ H, if tip-over is to be avoided:

kv(h)

kh(h)>

u

∆ρ gy

[u

∆ρ gyF (h)− gx

gy

]·[

φ(h) ∆S(h)

φ∆S

1

λo(h)− 1

λw(h)

+ F (h)

].

(4.99)

Here kv/kh is the endpoint vertical to horizontal per-meability ratio, assumed equal for both uids, butallowed to vary with y.The inequality can always be satised, for nonzero

kv/kh, provided that the average velocity u is smallenough. The inequality denes a second critical rate,qvisc, above which gravity dominance is asymptoti-cally impossible. We shall refer to qvisc as the criticalrate for onset of viscous tonguing.The accuracy of Eq. 4.99 has not been evaluated

theoretically, but some experimental and numericalevidence of its validity exists.8991

If F (h) > 0 for all h, downdip water injection isgravity-stable for all injection rates. It is then easyto see from Eq. 4.92 that the front slope will increasein magnitude as q increases. A steeper front implies abetter vertical sweep eciency. If q > qvisc, however,Eq. 4.92 is no longer applicable, because of onset of

viscous tonguing. The theory does then not allowestimation of front behavior, unless the rate is largeenough to allow gravity to be neglected entirely. Ifdisplacement with negligible gravity is unstable, i.e.,if vmax/vavr > 1 (compare Eq. 4.72), then there mustexist an optimal injection rate, for which the verticalsweep eciency is at its maximum. Ekrann77 conjec-tured that qvisc is close to this optimal rate. Thereexists91 some numerical evidence to support this con-jecture.A second application of the inequality in Eq. 4.99

arises in connection with VE pseudofunctions forreservoir simulation.66,73,75 VE pseudos require ver-tical equilibrium and gravity dominance. The in-equality addresses directly the question of gravitydominance.

Some Two-Layer Reservoirs

In this section we illustrate the theory with resultsfor some model two-layer reservoirs. The followingvariables are computed:

1. Critical rate qcrit for onset of gravity tonguing,by Eq. 4.94

2. Length l = x(h=0) − x(h=H) of a gravity-stablefront, by numerical integration of Eq. 4.92

3. Asymptotic velocity vtip of the leading tip of agravity tongue, Eq. 4.95

4. Critical rate qvisc for onset of viscous tonguing,Eq. 4.99

Item 3 above can be computed from Eq. 4.95 afterinserting ∂xh = 0 = ∂2

xh (∂xh)−1 if full or partial sta-bility does not prevail at h=0. Item 4 can be deducedfrom Eq. 4.99 in the case of full gravity stability. Withpartial stability, the computation of these two itemsrequires more information than given in the previousdiscussion.77

We consider three dierent two-layer geometries;one in which the top layer has 10% of the total reser-voir thickness, one in which the two layers are equallythick, and one in which the top layer has 90% of to-tal reservoir thickness. In all cases φ and ∆S areassumed to be constant with y.As will be shown, one can dene dimensionless ver-

sions of the four variables in question. In horizontalreservoirs, and for a given layer geometry, these di-mensionless variables are functions of two parame-ters only, the top-to-bottom layer mobility (perme-ability) ratio, λ2

o/λ1o, and the water-oil mobility ratio

M . For horizontal reservoirs, therefore, we displaycontour plots of the dimensionless variables, varyingM and λ2

o/λ1o. In dipping reservoirs, the variables,

except for qcrit, become functions of more than thesetwo parameters.The dimensionless critical average velocity, ucrit,

for onset of gravity tonguing is

ucritdef=

qcrit

∆ρ |gx|λoH=

ucrit

∆ρ |gx|λo. (4.100)

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96 CHAPTER 4. SOLUTIONS

For a given layer geometry, and with φ∆S constant,it can be seen from Eqs. 4.93 and 4.94 that ucrit

will be a function of M and λ2o/λ

1o only. Figs. 4.34

through 4.36 display contour plots of ucrit vs. theseparameters. Above the dotted curve in these gures,downdip water injection is predicted to be gravity-stable only if the average velocity u is less than ucrit.Below the dotted curve, downdip water injection willnever experience gravity tonguing. Updip water in-jection results in gravity-tonguing, however, unlessthe average velocity is larger than ucrit.

0.1

1

10

100

0.1 1 10

0.1 0.2 0.4 0.7 1 1.1

1.21.4

23 5

10

31.4

0.70.4

0.2

M ~ucrit =

λ o2

/ λ o1

Figure 4.34: Dimensionless critical velocity ucrit forgravity tonguing in a two-layer reservoir. Top layer10% of total height.

0.1

1

10

100

0.1 1 10

0.1 0.2 0.40.7 1

1.1

1.21.42

3 5 10

1.40.7

0.40.2

M~ucrit =

λ o2

/ λ o1

Figure 4.35: Dimensionless critical velocity ucrit forgravity tonguing in a two-layer reservoir. Top layer50% of total height.

The dimensional variables are obtained by inver-sion of Eq. 4.100:

qcrit = H ucrit = ucritH ∆ρ |gx|λo . (4.101)

The length l of a gravity-stable front can be deter-mined from Eq. 4.92 by integration. For a horizontalreservoir gx = 0 and l becomes inversely proportionalto the dimensionless average Darcy velocity,

udef=

u

∆ρ g λo. (4.102)

0.1

1

10

100

0.1 1 10

0.4 0.7

1 1.1

1.2

1.4

23 5 10

1.40.7

0.40.2

M ~ucrit =

λ o2

/ λ o1

Figure 4.36: Dimensionless critical velocity ucrit forgravity tonguing a in two-layer reservoir. Top layer90% of total height.

Note that the dip angle ϑ = arcsin(|gx|/g) does notenter into this denition, in contrast to what was thecase in Eq. 4.100.From Eq. 4.92, the dimensionless front length

ldef=

l

Hu (4.103)

becomes a function only of M and λ2o/λ

1o for a hori-

zontal reservoir with given layer geometry. Figs. 4.37through 4.39 display contour plots of l. Note thatthe front length increases as the limit of stability isapproached (compare Figs. 4.34 through 4.36).

0.1

1

10

0.1 1 10

0.1

0.3

13

10 30

M

~l =

λ o2

/ λ o1

Figure 4.37: Dimensionless length l of gravity-stablefront in a horizontal two-layer reservoir. Top layer10% of total height.

The dimensional front length l is obtained by in-version of Eq. 4.103:

l =l H

u. (4.104)

The length l of the gravity-stable front gives a roughindication of vertical sweep eciency. The shorter thefront, the better the sweep eciency. By Eqs. 4.102and 4.104, l will be inversely proportional to the in-jection rate. This is true only in horizontal reservoirs.In a dipping reservoir, Eq. 4.92 shows that the front

length l will be a function of the dip angle ϑ. Also, l

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4.2. CROSS-SECTIONAL DISPLACEMENT 97

0.1

1

10

0.1 1 10

0.1

0.3

131030

M

~l =

λ o2

/ λ o1

Figure 4.38: Dimensionless length l of gravity-stablefront in a horizontal two-layer reservoir. Top layer50% of total height.

0.1

1

10

0.1 1 10

0.1

0.3

1310

30

M

~l =

λ o2

/ λ o1

Figure 4.39: Dimensionless length l of gravity-stablefront in a horizontal two-layer reservoir. Top layer90% of total height.

will no longer be inversely proportional to u. Even fora two-layer reservoir with given geometry, therefore, lwill generally be a function ofM , λ2

o/λ1o, ϑ, and u. We

shall illustrate the dependence of l on u for downdipand updip water injection in one example reservoir,Figs. 4.40 and 4.41. The top layer has 10% of totalthickness, and the top to bottom layer permeabilityratio λ2

o/λ1o is 10. The reservoir dips 10

.For a horizontal reservoir, l would be constant with

u. The horizontal result is recaptured in Figs. 4.40and 4.41 as u approaches innity, when M is smallenough to allow stability in horizontal reservoirs.Note also that updip water injection, if stable, alwaysproduces a longer front than does downdip injectionfor the same velocity u.The relative leading tip asymptotic velocity,

vtipdef=

vtip

vavr, (4.105)

is the inverse of ER as dened in Eq. 4.98. It can beshown that vtip becomes solely a function of mobilityratio and stratication in horizontal reservoirs.77

Figs. 4.42 through 4.44 give contour plots of vtip

for the three example two-layer geometries, again as-suming reservoirs to be horizontal.

0.1

1

10

0.1 1 10

0.1

0.3

1

1.5

22.5310

~l

~u

M =

Figure 4.40: Dimensionless length l of gravity-stablefront vs. dimensionless average Darcy velocity u fora dipping two-layer reservoir. See text for reservoirdata. Downdip water injection.

0.1

1

10

0.1 1 10

0.1

0.3

1

1.5

~l

~u

M =

Figure 4.41: Dimensionless length l of gravity-stablefront vs. dimensionless average Darcy velocity u fora dipping two-layer reservoir. See text for reservoirdata. Updip water injection.

The relative tip velocity vtip is unity if conditionsfor stability are satised, compare Figs. 4.34 through4.36.For a dipping reservoir, vtip in addition becomes

a function of the dip angle ϑ and the dimensionlessDarcy velocity u. To illustrate the dependence on u,we choose the same example reservoir as in Figs. 4.40and 4.41. Figs. 4.45 and 4.46 show relative tip ve-locity vtip vs. dimensionless Darcy velocity u, fordowndip and updip water injection, respectively.Again one can read from these gures that downdip

and updip injection approach the same solution,namely that corresponding to a horizontal reservoir,as u approaches innity. However, the limiting so-lution is not identical to the gravity-free solution inEq. 4.67, because we force the front to be gravity-dominated, thus putting its leading tip at the bottomboundary. With the data given, Eq. 4.67 puts theleading tip in the top layer. When u approaches in-nity, gravity forces become negligible. Thus, the as-sumption of gravity dominance must eventually breakdown. Eq. 4.67 is the correct limiting solution.

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98 CHAPTER 4. SOLUTIONS

0.1

1

10

100

0.1 1 10

1

1.21.5

2

4

10

50M

λ o2

/ λ o1

~υtip =

Figure 4.42: Relative asymptotic tip velocity vtip

of gravity tongue in a horizontal two-layer reservoir.Top layer 10% of total height.

0.1

1

10

100

0.1 1 10

1

1.2

1.52410

50M

λ o2

/ λ o1

~υtip =

Figure 4.43: Relative asymptotic tip velocity vtip

of gravity tongue in a horizontal two-layer reservoir.Top layer 50% of total height.

The validity of the gravity dominance assumptioncan be judged by the critical rate qvisc for onset ofviscous tonguing. Eq. 4.99 suggests the denition ofa dimensionless critical velocity, uvisc, for onset ofviscous tonguing,

uviscdef=

qvisc

H ∆ρ g λo

√khkv

, (4.106)

assuming kh/kv to be constant with y. For horizontalreservoirs it can be shown that uvisc will be a functiononly of the mobility ratio and of the stratication.77

Figs. 4.47 through 4.49 display contour plots of uvisc

for our example two-layer reservoirs, varying M andλ2o/λ

1o.

Dimensional variables are obtained from uvisc byinversion of Eq. 4.106:

qvisc = H uvisc = uviscH ∆ρ g λo

√kvkh

. (4.107)

A high kv/kh is conducive to gravity dominance.In dipping reservoirs, uvisc will no longer be inde-

pendent of kv/kh. To illustrate this, consider againthe dipping two-layer example reservoir; top layer has

0.1

1

10

100

0.1 1 10

11.2

1.42

41050

M

λ o2

/ λ o1

~υtip =

Figure 4.44: Relative asymptotic tip velocity vtip

of gravity tongue in a horizontal two-layer reservoir.Top layer 90% of total height.

~υtip

1

10

100

0.1 1 10

100

10

5

3

~u

M =

Figure 4.45: Relative asymptotic tip velocity vtip vs.dimensionless average Darcy velocity u for a dip-ping two-layer reservoir. See text for reservoir data.Downdip water injection.

10% of total thickness and ten times the permeabil-ity of the bottom layer, and the dip is 10. Figs. 4.50and 4.51 show uvisc vs. kv/kh for downdip and updipwater injection, respectively.For downdip injection, uvisc is seen to increase sig-

nicantly with kv/kh, Fig. 4.50. From Eq. 4.107,the dimensional critical rate qvisc will then increasestrongly with kv/kh. Also, from the gure, uvisc in-creases with M because a large M decreases the ten-dency for local stability, see Figs. 4.34 and 4.45.For updip water injection the picture is dierent.

For the larger M 's, uvisc decreases with increasingkv/kh over the whole range. The M=0.3 and M=1curves show a maximum. Comparing Figs. 4.50 and4.51, uvisc is always larger for updip injection. Incases where viscous tonguing is the major problem,these gures therefore suggest that updip water in-jection may be advantageous over downdip injection.

Sample Problems

We shall illustrate application of the previous resultsthrough some sample problems.

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4.2. CROSS-SECTIONAL DISPLACEMENT 99

1

10

100

0.1 1 10

1 1.5

3

5

10

100

~u

M =

~υtip

Figure 4.46: Relative asymptotic tip velocity vtip vs.dimensionless average Darcy velocity u for a dippingtwo-layer reservoir. See text for reservoir data. Updipwater injection.

0.1

1

10

1

0.1

10

0.2

0.30.50.7

12

357102030

M

λ o2

/ λ o1

~uvisc =

Figure 4.47: Dimensionless critical velocity uvisc foronset of viscous tonguing in a horizontal two-layerreservoir. Top layer 10% of total height.

Problem A. Given a two-layer reservoir, or a reser-voir approximated with such a geometry, which is tobe waterooded. The following data apply:

Total reservoir thickness 50 mTop layer thickness 5 mDip 0

Top layer permeability 5 darcyBottom layer permeability 0.5 darcyVertical to horizontalpermeability ratio kv/kh 0.1Porosity 0.25Initial oil saturation 0.75Residual (after waterooding) oil saturation 0.35Endpoint oil relativepermeability 0.8Oil viscosity 0.75 cpOil density 800 kg/m3

Endpoint water relativepermeability 0.3Water viscosity 0.375 cpWater density 1000 kg/m3

Acceleration of gravity 9.8066 m/s2

0.1

1

10

100

1 10

0.10.20.3

0.5

0.7

1

2357

1020

30M

λ o2

/ λ o1

~uvisc =

Figure 4.48: Dimensionless critical velocity uvisc foronset of viscous tonguing in a horizontal two-layerreservoir. Top layer 50% of total height.

0.1

1

10

100

1 10

0.10.2

0.30.5

0.71

235

710

20

30

M

λ o2

/ λ o1

~uvisc =

Figure 4.49: Dimensionless critical velocity uvisc foronset of viscous tonguing in a horizontal two-layerreservoir. Top layer 90% of total height.

Consider cross-sectional displacement, with the dis-tance between injector and producer being 1000 m.Assume that no mobile water is present initially inthe cross-section.

1. Show that gravity tonguing will not occur

2. Determine the critical rate qvisc for onset of vis-cous tonguing

3. Determine the length of the gravity-stable frontat an injection rate of 90% of qvisc

4. Estimate relative recovery at breakthrough atthis rate

5. Estimate relative recovery at breakthrough whengravity can be neglected, both under VE and nocrossow assumptions

Problem B. Consider the same reservoir as inproblem A. The maximum obtainable recovery by wa-ter ooding is 0.4/0.75, or 53% of OOIP. Adding sur-factants to the injection water reduces the residual oilsaturation to 10%, and increases the endpoint waterrelative permeability to 0.9. Water viscosity is notaltered. For this new set of conditions

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100 CHAPTER 4. SOLUTIONS

0.1

1

10

10-4 0.001

0.3

0.01 0.1 1

100

1

3

10

M =

~uvisc

k υ / k h

Figure 4.50: Dimensionless critical velocity uvisc foronset of viscous tonguing vs. vertical to horizontalpermeability ratio kv/kh. Downdip water injectionin two-layer reservoir. See text for reservoir data.

0.1

1

10

10-4 0.001 0.01 0.1 1

1

2

3

0.3M =

k υ / k h

~uvisc

Figure 4.51: Dimensionless critical velocity uvisc foronset of viscous tonguing vs. vertical to horizontalpermeability ratio kv/kh. Updip water injection intwo-layer reservoir. See text for reservoir data.

1. Show that gravity tonguing will occur

2. Determine the critical rate qvisc for onset of vis-cous tonguing

3. Determine the relative asymptotic tip velocity ofthe gravity tongue, at an injection rate of 90%of qvisc

4. Estimate relative recovery at breakthrough atthis rate

5. Estimate relative recovery at breakthrough whengravity can be neglected, both under VE and nocrossow assumptions

Problem C. Consider a reservoir similar to the onegiven in Problem A, except that it dips 10. Theendpoint water relative permeability is now 0.4, andkv/kh=0.01. For downdip and updip water injection

1. Determine the critical rate qvisc for onset of vis-cous tonguing

2. Let the injection rate equal 0.9 qvisc. Determinethe length of a gravity-stable front or the rela-tive tip velocity vtip, according to whether thedisplacement is gravity-stable

Solution, Problem A. From the data given:

M = 0.3/0.375 cp0.8/0.75 cp = 0.75

∆ρ = 200 kg/m3

λo = 0.95 d· 0.8 / 0.75 cp = 10−9 m2/Pa sφ∆S = 0.25 · 0.4 = 0.1λ2o/λ

1o = 10

1. Fig. 4.34 is applicable in this case. With M =0.75 and λ2

o/λ1o = 10 one is safely below the dot-

ted curve, and horizontal (or downdip) water in-jection will never experience gravity tonguing.

2. From Fig. 4.47 one reads uvisc = 0.45. Withthe data above, and with H = 50m and√kv/kh = 0.316, Eq. 4.107 then delivers qvisc

= 1.39 10−5 m2/s = 1.2m2/d (per m of reservoirwidth). This corresponds to an average Darcyvelocity of 2.4 cm/d.

3. With λ2o/λ

1o = 10 andM = 0.75, one reads l = 0.6

from Fig. 4.37. With u = 0.9· 1.39 10−5 m2 s−1

/ 50m, Eq. 4.102 gives u = 0.128. By Eq. 4.104one then obtains l = 234m.

4. Assume that the front shape stabilizes prior tobreakthrough in the producer. For a rough cal-culation, assume the stabilized front to be linear.Relative recovery ER at breakthrough then be-comes (1000 − 0.5 · 234)/1000 = 0.88, i.e., 88%of mobile oil is recovered at breakthrough of wa-ter. Note by comparison with Fig. 4.30 that theapproximation of a linear front is probably pes-simistic.

5. The no crossow case is covered by Eqs. 4.53,4.59, and 4.60. The layers are ordered after in-creasing intrinsic front velocity, which requires aslight modication of superscripting in the equa-tions. The ratio F 1 between intrinsic front veloc-ities in layers 1 and 2 becomes 0.5/5 = 0.1, anda front position at breakthrough x1/L = 0.1145.Eq. 4.53 then gives ER = 20.3 %.

The VE case is covered by Eqs. 4.71 and 4.72.The maximum is obtained in Eq. 4.71 whenh1 = 45m and h2 = 50m. Then vmax =4.48 q/H φ∆S, from which ER = 0.223, i.e.,22.3 % relative recovery at breakthrough.

Solution, Problem B The changes from the pre-vious problem are M = 2.25, and φ∆S = 0.1625.

1. With λ2o/λ

1o = 10 and M = 2.25, one is slightly

above the dotted curve in Fig. 4.34. Grav-ity tonguing cannot then be prevented. For ahorizontal reservoir one has gx = 0, which byEq. 4.101 implies qcrit = 0.

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4.2. CROSS-SECTIONAL DISPLACEMENT 101

2. From Fig. 4.47, uvisc is 0.95, and from Eq. 4.107qvisc = 2.94 ·10−5 m2/s = 2.54m2/d, correspond-ing to an average Darcy velocity of 5 cm/d.

3. From Fig. 4.42, vtip = 1.2, independent of theinjection rate since the reservoir is horizontal.Since q = 0.9 qvisc, gravity dominance prevails.The absolute tip velocity is found by multiply-ing vtip by q/(H φ∆S).

4. Relative recovery ER at breakthrough can be es-timated as the inverse of vtip, i.e., ER = 83 %.This high recovery occurs even with gravitytonguing. The present way of estimating recov-ery is inconsistent with that used in the previousproblem, since the actual shape of the the locallystable front is now not taken into account. Withthe present method, recovery at breakthroughwith full gravity-stability would be 100%. Themethod therefore gives somewhat optimistic es-timates for short reservoirs.

5. The reasoning is as for the previous problem.The resulting recovery at breakthrough ER is16.6% in the no crossow case, and 8.4% in theVE case.

Remark. Gravity is seen to make a dramatic dier-ence in both problems. Gravity eects are generallyexpected to be important in high permeability/lightoil reservoirs, although not necessarily benecial.

Solution, Problem C. The altered water end-point relative permeability gives M = 1.

1. We now have a 10 dipping reservoir. Figs. 4.50and 4.51 apply. With kv/kh = 0.01, uvisc = 0.25(downdip injection) and uvisc = 0.98 (updip in-jection). The dimensional critical average Darcyvelocities uvisc become via Eq. 4.107 0.42 cm/dand 1.66 cm/d, respectively.

2. Comparing Eqs. 4.102 and 4.106, an injectionrate q = 0.9 qvisc corresponds to a dimension-less average velocity u = 0.9 uvisc

√kv/kh. For

downdip injection, u becomes 0.022, and 0.088for updip injection. u = 0.022 is outside therange in Fig. 4.40. Nevertheless, one can roughlyestimate l = 0.1. From Eq. 4.104, a dimensionalfront length l = 230m is found, correspondingroughly to 89% recovery at breakthrough.

For updip water injection, from Figs. 4.34 and4.41, u = 0.088 is too low to produce gravity sta-bility. The relative tip velocity vtip of a gravitytongue is about 1.4 from Fig. 4.46, correspond-ing to a relative recovery ER = 1/vtip at break-through of 71%.

Remark. Recovery at breakthrough is higher fordowndip injection in the example above. Further-more, the result for updip injection is computed by a

method giving optimistic results, as discussed previ-ously, reinforcing that conclusion.Note, however, that the updip result is achieved

with a rate four times that of downdip injection.Early oil is more valuable than late oil, and the ten-dency is to operate at high rates. For downdip injec-tion, the higher rate is supercritical and would leadto viscous tonguing, and possibly to a signicantlyreduced recovery at breakthrough. The theory doessuggest, therefore, that updip water injection may beadvantageous to downdip injection in this examplereservoir.

4.2.3 Field Examples

From the methods discussed above, the Ekrann ex-tensions to the Dietz74 method have been applied todata from two North Sea sandstone reservoirs.The reservoirs dier greatly in the level of hetero-

geneity. Reservoir 1 is very homogeneous and highlypermeable. Reservoir 2 is heterogeneous with largevariations in permeability vs. depth. This is illus-trated in Fig. 4.52 which shows a plot of permeabilityvs. depth for the two reservoirs.

0

50

100

150

0 2000 4000 6000

Reservoir 1

0 2000 4000 6000

Reservoir 2

Rel

ativ

ede

pth,

m

Permeability, md

Figure 4.52: Permeability vs. depth for reservoir 1and 2 (depth is relative to the top of the reservoir).

For both reservoirs, gas and water injection havebeen evaluated. Both reservoirs have a relatively highdip angle, indicating that gravity may play an impor-tant role for the displacement. Reservoir data includ-ing petrophysical data are given in Tables 4.3 and 4.4.

The vertical communication is dierent for the tworeservoirs. Reservoir 1 is expected to have a highdegree of vertical communication with relatively higheective vertical permeability. Reservoir 2 has a verylimited vertical communication due to high quantitiesof shales between the sandstone sequences and calcitecemented zones within the sands.Due to the low eective permeability for reservoir

2, this sequence may perform as a series of isolated,

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102 CHAPTER 4. SOLUTIONS

Table 4.3: Summary of reservoir data

Property Reservoir 1 Reservoir 2Sorg 0.10 0.10Sorw 0.23 0.30µo, cp 0.43 0.33µw, cp 0.30 0.27µg, cp 0.022 0.028

krw(Sorw) 0.15 0.30krg(Sorg) 0.60 0.70kro(Swi) 1.00 0.84Mwo 0.215 0.437Mgo 11.73 9.82ρo 670.0 612.0ρw 1000.0 1000.0ρg 178.0 240.0

Dip, 7.0 13.0Bo 1.45 1.56W 1250.0 500.0H 57.8 152.7L 2000.0 2000.0

decoupled layers. This situation is assumed in themethod of Dykstra-Parsons52 described in Sec. 3.2above. Therefore, the Dykstra-Parsons method hasalso been applied for comparison for reservoir 2.

Results

Based on the data given in Tables 4.3 and 4.4, thecriteria for stability as discussed in Sec. 3.2 have beeninvestigated, including

1. critical rate for gravity tonguing (Eq. 4.94), andvertical position (height from the bottom of thereservoir) where the criterion is violated, and

2. critical rate for viscous tonguing (Eq. 4.89), andvertical position where this occurs.

For the two reservoirs, the results have been sum-marized in Table 4.5 for both water and gas injection.In this table, the rate is calculated as

q = uWH/Bo, (4.108)

from data in Table 4.3 and 4.4. The area normal tothe displacement, WH, is based on an average wellspacing in the elds. In reality, this area is dependenton the distance from the well. The local displacementvelocity will therefore be dierent close to the wells,which may cause local instabilities such as coning andcusping in these areas. Such eects are ignored in thisevaluation.

Reservoir 1

The critical rate for viscous tonguing for reservoir 1was predicted to be very high as shown in Table 4.5.Viscous tonguing is therefore not expected. As shownin Table 4.5, the displacement will always be gravity

stable with water injection. The recovery at break-through will then be a function of the stabilized shapeof the front, which again is a function of the rate. Theshape of the front has been calculated for dierentdisplacement rates as shown in Fig. 4.53.

50

40

30

20

10

00 500 1000 1500 2000

Along dip, m

Dip

norm

al,

m u = 0.2 m/du = 0.1 m/d

Figure 4.53: Front position at breakthrough for reser-voir 1, water injection.

With gas injection, the mobility ratio is higher, anda critical rate to gravity tonguing exists, in this casevery high, higher than the rates which are feasibledue to other constraints such as tubing performance.One may then conclude that the displacement willbe performed under segregated conditions, and sincethe shape of the front in linear, the method of Di-etz74 may be used to predict recovery. Again, theshape of the front has been calculated and is shownin Fig. 4.54.

50

40

30

20

10

00 500 1000 1500 2000

Along dip, m

Dip

norm

al,

m u = 0.3 m/du = 0.15 m/d

Figure 4.54: Front position at breakthrough for reser-voir 1, gas injection.

Reservoir 2

Also in the case of Reservoir 2, water injection isgravity stable. However, in this case the critical rateto viscous tonguing is low, and the onset of viscoustonguing is predicted at 20.7 m above the base of thereservoir, i.e., at the top of the three lowermost lay-ers. This is a natural point where override of the wa-ter will occur, since the three lowermost layers havelow permeability. Note also that with a lower verti-cal to horizontal permeability ratio, viscous tonguingis developed at a very low rate. Since the verticalpermeability is expected to be low in this reservoir, astable displacement is not expected, and the other ex-treme of no communication between layers is a possi-

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4.2. CROSS-SECTIONAL DISPLACEMENT 103

Table 4.4: Summary of reservoir data

Reservoir 1 ∗ Reservoir 2Layer h k φ Swi h k φ Swi NTG

m md frac. frac. m md frac. frac. frac.1 1.51 1500.0 0.20 0.14 8.0 378 0.18 0.19 0.532 2.11 4380.0 0.24 0.10 2.7 402 0.16 0.18 0.273 0.60 1516.0 0.22 0.14 9.2 136 0.20 0.21 0.624 0.75 2127.0 0.23 0.13 5.9 249 0.17 0.19 0.715 2.41 4065.0 0.25 0.10 6.4 441 0.20 0.18 0.926 2.26 1686.0 0.21 0.14 7.4 200 0.17 0.20 0.677 1.32 3636.0 0.27 0.11 15.8 13 0.14 0.26 0.058 0.74 11970.0 0.27 0.07 5.0 1579 0.20 0.17 0.809 5.29 4851.0 0.26 0.10 2.0 2640 0.22 0.16 0.7510 4.56 3482.0 0.26 0.11 4.0 930 0.20 0.17 0.8811 1.03 3703.0 0.24 0.11 8.1 4388 0.22 0.16 0.5312 0.59 2277.0 0.23 0.13 2.8 232 0.22 0.20 0.9313 3.38 3276.0 0.25 0.11 3.7 2543 0.21 0.16 1.0014 2.50 3000.0 0.27 0.11 9.6 662 0.19 0.18 0.9415 0.29 3600.0 0.27 0.11 5.6 401 0.20 0.19 0.6016 2.28 5244.0 0.27 0.10 6.6 16 0.16 0.26 0.2317 3.72 3483.0 0.26 0.11 4.0 267 0.21 0.19 0.6918 0.57 3300.0 0.21 0.11 10.1 47 0.19 0.23 0.9519 0.29 21.0 0.16 0.47 15.1 250 0.21 0.20 0.9820 2.57 4441.0 0.27 0.10 9.6 30 0.18 0.24 0.6321 0.86 2730.0 0.24 0.12 3.5 28 0.17 0.24 0.9322 5.15 5075.0 0.28 0.10 7.6 98 0.19 0.21 0.9023 1.72 1893.0 0.19 0.1324 1.43 3583.0 0.27 0.1125 2.55 3876.0 0.27 0.1126 0.56 3300.0 0.28 0.1127 1.41 779.0 0.23 0.1828 1.69 4551.0 0.27 0.1029 0.28 27.0 0.23 0.4430 0.28 1560.0 0.16 0.1431 0.28 7330.0 0.22 0.0932 0.56 0.6 0.17 0.7433 1.41 4557.0 0.23 0.1034 0.85 1.0 0.16 0.71

Total ∗∗ 57.8 (3670) (0.25) (0.13) 152.7 (583) (0.19) (0.21)

* Net-to-Gross ratio = 1.0 for Reservoir 1

** Numbers in parenthesis are average values

Table 4.5: Critical rates for reservoir 1 and 2, accord-ing to Sec. 3.2

Res. Drive kvkh

qcrit hcrit qvisc hvisc

Mech. Sm3/d m Sm3/d m1 W.I. 1.0 - - - -1 G.I. 1.0 23000 40.8 - -2 W.I. 0.1 - - 240 20.72 W.I. 0.01 - - 45 20.72a W.I. 0.1 5500 35.8 390 35.82a W.I. 0.01 5500 35.8 100 35.82 G.I. 0.1 470 20.7 3300 132.8

ble approximation. Ignoring the viscous tongues, thegravity stable front positions are shown in Fig. 4.55.

Since the viscous tongue developed at the top ofthe three low-permeability, lowermost layers, addi-

120

80

40

00 500 1000 1500 2000

Along dip, m

Dip

norm

al,

m u = 0.03 m/du = 0.05 m/du = 0.07 m/d

Figure 4.55: Front position at breakthrough for reser-voir 2, water injection, ignoring viscous tongues.

tional calculations were performed omitting these lay-ers. The resulting reservoir description (2a) gave ahigher critical rate for viscous tonguing, but still be-low the expected production rates. The conclusion is

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104 CHAPTER 4. SOLUTIONS

that viscous tonguing will develop.Gas injection with a vertical to horizontal perme-

ability ratio of 0.1 predicted a higher critical rate forviscous tonguing. The critical rate for gravity tongu-ing is, however, below the rates at which this reservoiris expected to produce. A gravity tongue is thereforeexpected to occur at 20.7 m from the base of thereservoir, which is at the bottom of layer 3. This toppart of the reservoir may be excluded at some point inthe production phase by reperforating the productionwell. Gas injection may therefore present an interest-ing alternative also for this reservoir, due to the highmicroscopic sweep eciency of the gas (Sorg = 0.1and Sorw = 0.3).A Darcy velocity of 0.05 m/d corresponding to a

production rate of 2400 Sm3/d was assumed in thiscase. Four intervals of local stability were predicted,all moving with dierent velocities. Assuming a sta-ble displacement, the front positions are shown inFig. 4.56. In this gure arrows indicate the relativefront velocities. The dierent velocities are shown inTable 4.6Relative velocities are indicated by arrows and the

average front velocity was 0.541 m/d.

Along dip

Dip

norm

al

Figure 4.56: Front position at breakthrough forReservoir 2, gas injection, neglecting viscous tongu-ing.

Table 4.6: The four intervals of local stability, cfr.Fig. 4.56

Int. Bottom Top Velocity(m) (m) (m/d)

4 0.0 20.7 0.1083 20.7 45.9 0.2302 45.9 56.5 0.3741 56.5 152.7 0.789

Recovery at Breakthrough

In the stable displacement cases, a rough estimateof the recovery at breakthrough may be obtained byintegrating the shapes of the fronts. For rates aboveuvisc, this value is only a very rough estimate, and willbe optimistic, since tongues will develop and breakthrough earlier. How crucial this is for the eld be-havior is dependent on the development of the frac-tional ow after breakthrough, and will inuence thedesign for water or gas handling capacity. The two

limiting cases of (1) stable front and (2) noncommu-nicating layers will span the range of actual cases.The recovery determined by this method gives the

recovery of the movable oil (1-Swi-Sor). To get therecovery of the hydrocarbon pore volume, this recov-ery factor must be multiplied by (1-Swi-Sor)/(1-Swi).

For Reservoir 1, gravity-dominated stable displace-ment is expected both for water and gas injection, andthe recoveries at breakthrough are summarized in Ta-ble 4.7. The distance between wells is assumed to be

Table 4.7: Vertical sweep eciencies at breakthrough

Res. Drive u q Recovery in % ofMech. m/d Sm3/d Mov. Oil STOOIP

1 W.I. 0.10 5000 94.9 69.81 W.I. 0.20 10000 96.7 71.11 G.I. 0.15 7500 83.5 73.91 G.I. 0.30 15000 71.3 63.12 W.I. 0.03 1500 94.9 58.82 W.I. 0.05 2400 95.9 59.52 W.I. 0.07 3400 96.5 59.92 G.I. 0.05 2400 68.6 59.92 W.I. Dykstra-Parsons 52.6 32.62 W.I. Hearn method 33.4 21.0

2000 meters in all cases.For Reservoir 2, the recoveries for water injection

in Table 4.7 are optimistic, since the assumption is astable front, while the rates are above the critical ratefor viscous tonguing. Therefore, a displacement cal-culation using the Dykstra-Parsons method was per-formed for this reservoir. Recovery at breakthroughis shown in Table 4.7. Hearn76 proposed a methodto calculate pseudo relative permeability curves forstratied reservoirs. This method, as outlined byDake,92 gives the pseudo relative permeabilities pre-sented in Fig. 4.57, and the resulting fractional owcurve presented in Fig. 4.58. The recovery at break-

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Water saturation

Rel

ativ

epe

rmea

bilit

y

WaterOil

Figure 4.57: Pseudo relative permeability curves,Reservoir 2, using the Hearn76 method.

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4.2. CROSS-SECTIONAL DISPLACEMENT 105

1.0

0.8

0.6

0.4

0.2

0.00.1 0.2 0.3 0.4 0.5 0.6 0.7

Water saturation

Frac

tiona

lfl

owof

wat

erfw

Figure 4.58: Fractional ow curve, Reservoir 2 usingpseudo relative permeability curves in Fig. 4.57.

through is found by drawing the tangent to the frac-tional ow curve, Fig 4.58, as suggested by Welge,76

and is given in Table 4.7. In these cases, gravity isneglected, and these results are therefore pessimistic.The range of the recoveries at breakthrough for thesecases shows that large uncertainties exist, and thatmore sophisticated methods such as reservoir simula-tion should be used.A rough estimate of recovery at breakthrough as-

suming gas injection is given in Sec. 3.2 as the ratiobetween the average front velocity and the front tipvelocity. This value is given in Table 4.7, and is onlyan estimate of the recovery at breakthrough. In thecase of Reservoir 2 it will be optimistic, since the rateis above the critical rate for gravity tonguing. How-ever, it gives an indication of the potential for gasinjection into a heterogeneous reservoir like this one,even when all compositional eects are ignored.Finally, it should be noted that the recovery fac-

tors given in Table 4.7 are representative of two-dimensional cross-sectional displacement. The actualreservoir recovery factor will be lower due to arealsweep eciency less than unity.

Nomenclature

A = matrix or adsorption of surfactanta = retention term, intercept of tie lineB = formation volume factor, Rm3/Sm3

b = a(η) +A(α(η))C = overall volume fraction

C ′(S) = dC/dScij = phase volume fraction of component i in

phase jD = retardation factor or discriminant

Dip = dip angle, degreesdf , dg = Jacobians of f and gER = relative recovery at breakthrough of wa-

ter, fraction of total mobile oil initiallypresent

F = fractional ux= function dened in Eq. 4.93, Pa·s/m2

F i = ratio between intrinsic front velocities inlayers i and 1

f = fractional owf , g = general functions in rst-order terms of

conservation lawsG.I. = gas injectiong = acceleration of gravity, m/s2

H = thickness of reservoir, mh = height of front (Fig. 4.29), m= da/dC

hi = thickness of layer i (Fig. 4.20), mhi = height of front, (Fig.4.24), mh′= height of front adjusted for nite kv, m

I = identity matrixK = partition coecientk = permeability, m2

L = length (of reservoir), ml = along-dip length of gravity-stable front, m

M = mobility ratio (water-oil)N = number of layers or unknownsn = numberq = total ow rate, m2/s= owrate, Sm3/d

r = rarefaction curveS = saturation (water)T = temperature, Kt = time, su = Darcy velocity, m/s~u = state vectoru = intermediate state= average Darcy velocity, m/s

v = along-dip front velocity, m/s or m/dvavr = average front velocity, m/svf = intrinsic front velocity, m/s

vmax = maximum front velocity, m/svtip = asymptotic front velocity of leading tip of

gravity tongue, m/sW = width, m

W.I. = water injectionw = width of streamtube, mx = along-dip coordinate, mxi = along-dip position of front in layer i

(Fig.4.20), my = dip-normal coordinate, mα = tie line (α(η))γ = curve in (t, x)-spaceη = slope of tie lineϑ = dip angle, radiansκ = volumetric heat capacity, J/m3 Kκ = [(1− φ)/φ]κΛ = mobility integral (Eqs. 4.69 and 4.84

through 4.86), m3/Pa·sλ = eigenvalue or mobility= mobility, λ = kh kr/µ, kr at endpoint sat-

uration, m2/Pa·sµ = viscosity, Pa·s or cpξ = free variable, dimensionlessρ = density, kg/m3

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106 CHAPTER 4. SOLUTIONS

σ = shock velocity, dimensionlessΦ = ow potential, Paφ = porosity

Subscripts

c = connatecrit = criticalg = gash = horizontal or along-dipi = component or initialj = phasen = nonwettingo = oilp = phaser = residual or relatives = stationary or solid

visc = critical for onset of viscous tonguingw = water or wettingx = x-componenty = y-component− = limiting value when approaching x∗ from

the left+ = limiting value when approaching x∗ from

the right

Superscripts

i, j = layer number, increasing upwardsL = injected (left) conditionR = injected (right) condition+ = downstream− = upstream∗ = along-dip position of leading tip= common tie-line point′ = specic value or derivative

Symbols

˜= dimensionless¯= averaged over full reservoir thickness

∂h,x,y = partial dierentiation with respect to h,x, or y.

∆ = dierence

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Heating by Hot Fluid Injection, Trans., AIME(1959) 216 31215.

[37] Wingard, J.S. and Orr, F.M. Jr.: An Analyti-cal Solution for Steam/ Oil/ Water Displace-ments, paper SPE 19667 presented at 1989SPE Annual Technical Conference and Exhibi-tion, San Antonio, Oct. 57.

[38] Barkve, T.: The Riemann Problem for a Non-Strictly Hyperbolic System Modelling Non-Isothermal, Two-Phase Flow in a Porous Me-dia, SIAM J. Appl. Math., (1989) 49, 784798

[39] Delshad, M. and Pope, G.A.: Comparisonsof Three-Phase Relative Permeability Models,Transport in Porous Media, (1989) 4, 5984.

[40] Aziz, K. and Settari, A.: Petroleum ReservoirSimulation, Appl. Sci. Publ. London (1979).

[41] Bell, J.B., Trangenstein, J.A., and Shubin,G.R: Conservation Laws of Mixed Type De-scribing Three-Phase Flow in a Porous Media,SIAM J. Appl. Math., (1986) 46, 100017.

[42] Holden, L., Holden, H., and Risebro, N.H.:Some Qualitative Poperties of 2×2 Systems ofConservation Laws of Mixed Type:,The IMAVolumes in Math. and its Appl., 27, Nonlin-ear Evolution Equations That Change Type,Springer-Verlag (1989).

[43] Azevedo, A.V. and Marchesin, D.: MultipleViscous Prole Riemann Solutions in MixedElliptic-Hyperbolic Models for Flow in PorousMedia, Institudo de Matematica Pura e Apli-cada, Rio de Janeiro, (1990). Preprint.

[44] Fayers, F.J. and Matthews, J.D.: Evaluationof Normalized Stone's Methods for EstimatingThree- Phase Relative Permeabilities, SPEJ,(April 1984) 22532; Trans., AIME (1984) 277.

[45] Marchesin, D. and Medeiros, H.B.: A Note onthe Stability of Eigenvalue Degeneacy in Non-linear Conservation Laws of Multiphase Flow,Contemp. Math. Amer. Math. Soc. (1989) 100,21524.

[46] Trangenstein, J.A.: Three-Pase Flow withGravity, Contemp. Math. Amer. Math. Soc.(1989) 100, 14757.

[47] Shearer, M.: Loss of Strict Hyperbolicity forthe Buckley-Leverett Equations of Three-PhaseFlow in a Porous Medium, Numerical Simu-lation in Oil Recovery, Springer-Verlag, New-York-Berlin-Heidelberg (1988) 26383.

[48] Delshad, Mojdeh; Delshad, Mohammad;Pope, G.A., and Lake, L.W.: Two- and Three-Phase Relative Permeabilities of Micellar Flu-ids, SPEFE (Sept. 1987) 32737.

[49] Aanonsen, S.I.: Application of Fractional-Flow Theory to 3-Phase, 1-Dimensional Sur-factant Flooding., presented at the 1989 JointIMA/SPE Europ. Conf. on the Math. of OilRec., Robinson College, Cambridge UniversityJuly 2527.

[50] Glimm, J., Lindquist, B., Marchesin, D., andMcBryan, O.: Front Tracking for Hyperbolic

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108 CHAPTER 4. SOLUTIONS

Systems, Adv. in Appl. Math. 2 (1981) 91119.[51] Braester, C.: Simultaneous Flow of Immisci-

ble Liquids Through Porous Fissured Media,SPEJ (Aug.1972) 297305.

[52] Dykstra, H. and Parsons, R.L.: The Predictionof Oil Recovery by Water Flood, SecondaryRecovery of Oil in the United States, 2nd ed.,API (1950) 160174.

[53] Koval, E.J.: Method for Predicting the Per-formance of Unstable Miscible Displacement inHeterogeneous Media", SPEJ (June 1963) 145154; Trans., AIME (1963) 3.

[54] Orr, F.M. Jr., Pande, K.K., and White, C.D.:Relative Permeability in Heterogeneous PorousMedia, Interfacial Phenomena in PetroleumRecovery, N.R. Morrow, (ed.), Marcel Dekker,New York 1990.

[55] Pande, K.K, Ramey, H.J., Brigham, W.E., andOrr, F.M. Jr.: Frontal Advance Theory forFlow in Heterogeneous Porous Media, paperSPE 16344 presented at the 1987 SPE Califor-nia Regional Meeting, Ventura, April.

[56] Hewett, T.A. and Behrens, R.A.: Scaling Lawsin Reservoir Simulation and Their Use in a Hy-brid Finite Dierence/Streamtube Approach toSimulating the Eects of Permeability Hetero-geneity, presented at the 2nd Intl. Conf. onReservoir Characterization, Dallas, June 1989.

[57] Pope, G.A., Lake, L.W., and Helerich, F.G.:Cation Exchange in Chemical Flooding. Part I:Basic Theory without Dispersion, SPJ (1978)41834; Trans., AIME 18.

[58] Hirasaki, G.J: Ion Exchange With Clays inthe Presence of Surfactant., SPEJ (April 1982)18192.

[59] Johansen, T. and Winther, R.: The RiemannProblem for Multicomponent Polymer Flood-ing, SIAM J. Math. Anal. (1989) 20, No. 4,91829.

[60] Rhee, H.K., Aris, R., and Amundson, N.R.:On the Theory of Multicomponent Chromatog-raphy, Phil. Trans. Royal Soc., London (1970).

[61] Bryant, S.L., Schechter, R.S., and Lake, L.W.:Interactions of Precipitation/DissolutionWaves and Ion Exchange in Flow ThroughPermeable Media, AIChE J., (May 1986) 32,75164.

[62] Aris, R. and Amundson, N.R.: MathematicalMethods in Chemical Engineering, 2, PrenticeHall, Englewood Clis (1973).

[63] Zapata, V.J. and Lake, L.W.: A Theoreti-cal Analysis of Viscous Crossow, paper SPE10111 presented at 1981 SPE Annual TechnicalConference and Exhibition, San Antonio, Oct.57.

[64] Pande, K.K. and Orr, F.M. Jr.: Interaction ofPhase Behavior, Reservoir Heterogeneity andCrossow in CO2 Floods, paper SPE 19668,presented at 1989 SPE Annual Technical Con-ference and Exhibition, San Antonio, Oct. 57.

[65] Marle, C.M.: Multiphase Flow in Porous Me-dia, Gulf Publishing, Houston (1981).

[66] Coats, K.H., Dempsey, J.R., and Henderson,J.H.: The Use of Vertical Equilibrium in Two-Dimensional Simulation of Three-DimensionalReservoir Performance, SPEJ (March 1971)6371.

[67] Stiles, W.E.: Use of Permeability Distributionin Water Flood Calculations, Trans., AIME(1949) 913.

[68] El-Khatib, N.: The Eect of Crossow onWaterooding of Stratied Reservoirs, SPEJ(April 1985) 291302.

[69] Reznik, A.A., Enick, R.M., and Panvelker,S.B.: An Analytical Extension of the Dykstra-Parsons Vertical Stratication Discrete Solu-tion to a Continous, Real-Time Basis, paperSPE 12065 presented at the 1983 SPE An-nual Technical Conference and Exhibition, SanFrancisco, Oct. 58.

[70] Kufus, H.B. and Lynch, E.S.: Linear FrontalDisplacement in Multilayer Sands, ProducersMonthly (Dec. 1959) 3235.

[71] Roberts, T.G.: A Permeability Block Methodof Calculating a Water Drive Recovery Factor,The Petroleum Engineer (Sept. 1959) B45B48.

[72] Simon, A.D. and Koederitz, L.F.: An Im-proved Method for the Determination ofPseudo-Relative Permeability Data for Strati-ed Systems, paper SPE 10975 presented atthe 1982 SPE Annual Technical Conference andExhibition, New Orleans, Sept. 2629.

[73] Coats, K.H., Nielsen, R.L., Terhune, M.H.,and Weber, A.G.: Simulation of Three-Dimensional, Two-Phase Flow in Oil and GasReservoirs, SPEJ (Dec. 1967) 377388.

[74] Dietz, D.N.: A Theoretical Approach to theProblem of Encroaching and Bypassing EdgeWater, Proc., Kon. Ned. Akad. Wetensch.,Amsterdam (1953) B56, 8392.

[75] Thomas, G.W.: An Extension of Pseudofunc-tion Concepts, paper SPE 12274 presentedat the 1983 SPE Reservoir Simulation Sympo-sium, San Francisco, Nov. 1518.

[76] Hearn, C.L.: Simulation of Stratied Wa-terooding by Pseudo Relative Permeabilitycurves, JPT (July 1971) 80513.

[77] Ekrann, S.: An Analysis of Gravity-SegregatedPiston-Like Displacement in Stratied Reser-voirs, paper SPE 18598, SPERE (Feb. 1992)1438.

[78] Hiatt, W.N.: Injected-Fluid Coverage ofMulti-Well Reservoirs with Permeability Strat-ication, Drill. & Prod. Pract. , API (1958)165, 165194.

[79] Warren, J.E. and Cosgrove, J.J.: Prediction ofWaterood Behaviour in a Stratied System,SPEJ (June 1964) 149157.

[80] Ekrann, S.: On the Asymptotic Behaviour ofSharp Displacement Fronts in Layered Reser-

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REFERENCES 109

voirs, with Negligible Gravity Eects, SPORreport 6/90, Rogaland Research, Stavanger(Nov. 1990).

[81] Goddin, C.S., Craig, F.F., Wilkes, J.O., andTek, M.R.: A Numerical Study of WateroodPerformance In a Stratied SystemWith Cross-ow, JPT (June 1966) 765771.

[82] Root, P.J. and Skiba, F.F.: Crossow Eectsduring an Idealized Displacement Process in aStratied Reservoir, SPEJ (Sept. 1965) 229238.

[83] Sorbie, K.S., Wat, R.M.S., Rowe, T., andCliord, P.J.: Core Floods in Well Charac-terised Heterogeneous Systems; Experimentaland Simulation Results, paper SPE 16275 pre-sented at the 1987 SPE International Sympo-sium on Oileld Chemistry, San Antonio, Feb.46.

[84] Sorbie, K.S., Sheb, M., Hosseini, A., and Wat,R.M.S.: Scaled Miscible Floods in LayeredBeadpacks Investigating Viscous Crossow, theEects of Gravity, and the Dynamics of ViscousSlug Breakdown, paper SPE 20520 presentedat the 1990 SPE Annual Technical Conferenceand Exhibition, New Orleans, Sept. 2326.

[85] van Daalen, F. and van Domselaar, H.R.:Water Drive in Inhomogeneous ReservoirsPermeability Variations Perpendicular to theLayer, SPEJ (June 1972) 211219.

[86] Sheldon, J.W. and Fayers, F.J.: The Motion ofan Interface Between Two Fluids in a SlightlyDipping Porous Medium, SPEJ (Sept. 1962)275282.

[87] Fayers, F.J. and Muggeridge, A.H.: Exten-sions to Dietz Theory and Behavior of GravityTongues in Slightly Tilted Reservoirs, paperSPE 18438 presented at the 1989 SPE Sympo-sium on Reservoir Simulation, Houston, Feb.68.

[88] Ekrann, S.: Gravity-Assisted Displacement inStratied Reservoirs, paper SPE 20315, sub-mitted to SPE Dec. 1989.

[89] Ingsøy, P. and Ekrann, S.: Laboratory Studiesof Miscible Displacement in Two-DimensionalLayered Reservoirs, paper presented at the 3rdInternational Symposium on EOR, Maracaibo,Feb. 1989.

[90] Ingsøy, P. and Skjæveland, S.M.: Experimen-tal Validation of a New Method for OptimizingMiscible Flooding of Stratied Reservoirs, pa-per SPE 20241 presented at the 1990 SPE/DOESymposium on EOR, Tulsa, April 2225.

[91] Hovland, F.: Et studie av gravitasjons- ogviskøse-krefters innvirkning på fortrengnings-prosessen i lagdelte reservoarstrukturer vedhjelp av en analytisk og numerisk modell, the-sis, Bergen U., Bergen (Jan. 1990).

[92] Dake, L.P.: Fundamentals of Reservoir Engi-neering, Elsevier Schientic Publishing Com-pany, London (1978).

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110 CHAPTER 4. SOLUTIONS

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Part II

Reservoir Description

111

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Chapter 5

Geological Description

5.1 Geological Models inReservoir Description

The geological model provides the rock-framework forany reservoir model and is the necessary basis for thesubsequent petrophysical model, for reservoir simula-tion and reservoir management. Weak or erroneousgeological models, whether resulting in unpredictablehigh-permeability zones, shale barriers, sealing orleaking faults, or mineralogical/diagenetic hetero-geneity, will have serious consequences for reservoireconomics, production rates, and initiation/selectionof recovery processes. Reservoir heterogeneity, withits basis in the geological model, is the Achilles' heelof improved recovery eorts.Sound sedimentological and architectural input

(distribution, geometries, and internal congurationof sandbodies and shales) from analogous, recent en-vironments and from ancient outcrops is often a pre-requisite for providing the geological model with pre-dictive power.

5.1.1 Information Databases and Out-crop Analogues

Information databases containing data on thick-nesses, lengths, and orientations of potential owunits and architectural elements of outcroppingreservoir" complexes have become increasingly usedas data sources for improved reservoir description.They represent a means of increasing our under-standing of the variability of reservoir architec-ture/heterogeneity for specic depositional environ-ments as well as for quantifying this understanding.On the Norwegian shelf, there have been major re-

cent eorts in this direction through, for example, theSPOR reservoir description activity and through theSAFARI project. The former focussed on data collec-tion of delta-plain architectural elements,1 whereasthe latter is an ongoing program on uvial andshallow marine depositional environments.2 Thesedatabases provide quantied input to geomathemati-cal modelling systems such as DESIREE, SESIMIRA,or FLUREMO, which, in turn, provide the descrip-tion basis for reservoir simulation, Fig. 5.1. Outcropanalogues and geological information databases arenow being used at a variety of levels as aids in reser-

A B C1 x x x2 x x x3 x x x

RCA

Data Collection

SAFARIdatabase

AnalysisFunction

OtherAnalysis

Tools

MathematicalGeologicalModelling

ReservoirDatabase

OilProductionSimulation2

2

2

2

44

3

3

Figure 5.1: Outline of the character of the SAFARIdatabase and its role in reservoir evaluation.3

voir description, for example:(1) to reduce uncertainty in well-to-well correlation

and to improve prediction of petrophysical parame-ters between and beyond wells(2) as specic or general input to geomathematical

models and to reservoir simulation, Fig. 5.1(3) combined with high-resolution permeameter

data, to construct ow-unit models for specic reser-voirs and elds4

(4) general application to a suite of elds wherethe same depositional environment is important, forexample the ongoing eorts into data collection onoutcrops of submarine fans and turbidite systems toimprove knowledge on Cretaceous and Tertiary clasticreservoirs along the continental margin of north-westEurope.The major objective of the analogue data collection

is to systematically improve the operators' knowledgeof shale- and sandbody geometry/variability in dif-ferent depositional environments and structural set-tings. This creates more informed, and realistic (withmore deterministic input) geological reservoir mod-els. The database objective is a long-term one, andshould be treated as such. In the short term, anumber of statistical relationships, or at worst rulesof thumb, between various architectural parametershave been deduced. Assuming there is a reasonablematch (with respect to process, environment, andtectonics) between the analogue and the reservoir inquestion, these relationships can be used to make pre-dictions, reduce uncertainty and to build models as

113

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114 CHAPTER 5. GEOLOGICAL DESCRIPTION

Abandonedchannelsequence

Activechannelsequence

Ox-bow lakeabandonedmeanderChannel Levée

Pointbar

Backswamp

5m

(a) Meandering river channel

Mud depositionin abandonedchannel

Sand depositionin activebraided channels

Abandonedchannel sequence

Activechannel sequence

2m

(b) Braided river channels

Figure 5.2: Heterogeneity and architecture for uvial sequences in space and time.5

noted above.

5.1.2 Depositional Environments andHeterogeneity

Conditions within the original environment of deposi-tion exert the dominant control on macroscopic het-erogeneity within what is now the reservoir. Thereare two main factors which determine the vertical andlateral distribution of shales and variably permeablesandstone within the sedimentary packages depositedin dierent sedimentary environments. These are thespatial variability of energy and processes intrinsicto any environment the external conditions aectingbase level, as described in Sec. 5.1.3 below.The lateral variation in water depth and in current

and wave energy, inherent in specic depositional set-tings, will cause mud, silts, and variably sorted sandsto accumulate in lateral continuity with each other.In an alluvial environment, for example, uvial chan-nel sands with good reservoir quality will accumulateside-by-side with time-equivalent silts and mud (onoodplains, levees and abandoned channels as illus-trated in Figs. 5.2a and 5.2b), with poor reservoirquality. In general, the channel belts will be rela-tively narrow, elongate in a paleo-downslope direc-tion, and will be partially or completely enveloped bypoorly permeable sediments.One of the main challenges in reservoir sequences

of this type is to predict, on the basis of limitedwell data, the extent, geometry and orientation of

the channel-belt sandbodies. The widths of sandychannels and of channel-belt complexes can some-times be estimated from the thicknesses of sandstoneunits read directly from well logs. The success of thisevaluation, however, is dependent on having infor-mation about the sinuosity or character of the chan-nels, see Figs. 5.2a and 5.2b. This is by no means asimple interpretive task,6 and is further complicatedby the fact that the interconnectedness of channelsand units, or of belts of channel sand, is dependentnot only on channel pattern but also on avulsion fre-quency of the rivers and on the rate of subsidence ofthe sedimentary basin.7

Fig. 5.3 shows a recent graph for the relationshipbetween channel depth (can be deduced from channelsand thickness on well logs) and channel belt width(to be predicted) for various types of ancient andmodern uvial deposits.8 The plot distinguishes be-tween straight, nonmigratory channels (case 1A), me-andering channels (cases 1B-2B), and laterally unre-stricted (braided) channels (case 3). Plots of this typeare less useful in successions where channel patternsare dicult to deduce, where uvial runo is ood-dominated and channels are ill-established or wheresubsidence or base-level rise is minimal compared tosediment supply.Some of the Triassic uvial sandbodies at certain

levels in the Snorre Field (western ank of the NorthViking Graben) provide examples of channel beltswhose width is dicult to predict from the plot inFig 5.3. This is because of ashy or ephemeral ow

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5.1. GEOLOGICAL MODELS IN RESERVOIR DESCRIPTION 115

100

10

1

10 100 1000 10000 100000Channel belt width w, m

Cha

nnel

dept

hh,

m

1A

1B

2A

2B

3

w=64.6h1.54

w=513h1.36

w=0.01h2.9

w=0.95h2.05

w=12.1h1.95

Figure 5.3: Channel depth vs. channel-belt widthrelationships for various types of modern and an-cient uvial channel deposits, data from 45 publishedsources.8

conditions and probable low subsidence rates in cer-tain intervals during the deposition of the Lunde For-mation.In contrast to the downslope-oriented channel-belt

sandstone described above, shoreline or near shoresands often accumulate in belts perpendicular to pa-leoslope (shoreline parallel), as shown in Fig. 5.4. Thebarrier shoreline and shoreface sands will often be

Open sea

Barrierisland cutby tidalchannel

Lagoon andtidal flatFluvial coastal

plain

Lagoon andtidal flatsands, shalesand coals

Barriers islandsupward-coarseningsand

Open Marine shale

Fluvial coastalplain, interbeddedflood plain shalesand coals andupward-finingchannel sands

Figure 5.4: Barrier island and shoreline sandbodiesare narrow but extensive (perpendicular to slope).They grade gradually to shales below, and areabruptly capped by shales.5

well sorted because of the action of vigorous currents,waves, and tides, but will become more poorly sortedin an oshore direction as the amount of mud in-creases, and in a landward direction as lower energy,

muddy lagoonal areas are approached. In addition,shoreline-perpendicular sandbodies such as tidal in-lets or tidal deltas, Fig. 5.4, can be expected to cutthrough the barrier sands, connecting the oshoreand lagoonal areas. Shoreline progradation out intothe basin will generally cause a large-scale trend ofupwards coarsening as well as an upwards increasingpermeability prole.The third main domain of sand accumulation (ex-

cluding deposition on deltas which will be discussed indetail below) is from turbidity current and mass owdeposition on submarine fans in deeper water areasbeyond the continental terraces and shelves, Fig. 5.5.Reservoirs in Cretaceous (e.g., Agat Field) or Ter-

Turbidite fan atvalley mouth

Pelagicmuds onbasin floor

Slumping and slidingof slope muds

Sediment transport by tractioncurrents on shelf

Sedimenttransportdown valleyby grain flow

Figure 5.5: Deep water sands deposited as turbiditefans beyond the continental shelf. Basin-oor andslope muds are closely associated.5

tiary (e.g., Balder Field) strata are of this type. Aswith uvial sand belts, turbidite units tend to be no-toriously lensoid and lobate, elongate in a downslopedirection. The quality of such sands, often from re-worked materials on the adjacent shelf area, is com-monly good, but there can be signicant heterogene-ity due to interngering pelagic mud, Fig. 5.5.

5.1.3 Base-Level Changes and Hetero-geneity

It is being increasingly recognized that vertical sand-shale heterogeneity, and in particular certain shalybarriers, can be caused by external base-level changes,mainly rises of sea level or increased rates of basinalsubsidence. These relative sea level changes causemarine ooding events to sweep landward, causingshale deposition (heterogeneity) across wide basinalareas rather than merely within certain subenviron-ments as discussed above. As a consequence, suchchanges tend to vertically conne or bound reservoirs,or to create strongly layered reservoirs. The shaly lay-ers thus produced tend to bound low-order sedimen-tary units termed parasequences,9 which themselvesvary in facies laterally from marine shales at the basi-nal end to continental deposits at the landward end,Fig. 5.6.

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116 CHAPTER 5. GEOLOGICAL DESCRIPTION

Coastal belt sands

Marine fine-grained sediments

Coastal plain sediments

Alluvial plain sediments

Shaley intervalsboundingparasequences

Overall regressive

Snorre

Embla

Statfjord

GullfaksVisund

Brage

Oseberg

HeimdalFrigg

Agat

Mjølner

Overall transgressive

Figure 5.6: The basinward-landward-basinward shift of sediments belts during a sea-level cycle. The sequenceis markedly subdivided into shale-bounded parasequences.9

These ideas on heterogeneity and shale barriers ona scale larger than depositional environments are em-bedded in the concept of sequence stratigraphy. Thisconcept also shows the linkage of the spectrum of de-positional environments within a relative cycle of sealevel change, as illustrated in Fig. 5.6.The permeability layering resultant from base-level

(sea level) changes within a shallow marine envi-ronment is further illustrated in Fig. 5.7, from apart of the reservoir section of the Troll Field, o

Forward-steppingparasequences

Backward-steppingparasequences

Aggradingparasequences

Forward-steppingparasequencesof earliercycle

A cycleof

sea-levelchange

1675 m

Gamma ray

PARASEQUENCE ARCHITECTURE& HETEROGENEITY

PARTS OF HEATHER & SOGNEFJORD FMS.

1650 m

Figure 5.7: Parasequences and shale-prone parase-quence boundaries illustrated on a gamma-ray logfrom a reservoir zone in part of the Troll eld.10

western Norway.10 Note the abrupt, at-lying sur-faces which occur at the base of repeated, reduced-permeability intervals, as emphasized by sharp in-creases in the gamma-ray log response. These sur-

faces are the same as those referred to as oodingsurfaces. They bound small-scale, upward-coarseningunits which probably originate from repeated, abruptrises in relative sea level during accumulation of reser-voir sand. Their signicance within the context ofreservoir description is that they predictively under-lie shale units which thicken basinwards and that theyare sometimes associated with the development ofcarbonate cement.11

5.2 Macroscopic Hetero-geneities

5.2.1 Depositional Heterogeneity

The two main depositional controls on heterogeneityhave been introduced and discussed above. The mainresults, in terms of heterogeneity, of the environmen-tal or process control include:

• locally variable (but often predictable) sand andshale geometries, usually at a maximum in themost proximal (uvial) and most distal (subma-rine fans) environments

• local and small-scale trends of permeability vari-ation such as upward decreasing trends in chan-nel lls (both uvial and deep-sea fans), in tidalat units or in abandoned lobe sequences ofvarious types; upward decreasing permeabilitytrends occur as the result of lobe or sandatprogradation in almost all environments

• enhanced negative permeability trends (decreas-ing upwards) due to the accumulation of mi-caceous minerals, and clay clasts especially inmouth bar and storm wave-reworked, mouth-barunits.

The main eects of the uctuations in relative sealevel (sequence stratigraphic control), in terms of het-erogeneity, include:

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5.2. MACROSCOPIC HETEROGENEITIES 117

• more extensive shale barriers or permeability re-ductions, best developed during intervals of highrates of sea-level rise, i.e., along parasequenceboundaries and particularly during intervals ofmaximum landward ooding middle reaches ofFig. 5.6

• extensive coals, developed especially in situa-tions landward of the lower and middle reachesof Fig. 5.6

• large-scale, upward-increasing permeability tre-nds caused by the ubiquitous coarsening upwardmotifs of coastal progradation.

The range of depositional environments representedin the main sandstone reservoirs of the North Seais illustrated in Fig. 5.6. These range from alluvialfan and uvial at the landward end of the spectrum,e.g., Embla and Snorre elds, through coastal allu-vial and deltaic plain, e.g., Statfjord Fm. in Snorre,Oseberg and Statfjord elds; Ness Fm. in Oseberg,Brent and Gullfaks elds, through deltaic/shorelineor near shore reservoirs, e.g., Rannoch/Etive Fms. inGullfaks eld; Mjølner sand in Mjølner eld; HuginFm. in Sleipner eld; Ula Fm. in Ula and Gyda elds,and out to submarine fan and turbidity reservoirs,e.g., Brae Fm. in T-block and Brae elds; Heimdaland Frigg Fms. in Heimdal and Frigg elds.As regards reservoir sands deposited in coastal

belts, and heterogeneity imposed by the uctuationsof relative sea level, it is important to distinguishbetween sands deposited in a backstepping, trans-gressive, as opposed to a forward-stepping regressivemode, as shown in Fig. 5.6.Details of macroscopic heterogeneity within one

class of reservoirs, the deltaic reservoirs which areamong the most important on the Norwegian Shelf,are presented below.

5.2.2 Geometry and Heterogeneity inDeltaic Reservoirs

Introduction

A delta is, as dened geologically, a deposit, partlysubaerial, built by a river into or against a perma-nent body of water.12 The delta can be classiedon the basis of relative inuence of uvial, wave andtidal processes dening six principal delta types,13,14

Fig. 5.8. The deltaic environment comprises two dis-tinct subenvironments; the delta plain and the deltafront, Fig. 5.9.The delta plain covers that part of the delta

which contains both active and abandoned distribu-tary channels separated by interchannel areas.16 Im-portant interchannel subenvironments are crevassesplays, interdistributary bays, oodplains, levees, andlakes (ponds).The delta plain may be further subdivided into an

upper and lower delta plain,17 the upper delta plainbeing distinguished as that part of the system land-ward of the zone of regular tidal or marine inuence.18

(a) Fluvial dominated with lowwave and tide energy.

(b) Fluvial dominated with lowwave energy.

(c) Intermediate wave energy,high tide and low littoral drift.

(d) Intermediate wave energy,low tidal range.

(e) High wave energy, low lit-toral drift.

(f) High wave energy, strong lit-toral drift.

Figure 5.8: Conseptual models of preserved deltaicsandbodies for varying wave, uvial and tidal condi-tions.14

The delta front is the area in which sediment-laidenuvial currents enter the basin and are dispersedwhilst interacting with basinal processes. Althoughseveral subenvironments can be dened on the deltafront, sands are generally associated to the distribu-tary mouth bar, although often reworked and redis-tributed by waves and tidal currents.On the Norwegian Shelf, deltaic sandstone reser-

voirs, as the Middle Jurassic Brent Group, constitutea very important and prolic reservoir type. Deltaicsandstones are important also in a large number ofbasins worldwide, for example within the Gulf CoastBasin of the United States deltaic sandstones consti-tute up to 60 to 70% of oil and gas reservoirs.19

The delta plain sandstones often have a restrictedlateral extent and are encased in mud and siltstones.Generally, it is dicult to correlate the individualdelta plain sandstones between wells even in caseswhere the well distance is less than 1 km,2,20,21

Fig. 5.10. The delta front sandstones normally have alarger lateral extent, and are thus easier to correlate.In order to collect quantitative data for use in geo-

logical reservoir description, Mjøs et al.1 have studiedgeometrical aspects of delta plain sandstones alongthe Yorkshire coastal cli sections. These sedimentswere probably deposited in a delta plain setting sim-ilar to that of the Ness Formation.2,22 In another

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118 CHAPTER 5. GEOLOGICAL DESCRIPTION

Alluvial channel

Delta plain channel

and levee

Delta plain interchannel area

Delta front mouth bars

Delta front slope

Figure 5.9: Depositional facies of a high constructivelobate delta.15

0 150 0 150 0 150GR GR GR

1.2 km750 m

SW NE

Cretaceous

Ness II

Ness I

Rannoch/Etive Fm.

OsebergFm.

Nes

sFm

.

Coal or coaly shale

Unconformity

Figure 5.10: Reservoir scale correlation of the NessFormation in three wells from the Oseberg Field. TheNess Formation has been deposited in a delta plainsetting and shows discontinuous channel sandbodies.The correlation is approximately normal to the mainpalaeocurrent direction.2 (GR = Gamma Ray Log)

SPOR-project, quantitative geometry data from deltafront sediments in Kentucky and Utah have been col-lected.23

In this chapter, emphasis will be placed on quan-titative geometry and on architectural descriptionsof delta plain sandstones of the Ravenscar Group,mostly the Saltwick Formation, and from publisheddata on ancient delta front sediments, e.g., distribu-tary mouth bar sandstones, worldwide, in order todocument geometry and heterogeneity in deltaic sed-iments.

Distributary Channel Sands

Quantitative geometry data and channel ll charac-teristics of delta plain distributary channels resem-ble those of alluvial channels.16 Mud-rich suspended-load channels tend to have straight to low sinuos-ity reaches, with ribbon geometry, whereas sand-rich,bed-load channels, less than 5% mud, have low stabil-

ity and tend to have braided channels with a sheet-like geometry.24,25 Mixed-load rivers often seem toform channels which tend to migrate laterally (mod-erate stability). However, in spite of these generalstatements, the geometry of alluvial channels appearsto be dominantly controlled by ow and sedimentaryprocesses operating over a range of high discharges.6

The increase in discharge, width/depth ratio, andgrain size of transported sediment associated withexceptional overbank oods may cause a decrease insinuosity and increased braiding.6 Combinations ofchannel migrations and switching would be typicalof many channel belts,26 thereby producing complexarrangements of channel sandstones.On the delta plain, uvial channels are xed be-

tween each avulsive jump, but avulsions may be fre-quent and, moreover, multiple seaward bifurcating,rejoining or single channels may occur.14 Most likely,distributary channel patterns and lobe shifting pat-terns strongly inuence the delta plain channel char-acteristics. The type of distributary channel ll maybe dependent upon the rate of channel abandon-ment.27 Rapid abandonment of the channels maygive mainly mud-lled channels, whereas progressiveabandonment may produce sandstone-lled channels.Within the modern Mississippi lower delta plain, thethickest channel sandstones have been deposited atchannel bifurcation points. Elsewhere on the lowerdelta plain, only the lowermost few meters of thechannels are normally lled with sandstones.17

Geometry data from the modern Mississippi deltaindicate that the width/thickness ratio of the chan-nel ll changes downstream from the alluvial val-ley, width/thickness ratio 170,28 towards the deltaplain, width/thickness ratio 15.29 The upper andlower Mississippi delta plain areas have channels withsimilar width/depth ratios but both channel depthsand widths are decreasing downstream.29 From vari-ous uvial channels,8 there is a relationship betweenchannel sandstone thickness and channel depth. Amedian line between sandstone thickness and chan-nel depth can be drawn where channel depth is 0.55times the sandstone thickness.8

Examples of distributary channel sandstone geome-tries from the literature, Table 5.1, may suggest thattheir width/thickness ratios fall in the range 15 to1000.The highest width/thickness ratios in the modern

Rhone delta occur in places where channels have cutinto earlier deposited sandstones and in the alluvialvalley. The channels on the lower delta plain arelinear and have little tendency to migrate laterally,whereas the upper delta plain has channels whichtend to migrate laterally.32

Quantitative Geometries in the RavenscarGroup. Volumes of material within the channelbody which are separated by scour-surfaces are calledstoreys.33 The term multistorey sandbody is used asdened by Friend et al.33 and Blakey and Gubitosa,34

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5.2. MACROSCOPIC HETEROGENEITIES 119

Table 5.1: Published width/thickness data on distributary channel sandstones

Width Thickness Width/Thickness Comments1001000 Rhone delta30

< 50 Rhone and Rhine deltas, close to shoreline31

< 500 m < 15 m ca. 30 Ness Fm./Brent eld20

< 300 m < 18 m (ca. 15) Lower delta plain/Carboniferous32

4.511 km 1525 m (300440) Upper delta plain/Carboniferous32

and includes both vertical and lateral stacking ofstoreys. Channel sandstones with width/thicknessratios lower than 15 are called ribbons whereas chan-nel sandstones with width/thickness ratios in therange 15 to 100 and greater than 100 are called nar-row sheets and broad sheets, respectively, Fig. 5.11.Generally, xed channels produce mainly singlestorey

hSimple (single storey) Multistorey

Narrow: w/h = 15-100 Broad: w/h > 100

Simple

Multistorey

Channel sand body types

w

Ribbons

Sheets

Figure 5.11: Classication of channel cross-section ge-ometries.34

channel sandstones (ribbons) whereas xed channelbelts produce multistorey channel sandstones withribbon or narrow sheet geometry, Fig. 5.12. Mobilechannel belts may produce channel sandstones withbroad sheet geometry.The dominance of vertical accretion in the Saltwick

Formation channels1,2 may be related to channelswith relatively low sinuosity and minor lateral migra-tion. The storeys within one channel sandstone maybe the result of avulsions within a restricted elongatedchannel belt, as cutting new channels and lling oldones within the channel belt occurs in all rivers.6 Thesuperposition of dierent channel lls and lateral ac-cretion deposits produce a multistorey channel sand-stone. The multistorey character is also generated bysuperposition of dierent channel belts or by cluster-ing of distributary channels at the bifurcation pointon the delta plain.Generally, delta plain channel sandstones are elon-

gated bodies encased in interchannel mudrocks andhence have well-dened widths and lengths. Thethickness measurements of the channel sandstones inthe Saltwick Formation were made where the channel

Fixed channels(singlestoreyribbons)

Fixed channel belts(multistorey)

Mobile channel belts(sheet geometry)

w/h < 15

w/h > 15and up to more

than 200

w/h > 100and up to

several thousands

Figure 5.12: Characterization of channels and chan-nel belts based on stability and tendencies to migratelaterally.

sandstone sections were thickest.Width and thickness data from single- and multi-

storey channel sandstones in the Saltwick Formationof the Ravenscar Group are shown in Fig. 5.13. Mostof the width/thickness ratios are between 17 and 40,but ratios up to 67 are also recorded. Most of thesection widths recorded are suggested to be slightlyhigher than the true widths, since oblique cross sec-tions tend to record higher width values. The thick-nesses of the channel sandstones are from 4 m to morethan 22 m, the mean thickness is 9.5 m (n=45). Thewidths range from 100 m to 800 m, the mean is 280 m(n=23). The few quantitative data available on indi-vidual channel sandstone storeys indicate that theyare from 4 m to 13 m thick and the lateral extent isbetween 60 m and 480 m. The channel sandstoneswith the highest width/thickness ratios mainly dis-play laterally stacked storeys.A wide channel belt sandstone is located in the

northwestern part of the study area. This sandstone

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120 CHAPTER 5. GEOLOGICAL DESCRIPTION

25

20

15

10

5

0200 400 600 800 1000

Width w, m

Thi

ckne

ssh,

m

w/h = 60w/h

=30

w/h=

15

(a) Measured widths vs. measured thickness.

5

4

3

2

1

0200 400 600 800

Freq

uenc

y

Width w, m

(b) Histogram of measured widths.

8

6

4

2

00 5 10 15 20

Freq

uenc

y

Thickness h, m

(c) Histogram of measured thickness.

8

6

4

2

00 20 40 60 80

Freq

uenc

y

Width/thickness ratio

(d) Histogram of the width/thickness ratios.

Figure 5.13: Quantitative data for sections through uvio-deltaic channel sandstones from the Saltwick Forma-tion.38

comprises several mainly laterally stacked storeysforming a 3 km wide and about 15 m thick chan-nel belt sandstone. The palaeogeographical situationduring deposition of the Ravenscar Group indicates aclastic source area towards the north, and thereforethe wide channel sandstone belt is situated in themore proximal part of the depositional system. Inthe same manner as in the modern Mississippi delta,the large width/thickness ratio (200) recorded in theproximal part of the Saltwick Formation depositionalsystem may represent upper delta plain to alluvialplain channel geometry.

The geometry data, Fig. 5.13, indicate that thechannels in the Saltwick Formation should be clas-sied as narrow sheets34 since the width/thicknessratios are between 15 and 100. Literature data, how-ever, indicate that more than 50% of channel sand-stones on a humid delta plain have ribbon geometrieswith width/thickness ratios lower than 15.2 Distribu-tary channel geometry data from the Ferron sand-stone in Utah and from the literature23 indicate thatthe width/thickness ratios mainly fall in the range 8to 44.

Ancient upper delta plain channel sandstones areusually thick and display multistorey characteristics,whereas the lower delta plain channel sandstones arethin and singlestorey.16 Based on this, and the sedi-mentary characteristics of the oodplain mudrocks,the relatively thick and multistorey channel sand-stones in the Saltwick Formation seem to have formedon the upper delta plain and the upper part of thelower delta plain. This may explain the relatively

high width/thickness ratios recorded in the SaltwickFormation compared with channel sandstone ratiosrecorded in other delta plain sequences.2

Base (sea) level changes may inuence channelgeometry and channel stability on the delta plain.Broad sheet channel sandstone geometries may beformed due to a sea level lowering. This is seenat the base of the Scalby Formation in the Raven-scar Group where a 10 to 12 m thick sheet chan-nel sandstone has been formed.1 Nami and Leeder35

have reported that the channel sandstone width nor-mal to the palaeocurrent trend is about 70 km, al-though uncertainty is attached to whether the sand-stone is continuous within this width, indicating awidth/thickness ratio of about 5000 to 6000. Thissheet channel sandstone at the base of the ScalbyFormation has probably been formed in response toa base level lowering and seaward shift of the fa-cies tracts.1,35 Sea or base level lowering generatesless sediment accommodation space. Hence, channelstend to shift position frequently, forming mobile chan-nel belts which generate a very broad sheet sandstonegeometry.

Organization of Channel Sandstones. In broadcoastal plain settings, sequential channel avulsion willlead to a sequence with oset or diagonal channelstacking pattern as long as the width of oodplainor zone of inuence is signicantly larger than thechannel belt.7,36 The distribution of channel sand-stones in an alluvial suite is inuenced by laterallyvariable aggradation, compaction of ne sediment,

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5.2. MACROSCOPIC HETEROGENEITIES 121

tectonic movements and channel avulsion.7 Within adelta plain sequence, also the areal pattern and num-bers of coeval active distributary channels and typeof delta lobe shifts have important inuence on thedelta plain architecture.14 It is also of importance toconsider the depth of channel incisions27,37 since themagnitude of vertical scouring into pre-existing sedi-ments inuences the channel sandstone density withina sequence. When modelling the eect of channel in-cision depth on channel sandstone density, it becomesapparent that the channel sandstone density increaseswith increased channel incision.38 In that way vari-ations in depth of channel incision, due to allocyclicfactors like eustatic sea level changes, may cause ver-tical changes in channel sandstone density.

Crevasse Splay Sands

Crevasse splay sediments accumulate during oodswhen excess discharge in the uvial channel breachthe adjacent channel margin (levee), and coarse-grained material is transported into and depositedas fans in the oodplain areas adjacent to the uvialfeeder channel.It seems to be important to dierentiate between

two types of crevasse splay deposits:16,39,40 (1) small-scale single crevasse splay lobes attached to thelevee or alluvial ridge of the uvial channel, and (2)large-scale crevasse subdelta lobes forming compos-ite crevasse splay sandstones which inll bays andlagoons. The literature40 indicates that the singlecrevasse splay sandstone thicknesses are in the range0.3 to 4.5 m, but usually less than 2 m. Moderncrevasse subdelta deposits may be up to 15 m thick.17

The few data available on quantitative geometriesof single crevasse splay sandstones indicate that thewidth/thickness ratios mostly are in the range 150 to1000.

Crevasse Splays in the Ravenscar Group.Crevasse splay sandstones in the Saltwick Formationof the Ravenscar Group constitute about 20 to 40% ofthe interchannel delta plain sediments. The crevassesplay sandstones have a sheet-like geometry and areinterbedded with delta plain mud- and siltstones.Normally they thin rapidly at their outer margins andalso outwards from the proximal channelized part to-wards the unconned crevasse splay lobe. However,they may also die out more gradually as a wedge giv-ing way laterally to interstratied thin mud-, silt- andsandstones. Thicknesses of single crevasse splay sand-stones range from 0.3 to 2.5 m, and within the cen-tral parts of the crevasse splay sandstones thicknessescommonly vary by a factor of two. Most of the mea-sured sections through crevasse splay sandstones havethicknesses less than 1 m, but thicknesses up to 2.5 mare not uncommon.Two-dimensional sections through crevasse splay

sandstones can be classied as longitudinal (length),transverse (width) or oblique, Fig. 5.14. Length,width and thickness data for single crevasse splay

Length

Length

Width

Width

chan

nel

Distributary

Distributa

ry

mouth

bar

Crevassesplay

Figure 5.14: Denition of distributary mouth bar andcrevasse splay width, w, and length `.23

sandstones from the Saltwick Formation are shownin Fig. 5.15. These data are collected from crevassesplay sandstone units which were deposited during asingle ood event or period. Sandstone units sepa-rated by a few centimeters of coal or shale or only bya horizon with rootlets are considered as two crevassesplay sandstones. Any dierence between widths andlengths of crevasse splay sandstones in the SaltwickFormation is not revealed. The recorded widths areup to 2200 m and lengths are up to 1750 m. Severalof the data points in Fig. 5.15 represent minimumvalues since many crevasse splay sandstones extendbeyond the exposure.There is no linear relationship between width and

thickness of the single crevasse splay sandstones,Fig. 5.15. Maximum observed width/thickness ra-tios are, however, not greater than 1500. Thelength/thickness ratios do not exceed 2000, Fig. 5.15,and are consequently within the same order-of-magnitude as the width/thickness ratios. The sim-ilarity in width/thickness and length/thickness ratiosof the crevasse splays in Yorkshire is consistent with alobe-shaped geometry. This is also in accordance withmost of the literature40 data, although tongue-shapedand channelized geometries are also reported.28,41,42

Several of the transverse sections through singlecrevasse splay sandstones in the Saltwick Forma-tion display a lensoid geometry comprising very lowwidth/thickness ratios, 10 to 100, large thicknesses,1.0 to 2.5 m, and small widths, 18 to 200 m. The lowerboundaries of these sections are commonly concaveupwards and seem to be erosional. However, the up-per boundaries are in some cases also convex upwardsand may thus suggest that bedforms formed positivefeatures. These lensoid crevasse splay sandstones arelocated close to a uvial channel margin and probablyrepresent the channelized proximal part of a crevasse

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122 CHAPTER 5. GEOLOGICAL DESCRIPTION

Saltwick Fm.Gristhorpe Mb.Scalby Fm.

3.0

2.0

1.0

0500 1000 1500 2000

Width w, m

Thi

ckne

ssh,

m

w/h

=10

0

w/h=

500

w/h = 1000

w/h = 2000

(a) Plot of measured widths against measured thicknesses.Points with width/thickness ratios less than 100 and thick-nesses greater than 1.5 m represent sections through the prox-imal, channelized part of the crevasse splay sandstones.

3.0

2.0

1.0

0500 1000 1500 2000

Thi

ckne

ssh,

m

Length l, m

Saltwick Fm.Gristhorpe Mb.Scalby Fm.

l/h=

100

l/h=

500

l/h= 1000

l/h = 2000

(b) Plot of measured lengths against measured thicknesses.

4

2

00 500 1000 20001500

Length/thickness ratio

Num

ber

ofob

serv

atio

ns

(c) Histogram of the length/thickness ratios.

20

16

12

8

4

00 500 1000

Num

ber

ofob

serv

atio

ns

Width/thickness ratio

(d) Histogram of the width/thickness ratios.

Figure 5.15: Quantitative data for sections through crevasse splay sandstones from the Ravenscar Group. Within(a) and (b), open symbols represent minimum values where the full width of the crevasse splay sandstone wasnot observed due to termination of the exposure.40

splay lobe.The composite crevasse splay sandstones, 3 to 6 m

thick, may have a lateral extent of up to more than20 km implying a width/thickness ratio larger than3000. The width/thickness ratios of the individualcrevasse splay sandstone units that form the build-ing blocks of the composite crevasse splay sandstone,are comparable with what is elsewhere found for sin-gle crevasse splay sandstones in the Ravenscar Group.The large extent of the composite crevasse splay sand-stones must be the result of both lateral and verti-cal stacking of single crevasse splay sandstones. Inthe proximal parts of the composite crevasse splaysequence, a conned or channelized sandstone is lo-cated in the axial part grading gradually outwardsinto more sheet-like sandstones and siltstones. Thegeometries of the composite crevasse splay sequencesmay be the result of similar crevasse splay subdeltasettings as proposed by Fielding,36 Fig. 5.16, where amajor crevasse channel forms the proximal axial partfringed by crevasse sheets and where the lobes withsheet geometries are most prominent in the medialand distal parts of the crevasse subdelta.

Application of the Data in Reservoir Mod-elling

The data from the Ravenscar Group in Yorkshire mayhave general relevance to deltaic reservoir rocks ofsimilar type in other areas. In the northern NorthSea, delta plain reservoir sandstones occur in theNess Formation of the Brent Group.21,43,44 Whetherthe Yorkshire data can be applied to reservoir mod-

∼ 15 km

Figure 5.16: Crevasse splay complex that forms acrevasse subdelta. The proximal part is dominatedby channelized, erosively based sandstones. The me-dial part is dominated by interbedded, sharply basedsandstones forming irregular areas, whereas the dis-tal part is dominated by single or multiple coarsen-ing upward sequences forming lobes. (Modied, afterFielding.36)

elling of, for instance, the Ness Formation, Fig. 5.10,must be considered for each specic oil and gas eld.When considering similarities between the Yorkshireoutcrops and the petroleum reservoir, attention mustespecially be paid to factors like sandstone/shale ra-tio, types and arrangement of lithofacies groups andsubsidence rate (or accommodation rate).

Assuming the geometry data from the RavenscarGroup are relevant for a specic reservoir, geologicalmodelling of the reservoir would have to take into

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5.2. MACROSCOPIC HETEROGENEITIES 123

account the collected outcrop data. Some exampleson data and rules that can be applied to delta plainreservoir modelling are listed below:

1) Crevasse splay sandstones are lobate with lengthnot more than 1 to 2 times the width.

2) The thickness commonly varies by a factor oftwo, hence, thickness can in most cases be setto a constant value within each crevasse splaysandstone.

3) Width/thickness and length/thickness ratios ofcrevasse splay sandstones are less than 2000.Hence, a single crevasse splay sandstone encoun-tered in a well must be derived from a uvialchannel located closer than 2000 times the thick-ness of the drilled crevasse splay sandstone.

4) Crevasse splay channel sandstones havewidth/thickness ratios less than 100 andthicknesses larger than 1.5 m, and are located ina proximal position in contact with the uvialfeeder channel.

5) Composite crevasse splay sandstones may havegreat lateral extent and hence, they are correlat-able for several kilometers (up to 20 km).

6) More than 50% of the channel sandstones onthe delta plain have width/thickness ratio in therange 5 to 30, whereas more than 90% have ratiosin the range 5 to 70.

7) Formation of broad sheet channel sandstones(w/h > 100) preferentially takes place at specicstratigraphic levels and in proximal positions onthe uvio-deltaic plain.

8) Channel sandstones are usually multistorey. Ver-tical stacking of storeys implies a decrease inwidth/thickness ratio, whereas lateral stackingof storeys implies the opposite.

9) The thickness of a channel sandstone can be setto a constant value since the major and centralpart (more than 80%) of the channel sandstonebody is relatively tabular.

Delta Front Deposits

Factors Controlling Sediment Distribution.In the geological record, the delta front sedimentsgenerally comprise large-scale coarsening upward se-quences, Fig. 5.17, which represent the transitionfrom ne-grained oshore or prodelta facies upwardsinto shoreline facies which is usually sandstone dom-inated. These delta front sequences result fromprogradation of the delta and may be truncated byuvial- or tidal-distributary channels as progradationcontinues. The delta front sequence varies consider-ably within deltas in relation to the proximity of thedistributary channel mouth, and between deltas ac-cording to the regime of the former delta front and thenature and extent of synsedimentary deformationalprocesses.Sandstone geometries within the dierent delta

front associations are also highly variable, but cannevertheless be classied on the basis of the sedimen-

30

25

20

15

10

5

0

Met

ers

Dis

trib

utar

ym

outh

bar

Dis

tal

bar

Pro

delta

Figure 5.17: Vertical sequence of facies within a dis-tributary mouth bar sandstone.23

tary environment they were deposited in.The uvial dominated deltas are generally highly

constructive features, where the main sediment loadon the delta front is deposited in the distributarymouth bar. According to Elliott,16 three main typesof uvial dominated deltas can be dened:

1) The deep water, uvial dominated deltas are gen-erally characterized by up to 100 m thick deltafront coarsening-upwards sequences with thickprodelta muds reecting progradation into rel-atively deep water. In several cases, mouth barsandstones are laterally discontinuous, display-ing lensoid geometries, implying that they werelocated around widely spaced distributary chan-nels. However, several deep water deltas con-tain thick, laterally continuous sandstones dueto partial marine reworking of sands from thedistributary mouths.45,46 In either cases, partialmarine reworking of the distributary mouth barsprobably results in relatively higher `/h ratios,as dened in Figs. 5.11 and 5.14.

2) The shallow water, uvial-dominated deltas aregenerally characterized by thinner delta front se-quences with a high sandstone/shale ratio andthey generally have a lobate geometry, forminga continuous delta front sandstone. These deltafront sheet sands47 can be traced for more than800 km2 attaining thicknesses up to 30 m.

3) The uvial-dominated, turbidite-fronted deltasgenerally comprise thick coarsening upwards se-quences resting on a semicontinuous package ofturbidite sandstones and heterolithics. Theseturbidites are generated either directly from thedistributary mouth, or by slumps and slides onthe upper delta front.48,49 The latter may indi-cate periods with relatively steep delta front gra-dients. The distributary mouth bar sandstonesin deltaic systems of this type are probably mostoften discontinuous. Strong marine reworking of

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124 CHAPTER 5. GEOLOGICAL DESCRIPTION

the distributary mouth bar sandstones is com-mon, and may result in laterally extensive sheetsof sands, suggesting high `/h ratios.

In other deltaic settings, however, reworking dueto waves and tidal processes strongly inuences thedistribution, and thus the geometries, of the deltafront deposits. The importance of these processes andtheir associated delta types is briey outlined below.In wave-dominated deltas, wave action may re-

work sediment deposited at the distributary mouth.High wave energy also impounds the discharge andincreases mixing of the water masses, and most ofthe sand is transported alongshore by longshore cur-rents to produce a continuous fringe of beach sands,Fig. 5.18. The delta front facies of wave-dominateddeltas resemble those of prograding beach fronts,

channel

bank

barcrest

barfront

swashbars

subaqueouslevee

wave

approach

beach

beach

(a) Direct onshore wave approach.

channel

swashbars

subaqueouslevee

waveapproach

beach

beach

(b) Oblique wave approach and associated dominant longshoredrift direction.

Figure 5.18: Wave-dominated river mouth setting.16

where the sandstone facies are well sorted and ex-hibit evidences of strong wave reworking. In somecases, strike-aligned sandstone bodies of beach faceand distributary mouth bar origin connect laterallyto sandstone bodies representing uvial distributarychannels.47

In tide-dominated deltas, tidal processes may in-uence the distributary mouth areas by increasingthe mixing between the water masses and there-fore promote sediment deposition in the river moutharea. Bidirectional sediment transport along pre-ferred ebb- and ood-dominated pathways prevails inthe channels and the river mouths, and these regionsare dominated by elds of linear current ridges gener-ally running oblique/normal to the coastline, whereasno proper distributary mouth bar sandstones are de-veloped, Fig. 5.19.

A A1

B B1

C

C

C1

C1

Lineartidalshoals

Cross-section A

Cross-section B

Cross-section C

bank

A

A1

B

B1

Sand filled channel

Figure 5.19: Tide-dominated river mouth illustratingthe funnel shape of the lower distributary channel, thepredominance of linear tidal current ridges or shoalsin the channel, and a zone of intense meandering atthe head of the funnel-shaped channel.16

In addition to waves and tides which often partlyor completely redistribute the sediments introducedto a basin, also delta front instabilities may stronglyinuence the sediment distribution.5052 Studies ofmodern delta fronts show that these sites of high sed-imentation rates are commonly inherently unstable,and that several types of delta front instabilities oc-cur. These are either shallow features involving theuppermost few meters of the sediment surface on thedelta slope; slides, slumps, debris ows, turbidites,grain ows; or, alternatively, deeper penetrating fea-tures including deep rotational slides, growth faults,mud diapirs, etc.The importance of the mass movement processes

on the delta front is at least threefold. Firstly, slopefailure (which causes sediment masses to move down-slope) is a destructional process which redistributesediments from the upper delta front and down thedelta slope. These processes are most common inuvial-dominated deltas where the rate of seawardgrowth tends to be more rapid. For the Mississippidelta, it is estimated that 40% of the sediment sup-plied is involved in some kind of mass movement af-ter deposition.53 The frequent occurrence of synsed-imentary deformation in many deltaic systems maysuggest that the sandbodies are more discontinuousand irregular in shape than what is expected whensynsedimentary deformation is not taken into consid-

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5.2. MACROSCOPIC HETEROGENEITIES 125

eration.Secondly, scars and gullies produced as a result of

mass movements may control later ow of sediment-laiden water down the slope.52

Thirdly, large volumes of sands may be trapped inreliefs produced by normal faults moving contempo-raneously with sedimentation. Very often these nor-mal faults have a listric geometry aecting deltaic se-quences up to several kilometers thick involving sev-eral deltaic systems.54 However, they may as wellaect sequences on the scale of one deltaic system.50

Fault movement may be incremental with sedimenta-tion more or less smoothing out the reliefs produced.Alternatively, fault movement may be considerableand rapid (delta front collapse) creating an irregu-lar relief where previously continuous sandstone bod-ies are disrupted. Whatever the type of delta frontinstabilities, the corresponding sandbody geometriesare generally complex and hard to predict.

Heterogeneities of Distributary Mouth BarSediments. The distributary mouth bar facies nor-mally represent the most signicant delta front reser-voir facies. A cross section through a eld producingfrom sediments of this type is presented in Fig. 5.20.These deposits originate as bars which form at themouth of the distributaries. Extremely high sedimen-tation rates are recorded from this particular suben-vironment.A vertical prole through a distributary mouth bar

deposit is shown in Fig. 5.17. The coarsening upwardstrend, the change in types of sedimentary structuresand the decrease in amount of bioturbation are char-acteristic features.The distributary mouth bar sandstones gradually

thin laterally, exhibiting an overall lensoid form. Thelateral decrease in thickness is most pronounced inthe dip direction, and more gradual in the strike di-rection.55

The distributary mouth bar complexes may appearrelatively massive, and can thus generally be charac-terized by a relative absence of discontinuous shaleswithin the main sandstone interval. However, dis-tributary mouth bar complexes may comprise highlystratied sandstones and shales.55 Field data suggestthat three types of shales predominate in distributarymouth bar complexes:

1) thick (up to several meters) shales extendingacross the eld dening the interval with dis-tributary mouth bar sandstones;

2) shales (up to 30 to 50 cm thick) with lateral ex-tent between 100 to 3000 m. These shales sepa-rate individual lobes;

3) thin shales with lateral extent below 100 m.

Shales of this type located within the proximal re-gion of the distributary mouth bar sandstone are thinand short compared to those occurring in the moredistal parts of the distributary mouth bar. In both in-stances, however, eld data55 suggest that a relatively

thick, 40 cm, shale rarely exceeds a lateral extent of100 m.Relatively steep foresets, 5 to 15 degrees, may oc-

cur within the distributary mouth bar sandstone.52

However, most distributary mouth bar sediments gen-erally display horizontal stratication due to the ex-tremely low dip of the slope in this particular suben-vironment.17

Application of Quantitative Geometry. Quan-titative geometry data of distributary mouth barsandstones have been collected from the literature,Table 5.2.Despite the considerable variation in both thick-

nesses, widths and lengths of the distributary mouthbar sandstones as evidenced by the eld data, thereare reasons to believe that this data base can act as apredictive tool and thus be applied in reservoir mod-elling. The most important geometrical aspects of thedistributary mouth bar sandstones are as follows:

1) Distributary mouth bar sandstones may occurinterbedded with shales. The shales are gener-ally discontinuous, and a 40 cm thick shale rarelyexceeds a lateral extent of 100 m.

2) Distributary mouth bar sandstones have alensoid form, often asymmetrically developed rel-ative to the distributary channel.

3) The length of the distributary mouth bar sand-stones falls in the range 2.4 km to 9.6 km, witha mean of 6.3 km.

4) The width of the distributary mouth bar sand-stones falls in the range 1.5 to 8 km, with a meanof 3.5 km.

5) The fact that some distributary mouth bar sand-stones are known to cover several 100's of kilo-meters may suggest that individual distributarymouth bar sandstones merge to form a compos-ite distributary mouth bar sandstone.

6) The thickness range is considerable, varying froma few meters and up to several tens of meters,with a mean of 18.5 m.

7) The thickness of individual distributary mouthbar sandstones generally shows a gradual de-crease in thickness laterally. The decrease inthickness is most rapid in the dip direction andmore gradual in the strike direction of the depo-sitional system.

8) Three types of shales can be dened in a distribu-tary mouth bar system: (1) thick shales, up to afew meters, extending across the eld; (2) thin-ner shales separating individual lobes and withlateral extent between 100 and 3000 m; (3) thin,up to 40 cm, shales with lateral extent less than100 m.

5.2.3 Structural Heterogeneity

Natural fracture systems in sandbody reservoirs mayeither enhance permeability,56 or reduce permeabil-ity.57,58 Some reservoirs, e.g., the Gullfaks and Hild

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126 CHAPTER 5. GEOLOGICAL DESCRIPTION

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Figure 5.20: Generalized reservoir-geological model of the Tilje Formation in an area where the formation isdominated by fan-deltaic sediments. The model is based on a combination of well data and the quantiedgeological data collected in the Ridge Basin. The scale bar is only valid for the interwell areas.3

Fields, are heavily fractured in places, causing signif-icant compartmental subdivision of the reservoir andits hydrocarbon reserves. For such reservoirs, it isimportant that a thorough structural model of thefractured rock-body is constructed. This requires ananalysis which takes into account fracture frequency,fracture geometry, three-dimensional conguration,sense of displacement, texture of fracture-ll, defor-mation style and relative ages of the various fracturesystems.59 A range of fracture geometries is illus-trated in Fig. 5.21, and an example of fracture typesin part of the Troll Field is shown in Fig. 5.22.

5.2.4 Diagenetic Barriers

Introduction

The origin of calcite cement in shallow marine sand-stones has been discussed by several authors,6171

but surprisingly few papers have discussed the ge-ometry of calcite cemented zones in shallow marinesandstones.68,7072 In this section, we will present amodel for calcite cementation developed by Bjørkumand Walderhaug.70,71

Calcite cementation in shallow marine sandstones

typically occurs as: (1) continuously cementedlayers; (2) discontinuously cemented layers con-sisting of calcite-cemented (attened) concretionsand intervening areas of noncemented sandstones(stratabound concretions); (3) concretions scatteredwithin the sandstones, Fig. 5.23. Patchy calcite ce-mentation,70 the case when calcite cement only oc-cludes some of the pores, will not be discussed furtherin this context.Thickness of calcite cemented zones varies from a

few cm to several meters. In the calcite cementedzones, the pores are lled with calcite cement and theporosity is normally less than 1% and the permeabil-ity much less than one millidarcy. Thus, calcite ce-mented zones may, if they are laterally extensive andcontinuous, form permeability barriers in reservoirs.In the North Sea, up to 10% of marine sandstones aretypically cemented by calcite. In outcrop studies, ithas been proven that calcite cemented layers in theBridport Sands61,73 have lateral extent greater thanthe length of the exposures, i.e., greater than 3 km.Both continuous and discontinuous calcite-cementedzones are common in the exposures of the BridportSands in Southern England.61,68,69,71,73 The verticalthickness of the continuously cemented calcite layers

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5.2. MACROSCOPIC HETEROGENEITIES 127

Table 5.2: Descriptive statistics of the distributary mouth bar sandstone bodies.23

Mean Std. D. Min. Max. Range CountWidth, m 3.477 1.921 1.5 8 6.5 9

Thickness, m 18.589 12.86 1.2 42 40.8 10Length, km 6.318 2.409 2.4 9.6 7.2 8

Length/Width ratio 2.106 0.505 1.5 3 1.5 7Width/Thickness ratio 367.686 378.789 83.33 1333.33 1250 9Length/Thickness ratio 622.906 592.55 205.71 2000 1794.29 8

2

7

6

1

3

4

8

5

2

Figure 5.21: Schematic representation of fracture ge-ometries: (1) conjugate fracture set; (2) conjugatesets related to true three-dimensional stress; (3) de-formation band; (4) shear band; (5) symmetrical bi-furcation; (6) asymmetrical bifurcation; (7) bifurca-tion related to tip of fracture; (8) high-angle conju-gate fracture set related to competent rock unit.59

varies from more than 0.5 meter to less than 0.2 me-ter. The upper and lower boundaries of the cementedlayers are wavy.

Scattered concretions, i.e., calcite-cemented con-cretions that are not conned to a plane or thin zone,are probably also common in North Sea reservoirssuch as the Rannoch and Oseberg Formations. Inthe Valtos Formation74,75 in Skye in Scotland andat Osmington Mills76,77 in Southern England, scat-tered concretions are observed. Their sizes vary fromless than 0.5 meter in diameter to more than two me-ters in diameter and the distance between concretionsvaries from zero for merged concretions to several me-tres. The scattered concretions are often spherical,but attened concretions with the longest axes in thehorizontal plane are also common.

Oxygen isotope data from calcite cement in theNorth Sea sandstones suggest that the cementationprocess is continuous and may start soon after depo-sition and prevail until present burial depth, at tem-peratures ranging from 20 to nearly 90 C.69 Thissuggests that cementation is not an event but rathera process taking place over several millions to tens ofmillions of years. In the subsurface, the lateral extentand continuity have to be predicted based on infor-mation extracted from cores and petrophysical logs.To predict the lateral extent and continuity of calcitecemented intervals, it is necessary to study the two-

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Figure 5.22: Main fracture types in the Troll Fieldreservoir rocks: (1) single fractures: (2) anastomos-ing fractures; (3) fracture swarm; (4) macrofault;(5) contrast in fracture geometry between carbonate-cemented zone and sandstone without carbonate ce-ment; (6) subhorizontal fractures of possible tectonicorigin in poorly cemented sandstone. Triangles in-dicate tectonic breccia, cc carbonate-cemented sand-stone, and m possible other type of mineralization60

and three-dimensional geometry of calcite-cementedzones and concretions in outcrops. A successful pre-diction also depends on a growth model that is ableto explain the geometrical arrangement of calcite ce-mentation.A summary of the model presented by Bjørkum

and Walderhaug70 is given below.

Growth Model for Calcite Cement in ShallowMarine Sandstones.

Calcite-cemented layers and concretions in shal-low marine sandstones are formed by a couplednucleation-diusion controlled redistribution of car-bonate fossils (CaCO3). Calcite cement is precipi-tating while the carbonate fossils are being dissolved.This is possible since the equilibrium concentration

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128 CHAPTER 5. GEOLOGICAL DESCRIPTION

a

b

c

Figure 5.23: a) continuously calcite cemented layer;b) stratabound concretions; c) scattered concretions.

of dissolved calcium carbonate is higher than for cal-cite cement. Few of the carbonate-cemented layers inNorwegian sandstone reservoirs contain hardgrounds.This type of calcite cement will therefore not be fur-ther discussed here.Since the main source of the calcite cement in North

Sea shallow marine sandstones is believed to be car-bonate fossils from within the sandstones,67,70,71 andsince the redistribution process is local,70 the geom-etry of calcite cemented zones is controlled by: (1)the primary distribution of carbonate fossils in thesandstonesdetermines the large scale geometry; (2)the spatial arrangement as well as the number of nu-cleation points in the systemdetermines the geom-etry on a scale of meters to centimeters.

Primary Distribution of Carbonate Fossils.The lateral extent of calcite-cemented layers is con-trolled by the primary distribution of carbonate fos-sils, again controlled by the depositional environment.Carbonate fossil-rich layers in shallow marine sand-stones include the following types:

1. Layers rich in biogenic carbonate formed by theaccumulation in situ of carbonate fossils dur-ing periods of low siliclastic input. The lay-ers may experience short-term, in-situ reworkingand winnowing.61,72

2. Carbonate fossil lags at the base of storm-deposited layers.7880

3. Carbonate fossil lags located at the base of sub-marine channels.62

4. Carbonate fossils lags located above submarineerosion surfaces.81

The lateral extent of these dierent types of carbon-ate fossil-rich layers is controlled by the depositionalenvironment and varies from a few meters to morethan 10 km.

Nucleation and Growth of Concretions andLayers. For nucleation of a mineral to occur, afree energy barrier must be surpassed. This impliesthat nucleation of calcite requires a certain degreeof supersaturation. The pore water is likely to be-come supersaturated with respect to calcite cementupon burial, since the solubility of carbonate fossils,i.e., aragonite and Mg-rich calcite, is higher than forcalcite cement. The critical supersaturation is mostlikely to occur rst in the shelly part of the sediment.Also, carbonate fossils may be favorable substrates fornucleation of calcite. In both cases, the calcite nucleiare likely to form within the biogenic carbonate-richsands.The degree of supersaturation in the pore water

in the immediate vicinity of the nucleus will be low-ered, and the probability of forming a new nucleusis signicantly reduced. Neighboring concretions arethus likely to form a certain distance away from thenucleus/concretion already formed, i.e., where theamount of dissolved calcite not is lowered. The ra-dius of the region adjacent to the nucleus/concretionwhere the concentration of dissolved calcite is low-ered, is referred to as the range of inuence. It in-creases with time, Fig. 5.24.Formation of new concretions will stop when the

Transientstate

Semi-steadystate

Time after nucleation

Ran

geof

infl

uenc

e

Figure 5.24: Range of inuence of a concretion as afunction of time.

ranges of inuence overlap. Immediately after nucle-ation, the range of inuence is small but increasesrapidly until the rate of carbonate fossils dissolvedequals the rate of calcite cement precipitating. Dur-ing the transient period, the amount of calcite precip-itated per unit time at the surface of the concretionexceeds the amount of carbonate dissolved within therange of inuence, and the range of inuence conse-quently increases rapidly, Fig. 5.24.When the range of inuence has expanded to a

value where the amount of calcite precipitated perunit time equals the amount of carbonate dissolvedper unit time within the range of inuence, the con-cretion enters the semi-steady state where the rangeof inuence only increases slowly as clastic carbon-

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5.2. MACROSCOPIC HETEROGENEITIES 129

ate is consumed. A critical factor in the nucleationprocess is thus the range of inuence during the timeof nucleation of the neighboring concretion. If it issmall, the concretions are close.If the concretions form in a plane, Fig. 5.23, the

distance between concretions may determine whetherthey will merge and form a continuous layer, Fig. 5.25.In case A, a relatively small number of nuclei form andthe biogenic carbonate within the layer gives rise to

A B

Figure 5.25: Inuence of number of nucleation pointson the continuity of a calcite-cemented layer.

a layer of stratabound concretions. In case B, morenuclei form, and the same amount of calcite cementforms a continuous layer. The distance between con-cretions formed in a plane will be semiregular andvary between one and two ranges of inuence.70 Therange of inuence may vary from a few centimetersto more than one meter. Shortest range is expectedin sandstones containing high concentrations of car-bonate fossils with high specic surface areas.If nucleation originally takes place within the cen-

tral part of the layer, Fig 5.26, a layer of strataboundconcretions forms. However, if the supply of clasticcarbonate is not exhausted, the concretions will con-tinue to grow and may eventually merge and from acontinuously cemented layer.Whether a biogenic, carbonate-rich layer will turn

into a continuously or discontinuously cemented layeris determined by the concentration of carbonate fos-sils and the thickness of the carbonate-rich sand unit.If the thickness of the shelly sand unit is less than tworanges of inuences, the concretions are likely to beconned to the central part of the shelly sand unit.Hence, the concretions are likely to be conned to aplane and are therefore more likely to form a con-tinuous layer, Fig. 5.26, than if the concretions hadnucleated at dierent levels. If the thickness of thesand unit is several meters, the level of nucleation is

Figure 5.26: Formation of continuous calcite-cemented layer.

not likely to be a single plane, but random.70 Whenconcretions nucleate at dierent levels within a thick,shelly sand unit with low shell content, they are lesslikely to merge laterally and form continuously ce-mented layers. If the initial concentration of carbon-ate fossils is very high, > 30%, a continuously ce-mented sandstone unit is likely to form, independentof the nucleation pattern.Bjørkum and Walderhaug70 have calculated the

growth rate of concretions assuming diusion-controlled growth. Their calculations suggest that itmay take from a few million to several tens of millionyears to form a concretion with a radius of 0.5 meter.

Geochemical Implications. Walderhaug et al.69

have shown that two vertical geochemical proles ina continuously cemented layer are very dierent al-though the proles are separated by only 30 cm. Thisis hardly surprising in the case where the calcite ce-mented layer forms by merging of concretions.7072,82

Some vertical proles would then transect the earlygeneration of calcite cement, whereas others mightonly encounter later generations of calcite, i.e., morenegative oxygen isotope values reecting high temper-atures. Also, correlation studies of calcite-cementedzones in the Fensfjord Formation in the Brage Field69

suggest that geochemical data, i.e., stable isotopesand trace element, are of limited value for correlationof calcite cemented layers.

Predicting Lateral Extent of Cored Calcite-Cemented Layers

To predict the lateral extent of a calcite cementedlayer from cores, we rst have to identify the sedimen-tological process controlling the distribution of theshelly sand unit, i.e., types 14, as described above.Calcite cemented layers of type 1 can often be rec-

ognized in cores by extensive bioturbation due tolow siliclastic input (nondeposition), by absence ofa coarse siliclastic lag, by absence of fossils trans-ported from other environments, and by not being

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130 CHAPTER 5. GEOLOGICAL DESCRIPTION

located above an erosion surface. Many of the calcitecemented layers in the shallow marine Fensfjord For-mation in the Brage Field may be classied as type1 layers. The lateral extent of such layers is likely tobe of the order of a kilometer or more.Type 2 layers may be recognized by their location

at the base of a storm-deposited layer,78 by contain-ing a coarse siliclastic lag, and sometimes by contain-ing fossils transported from other environments.79

The lateral extent is from meters to hundreds of me-ters.Layers of type 3 and 4 are located above erosion

surfaces and are coarser grained than the underlyingsediments. The lateral extent is likely to be frommeters to some hundreds of meters for type 3, whereasthe lateral extent of type 4 layers is uncertain.

Predicting Continuity From Cores

Wavy upper and lower boundaries of calcite-cementedlayers do not imply that the layer consists of individ-ual concretions, i.e., discontinuous layer, as this isalso likely to be observed in layers where the concre-tions have merged and formed laterally continuous,cemented layers. Also, the thickness of the calcite-cemented layers does not give any indication of thecontinuity of the layers since thin layers (10 cm) maybe continuous for several km, as seen in the Brid-port Sands. Also, high content of preserved carbon-ate fossils implies that the content of carbonate fossilswithin the sand unit was higher than required to llthe pores with calcite cement. Therefore, high con-tent of preserved fossils indicates that the calcite ce-mented layer is continuous. On the other hand, lackof preserved carbonate fossils does not imply that thelayer is discontinuous. A continuous layer may formfrom source material supplied by the sandstones im-mediately below and above the plane of nucleation,Fig. 5.26.

5.3 Microscopic Hetero-geneties

5.3.1 Introduction

Pore-scale heterogeneities may aect permeability,relative permeability and chemical ooding perfor-mance.North Sea sandstone reservoirs (including mainly

sandstones of the Brent Group and the Statfjord For-mation) typically contain 40 to 90 wt% quartz, 5 to30 wt% feldspar, 2 to 20 wt% kaolinite, 0 to 20 wt%mica/illite, and 0 to 5 wt% carbonate, and minoramounts of chlorite, pyrite and heavy minerals.The mineralogical composition at the time of depo-

sition is one of the main factors controlling the sec-ondary (authigenic) minerals that form. Especiallyimportant is the presence of feldspar and mica forformation of authigenic clays and secondary porosity.

Clay minerals have a very high specic surfaceareatypically between 10 and 100 m2/g. Most ofthe surface area in the sandstones is therefore associ-ated with the clays.The eect of the clay minerals on permeability,

residual oil saturation and the success of chemicalooding will depend on the amount of clay, the typeof clay, and the microspatial distribution of the clay.We will concentrate on the eect of kaolinite and il-

lite. Only minor amount of smectite is reported fromthe North Sea sandstones and is therefore not dis-cussed.

5.3.2 Kaolinite

Kaolinite is the most commonly occurring mineral inthe North Sea sandstones. The amount varies fromnearly zero to ca. 30 wt%. Both authigenic, Fig 5.27,and detrital kaolinite, Fig. 5.28, are present. The rel-ative amount of these genetically dierent kaolinite

Figure 5.27: Authigenic kaolinite.

Figure 5.28: Detrital kaolinitic, clay clast.

types is dicult to quantify exactly due to lack ofobjective criteria for their identication. It seems ev-ident, however, that the relative and absolute volumeof detrital kaolinite is highest in the ne-grained sand-stones. The total volume of kaolinite in the mediumto coarse grained sandstones is generally less than10%. Most of the kaolinite in this size bracket is,however, authigenic. The specic surface area of au-thigenic kaolinite is typically of the order of 10 m2

per gram.The specic surface area of detrital kaolinite varies,

and the eective surface area, i.e., the area accessible

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5.3. MICROSCOPIC HETEROGENETIES 131

to a given chemical, depends on the size and tortuos-ity of the pore networks within the clasts.It is important to dierentiate between detrital

and authigenic kaolinite because they are likely tohave dierent eects on porosity, permeability, rela-tive permeability, pore size distribution, and adsorp-tion of surfactants.Authigenic, pore-lling kaolinite normally only lls

some of the pores, while surrounding pores may con-tain no kaolinite. The porosity, i.e., the microporos-ity, within the pores containing authigenic kaolinite istypically 30 to 50%. The individual kaolinite crystalsare pseudohexagonal platelets with a length varyingfrom a few µm to a few tens of µm. The kaoliniteplatelets may also be stacked like an expanded deckof cards, vermicular morphology. The pores lledwith authigenic kaolinite will contain micropores withpore-throat diameters of typically one to a few µm.Detrital kaolinite occurs as clasts and is often grain-

supporting, which means that the kaolinite does notll the pores, but occurs as grains. The kaolinitic clayclasts may be identied by being a part of the grain-supporting matrix, containing kaolinite with poorlydeveloped crystal surfaces and having microporosityon the order of 10 to 20%. Also, the microporeswithin the kaolinite clay clasts are small. The pore-throat diameter is typically much less than 1 µm,whereas the pores lled with authigenic kaolinite havepore throat diameters in the order of 1 µm or more.

Origin of Kaolinite

In order to predict the distribution of authigenic anddetrital kaolinite in a reservoir, it is necessary to un-derstand its origin. It is generally agreed that most ofthe authigenic kaolinite in the North Sea sandstonesformed when the sediments were exposed to meteoricwater,8387 by transformation of feldspars and micaaccording to the following reactions:

2 Muscovite + 2H+ + 3 H2O

3 Kaolinite + 2K+ (5.1)

2K(Na)-feldspar + 2H+ +H2O

Kaolinite + 2K+(Na+) + 4 SiO2 (5.2)

The protons in the reaction are provided by CO2 dis-solved in the meteoric water. The Middle Jurassicsandstones of the North Sea have experienced mete-oric water ushing during and after deposition andduring subaerial exposure when the late Cimmerianunconformity was formed.86

Most of the authigenic kaolinite is likely to haveformed due to meteoric water ushing during or im-mediately after deposition, while the late Cimmerianunconformity has had an insignicant impact on thedistribution of the authigenic kaolinite.88,89 This im-plies that the distribution of authigenic (and detrital)kaolinite in North Sea reservoirs is related to sedimen-tary facies.

5.3.3 Illite

Here we will discuss the eect of authigenic illite onreservoir properties. Detrital illite has eects on per-meability which is similar to the detrital kaolinite dis-cussed above.Authigenic illite has high specic surface area (typ-

ically on the order of 100 m2/g) and occurs as brousand pore-bridging cement in sandstones, Fig. 5.29.

Figure 5.29: Authigenic illite.

Formation of Illite

Authigenic illite is often formed where kaolinite andK-feldspar are transformed to illite by the reac-tion90,91

K-feldspar + kaoliniteillite + 2SiO2 + H2O. (5.3)

Illitization of kaolinite is common only in reservoirswhere the temperature is or has been higher than120 C..92 Based on thermodynamical considera-tions, illitization may take place at temperatures aslow as 70 C..93 Only an insignicant amount of illiteis, however, observed to form at temperatures lowerthan 120 C. This suggests that illitization of kaolin-ite is kinetically controlled.93 Also, formation of illiteby transformation of kaolinite and K-feldspar may beinhibited or retarded by the presence of hydrocar-bons94 suggesting that inll of HC at temperaturesbelow 120 C may help to preserve high permeabili-ties.During incipient illitization, nucleation and growth

seem to take place on the kaolinite crystals.95 Thisimplies that the microspatial distribution of illite re-sembles the distribution of kaolinite which tends tobe patchy, i.e., lling some pores. This microspatialdistribution of illite may be less detrimental to thepermeability than if the illite were evenly distributed.In most North Sea reservoirs, authigenic illite has

not formed. The main reason is that most of the reser-voirs have not experienced temperatures required forillitization to occur.

5.3.4 Chlorite

Chlorite is reported to be present in the intra Dun-lin Cook sandstone as 1 to 2 µm sized platy, grain-

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132 CHAPTER 5. GEOLOGICAL DESCRIPTION

coating (pore-lining) cement, Fig. 5.30. The eect ofthe chlorite coating on permeability depends on the

Figure 5.30: Authiqenic chlorite.

thickness of the coating relative to the pore-throat di-ameter. Presence of surface-coating chlorite may alsoinhibit nucleation and growth of quartz cement andmay thus help to preserve porosity.96

5.3.5 Nonclay Cements

Carbonate Cements

In most shallow marine sandstone reservoirs on theNorwegian Continental Shelf, some of the porosityis destroyed by pore-lling carbonate cement, i.e.,calcite, dolomite/ankerite, siderite. Siderite occursmainly as patchy pore-lling cement, which meansthat the porosity is only reduced locally, i.e., on apore scale. The carbonate cement is often associatedwith biotite86 and clay clasts. Dolomite/ankerite alsotends to occur as patchy cement, but is also observedto have completely lled the pores in a greater volumeof the sandstones, giving rise to concretions and/orlayers of carbonate-cemented sandstones. Calcite ce-ment, which is the dominating carbonate cement,tends to occur as cemented concretions or layers.71

This type of cementation is discussed on page 129.Patchy calcite cement is only of minor importance,

except in the Oseberg Formation in the VeslefrikkField where patchy calcite cement is common.

Quartz Cement

Quartz cement grows on the quartz grains and may,if buried to depth greater than 5 km, reduce theporosity to typically less than 5%. Only insignicantamounts of quartz cement are observed in sandstoneswhich have not been buried deeper than 2 km. Pres-ence of quartz cement is not likely to aect the per-meability of the sandstones signicantly, unless thethe porosity is less than 15 to 20%.

5.3.6 Secondary Porosity

Secondary porosity is formed when unstable grainsdissolve. In the North Sea Jurassic reservoirs, sec-

ondary porosity is mainly related to dissolution offeldspars, K-feldspar and plagioclase, mica and heavyminerals. When the unstable grains are being dis-solved, clay minerals, mainly kaolinite, precipitate aspore-lling cements in the sediment. The distribu-tion of secondary porosity is therefore genetically andspatially related to the distribution of authigenic clayminerals.Secondary pores may be identied by being over-

sized and by containing rims or relics of the dissolvedgrain. Formation of secondary porosity by dissolu-tion of feldspars is mainly related to meteoric waterushing during or soon after deposition91 and duringillitization.91 The volume of new minerals formed issimilar to the volume of the minerals that is beingdissolved. Although the porosity is not signicantlyaected during illitization, the permeability is likelyto be signicantly reduced.87

5.3.7 Permeability and Petrology

The permeability of a sandstone is determined byporosity, surface area, and the distribution of the sur-face area within the pore network in the sandstone.The surface area is related to the grain size. Decreas-ing grain size is positively correlated with reduction inpermeability. For a given grain size and porosity, thepermeability will, however, also be inuenced by thetype of clay and the amount and distribution of clays.The eect of dierent types of clay on the permeabil-ity is not yet quantitatively determined. Hence, thediscussion of the relationship between permeabilityand clay is only qualitative.Authigenic kaolinite is pore-lling and therefore

will occur in pores that are part of the pore network.Since the pores that are (partly) lled with authigenickaolinite mainly contribute to microporosity, the per-meability will rst be reduced in these pores. How-ever, if the number of pores containing pore-llingkaolinite becomes high, i.e., when the communicationbetween the clean pores is interrupted, the (bulk) per-meability will be reduced. This situation is not likelyto occur unless the amount of authigenic kaolinite ex-ceeds 10 wt%. Such high values for authigenic kaoli-nite are uncommon in the North Sea sandstones.Detrital kaolinite does not aect the pore network

in the same way as authigenic kaolinite because it oc-curs (partly) as grain-supporting clasts and not as apore-lling cement. The detrital kaolinite clay clastsare less resistant to mechanical compaction thanquartz and feldspar grains and are thus likely to besomewhat deformed/squeezed during burial due to in-crease in the eective stress. Deformation/squeezingof the clay clasts will therefore partly ll the neigh-boring pores. Presence of detrital kaolinite may thusbe detrimental to the permeability.Detrital kaolinite is, however, not likely to be as

detrimental to the permeability as the same amountof authigenic kaolinite would have been. The eect ofsqueezing the kaolinite clay clasts is only likely to bemore detrimental to the permeability in sandstones

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5.4. MINIPERMEAMETER USE 133

where the clay clasts constitute a signicant part ofthe grain-supporting matrix.In reservoirs where authigenic illite has formed, the

permeability is typically reduced by more than oneorder-of-magnitude.87 The main reason for the reduc-tion is the very high specic surface area of the illitewhich often occurs as pore-bridging cement tendingto block the pore throats.Since incipient illitization of kaolinite and K-

feldspar seems to take place on the kaolinite crys-tals,95 which only ll some of the pores while mostof the pores are clean, the eect of illite growth onpermeability is likely to be minor until the amount ofillite formed has exceeded some critical value. Also,the amount of illite likely to form will be controlledby the amount of kaolinite and K-feldspar present inthe sandstone prior to illitization.The eect of pore lining, i.e., grain coating chlorite

on the permeability is only likely to be signicantwhen the thickness of the chlorite layer is comparableto the pore-throat size.

5.3.8 Relative Permeability

Relative permeability and residual oil saturation aredependent on the pore-size distribution, determinedby Hg-injection curves, and the wettability, deter-mined by the Amott test. The pore-size distributionmeasurements and the wettability tests do not, how-ever, give any information on the spatial distributionof the dierent pore sizes and the wettability. It isimperative to relate the geometrical arrangement ofthe pores and the minerals to the relative permeabil-ity and residual oil saturation measurements, sincethe spatial distribution of the pore size and the wet-tability is likely to have impact on these parameters.Improved understanding of these relationships callsfor a close cooperation between geologists and reser-voir engineers.

5.3.9 Retention of Chemicals

Surfactant molecules are much less than 1 µm, poly-mers greater than 1 to 10 µm. The accessible sur-face area in a sandstone will therefore be dierentfor surfactants and polymers. The size of the pore-throats in both authigenic kaolinite and clay clasts isso small that polymers are not likely to have accessto the surface area within these minerals.97 Surfac-tants, however, are likely to have access to a greaterportion of the surfaces associated with clay clasts andauthigenic kaolinite.98

As opposed to pores associated with loosely packedauthigenic kaolinite, the pores within the detrital clayclast are not likely to be eectively ushed duringchemical ooding.98 Adsorption of surfactants tomineral surfaces associated with the detrital clay clastis therefore likely to take place by diusion of surfac-tants to the interior of the clast.98 Diusion is a slowprocess and it may take weeks to achieve adsorptionequilibrium.98 It is therefore important to determine

the amount and type, as well as the spatial distribu-tion of the kaolinite present, in order to understandthe adsorption of surfactants in a sandstone.In sandstones containing authigenic illite, which

has a specic surface area that is ten times that of au-thigenic kaolinite, the adsorption of surfactants ontothe mineral surface is likely to be very high. This, to-gether with the low permeability in these sandstones,may be detrimental to the economy of a surfactantproject.

5.4 Minipermeameter Use

The minipermeameter can be applied for two dier-ent aspects of reservoir characterization. The rstand most obvious application is the quantication ofsmall-scale heterogeneities to study their inuence onuid ow. The other application is collection of alarge amount of facies-related permeability data forassigning more realistic permeability values to grid-blocks.The importance of small-scale (0.001 to 0.1 m)

heterogeneities on reservoir behavior99102 has stimu-lated the eort to describe reservoirs in greater detail.Traditionally, quantitative assessment of small-scaleheterogeneities depended mainly on core plug mea-surements. However, minipermeameter data are in-creasingly used to supplement core plug data, and themethod has the advantages of rapid data acquisition,low expenses, being nondestructive, and sampling canbe made virtually anywhere on the rock surface. Anumber of studies4,103117 have used the miniperme-ameter successfully to explore small-scale variationson outcrop surfaces as well as cores.The signicant increase in data permits more re-

liable calculations of facies-related statistical param-eters, e.g., mean, standard deviation, coecient ofvariation, for use in reservoir modelling. In addi-tion, the increased database provides more valid per-meability distribution functions (PDF's) and vari-ograms. Traditionally, permeability has been as-sumed to be log-normally distributed, but recentwork has indicated facies dependency,4 and probablyscale dependency for the PDF's. The permeameterhas the potential to enable acquisition of data vol-umes large enough to sort out this question.Permeameter analyses will also help with more con-

dent subdivision of a reservoir in ow units.118,119

Especially important is the potential for detection ofthin, high and low permeable zones that are criticalfor ow in reservoirs.

5.4.1 The Electronic Field and Labo-ratory Permeameter

Scientists from Department of Mineral Resources atImperial College, have constructed a state-of-the-art electronic eld permeameter for making nonde-structive permeability measurements on outcrops and

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134 CHAPTER 5. GEOLOGICAL DESCRIPTION

cores.113,120 The technical principle for permeabil-ity estimation involves injection of gas (nitrogen)into a porous rock such that ow in the combinedpermeameter-rock system is laminar and steady state.For a given pressure dierential, the ow rate will beproportional to the eective permeability of the rockvolume sampled. The injection pressure is measuredusing an electronic pressure transducer positioned in-side the probe at the point of injection, and the pres-sure dierential is simply the transducer reading mi-nus the ambient atmospheric pressure. Flow rate isdetermined using a series of laminar ow elements(LFE's) and a micromanometer. Having experimen-tally derived a relationship between micromanome-ter reading and permeability (Hassler-calibrated coreplugs), it is possible to calculate permeability val-ues from measured pressure transducer and micro-manometer readings.Statoil has developed an automatic laboratory ver-

sion of the eld permeameter.115 This laboratory per-meameter is coupled to a computer-controlled sampletable which can be programmed to move the core slabsamples within a predened grid. A laser is attachedto the permeameter probe to detect and avoid frac-tures and holes in the core material. The recording ofmeasurements is also computer controlled. The labo-ratory permeameter can therefore be programmed tocollect detailed permeability data in a very ecientmanner. Hundreds of measurements can be takenovernight. The computer control of the sample tablealso enables a much higher precision in the locationof the measurement probe than can be achieved usingthe hand-held eld permeameter, and so the labora-tory permeameter is ideal for collecting detailed dataon small-scale heterogeneities using a high resolutionsampling grid.Empirical calibration of the permeameters usually

follows the procedure outlined by Cadman,121 basedon standard homogenous core plugs with a diameterof 1.5 inches. Homogeneity of the core plugs is eval-uated by visual inspection and CAT-scanning, andtheir permeability is derived by measurements in aHassler cell. The calibration procedure involves vepermeameter measurements from each end of the coreplugs. The net ow rate for each plug is estimatedby taking the geometric mean of the measurementsfrom the two ends, followed by harmonic averagingof the two means. Regression analysis on the Hasslercell permeabilities and the permeameter ow rates iscarried out to dene the linear relationship and corre-lation coecients between the two parameters. Thisprocedure is carried out for all the LFE's, so the in-strument is recording accurate measurements of owrate over the 1 md to 15 darcy range.

5.4.2 Data Collection

Sampling grids both for outcrop and core studiesmust be designed so that the data can be subjectedto statistical analysis. A regular grid oriented or-thogonally to the primary depositional bedding is

most commonly used. The rock face must be rathersmooth, dry, and free from dust to obtain reliablepermeability values. For outcrop studies, it is alsoessential to remove the outer weathered crust.Geological features like grainsize, sorting, sedimen-

tary structures and genetic facies have to be recordedat each sampling point. Then the measured ow rateand geological data are entered into a spreadsheet ona portable PC. The empirical relationships betweenow rates and permeability are included as macrofunctions in the spreadsheet so that permeability val-ues are calculated directly from the ow rate readings.Each measurement takes about 30 to 60 seconds de-

pending on the permeability. The ow needs longerstabilization time for low permeabilities. One personmay sample 300 to 500 permeability values per day.However, measurements on outcrop require properface preparation, and this usually controls the overallsampling eciency for such work.

5.4.3 Scale

The volume of investigation per measurement is ofthe order of 1 to 2 cm3, and the data should besubjected to a scaling procedure for application inreservoir modelling. The scaling procedure depend onthe sedimentary facies.100,122 Since the statistical pa-rameters (e.g., mean, standard deviation, coecientof variation) and functions (PDF's and variograms)probably are scale dependent, the statistics have tobe recalculated on each scale.In addition to volume considerations, one also must

consider the hemispherical permeameter ow patternwhen comparing with permeability data from coreplugs. However, there seems to be signicant correla-tion between permeameter data and horizontal per-meability data from core plugs.4

Nomenclature

h = thickness, m` = length, mw = width, m

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[68] Kantorowicz, J.D., Bryant, I.D., and Dawans,J.M.: Controls on Geometry and Distributionof Carbonate Cements in Jurassic Sandstones:Bridport Sands, Southern England and theViking Group, Troll Field, Norway,Diagenesisof Sedimentary Sequences, J.D. Marshall (ed.),Blackwell, Oxford (1987) 103118.

[69] Walderhaug, O., Bjørkum, P.A., and NordgårdBolås, H.M.: Correlation of Calcite-CementedLayers in Shallow-Marine Sandstones of theFensfjord Formation in the Brage Field, Corre-lation of Hydrocarbon Exploration, J.D. Collins(ed.), Graham & Trotman, London (1989) 367375.

[70] Bjørkum, P.A. and Walderhaug, O.: Geo-metrical Arrangement of Calcite Cementationwithin Shallow Marine Sandstones, Earth-Science Reviews (1990) 29, 145161.

[71] Bjørkum, P.A. and Walderhaug, O.: LateralExtent of Calcite Cemented Layers in Shal-low Marine Sandstones North Sea Oil and GasReservoirs - II , A.T. Buller et al. (eds.), Gra-ham & Trotman, London (1990) 331336.

[72] Fursich, F.T.: Rhythmic Bedding and ShellBed Formation in the Upper Jurassic of EastGreenland, Cyclic and Event Stratication, G.Einsele and A. Seilacher (eds.), Springer, Berlin(1982) 208222.

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[76] Melville, R.V. and Freshney, E.C.: The Hamp-shire Basin and Adjoining Areas, Inst. of Geol.Sci., London (1982) 146.

[77] Allen, P.A. and Underhill, J.E.: Swa-ley Cross-Stratication Produced by Unidirec-tional Flows, Bencli Grit (Upper Jurassic),Dorset, UK, J. Geol. Soc. (1989) 146, 241252.

[78] Kumar, N. and Sanders, J.E.: Characteristicsof Shoreface Storm Deposits: Modern and An-cient Examples, J. Sed. Pet. (1976) 46, 145162.

[79] Aigner, T. and Reineck, H.E.: ProximityTrends in Modern Storm Sands from the Hel-goland Bight (North Sea) and Their Implica-tions for Basin Analysis, Senkenbergiana Mar-itima (1982) 14, 183215.

[80] DeCelles, P.G.: Variable Preservation of Mid-dle Tertiary Coarse-Grained, Nearshore toOuter Shelf Storm Deposits in Southern Cal-ifornia, J. Sed. Pet. (1987) 57, 250264.

[81] Nummedal, D. and Swift, D.J.P.: Transgres-sive Stratigraphy at Sequence-Bounding Un-conformities: Some Principles Derived fromHolocene and Cretaceous Examples, Soc.Econ. Paleo. Min. Spec. Publ., Tulsa (1987) 41,241260.

[82] Kennedy, W.J. and Garrison, R.E.: Morphol-ogy and Genesis of Nodular Chalks and Hard-grounds in the Upper Cretaceous of SouthernEngland, Sedimentology, (1975) 22, 311386.

[83] Hancock, N.J. and Taylor, A.M.: Clay Min-eral Diagenesis and Oil Migration in the MiddleJurassic Brent Sand Formation, J. Geol. Soc.(1978) 135, 6972.

[84] Sommer, F.: Diagenesis of Jurassic Sandstonesin the Viking Graben, J. Geol. Soc. (1978)135, 637.

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138 CHAPTER 5. GEOLOGICAL DESCRIPTION

[85] Bjørlykke, K., Elverhøi, A., and Malm, A.D.:Diagenesis in Mesozoic Sandstones from Spits-bergen and North Sea: A Comparison, Geol.Runds. (1979) 68, 115171.

[86] Bjørlykke, K. and Brendsdal, A.: Diagene-sis of the Brent Sandstones in the StatfjordField, North Sea, Roles of Organic Matter inSediment Diagenesis, D.L. Gautier (ed.), Spec.Public. Soc. Econ. Paleont. Miner. (1986) 38,15767.

[87] Bjørlykke, K., Nedkvitne, T., Ramm, M., andSaigal, G.C.: Diagenetic Processes in the BrentGroup (Middle Jurassic) Reservoir of the NorthSea - An Overview, Geology of the BrentGroup, Morten et al., (eds.), Geol. Soc. london.(1992) (in press).

[88] Bjørkum, P.A., Mjøs, R., Walderhaug, O., andHurst, A.: The Role of the Late CimmerianUnconformity for the Distribution of Kaolin-ite in the Gullfaks Field, Northern North Sea,Sedimentology (1990) 37, 395406.

[89] Bjørkum, P.A., Mjøs, R., and Walderhaug, O.:Kaolinittfordeling og bergartsegenskaper medeksempler fra Gullfaksfeltet, SPOR report No.2, Rogaland Research, Stavanger (1988).

[90] Bjørlykke, K.: Diagenetic Reactions in Sand-stones, Sediment Diagenesis, A. Parker andB.W. Sellewood (eds.), NATO ASI Series, Rei-del Publishing Company (1983), 169213.

[91] Bjørlykke, K.: Formation of Secondary Poros-ity. How Important Is It? AAPG Mem. (1984)37, 28592.

[92] Hower, J., Eslinger, E.V., and Perry, E.A.:Mechanism of Burial Metamorphism ofArgillaceous Sediments: 1. Mineralogical andGeochemical Evidence, GSA Bull. (1976) 87,72537.

[93] Bjørkum, P.A. and Gjelsvik, N.: An Isochemi-cal Model for Formation of Authigenic Kaolin-ite, K-feldspar and Illite in Sediments, J. Sed.Pet. (1988) 58, 50611.

[94] Thomas, M.: Diagenetic Sequences and K/ArDating in Jurassic Sandstones, Central VikingGraben: Eect on Reservoir Properties, ClayMinerals (1986) 21, 695710.

[95] Ehrenberg, S.N. and Nadau, P.: Formation ofDiagenetic Illite in Sandstones of the Garn For-mation, Haltenbanken Area, Mid-NorwegianContinental Shelf, Clay Minerals (1989) 24,23353.

[96] Hield, M.T. and Larese, R.E.: Inuence ofCoatings on Quartz Cementation, J. Sed. Pet.(1974) 44, 1296374.

[97] Lund, T. et al.: Polymer Retention and In-accessible Pore Volume in North Sea ReservoirMaterial, paper presented at the 1991 Euro-pean Symposium on IOR, Stavanger, May 2123.

[98] Austad, T. Bjørkum, P.A., and Rolfsvåg, T.A.:Adsorption II. Nonequilibrium Adsorption of

Surfactants onto Three Reservoir Cores fromthe Norwegian Continental Shelf. The Eect ofClay Minerals, J. Pet. Sci. Eng. (1991) 6, 12535.

[99] Kortekaas, T.F.M.: Water/Oil Displace-ment Characteristics in Crossbedded ReservoirZones, SPEJ (Dec. 1985) 91726.

[100] Weber, K.J. and van Geuns, L.C.: Frameworkfor Constructing Clastic Reservoir SimulationModels, paper SPE 19582 presented at the1989 SPE Annual Technical Conference and Ex-hibition, San Antonio, Oct. 811.

[101] Muggeridge, A.H.: Generation of Pseudo Rela-tive Permeabilities from Detailed Simulation ofFlow in Heterogeneous Porous Media, Reser-voir Characterization II , L.W. Lake et al.(eds.), Academic Press, New York (1989) 197225.

[102] Ringrose, P.S. et al.: Relevant Reservoir Char-acterization: Recovery Process, Geometry andScale, paper presented at the 1991 EuropeanSymposium on IOR, Stavanger, May 2123.

[103] Weber, K.J.: Inuence of Common Sedimen-tary Structures on Fluid Flow in ReservoirModels, JPT (March 1982) 66572.

[104] Goggin, D.J., Chandler, M.A., Kocurek, G.A.,and Lake L.W.: Patterns of Permeability inEolian Deposits, paper SPE/DOE 14893 pre-sented at the 1986 SPE/DOE Symposium onEnhanced Oil Recovery, Tulsa, April 2023.

[105] Stalkup, F.I.: Permeability Variations Ob-served at the Faces of Crossbedded SandstoneOutcrops, Reservoir Characterization, L.W.Lake, and H.B. Carrol Jr. (eds.), AcademicPress, New York (1985) 14179.

[106] Stalkup, F.I. and Ebanks, W.J.Jr.: Perme-ability Variation in a Sandstone Barrier Island-Tidal Channel-Tidal Delta Complex, FerronSandstone (Lower Cretaceous), Central Utah,paper SPE 15532 presented at the 1986 SPEAnnual Technical Conference and Exhibition,New Orleans, Oct. 58.

[107] Goggin, D.J., Chandler, M.A., Kocurek, G.A.,and Lake L.W.: Patterns of Permeability inEolian Deposits: Page Sandstone (Jurassic),North-Eastern Arizona:, SPEFE (June 1988)297306.

[108] Chandler, M.A., Kocurek, G., Goggin, D.J.,and Lake, L.W.: Eects of Stratigraphic Het-erogeneity on Permeability in Eolian SandstoneSequence, Page Sandstone, Northern Arizona,AAPG Bull. (1989) 73, No. 5, 65868.

[109] Goggin, D.J., Chandler, M.A., Kocurek, G.,and Lake, L.W.: Permeability Transects in Eo-lian Sands and Their Use in Generating Ran-dom Permeability Fields, paper SPE 19586presented at the 1989 SPE Annual TechnicalConference and Exhibition, San Antonio, Oct.811.

[110] Kittridge, M.G., Lake, L.W., Lucia, F.J., and

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REFERENCES 139

Fogg, G.E.: Outcrop-Subsurface Comparisonsof Heterogeneity in the San Andreas Forma-tion, SPEFE (Sept. 1990) 25973.

[111] Dreyer, T., Scheie, Å., and Walderhaug, O.:Minipermeameter-Based Study of Permeabil-ity Trends in Channel Sand Bodies, AAPGBull. (1990) 74, 3794.

[112] Jensen, J.L.: A Model for Small-Scale Per-meability Measurement With Applications toReservoir Characterization, paper SPE/DOE20265 presented at the 1990 SPE/DOE Sym-posium on EOR, Tulsa, April 22-25.

[113] Lewis, J., Lowden, B., and Hurst, A.: Per-meability Distribution and Measurement ofReservoir-Scale Sedimentary Heterogeneities inSubsurface Exposures of a Shallow MarineSand-Body, Field Guidebook A11, the 1990International Sedimentological Congress, Not-ingham, Sept. 13.

[114] Corbett, P.W. and Jensen, J.L.: A Compar-ison of Small-Scale Permeability MeasurementMethods for Reservoir Characterization, paperpresented at the 1991 Conference on Advancesin Reservoir Technology, Edinburgh, Feb. 2122.

[115] Hurst, A. and Rosvoll, K.J.: Permeabil-ity Variations in Sandstones and Their Re-lationship to Sedimentary Structures, Reser-voir Characterization II , L.W. Lake, H.B. Car-roll, T.C. Wesson (eds.), the 1989 InternationalReservoir Conference, Dallas, Academic Press,16696.

[116] Jacobsen, T. and Rendall, H.: Permeabil-ity Patterns in Some Fluvial Sandstones. AnOutcrop Study from Yorkshire, North EastEngland, Reservoir Characterization II , L.W.Lake, H.B. Carroll, and T.C. Wesson (eds.),Academic Press, New York (1989) 31538.

[117] Lowry, P. and Jacobsen, T.: Sedimentologi-cal and Reservoir Characteristics of a Fluvial-Dominated Delta Front Sequence: Ferron Sand-stone Member (Turonian), East-Central Utah,paper presented at the 1990 Conference on Ad-vanced Reservoir Geology, London, Jan. 2930.

[118] Hearn, C.L., Ebanks, W.J., Tye, R.S., andRanganathan, V.: Geological Factors Inuenc-ing Reservoir Performance of the Hartzog DrawField, Wyoming, JPT (Aug. 1984) 133544.

[119] Ebanks, W.J.: Flow Unit Concept-IntegratedApproach to Reservoir Description for Engi-neering Projects, AAPG Bull. (1987) 71, 5511.

[120] Daltaban, T.S., Lewis, J.J.M., and Archer, J.S.:Field Mini-Permeameter Measurements-TheirCollection and Interpretation, paper presentedat the 1989 European Symposium on IOR, Bu-dapest, April 2527.

[121] Cadman, M.: Nondestructive PermeabilityMeasurement, MS-thesis, Heriott-Watt U.(1984).

[122] Holden, L., Høiberg, J., and Lia, O.: An Es-timator for Eective Permeability, paper pre-sented at the 1990 SPE European Conferenceon the Mathematics of Oil Recovery, Paris,Sept. 1114.

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Chapter 6

Heterogenity Models

6.1 Stochastic Models for Re-servoir Characterization

6.1.1 Introduction

Fluid ow in petroleum reservoirs is usually modeledby a set of dierential equations. Various character-istics of the reservoir are included in the equations ascoecients or initial and boundary conditions. Ex-amples are porosity, initial saturations and perme-ability. These characteristics have to be specied ev-erywhere in the reservoir. Given that the characteris-tics are observed in the wells only, one faces the prob-lem of dening a spatial representation based on theseand additional general geological knowledge. If tradi-tional spatial interpolation is used, the representationof these characteristics will appear as smoothed. Thiswill entail biased results when solving the ow equa-tions. By utilizing a stochastic model for reservoircharacterization, various pieces of knowledge and ob-servations can be integrated. Stochastic simulationcan provide realizations which reproduce the hetero-geneity in the characteristics. By solving the set ofequations for one such realization, less biased resultscan be obtained. The results obtained from a set ofrealizations of reservoir characteristics will representthe uncertainty associated with the stochastic model.For reservoirs with considerable heterogeneity, three-dimensional stochastic models have to be used in or-der to produce reliable results.It is convenient to operate on two dierent scales

when representing reservoir characteristics. Thereservoir characterization scale will normally be neso that geological experience can be used in verica-tion of the results. Grids with 105 to 1011 cells shouldbe used. In order to provide input to the reservoirproduction simulators, a change of scale to a systemof 103 to 105 blocks must be made. This is denotedhomogenization and it is discussed in Sec. 8.3.In this section quantitative modelling of reservoir

characteristics will be discussed. Emphasis will beput on stochastic modelling and simulation since thisprovides a tool for ecient integration of dierenttypes of information, can reproduce the heterogeneityand makes assessment of uncertainty possible. Sta-tistical methodology is relevant for several aspects ofreservoir evaluation, but in this section only spatial

models for reservoir description will be presented.

Stochastic models for reservoir characteristics maybe process-related or descriptive. The former entailsmodelling of physical phenomena involved in the tec-tonic and sedimentary processes creating the reser-voir. Hence the time dimension constitutes an impor-tant part of the model. Some studies in two dimen-sions are reported, see Jacod and Joathon,1 Bridgeand Leeder2 and Alexander.3 The extension to threedimensions is complicated and the number of param-eters required increases dramatically. Few, if any,studies of particular reservoirs are reported, althoughthe work by Tetzlaf and Harbaugh4 looks promising.The resemblance to basin modelling except for thescale, should be recognized. In descriptive modelling,the time dimension is usually ignored and emphasisis put on describing the reservoir the way it presentlyappears. Structural and sedimentary characteristicsare often modelled independently and then merged.This approach encourages simple models with rela-tively few parameters. Using descriptive modelling inreservoir characterization has gained increasing popu-larity in recent years and numerous articles have ap-peared in journals and proceedings. Unfortunately,most of the studies are for two dimensions, althoughthe need for three-dimensional models of heteroge-neous reservoirs are recognized. Some key referenceson the applied side are Haldorsen and Lake,5 Begg etal.,6 Mathews et al.,7 Nybråten et al.,8 Rudkiewicz etal.,9 Høiberg et al.,10 Alabert and Massonnat,11 andDamsleth et al.12

A stochastic model for the characteristics of a par-ticular reservoir must be dened subject to: (1) thetype of depositional and post-depositional processthat formed the reservoir; (2) the question concerningthe reservoir sought answered; (3) and the amountand type of information available. Reservoirs de-posited in uvial, deltaic and marine environmentsdo usually require quite dierent model formulations.The model should, regardless of type, be kept sim-ple, although oversimplication can cause nonrepre-sentative results. The art of modelling is to denea simple but still representative model. The repre-sentation depends on the problem to be solved. Inevaluation of uid ow heterogeneities is the charac-teristics that have proven important; while in the pre-diction of hydrocarbon in place it appears to be less

141

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142 CHAPTER 6. HETEROGENITY MODELS

important. The fact that the representation must berescaled into a system of blocks suitable as input tothe production simulator encourages simplicity. Usu-ally, few observations but extensive geologic experi-ence are available for the reservoirs under study. Anapproach based on a qualied guess of the charac-teristics which later is updated by the informationin the observations as they appear, looks attractive.If direct observations were abundant, a much moredata-adaptive procedure should be used. The inter-pretations of the depositional origin of the reservoirare often associated with uncertainty, hence it couldbe convenient to evaluate several stochastic models inparallel.The models for reservoir characteristics dened

here will be used for evaluation of recovery, oftenincluding the application of enhanced recovery meth-ods. Hence, heterogeneity appears as important. Theheterogeneity structure is usually very complex andstochastic models based on strong assumptions aboutstationarity tend to fail. Experience from full-eldmodelling seems to have resulted in a consensus thattwo-stage models must be used. The rst stage es-tablishes the facies architecture of the reservoir, whilein the second stage, petrophysical characteristics likeporosity, saturation and permeability properties, areassigned to each facies element. The petrophysicalmodel will depend on the facies present. This ap-proach to modelling makes it possible to segregatestrongly dierent petrophysical populations, and itenables the user to apply general geological knowl-edge in specifying the model parameters.Key references to two-stage models are Omre et

al.,13 Aasen et al.14 Rudkiewicz et al.,9 Alabert andMassonnat,11 and Damsleth et al.12

A two-stage stochastic modelling approach to reser-voir characterization will be used in this section. Thediscussion will be based on the following notationwith random variables being indicated by capital let-ters, as usual in statistics:

• L(x);x ∈ D random eld having discreteoutcomes, with x being the spatial reference vari-able in the domain D, representing the reservoir,and the L(x) takes labels, l1, . . . , n, represent-ing the facies types. A model for the spatialdependence of L(x) in D will dene the faciesarchitecture.

• Y(x)|L(x) = l(x);x ∈ D random func-tion having multivariate, continuous outcomeswith x and D as above. The random func-tion is conditioned on the actual facies archi-tecture, L(x) = l(x);x ∈ D, and Y(x) =(φ(x), s(x), kv(x), kh(x), . . .) represents porosity,saturation, permeability properties etc. A modelfor the intervariable and spatial dependence ofY(x) in D for each possible outcome of L(x)will dene the petrophysical characteristics ofthe reservoir.

The rest of this section will be devoted to the dis-

cussion of possible spatial stochastic models for thefacies architecture and the petrophysical characteris-tics. These models will of course be dependent on aset of model parameters to which values must be as-signed in order to make the model operable. Proce-dures for estimating these parameters based on avail-able information from the reservoir under study willnot be presented here. Parameter estimation in spa-tial settings is normally quite complicated, however,and the associated uncertainty is often large.A more thorough discussion of models and of pa-

rameter estimation, with numerous other applica-tions, can be found in Hjort and Omre.15 Otherreferences to general methodology are Journel andHuijbregts,16 Ripley,17,18 Yaglom19 and Stoyan etal.,20 while petroleum-specic references are Hal-dorsen and MacDonald,21 Dubrule,22 and Haldorsenand Damsleth.23

6.1.2 Facies Architecture Models

The facies architecture is represented by the discreterandom eld L(x);x ∈ D. Two classes of phenom-ena can be dened:

• event phenomenarepresenting the occurrenceof smaller facies units in a background facies.In Fig. 6.1, from Haldorsen and Lake,5 the out-come of simulation of shale units in a predomi-nant sand matrix is presented. Other examplesare river beds and uvial fans in a tidal at back-ground and micro faults in a reservoir.

• mosaic phenomenarepresenting a packing ofdierent facies with no facies constituting a back-ground. In Fig. 6.2, from Fält et al.,24 the out-come of simulated facies in a complex deltaic en-vironment is presented. Another example maybe calcite-cemented sheets in certain horizons.

Figure 6.1: Example of event phenomenon, from Hal-dorsen and Lake.5

Note that choice of model class depends on both thetype of phenomenon and the scale. Hence, in manycases mixed models can be of interest. In Fig. 6.3,from Høiberg et al.,10 the shales in a sand matrix aregenerated by a mixed model.

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6.1. STOCHASTIC MODELS FOR RESERVOIR CHARACTERIZATION 143

Figure 6.2: Example of mosaic phenomenon, fromFält et al.24

Figure 6.3: Example of mixture of event and mosaicphenomena, from Høiberg et al.10

Event phenomena

Event phenomena are usually modelled by markedpoint processes, see Stoyan et al.20 The term pointprocesses concerns the distribution of points in a do-main and the prex, marked, says that a set of at-tributes is assigned to each point. A marked pointmay be denoted

U = (X,L,S), (6.1)

with X being a random reference location in D, Lthe facies label and S = (S1, . . . , Sm) a multivariate,random variable, discrete or continuous, representingshape, size, orientation etc. In Fig. 6.1, each shaleunit is represented by one marked point. The modelmay also represent reservoirs with several facies.The stochastic model for a specic reservoir is de-

ned over the domain D by the background faciespresent, the number of facies units, n, and their rela-tions through the joint probability distribution

Prob U1 = u1, . . . ,Un = un . (6.2)

This species the probability for all possible outcomesof the n marked points in D, eg. all possible cong-urations of shale units. It is easy to see that thedimension of the joint probability is n× (3 + 1 +m)which often is high. In practice, a parametrization ofthe joint probability is used,

Prob U1 = u1, . . . ,Un = un =

c0 exp

n∑i=1

β(ui) +

n∑i=1

n∑j=1

γ(ui,uj)

,

(6.3)

with co being a normalizing constant, β(·) a speci-ed function for the relation between references andattributes for each marked point, and γ(·, ·) a speci-ed function for pairwise interaction between marked

points. The function β(·) may be specied such thatlarger facies units tend to be located in the lower partof the reservoir. The interaction function γ(·, ·) mayprohibit overlaping of two facies units with the samefacies or it may impose attraction between certaintypes of facies. This parametrization is of course verysimple since only pairwise interactions are considered,and one could imagine interactions up to order n. Inthe example below, it is shown that even a simpleparametrization like this is surprisingly rich.Few analytical results are available based on this

model since the mathematics usually get very com-plex. One result is directly obtainable, however,

Prob Uk = uk|Ui = ui; i = 1, . . . , n; i 6= k =

Prob Ui = ui; i = 1, . . . , nProb Ui = ui; i = 1, . . . , n; i 6= k

=

c0 exp

β(uk) +

n∑i=1

γ(uk,ui)

. (6.4)

The simple form of the conditional probability is theprimary motivation for this form of parametrizationand it is extensively used for dening an ecient sim-ulation procedure for the stochastic model. Note thatif γ(uk,ui) is a constant for all pairs of marked pointswith |xk−xi| larger than a distance r, the conditionalprobability of the marked point in an arbitrary loca-tion, given the other marked points, will only dependon the points located closer than r. Then some sortof Markov behavior is said to appear, see Baddeleyand Møller.25 Markov properties will be more thor-oughly discussed later. It is worth noting that byignoring the marks and considering the point processonly, and by setting β(·) and γ(·, ·) to constants, allpoint congurations in D will have the same proba-bility. This corresponds to the familiar Poisson dis-tribution of points within D.From observations of the reservoir under study, the

specic values of some characteristics in certain loca-tions and more precise estimates of global parameterslike volume fraction of facies and petrophysical vari-ability, can be obtained. Seismic data may also beavailable. The reservoir realizations generated fromthe stochastic model must reproduce this informa-tion to the precision that they are determined. Thestochastic model can be extended in order to enforcethese constraints:

Prob Ui = ui; i = 1, . . . , n =

c0 exp

n∑i=1

β(ui) +

n∑i=1

n∑j=1

γ(ui,uj)

× exp

−r∑j=1

σj [wj − gj(uk; k = 1, . . . , n)]2

.

(6.5)

The last multiplication factor is a penalty function fordeviations between the r constraints wi; i = 1, . . . , rand the corresponding property of the realization

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144 CHAPTER 6. HETEROGENITY MODELS

specied by the functions gi(·); i = 1, . . . , r. Eachconstraint is associated with a strength parameter σi.An increase of σi reduces the tolerance of the corre-sponding constraint. By setting the strength param-eter to zero the corresponding constraint is ignored.Realizations of marked-point processes can be gen-

erated by the Ripley-Kelly death/birth procedure, seeRipley.18 The procedure is dened for a given num-ber of marked points, n, and is initiated by a randomgeneration of n marked points in D having positiveprobability. An iterative algorithm is used, such thatin each iteration one marked point is randomly drawn,say uk, and an arbitrary new marked point u0 is gen-erated from ProbU0 = u0|Ui = ui; i = 1, . . . , n; i 6=k. The identied uk is replaced by u0. This it-erative procedure will, after convergence is reached,have generated a realization according to the modelof the marked point process. The number of iter-ations required depends crucially on the complexityof the model, but 10 × n iteration can be used as arule of thumb. Clever implementations can usuallyimprove the rate of convergence considerably, partic-ularly when constraints are present.The actual realization of facies architecture,

l(x);x ∈ D, is dened by the conguration ofmarked points, each of them representing a faciesunit. In cases with overlap of units with dierentfacies labels, a priority rule has to be dened in orderto obtain a unique realization.

Example from Høiberg et al.10 Consider themarked point, which may represent one shale unit,

U = (X,L,W, T ) (6.6)

whereX is a three-dimensional reference in D, L is la-bel for shale, W is the horizontal width, length beingequal, and T the thickness. As previously discussed,the marked-point process denes the joint probabilityfor a given conguration of shale units. The numberof shale units is xed and the joint probability hasβ(·) = 0.0. In Fig. 6.4, three vertical cross sectionsfor the three dierent cases are presented:(1) no interaction, which can be obtained by setting

γ(·, ·) = 0.0. (6.7)

Note that no interaction produces relatively frequentoverlap of shales.(2) no spatial overlap, which can be obtained by

setting

γ(ui,uj) =

−∞ if ui and uj overlap,

0 else. (6.8)

Note that the shale units may be located arbitrarilyclose and that overlap is prevented.(3) spatial repulsion, which can be obtained by set-

ting:

(a) No interaction.

(b) No spatial overlap.

(c) Spatial repulsion.

Figure 6.4: Example of realizations from a marked-point process with varying parameter values.

γ(ui,uj) =−∞, if |xi − xj | ≤ rmin

ln

[|xi − xj | − rmin

rmax − rmin

], else

0, if |xi − xj | > rmax

(6.9)

Note that no centers of shale units are located closerto each other than rmin and that the Markov propertypreviously mentioned does appear with an inuencedistance rmax.This simple pairwise interaction parametrization

appears surprisingly rich and can be used for mod-elling many natural phenomena.Models for event phenomena are frequently ap-

plied. In Haldorsen and Lake,5 a two dimensionmarked-point process without the interaction termwas used for modelling shale units. This paper is re-garded as the pioneer work in descriptive, stochasticmodelling in reservoir description. Augedal et al.26

present a three-dimensional model for a reservoir ofuvial origin. In Haldorsen and MacDonald21 severalapplications of marked-point processes are suggestedand presented. Production forecasting in a meander-ing river system modelled by a marked-point processis evaluated in Meling et al.27 Gundesø and Ege-land28 present a program system based on a three-

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6.1. STOCHASTIC MODELS FOR RESERVOIR CHARACTERIZATION 145

dimensional, marked-point process formalism. Theirwork seems to be the rst serious attempt to modelseveral facies, although the actual interaction modeland simulation procedure look somewhat ad hoc. InClemetsen et al.29 a three-dimensional model for u-vial sand based on hierarchical generation and formalspecication of interaction between channel belts isdened. In a related work,30 it is demonstrated thatthe interaction term can be used to reproduce the in-terconnectedness ratios found in Brigde and Leeder.2

Marked-point processes are used for modelling mi-crofaults in a fault zone,31 and this can be used forevaluating uid ow across the zone. In Høiberg etal.,32 a mixed model between marked-point processesand Markov random elds for shale distribution ispresented.The experience is limited with marked-point pro-

cess models for facies architecture. This model typeappears exible and each marked point can easily beinterpreted as dierent facies units. Hence, size distri-butions and varying volume fractions in the reservoircan be specied from qualied guesses by the geolo-gists. The interaction functions are more complicatedto interpret. The Ripley-Kelly simulation procedureis reliable and it seems to converge reasonably fast.It should be mentioned that multifacies models

with a large number of constraints often cause con-vergence problems. The experiences are generally fa-vorable, however, and marked-point process modelsare probably underused in reservoir characterization.

Mosaic Phenomena

Mosaic phenomena are usually modelled by some sortof Markov random-eld model.17,33,34 The name canbe associated with the Markov property in time serieswhere the probability for next transition is dependenton the present state only, not the former history. TheMarkov random eld is dened on a grid covering thedomain D, Lh;h ∈ DG, with h being the grid nodereference (i, j, k), DG being the grid covering D andNG being the number of nodes. Since there is nospatial ordering, the Markov property states

Prob Lj = l|Li = li; i ∈ DG; i 6= j =

Prob Lj = l|Li = li; i ∈ Nj , (6.10)

with Nj being a neighborhood centered at, and ex-cluding, node j. Hence, the conditional probabilityfor facies type in grid node j, given the facies in therest of D, is dependent on the facies distribution ina neighborhood around node j only. This neighbor-hood, Nj , is usually chosen to be quite small relativeto DG. Note, however, that even for small neighbor-hoods, say 3 × 3 × 3, and a small number of faciestypes, say 5, the number of combinations, and hencethe probabilities to be specied, is enormous, namely527. To make the model operable, a parametrizationmust be dened, and in its simplest form it may be

Prob Lj = l|Li = li; i ∈ Nj =

c0 exp

β ∑i∈Nj

δlli

, (6.11)

with β being a user-specied parameter and δlk be-ing equal to one wherever l = k and zero elsewhere.Hence, the conditional probability for facies l de-pends on the proportion of facies l present in theimmediate neighborhood. In most cases, more com-plicated parametrization better adapted to the phe-nomenon at hand should be used. There are certainconstraints on valid parametrization to ensure the ex-istence of the joint probability distribution for all gridnodes, however. These constraints are specied in theHammersley-Cliord theorem.33,35

Observations made on the reservoir under studywill often provide constraints to the realizations thatcan be generated. The stochastic model above can beextended in order to enforce these constraints:

Prob Lj = l|Li = li; i ∈ DG; i 6= j =

c0 exp

β ∑i∈Nj

δlli

× exp

r∑k=1

σk [wk − gk(li; i ∈ DG)]2

,

(6.12)

with wi, σi and gi(·) being dened as previously. Notethat the strict local dependence on Nj is eliminated ifglobal constraints are enforced. This does not createproblems as long as the corresponding gi(·)'s can becomputed or updated quickly.At rst glance, the Markov and the marked-point

process models look similar. The fact, however, thatfor the latter model the spatial reference is a randomvariable, makes the models philosophically dierent.Realizations of Markov random elds can be gen-

erated by the Metropolis procedure or by the closelyrelated Gibbs sampler.17,35 The latter is based onan initial state having positive probability. An itera-tive algorithm is used. In each iteration, an arbitrarynode j is drawn, and a new label is assigned to j fromthe conditional probability, ProbLj = l|Li = li; i ∈DG; i 6= j. This iterative procedure will, after con-vergence is reached, generate realizations according tothe model for the Markov random eld. The numberof iterations required depends crucially on the com-plexity of the model, but 200×NG may be sucient,although complete convergence is seldom reached. Alarge number of exact constraints in nodes in the lat-tice and clever implementation will in general improvethe rate of convergence considerably.The actual realization of facies architecture,

l(x);x ∈ D, is dened by assigning the label inthe grid node to the corresponding grid cell.

Example from Omre et al.13 Consider a mo-saic phenomenon with two facies types only, hencel(x) ∈ 0, 1, and let for example 0 indicate sand

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146 CHAPTER 6. HETEROGENITY MODELS

and 1 indicate calcite cementation. The phenomenonis dened in a two-dimensional time horizon in thereservoir and the facies occurrences are observed inten wells penetrating the horizon. The set of well lo-cations is denoted B and the actual facies observationsl0i ; i ∈ B. Based on the observations the areal frac-tion of sand, ρs, is set to 0.7. The following Markovrandom eld model in two dimension is used:

Prob Lj = 1|Li = li; i ∈ DG; i 6= j =

c0 exp

−∑i∈Nj

βij · δ0li

× exp

−∑k∈B

σk[l0k − lk]2

−σt[ρs −1

NG·∑i∈DG

δ0li ]2

, (6.13)

where βij is a continuity parameter dependent on thedirection i − j, and δ0l is equal to one for l = 0 andzero otherwise. The last factor in the model is dueto the constraints and the strength parameters σkand σt are assigned values suciently large so thatthe probability for deviations from the specied con-straints can be ignored.In Fig 6.5, four realizations from the model gener-

ated by a slightly extended Metropolis algorithm arepresented. Observe that in all realizations the sameobservations are reproduced; the sand gross ratio is0.7 while the βij parameters are varied. The numberof iterations is 300×NG. The actual parameters aregiven in the following table:

case Nj βE,W βN,S βNE,SW βSE,NWa 7× 7 2.0 2.0 2.0 2.0b 5× 5 0.01 0.01 0.01 0.01c 5× 5 2.0 2.0 2.0 2.0d 5× 5 0.05 0.05 0.05 2.05

Even this simple model can, with varying parame-ters, generate realizations with a wide range of calcitecontinuity.Markov random elds as models for mosaic phe-

nomena have not been in frequent use. In Omre etal.,13 two-dimensional models for calcite cementationin several geologic horizons in a reservoir are dened.Fält et al.24 present a Markov model for several fa-cies in a deltaic environment. A particular type ofparametrization is used so that the size distributionfor each facies unit can be specied. In Høiberg etal.,32 a mixed model based on marked-point processesand Markov random elds for shale distribution ispresented. The Markov model is used for both shaleextent and thickness variation.Experiences from use of Markov random elds

as models for facies architecture are limited. Themodel formulation is extremely exible, but theparametrization is often complicated to interpret,hence dicult to assess from experience. The sim-ulation procedures have not proven very ecient in

(a) (b)

(c) (d)

Figure 6.5: Example of realizations from a Markovrandom eld model with varying parameter values.

three dimensions, unless a relatively large number ofexact constraints in nodes in the lattice is dened.Further research on more convenient parametrizationof Markov random elds with associated simulationprocedures should be pursued. Ecient simulationalgorithms seem to be the limiting factor.For mosaic phenomena, several alternative models

to Markov eld exist, and a brief outline of some ofthem follows.The two-point histogram approach is dened in

Farmer.36 The parametrization is by frequencies offacies labels in the grid DG. For a shift vector h, thefrequencies for all pairwise combinations of facies la-bels in (i, i + h); i, i + h ∈ DG must be specied.Normally, shift vectors to the neighboring grid nodesare used, hence the neighbor relations in the modelare given. The valid realizations should reproduce allof the two-point frequencies specied and can in theterminology previously dened be written as

Prob Lj = l|Li = li; i ∈ DG; i 6= j =

limσk→∞

c0 exp

r∑k=1

σk [wk − gk(li; i ∈ DG)]2

,

(6.14)

with wk being the label frequencies to be reproduced,

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6.1. STOCHASTIC MODELS FOR RESERVOIR CHARACTERIZATION 147

gk(·) the realization of the frequencies and σk be-ing the strength parameter. The expression ensuresthat all the constraints, wk, are being exactly repro-duced and allows the realizations to be generateduniformly random beyond these constraints. Onemay say that co implicitly denes a uniform, ran-dom stochastic model. This makes the two-point his-togram approach a particular, asymptotic case of theconstrained Markov random eld model. Farmer36

suggests using an annealing procedure, frequentlyused in physics, for simulation purposes. A cleverimplementation is required in order to make the sim-ulation algorithm feasible. No applications of the ap-proach to case studies have been found in the litera-ture, hence practical experience seems to be lacking.

The truncated random function approach is denedin Rudkiewicz et al.9 and Galli et al.37 This pro-cedure is based on the model for a continuous ran-dom function Y (x);x ∈ D and a set of thresholds(y1, . . . , yn−1) dening intervals on the real line corre-sponding to the n facies types. Models for continuousrandom functions will be more thoroughly discussedlater. The discrete random eld representing the fa-cies architecture is then dened by

L(x) =

l1 if Y (x) ≤ y1

l2 if y1 < Y (x) ≤ y2

...ln if yn−1 <Y (x).

; x ∈ D (6.15)

It is obvious that the n− 1 thresholds should be de-ned so that the required volume fractions of each fa-cies are reproduced. The authors recommend lettingthe threshold values be location-dependent. From thedenition, it is easy to see that the facies types willbe ordered in the sense that the facies lj will alwaysbe surrounded by lj−1 and/or lj+1. In many casesthis is inconvenient. The simulation procedure is de-ned by simulation of the continuous random func-tion followed by the truncation. Numerous articleson applications of the approach exist, although theyall seem to come from the Heresim group from Centrede Geostatistique and IFP, France. Most case stud-ies are based on the same data set, see Ravenne andBeucher38 and Rudkiewicz et al.9 Experience sug-gests that the model can be useful for reservoirs forwhich the facies occur in a sequence. If that is not thecase, the model should not be used. The simulationalgorithm is extremely fast.

The indicator approach to facies modelling is de-ned in Suro-Pérez and Journel.39 The model isbased on the specication of spatial covariance func-tions for indicators of dierent facies types. The ap-proach is more thoroughly discussed in the next sec-tion since it was originally used for modelling con-tinuous petrophysical characteristics. In Alabert andMassonnat,11 a case study based on this approachis presented. More experience is required before thepotential of the procedure can be fairly judged.

6.1.3 Petrophysical Models

The transformed petrophysical characteristics arerepresented by a multivariate, continuous randomfunction conditioned on the facies architecture,

Y(x) =

(Y1(x), Y2(x), . . . , Yn(x))|L(x) = l(x);x ∈ D,(6.16)

with the elements representing porosity, horizontaland vertical absolute permeability, initial saturationetc. The transformations are usually univariate, forexample logarithms of permeability, and they aremade to better fulll the assumptions of Gaussian-ity in the random function. This assumption makesa well established theory available, see Yaglom.19

Gaussianity entails that the random function is fullyspecied by the conditional expectations and covari-ances. A convenient representation is obtained by

EY(x)|L(x) = l = µl, (6.17)

CovY(x′),Y(x′′)|L(x′) = l′,

L(x′′) = l′′ = Σl′l′′ ρ(x′ − x′′), (6.18)

with µl being the vector of, µi|l; i = 1, . . . , n, the ex-pected values for the components in Y(x) given thefacies l. The intervariable covariance matrix Σlk con-tains the elements σi|lσj|kρij ; i, j = 1, . . . , n, with σi|lbeing the standard deviation of component i given fa-cies l, and ρij being the correlation between the com-ponents i and j. The spatial correlation function,ρ(x′ − x′′), is independent of both component andfacies. It is worth noting that the covariance struc-ture contains one intervariable and one spatial factor.The model can easily be extended to more complexexpectation structures, and made spatial correlation-dependent on both components and facies.This model corresponds to a random function hav-

ing the following decomposition,

(Y(x)|L(x) = l) = µl + Σl · Λ ·U(x), (6.19)

with µl as above, Σl being a diagonal matrix contain-ing σi|l, Λ being a Cholesky-factor of the correlationmatrix [ρij ] and U(x) being a vector of independentUi(x), all with expectation zero and spatial correla-tion function ρ(·). Consequently, the facies depen-dence is in µl and Σl only, the intervariable depen-dence in Λ, and the spatial dependence in Ui(x). Thismakes both statistical inference and stochastic simu-lation much simpler.Observations from the reservoir under study will

often provide constraints on the realizations that canbe generated. Consider a set of point observationsy(xi); i = 1, . . . ,m and assume that they shall beexactly reproduced. Extension to constraints of lin-ear combinations of the observations is simple. Thiscan be used for conditions of more spatially aggre-gated observations like seismic data and test produc-

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148 CHAPTER 6. HETEROGENITY MODELS

tion data. Exact reproduction entails that the condi-tional random function,

Y(x)|L(x) = l(x);Y(xi) = y(xi);

i = 1, . . . ,m; x ∈ D (6.20)

is of interest. It follows from standard probabilitytheory that the function also is Gaussian.In order to simulate the conditional random func-

tion, it is convenient to consider

Uj(x)|Uj(xi) = uj(xi); i = 1, . . . ,m;x ∈ D (6.21)

with zero expectation and spatial correlation functionρ(·). The conditioning values can be determined fromthe decomposition above.The realization of the random function is normally

stored on a grid, DG. Hence the random function maybe considered as a multi-Gaussian distribution anda traditional inversion-of-covariance-matrix approachto simulation of conditional distributions could beused. The grid size in reservoir characterization canbe up to 107, however, and this makes the tradi-tional approach prohibited from a processing pointof view. An alternative simulation procedure is pre-sented in Journel and Huijbregts.16 This procedureis based on a decomposition of the conditional ran-dom function into a nonconditional random functionand a conditional expectation function. By gener-ating the two components separately and thereaftercombining them, a fast and reliable procedure can beobtained. Nonconditional simulation of a Gaussianrandom function with a specied spatial correlationfunction can be done by various procedures. Mostfrequently used are the turning band procedure, seeJournel and Huijbregts,16 procedures in the frequencydomain, see Borgman et al.40 and lter smoothingprocedures, see Journel and Huijbregts.16 In a recentarticle, Omre et al.,41 the screening sequential proce-dure is dened and evaluated. It appears as an ex-tremely fast and reliable procedure for certain classesof correlation functions. The conditional expectationfunction can be obtained by using a simple krigingprocedure, see Journel and Huijbregts.16 Based onthis, a realization of the conditional random functioncan be generated, uj(x)|uj(xi), i = 1, . . . ,m;x ∈D.The transformed petrophysical characteristics are

represented by the realization,

(y(x)|l(x);y(xi); i = 1, . . . ,m) =

µl(x) + Σl(x) · Λ·(u(x)|u(xi); i = 1, . . . ,m);x ∈ D, (6.22)

and the petrophysical characteristics used in the owequations are obtained by the inverse transform.In Fig. 6.6, from Fält et al.,24 a vertical cross sec-

tion of one realization of the petrophysical character-istics, horizontal absolute permeability, is presented.The underlying facies architecture can be observedand the exibility of the two-stage model for repre-senting heterogeneities is exposed.

Figure 6.6: Example of realization from a two-stagemodel with Markov random eld model for facies ar-chitecture and Gaussian random function model forpetrophysical variables, from Fält et al.24

Most applications of two-stage models seem to uti-lize Gaussian random functions in the petrophysi-cal stage. The rst attempt using two-stage modelsseems to be in Omre et al.,13 where a marked-pointprocess is used as facies model. The same type ofmodel is used in Aasen et al.14 Rudkiewicz et al.9

present a Gaussian model mixed with a truncatedrandom function model for facies distribution. Thework of Alabert and Massonnat11 is based on an in-dicator approach to facies modelling and a Gaussianrandom function. In Damsleth et al.,12 a markedpoint process inspired model for facies distributionis used while Fält et al.24 present a Markov randomeld as a facies model.The experience from the use of Gaussian random

functions as models for petrophysical characteristicsin a two-stage approach is extensive. In spite ofthe lack of exibility in the Gaussian model itself, itseems to contain sucient exibility when being con-ditioned on the facies architecture. The simulationprocedures are extremely fast and reliable and thisis important when the representations are on largegrids. The fact that Gaussian random functions arebased on a well-established theory will be importantwhen statistical inference from observations is per-formed.There exist alternatives to Gaussian random func-

tions as model for petrophysical characteristics. Abrief presentation of a couple of alternatives follows.Fractal processes have gained popularity in recent

years, see Hewett42 and Feder.43 Continuous, self-ane fractal processes have been used for reservoirdescription. Note that they are closely related to theGaussian model in the sense that the increments andconditional distributions of the fractal process mostfrequently used are Gaussian. Hence, the discussionabove can easily be extended also to include the frac-tal model. The literature on fractals provides numer-ous algorithms for simulation. One should be aware,however, that not all of them are reliable in reproduc-ing a model in three dimensions with specied param-eters, see Omre et al.41 A couple of very interestingcase studies, reported in Behrens and Hewett44 andMathews et al.,7 are performed. In these studies, no

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6.2. FIELD EXAMPLES 149

facies architecture model was used. The main advan-tage of a fractal model over a Gaussian model willprobably disappear if two-stage models are applied.The indicator approach is advocated by Journel

and Alabert.45 For notational convenience, considera univariate random function not conditioned on fa-cies, Y (x);x ∈ D. In the indicator approach thecontinuous variable is coded into a number of indica-tor variables, say s, which are dened through a setof thresholds y1, . . . , ys,

Ii(x) =

1 for Y (x) ≤ yi0 else. ;i = 1, . . . , s

(6.23)By probability theory, it is simple to demonstratethat the expected value and covariance of the indica-tors are related to the univariate and bivariate char-acteristics of Y (x):

EIi(x) = ProbY (x) ≤ yi, (6.24)

EIi(x′)Ij(x′′) =

ProbY (x′) ≤ yi and Y (x′′) ≤ yj. (6.25)

Consequently, by letting s grow, a more rened modelfor the complete univariate and bivariate characteris-tics is obtained. Conditional simulation in an indica-tor setting is based on linear combinations of indica-tors and models for spatial indicator covariance andcrosscovariance functions:

CovIi(x′), Ij(x′′) = Cij(x′ − x′′) (6.26)

Note that the number of parameters required oftenis large. The underlying model for which this is cor-rect is not fully dened and understood. The simula-tion is performed with the SIS-algorithm, see Journeland Alabert,45 and a realization of Y (x)|Y (xi) =y(xi); i = 1, . . . ,m;x ∈ D is obtained. Note that theindicator approach is richer than the Gaussian modelsince each quantile of the variable is modelled sep-arately. No case studies seem to be available usingthis technique for the petrophysical variables, hencelittle experience has been gained. The SCRF groupat Stanford University, California, advocates the ap-proach strongly, however.The Markov random eld approach can also be

used for modelling of continuous petrophysical char-acteristics if discretization into a large number ofclasses is accepted. As previously mentioned, theMarkov random-eld model appears as extremelyexible in the sense that all higher order interactionscan be dened. The challenge is usually to denea suitable parametrization. Simulation can be per-formed by the Metropolis procedure or the Gibbssampler, although neither of these have proven veryecient.

6.1.4 Final Remarks

Stochastic modelling in reservoir characterizationserves two purposes. First, it makes reproduction

of the heterogeneity of the characteristics possible.This contributes to a more reliable evaluation of re-covery. Second, stochastic modelling is required inorder to assess the uncertainty in the predicted pro-duction volumes. The latter will be more thoroughlydiscussed in Sec. 8.5. The introduction of stochasticmodelling and simulation in reservoir characteriza-tion entails an increasing computer dependency. It isexpected that reservoir characterization will developinto a computer-intensive discipline just like seismicprocessing and ow simulation are today.

6.2 Field Examples

6.2.1 Introduction

The North Sea contains a number of oil and gas eldswith large, heterogeneous reservoirs, high develop-ment and drilling costs and extensive infrastructure,existing or planned, that can support extended eldlife. These parameters encourage the use of hetero-geneity modelling as a tool for predicting eld perfor-mance and improving oil recovery.The potential of such modelling was recognized

early, as exemplied by Johnson and Krol's descrip-tion of a sedimentary process-driven log correlationmodel of the Statfjord Formation in the Brent eld.47

This approach is commonly used also for other elds,and is well suited for homogeneous reservoirs and forheterogeneous reservoirs in mature elds with goodwell-control. Also, correlation models can be ex-panded by the inclusion of stochastic elements, asdemonstrated by stochastic modelling of noncorrelat-able shales within a well correlation framework forthe Frigg eld.48

However, for heterogeneous reservoirs with widelyspaced wells, as North Sea elds in the appraisal andeld development planning phase, typically having in-terwell distances of more than 2 km, well correlationis associated with signicant uncertainties. For thiskind of reservoirs, stochastic modelling provides anattractive and promising alternative.23

Most stochastic modelling studies published so farconcern outcrop data, synthetic reservoirs,28,38,49 orunspecied reservoirs.12 An example of extensive oil-eld application is the modelling performed of theSnorre eld.50,51 The eld is located in the north-ern North Sea, in the prolic hydrocarbon provincethat contains the giant Brent, Statfjord and Gullfakselds, Fig. 6.7. The Snorre reserves occur in the up-per Triassic Lunde and the upper Triassic to lowerJurassic Statfjord formation. Both formations areinterpreted to be uvial, deposited by river systemsthat varied from braided to low sinuous.52 This re-sulted in heterogeneous networks of sandbodies withvarying sandbody dimensions and net-to-gross ratios.Although the productivity of the Statfjord forma-

tion is demonstrated by the nearby Statfjord eld,production performance from that eld does not nec-essarily apply directly to the Snorre eld, due to lat-

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150 CHAPTER 6. HETEROGENITY MODELS

61°

60° N

1° E 2° 3° 4° 5°

60° N

61°

1° E 2° 3° 4° 5°

Bergen

FlorøSnorre

DunlinStat-fjord

Gullfaks

Troll

Oseberg

U. K

ingd

om

Nor

way

Ninian

Brent

Cormorant

0 50 100 km

North Sea

Northern

Figure 6.7: Oil and gas elds of the northern North Sea.

eral variation of reservoir properties. Also, no pro-duction experience currently exists for the Lunde For-mation, beyond testing of exploration and appraisalwells.The development of a computer program that could

model the large-scale heterogeneities of the uvialsandbody network was therefore initiated, in coop-eration with the Norwegian Computing Center. Thisresulted in a program called SISABOSA,26 which wasused in the Snorre eld appraisal and developmentplanning phase, to predict average well and eld be-havior.53 The program was further developed and re-ned, with contributions also from Statoil, resultingin the FLUREMO program,29 which later has beenused to remodel parts of the reservoir sequence.

6.2.2 Large-Scale Heterogeneity Mod-elling

For the Snorre Field Development Plan, individ-ual reservoir units, typically 80200 m in thickness,were modelled over selected parts of the eld. Foreach unit, a simplied 3D representation of the u-vial sandbody network was generated by the SISA-BOSA program, by stochastically distributing per-meable sand parallelepipeds in a nonpermeable mud-stone matrix, Figs. 6.8 and 6.9, until a predenedsand content variation was obtained. Prior to theSIASBOSA-distribution, each unit was subdividedinto 24 subunits with dierent sand content, in or-der to reect the systematic variation of sand contentwith depth observed in the wells.

Figure 6.8: 3D-representation of stochastic distribu-tion of sandstone bodies in SISABOSA (schematic).

The sand content used in the modelling was de-termined by log evaluation. Since the purposewas to model the large-scale sandbody geometry,sand content was not restricted to net sand, asdened by petrophysical cut-o values, but in-cluded also tight, carbonate-cemented sand and low-permeability, abandonment-facies sand, interpretedto be parts of larger sandbodies. On the other hand,sandbodies with a log thickness of less than 1.5 mwere excluded, on the assumption that such bodieswere not likely to have a lateral extent that allowedsignicant contribution to uid ow. Also, sands in-terpreted to be crevasse splay sands were excluded

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6.2. FIELD EXAMPLES 151

-60 m

-40

-50

-30

-20

-10

0

2000 300100 400 m

Figure 6.9: Cross section through SISABOSA 3D-representation.

from the sand content, due to their virtually nonper-meable properties, resulting from extensive pedogeniccementation.All parallelepipeds were distributed as horizontal

bodies, which were allowed to cut into partly under-lying bodies, in order to mimic erosion. The paral-lelepipeds had innite length, while thicknesses andwidths were specied to reect thicknesses and widthsof real uvial sandbodies as determined from analogstudies and from the literature.54 The average longaxis direction, and the variability around that direc-tion were also predened, based on measurements onoriented cores and dipmeter data from cored inter-vals.For models covering an area that contained an ex-

ploration or appraisal well, the sandbody distributionwas conditioned on that well, in such a way that thenumber, thickness and depth of the sandbodies ob-served in the well were reected in the model.Stochastic modelling will by its nature produce a

range of dierent representations. In order to ensurethat the model used for subsequent reservoir simula-tion corresponded to an average network congura-tion, ten representations were generated from the setof most likely input parameters. For each representa-tion, the degree of sandbody interconnectedness wasdetermined, and the representation with an intercon-nectedness value closest to the mean of the ten repre-sentations was selected for simulation.53 In addition,ve representations for each of a set of low-case andhigh-case parameters were generated for sensitivitypurposes.The 3D-models generated by the above procedure

contain detailed information on the restrictions touid ow caused by the mudstone interspaced be-tween adjacent but disconnected sandbodies. In orderto transfer this essential information to the grid-cellsystem used in standard reservoir simulators, trans-missibility values in the x, y and z directions werecalculated for all grid cells, using the procedure de-

scribed by Clemetsen et al.29

The nonhorizontal orientation and crestal erosionof the units observed from the seismic data were ob-tained by the appropriate tilting and truncation ofthe reservoir simulation grid.Grid-cell size sensitivity studies showed that grid-

cell sizes of 50 × 50 × 3 m to 100 × 200 × 3 mgave an optimal balance between the need to repro-duce geometrical details and the need for a manage-able number of simulation grid cells. The actual sizeselected depended upon the properties of each indi-vidual reservoir, with sandbody width and thicknessbeing the most critical parameters. Due to the size ofthe eld, approximately 100 km2, with an oil columnof some 200 m, full-eld modelling with such grid-cellsizes was considered impractical. Instead, individ-ual reservoir units were modelled based on elementmodels. These models covered areas of 4 to 10 km2,located in such a way that uid ow within majorfault compartments could be simulated.In addition to the simulation of element models,

full-eld simulation with larger grid-cell size was per-formed to evaluate the eects of ow between thefault compartments. The element models were usedto generate pseudofunctions for the full-eld simula-tion.In SISABOSA, each sandbody was treated individ-

ually, and stochastically distributed as such. In theFLUREMO procedure, each sandbody belongs to asandbody family, the model equivalents to individualriver channels and composite channel belts, Fig. 6.10.As for SISABOSA, a width to thickness relationship isgiven for the sandbodies. In addition, the number and

0 500 1000 1500 2000 2500 m

-60 m

-30

-20

-10

-40

-50

Figure 6.10: Cross section through FLUREMO 3D-representation. Dierent hatching represents dier-ent sandbody families.

distribution of channel sandbodies within the channelbelt are guided by user supplied probability distribu-tions. The eect of shifts in channel belt positionresulting from river avulsion is modelled by the useof a repulsion function. Sinuosity of individual chan-nels and of channel belts can be introduced, togetherwith the main direction of the river system, Fig. 6.11.Also, lateral and vertical trends in sand content canbe included in the modelling.In the FLUREMO modelling, the selection of a 3D-

representation that corresponds to an average reser-voir conguration is based on the more time consum-ing procedure of performing reservoir simulation on a

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152 CHAPTER 6. HETEROGENITY MODELS

0 1000 2000 3000 4000 m0

1000

2000

3000

4000

5000 m

Figure 6.11: FLUREMO channel belt representationwithin a 10 m depth interval.

number of representations.

6.2.3 Modelling of Heterogeneity Dis-tribution Within Single Sand-bodies

The above modelling procedures result in a sandbodynetwork, reecting the large-scale reservoir hetero-geneities. The eect of heterogeneities of individ-ual genetic sandbodies, i.e., grain-size variations, ero-sional surfaces, shale drapes, sedimentary structuresand pore-textural features, as well as minor faults andtectonic fractures,55 must be modelled on a smallerscale. Also, information on whether a grid block is lo-cated in the upper or lower part of a sandbody is notprovided by the output of SISABOSA or FLUREMO.Hence, it has not been possible to model verticaltrends in permeability distribution observed withinchannel sandbodies with the above procedures.Such modelling can be performed as a continuous

modelling, based on geostatistics.12,38 An alternativemethod is the use of small-scale reservoir data in amodel based on eld analogs. This method can be il-lustrated by the simulation on 2D-models of a braidedstream channel sandbody from the middle reservoirunit of the Statfjord formation (MSF), performed toinvestigate the eect on recovery caused by internalinhomogeneities.56

Three two-dimensional heterogeneity models wereconstructed by combining reservoir data obtainedfrom exploration and appraisal wells in the Statfjordformation with the internal geometry of a multistoreybraided-stream channel sandbody from the Permo-Triassic Maroon Formation in Colorado. The storeysof the sandbody are bounded by erosional surfacesand discontinuous shale drapes, and represent smallsub-channels or channel bars. The Maroon sandbody,

which was mapped in detail in a road section, hasgrainsize distribution, thickness, facies and genesisthat correspond to that of typical MSF channel sand-bodies and is thus considered to be a suitable analogto the Statfjord formation sandbodies.In accordance with observations from the MSF

sandbodies, all three heterogeneity models had a gen-eral decrease in grainsize and permeability from baseto top, and also a similar vertical trend of theseproperties within individual storeys, Fig. 6.12. Ver-tical and horizontal permeability variation was mod-elled according to empirical data from the relation-ship between sedimentary lithofacies and permeabil-ity.57 Nine permeability classes were dened, withtheir related capillary pressure, irreducible water sat-uration and relative permeabilities, as derived fromconventional and special core analysis of the MSFsandbodies.Various geological assumptions were introduced by

constructing three dierent basic models. In model 1,the lowermost beds were modelled as an open, high-permeability zone, except for the basal bed, whichhad reduced permeability, reecting a clay-rich ma-trix and contained shale clasts. Model 2 was similar,except that all shale drapes were removed, while inmodel 3, the high-permeability lower zone was mod-elled as a discontinuous layer. All models were ap-proximately 100 m in length and 10 m high, and weregridded with blocks measuring 1 × 0.25 m.In order to expand the modelling to reect Snorre

production well spacing, 2D simulation was also per-formed on three extended versions of model 1, one300 m long and two 900 m long. For comparison pur-poses, two reference models were also simulated, onehomogeneous and one layer-cake model. The volume-weighted, arithmetic average permeability of the MSFsandbodies was assigned to all models.

6.2.4 Results and Discussion

Full validation of the modelling results will have toinclude history matching of eld production data,which for the Snorre eld will not be available before1992. However, early evaluation and validation ofrepresentations can be obtained already at the mod-elling stage. Synthetic wells, generated from the rep-resentations at random locations, can be comparedwith actual eld logs and cores. Such comparisonhas a statistical element, e.g., to check that the sandcontent variation generated corresponds to that ob-served in the wells. Visual inspection by the geolo-gist is also indispensable to ensure reasonable repre-sentations. Similarly, synthetic cross sections can becompared with actual cross sections observed in rockoutcrops in areas with analogous depositional envi-ronments.Such evaluation of the Snorre eld modelling

showed that the discrete modelling approach com-bined with actual well data and data from eld ana-logues resulted in reasonable reproductions of the u-vial sandbody network of the Snorre eld reservoirs.

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6.2. FIELD EXAMPLES 153

0-1

1-10

10-50

50 -100

100-200

200-400

Horizontal Permeability (md)

0

5

10 m

10 m 0 5

400-600

600-1000

1000-2000

2000-4000

Figure 6.12: Permeability distribution within a model of a Statfjord formation sandstone, based on analog data.

This was supported by the ability of a reservoir sim-ulation based on a SISABOSA model to predict theresults of a 18 day production test.53 A comparisonbetween the prediction and the actual result is shownin Fig. 6.13.The statistical variations that must be expected

within a heterogeneous reservoir were evaluated bysimulation of a number of representations generatedfrom repeated modelling of selected sets of input data,Fig. 6.14. The gure illustrates that reservoirs withlow sand-to-gross ratios are most strongly aectedby the large-scale heterogeneities of the uvial sand-body network. These reservoirs also show the great-est variability in recovery eciency, thus making re-liable production predictions dicult. The eect de-creases with increasing sand-to-gross ratio, with ex-pected eld performance approaching that of a ho-mogeneous reservoir for high ratios.The eect of the small-scale heterogeneities within

single sandbodies was obtained by 2D simulations ofthe single sandbody model, showing that the produc-tion performance was strongly dependent on the typeand degree of heterogeneity. Thus, the heterogeneousmodels showed a reduction in recovery eciency atwater breakthrough from some 10 to 30 % comparedto homogeneous models. The results were used tomodify the recovery eciencies obtained by the ele-ment model simulation.The elongated versions generally gave higher recov-

ery eciencies than the corresponding short models,showing that in this case, the eect of the small-scaleheterogeneities is reduced with increased well spacing.The results also illustrated the importance of usingindividual capillary pressure curves for each perme-ability class.

The modelling procedure of selecting average 3D-representations can be used to predict average eldand well behavior. Based on this, well spacing can beoptimized by modelling dierent inter-well distances.By its nature, however, the described procedure doesnot provide the site-specic information needed to op-timize well locations.Thus, although the stochastic modelling can con-

tribute signicantly to eld planning at an early stagein eld development, modications to include moredeterministic aspects are needed at the more ma-ture stages, when more closely located wells providea denser grid of well information. Also, improve-

1.50

1.00

0.50

05 10 15 20

Test Observation

SISABOSA Prediction

Infinite, Homogeneous Reservoir

Pro

du

ctiv

ity

Ind

ex (

std

m3 /

d/k

Pa)

Flow Period (days)

Figure 6.13: Comparison between actual well test andprediction based on a SISABOSA 3D-model.

ment in small-scale modelling is obtained by expand-ing such modelling to 3D,12 although gridding andhomogenization constitute major constraints for us-

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154 CHAPTER 6. HETEROGENITY MODELS

Sand to gross ratio

Sim

ula

ted

rec

ove

ry e

ffic

ien

cy

Representing three different thickness/width distributions

Homogeneous model

Pure depletion homogeneous model

0.60.50.40.30.20.10.0

Figure 6.14: Simulated recovery eciencies of dier-ent FLUREMO representations.

ing this heterogeneity scale in 3D-simulations.

6.2.5 Concluding Remarks

The modelling performed for the Snorre eld illus-trates the importance of including information on thelarge-scale heterogeneity when simulating the produc-tion performance of uvial reservoirs. This is particu-larly the case for reservoirs with moderate to low sandcontent, where the degree of sandbody interconnect-edness becomes critical. It also illustrates the eectsof smaller scale heterogeneities and the importance ofincluding capillary forces when simulating how suchheterogeneities aect sweep and water production.

Nomenclature

C(·) = spatial covariance functionc0 = normalizing constantD = geographical domainDG = grid over geographical domaing(·) = realization of constant characteristic

I(·)/i(·) = random variable/realization indicatorvariable

h = shift vectorL(·)/l(·) = random variable/realization facies types

m = dimension of vectorn = number of marked points

NG = number of nodesNi = neighbourhood centred at and excluding

gridnode ir = distance

S/s = random variable/realization attributes ofmarked point

U/u = random variable/realization marked pointw = constraint

X/x = random variable/realization, reference tolocation

Y(·)/y(·) = random variable/realization petrophysi-cal variables

β(·) = relation between location and attributesof marked point

β = neighbourhood parameterδij = kronecker delta

γ(·, ·) = relation between locations and attributesof two marked points

Λ = Cholesky-factor of correlation matrixµ = expected valueρs = coverage ratio of shaleρ(·) = spatial correlation functionρij = correlation between variables number i

and jΣ = covariance matrixσ = strength parameter

σi|j = variance variable number i given faciestype j

References

[1] Jacod, J. and Joathon, P.: Use of Random-Genetic Models in the Study of SedimentaryProcesses, Mathematical Geology (1971) 3,No. 3, 26579.

[2] Bridge, J.S. and Leeder, M.R.: A simulationmodel of alluvial stratigraphy, Sedimentology(1979) 26, 61744.

[3] Alexander, J.: Idealised ow models to pre-dict alluvial sandstone body distribution in theMiddle Jurassic Yorkshire Basin," Marine andPetroleum Geology (1986) 3, 298.

[4] Tetzlaf, D.M. and Harbaugh, J.W.: SimulatingClastic Sedimentation, Van Nostrand Reinhold,(1989).

[5] Haldorsen, H.H. and Lake, L.W.: A New Ap-proach to Shale Management in Field-ScaleModels, SPEJ (Aug. 1984) 44757.

[6] Begg, S.H., Carter, R.R., and Draneld, P.:Assigning Eective Values to Simulator Grid-block Parameters for Heterogeneous Reser-voirs, SPERE (Nov. 1989) 45563.

[7] Mathews, J.L., Emanuel, A.S., and Edwards,K.A.: Fractal Methods Improve Mitsue Misci-ble Predictions, JPT (Nov. 1989) 113642.

[8] Nybråten, G., Skolem, E., and Østby, K.:Reservoir Simulation of the Snorre Field,North Sea Oil and Gas Reservoirs - II, A.T.Buller et al. (eds.), Graham & Trotman, Lon-don (1990) 10314.

[9] Rudkiewicz, J.L., D. Guerillot, D., Galli, A.,and Heresi Group: An Integrated Softwarefor Stochastic Modelling of Reservoir Lithologyand Property with an Example from the York-shire Middle Jurassic, North Sea Oil and GasReservoirs - II, A.T. Buller et al. (eds.), Gra-ham & Trotman, London (1990) 399406.

[10] Høiberg, J., Omre, H., and Tjelmeland, H.:Large-Scale Barriers in Extensively DrilledReservoirs, Proc., 2nd European Conferenceon the Mathematics of Oil Recovery, Arles(1990) 3141.

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REFERENCES 155

[11] Alabert, F.G. and Massonnat, G.J.: Het-erogeneity in a Complex Turbiditic Reservoir:Stochastic Modelling of Facies and Petrophysi-cal Variability, paper SPE 20604 presented atthe 1990 SPE Annual Technical Conference andExhibition, New Orleans, Sept. 2326.

[12] Damsleth, E., Tjølsen, C.B., Omre. K.H., andHaldorsen, H.H.: A Two-Stage StochasticModel Applied to a North Sea Reservoir, paperSPE 20605 presented at the 1990 SPE AnnualTechnical Conference and Exhibition, New Or-leans, Sept. 2326.

[13] Omre, H., Halvorsen, K.B., Holden L., andHøiberg, J.: Reservoir Heterogeneity, Geo-logical Description and Eects on Fluid Flow- Models for Hetereogeneity, Proc., SPOR-Seminar, Stavanger (1988) 296308.

[14] Aasen, J.O. et al.: A Stochastic ReservoirModel and its Use in Evaluations of Uncer-tainties in the Results of Recovery Processes,North Sea Oil and Gas Reservoirs - II, A.T.Buller et al. (eds.), Graham & Trotman, Lon-don (1990) 42536.

[15] Hjort, N.L. and Omre, H.: Topics in SpatialStatistics, to appear in Scandinavian Journalof Statistics (1991).

[16] Journel, A.G. and Huijbregts, C.J.: Min-ing Geostatistics, Academic Press New York(1978).

[17] Ripley, B.D.: Spatial Statistics, John Wiley,New York (1981).

[18] Ripley, B.D.: Stochastic Simulation, John Wi-ley, New York (1987).

[19] Yaglom, A.M.: Correlation Theory of Sta-tionary and Related Random Function. VolumeI: Basic Results, Springer-Verlag, New York(1987).

[20] Stoyan, D., Kendall, W.S., and Mecke, J.:Stochastic Geometry and Its Applications,Akademie-Verlag, Berlin (1987).

[21] Haldorsen, H.H. and MacDonald, C.J.: Sto-chastic Modelling of Underground Reservoir Fa-cies, paper SPE 16751 presented at the 1987SPE Annual Technical Conference and Exhibi-tion, Dallas, Sept. 2730.

[22] Dubrule, O.: A Review of Stochastic Modelsfor Petroleum Reservoirs, paper presented atthe 1988 BSRG Meeting on Quantication ofSediment Body Geometries and Their InternalHeterogeneities, London, March 12.

[23] Haldorsen, H.H. and Damsleth, E.: StochasticModelling, JPT (Apr. 1990) 40412.

[24] Fält, L.M., Henriquez, A., Holden, L., andTjelmeland, H.: MOHERES - A program forreservoir modelling, combining sedimentary ar-chitecture and petrophysical properties vari-ability, paper presented at the 1991 EuropeanSymposium on IOR, Stavanger, May 2123.

[25] Baddeley, A. and Møller, J.: Nearest-neighbour Markov point processes and random

sets, Int. Statist. Rev. (1989) 57, 89121.[26] Augedal, H.O., Stanley, K.O., and Omre, H.:

SISABOSA, a programme for stochastic mod-elling and evaluation of reservoir geology,Proc., Conference of Reservoir Description andSimulation with Emphasis on EOR, Oslo, Sept.1986.

[27] Meling, L.M., Mørkeseth, P.O., and Langeland,T.: Production Forecasting for Gas FieldsWith Multiple Reservoirs of Limited Extent,paper SPE 18287 presented at the 1988 AnnualTechnical Conference and Exhibition, Houston,Oct. 25.

[28] Gundesø, R. and Egeland, O.: SESIMIRA - ANew Geological Tool for 3D Modelling of Het-erogeneous Reservoirs, North Sea Oil and GasReservoirs - II, A.T. Buller et al. (eds.), Gra-ham & Trotman, London (1990) 36371.

[29] Clemetsen, R., Hurst, A.R., Knarud, R., andOmre, H.: A Computer Program for Evalua-tion of Fluvial Reservoirs, North Sea Oil andGas Reservoirs - II, A.T. Buller et al. (eds.),Graham & Trotman, London (1990) 37385.

[30] Øverland, K.M.: Stokastisk modellering av etuvialt Nordsjøreservoar, MS thesis, Høgskole-senteret i Rogaland, Stavanger (1990) (in Nor-wegian).

[31] Omre, H. and Sølna, K.: Stochastic Mod-elling and Simulation of Faults Zones, Proc.,The GEODATA Conference on Geomathemat-ics and Geostatistics, Leeds, (1990) to appearin Sci. de la Terre.

[32] Høiberg, J., Omre, H., and Tjelmeland, H.:A Stochastic Model for Shale Distribution inPetroleum Reservoirs, Proc., The GEODATAConference on Geomathematics and Geostatis-tics, Leeds, (1990) to appear in Sci. de la Terre.

[33] Besag, J.: Spatial interaction and the statisti-cal analysis of lattice systems [with discussion],J. Royal Statist. Soc. B (1974) 36, 192236.

[34] Derin, H. and Kelly, P.A.: Discrete-IndexMarkov-Type Random Processes, Proc., IEEE(1989) 77, No 10, 1485510.

[35] Geman, S. and Geman, D.: Stochastic relax-ation, Gibbs distributions, and the Bayesianrestoration of images, Trans., IEEE (1984)PAMI 6, 72141.

[36] Farmer, C.: The Mathematical Generation ofReservoir Geology, paper presented at the 1989joint IMA/SPE conference on the Mathematicsof Oil Recovery, Cambridge, July 2527.

[37] Galli, A., Guerillot, D., Ravenne, C., and HerisiGroup: Combining Geology, Geostatistics andMultiphase Fluid Flow for 3D Reservoir Stud-ies, Proc., 2nd European Conference on theMathematics of Oil Recovery, Arles (1990) 119.

[38] Ravenne, C. and Beucher, H.: Recent Develop-ment in Description of Sedimentary Bodies in aFluvio-Deltaic Reservoir and Their 3D Condi-

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156 CHAPTER 6. HETEROGENITY MODELS

tional Simulations, paper SPE 18310 presentedat 1988 Annual Technical Conference and Ex-hibition, Houston, Oct. 25.

[39] Suro-Pérez, V. and Journel, A.G.: StochasticSimulation of Lithofacies: an Improved Sequen-tial Indicator Approach, Proc., 2nd EuropeanConference on the Mathematics of Oil Recov-ery, Arles (1990) 310.

[40] Borgman, L., Taheri, M., and Hagan, R.:Three-Dimensional, Frequency-Domain Simu-lations of Geological Variables, Geostatisticsfor Natural Resources Characterization, Part I,G. Verly et al. (eds.), Reidel Publ. Co. (1984)51741.

[41] Omre, H., Sølna, K., and Tjelmeland, H: Simu-lation of Random Functions on Large Lattices,submitted to Mathematical Geology (1991).

[42] Hewett, T.A.: Fractal Distributions of Reser-voir Heterogeneity and Their Inuence on FluidTransport, paper SPE 15386 presented at 1986SPE Annual Technical Conference and Exhibi-tion, New Orleans, Oct. 58.

[43] Feder, J.: Fractals, Plenum Press, New York(1988)

[44] Hewett, T.A. and Behrens, R.A.: ConditionalSimulation of Reservoir Heterogeneity WithFractals, SPEFE (Sept. 1990) 21725; Trans.,AIME, 298.

[45] Journel, A.G. and Alabert, F.G.: Focusingon Spatial Connectivity of Extreme-Valued At-tributes: Stochastic Indicator Models of Reser-voir Heterogeneities, paper SPE 18326 pre-sented at 1988 Annual Technical Conferenceand Exhibition, Houston, Oct. 25.

[46] Omre, H., Sølna, K., and Tjelmeland, H.: Cal-cite Cementation: Description and ProductionConsequences, paper SPE 20607 presented atthe 1990 SPE Annual Technical Conference andExhibition, New Orleans, Sept. 2326.

[47] Johnson, H.D. and Krol, D.E.: GeologicalModeling of a Heterogeneous Sandstone Reser-voir: Lower Jurassic Statfjord Formation, BrentField, paper SPE 13050 presented at the 1984SPE Annual Technical Conference and Exhibi-tion, Houston, Sept. 1619.

[48] Skaug, M. and Gundesø, R.: Geological Mod-elling of the Frigg Field With Special Emphasison Shale Mapping, paper SPE 15859 presentedat the 1986 European Petroleum Conference,London, Oct. 2022.

[49] Matheron, G. et al.: Conditional Simulationof the Geometry of Fluvio-Deltaic Reservoirs,paper SPE 16753 presented at the 1987 SPEAnnual Technical Conference and Exhibition,Dallas, Sept. 2730.

[50] Hollander, N.B.: Snorre, Geology of the Nor-wegian Oil and Gas Fields, A.M. Spencer etal. (eds.), Graham & Trotman, London (1987)30718.

[51] Jorde, K. and Diesen, G.W.: The Snorre Field

- a Major Field in the Northern North Sea,Giant Oil and Gas Field of the Decade 1978-1988, AAPG Memoir 54, (in print)

[52] Nystuen, J.P., Knarud, R., and Jorde, K.: Cor-relation of Triassic to Lower Jurassic Sequences,Snorre Field and Adjacent Areas, NorthernNorth Sea, Proc., Correlation in HydrocarbonExploration, Bergen (1988), Graham & Trot-man, London (1989) 27389.

[53] Stanley, K.O., Jorde, K., Ræstad, N., andStockbridge, C.P.: Stochastic Modelling ofReservoir Sand Bodies for Input to ReservoirSimulation, Snorre Field, Northern North Sea,Norway, North Sea Oil and Gas Reservoirs -II, A.T. Buller et al. (eds.), Graham & Trot-man, London (1990)91101.

[54] Collinson, J.D.: Vertical Sequence and SandBody Shape in Alluvial Sequences, FluvialSedimentology, A.D. Miall (ed.), Can. Soc.Petrol. Geol. Mem. (1978) 5, 57786.

[55] Weber, K.J.: How Heterogeneity Aectsthe Oil Recovery, Reservoir Characterization,L.W. Lake and H.B. Carroll (eds.), AcademicPress, Orlando (1986) 487544.

[56] Høimyr, Ø., Kleppe, A., and Nystuen, J.P.:The Eect of Heterogeneities in BraidedStream Channel Sandstones on the Simulationof Oil Recovery: A Case Study, Advances inReservoir Geology, Geol. Soc. Spec. Publ. (inpress).

[57] Pryor, W.A.: Permeability-Porosity Patternsand Variations in some Holocene Sand Bodies,AAPG Bulletin (1973) 16289.

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Chapter 7

Tracer Testing

7.1 Tracer Types

7.1.1 Introduction

In this section is given a review of use of tracers forreservoir description. For a more complete treatment,especially for the radioactive tracers, see Bjørnstad.1

The present state of tracer knowledge is gainedthrough dedicated laboratory investigation pro-grams,213 through oil eld experience,1447 ground-water movement investigations,4850 atmospherictracing experiments,51,52 and also to a signicantdegree through the work carried out on migrationof radioactive species in soil for the purpose ofevaluating radioactive waste repository sites.53

Although the integrated knowledge from thesestudies is substantial, the information achieved is notalways consistent. Results from one area of inves-tigation cannot readily be transferred to new eldsbecause of both scaling problems and changing ex-perimental conditions. However, the views conveyedhere on tracer behavior are based on an extract ofthe most reliable and consistent investigations fromthe elds referred above, and constitute as such thepresent state of tracer behavior knowledge.The primary goals of the interwell tracer tests re-

ported in the literature may be summarized as fol-lows:

a) Depiction of ow directions and interwell con-nections.

b) Evaluation of volumetric sweep eciency be-tween injector and producer.

c) Detection and evaluation of permeabilitystratication.

d) Detection of faults and barriers to ow.

An increasing interest has been registred the last twoyears for studies involving tracer test for evaluationof complex relationship, such as

e) Estimation of average residual oil saturation.f) Estimation of average ion exchange capacity.g) Evaluation of changing imbibition potential.h) Evaluation of the eect of reservoir chemical

treatment.

For unambiguous single-phase tracing of either thewater or the gas phases in enhanced oil recovery, the

following tracer selection criteria apply to added trac-ers:

a) Insignicant degradation under injection,reservoir and production conditions (i.e., highthermal, chemical, physical and microbiologi-cal stability).

b) Must follow passively the labelled phase with-out signicant interactions.

c) Insignicant natural occurrence in involveduids.

d) Detectable at very low concentrations.e) Toxicity/radiotoxicity at an environmentally

acceptable level.f) Nonproblematic logistics and handling.g) Sucient commercial availability.h) Acceptable cost.

There are in principle three types of tracers avail-able:

a) Stable-isotope ratios.b) Radioactive atoms or molecules.c) Nonradioactive chemical compounds.

The main characteristics of each of these types arebriey given below.Stable-isotopic ratios. A prerequisite for appli-

cation of isotopic ratio tracer is the existence of a sig-nicant (and in reality naturally occurring) dierenceof the isotopic ratio in question between the injecteduid and the reservoir uids. Accordingly, the accu-racy and sensitivity of the method depends on a goodcharacterization of the initial uids and on the actualsize of the isotopic ratio dierence, respectively.The isotopic ratios are readily measured with sen-

sitive mass spectroscopy techniques.Radioactive species are probably the most fre-

quently used tracers in well-to-well tests. Their mainadvantage is their extremely low detection limits. De-pending on the particular radioactive species in ques-tion, they may be detected in concentrations from afew thousand atoms per liter and upwards.For beta-particle and electron emitting tracers,

the main detection method will be liquid scintil-lation counting or gas proportional counting (forgaseous species). For tracers which emit gamma or x-rays, the detection technique is high-energy (gamma,

157

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158 CHAPTER 7. TRACER TESTING

x-ray) electromagnetic radiation spectroscopy withsolid scintillation or semiconductor detectors.Nonradioactive molecules with good survival

characteristics at reservoir conditions may also be ap-plied for uid tracing. However, the requirement ofhigh detection sensitivity also apply here. Althoughthe sensitivity for chemical species normally is sev-eral orders of magnitudes lower than for radioactivespecies, the lower cost of the nonradioactive materialallows injection of higher quantities to partly compen-sate for the lack of sensitivity. However, if moleculesshall comply with the requirement of noninteractionwith uids and rocks, the amount injected can obvi-ously not be unlimited. Therefore, the main targetsfor chemical tracers have traditionally been smallerland-based reservoirs with short (less than a few hun-dred meters) well spacing.The analysis of the chemical tracers has been re-

ported to been carried out by gas and liquid chro-matography (GC and HPLC), nuclear magnetic reso-nance (NMR), ion chromatography (IC), colorimetry,uorometry and neutron activation analysis (NAA).

7.1.2 Tracers for Injected Water

A substantial number of eld tests involving wa-ter tracers have been reported in the litera-ture.1430,39,45,46 This chapter will discuss the ex-perimental experiences with petroleum reservoir wa-ter tracers.

Isotopic Ratios

To the best of our knowledge, none of the reportedeld tests has included isotopic ratio analysis as amain tool.The isotopic ratios that are directly applicable as

tracers are the naturally occurring 3H/1H (D/H) andthe 18O/16O ratios in the water molecules measuredas the δD and the δ18O values. For some North Seareservoirs, a lower sensitivity limit of 0.5 to 1% con-tent of injection water into formation water has beenestablished. This sensitivity limit could in principlebe lowered by addition of enriched heavy water, D2O,to the injection water. However, due to the relativelyhigh natural occurrence of deuterium (0.015%), theneeded amount of heavy water to give a substan-tial sensitivity increase would be economically pro-hibitive. In general, the cost of isotopically enrichedmaterial is too high to be of real interest for interwelltests. However, this method has higher actuality insingle-well push-and-pull operations.

Radioactive Atoms or Molecules

Most of the tests referred to above involve radioac-tive tracers.1418,2123,25,26,2830,39,45,46 These trac-ers may be subdivided into three classes, neutral, an-ionic, and cationic tracers. These have dierent be-havior and interactions on the microlevel in a porousmedium.

Experience indicates in general that neutral andanionic species have a dynamic ooding behaviorwhich approaches that of water while the cations tendto be slowed down relative to the water ow mainlydue to reversible ion exchange to the rock material.Applicable tracers have, however, been identied inall three classes. For a detailed discussion see Bjørn-stad.1

Chemical Tracers

Several reported eld tests have included non-radioactive chemical tracers,1520,23,24,26,29 mainlybeing anions and noncharged watersoluble organicmolecules. Examples of anions are NO−3 ,

16,1820,23,39

Cl−,24,27 Br−,18,19 I−,18,20,26,27 SCN−,1720,23,24

HBO−3 ,15 and B4O−2

7 ,.3,19,20 Examples of or-ganic molecules are uorescein,15 rhodamineB,;15 methanol,17 ethanol,17,18 and isopropylalco-hol.16,18,23,39

There are also indications from a eld test29 thatcations may be used under certain conditions. Onesuch cation is potassium, K+. The accuracy dependson the maturity of the ood: Chromatographic delayof the tracer due to reversible sorption on rock sur-faces is most prominent in the start of the ood andis thereafter gradually reduced.The amount of tracer chemical needed for injec-

tion depends not only on the sensitivity of the ana-lytical technique and on the total estimated dilutionvolume for the tracer between injector and producer,but also to a signicant degree on the level of nat-urally occurring tracer species. The conditions maybe rather dierent from reservoir to reservoir. There-fore, no specic rules or limits can be given, and eachplanned tracer test has to be evaluated individually.However, as a general guide, a few examples on thistopic are given below for the most interesting tracerchemicals, mainly using published datafrom smallerland-based reservoirs with well spacing in the rangeof 30 to 100 m.In pilot tests reported by Widyer et al.,20 0.3 pore

volumes (PV) NO−3 has been injected continuously inconcentrations of approximately 300 ppm as NH4NO3

and by Wagner18 as slug injection in a total amountof 4.5 tons. Although biodegradation had apparentlyreduced the total concentration, the ion was clearlydetected in producers.Cl− and Br− are normally of less use as added trac-

ers due to high natural concentrations in formationand injection brine. In specic cases, these tracershave been used in continuous concentrations of 5000ppm,24 as NaCl, and 350 ppm,20 as NaBr, respec-tively, with positive results.I− as KI has been injected in amounts of 170 to 635

kg18,20 and been positively identied in productionwells without any noticeable sorption or degradation.SCN− has been injected continuously in concen-

trations of 300 to 1000 ppm, as NH4SCN, to a totalmass of up to 1150 kg.17,18,20 The performance hasin general been satisfactory.

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7.1. TRACER TYPES 159

HBO−3 injected in concentrations of approximately100 ppm, as H2BO3, over a few hours and B4O2−

7 inconcentrations of 700 ppm for 0.3 PV, in small land-based reservoirs, was never detected in a productionwell.15,20

Fluorescein and rhodamine B were simultaneouslyinjected in a small land-based reservoir in amounts of55 to 500 g.15 The rst was positively identied in aproduction well while the latter was never regained.Methanol and ethanol injected continuously in a

concentration of 1400 ppm17 or as a slug with asmuch as 47.3 m3, (well distance or dilution volumenot given),18 were clearly produced in good yieldalthough some degree of bacteriological degradationwas experienced.Isopropanol, also with an injected amount of 47.3

m3, seems less prone to bacteriological attack, andwas readily produced in good yield in productionwells.From this brief review, one may conclude that a

number of eld-tested inorganic and organic ions andmolecules are conditionally useful as reservoir watertracers in small reservoirs or in pilot areas where thewell spacing or the total dilution volume is relativelysmall.What are the prospects for these tracer chemicals

in larger reservoirs with typical well spacings around1000 m? In a landbased reservoir with well spacings800 to 1000 m, 109 tons of NH4NO3 were injected.16

Breakthrough had not occurred at the time of pub-lishing. In any case, with such large amounts of chem-icals a number of practical operations become moredicult, especially on oshore platforms with weightand space limitations. In addition, NO−3 being oneof the major components in the injection water, willhardly comply with the requirement of behaving pas-sively in the water ow, at least in the vicinity of thewells.If the use of SCN− were to be scaled up from

the previously mentioned smaller-scale reservoir ex-periences to the North Sea conditions, 115 tons ofNH4SCN would be needed for one injection. Thislarge amount is impractical to handle and its use astracer will in practice be prohibitive.However, better analytical techniques may turn

this conclusion around. With the enrichment tech-nique developed by Bjørnstad54 for SCN−, a reduc-tion by a factor of 10 to 50 is probably already withinreach for SCN−.

Summary

Table 7.1 summariezes the present state of knowledgeof the applicability of water tracers, and gives recom-mendations for use. Some of the tracers mentionedshould not be applied in the same test since their in-dividual analysis may be precluded. This is true for35SCN− + S14CN− where both must be counted byliquid scintillation technique and have very nearly thesame beta energy. For the radiolabelled thiocyanates

and the radiolabelled cobalthexacyanides the sepa-ration technique now applied for SCN− does not ex-clude a simultaneous collection of the Co(CN)3−6 . ForCo(CN)5(14CN)3− + gamma-radioactive cobalthexa-cyanides, the rst tracer will have to be counted withliquid scintillation spectroscopy which will also re-spond to the presence of the others. At present, it isunclear whether the spectral separation in the scintil-lation spectra is suciently large to allow simultane-ous counting with good eciency. These problems arelargely a matter of analytical development, and mayprobably be resolved without substantial diculty.

7.1.3 Tracers for Injected Gas

Injection gases may roughly be divided into threetypes: natural or lean hydrocrabon gas, composedmainly of methane and its light homologous gasmolecules; nitrogen; and carbon dioxide. These gasesbehave dierently in the reservoir. The behavior ismainly determined by their dierent critical values oftemperature and pressure and their dierent solubil-ity in the oil and water phases at reservoir conditions.A further complexity is added by the fact that even

the individual components in the natural gas behavedierently. This is the reason why one can hardlyclaim that there exists an ideal gas tracer for injectedgas in general. For pure CO2-injection, however, theradioactive 14CO2 may serve as an ideal tracer.Much of the existing knowledge of gas tracers has

been gained through eld experiments. Most of thesehave been of the inject-and-observe type withoutmuch a priori information on their relative behavior.Several such eld experiments have been reported inthe literature.16,3147 Interesting and potentially use-ful information has emerged from atmospheric trans-port experiments.51,52 Recently, controlled labora-tory experiments under simulated reservoir condi-tions have added important and detailed informationon gas tracer relative behavior.12,13

In principle, there are three types of gas tracersavailable: nonradioactive isotopic tracers, radioactivemolecules, and nonradioactive chemical molecules.The various types are treated briey below.

Nonradioactive Isotopic Tracers

One potential isotopic tracer for oil reservoirs is themethane molecule 13CD4 where all hydrogen atomsare exchanged with deuterium. This molecule is prac-tically nonexistent in natural methane. Thus, oneof the main tracer requirements is fullled. It hasnever been reported used as a tracer in subsurfacegas tracing, but has successfully been applied to mon-itor long-range, 155 to 2500 km, air transport in USAafter release of only 84 g of the substance in the at-mosphere.55 The practical implementation for gastracing by 13CD4 in oil reservoirs depends probablyon the cost of sucient amounts of the gas whichagain is determined by the analytical sensitivity. Up-dated values of these parameters are not known at

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160 CHAPTER 7. TRACER TESTING

Table 7.1: Interwell water tracer.

Test Reservoir Radia- Approx. Approx. Approx. Ana-

Half- duration type tion detect. amount tracer lytical

Tracer life < 1.5 > 1.5 Sand- Lime- type limit h) needed k) cost l) method

year year stone stone GBq(Ci) nok (103)

A. Isotopic ratios

D/H - Y a) Y Y Y - SδD = 0.5 % i) - - MS m)

87SR/86SR - Y Y (Y) (Y) - SR = 2× 10−5 j) - - MS

B. Radioactive species

HTO 12.32 y Y Y Y Y β− d) 1.52 Bq/l 3700 (100) 30 LSC n)

22Na+ 2.6 y Y Y Y P β+ e),γ f) 0.1 Bq/l 18.5 (0.5) 500 LSC, GAM o)

36Cl− 3× 105 Y Y Y Y β− < 10−15 at/at 1.85 (0.05) 100 AMSp)

(∼ 2× 1022

atoms )125I− 60 d Y N b) Y Y(P) ε g), γ < 0.05 Bq/l 37 (1) 150 LSC, GAM

S14CN− 5730 y Y Y Y P β− 0.005 Bq/l 37 (1) 200 LSC35SCN− 87 d Y N Y P β− 0.005 Bq/l 37 (1) 200 LSC35SO2−

4 87 d Y N Y P β− 0.05 Bq/l 37 (1) - LSC

Probable tracers, but awaiting experiment clarication129I− 1.6× Y Y Y Y(P) β−, γ < 10−15 at/at 0.03 (8× 10−4) 150 AMS

107 y (2× 1022 atoms)56Co(CN)3−6 78 d Y Y P c) P ε,β+, γ ∼ 0.01 Bq/l 1837 (0.51) - LSC, GAM57Co(CN)3−6 270 d Y Y P P ε, γ ∼ 0.01 Bq/l 1837 (0.51) - LSC, GAM58Co(CN)3−6 71 d Y N P P ε,β+, γ ∼ 0.01 Bq/l 1837 (0.51) - LSC, GAM60Co(CN)3−6 5.2 d Y Y P P β−, γ ∼ 0.01 Bq/l 1837 (0.51) - LSC, GAM

Co(CN)5 5730 d Y Y P P β+ ∼ 0.01 Bq/l 1837 (0.51) - LSC, GAM

(14CN)3−

C.Non-radioactive species

Fluorescein - Y ? P(SR) P(SR) - - > 10 kg - FM q)

Methanol - Y ? P(SR) P(SR) - - > 10 m3 - GC r)

Ethanol - Y ? P(SR) P(SR) - - > 10 m3 - GC

Isopropanol - Y Y P(SR) P(SR) - - > 10 m3 - GC

I− - Y Y Y(SR) P(SR) - - > 1 ton - -

SCN− - Y Y Y(SR) P(SR) - - > 1 ton - -

NO−3 - Y ? Y(SR) P(SR) - - > 1 ton - CM s)

Co(CN)3−6 - Y Y P(SR) P(SR) - - > 1 ton - HPLC t)

present. A somewhat poorer though cheaper alterna-tive to 13CD4 is CD4, with naturally occurring car-bon.Tracers of this kind are not yet ready for reservoir

applications, but should be investigated as possiblefuture alternatives.

Radioactive Molecules

A number of tracer tests incorporating radioactivelytagged gas tracers have been reported in the litera-ture.3241,4347 The tracers fall into two groups. Therst comprises the inorganic gases with tritiated hy-drogen and the noble gases. The second group is com-posed of organic gases including radiolabelled, mainlytritiated, methane and its heavier homologous ethane,propane and butane. See details in Bjørnstad.1

Chemical Tracers

Except for a very early report on the use of heliumgas in a reservoir,31 the main interest has been con-

centrated on the poly- and perhalogenated hydro-carbons including one or more uorine atoms in themolecule.41,42,46 The main reasons for this are theirchemical, thermal and microbial stability, their lownatural background and their high detectability bygas chromatography separation followed by electroncapture detection (GC/ECD). See details in Bjørn-stad.1

Summary

Table 7.2 concentrates the present state of knowledgeof oil reservoir gas tracers and gives a few guidelinesand recommendations for use.Some of the tracer molecules listed have not

been applied (not reported) in actual interwell eldtests. These are 13CD4, 127Xe, 14CO2, and the 14C-labelled hydrocarbons. However, the chemistry forthe carbon-containing tracers is not signicantly dif-ferent from the tritiated or the nonlabelled versionsof the respective molecules, nor are their physical in-teractions. The same is true for the xenon isotopes.

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7.2. RESERVOIR CHARACTERISTICS 161

Footnotes for Table 7.1 and 7.2

a) Y = yesb) N = noc) P = probably, not sucient testedd) β− = negative beta particles (negatrons)e) β+ = positive beta particles (positrons)e) For Table 7.2

Conv. = conversion of the radioactive species into an-other molecular form before detection

f) γ = gamma radiationsg) ε = decay by electron captureh) Very approximate limits given only as a guide. Vary

with complexity of the analysisi) Standard deviation on δD. Implies a sensitivity of 0.5

to 1 % inmix of injection water into formation water, orvice versa, for normal North Sea reservoirs

i) For Table 7.2ECD = Electron Capture Detector

j) Standard deviations of the ratio. The method is hardlyapplicable in North Sea reservoirs due to high concen-tration of Sr in formation water and low concentrationin injection water

k) May vary a factor 10 from test to testl) Vary with timem)MS = Mass spectrometryn) LSC = Liquid scintillation countingo) GAM = Gamma spectrometryp) AMS = Accelerator mass spectrometryq) FM = Fluorometryr) GC = Gass chromatographys) CM = Colorometryt) HLPC= High-pressure liquid chromatography

Their dynamic interactions are therefore consideredlargely known, and, although some dierences may beexpected in diusion-controlled interactions, no sur-prises are expected in their gross rate of movement inreservoirs.

7.1.4 Future Use

Although the tracer technique in reservoir descrip-tion is a mature idea, its rate of development into aunique and useful instrument has not been particu-larly impressive. In the later years, however, bothdedicated laboratory experiments and well-plannedand executed eld tests have added positively to theknowledge of tracer behavior under extreme condi-tions, like those encountered in oil reservoirs. Hence,the present status is a collection of well examinedtracers for water and gas movement, as given in Ta-bles 7.1 and 7.2.However, the existing limited collection does not

oer sucient freedom, versatility, and capacity tothe user, especially for complex reservoirs with manywells. Hence, there is a continued strong need to en-large the collection of well-characterized water andgas tracers.The present development concentrates on various

tasks:

• Ideal or near-ideal tracers for the purposes listedabove.

• Water/oil and gas/oil partitioning tracers forpossible interwell determination of residual oil(or rather water contactable, remaining oil) sat-uration. Crucial here is the determination of K-values as a function of changing reservoir param-eters like pressure, temperature, water pH andsalinity, oil and gas composition etc.

A eld test with positive conclusion has alreadybeen conducted using phenol and ortho-cresol aspartitioning tracers.56

• Reversibly rock-sorbing tracers for determina-tion of the average interwell ion exchange capac-ity which is mainly associated with the clay con-tent. This parameter is of interest in tertiary

oil recovery when polar or ionic surfactants areplanned used.

• Water/oil partitioning tracers for signalling ap-proaching waterfront before water breakthrough.It is recently demonstrated theoretically thatsuch measurements are feasible.57

• Gas/oil partitioning tracers for signalling gasconing before gas breakthrough.

The new tracers are sought both among the ra-dioactively labelled molecules, the non-radioactivemolecules (in particular the poly- and peruorinatedchemicals) and the molecules labelled with stable iso-topes with low natural abundances.58

7.2 Reservoir Characteristics

7.2.1 Introduction

In this section we will discuss how tracers can be usedto unveil reservoir characteristics, i.e., reservoir het-erogeneities and in-situ residual oil saturations.

7.2.2 Reservoir Heterogeneities

Tracer testing of a reservoir may give informationabout layer structure and channeling, barriers to ow,and dispersivity of the medium.Brigham and coworkers5962 have described the

possibilities of determining the number of layers andgeological parameters by well-to-well tracer testing.They considered a three-dimensional reservoir con-sisting of non-communicating layers. Only one ow-ing phase was in fact treated. The tracer movementis accounted for by convection and dispersion eects.For several well patterns and values of the Pecletnumber, they have published tracer response curvesgoverning two-dimensional tracer ow.We will briey outline the main idea by giving a

simplied example where dispersion is neglected andwhere water is the mobile phase and oil is immobile.Assume that the dimensionless tracer response of

a production well is given by the curve in Fig. 7.1.This may be the result of a tracer pulse injected in a

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162 CHAPTER 7. TRACER TESTING

Table 7.2: Interwell gas tracer.

Test Radia- Approx. Approx. Rel. Ana-

Tracer Half- duration tion detect. amount tracer lytical Comments

life < 1 y > 1 y type limit h) recom- cost method

mended k)

A. Isotopic ratios13CD4 + - Y a) Y - < 10−15 l/l < 100 g High GC r)/ Possible,

Similar MS n) but not

Molecules applied

B. Radioactive species

HT 12.32 y Y (Y) β− d) ∼ 10−3 Bq/l 1110 GBq Medium Conv. e) Caution, isotope

(30 Ci) + LSC n) exchange degradiation85Kr 10.76 y Y Y β−,(γ) ∼ 2× 10−4 1110 GBq Medium LSC Well documented

Bq/l (30 Ci) performance127Xe 36.4 y Y N b) ε g),(γ) ∼ 2× 10−4 1110 GBq High LSC, Special cases, not

Bq/l (30 Ci) GAM o) previously reported133Xe 5.2 y Y N β−,(γ) f) ∼ 2× 10−4 1110 GBq Medium LSC, Special case

Bq/l (30 Ci) GAM only14CO2 5730 y Y N β− < 5× 10−4 370 GBq High/ Conv. Not

Bq/l (10 Ci) Medium + LSC reported (?).

CH3T 12.32 y Y Y β− ∼ 10−3 Bq/l 1850 GBq (50 Ci) Medium Conv. +LSC Well documented use

C2H5T 12.32 y Y Y β− ∼ 10−3Bq/l 1850 GBq (50 Ci) Medium Conv. +LSC Well documented use

C3H7T 12.32 y Y Y β− ∼ 10−3 Bq/l 1850 GBq (50 Ci) Medium Conv. + LSC Well documented use

C4H9T 12.32 y Y Y β− ∼ 10−3 Bq/l 1850 GBq (50 Ci) Medium Conv. + LSC Well documented use14CH4 5730 y Y Y β− ∼ 5× 10−4 370 GBq High Conv. Not

Bq/l (10 Ci) + LSC reported14CH3 * 5730 y Y Y β− ∼ 5× 10−4 370 GBq High Conv. Not

CH3 Bq/l (10 Ci) + LSC reported14CH3 * 5730 y Y Y β− ∼ 5× 10−4 370 GBq High Conv. Not

C2H5 Bq/l (10 Ci) + LSC reported14CH3 * 5730 y Y Y β− ∼ 5× 10−4 370 GBq High Conv. Not

C3H7 Bq/l (10 Ci) + LSC reported

C. Non-radioactive species

He - Y Y - - - Low/medium MS Spesial cases

SF6 - Y Y - ∼ 10−12 l/l > 102 kg Low GC/ECD i) Well documented

Freon-11 - Y Y - < 10−9 l/l > 102 kg Low GC/ECD Probably restricted (?)

Freon-12 - Y Y - < 10−9 l/l > 102 kg Low GC/ECD Probably restricted (?)

PMCP - Y Y - < 10−12 l/l 5 kg Low GC/ECD Promising eld response.

PMCH - P c) P - < 10−12 l/l 25 kg Low GC/ECD Promising, but not

suciently tested.

nearby well. The total ow rate between the wells isassumed to be known and is denoted by q. The max-imum dimensionless tracer concentration for a one-layer model of a reservoir with the same well pat-tern may be obtained from the results of Brighamand coworkers, or by a 2D simulation. This peakvalue is denoted by Cp. The arrival times of the fourtracer peaks are denoted by t1, . . . , t4 and the respec-tive dimensionless tracer concentration peak valuesare Cp1, . . . , Cp4. Since the response curve consists offour isolated peaks, the results may be explained asthe response from four separate layers. The fractionalow in each layer is proportional to the permeability-thickness product and may be written

qiq

=kihi∑4j=1 kjhj

. (7.1)

Based on the assumptions above, we see that the dif-ference in peak values, Fig. 7.1, relative to the orig-inal Cp, is due to dilution caused by the tracer-freewater produced from the remaining layers. This isexpressed by

qiq

=CpiCp

. (7.2)

Thus, the quantity

kihi∑4j=1 kjhj

may be determined for each layer i.The volume swept is related to the parameters by

the expression

tiqi = βφi(1− Sor,i)hi, (7.3)

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7.2. RESERVOIR CHARACTERISTICS 163

Pore Volume Injected

Tra

cer

Con

cent

ratio

n

Figure 7.1: Tracer response for layered reservoir

where β represents the areal sweep eciency, whichmay be assumed known for a specic well pattern.Thus, the value φi(1 − Sor,i)hi for each layer is alsodetermined by this method.The inclusion of longitudinal dispersion makes an

analytical treatment more dicult. Although themethod described strictly applies for quite idealizedconditions only, it has provided useful information forrealistic reservoir problems.63,64

Abbaszadeh-Dehghani and Brigham59 describe amethod which optimizes the results when the tracerresponse curves from the layers interfere, a phe-nomenon which usually occurs. Hagoort65 investi-gated the tracer response in a similar model with con-tinuous stratication.A comparison between tracer testing and pressure

testing concerning the determination of layer struc-ture has been performed by Mishra and Ramey.66

They concluded that tracer testing is much more sen-sitive to the geological structure of the reservoir thanpressure testing. The tracer response curves have alsobeen utilized to determine appropriate tracer injec-tion volumes.18,67

Hagoort65 discussed the relative importance of lon-gitudinal versus transversal dispersion for a tracer sig-nal in a watered-out layered reservoir. He concludedthat, in his treatment, transverse dispersion is themost important factor.Single-layer heterogeneous reservoirs have been

considered by Mishra et al.68 In this work, a het-erogeneity index was introduced related to tracer re-sponses. Reservoirs characterized by a high indexshowed distinct preferential ow paths. These resultsin tracer response functions are impossible to matchby the simplied models described above. Tracer re-sponses from reservoirs with a simple heterogeneityhave also been studied by utilizing a general numeri-cal streamtube model by Sagen et al.69

Tracer testing may have potentials in a method fordetermining the importance of imbibition in reservoirow. The work of Sylte et al.70 and Skilbrei andSylte29 describe a signicant reduction in tracer res-idence time in the reservoir, when comparing tracerinjection at late and early stages during continuouswater injection in the Ekosk reservoir. Rogde30 hasrecently reported eects on the tracer responses fromthe Gullfaks eld, also attributed to imbibition.

As part of remedial work prior to a CO2 injec-tion, Beier and Sheely47 used the result of well-to-well tracer tests to compute volume sweep eciencybetween injection and production wells. An analyti-cal method was applied. They were able to identifychannels between several injectors and producers.

7.2.3 Residual Oil Saturation

Chemicals which are introduced in small amounts inmultiphase ow move at a velocity which depends onthe degree of partitioning between the phases and thefractional ow of each phase. Certainly, also otherphenomena may be important, but are neglected inthis treatment.71 In tracer testing this is utilized todetermine residual oil saturation. The process is con-sidered by means of the following conservation equa-tion for the tracer mass in 1D, two-phase ow:

∂t((SwCwρw + SoCoρo)φ)

+∂

∂x(Cwρwuw + Coρouo) = 0. (7.4)

We have here neglected adsorption of tracer on therock and tracer diusion/dispersion. Assume, in ad-dition, that the uids are incompressible. The parti-tioning is often treated by a linear equilibrium equa-tion,

Co =ρwρoKCw.

Also nonlinear models have been used.72,73 For sim-plicity, we denote Cw by C. At residual oil saturation,Sor, Eq. 7.4 may be written in terms of concentrationin the water phase

Ct +uw

φ(1− Sor +KSor)Cx = 0. (7.5)

This is a rst-order wave equation which is easilysolved. The velocity of a tracer slug is

v =uw

φ(1− Sor +KSor), (7.6)

and a partitioning tracer, K > 0, is retarded relativeto a water tracer, K = 0, by

vP =1

1 +K Sor1−Sor

vW . (7.7)

Cooke74 proposed to determine residual oil saturationby well-to-well tracer testing by recording the arrivaltimes of two tracers with dierent partitioning coef-cient. According to Chang et al.,75 the applicationof this method has never been reported.In single-well tracer testing, a chemical which re-

acts with the reservoir uids and produces a newchemical with a dierent partitioning coecient, isutilized. After injection of the primary tracer A, ashutin period is necessary to allow for production ofthe secondary tracer B. The well is backproduced un-til sucient amounts of tracer A and B have arrived.

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164 CHAPTER 7. TRACER TESTING

The single-well chemical tracer method was proposedby Deans76 and had been utilized in more than 140tests until 1986, .77 It is considered to perhaps be themost reliable means for determining in-situ Sor.75,78

Connate water saturation may be estimated by asimilar method.79 For the determination of resid-ual oil saturation, an oil- and water-soluble ester isusually applied as the primary tracer. The ester hy-drolyzes and produces a secondary tracer which is al-most insoluble in oil. At ideal conditions, i.e., singlelayer, radial ow, negligible dispersion, and secondarytracer only created during shut in, the following ex-pression for determination of the residual oil satura-tion may be obtained from Eq. 7.7,

Sor =VA − VB

VA + (KA − 1)VB. (7.8)

Here VA and VB are the amounts of uid produceduntil the arrival of tracer A and B respectively, andKA is the partitioning coecient for tracer A denedabove.Since the eect of dispersion is to reduce concentra-

tion gradients, an injected tracer pulse may becomequite smeared in the backow process. To determinethe arrival time, or more precisely, the produced uidvolume for such a smeared tracer prole, Deans andcoworkers applied the following integral mean,

VI =

∫∞0CIV dV∫∞

0CIdV

. (7.9)

Here V is the volume produced and CI is the pro-duced concentration of tracer I, as a function of pro-duced volume. The quantity VI is called the mean re-tention volume. Deans and Majoras80 demonstratedthat, under certain conditions, the volumes in Eq. 7.8may be replaced by the mean retention volumes tocover the eect of dispersion. They also discussedhow the eects of uid drift and hydrolysis duringuid ow may be included in Eq. 7.8. Optimal con-ditions concerning the application of their extendedformula are also discussed. The models have beenquite successful for estimation of residual oil satura-tion in homogeneous sandstone reservoirs.Seetharam and Deans77 treated the eect of ow

irreversibility in reservoirs consisting of multiple lay-ers. A computer program is described which permitsestimation of residual oil in multiple layer reservoirs.Residual oil saturation in carbonate reservoirs has

not been satisfactory estimated by the simple mod-els discussed above. Deans and Carlisle81 developeda model which included diusive tracer transport be-tween the owing fraction and the stagnant fractionof the pores. By their extended model, some charac-teristic features of single-well tracer testing performedin carbonate reservoirs were reproduced.Tang and Harker78 reported a new method called

the Mass Balance Method for determination of Sorby tracer testing. They claimed that this method iswell suited for single-well tracer testing in carbonatereservoirs. The method utilizes the fact that the ester

hydrolysis is a function of water saturation. Reducedwater saturation implies reduced hydrolysis.

7.3 Field ExamplesRadio-active Tracers

7.3.1 Introduction

Various types of tracers have for many years beenused in reservoir ooding with the main objective todetermine interwell ow characteristics such as wellcommunication, ow barriers, preferential ow path,rate of movement of injected uid, and sweep ecien-cies. The well-to-well tracer technology is recently re-viewed by Agca et al.82 They also present tracer re-sults from the Big Muddy Field, Wyoming, and claimthe UTCHEM simulator to be a suitable tool for in-terpreting the eld observations.The properties of noncommunicating layered reser-

voirs have been studied by many investigators. Themodel study by Mishra and Ramey66 on relation-ship between well-to-well tracer and buildup pressure-transient responses shows the computed tracer testresponse to be sensitive to the degree of layering andpermeability contrast, while the pressure response isnot. A corresponding eld example was carried outby Brigham and Abbaszadeh-Dehghani.60

The use of well-to-well tracers in the North Seahas increased rapidly in the last ten years. The ob-jectives have mainly been qualitative, i.e., to conrmthe ood behavior provided by the reservoir monitor-ing program and simulation.In the future, we expect the well-to-well testing to

be an important part of the general reservoir monitor-ing program for all new elds to be put on production.For producing elds in the decline period, the abilityof tracer testing to determine sweep eciency will beimportant for nding unswept parts of the reservoir,i.e., the locations for inll wells.Some of the major applications of tracer in the

North Sea will be reviewed in the following.

7.3.2 Monitoring the Gullfaks Field

The heavily faulted Gullfaks eld is considered one ofthe most geologically complex elds in the North Sea.Despite repeated acquisition of three-dimensionalseismic data, the interpretation of these data havein many parts of the eld failed to forecast pres-ence of minor faults and the exact position of themajor faults. In addition to the uncertainty in geo-logical mapping, the degree of communication acrossthe faults was unknown up to the time when the rstwell was put on stream. Therefore, tracer technologywas included from production start as one of severalmethods for reservoir monitoring the Gullfaks eld.A screening of possible water tracers was rst per-

formed. The logistical and safety aspect of tracerinjection pointed to beta-emitting radioactive tracerswith low detection limits in the water phase. Possible

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7.3. FIELD EXAMPLESRADIOACTIVE TRACERS 165

Figure 7.2: Gullfaks A/B area, top Rannoch. Wells included in tracer program.

retention in the reservoir reduced the number of pos-sible chemical forms. Additionally, considerations ofcost and availability of the radioactive material, andanalysis and injection service, limited the total num-ber of prequalied tracers to two, namely tritiatedwater (HTO) and C-14-labelled thiocyanate (C-14).Later, as the detection resolution was improved, thephysical and economical requirement were changed,and Cl-36 in the form of chloride was also prequali-ed.The major incentive to perform the rst tracer in-

jection in the Gullfaks eld was to detect water owacross a major fault where sand to sand contact wasmapped. The second objective was to test the abilityof the tracer to follow the water front. The third ob-jective was to acquire information on the ow patternwithin the fault block.The rst tracer batch, 15000 GBq HTO, was in-

jected in the rst injection well (A-5H), Fig. 7.2, April15. 1987, ten days after it was put on stream. Thisinjector is completed subsea and perforated in the oilzone. It is situated close to a major fault, straightacross from a planned producer (A-10) in the adja-cent fault block. The horizontal distance to this pro-

ducer and to the closest producer in its own faultblock (A-2AH) is 450-500 m.Breakthrough of tracer and water occurred simul-

taneously in A-2AH after exactly one year. At thispoint, there were doubts whether tracer movementacross the fault was possible, because of the large,measured pressure gradient in the opposite directionin the early production phase. To avoid missing thedetection of any ow across the fault, a second slug of5000 GBq HTO was injected into A-5H, August 16.1988. At the same time, 15 GBq C-14 was injectedinto well A-12 further north. Well A-12 has a similarposition as A-5H, close to a major fault, across froma producer. The well is perforated in the water zone,and was put on stream 6 months prior to the tracerinjection.The three tracer slugs were never detected across

their neighbouring faults. This means that water owis negligible across these faults in this particular for-mation and does not aect well placement in the area.The tracer movements within each fault block have

yielded valuable information. The second HTO slugfrom A-5H arrived at the closest producer (A-2AH)only 4 months after it was injected, Fig. 7.3. Also,

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166 CHAPTER 7. TRACER TESTING

the total recovery of tracer from the rst tracer slugis much lower than that for the second injection. Thedierence in observed travelling time for the two slugshas served as an important parameter in the historymatching of the eld. The loss mechanism has beensuggested to be caused by imbibition of front water.30

The C-14 injected in A-12 arrived at the closestproducer A-3H simultaneously with the water break-through, Fig. 7.4. This supports the theory of im-bibition. However, some ambiguity exists, since theGullfaks reservoir is not known to be strongly waterwet.

4

3

2

1

088 89 90 91

HTO in A-1H

HTO in A-17

HTO in A-2AH

HTO in A-19

X10

3(B

q/l)

Figure 7.3: Gullfaks HTO results.

88 89 90 91

4

3

2

1

0

C-14 in A-3H

C-14 in B-4

C-14 in B-7

C-1

4co

ncen

trat

ion

(Bq/

l)

Figure 7.4: Gullfaks C-14 results.

A completely dierent objective for tracer injectionin the Gullfaks eld was formulated for the WAG pilotstarted in March 1991. The tracer study is part ofan extensive data acquisition program designed foruse in the history matching of reservoir simulationmodels built for the WAG pilot and mainly coveringthe central fault block. The appointed WAG injectorwas labelled with 7400 GBq HTO in July 1990, twoand a half years after it was put on stream as a waterinjector. It was relabelled with 0.7 GBq Cl-36 a shorttime before the gas injection period, and with gastracers during the rst gas injection periods.In summary, valuable reservoir information has

been obtained from the Gullfaks eld by the use oftracers. Tracer response data is regarded as impor-tant for understanding the ood pattern and hencefor the reservoir management of the eld.

7.3.3 Veslefrikk Tracer Implementa-tion Plan

Despite a signicantly higher quality of the seismicdata of the Veslefrikk eld, compared to many otherelds, essential information on the communication

pattern in the reservoir was unavailable before pro-duction start. This, together with a strategy ofcommingled production from two separate reservoirs,brought forward the importance of extensive earlymonitoring of the ood pattern of the two reservoirs.Prior to production start, together with many otherreservoir monitoring methods, a eld-wide tracer im-plementation plan was made, with the intention tolabel each water injector uniquely. This was doneby employing the three available tracers, previouslyprequalied for Gullfaks, in dierent combinations infour of the ve wells, and leaving the remaining wellunlabelled. The tracers were injected soon after eachwell came on stream, early in 1991. To avoid prob-lems with interpretation due to the mechanism of im-bibition, a partitioning tracer was injected togetherwith the water tracer in the rst well. The partition-ing tracer is C-14-labelled butanol. To date, no dataare available since water breakthrough has not yetoccurred. The tracer program is expected to provideessential information for waterood monitoring.

7.3.4 Ekosk Field Application

The rst published data from the North Sea on inter-well tracers are from the Ekosk water injection pilotin the Lower Ekosk formation, performed in 1986 to1988.70 Two dierent radioactive tracers were used.Tritiated water was injected from the start of the wa-ter injection period. The second tracer, Iodine-125,was injected in one batch, approximately six monthslater.The tracer observations in the surrounding produc-

tion wells show a steeper increase in the 125-Iodineresponse, compared with the corresponding resultsfor tritiated water. The transport times through thereservoir were only slightly dierent. These observa-tions were qualitatively interpreted as evidence of anecient imbibition ooding mechanism. However, aquantitative evaluation is wanting since no appropri-ate simulation tool is available.

7.3.5 Murchison Field application

Tritiated water, Cobalt-60 and Carbon-14 were in-jected into three injection wells in 1981, one yearafter the eld was brought on stream. The tracerswere injected soon after the injection wells had beencompleted. The objective was to determine whetherinjected water remained in the Rannoch-type (Micasand) formation or moved upward into the Etive-type(Massive sand) formation.Massie et al.83 summarize the eld behavior four

years later. The tracers had not been detected al-though injection water had broken through in thesurrounding updip producers. The complete tracerloss in the reservoir was surprising, since the sametracers had been used with success in a previous eldtracer test.21 One possible explanation may be tracerloss at the water front due to imbibition, as indicatedfor the Gullfaks eld above.30

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REFERENCES 167

In 1988, a new comprehensive tracer test programwas implemented in the Murchison eld.84 Tritiatedwater, Carbon-14 and combinations of the two wereinjected into various injection wells to investigate in-terwell and fault transmissibilities and areal/volumesweep eciency. No results are yet reported.

Nomenclature

C = tracer concentration, mass fractionCp = tracer peak valueh = thickness, mK = partitioning coecientk = permeability, mdq = ow rate, m3/sS = saturationt = arrival time, su = Darcy velocity, m/sv = velocity, m/s

V, V = volume, m3

w = velocity of tracer slug, m/sβ = areal sweep coecientφ = porosityρ = density, kg/m3

Subscripts

A = reactive tracerB = product tracerI = tracer, I = A,Bi = layer numbero = oilP = partitioning tracerr = residualt = dierentiation with respect to time

W = water tracerw = waterx = dierentiation with respect to distance

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170 CHAPTER 7. TRACER TESTING

Technical Conference and Exhibition, Housten,Oct. 25.

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[72] Tomich, J.F., Dalton, R.L., Deans, H.A.,and Shallenberger, L.K.: Single-Well TracerMethod to Measure Residual Oil Saturation,JPT (Feb. 1973) 2118.

[73] Sheely, C.Q.: Description of Field Tests to De-termine Residual Oil Saturation by Single-WellTracer Method, JPT (Feb. 1978) 194202.

[74] Cooke, C.E.: Method of Determining ResidualOil Saturation in Reservoirs, U.S. Patent No.3,590,923 (July 1971).

[75] Chang, M.M., Maerefat, N.L., Tomtsa, L., andHonarpour, M.M.: Evaluation and Compari-son of Residual Oil Saturation DeterminationTechniques, SPEFE (March 1988) 25162.

[76] Deans, H.A.: Method of Determining FluidSaturations in Reservoirs, U.S. Patent No.3,623,842 (Nov. 1971).

[77] Seetharam, R.V. and Deans, H.A.: SimulatorModels for Single-Well Tracer Test ParameterEstimation and Model Selection in Heteroge-neous Formations, paper SPE 15435 presentedat the 1986 Annual Technical Conference andExhibition, New Orleans, Oct. 58.

[78] Tang, J.S. and Harker, B.: Mass BalanceMethod to Determine Residual Oil Saturationfrom Single Well Tracer Test Data, JCPT(March-April 1990) 11524.

[79] Deans, H.A. and Shallenberger, L.K.: Single-Well Chemical Tracer Method to Measure Con-nate Water Saturation, paper SPE 4755 pre-sented at the 1974 SPE Symposium on IOR,April 2224.

[80] Deans, H.A. and Majoros, S.: The Single-Well Chemical Tracer Method for Measur-ing Residual Oil saturation, Final Report,DOE/BC/20006-18 (Oct. 1980).

[81] Deans, H.A. and Carlisle, C.T.: Single-WellTracer Test in Complex Pore Systems, paperSPE 14886 presented at the SPE/DOE Sympo-sium on EOR, Tulsa, April 2023.

[82] Agca, C., Pope, G.A., and Sepehrnoori, K.:Modelling and Analysis of Tracer Flow in OilReservoirs, J. Pet. Sci. Eng. (1990) 4, 319.

[83] Massie, I., Beardall, T.J., Hemmens, P.D., andFox, M.J.: Murchison: A Review of ReservoirPerformance during the First Five Years, pa-per SPE 14343 presented at the 1985 SPE An-nual Technical Conference and Exhibition, LasVegas, Sept. 2225.

[84] King, W.R. and Leibrecht, R.L.: The Impor-tance of Data Aquisition in the Developmentof the Murchison Field, paper presented atthe 1990 third Conference on Reservoir Man-

agement in Field Development and Production,Kristiansand, June 1112.

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Chapter 8

Reservoir Modelling

8.1 Review of Numerical Mod-els

8.1.1 Introduction

Numerical reservoir simulation models are today es-sential tools in eld development planning and reser-voir management, and a wide variety of mathe-matical formulations and numerical solution tech-niques has been described in the literature duringthe past decades. This review is mainly concernedwith the state-of-the-art level represented by themost widespread, commercially available simulators.Model equations, solution techniques and typical nu-merical errors are reviewed, and the applicability ofdierent formulations for dierent recovery processesis discussed. Simulation models for thermal recoverymethods and fractured reservoirs are not included,and no attempt has been made to reference all rele-vant literature.Commercial reservoir simulators are generally

based on block-centered nite dierence methods fornumerical discretization, and the present expositionis with few exceptions limited to this approach. Thefundamentals of reservoir simulation with nite dier-ence methods are described in several textbooks.13

8.1.2 Black-Oil Models

Volumetric models encompass the widely appliedblack-oil model and some variants and extensionsthereof where the conservation equations are ex-pressed in terms of oil, gas and water volumes at stan-dard conditions, i.e., after separation at the surface.In nite dierence form, a fairly standard black-oil

model can be summarized as follows,4

Mn+1mi −M

nmi = −∆t(Qmi +

∑j

qmij) (m = o, g, w),

(8.1)where the gridblock accumulation terms are given by

Mo = V φ(boSo + rsbgSg), (8.2a)

Mg = V φ(bgSg +RsboSo), (8.2b)

Mw = V φbwSw, (8.2c)

and the interblock ow terms are given by

qoij =bokroµo

Tij ∆Φoij +rsbgkrgµg

Tij ∆Φgij , (8.3a)

qgij =bgkrgµg

Tij∆Φgij +Rsbokroµo

Tij∆Φoij , (8.3b)

qwij =bwkrwµw

Tij∆Φwij , (8.3c)

where

∆Φmij = (pmi−pmj)−ρmg(Di−Dj) (m = o, g, w).(8.4)

All variables are understood to be averages for a grid-block or a pair of coupled gridblocks in the usualsense. This means, for instance, that the nonindexedterms of Eqs. 8.3a-c are evaluated in gridblock i or jdepending on the direction of ow (upstream mobili-ties).The gridblocks are assumed here to be numbered

by a single index regardless of the number of spatialdimensions, and the summation in Eq. 8.1 is over allneighboring (or coupled) gridblocks. The interblocktransmissibilities, Tij , the net gridblock volumes, Vi,and the block center depths, Di, depend on the ge-ometry of the gridblock system and the absolute per-meabilities, and they all can usually be considered asconstant during a simulation. The above formulationis valid both for purely rectangular gridblock systemsand for more general gridblock congurations as forinstance corner-point geometry,5 grids with local re-nements, and certain nite-element formulations,6

but the number of couplings per gridblock and theexpressions for Tij , Vi and Di will of course vary.For each timestep, the primary variables are the

pressures and the saturations in each gridblock.Eq. 8.1 gives three equations per gridblock, but thenumber of primary unknowns are reduced to three,e.g., po, So and Sw if all phases are present, by thefollowing three auxiliary equations for each gridblock,

So + Sg + Sw = 1, (8.5a)

po − pw = pcow, (8.5b)

pg − po = pcgo. (8.5c)

A variety of solution techniques is available foradvancing the solution in time. With the IMPES

171

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172 CHAPTER 8. RESERVOIR MODELLING

(implicit pressure - explicit saturation) method,2 allterms of the right-hand side of Eqs. 8.3a-c are takenat the old time level, n, except the selected primarypressure variable, and a comparatively rapid, sequen-tial solution for pressure and saturations is possible.With a fully implicit method,4 all ow terms aretaken at the new time level n + 1, and a simulta-neous solution for all three primary unknowns gen-erally must be performed. More computer time pertimestep than with IMPES is required, but in manycases longer timesteps can be taken since the fully im-plicit method is unconditionally stable. Intermediatesolution techniques are also employed in many simu-lators, i.e., with some implicitness that still permitsa sequential solution for pressure and saturations.

The solution of the sets of linear equations thatarise for each timestep (or rather for each outer iter-ation if for instance a Newton method is applied tohandle the nonlinearities) generally takes a large frac-tion of the computer time for a simulation, especiallyif a large number of gridblocks are used. Again, manyalternative methods exist. Nested factorization7 andsuccessive overrelaxation methods1 are among themost popular ones.

The black-oil model described above is applicablefor many recovery processes in both oil, gas and gascondensate reservoirs: (1) depletion processes, heretaken as all recovery processes not involving injection;(2) water injection; (3) many cases of gas injection.

The inclusion of the rs-terms (which are not in-cluded in all black-oil simulators) makes the modelapplicable also for depletion of volatile oil reser-voirs, where the free gas produced from the reservoirmay contribute signicantly to the surface-oil pro-duction, and for depletion of gas condensate reser-voirs. The preparation of uid data is more di-cult for these cases than for standard black-oil ap-plications, however, and generally requires numericalPVT-simulation.8

The applicability of the model to gas injection pro-cesses depends very much on the type of gas injectedand what phase-behavior eects are important in thereservoir. With both the Rs- and rs-terms included,both swelling and vaporization of reservoir oil can inprinciple be modelled, but the model is generally notapplicable if the injection gas diers signicantly fromthe solution or equilibrium gas in the reservoir. Also,although the swelling/vaporization eects to some ex-tent can mimic a multicontact miscible process, othertypes of models are recommended for such processes.

Another important aspect of simulating gas injec-tion processes should be mentioned. The assumptionof instantaneous phase equilibrium within entire grid-blocks can be very optimistic, and therefore it is oftenbetter to disregard swelling and vaporization. Somesimulators have the possibility to put a time delay onthese mechanisms.

For simulation of WAG injection processes, manyof the same considerations as for gas injection apply.

8.1.3 Grid Eects

The most important numerical errors associated withnite dierence type simulation models are numericaldispersion and grid orientation eects.Numerical dispersion has the eect of smearing dis-

placement fronts and arises because of the upstreamevaluation of the coecients in the interblock owterms (which generally is necessary to achieve stabil-ity). The eect is illustrated in Fig. 8.1, and someimportant characteristics can be obtained from an

80

60

40

20

00 500 1000

Distance, ftW

ater

satu

ratio

n,%

Analytical solution

Numerical solution

Figure 8.1: The numerical dispersion eect illustratedfor a one-dimensional displacement of oil by water.3

analysis of the equation for one-dimensional, incom-pressible, oil-water ow corresponding to Eqs. 8.18.4. This equation can be written

φ∆x(Sn+1w − Snw) = −v∆t(fi − fi−1), (8.6)

where v is the total ow rate (oil + water), and fis the fractional ow function for water. For a linearfractional ow function, f ′ = df/dSw = constant, thenumerical dispersion level is given by1

Dnum =1

2vf ′∆x

(1− vf ′∆t

φ∆x

)(8.7a)

or

Dnum =1

2vf ′∆x

(1 +

vf ′∆t

φ∆x

). (8.7b)

Eq. 8.7a applies if the right-hand side of Eq. 8.6 istaken at time level n (corresponding to an IMPESmethod), while Eq. 8.7b applies if time level n + 1is chosen (corresponding to a fully implicit method).There are three important implications:(1) For a given grid and the same timestep sizes,

the numerical dispersion is always larger with a fullyimplicit method than with IMPES.(2) For a fully implicit method, the numerical dis-

persion eect is increasing with timestep size.(3) For IMPES, the numerical dispersion is de-

creasing with timestep size. Under the given idealassumptions, the numerical dispersion eect is van-ishing when ∆t is taken at the stability limit.With a nonlinear fractional ow curve, the dier-

ence between the so-called Welge tangent and the

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8.1. REVIEW OF NUMERICAL MODELS 173

fractional ow curve gives rise to a self-sharpeningeect which in eect reduces the numerical disper-sion. This can in some cases be used to constructpseudorelative permeabilities which reduce the nu-merical dispersion. It also means that single-phaseconcentration fronts and gas-oil displacement fronts(not stabilized by gravity) in general will be muchmore inuenced by numerical dispersion than water-oil displacement fronts.The numerical dispersion can be reduced by a two-

point upstream mobility weighting scheme9 whichis available in many simulators. There are alsomany other ways to reduce numerical dispersion,both within the context of nite-dierence meth-ods,10,11 and with dierent categories of numericalmethods.1214

The grid-orientation eect is another importantcharacteristic of conventional nite-dierence meth-ods. The eect is illustrated in Fig. 8.2 where theresults of simulations with similar gridblock sizes butwith dierent grid orientations with respect to themain ow direction are compared. The eect is mostpronounced for unfavorable mobility ratio displace-ments, and it arises because the usual 5-point (2-D) and 7-point (3-D) nite-dierence schemes give apreference for ow parallel to the gridblock axes, i.e.,they are not rotationally invariant. This problem canbe much reduced by including couplings (transmis-sibilities) diagonally between gridblocks, i.e., with a9-point scheme in 2-D.15,16

8.1.4 Other Volumetric Models

One way to account more properly for phase behavioreects in black-oil type models is to correlate the oiland gas properties with some measure of the degreeof contact there has been between the oil and theinjection gas.17,18 For instance, a decreasing abilityof the injection gas to vaporize residual oil as thelighter components of the residual oil get stripped ocan be represented in this fashion. The approach hasbeen implemented in several commercial simulators,and some good results have been reported. The uiddata preparation can be cumbersome, however, andthe accuracy is uncertain if the pressure/compositionpath followed in the reservoir diers too much fromthe one for which the uid data are generated.A dierent volumetric model is the Todd-Longsta

miscible ood model,19 which has gained wide pop-ularity. Basically, this model is referred to a owsituation with rst-contact miscibility, i.e., where allhydrocarbon compositions are linear combinations ofthe reservoir oil and injection gas, and it is designedto represent a mixing process of oil and gas dominatedby viscous ngering without resolving the details ofthe ngering geometry.The Todd-Longsta model diers from the basic

black-oil model mainly in that (1) So and Sg nowtake the meaning of the fraction of reservoir oil andinjection gas (whether fully mixed or in separate n-gers) in a gridblock, and (2) the mobility, λ = kr/µ,

of each component is given as

λo =SoShkrh/µoe, (8.8a)

λg =SgShkrh/µge, (8.8b)

where Sh = So + Sg, krh is the relative permeabil-ity to a hydrocarbon phase as a function of watersaturation, and the eective viscosities are taken asfunctions of the local degree of mixing,

µoe = µ1−ωo µωa , (8.9a)

µge = µ1−ωg µωa . (8.9b)

µo and µg are pure-component viscosities, and µa isa fully mixed viscosity,

µ−1/4a =

(SoShµ−1/4o +

SgShµ−1/4g

). (8.10)

This model is applicable both for rst-contact andmulticontact miscible processes, but one main di-culty is to select a suitable value for the mixing pa-rameter ω; ω = 0 (1) corresponds to no (full) mixing,and ω = 0.5 is often quoted as a suitable value forfull-eld reservoir simulation. In reality, several fac-tors such as the presence of small-scale reservoir het-erogeneities need to be taken into consideration andthe choice of ω requires great care, see for instanceRefs. 20 and 21.A four-component version of the Todd-Longsta

formulation is required if parts of the reservoir hasfree gas dierent from the injection gas,19 but itshould be noted that the formulation has no provi-sion for a transition between miscible and immiscibleconditions.

8.1.5 Compositional Models

A compositional model corresponding to the black-oilmodel described in Sec. 8.1.2 is given by:

Mn+1wi −M

nwi = −∆t(Qwi +

∑j

qwij), (8.11a)

Mn+1αi −M

nαi = −∆t(Qαi +

∑j

qαij), (8.11b)

where α now represents each of the hydrocarbon com-ponents (possibly including N2, CO2 and H2S). Mw

and qwij are given by Eqs. 8.2c and 8.3c, but the oiland gas terms are replaced by

Mα = V φ(xαCoSo + yαCgSg), (8.12)

and

qαij =xαCokro

µoTij∆Φoij +

yαCgkrgµg

Tij∆Φgij .

(8.13)Eqs. 8.4 and 8.5a-c still apply. Furthermore, we nowhave component material balance equations,

xαL+ yα(1− L) = zα, (8.14)

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174 CHAPTER 8. RESERVOIR MODELLING

(a)

(b)

1.0

0.9

0.8

0.7

0.6

0.5

0.40.5 0.7 0.9 1.1 1.3 1.5

PV injected

Oil

reco

very

,PV

Diagonal

Parallel

Yanosik-McCracken, 9-point∆x = 0.2∆x = 0.1∆x = 0.05Standard 5-point

scheme

(c)

.

.

Figure 8.2: The grid orientation eect illustrated for an unfavorable mobility ratio displacement in a 5-spotpattern: (a) parallel grid, (b) diagonal grid, and (c) comparison of simulation results.3

and phase equilibrium equations,

Kα =yαxα

or fαo = fαg, (8.15)

as additional auxiliary equations for each gridblock.L is the total mole fraction of hydrocarbon liquid inthe gridblock and is connected to the hydrocarbonsaturations by

L =CoSo

CoSo + CgSg. (8.16)

Some simulators include a dispersive ow term, i.e.,with concentration gradients as the driving force forow, but in most applications this would be overshad-owed by numerical dispersion. In some simulators,the dissolution of hydrocarbon components in wateris included, which may be important for applicationsto CO2-ooding.The hydrocarbon phase behavior is today in most

compositional simulators described by a cubic equa-tion of state (EOS), e.g., the Peng-Robinson EOS orvariants of the Redlich-Kwong EOS.2224

Eqs. 8.11a-b now represent NC + 1 equations, andthe primary variables for each timestep could be po,Sw and zα, α = 1, . . . , NC − 1. NC is the number ofcomponents. An IMPES-type solution technique ismost often applied in compositional reservoir simula-tors despite the associated stability restrictions. Im-plicit compositional simulators are also available andmay oer great advantages for some types of prob-lems. It should be noted, however, that the ratio be-tween computer time per timestep for implicit contrathe IMPES method generally is considerably largerfor a compositional model than for a black-oil model,especially when the number of components is high.

The computational eciency of compositional simu-lators has increased a lot during the last decade, butcompositional simulation still generally requires con-siderably more computer time than black-oil simula-tion.An important aspect of compositional simulation

is the choice of components and the tuning of thephase behavior model to experimental PVT-data, i.e.,the assignment of appropriate values for the relevantcomponent properties. Typically, 5 to 12 componentsare used.25

A compositional model is most often applied forreservoir processes where a volumetric model is in-sucient, typically:

• gas injection in gas condensate reservoirs

• gas injection with near or developed miscibility

• injection of gases like N2 and CO2

• gas cycling for vaporization of residual oil.

Although a compositional model may seem like (andindeed in some respects is) a most general and accu-rate tool, some important shortcomings can be men-tioned:

• As mentioned in connection with black-oil mod-els, the assumption of instantaneous phase equi-librium in gridblocks can be overly optimistic re-garding swelling and vaporization, and with acompositional simulator it is dicult (or impos-sible) to relax this assumption as with black-oilmodels.

• A typical compositional simulator has no built-inmechanism for ngering as in the Todd-Longsta

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8.2. GEOLOGICAL STRUCTURES 175

formulation. Thus for miscible processes a Todd-Longsta model may represent a better approachdespite the simplistic phase behavior treatment.

• The processes simulated with compositional sim-ulators are often more prone to numerical disper-sion and grid orientation eects than black-oilproblems.

8.1.6 Chemical Flood Models

The simulation of chemical ood processes is a verychallenging task, and this category of models has notreached the same level of maturity as the previoustypes. Areas of application for a chemical ood simu-lator may include polymer ooding, surfactant ood-ing with or without the use of a polymer, and deeply-set polymer gels.In the same nite dierence form as used previ-

ously, a general chemical ood model can be written:

Mn+1αi −M

nαi = −∆t

Qαi +∑j

qαij

, (8.17)

where

Mα = V

(φ∑m

ξαm ρm Sm + Γα

), (8.18)

and

qαij =∑m

ξαm ρm krmµm

Tij ∆Φmij

+ dispersion terms. (8.19)

Here α represents the set of components necessary todescribe the target processes. For surfactant oodingthese could include reservoir oil, water, surfactant,cosurfactant polymer, Na-ions and Ca-ions. Salt-ionconcentrations generally must be modelled becauseof the strong inuence on phase behavior. Typically,three uid phases must be taken into account: anoleic (oil-rich), an aqueous (water-rich) and a (mid-dle) microemulsion phase.The above equations do not fully reveal the com-

plexity of chemical ood modelling because the com-plexity mainly lies in the relationships between thevarious coecients, e.g., in

• the phase behavior (concentration) relationships

• the dependence of viscosities, densities and in-terfacial tension on concentrations

• the dependence of relative permeabilities on in-terfacial tension.

Fully satisfactory models for all these relationshipshave not yet been established, and all the experimen-tal data that currently are needed may be very di-cult to obtain.Chemical ood simulators are generally based on

an IMPES-type solution technique, because of the

high number of components and the very complicatedrelationships between the parameters. Dispersion canbe an important aspect because of strong dependencyof process eciency on concentration levels. For thisreason, physical dispersion terms often are included,and great care should be taken to ensure that thecombined eect of numerical and physical dispersionin the simulation is representative.Examples of chemical ood models are given by

several authors.2629 Scott et al.26 include equationsfor chemical reactions that can be used for instance tomodel a polymer gelling process. They also include atemperature equation because the eect of the tem-perature dierence between injection water and thereservoir can be important in some cases.Modelling of pure polymer ooding must in this

context be regarded as comparatively simple, and canbe done by some extension of a black-oil model.30

8.2 Geological Structures

The main concern of this section is clastic sandstonereservoirs, which represent most of the reservoirs inthe North Sea. Chalk reservoirs present challenges oftheir own.It will be focussed on the application of conven-

tional mapping procedures as employed by IRAP31

or GRID,32 since stochastic models are discussed ina previous chapter.The geological model, as built by a surface map-

ping program or by other means, is generally an arrayfor each layer with equidistant points in the x and ydirections. To each (x, y) point particular data is cou-pled, such as layer depth, permeability, porosity, orother petrophysical data. In order to save space, theorigin coordinates and the increment in the x- andy-directions may be given, followed by the associatedordered data. As a rule, the data is tied to a layeredrepresentation of the reservoir. Historically, this hasbeen convenient, in that arithmetical or logical oper-ations (like merging of layers), are easier to perform.However, it has become a constraint on the abilityto deal with heterogeneous reservoirs. Fig. 8.3 showsthe top of the upper formation of the Troll eld,33

which will be used here as an example. This imageis created from a three-dimensional array, where thedistance between the x- and y-points is 100 m, andthe data coupled to each of these (x, y) points is thedepth to the top of the reservoir.Onto this geological model, a simulation grid is

overlaid. The grid blocks may be cartesian, irregu-lar cartesian (corner grid point geometry), triangular(as appropriate to nite element methods), or as ad-equate to the reservoir simulator to be used. In anycase, a representative value of the geometrical andpetrophysical quantities in the geological model hasto be assigned to each simulation grid block.Here, corner-point geometry34 will be discussed

since it has become the standard representation formany North Sea simulation studies. Each gridblock is

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176 CHAPTER 8. RESERVOIR MODELLING

Figure 8.3: Top of the upper formation of the Troll eld.33

represented by its 8 corners. The corners are denedas depth measured along a vertical axis, and lie alonga so-called coordinate line which is specied by the x,y and z coordinates of its top and bottom, Fig. 8.4.In this way, gridblocks may have arbitrary shapes andreservoir boundaries, faults, erosion, pinch-outs andother geometrical features may be described withinthe simulation grid. The depth value of each corner

faultlowertrace

fault upper tracecornerpointsfault throw

coordinateline

Figure 8.4: Denition of a gridblock in corner-pointgeometry.

point of a given gridblock is calculated by interpola-tion between the nearest points in the geological gridof the layer being gridded.Fig. 8.5 shows a simulation grid with corner-point

geometry overlaid on the geological model of Fig. 8.3.In the following, several reservoir engineering aspects

Figure 8.5: Simulation grid overlaid on the geologicalmodel.33

of the transfer of the geological to the reservoir sim-ulation model will be discussed taking Figs. 8.3 and8.4 as examples.

8.2.1 Faults

Faults are usually represented in the geological modelby one trace for the upper and another trace for thelower limits (see Figs. 8.4 and 8.5). The throw isthen determined by the discontinuity of the contoursat the traces. The transfer to the simulation gridis straightforward: the upper trace is chosen as the

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8.2. GEOLOGICAL STRUCTURES 177

boundary of one row of simulation gridblocks at thetop. At the downwards side of the fault, the lowertrace is chosen as the boundary of the neighbouringrow of simulation gridblocks at the bottom. The in-tersection of the surfaces delineating the layers withthe fault plane denes the faces of the simulation gridat the fault plane.Care has to be taken with sloping faults, because

the values in the geological grid will not, as a rule, fallon corner points of the simulation gridblocks. Manualintervention is then usually needed to control the in-terpolation algorithms used in the gridding package.Fig. 8.5 shows how corner-point geometry can

be designed to follow faithfully the most importantfaults, giving correct volumes in place, ow bound-aries and reservoir geometry.

8.2.2 Assignation of PetrophysicalValues

Petrophysical values must be assigned to each simu-lation grid block. Permeability will be discussed hereas an example. Generally, reservoirs are considered tobe layered. Therefore, gridding programs calculate bydefault the horizontal permeability as the arithmeticaverage of the values in the geological grid within thesimulation gridblock. The vertical permeability is cal-culated as the harmonic average. Other values, likeporosity, uid saturation, net-to-gross ratios are av-eraged arithmetically.

8.2.3 Gridblock Size and Representa-tion of Physical Flow Mecha-nisms

The ow of gas is usually not hindered by hetero-geneities, due to its low viscosity. Pressure gradi-ents far from the production wells are generally small.The production mechanism is usually gas expansion,sometimes supported by active aquifers. Further-more, PVT properties, as for example gas formationvolume factor, change smoothly in an areal view, ifno seals are present. The wells, except in very lowpermeability reservoirs, are clustered in groups.As a result of these considerations, the uid trans-

port equations may be solved with large simulationgridblocks. In the Troll East and Troll West gasprovinces, some blocks are 2× 2 km2 in area.Oil ow is usually more complicated. Viscous, cap-

illary and gravitational forces may be important todierent degrees in dierent parts of the reservoirs.Furthermore, heterogeneities may also dictate owpaths. The production mechanism is usually providedby either dissolved gas coming out of solution or bydrive from gas or water columns. Pressure gradients,even in high permeability reservoirs, are signicant.Saturation may change both areally and vertically.The wells, which are pressure discontinuities, are usu-ally distributed throughout the reservoir.

As a result, numerical calculations require smallsimulation grid blocks in oil-producing areas. Thismay be seen in the Troll oil province at the left-handside of the grid shown in Fig. 8.5, which is too coarseto simulate oil production properly.Grid renement is then necessary to capture the

details of the uid movement in the wells. A problemarises when the geometry of the eld has been repre-sented in a grid with a distance larger than the detailrequired in ow calculations in the grid renement.In this example, the distance between geological gridpoints is 100 m and only a few meters in the gridrenement. If heterogeneities of size of the same or-der as the geological grid are present, and they areimportant in determining the ow pattern, then theyhave to be introduced by hand by means of transmis-sibility multipliers between simulation gridblocks, orby specially adapted programs.Lien et al.2 show how impermeable, thin calcite

layers may be introduced stochastically. Here, thecalcites found in a horizontal well are introduced byhand, and a probability function of other calcites isused to distribute random barriers in the rest of thesimulation model. Since these calcites have a dipwhich diers from the dip of the layers in the sim-ulation model, transmissibility multipliers in the ap-propriate x-, y- and z-directions are introduced, ap-proximating the eect of the the calcites in reducinguid ow. Fig. 8.6 shows an example. It should be

40m

1000m200m

Figure 8.6: Dipping calcites with small lateral extentsuperimposed on the simulation grid as transmissibil-ity barriers (modied from Lien et al.2)

realized that the dipping calcites in the reservoir donot have the same eect as broken surfaces followingthe simulation grid, and that some uid may be re-tained against these surfaces in the simulation whichwould not be hindered in the reservoir. However, noother method has been envisaged giving a better ap-proximation.An important issue is how the ow is calculated. If

high permeabilities and large density dierences be-tween reservoir uids are present, the ow calcula-tions may assume uid segregation (vertical equilib-rium). Fewer simulation gridblocks are then neededthan in the case of disperse ow.36 A more detailedrepresentation of the physical phenomena in the reser-voir is able to circumvent the need of a rened grid.

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178 CHAPTER 8. RESERVOIR MODELLING

8.2.4 Comunication Channels

Fig. 8.5 shows three of the communication channelsbetween Troll West gas province and the Troll Eastprovince, one in the north and two in the east-westdirection. The channel in the north has sand to sandcontact in the gas zone, whereas the other two havecontact only in the water zone.In some cases, pressure gradients arise because one

province of a eld is produced earlier, as in the case ofTroll where the East province will be produced from1996 while the production of the West gas provinceprobably will start later than year 2000. Then it isimportant to represent faithfully such communicationchannels. Since the geological grid shown in Fig. 8.3has a resolution of 100 m, and the channels may bethinner, manual intervention is required. Since thedepth conversion of the seismic data usually has anaccuracy of a few tens of meters, the thickness of thesand to sand contact in the hydrocarbon zones maynot be well determined. As the inuence on uidow from one region to another, and therefore on hy-drocarbon recovery, may be signicant, various caseshave to be run with dierent possible congurationsof such channels. In the case of Fig. 8.3, a dierencein thickness of a few tens of meters may representdierence in gas ow of several billions of standardm3.

8.2.5 Heterogeneities

Conventional mapping and gridding packages encour-age the construction of geological models where therole of heterogeneities is inhibited. In the last fewyears, a number of stochastic models have been de-scribed which introduce explicitly heterogeneities inthe geological model, as discussed previously in thismonograph.The inclusion of dipping calcite cemented layers or

other low-permeability features has been referred toearlier in this chapter. Stochastic shales of small lat-eral extent may be introduced by calculating an eec-tive vertical permeability,37 with the average lengthof the shales, how many are found per meter in thewells and the fraction of shales in the reservoir asparameters.Recent developments tends to incorporate both

sedimentological and geological knowledge togetherwith stochastic elements38 (and references therein).

8.2.6 Number of Layers

Increasing the number of blocks increases the com-putation time nonlinearly. Increasing the number oflayers is usually very onerous. However, it is very im-portant in most reservoir studies to represent perme-ability contrasts properly. The injection of uids inorder to maintain pressure or to improve the sweep ef-ciencies requires enough layers to study the interplaybetween viscous, gravitational and capillary forces.

Even in an homogeneous reservoir, the density dif-ference between the injected uids and the uids inplace will induce gravitational eects which require anumber of model layers to study hydrocarbon recov-ery.Vertical cross-section studies with varying numbers

of layers will indicate the right layering for a three-dimensional model.

8.2.7 Quality Control: Three-Dimen-sional View of the Reservoir

A very important step when the geological model hasbeen transferred to a reservoir simulation grid is toverify that the software used has performed satisfac-torily. Examples of zero or negative thickness, poros-ity, permeability, etc., have been found in reservoirmodels which have been used to plan eld develop-ment in places where they are physically not possi-ble. The reasons are rounding errors in the algorithmsperforming the data transfer, bugs caused by dicul-ties in imagining all possible cases, or operator errors.Fault throws, geometry on both sides of a fault, pinch-outs, etc., are typical traps for the reservoir engineer.The best quality control is provided by a three

dimensional view of the reservoir, its geometry andpetrophysical values. The software used to performthis, as the one which was used to generate Figs. 8.3and 8.5, should be able to rotate, zoom and pan theimage in real time, as it then provides the easiest wayfor the reservoir engineer to assure the data transferquality.Modern graphical workstations have proved to be

time and cost eective in quality assurance of thetransfer of the geological model to the reservoir sim-ulation grid.

8.3 Eective Properties

Naturally occurring porous media are never homoge-neous. Signicant heterogeneity may exist on spa-tial scales down to say, the cm or mm scale. Inpetroleum reservoirs, the processes of interest nor-mally take place on a much larger scale. It becomesimpractical, and even unnecessary, to take individ-ual small-scale heterogeneities into account. One isinterested only in their average eect.The need to determine the average, or large-scale,

eects of small-scale phenomena arises in a multitudeof scientic disciplines, one example being that con-cerned with the mechanical properties of compositematerials.39

The derivation of Darcy-scale equations from porelevel Navier-Stokes type equations is a second exam-ple.4044 In the present treatment, we shall not con-sider the underlying pore level problem. We shall as-sume that the conventional macroscopic equations arevalid locally. Our concern is small-scale spatial varia-tion of macroscopic parameters; absolute and relativepermeabilities, and capillary pressure.

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8.3. EFFECTIVE PROPERTIES 179

The problem outlined lends itself to solution interms of eective properties, i.e., eective absoluteand relative permeabilities and eective capillarypressure. The general concept of eective propertiesis relatively well dened.45,46 In the petroleum lit-erature, however, the usage of this and similar termsoften tends to be somewhat qualitative. Here, weshall try to comply with the more strict denition.Our main aim is to bring out clearly a set of condi-tions that guarantee the applicability of the concept.Methods to practically determine eective propertieswill be discussed in subsequent sections.As can be inferred from the previous discussion,

the concept of eective properties is generally appli-cable only on a spatial scale much larger than thatof the heterogeneities. Let us assume for the timebeing that we can dene large-scale dependent vari-ables (saturations, pressures, velocities), and that thelarge-scale variables satisfy a set of partial dierentialequations (pde's). Then we require that

• Eective properties are parameters in the pde'sfor the large scale variables.

• Eective properties are determined exclusivelyby the spatial distribution of properties in theoriginal heterogeneous medium.

Eective properties become large-scale physical prop-erties of the porous medium itself. This implies thatthey should be case-independent, i.e., they should notdepend on the characteristics of the particular owsituation under investigation. Eective properties areallowed to be functions of large scale variables.An intuitively appealing way of dening large-scale

variables is via volume averaging (see Dagan43 for adiscussion of the relationship between volume averag-ing and the alternative technique of ensemble averag-ing). A careful analysis employing the volume averag-ing technique was given by Quintard and Whitaker,44

applied to the problem of two-phase ow in porousmedia. A somewhat less detailed treatment can befound in Ekrann and Dale,47 on which the brief ex-position below is based. Only the one-phase case istreated in some detail.Let < > denote volume averaging, i.e.,

<f >def=

1

V

∫ ∫ ∫f dV , (8.20)

where f is some function of space and V is the nitevolume over which integration takes place. Let <f >be associated with the centroid ~r of V . Then anyvariable or parameter f can be decomposed accordingto

f(~r )def= <f > (~r) + f(~r ), (8.21)

where ˆ signies deviation from the average. If anaveraged quantity <f > can be regarded as constanton the scale of averaging, then we obtain by averagingEq. 8.21

<f >= 0. (8.22)

Consider now one-phase incompressible ow governedby Darcy's law

−µui = kij p,j , (8.23)

where ui and kij are components of the Darcy velocityand of the permeability tensor, respectively; p,j de-notes the j'th component of the pressure gradient. µis the viscosity. The summation convention (summa-tion over repeated indices) is employed. Decomposenow ui, kij , and p,j as specied in Eq. 8.21. Afterintroduction into Eq. 8.23 and averaging, one obtains

−µ <ui> = <<kij><p,j>>

+ <<kij> p,j >

+ < kij <p,j>>

+ < kij p,j > . (8.24)

If averaged quantities can be regarded as constants,then the outermost averaging in the rst rhs-term canbe removed. Furthermore, Eq. 8.22 applies, and thesecond and third term disappear. The fourth term isthe important one. For Eq. 8.24 to be a pde for theaveraged variables <ui> and <p,j>, this term mustbe expressible in terms of these variables. This is theclosure problem. It is not obvious from the outsetthat closure is possible. Fortunately, however, it canbe shown that

p,l = αlj <p,j> (8.25)

if boundaries are suciently far removed for small-scale boundary conditions not to inuence the small-scale pressure distribution p,l; αlj is determinedby the small-scale permeability distribution (hencethe ). Arguments for Eq. 8.25 are given by Ekrannand Dale.47 Detailed proofs are given by Quin-tard and Whitaker.44,49 Introducing Eq. 8.25 intoEq. 8.24, one obtains

−µ <ui> = (<kij> + <kil αlj>)

<p,j>

def= keff

ij <p,j> (8.26)

keffij now satises all requirements to be (a componentof) the eective absolute permeability tensor. Notethat Eq. 8.26 in the large-scale variables <ui> and<p,j> has the same form as the original small-scaleEq. 8.23.Consider now two-phase incompressible immiscible

ow, i.e., with capillarity but without dispersion. Forany given average (large-scale) saturation, supposethat the corresponding small-scale saturation distri-bution is known. Then the development above can berepeated for each phase, for any given large-scale sat-uration. Thus, eective individual phase permeabili-ties can be dened for a full range of large-scale satu-rations. Similarly, an eective capillary pressure canbe dened as the dierence between averaged pres-sures for the two phases.

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180 CHAPTER 8. RESERVOIR MODELLING

Although not complete, we take this brief discus-sion to indicate that eective properties may existalso in the two-phase case. If so, large-scale variablesshould be governed by pde's of the same mathemati-cal form as the original pde's.The essential complication arising when going from

one to two phases is the added need to know thesmall-scale saturation distribution corresponding toa given large-scale saturation.47,48 This distributionwill generally be inuenced not only by the small-scale heterogeneity pattern, but also by the prevailingbalance between viscous, gravitational and capillaryforces, which in turn is determined by the local valueof the large-scale variables. Thus, one must generallyexpect two-phase eective properties to be functionsof all large-scale variables (not only saturation), evenif the original small-scale properties are not.It is time to examine with some care the assump-

tions made when developing the results above. Theseassumptions imply conditions whose satisfaction issucient to guarantee applicability of eective prop-erties. Eq. 8.26 is central to the whole development.It relies on the assumption that <kij > and <p,j >can be regarded as being constant on the spatial scaleL (∼ V 1/3) of averaging. This is satised if the spa-tial scale lk of the heterogeneities is much smallerthan the averaging scale, and if simultaneously thespatial scales Lk and Lp over which averaged perme-abilities and averaged pressure gradients change, aremuch larger than L. In the two-phase case, satura-tion plays a similar role to permeability. Let lS andLS denote the spatial scales of small-scale saturationvariation and of variation of averaged saturations, re-spectively. Then the remarks above can be summa-rized as

lk , lS L Lk , LS , Lp. (8.27)

Eq. 8.27 is a separation of scales condition. Someauthors50 claim that petroleum reservoirs containheterogeneities on all scales. If so, the concept ofeective properties may not be applicable, since theinequality lk L cannot then be satised for anychoice of averaging scale L.To develop Eq. 8.25, we had to argue that bound-

aries be far enough removed that their small-scaleeects not be felt. In other words, for a given large-scale pressure gradient, the small-scale pressure dis-tribution should be completely determined by theheterogeneous medium itself. Similarly, in the two-phase case, the small-scale saturation distributioncorresponding to a given large-scale saturation shouldbe completely determined by the properties of themedium and, possibly, by the local value of large-scale velocities and pressure gradients. Again, thisrequires some distance from boundaries. In addi-tion, one must require history to be forgotten, inthe sense that the small scale saturation distributionmust have had time to obtain its nal shape. Thiscondition is sometimes referred to as one of devel-oped ow. Eective properties are not strictly appli-cable in situations with undeveloped ow, since they

would then have to vary with the stage of the process,in conict with requirements previously put down. Tosummarize, for eective properties to be applicable,

1. Boundaries must be suciently remote,

2. The ow must be suciently developed.

As formulated, these conditions are rather qualita-tive, as is the basic separation of scales condition.This reects the fact that formulation in terms ofeective properties can in practice only be approx-imate. Quantication would depend on the accuracyrequired in an actual application, of course, but alsostrongly on the type of problem considered. Gen-eral guidelines do not seem to exist. Nevertheless,one should keep these conditions in mind, and tryto quantify the degree of satisfaction needed in eachparticular case.A more stringent treatment is possible via a limit-

ing process in which one lets the scale of the hetero-geneities approach zero.5153 If dependent variablesconverge weakly in this process to dependent vari-ables in a homogeneous medium subject to similarinitial and boundary conditions, then the propertiesof that homogeneous medium is said to be the homog-enized properties of the original medium. Homoge-nization has the distinct advantage of allowing strictmathematical analysis. Also, being limiting values,results are often simpler, even in some cases to thepoint of delivering eective two-phase properties inclosed form. However, eective two-phase propertieswill in general depend on the absolute scale of theheterogeneities, making homogenization not alwaysapplicable.As an example of violation of the separation of

scales condition, consider one-phase incompressibleow around a fully penetrating well in a heteroge-neous slab reservoir. Suppose that one wants to sub-stitute a homogeneous medium for the real one, i.e.,one wants to determine the eective absolute perme-ability. The large-scale pressure gradients vary as 1/r,with r being the distance from the well. We can take ras a measure for Lp. But then lk L Lp cannotbe satised close to the well, regardless of choice ofL, unless the scale lk of the heterogeneities is muchsmaller than the well radius. This all amounts tosaying what may be intuitively clear from the out-set, namely that accurate treatment of near-well ef-fects requires individual heterogeneities to be takenexplicitly into account.For a second example, consider two-phase ow in a

heterogeneous cross section, as illustrated in Fig. 8.7.Also indicated in the gure is a cross-sectionally aver-aged saturation distribution which varies on the samescale as the permeability heterogeneity. Assume thataveraging is performed over the full reservoir height,in which case one only needs to worry about satisfac-tion of conditions horizontally. With the averagingscale L chosen such that lk LS , then LS ≈ L.We conclude that the separation of scales conditioncannot be well satised. In this geometry then, one

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8.3. EFFECTIVE PROPERTIES 181

Satu

ratio

n

Figure 8.7: Heterogeneous cross section and cross-sectionally averaged saturation.

should expect eective properties to be applicableonly in situations where the cross-sectionally aver-aged saturation is reasonably constant on the (hori-zontal) scale of the heterogeneities.Suppose, in the geometry illustrated, that one has

an initial no-ow situation. The vertical saturationdistribution is then governed initially by a balancebetween capillary and gravitational forces. Turn ona constant injection (and production) rate, and as-sume for simplicity that the injection well is so farremoved that we do not have to take the injecteduid into account. Suppose that the injection rate isso high that the saturation distribution will eventu-ally be governed dominantly by viscous forces. Evenif the injection rate is felt instantly, due to incom-pressibility, the uids will need some time to redis-tribute. Eective properties will have to be based onthe eventual stabilized small-scale saturation distri-bution. The use of eective properties to model thetransition period would be incorrect, and would beequivalent to assuming instant redistribution of u-ids. This is an example of a situation intended to beexcluded by condition two above.

8.3.1 Single Phase

The Variables

In single-phase ow it is necessary to dene porosityand permeability. The eective porosity must be de-ned as the arithmetic average of the ne scale poros-ity in order to conserve mass of the uids. The eec-tive permeability is much more complex and the restof this section is devoted to the eective permeability.

The Scale Problem

From a strictly mathematical point of view, eectivepermeability is only dened if the separation of scaleassumption, Eq. 8.27, is satised. Fortunately, forthe absolute permeability this assumption needs onlybe satised in a very weak sense. Except in reser-voirs where the permeability variation is very com-

plicated, e.g., in reservoir of labyrinth type with per-meable sand bodies in a nonpermeable matrix, theseparation of scale assumption is satised sucientlyfor single phase ow. The ow depends usually onlyon the ne scale permeability locally and on averagevalues of the permeability in distant volumes. It isthen reasonable to replace the ne scale variation byareas with eective large-scale permeabilities.The permeability is dened as an average over a

volume. The properties of the permeability dependheavily on the size of this support volume. The per-meability is measured on core scale and estimatedfrom well tests. But in the reservoir simulator, thepermeability is used as constant values in reservoirsimulator blocks. This means that it is not possi-ble to use the measured data directly. The transfor-mation from measured data to block values is verycomplicated. If the permeability varies smoothly, theblock value will be close to the measured data. If thevariability in the permeability is large, e.g., due toseveral facies with dierent permeability properties,it is dicult to estimate the block value of the per-meability. This may be done by rst modelling theblock stochastically. In some cases the block consistsof one facies, but the variability in the permeability islarge. The permeability estimate should then dependon all available data about that facies, also data fromother blocks containing the same facies. The eect ofthe size of the support volume is often neglected, seeHaldorsen54 and Holden et al.55

The permeability is a tensor. In reservoir simula-tion, it is usually assumed that the grid is orientedsuch that the permeability is a diagonal matrix. Thisassumption is often impossible to satisfy when eec-tive permeabilities are estimated from a ne scale per-meability. Eective full permeability tensors are dis-cussed in Kasap and Lake56 and Holden et al.55 Inthe following we will discuss the eective permeabilityin a certain direction.The eective permeability is estimated from the

ne scale permeability. The ne scale permeabil-ity may be generated by stochastic modelling, seeChap. 6 page 143. An estimator of the eective per-meability then generates a coarse-grid representationof the permeability which gives approximately thesame ow properties as a ne-grid representation ofthe permeability. For an overview over the dierentmethods and a comparative test on dierent hetero-geneity models, see Grindheim.57

Analytic Methods

The majority of the analytic methods is simple andfast. Unfortunately, they are in most cases inaccu-rate because they do not use the spatial structure ofthe permeability. They are therefore often of littlepractical use. But they are important in order tounderstand the phenomena of eective permeability.Assume the reservoir has n equal thick layers with

ki as the constant permeability in each layer. Thenthe eective horizontal permeability is the arithmetic

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182 CHAPTER 8. RESERVOIR MODELLING

average

ka =1

n

∑i

ki. (8.28)

The eective vertical permeability is the harmonicaverage

kh =n∑i

1ki

. (8.29)

It is possible to generalize these estimators a little.Assume that the permeability is known in a gridki,j,l with equal size gridblocks. The arithmetic av-erage of the harmonic average of each row in the mainow direction (the arithmetic/harmonic average) is

kah =1

nm

∑i,j

p∑l

1ki,j,l

, (8.30)

where n,m, p is the number of blocks in the i, j, l di-rections respectively; kah is an estimate for the ef-fective permeability in the l direction. The harmonicaverage of the arithmetic average of each cross sectionnormal to the ow (the harmonic/arithmetic average)is

kha =p∑

l

((n m)/(

∑i,j ki,j,l)

) . (8.31)

These estimates satisfy

kh ≤ kah ≤ keff ≤ kha ≤ ka,

where keff is the correct eective permeability. Unfor-tunately, the dierence between kha and kah is usu-ally large. Le Loc'h? has followed the above strategywith under- and overestimates which nally convergeto the eective permeability in the block. Her methodis however very computer intensive in most applica-tions. She expands on some of the results of Math-eron.59

Warren and Price60 are the rst to attack the prob-lem of eective permeability. They concluded that inthe case of small spatial correlation and for a widerange of univariate distributions, the geometric aver-age is a good estimate for the eective permeabilitythrough the block, i.e.,

kg = [∏i,j,l

ki,j,l]1

nmp . (8.32)

Usually, there is a large spatial correlation, and this isvery important for the eective permeability throughthe block.It is possible to generalize these estimates in the

formula59

kf (λ) = (1

nmp

∑i,j,l

kλi,j,l)1λ . (8.33)

This function is strictly increasing in λ and equal tothe harmonic average for λ = −1, the geometric av-erage for λ = 0 (through a limiting process) and thearithmetic average for λ = 1. If there is larger hor-izontal correlation than vertical, one should expectto get a good estimate for the vertical permeability

using −1 < λ < 0 and a good estimate for the hor-izontal permeability using 0 < λ < 1. Unfortunatly,this formula is very sensitive to extreme values of thepermeability. The λ is therefore very unstable anddicult or in some cases meaningless to estimate.

The Streamline Methods

In many reservoirs, discontinuous barriers embeddedin the porous medium is the dominant permeabilityreduction agent. Streamline techniques are developedfor calculating this reduction, by estimating the in-creased length of the streamline due to barriers. Hal-dorsen and Lake61 propose the formula

keff =k(1− F )

(1 + sd)2, (8.34)

where k is the background permeability, F is the ra-tio of the volume of the barriers to the total volume,s is the expected number of barriers per length unitand d is the expected length around the barrier. Beggand King37 extend the method using statistical distri-butions for the shales. They also propose a methodwhere the estimate depends on the location of theshales. These methods are further developed,6264 re-ducing some of the assumptions further.

Renormalization Methods

The idea behind the renormalization technique is toestimate the eective permeability over local regionsto form a new renormalized permeability distributionwith lower variance than the original. The techniquemay be repeated until a stable estimate is found.King65 has given an algorithm where the eectivepermeability of 2× 2(×2) blocks is calculated in eachstep. He uses the similarity with resistor networks.The method is fast. The accuracy is usually betterthan the analytic methods and in many cases suf-cient. These techniques are further developed byMohanty and Sharma.66

Numerical Methods

These methods are based on the solution of the single-phase ow equations, Darcy's law and conservation ofmass. Since the methods are based on the ow equa-tions, they are accurate. Most authors54,59,61,63,67,68

seem to accept this as the correct estimate.The solution of the ow equations in a region de-

pends on the boundary condition to the region whichusually is unknown. A neutral choice of boundaryconditions in a block37 is constant pressure on the in-ow and outow boundaries and no ow at the otherboundaries. These boundary conditions are used inmost other estimators either explicitly or implicitly.It is possible to reduce the eect of the boundaryconditions by using a skin around the gridblock, seeGomez-Hernandez and Journel.68

The single-phase ow equations result in a systemof linear equations. Holden et al.67 have shown that

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8.3. EFFECTIVE PROPERTIES 183

only a few iterations with an iterative linear solver isrequired in order to get an accurate estimate. Thepermeability is estimated from an average of the es-timated ow through each layer normal to the mainow direction. This reduces the need for computerresources drastically. However, this method is stillsignicantly more computer intensive than the ana-lytical estimates or the renormalization techniques.

Practical Use

In some cases it is sucient to use analytical meth-ods. This is the case if the variability in the perme-ability is small or if it is sucient with very roughestimates. Analytical methods may also be sucientif the blocks have extreme shapes, e.g., estimates ofthe eective vertical permeability when the blocks arevery thin. Then the arithmetic and harmonic aver-ages are accurate enough.If the block contains discontinuous barriers and the

permeability variation in the matrix is small, stream-line methods are usually the best estimators. Therenormalization method is not good for these kindsof heterogeneities and the numerical methods need ane grid.The renormalization methods are good for a large

range of heterogeneities. The increase in computerresources compared to analytical methods is usuallynot that large. However, these methods are quitecumbersome to implement.The numerical methods usually yield the best es-

timates and require most computer resources. If thegrid is rened, these methods converge to the correctsolution in some sense. Also, these methods are quitecumbersome to implement. In most cases it is su-cient to take one gridblock at a time, neglecting theboundary eect. If there are large heterogeneities atthe gridblock size, it may be necessary to introduce askin around the block.In Fig. 6.6 page 150 the ne scale horizontal perme-

ability in a cross section through a 129× 129× 129 ≈2 150 000 blocks is shown. Fig. 8.8 shows the eec-tive horizontal permeability in the cross section using

1

Figure 8.8: Homogenized horizontal permeability.69

16 × 16 × 16 = 4096 blocks. In this example the nu-merical method67 is used with only a few iterations

in the linear equation solver. The ow properties areapproximately the same in the two grids.

8.3.1 Two Phases

Introduction

The mathematical analysis of saturation-dependenteective parameters for two-phase ow in heteroge-neous porous media started quite recently. Shvi-dler's paper71 is the rst study on the subject pub-lished since the paper by Condreanu et al.70 in 1966.Several later theoretical studies have presented an-alytical closed form solutions for, or algorithms forthe numerical calculation of saturation dependenteective parameters. Dierent mathematical tech-niques have been applied, such as traditional vol-ume averaging,44,7275 asymptotic analysis and per-turbation techniques (homogenization theory, mul-tiple scales),52,76,77 and stochastic analysis.71,7885

Within the petroleum literature proper, mainly deter-ministic (periodic) heterogeneities have been treated.All these studies are essentially based on the assump-tion that the scale conditions, Eq. 8.27, are satised.Thus, there remains a need for numerical testing ofthe range of validity of the parameters constructedhitherto, when scale conditions are not fullled. How-ever, theoretical work has started in order to removethe scale constraints, which are thought to be overlysevere.72

An Example

In order to present as clearly as possible some mainissues within the study of two-phase eective prop-erties, we rst discuss a well-known elementary ex-ample, namely one-dimensional Buckley-Leverett dis-placement in a heterogeneous core (see Marle86 for apresentation of this theory in the homogeneous case).Consider a one-dimensional heterogeneous reser-

voir model consisting of a periodic or random dis-tribution of two dierent rock types. Each rock typeis characterized by its absolute permeability, poros-ity and relative permeability curves. The capillarypressure is assumed negligible. Thus, the parametervariation in the model is piecewise constant, with adiscontinuity at the boundary between two dierentrock samples. For simplicity, we will assume that thesamples have a common length and the same proba-bility of occurrence.Now consider standard incompressible, immiscible

Buckley-Leverett type displacement in this model.Initially, the core is saturated with oil, with water ev-erywhere at its irreducible saturation. At time t = 0,one starts injection of water into the model with totalDarcy velocity remaining constant in time. The evo-lution of the water saturation distribution S(x, t) inthe core is now uniquely determined by the Buckley-Leverett theory for homogeneous media, coupled withthe requirement of continuity of ow velocities acrossboundaries between dierent rock samples. Then, the

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184 CHAPTER 8. RESERVOIR MODELLING

water saturation distribution will have to be discon-tinuous across these boundaries, since the saturationvs. fractional ow relationships in general will be dif-ferent for the two homogeneous parts.Fig. 8.9 shows a typical Buckley-Leverett discon-

tinuous saturation distribution in a periodic medium,

TrueEff.Av3Av4

1.0

0.8

0.6

0.4

0.2

0.00 5 10

Wat

ersa

tura

tion

Number of periods

Figure 8.9: True, average and eective saturation dis-tribution in a periodic medium.

after water has advanced about 10 periods into themedium. The two rock types are here chosen to be ho-mogeneous with respect to absolute permeability andporosity, and irreducible saturations are zero. Therelative permeability curves are shown in Fig. 8.10.The water/oil viscosity ratio is 0.5. The eective

Rock 1Rock 2Effective

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Water saturation

Rel

ativ

epe

rmea

bilit

y

Figure 8.10: Rock and eective relative permeabili-ties.

curves are computed according to Eq. 8.35 below, in-terpreted for the negligible capillary pressure case.Note that the eective curves not necessarily fall be-tween the rock curves.Also shown in Fig. 8.9. are two averages of the true

saturation distribution, based on an averaging inter-val of 1.5 and 2 period lengths, respectively. As thelength of the averaging interval increases, one expectsthe resulting average saturation prole to stabilize. Itis this limiting prole, suciently far from boundariesand initial time, that should be captured by the singleset of eective relative permeability curves.

In this case, a good t between average and eec-tive saturation distribution is obtained for an averag-ing volume on the same scale as the heterogeneities.This is due to the ordered structure of the hetero-geneities. In a randomly heterogeneous medium, oneexpects averaging on a larger scale to be necessary, inaccordance with the scale conditions.

Methods for Calculation of Eective RelativePermeabilities

As discussed previously, two-phase ow in a hetero-geneous porous medium, satisfying the scale condi-tions, can be approximately described by eectiverelative permeability curves and an eective capillarypressure curve. The explicit calculation or analyti-cal determination of eective curves requires knowl-edge of the small-scale saturation distribution corre-sponding to a given average saturation and averagepotential pressure gradient value. By this procedure,the calculation of eective relative permeability be-comes mathematically equivalent to a series of cal-culations of eective absolute permeabilities. Threedierent main ow cases are treated in the literature,namely those of capillary equilibrium, negligible cap-illary pressure and vertical equilibrium. The practicalmethods reported in the literature for computing ef-fective parameters, are so far restricted to periodicheterogeneities.

Deterministic HeterogeneitiesCapillary Equilibrium. The most thorough

study of eective parameters for the capillary equi-librium, or quasi-static case, was given by Quintardand Whitaker.44 They studied a two-region, deter-ministically heterogeneous medium, and applied tra-ditional volume averaging techniques. Special atten-tion was given to the problem of uid trapping. Theirkey assumption in determining the local uid distri-bution was that the local capillary pressure, every-where in the averaging volume, can be set equal tothe large-scale capillary pressure evaluated at the cen-troid of the averaging volume. For periodic local het-erogeneities, the equilibrium condition also allowedthem to completely solve the closure problem, therebyproviding a procedure for calculating the eective rel-ative permeabilities and capillary pressure curves. Inthe one-dimensional case, this method gives the fol-lowing expression for the eective relative permeabil-ity,

2

kkeffr (S)

=1

k1kr1(S1)+

1

k2kr2(S2). (8.35)

Here S1 and S2 denote the local saturations for whichthe local capillary pressures are equal, and S is theirporosity-weighted arithmetic average. For this exam-ple, we assume a rock period consisting of two rocktypes of the same length, with absolute permeabili-ties k1 and k2 having k as their harmonic mean, andwith relative permeability curves kr1 and kr2.

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8.4. DYNAMIC PSEUDOFUNCTIONS 185

Also essentially based on the capillary equilibriumcondition is the paper by Saez et al.77 applying themethod of multiple scales, a perturbation techniqueto obtain the large-scale ow equations with eec-tive parameters. They presented an explicit analyti-cal procedure to calculate the eective parameters fora periodic medium, assuming a detailed knowledge ofthe heterogeneities inside the period. This procedurecorresponds to the solution of the closure problem byQuintard and Whitaker.44

The multiple-scale method was also used byBourgeat52,76 to obtain eective properties for pe-riodic media in which relative permeability and cap-illary curves do not vary spatially. In addition, heproved in a rigorous way the convergence of small-scale equations to large-scale equations with eectiveproperties coecients when the small-scale hetero-geneity tends to zero. However, in the general case,with spatially varying saturation dependent param-eters, he was able to prove existence of the limitingeective properties only, not to identify them.Negligible Capillary Pressure. At the other

extreme, assuming negligible capillary pressure, theone-dimensional Buckley-Leverett case has been an-alytically treated by Dale.53 Here, the eective rel-ative permeability for a two-region medium is givenby an expression identical to that above, if S1 andS2 are now interpreted as local saturations for whichthe local fractional ow functions have the samevalue. Also, convergence of the small-scale dynamicBuckley-Leverett solution to the large-scale dynamicsolution with eective parameters, as the heterogene-ity scale tends to zero, is formally proved.Both conditions are nicely discussed by Smith.48

He computes the steady-state eective relative per-meabilities numerically, assuming a 2D random dis-tribution of absolute permeability. The variation ofcapillary pressure curves is correlated to this distri-bution through the Leverett J-function.Vertical Equilibrium. The vertical equilibrium

eective relative permeabilities for stratied reser-voirs (classically termed pseudorelative permeabil-ity,87,88 are based on the assumption that the verticaluid distribution is uniquely determined by equilib-rium of gravity and capillary forces. To constructthem, one averages over the full reservoir height,thereby also reducing the dimensionality of the prob-lem. Detailed expositions of this theory can be foundin all reservoir engineering textbooks, see for instanceDake.89

General. In the 1D-case, an analytical treatmentof saturation-dependent eective parameters, includ-ing the eects of capillarity, gravity, and rate is givenby Dale and Ekrann.90 The construction of eectiveparameters is based on steady-state periodic satura-tion and pressure solutions in a periodically hetero-geneous medium. It turns out that eective relativepermeabilities depend on the absolute length scale ofthe period. The capillary equilibrium and the negli-gible capillary cases are recovered as the rate tends

to zero or innity, respectively.

Random Heterogeneities. All results mentionedso far depend on an periodic model of the local het-erogeneities in order to obtain analytical expressionsfor the eective parameters. This is a somewhat in-exible model of the small-scale reservoir parametervariation, which is normally assumed to be random,with a specic correlation structure. To overcomethis problem of limited information about the localdetails of small-scale heterogeneity, there is a needto develop techniques which make possible an esti-mation of saturation-dependent eective parameterson the basis of a statistical description of the reser-voir only. In the groundwater ow literature, such astochastic modeling framework for unsaturated owsystems has already been developed.79,80,8284

The rst papers to apply statistical concepts in thestudy of two-phase ow in heterogeneous media, arethose of Smith and Brown81 and Shvidler.78 Smithand Brown used Monte Carlo simulations to deter-mine expected ow behavior in a random 1D-modelwith varying relative permeability curves, for dierentcorrelation structures. However, they did not quan-titatively identify eective parameters.Shvidler studies multiphase ow in a randomly het-

erogeneous medium consisting of inclusions of dier-ent shapes (spheres and ellipsoids) and absolute per-meabilities in a homogeneous matrix. Relative per-meability curves are not varying. He assumes thatuid phases are distributed in such a way that an in-dividual inclusion contains only one uid phase. Theform of the inclusions may result in anisotropic ef-fective relative permeabilities for this medium. Todetermine the components of the eective relativepermeability tensor, the self-consistent technique isused. However, the results depend strongly on the(unknown) distribution of the phases over the inho-mogeneity elements.

8.4 Dynamic Pseudofunctions

Dynamic pseudofunctions were originally devisedto obtain improved numerical accuracy in coarse-grid reservoir simulations,91 and as a means of ef-fectuating a quasi-3D computation via two sepa-rate 2D computations.91,92 The pseudofunctions inquestion are most often relative permeabilities, butmust also generally include capillary pressure.47,88,91

The term dynamic was coined to distinguish fromso-called vertical equilibrium (VE) pseudofunc-tions.87,93 Dynamic pseudos require no assumptionsof equilibrium ow.Dynamic pseudofunctions are constructed by back

calculation from some single (normally numerical)solution of the ow equations.47,88,91,92 FollowingEkrann and Dale,47 let us henceforth refer to thisunderlying solution as the parent solution. In prin-ciple, there are no restrictions on the parent solu-tion. It can describe, for instance, a highly tran-

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186 CHAPTER 8. RESERVOIR MODELLING

sient or dynamic process. To compute dynamic pseu-dos, a coarse grid, related to or identical to the gridsubsequently to be used in the simulation of ulti-mate interest, is overlaid the parent computationaldomain. Pseudorelative permeabilities require indi-vidual phase uxes over coarse gridblock boundaries,produced from the parent solution by integration.One also needs to compute coarse gridblock pres-sures, e.g., by some averaging of parent solution pres-sures. Given these data, one can construct individualphase transmissibilities as quotients between coarsegridblock uxes (times viscosity) and the appropri-ate pressure dierences. Dividing through with somemeasure of the absolute transmissibility, one obtainsdynamic pseudorelative permeabilities for interblockboundaries. Relating these pseudos to the upstreamblock saturation, individual block pseudos as func-tions of saturation are obtained. By construction,dynamic pseudos are directional, a fact which is of-ten neglected in practice.The general idea behind dynamic pseudos is to

transfer the accuracy characteristics of the parentsolution to the coarse-grid simulation. It can beshown for immiscible two-phase incompressible owthat dynamic pseudos, if properly constructed, allowthe parent solution to be reproduced exactly, in a cer-tain sense, in the subsequent coarse grid simulation.47

The dynamic pseudos generally become functions ofthe particular parent solution chosen, of the exactlayout of the coarse grid, and even of the numericalmethod to be employed.47 Thus, dynamic pseudosare conceptually dierent from eective properties,Sec. 8.3, which are not allowed to depend on suchnonphysical parameters.To illustrate these points, follow Ekrann and Dale47

and consider simple Buckley-Leverett type displace-ment in a 1D homogeneous reservoir, with rectilinearrock relative permeabilities. In this case, due to ho-mogeneity, eective relative permeabilities are identi-cal to their rock counterparts. Let dynamic pseudosbe generated from analytical parent solutions, andconstructed in such a way as to allow exact reproduc-tion. Fig. 8.11 displays dynamic pseudos generatedfrom three parent solutions, diering only in theirdisplaced to displacing uid viscosity ratio M . Thecoarse grids are identical in the three cases. Pseudosdisplayed are for the central of three blocks. As canbe seen, the parent solution strongly inuences theend result, even if rock properties are identical in thethree cases. Only for largeM do the dynamic pseudosbegin to resemble their eective counterparts.For a second example, consider the same physical

system as above, but with M xed at 1.1, and withfour dierent coarse-grid discretizations. Displayedpseudos in Fig. 8.12 all correspond to approximatelythe same observation point. Pseudos are seen to de-pend very strongly on discretization in this case, ap-proaching their eective counterparts as the blocksizediminishes.Fig. 8.13 demonstrates the dependence of pseudos

1.2

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Water saturation

Pseu

dore

lativ

epe

rmea

bilit

y

M = 5.0M = 2.0M = 1.01

Figure 8.11: Dynamic pseudorelative permeabilitiesfor a 1D homogeneous medium, dierent viscosity ra-tios M .

1.2

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Water saturation

Pseu

dore

lativ

epe

rmea

bilit

y

2 blocks5 blocks10 blocks20 blocks

Figure 8.12: Dynamic pseudorelative permeabilitiesfor a 1D homogeneous medium, dierent coarse-griddiscretizations.

on the numerical method to be employed in the sub-sequent coarse-grid simulation. The system is as out-lined in the rst example, except thatM is xed at 2.Two numerical schemes are considered, diering onlyin their weighting of relative permeabilities.It seems reasonable to assume that dynamic pseu-

dos will produce accurate answers when employed incoarse-grid simulations of cases similar to the parentone. For one of the previous simple examples sucha similar case could be one obtained by slightly per-turbing the viscosity ratio M . Under more realisticcircumstances, perturbation of well rates, say, couldbe of more interest.In fact, this perceived increased accuracy is the rea-

son for taking the trouble to generate dynamic pseu-dos in the rst place, of course. The general feel-ing appears to be that the accuracy enhancement islarger the closer a case to be simulated is to the par-ent case. The literature, however, is surprisingly voidof attempts to quantify such ideas. In fact, some au-thors88,91,92 check the validity of their dynamic pseu-dos by conrming that their use implies a reasonablygood reproduction of the parent solution, while oth-

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8.4. DYNAMIC PSEUDOFUNCTIONS 187

1.2

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Water saturation

Pseu

dore

lativ

epe

rmea

bilit

y 1-point upstream2-point upstream

Figure 8.13: Dynamic pseudorelative permeabilitiesfor a 1D homogeneous medium, two dierent numer-ical weighting schemes.

ers make no eort whatsoever at verication. As men-tioned previously, carefully constructed pseudos allowexact reproduction of the parent solution. No conclu-sions can be drawn from this, however, with regardto the accuracy produced in nonparent cases. Theproper test is a comparison between the coarse-gridsimulation result in the nonparent case of interest,and an independent accurate solution of that sameproblem.In the second mode of operation alluded to above,

dynamic pseudos are constructed from a 2D cross-sectional parent solution, and subsequently employedin a 2D coarse-grid areal simulation. In this way, bothcross-sectional and areal eects contribute to the nalresult, and the results are therefore expected to be anacceptable approximation to the 3D solution sought.Again, quantitative evidence to support the validityof this procedure seems to be largely absent from theliterature. An exception is oered by Jacks et al .92

In recent years, dynamic pseudos have been em-ployed for a purpose somewhat dierent from thoseenvisaged by the original authors. It has long beenrealized that petroleum reservoirs are far from homo-geneous, and that signicant heterogeneity may ex-ist on spatial scales too small to resolve in a typicalreservoir simulation. This has spurred much interestin methods to capture the large-scale eects of suchsubgrid scale heterogeneity.Most commercial reservoir simulators have options

allowing some previous simulation result to serve asa parent solution from which to derive dynamic pseu-dos. This wide availability has tempted many work-ers to use dynamic pseudos for the problem at hand,generated from a ne-grid parent solution in somerepresentative, detailed heterogeneity pattern.9497

By the discussion above, however, dynamic pseudosso generated will generally reect not only the under-lying heterogeneity pattern itself, but also other char-acteristics of the parent solution, the grid, and thenumerical method to be employed. Thus, the gen-eral applicability may be questionable, unless special

precautions are taken.Capturing the large-scale eects of small-scale het-

erogeneity is exactly what eective properties are allabout. There appears to be some relationship be-tween the two concepts, therefore. Both are based onspatial averaging. However, averaging procedures aresomewhat dierent, since the construction of eectiveproperties relies on moving averages, thus producingaveraged quantities which are continuous functionsof position. The averaged quantities needed for con-struction of dynamic pseudos, on the other hand, aregrid functions, with one value per gridblock or pergridblock boundary. We shall disregard these dier-ences in the following brief discussion.Let the small-scale heterogeneity pattern and the

parent solution be such as to satisfy the conditionsdiscussed in Sec. 8.3, in particular the separation ofscales condition Eq. 8.27, with L now taken to signifythe linear size of coarse gridblocks. One should thenexpect reasonably dened averaging procedures toproduce averaged quantities which are approximatelyconstant on the spatial scale L. Averaged quanti-ties, therefore, should not vary signicantly with rea-sonable variations in L. Dynamic pseudos are con-structed from such averaged quantities, and shouldtherefore also not vary signicantly with the gridblocksize L. By similar arguments, this near constancy ofaveraged quantities should imply that the dependenceon the numerical method to be used in the ensuingcoarse-grid simulation should disappear.We take the above discussion as an indication that

dynamic pseudos may coincide with their eectivecounterparts under certain conditions, in particularthat of separation of scales. The complete determina-tion of eective properties via dynamic pseudos wouldgenerally require a set of parent solutions, to capturethe dependence on e.g., large-scale pressure gradients.It should be emphasized that this possible equiva-

lence with eective properties can only be true withstrict restrictions imposed on the parent solution.These restrictions are such that the term dynamiclooses much of its relevance, since they imply well de-veloped ow with very smooth variation of averagesaturations.It is perhaps instructive to discuss briey our pre-

vious examples in the light of separation of scales.Figs. 8.11 and 8.12 reveal that the direction of changeof the dynamic pseudos is consistent with the condi-tion. As L LS becomes better satised, i.e., asthe saturation becomes more constant on the linearscale L of gridblocks, dynamic pseudos approach theirrectilinear eective counterparts.The pseudos in Fig. 8.14 have been produced from

the same parent solution as those in Fig. 8.13, exceptthat more coarse blocks have been added to make thereservoir physically longer. Pseudos corresponding toa far downstream block are displayed. The parentsolution saturation gradient is much smaller in thisblock, due to the unfavorable viscosity ratio, provid-ing for better satisfaction of L LS . Therefore,

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188 CHAPTER 8. RESERVOIR MODELLING

0.0 0.2 0.4 0.6 0.8 1.0

1.2

1.0

0.8

0.6

0.4

0.2

0.0

1-point upstream2-point upstream

Water saturation

Pseu

dore

lativ

epe

rmea

bilit

y

Figure 8.14: Dynamic pseudorelative permeabilitiesfor a 1D homogeneous medium, two dierent numer-ical weighting schemes.

the dependence on the numerical weighting scheme isdiminished, and the dynamic pseudos become moresimilar to their eective counterparts.To summarize, the following views have been ad-

vocated:

• Dynamic pseudos are essentially devices for re-production of a parent solution.

• Dynamic pseudos approach their eective coun-terparts only when strict conditions are imposedon the parent solution.

8.5 Uncertainty in Forecasting

8.5.1 Introduction

The ultimate goal of all geological, geophysical andreservoir analyses of a petroleum reservoir located inthe underground is to gain the ability to predict theoutcome of any recovery method applied to the reser-voir. The rst prerequisite is then a knowledge aboutthe amount and distribution of oil and gas in place.The second is the ability to model correctly the dy-namics of the owing phases under various displace-ment processes. The total body of knowledge is usu-ally formalized into a numerical reservoir simulationmodel. The art and science of formalizing reservoirknowledge into a simulation model and performingpredictions of reservoir performance is summarizedin e.g., Mattax et al.3

The fact that the results of a reservoir simulationshould be looked at with caution is probably clearto any reservoir engineer. There is a good reasonwhy the results should have associated uncertainties:All parameters entered into the model are uncertain.As an illustration we choose Table. 8.1, taken fromDeSorcy.98 The table lists various static reservoir pa-rameters with ranges of accuracy depending on thesources of the estimates.The paper does not make clear how the accuracy

ranges were determined. The task of making realistic

Table 8.1: Source and accuracy of reserve factors.98

Approximaterange of

Typical source expectedFactor of estimate accuracy (%)

Area Drillholes ± 1020Geophysical data ± 1020Regional geology ± 5080

Pay Cores ± 510thickness Logs ± 1020

Drilling time recordsand samples

± 2040

Regional geology ± 4060Porosity Cores ± 510

Logs ± 1020Production data ± 1020Drill cuttings ± 2040Correlations ± 3050

Interstitialwater

Capillary pressuredata

± 515

saturation Oil based cores ± 515Saturation logs ± 1025Routine core analysiswith adjustments

± 2550

Correlations ± 2560Formationvolumefactor

Pressure-volume-temperature analysisof uid samples

± 510

Correlations ± 1030

estimates of the uncertainties in reservoir parametersis of course fundamental in any analysis of predictedperformance.Uncertainties are introduced into the modelling of

ow processes in other ways: The dynamic modelitself may not be an adequate idealization of the owprocesses, and the numerical methods employed tond approximate solutions of the model equations willalso introduce uncertainties.We shall here ignore these sources of uncertainty.

To formulate the problem of conducting an uncer-tainty analysis we use the equations of two-phase owdescribing e.g., a displacement of undersaturated oilby water.The transport equations are,

∂t(φρjSj) =

div(kλjρj(grad(pj)− ρjggrad(D))− qj ,(8.36)

λj =krj(Sj)

µj, j = w, n, (8.37)

Sw + Sn = 1, pn − pw = pc. (8.38)

In addition to the equations, we have boundary con-ditions across internal and external boundaries withconditions at well boundaries included in the sourceor sink terms qj .All data entered into the equations will be uncer-

tain, but with varying degrees of uncertainties: Fluid

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8.5. UNCERTAINTY IN FORECASTING 189

properties may be given with relatively high accu-racy. Permeability can have a large variability. Posi-tions, extents and transmissibilities of internal reser-voir boundaries may also be largely unknown.The problem of performing an analysis of predicted

results of a displacement process due to uncertaintyin, say, the permeability k, contains two tasks:Our ideas about the possible variability of k must

be formalized. A natural approach is to regard k asa random variable, or more correctly, a random func-tion. The apparatus of stochastic reservoir descrip-tion99 is at our disposal for this task.The next task is then to nd relationships between

the statistical properties of k and the resulting sta-tistical properties of saturations, pressures or, ulti-mately, recovery.The rst task, to formalize and parameterize, the

uncertainties in the reservoir parameters is of coursethe crucial task, where all available typical and spe-cic information about the reservoir must be utilized.The second task is largely mathematical, since thepressures and saturations are uniquely determined bythe data. In practice, however, the calculation of un-certainties in production results is a formidable prob-lem.An uncertainty analysis can easily involve more

work than the prediction itself. Consequently, suchanalyses have traditionally been performed on sim-plied reservoir models (and the art of simplicationis still essential to the feasibility of an uncertaintyanalysis). What has changed in recent years is ourability to create highly detailed, complex reservoirdescriptions by stochastic methods. Coupled withthe rapidly increasing capabilities of computers, thesemethods will be essential tools in future uncertaintyanalyses.Fig. 8.15 taken from a paper by Garb,100 shows the

traditional view of the decreasing range of recoveryestimate with time during the life of a reservoir, withthe methods used for obtaining the estimates. Themethods we describe include volumetric and simula-tion studies. Descriptions of so-called material bal-ance methods and decline analysis can be found else-where.98,100

The diculty of making precise estimates of recov-erable reserves has been well illustrated in papers an-alyzing predicted reserves versus those actually re-covered.101103 Franzen et al.103 investigated 40 elddevelopment projects in the Gulf of Mexico. One oftheir results is that the relative deviation in predictedreserves E from actual values A varied from -9 to +12:

−9 <E −AA

< 12.

In a paper on 3D seismics, Ruitenberg et al.104 quotea survey of 10 North Sea elds showing changes inreserves from 7% to 81% of the amounts predictedfrom 2D surveys.The literature on uncertainty analyses of predicted

reservoir performance is large, and will certainly con-tinue its growth considering the importance of the

Range ofestimate

A B C D E F

A B C D E F

Actual recovery

Ulti

mat

ere

cove

ry

Range of recovery estimate

Relativerisk

High

LowTime

Barrelsperacre

Barrelsper

acre-ft. Barrels period

Ris

k

(a) Range in estimates of ultimate recovery during life of reser-voir.

A B C D E F

A B C D E F

Time

Cumulative

Rate

Productionprofile

Log

prod

uctio

nra

te

Cum

ulat

ive

prod

uctio

n

Studymethod

Performance

VolumetricAnalog

Propertystatus

Pre -drillingperiod

Abandonment

Production operationsDevelop-

mentperiod

(b) Study methods during life of reservoir.

Figure 8.15: Recovery estimates and methods.100

problem. Our list of references is far from com-plete but should furnish a starting point for furthersearch.105,106

8.5.2 General Techniques of Uncer-tainty Analysis

In general terms, the problem of performing an un-certainty analysis may be formulated as follows: Wehave given a functional relationship

Y = r(X) (8.39)

between a data vector

X = [X1, . . . , Xn]T (8.40)

and a result vector

Y = [Y1, ..., Yk]T . (8.41)

Uncertainties in the data are specied through a prob-ability distribution with a density fX for X. At leastthe expectation and covariance of X are given:

EX = mX , CovX = CX . (8.42)

We want to calculate the corresponding uncertaintyin Y expressed by a probability distribution FY with

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190 CHAPTER 8. RESERVOIR MODELLING

a density function fY , or at least

mY = EY , CY = CovY . (8.43)

The function r may have a simple form, e.g.,

r(X) = X1X2 · · ·Xn, (8.44)

or be inexpressible by analytic means if e.g.< X is apermeability and r is recovery.The techniques used in solving the problem have

been classied by Smith and Buckee107 in two groups:Parametric methods and Monte-Carlo methods.In a parametric method one seeks to calculate fY

or at least mY and CY by analytical means. In manycases of interest, however, all we can hope to get areapproximations of mY and CY . In a Monte-Carlo(MC) method, X is randomly sampled, and an em-pirical distribution is built up from the correspondingsample of Y -values.

8.5.3 Parametric Methods

In simple cases, statistical characteristics for Y maybe calculated directly from those of X. Suppose e.g.that

Y = X1 · · ·Xn (8.45)

and that the Xi's are independent. Then, withEXi = mi and VarXi = σ2

i ,

mY = m1 . . .mn, (8.46)

σ2Y = (σ2

1 +m21) · · · (σ2

n +m2n)−m2

1 · · ·m2n, (8.47)

and any moment EY q can be written explicitly interms of moments EXp

i . The distribution functionfor Y may not be so easily calculated, however.In general, parametric methods only provide low-

order approximations of mY and CY . A Taylor ex-pansion of r in terms of a disturbance δx around apoint x gives

r(x+ δx) =

r(x) + r′(x)δx+ (1/2)r′′(x)(δx⊗ δx) + · · ·(8.48)

Assuming that x and r have respectively n and kcomponents, r′ is a k × n-matrix

r′(x) =

[∂r

∂x1, . . . ,

∂r

∂xn

], (8.49)

r′′ is a k × n2-matrix

r′′(x) =

[∂r′

∂x1, . . . ,

∂r′

∂xn

], (8.50)

and δx⊗ δx is a Kronecker product of δx with itself,i.e., an n2-vector

(δx⊗ δx)T = [δx1δxT , . . . , δxnδxT ]. (8.51)

This notation is used by Dettinger and Wilson108 ina rst- and second-order analysis of uncertainties inmodels of ows in aquifers. If we let

x = mX , δx = X −mX , (8.52)

the Taylor expansion gives to rst order

mY ' r(mX), (8.53)

CY ' r′(mX) CX r′(mX)T , (8.54)

and a second order approximation of mY is given by

mY ' r(mX) + (1/2) r′′(mX) cX , (8.55)

where cX is a vector of n2 components obtained byplacing the columns of CX underneath each other, inorder from left to right.A rst-order analysis may be extended to the trans-

port equations of a displacement process. After dis-cretization in space and time, the equations have thegeneral form

v(yn+1) = v(yn) + ∆tnf(yn, yn+1), (8.56)

where yn contains saturations and pressures at timetn. Suppose that the equation above contains a ran-dom parameter X,

v(X,Y n+1) = v(X,Y n) + ∆tnf(X,Y n, Y n+1),(8.57)

and that we are interested in the expectation andvariance of a function

Un = r(X,Y n), (8.58)

given expectation and variance of X; Un may repre-sent recovery or a well rate or some other function ofsaturation and pressure at time tn. A Taylor expan-sion of the equation gives to rst order

mnU = r(mX , y

n), (8.59)

and

CnU = r′(mX , yn) CX r′(mX , y

n)T , (8.60)

where the derivative of r with respect to x is given by

r′(x, y) = ∂xr + ∂yr ∂xy. (8.61)

Here yn is the rst order approximation of mYn , ob-

tained by solving the transport equations

v(x, yn+1) = v(x, yn) + ∆tnf(x, yn, yn+1) (8.62)

with x = mX . To nd the derivatives ∂xyn, we dif-ferentiate the transport equations with respect to x.The result is a set of equations

vn+1x + vn+1

y yn+1x =

vnx + vny ynx + ∆tn(fn+1

x + fn+1y ynx + fn+1

z yn+1x ),

(8.63)

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8.5. UNCERTAINTY IN FORECASTING 191

where the indices x, y and z denote dierentiationswith respect to x, y and z of v(x, y) and f(x, y, z),and yx = ∂xy. If the initial saturations and pressuresare independent of X, y0

x will be zero.The equations for ynx can be solved in parallel with

the equations for yn. According to Anterion et al.,109

the simulator SCORE contains an option for doingthis and may thus be used for calculating rst or-der approximations of variances. Dettinger and Wil-son108 propose a similar method for one-phase ow.No numerical results are given.If the parameter vector X contains L components,

and the solution vector Y contains N pressures andsaturations, the equations for the derivatives contain2NL component equations. If L is large, say L = N ,which it would be if X represents e.g. the full set ofpermeabilities, it will probably be too cumbersometo solve these equations even though they are linear.There exists, however, a way of circumventing theproblem. To compute CkU at the time tk, we need ∂xrand ∂yr∂xy calculated at tk. The last quantity can becalculated directly using a set of equations that areadjoint to the equations for ynx . The method worksas follows: Calculate un, n = k, · · · 1, from

uk(vky −∆tk−1fkz ) = ∂yr

k, (8.64)

unvny = un+1vn+1y + ∆tn(unfnz + un+1fn+1

y ), (8.65)

n = k − 1, . . . , 1.

Then

∂yrk ∂xy

k =

k∑n=1

un(vn−1x + ∆tn−1 f

nx − vnx ). (8.66)

The method is well known from automatic historymatching.3 If r has J components, the adjoint equa-tions for the J×2N -matrices un have 2NJ componentequations. If J L, the work saved by solving theseequations instead of the equations for ynx is substan-tial. Observe, however, that all yn must be stored inorder to calculate the coecients of the adjoint equa-tion, Eq. 8.65, and that the whole set of calculationsmust be repeated for each value of k.We now assume that Y is a scalar function r(X) of

X.The cumulative distribution function for Y is given

byFY (y) =

∫R(y)

fX(x)dx, (8.67)

whereR(y) = x : r(x) ≤ y. (8.68)

IfX has a multivariate, standardized normal distribu-tion, and r is a linear function of x, the integral canbe calculated analytically. Within the discipline ofstructural safety theory, methods called FORM andSORM (First and Second Order Reliability Method)have been developed that use this fact to approximateFY in the general case.

Details of the methods can be found in Ref. 110.Selvig111 and Langtangen112 have applied the meth-ods in stochastic analyses of two-phase ow in onedimension. Further investigations of the methodswould be interesting.

8.5.4 Monte Carlo Methods

The great majority of papers dealing with prob-abilistic uncertainty analysis use MC methods astheir main tool of investigation. A large no of pa-pers.81,85,105107,113133 apply MC methods in stud-ies of predicted behavior of ground-water ow, re-serves estimates, and EOR methods.Formulated in terms of a functional relation be-

tween a vector X with components Xk, k = 1 · · ·nand a scalar Y ,

Y = r(X), (8.69)

the method is conceptually simple: Once a proba-bility distribution for X, expressed through a distri-bution function FX(x) or a density fX(x), has beendecided upon, the method uses a random number gen-erator to generate a sample

xi, i = 1 : N.

Statistical characteristics of Y are then estimatedfrom the sample

yi = r(xi), i = 1 : N. (8.70)

Examples are estimates of mean and variance,

mY =∑i

yiN, (8.71)

σ2Y =

∑i

(yi − mY )2

N − 1, (8.72)

or an approximation of the distribution functionFY (y) by the empirical distribution function

FNY (y) =(number of yi ≤ y)

N. (8.73)

The main disadvantage of the method is that the sam-ple size must be large because the estimated charac-teristics converge slowly towards the true values, witherrors of the order of 1/

√N . This behavior is reected

in commonly used condence intervals, e.g.

mY ±tpσy

(N − 1)12

, (8.74)

for the estimated mean. A further illustration is pro-vided by a condence band for FNy obtained by in-verting the Kolmogorov-Smirnov134 test:A εN such that

PFNY (y)− εN ≤ FY (y) ≤ FNY (y) + εN = 0.95(8.75)

is given by

εN =1.358√

N + 0.12 + 0.11/√N

(8.76)

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192 CHAPTER 8. RESERVOIR MODELLING

for N > 80. If we want εN = 0.05, we mustchoose N = 740. Normally, several hundreds oreven thousands of realizations are generated in MC-calculations. The large value of N that may be neces-sary limits the complexity of the function r(x). Per-forming 740 full reservoir simulations to nd a proba-bility distribution for recovery is probably out of thequestion.Various methods for obtaining faster convergence

have been introduced. These methods are usuallycalled variance reduction methods (and sometimes`swindles'135). McKay et al.136 compare two suchmethods with the standard MC method. The rst,called Stratied Sampling (SS), may be formulatedas follows:Divide the range of each component Xk into J in-

tervals Ikj , so that the marginal probability of Xk

occurring in Ikj is 1/J . Sample each Xk once fromeach interval Ikj , giving the value ξkj . A sample ofIn vectors is obtained from the set ξkj by includ-ing all vectors x = [xk] with xk = ξkj for some j in1 . . . J. The second method, called Latin Hyper-cube Sampling (LHS), generates a set of J vectors.The k-th component of each vector is obtained by ran-dom sampling without replacement from the set ξkj .Observe that in SS the whole, n-dimensional range ofX is covered, while LHS results in a set of vectors suchthat each set of component values span the range ofthat component. McKay et al.136 demonstrate byanalysis and numerical experiments on a problem ofow in pipes that of the three methods compared,LHS was the most ecient. Ding et al.124 arriveat the same conclusion in a study of a cyclic steaminjection process. Startzman and Wattenbarger127

compare ordinary MC with LHS in a risk analysisinvolving an integral expressing Present Worth Protcontaining 5 random parameters. They found thatthe LHS method gave acceptable estimates of meanand variance at about 1/10 of the cost of the MCmethod. The latter two papers are the only ones wehave found that use the LHS technique in problemswithin petroleum technology. The method obviouslydeserves further study.Variance reduction may also be obtained by using

so-called control variates. Suppose that the expec-tation mY of a scalar function Y = r(X) is to beestimated from a sample xi. Suppose also that an-other function Z = s(X) can be found that correlateswell with Y and whose expectation mZ is known ex-actly. Then, instead of estimating mY by a samplemean

mY =∑i

yiN, yi = r(xi), (8.77)

we can use the estimator

mY =∑ yi − zi +mZ

N, zi = s(xi). (8.78)

The variance of mY will be smaller than that ofmY if VarZ < 2 CovY,Z. In applications, s isan approximation of r with a functional form that

allows mZ to be calculated analytically. Smith andBuckee107 recommend the method for reserves calcu-lations. An extended version of the method can befound in an article by Ripley.135

8.5.5 Use of Geostatistical Methods inUncertainty Analysis

With the introduction of geostatistical methods inreservoir description, one would think that thesemethods immediately would be put to use in uncer-tainty analyses, and in particular that the emerg-ing stochastic description simulators would be usedfor such purposes. From the papers published onthe subject, we conclude that so far this has notbeen the case. The explanation is that applying theMC method with repeated production simulations onstochastically generated reservoirs is too time con-suming to be practical unless the reservoir modelsare highly simplied. Static calculations of e.g. oilin-place should be more amenable to this approach.We have listed 5 papers98,107,116118 that deal with

reserves estimates. The rst three use a simpliedanalysis and the last two, especially the last one,apply geostatistical techniques. A simplied uncer-tainty analysis of oil in place employs the equation

N = V αφSoiBoi

, (8.79)

where V is the gross volume, α is a net-to-gross ratioand the other variables denote porosity, initial oil sat-uration and formation volume factor. Based on seis-mics, well data, and other information, probabilitydistributions can be assigned for each variable. Rect-angular, triangular or normal distributions are usu-ally employed. Thereafter, a parametric method oran MC method gives the statistical characteristics ofN . Smith and Buckee107 compare the two approacheson this problem. In contrast, Berteig et al.118 basetheir analysis on the following formula for volume ofhydrocarbons in-place (in reservoir units)

V =∫RI(u)φ (u)(1− Sw(u))du. (8.80)

I is an indicator function dened by

I(u) =

1 Dt(ux, uy) < uz ≤ Db(ux, uy)0 else, (8.81)

where Dt and Db denote the depths to the top andbottom bounding surfaces of the reservoir, and R isa spatial domain large enough to contain any possi-ble reservoir. The depths, as well as the porosity φand the water saturation Sw, are regarded as randomfunctions. A Bayesian approach is used to dene thebounding surfaces. The top depth, Dt, is dened as agaussian random function with an expectation func-tion and a covariance function,

EDt(ux, uy) =∑i

Aidi(ux, uy) (8.82)

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8.5. UNCERTAINTY IN FORECASTING 193

CovDt(ux, uy), Dt(u′x, u′y) = C(ux − u′x, uy − u′y).

(8.83)The functions di come from a seismic depth-conversion model and are regarded as known. Thecoecients Ai are regarded as random variables withprior distributions expressing the uncertainty in thetop depth in the absence of well observations. Withobservations of Rt in a number of wells, posterior dis-tributions for Ai can be calculated, thereby reducingthe variability of Dt. The statistical properties of φand Sw are dened through the distributions obtainedfrom measured values in wells. The MC method cannow be applied to estimate statistical properties ofV . A valuable aspect of the Bayesian approach isthat it allows a quantication of increased informa-tion. In a case study with data from the Troll Field,the authors conclude that the 0.92 condence inter-val is about 95% of the average value of V with onewell present and about 24% of the average with eightwells present.

We now turn to analyses of dynamic reservoir re-sponse. Apart from the classical paper byWarren andPrice,60 most of the early papers come from groundwater research. Some contributions from this eldcan be found the reference list.82,84,85,108,113115 Areview of methods developed in ground water researchprior to 1982 can be found in Brown and Smith.137

An interesting application appears in a paper by Nel-son et al.115 The paper reports an analysis of con-taminant ow in an aquifer in the Avra Valley, nearTucson, Arizona. The rst part of the work concernsestimating transmissibilities of the aquifer. By use ofkriging and inverse modelling based on observationsof hydraulic head, a set of expected transmissibili-ties and a covariance matrix for the set is calculated.Next, an MC study is conducted, involving the fol-lowing steps: A number of realizations of transmis-sibilities (600) are generated. For each realization,a one-phase ow equation is solved and a velocityeld is calculated. The ow eld is used to computetransport times of a contaminant from a line sourcelocated inside the ow domain to the boundary ofthe domain, resulting e.g. in distributions of the cu-mulative amount of contaminant transported to theboundary as function of time. The large number ofrealizations is made possible by the fact that the owis one-phase and stationary.

Many of the papers that report applications of geo-statistical methods in forecasts of petroleum reservoirperformance seem to have an objective that is dier-ent from the one that is adopted here. In severalpapers,138146 stochastic reservoir models are used tocreate one or a few `typical' reservoirs rather than asample of possible reservoirs that is suciently largeto allow statistical analysis. The interesting pointis that the authors claim that a `typical' reservoircreated by statistical methods conforms better withobserved production history than reservoirs modelledby conventional means. If the `typical' reservoir is se-lected randomly, the implication is that the random-

ness has little inuence on the recorded productionresults. In less complicated ows, this behavior maybe explained: The variance of an eective permeabil-ity decreases with the correlation length of the pointpermeability of a porous medium.An important aspect of stochastic reservoir mod-

elling is that it attempts to quantify our uncertaintyabout a reservoir in a precise way, thus reducing sub-jectivity in assessments of that uncertainty. Hencea stochastic reservoir model should provide the bestpossible basis for uncertainty analyses of predictedreservoir performance. More sophisticated techniquesand/or greater computational power are, however,needed before such analyses can be undertaken. Afew of the references120125 report variabilities in pro-duction results, calculated from samples of stochasticreservoirs. The reported results are, however, mostlyintended to show that variability can be signicantrather than making a systematic study of the vari-ability.Most uncertainty analyses of predicted perfor-

mance of `real' reservoirs must rely on prudent simpli-cations of reservoir behavior. Important examples ofstudies using simplied models have appeared.105,106

The suggested procedure consists of three steps. Inthe rst step, sensitivity analyses are performed toidentify which reservoir parameters are the most crit-ical in terms of their inuence on production results.To x ideas, suppose that the parameters xi des-ignate e.g. horizontal permeability, kh, and verticalpermeability, kv, sealing capacity of faults, shale con-tinuity, and relative permeability, kr, and that the un-certainty in a recovery factor F , due to uncertaintiesabout the parameters, is to be estimated. Assumingthat the parameters are independent, the second stepconsists of assigning probability distributions for theeect of each parameter. A base case is run where allthe parameters are given reference values x0

i , givinga reference recovery factor F 0. Next, F is expressedas a product

F = F1 × f2 × f3 · · · , (8.84)

whereF1 = F (x1, x

02, x

03, · · ·), (8.85)

f2 = F (x01, x2, x

03, · · ·)/F 0, (8.86)

with similar denitions for the other fi's.Now, each parameter is varied in turn, while the

others are kept at their reference values, such thata pessimistic (L) and an optimistic (H) recovery re-sults. For each factor we then have a triplet of values:

FL1 , F0, FH1 ,

fL2 , 1, fH2 ,

etc.Based on these triplets, a subjective probability

density is assigned to each factor. A piecewise lin-ear distribution appears to be a natural choice.

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194 CHAPTER 8. RESERVOIR MODELLING

The third step of the analysis is then to use a para-metric method or an MC method to compute the re-sulting probability distribution for F .Renements of the method are possible, e.g. by ex-

tending the method to include interdependent reser-voir parameters.106

No detailed examples have been given. The suc-cess of the method depends obviously on the skill ofthe analyst in selecting the most signicant reservoirparameters as well as assigning probability distribu-tions for the terms in the recovery factor. It is tobe expected that such skills eventually will be madeless decisive by the introduction of more rened andpowerful statistical techniques. However, subjectivitywill certainly continue to be an element of uncertaintyanalysis.

8.6 Field Examples

8.6.1 Introduction

Reservoir simulation has been an important tool inthe stage of planning for development and operationand later during production for all North Sea reser-voirs.36,147150

A characteristic of these studies is that a combi-nation of models is used, from single-well to full-eldmodels. The coarse grids of full-eld simulation mod-els cannot capture the degree of detail needed for un-derstanding near-well phenomena like coning, cusp-ing, condensate dropout, the eect of barriers to ow,etc. Single-well models are then used to study theseeects, which are then incorporated in the full-eldmodels by means of pseudofunctions or other spe-cially designed tools.151 Element models incorporat-ing several wells are also generally needed in order tostudy the interaction amongst wells, communicationacross faults, the eect of interwell heterogeneities,permeability contrasts, etc.Recent developments allow the study of regions of

the reservoir while taking into account the interac-tion with the rest of the reservoir33 and near-wellphenomena with local grid renements and smalltimesteps.152 In the following, these introductorycomments will be expanded with examples from theStatfjord and Troll elds.33,150152

8.6.2 Troll

The Troll eld covers a large area, approximately 750km2, and contains approximation 600 millions Sm3

of oil and 1700 billions Sm3 of gas. An areal viewis shown in Fig. 8.16. The reserves are distributedin three provinces: (1) the Troll East gas province,containing approximately 2/3 of the gas reserves in agas cap of up to 250 m height; (2) the Troll West gasprovince, with a gas cap up to 200 m thick and a thinoil layer between 12 and 15 m; (3) the Oil province,with a 22 to 28 m thick oil column beneath a 10 to50 m thick gas cap.

0 10 km31/6

31/2 31/3

Horizontal well31/2 - T1

Horizontal well,31/5 - T1

Troll eastTroll west gas provinceTroll west oil province

Northerncommunicationchannel

Middlecommunicationchannel

Southerncommunicationchannel

Figure 8.16: Area view of the Troll eld.33

Reservoir simulation of the Troll eld has to tackleseveral challenges: (1) large size, (2) two phases inTroll East and three phases in the western provinces,(3) dierent uid contacts, (4) communication chan-nels between the provinces, (5) conditions for verticalequilibrium except in the well neighborhood wheredisperse ow dominate, (6) tens of gas productionwells a few hundred meters from each other, (7) exis-tence of laterally extensive barriers to ow, (8) com-bined oil and gas development and hydrocarbon re-covery as a function of relative timing of gas and oilproduction must be studied, (9) a large number ofhorizontal wells is needed for oil recovery, (10) de-tailed study of gas and water coning is required fordetermining the oil production potential.Since the oil production is inuenced by the timing

and level of gas production, both oil and gas modelshave to be coordinated. For simplicity, they will bepresented separately in the following.

8.6.3 Troll Gas Model

The gas production has been studied with a coarse-grid system with gridblocks ranging from 1000 to3000 m, as shown in Fig. 8.5. The simulation gridshown there, overlaid the geological model, was dis-torted areally to t the main faults. Care was takento design the grid with nearly orthogonal blocks andfollowing the expected ow pattern. The three com-

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8.6. FIELD EXAMPLES 195

munication channels between the main fault blockswere represented as faithfully as possible since thegas and water uxes between the provinces inuencethe total gas recovery factor.Since it was expected that many runs would be nec-

essary in order to simulate the requirements of the theongoing gas sale contract negotiations, as few blocksas possible were used. One-dimensional, numericalank aquifers were attached to the simulation model,with buer blocks along the sides and on the bottom,in order to get the correct time response of the aquiferand to give a total water volume of approximately 40hydrocarbon pore volumes.The resulting grid had 27 rows in the west-east di-

rection, 39 rows in the north-south direction, sevenlayers to represent the 13 geological zones, and oneextra layer for representing the underlying aquifer.The total number of blocks was then 8424, with 5351active blocks, including the blocks used to representthe ank aquifers.As many as 24 wells were represented in one grid

simulation block of 1600 × 1600 m. For simplication,6 groups of 4 eective wells were used, and Peace-man's formulation154 for taking into account the wellinterference was introduced. This approximation wasveried by rening the simulation gridblock where thewells are perforated. The renement consisted of 11× 11 blocks in the I- and J-directions, and 5 layers inthe second global layer from the top, in order to repre-sent the partial perforation of this global block. Theresults were approximately the same.151 The CPU-time was of the order of 15 min on an IBM 3090 withvector processor.

8.6.4 Results of the Gas Model

The drive mechanisms are mainly gas expansion andaquifer inux. Although the Troll sand is loosely con-solidated and fairly compressible, the contribution ofpore volume reduction to gas recovery is small. Rockcompaction is, however, signicant in the aquifer andmay generate a strong aquifer inux.The two main gas accumulations, the Troll East

and the Troll West gas provinces, communicatethrough three channels as shown in Fig. 8.16 andFig. 8.3, and through the water zone. The total eldrecovery is optimized by depleting the gas caps insuch a way that the pressure gradients across the eldand the reservoir pressure at shut-in time are mini-mized. As 2/3 of the gas reserves is contained in theeastern gas cap, recovery is enhanced by the currentproduction plan of commencing the gas production inTroll East in october 1996.Development of the western gas cap depends on

the gas market in the last years of this century. Theimpact of timing of Troll West gas development onultimate eld recovery is shown in Fig. 8.17. Delayeddevelopment of Troll West causes gas expansion inthe Troll West area and increased gas ux betweenWest and East through the communication channels.The pressure drop in the East will be transmitted to

Delayed Troll West production start-up, years

Rec

over

yfa

ctor

gas

Figure 8.17: Decrease in gas recovery caused by delayof gas production from Troll West gas province.33

the Troll West province, tilting he contacts and stim-ulating the natural waterdrive in Troll East. Largerresidual gas volumes will then be left behind the ad-vancing water front at high pressures.Due to the large aquifer, pressure decline will be

sensitive to plateau level. Fig. 8.18 shows the possibleincrease in reserves by accelerating the production.

Total annual contract quantity

Rec

over

yfa

ctor

gas

Figure 8.18: Increase in gas recovery caused by in-creased gas production.33

In addition to the factors already mentioned, ulti-mate gas recovery will depend on aquifer strength,which cannot yet be determined accurately, water-gas displacement eciency and minimum achievabletubing head pressure. The studies carried out showthat both minimum reservoir pressure and wateringof the gas wells compete in determining shut-in time,depending on the exact gas plateau rates and timingof production, and showing a good reservoir manage-ment strategy.Anticipated ultimate eld recovery is in the range

of 70 to 80%. The plateau length and the need for gascompression at the oshore platform are determinedby the minimum inlet pressure at the onshore pro-cessing plant and the productivity of the gas wells.High-rate production tests in the eld indicate largeturbulent pressure loss across the perforations, whichmay be reduced by a longer perforation interval withfor example horizontal wells. In the case of well shut-in due to water breakthrough, horizontal wells mayincrease ultimate recovery if placed closer to the topof the reservoir. The reduced ow velocities in thiscase may also reduce the risk of sand production.

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196 CHAPTER 8. RESERVOIR MODELLING

Region 1

Region 2

Figure 8.19: Use of the ux boundary condition.33

8.6.5 Troll Oil Model

Oil production with horizontal wells in the thin-oilcolumns of Troll is controlled by the coning of gasand water and by the movement of the contacts in-duced by gas production. Local grid renement withsmall timesteps is used to resolve the uid movementinto the wells. The inuence of the gas productionin remote parts of the eld on the movement of thewater-oil and gas-oil contacts has to be taken intoconsideration. This is achieved by running the full-eld model with only gas and water, which are thephases which can ow through the communicationchannels, and storing these uxes in a le. Then,the region of interest can be run with with threephases and with local grid renements for the hori-zontal wells and the uxes in the region are read fromthe history le, see Fig. 8.19 for an example of tworegions. Running region 2 with ve local grid renedareas of 1200 blocks each, with ve horizontal wells,takes 211 min of CPU-time. During the rst produc-tion period, the global grid (region 1) advanced withtimesteps of 30 days, whereas the local grids usedtypically 3 to 5 days. At gas breakthrough, the localgrid needs shorter times (under one day), since thereservoir management strategy calls for cutting backthe oil rate each time gas breakthrough was experi-enced. When the gas cone stabilizes and the oil rate

is close to critical, the global and local grid use largetimesteps of typically 90 and 30 days, respectively.One of the results to be studied was the inuence

of simultaneous gas production on cumulative oil pro-duction from region 2, Fig. 8.20. The drive mecha-

1.2

1.0

0.8

0.6

0.4

0.2

0.00 1000 2000 3000 4000

Time, days

Nor

mal

ized

cum

ulat

ive

oil

prod

uctio

n

Separate oil and gas development

Simultaneous oil and gas development

Figure 8.20: Oil recovery with and without simulta-neous gas production.33

nism shifts from gas drive to water drive, the distancebetween the gas-oil contact is larger and the cumu-lative production is slightly higher. Since this is adelicate balance depending on production rates andrelative timing (and on the presence of local barri-ers), the results only indicate the range of productionwhich can be expected.

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8.6. FIELD EXAMPLES 197

Upper Brent

Upper Brent

Lower Brent

Lower Brent

StatfjordTarbert

Ness

Tarbert

Ness

NessEtive

RannochBromm

NessEtive

RannochBromm

NansenErikssonRaude

Full field model

58

58 5858

58

58

30

30 3030

30

30

3

123456

12

10 97

2

Statfjo

rdLower

Brent

Upper

Brent

Figure 8.21: Schematic simulation model of the Statfjord eld.149

8.6.6 Statfjord Field.

Reservoir simulation has been extensively used for theStatfjord eld since its discovery in 1974 and afterthe production start in 1979, and has been the basisfor all decisions made in its management. Papers byHaugen et al.149 and Dixon150 are used as sources inthe following considerations.The dimensions of the model have grown from two-

dimensional 2500 blocks to the present full eld modelwith approximately 40 000 blocks, reecting both theincreased knowledge of the eld and the advances incomputing power.The previous full-eld model consisted of three de-

tailed models, Fig. 8.21. The models for Upper Brent(with ten layers) consisted of around 11 000 grid-blocks, for Lower Brent (with nine layers) of around10 000 and for the Statfjord reservoir (with seven lay-ers), approximately 11 000. The detailed models werehistory-matched and used to study sweep eciency,perforation strategy, well recompletion, and produc-tion rates. In order to study platform process con-straints, water and gas injection limitations, well con-straints and gas sales rates, a full-eld model was es-tablished of around 21000 blocks by merging layers inthe Brent reservoirs and the ne model for the Stat-fjord reservoir. The reduction of the detailed modelswith dynamic pseudorelative permeability and capil-lary pressure curves and using these in the full-eldmodel to reproduce the history match was a time-consuming process. The model was successfully usedto determine reservoir performance with dierent gassales levels and updating of platform capacity. Forthe present model of around 40 000 gridblocks, it wasdecided that the detailed models would be amalga-mated without reduction. Use of parallelization ofthe three isolated reservoirs and vectorization of theblack-oil simulator had allowed a reduction in com-

puter time by a factor of between two and three, andthis fact together with faster machines made it pos-sible to save the engineering time spent in buildingsmaller models, generating pseudofunctions and re-matching the full eld history match.

Nomenclature

A = actual reserves, m3

= random variableB = formation volume factor, Rm3/Sm3

= random variableb = inverse volume factor, Sm3/Rm3

C = molar density, mol/m3

= spatial correlation function= covariance matrix

D = depth, md = expected length around a barrier, m= unspecied function

E = estimated reserves, m3

= expectation valueF = ratio of volume of barriers to total volume= distribution function

f = fugacity= fractional ow function= generic function of space= unspecied function

fX = density function of Xg = acceleration of gravity, m/s2

i, j, k = block numberK = equilibrium K-valuek = absolute permeability, md or m2

L = mole fraction liquid in hydrocarbon mix-ture

= spatial scale of averaging= number of vector components

L = spatial scale of large-scale variationl = spatial scale of small-scale variation

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198 CHAPTER 8. RESERVOIR MODELLING

M = gridblock content, Sm3, mol or kg= viscosity ratio

mX = expected value of XN = size of sample, STOIIP, Sm3

= numbern, k = number of components

n,m, p = number of blocks in i, j, l directionP = probabilityp = pressure, Pa

p,j = component j of pressure gradient, Pa/mp, q = arbitrary numbersQ = rate of production from a grid block,

Sm3/d, mol/d or kg/dq = ow rate between connected grid blocks,

Sm3/d, mol/d or kg/d= source or sink term, kg/(m3·s)

R = integration domainRs = solution gas/oil ratio, Sm3/Sm3

r = unspecied functionrs = dissolved oil/gas ratio, Sm3/Sm3

~r = radius vectorS = saturation (water), fractions = unspecied function= expected number of barriers

T = interblock specic transmissibility, m3/dt = time, s or d

U, Y, Z = result vectorsu = Darcy velocity, m/s or m/d= coordinate vector= solution of dierence equation

V = net grid block volume, m3

= volume of averaging= gross reservoir volume, m3

v = velocity, m/d= unspecied function

X = data vectorx = mole fraction in hydrocarbon liquid phase= spatial coordinate= distance, m= independent variable (vector)

y = mole fraction in hydrocarbon vapor phase= dependent variable (vector)= solution of dierence equation

z = mole fraction in total hydrocarbon mix-ture

α = net-to-gross ratio, fractionα = proportionality constantΓ = adsorbed mass per bulk reservoir volume,

kg/m3

Φ = ow potential, Paε = toleranceλ = mobility, (Pa·s)−1

= parameter between -1 and 1µ = viscosity, Pa·s or cpξ = mass concentration, dimensionlessρ = density, kg/m3

σ = standard deviationφ = porosity, fractionω = mixing parameter

Subscripts

a = apparent or averageC = componentc = capillary

e = eectivef = estimateg = gas or geometrich = hydrocarbon or horizontal or harmonic

i, j = indicesk = permeabilitym = uid phasen = time level or nonwetting

num = numericalo = oilp = pressurer = relativeS = saturationv = verticalw = water or wettingα = component

Superscripts

e = eectiveH = highk = time level, component numberL = lowN = sample size of stochastic variablen = time level

Operators

<> = spatial averagingˆ= spatial deviation or mean¯= average or harmonic mean′ = dierentiation

˜ = estimatorδ = perturbationfx = ∂f/∂x

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[127] Startzman, R.A. and Wattenbarger, R.A.: AnImproved Computation Procedure for RiskAnalysis Problems With Unusual ProbabilityFunctions, paper SPE 13772 presented at the1985 SPE Hydrocarbon Economics and Evalu-ation Symposium, Dallas, March 1415.

[128] MacDonald, R.C. and Campbell, J.E.: Valua-tion of Supplemental and Enhanced Oil Recov-ery Projects With Risk Analysis, JPT (Jan.1986).

[129] Gittler, W.E. and Krumrine, P.H.: A NovelApproach for Risk Assessment in ChemicalEOR Projects, paper SPE 13767 presentedat the 1985 SPE Hydrocarbon Economics andEvaluation Symposium, Dallas, March 1415.

[130] Barua, J., Prescott, T., and Haldorsen, H.H.:Financial and Technical Decision Making forSurfactant Flooding, paper SPE 15074 pre-sented at the 1986 SPE California RegionalMeeting, Oakland, April 24.

[131] Walstrøm, J.E., Mueller, T.D., and McFarlane,R.C.: Evaluating Uncertainty in EngineeringCalculations,JPT (Dec. 1967).

[132] Brown, C.E. and Smith, P.J.: The Evaluationof Uncertainty in Surfactant EOR PerformancePrediction, paper SPE 13237 presented at the1984 SPE Annual Technical Conference and Ex-hibition, Houston, Sept. 1619.

[133] Bu, T. and Aanonsen, S.I.: Surfactant Flood-ing Uncertainty Analysis, paper presented atthe 1991 European Symposium on IOR, Sta-vanger, May 21-23.

[134] Bickel, P.J. and Doksum, K.A.: MathematicalStatistics, Holden-Day, Oakland (1977).

[135] Ripley, B.D.: Stochastic Simulation, John Wi-ley & Sons, New York (1987).

[136] McKay, M.D. and Beckman, R.J.: A Compar-ison of Three Methods for Selecting Values ofInput Variables in the Analysis of Output froma Computer Code, Technometrics (May 1979)21, No. 2.

[137] Brown, C.E. and Smith, P.J.: The Applicationof Stochastic Modeling to Petroleum Engineer-ing Problems, paper presented at the ASMEConference, Paris (July 1982).

[138] Emanuel, A.S., Alameda, G.K., Behrens, R.A.,and Hewett, T.A.: Reservoir Performance Pre-diction Methods Based on Fractal Geostatis-tics, SPERE (Aug. 1989).

[139] Mathews, J.L., Emanuel, A.S., and Edwards,K.A.: Fractals Improve Mitsue Miscible Pre-dictions, JPT (Nov. 1989).

[140] Tang, R.W., Behrens, R.A., and Emanuel, A.S.:Reservoir Studies Using Geostatistics To Fore-cast Performance, paper SPE 18432 presented

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204 CHAPTER 8. RESERVOIR MODELLING

at the 1989 SPE Symposium on Reservoir Sim-ulation, Houston, Feb. 68.

[141] Keijzer, J.H. and Kortekaas, T.F.M.: Compar-ison of Deterministic and Probabilistic Simula-tion Models of Channel Sands In the StatfjordReservoir, Brent Field, paper SPE 20947 pre-sented at the 1990 Europec, The Hague, Oct.2224.

[142] Hazeu, G.J.A., Krakstad, O.S., Rian, D.T.,and Skaug, M.: The Application of New Ap-proaches for Shale Management in a Three-Dimensional Simulation Study of the FriggField, SPEFE (Sept. 1988).

[143] Carr, L.A., Bentreau, R.I., Corrigan, M.P., andvan Doorne, G.G.: The Successful Characteri-zation of a Complex Reservoir Using 3-D Seis-mic, Geostatistical Reservoir Description andSponge Core Analysis, paper SPE 16780 pre-sented at the 1987 SPE Annual Technical Con-ference and Exhibition, Dallas, Sept. 2730.

[144] Crane, O. and Tubman, K.M.: Reservoir Vari-ability and Modeling With Fractals, paperSPE 20606 presented at the 1990 SPE AnnualTechnical Conference and Exhibition, New Or-leans, Sept. 2326.

[145] Stanley, K.O., Jorde, K., Ræstad, N., andStockbridge, C.P.: Stochastic Modelling ofReservoir Sand Bodies for Input to ReservoirSimulation, Snorre Field, Northern North Sea,Norway, North Sea Oil and Gas Reservoirs-II ,A.T. Buller et al. (eds.), Graham & Trotman,London (1990).

[146] Nybråten, G., Skolem, E., and Østby, K.:Reservoir Simulation of the Snorre Field,North Sea Oil and Gas Reseroirs-II , A.T.Buller et al. (eds.), Graham & Trotman, Lon-don (1990).

[147] Ånes, H.M., Haga, O., Instefjord, R., andJakobsen, K.G.: The Gullfaks Lower BrentWaterood Performance, paper presented atthe 1991 European Symposium on IOR, Sta-vanger, May 21-23.

[148] Petterson, O., Storli, A., Ljosland, E., andMassie, I.: The Gullfaks Field: Geology andReservoir Development, North Sea Oil andGas Reservoirs A.T. Buller et al. (eds.), Gra-ham & Trotman, London (1989).

[149] Haugen, S.A., Lund, Ø. and Høyland, L.A.:Statfjord Field, Development Strategy andReservoir Management, JPT (July 1988) 86373.

[150] Dixon, R.: History Matching of SimulationModels of Oil and Gas Fields in the North Sea,paper presented at the 1987 Latin AmericanPetroleum Engineering Conference, Punta Are-nas.

[151] Lie, Ø. and Henriquez, A.: Practical Applica-tion of New Simulation Techniques for LargeOil Fields, paper presented at the 1989 Inter-national Forum on Reservoir Simulation, Alp-

bach, Sept. 4-8.[152] Cheshire, I. and Henriquez, A.: Local Grid Re-

nement, paper presented at the 1989 JointIMA/SPE European conference on the Mathe-matics of Oil Recovery, Cambridge, July 2527.

[153] Henriquez, A., Apeland, O.J., and Ørke, T.:Development of a Large Gas Reservoir withOil Rim, paper presented at the InternationalConference on Development of Gas and Con-densate Fields, Krasnodar, May 28June 2.

[154] Peaceman, D.W.: Near-singularities of Pres-sure and Concentration of the Wellbore inReservoir Simulation, paper presented at the1988 International Forum on Reservoir Simula-tion, Alpbach, Sept. 1216.

[155] Henriquez, A., Kårstad, T., and Steihaug, T.:Practical Use of Supercomputing in Black-OilReservoir Simulation, North Sea Oil and GasReservoirsII , A.T. Buller et al. (eds.), Graham& Trotman, London (1989).

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Part III

Methods

205

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Chapter 9

Gasooding

9.1 Displacement Mechanisms

9.1.1 Introduction

Many oil elds in the North Sea contain high-permeability sandstone formations with dippingreservoirs containing light oil and some of the reser-voirs comprise a gas cap. Most of the elds are de-veloped by waterooding and pressure supported bywater injection. Hydrocarbon gasooding in some ofthe reservoirs, however, could yield higher oil recover-ies compared to forecasted waterooding recoveries.1

A gravity-stable gas injection scheme with an ac-ceptable sweep eciency could be maintained in someof these highly permeable, light-oil reservoirs pro-vided that the reservoir dip is suciently high. Highpressure and high temperature in combination withthe above-mentioned reservoir properties contibute tofavorable phase behavior eects between the reser-voir oil and the injection gas. This can contributesignicantly to the recovery of oil both in the caseof immiscible displacement and in processes closer tomiscibility.2

This section focuses on the opportunities and con-cerns related to gasooding in North Sea oil elds.Among the opportunities are natural gas resources ofwhich sucient volumes might become available forgasooding in selected reservoirs, along with the factthat light crude oil promotes extraction of valuableoil fractions during gasooding. Signicantly dippingformations of permeable sands provide favorable con-ditions for oil recovery by gravity-stable gas injec-tion schemes. Among the concerns are unfavorablegas-oil mobility ratios and formations consisting ofan overall upwards-coarsening permeability sequencepromoting gas override. Both factors contribute topoor volumetric sweep eciencies. Reservoir proper-ties classied as opportunities and concerns will rstbe discussed.

9.1.2 Opportunities and Concerns

High-permeability sandstone formations are presentin many oil elds in the North Sea. Many of these for-mations have a signicant dip angle, and some of thereservoirs contain an original gas cap. The reservoirscontain light oil and pressure and temperature arehigh. Hydrocarbon gas reserves are generaly avail-

able at varying distances from the oil elds or as as-sociated gas.

High Local Displacement Eciency

Laboratory experiments have demonstrated that gas-ooding in high permeability sandstone cores undergravity-stable conditions yields very high local dis-placement eciency of oil. The term local displace-ment refers to reservoir zones being swept or con-tacted by the injection gas.Residual oil saturation (Sor) values around 15%

of total porevolume are reported by Haugen andHaaland3 from hydrocarbon immiscible gasoodingexperiments at 206 barg pressure in a verticallymounted assembly of cores from the Statfjord forma-tion of the Statfjord eld. These saturations resultedfrom updip injection of one pore volume separator gasat optimal conditions for gravity-stable coreoods.They assumed that the ongoing gasood in the Stat-fjord reservoir is a miscible process, so that residualoil saturations even lower than the measured 15% areexpected in ooded reservoir zones. As a result, therecoverable reserves of the reservoir have been up-graded signicantly as compared with expected re-covery by water injection.Torvund and Nipen4 reported residual oil satura-

tion values around 10% based on hydrocarbon gasdisplacement experiments at reservoir conditions inOseberg core samples. These low Sor values wereobtained by various immiscible gasoods, includinggravity-stable displacement tests. The gasoods wereperformed in medium (40 cm) and long (160 cm) coreassemblies containing very high permeability (rangingfrom 1 to 10 darcy) Etive sandstone cores.These high local-displacement eciency values en-

couraged the operator to evaluate eld-scale gasood-ing of the Alpha structure by external gas supply,and to reconsider the original development plan whichwas based on water injection as the drive mechanism.Torvund5 reported an estimated additional oil pro-duction of 12 million Sm3 from the Alpha structure.This adds to the expected additional oil recovery fromthe Gamma structure, also as a result of gas injection.Hustad and Holt6 reported that injection of sepa-

rator gas into a one-piece, 1.2 m long sandstone corecontaining a North Sea crude oil under reservoir pres-

207

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208 CHAPTER 9. GASFLOODING

sure and temperature gave a residual oil saturation of13%. This residual saturation resulted from injectionof 1.7 pore volumes of immiscible gas. Even lowervalues of residual oil saturation are measured if moregas is injected. The permeability of the sandstonewas 2.7 darcy. The core was vertically mounted andgas injected at the top. Prior to gasooding, the corewas waterooded from the bottom to a residual oilsaturation of 39%.Low residual oil saturations in gravity-stable ni-

trogen displacements are reported by Glasø7 in core-oods of one-piece, 50 cm long Bentheimer and Bereacores containing North Sea oil under high tempera-ture and pressure. A residual oil saturation of 24%is reported by Naylor et al.8 from high pressure, im-miscible, gravity-stable nitrogen displacement of syn-thetic oil in presence of connate water in a 1.15 darcyClashach sandstone core of 83.7 cm length, in verti-cal position. Niko et al.9 conducted gravity-drainageexperiments by nitrogen injection into a vertical col-umn of cores from the Statfjord eld following water-ooding. They reported a reduction in the wateroodresidual oil saturation from 31% to 14%.

High Permeability

Permeabilities of many sandstone formations in theNorth Sea sucently high to achieve close to gravity-stable displacement in the reservoir even at the highproduction rates that are required to ensure theeconomics of oshore production operations. Themaximum along-dip displacement rate, to maintaingravity-stable displacement is given by the followingequation,2

umax =kh∆ρ g sinα(µokro− µgkrg

) , (9.1)

where umax is maximum gravity-stable displacementvelocity, kh is absolute horizontal permeability, kroand krg are endpoint oil and gas relative permeabili-ties, µo and µg are oil and gas viscosities, ∆ρ is oil-gasdensity dierence, g is acceleration of gravity, and αis reservoir dip angle.Gravity-stable displacement is described in

Sec. 9.4. Velocity is the only parameter in Eq. 9.1that the reservoir engineer may adjust in a technicalsense. From a reservoir management point of view,however, the production rate generally is xed bythe development plan.

Favorable Component Exchange

Vaporization of intermediates (mainly C2 to C6) takesplace at the leading edge of the gas front. High pres-sure and high temperature promote such vaporiza-tion until the displacing gas phase is saturated withintermediates. This results in changes of the gas/oilviscosity ratio, density ratio and interfacial tensionbetween the two phases. Gas viscosity and density inthe displacement front are increased, improving the

microscopic displacement eciency. The density dif-ference is reduced, however,resulting in a lower stabledisplacement velocity.

Unfavorable Mobility Ratio and Poor SweepEciency

One of the main concerns in gasooding is the highgas mobility that results in poor volumetric sweep ef-ciency. The gas/oil mobility ratio is generally unfa-vorable in processes where low viscosity gas displacesreservoir crude, including light oil.The vertical sweep eciencies at breakthrough for

various gas/oil mobility ratios, M , as function ofviscous-to-gravity force ratios are reported by Craig10

and discussed by Stalkup,11 and shown in Fig. 9.1.Sweep eciencies for unfavorable mobility ratios are

100

80

60

40

20

00.1 1 10 100

Ver

tical

swee

pef

fici

ency

atbr

eake

thro

ugh,

%

Fvg =u µ o L

k g ∆ ρ h

M=1.0

M=2.1

M=5.76

M=50

Figure 9.1: Vertical sweep eciencies at break-through, linear uniform systems, after Craig10 andStalkup.11

reproduced in this gure. Breakthrough sweep e-ciency is reduced as the mobility ratios become moreunfavorable. The viscous-to-gravity force ratio is de-ned as

Fvg =uµoL

k∆ρ gh, (9.2)

where u = q/A (q is volumetric injection rate and Ais area perpendicular to ow), µo is oil viscosity, L islength of linear reservoir model, k is permeability inow direction, ∆ρ is oil-gas density dierence and his height of ow model.Gravity override is conceptually shown in Fig. 9.2a.

A single gravity tongue overrides the denser oil phasein a vertical cross section where the local displace-ment eciency is insensitive to the gas-oil capillarypressure (miscible displacement). Fig. 9.2 illustratesconceptually the dierent ow regimes at dierentFvg values. The lowest Fvg value corresponds toFig. 9.2a (Region I), the highest value to Fig. 9.2c(Region IV).Viscous instabilities strongly inuence behavior of

the displacement front and ngering of gas into the oilphase may be the result. This is illustrated in Fig. 9.2where Fvg increases from (a) to (c). Viscous ngering

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9.1. DISPLACEMENT MECHANISMS 209

Solvent Solvent Solvent

Oil Oil

(a) Region I & II (b) Region III (c) Region IV

Oil

Figure 9.2: Illustration of ow regimes in uniformvertical cross section, after Stalkup.11

will be amplied when a gasood approaches misci-bility. Fingering acts to destroy the integrity of misci-ble gas slugs and strongly inuences the displacementbehavior and sweep eciency of the reservoir.In Fig. 9.1, sweep eciency versus Fvg is shown

for linear ow models where the area perpendicularto the ow direction is constant and the uid veloc-ity is approximately constant. In reservoir ow sit-uations, however, the uid front velocity may varysignicantly. This variation complicates the determi-nation of an appropriate Fvg value.Craig et al.12 suggested to correlate breakthrough

sweep eciency for ve-spot models with a dimen-sionless scaling group,

F′

vg =qµo

4k∆ρ gh2, (9.3)

where the terms are as dened above for Fvg, Eq. 9.3.In other words, F

vg is based on volumetric injectionrate rather than average linear velocity. It expressesthe horizontal viscous pressure dierential divided bygravity dierential. Fig. 9.3 shows results from With-jack et al.13 In this gure, the mobility ratios of 7.5and 22.4 are applicable for gasooding.Transitions between displacements characterized

by gravity override, override with viscous ngering,and ngering are shown in Fig. 9.4 where F

vg isplotted versus pore volumes of solvent injected fortwo series of miscible displacement tests at gas/oilmobility ratios of 7.5 and 22 in a uniform perme-ability, ve-spot physical model. The propagation ofthe ood front was surveyed by X-ray CT images.At low values of F

vg, the displacement mechanism isdominated by override. At high values, the displace-ment changes into the gravity-dominated ow, goingthrough regimes of viscous ngering and gravity over-ride with ngering.Some of the sandstone formations in the North

Sea have an upward-coarsening permeability sequencethat is unfavorable for gasooding since it promotesgravity override of the gas.Aasen et al.14,15 calculated variations in gas

breakthrough time near gravity-stable conditionsfor upward-increasing and decreasing permeabilitytrends in a 15-layer, stochastic reservoir. They stud-ied 50 dierent reservoir model realizations. A regres-sion analysis of the gradients versus gas breakthroughtimes gave a signicant, positive correlation. A cross

M=0.178, 0.087 and 0.057

M=1.85

M=7.5

M=22.4

1.0

0.8

0.6

0.4

0.2

0.00.1 1.0 10.0

F v g’ = (q µ o ) / (4 kg ∆ ρh 2 )

Rec

over

yat

brea

kthr

ough

,PV

frac

tion

Craig et al.

Withjacket al.

Figure 9.3: Correlation of breakthrough sweep e-ciency, ve-spot uniform systems, from Craig et al.12

and Withjack et al.13

plot of the average gradient and gas breakthroughtime for the cross section study is shown in Fig. 9.5.The cross section studied had a length of 3600 m, aheight of 45 m, a dip of 7, and included gas cap, oilzone, and aquifer.

9.1.3 Factors Aecting Local Dis-placement Eciency

Local displacement eciency refers to recovery of oilin reservoir zones being swept or contacted by theinjection gas. Vaporization of intermediates (mainlyC2-C6) from the reservoir crude takes place at thefront of a lean hydrocarbon gas displacing light oilwith a high content of intermediate hydrocarbon frac-tions. Component exchange across the displacementfront occurs both in immiscible and miscible processesin such system. The amount of component exchangeincreases from immiscible to miscible conditions andthen inuences the displacement eciency.Gas-oil miscibility may be developed through mul-

tiple contacts between the crude and the injected hy-drocarbon gas in a dynamic process in the reservoir.Haugen and Haaland3 reported immiscible hydrocar-bon gasood at 206 barg pressure and miscible gas-ood in the pressure range from 317 to 352 bars usingStatfjord reservoir crude and separator gas in boththe immiscible and miscible coreoods.Miscible gasoods clearly improve the local dis-

placement eciency as compared to immiscible gas-ood. Residual oil saturation after miscible gasoodin Statfjord reservoir cores is reported to be fromzero to 3% after one pore volume of gas is injected,while immiscible gasood in a similar coreood butnow at lower pressure, results in a residual satura-tion of 15%.3 Permeabilities of the uppermost unit ofthe Statfjord reservoir are reported from two to ve

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210 CHAPTER 9. GASFLOODING

5.0

4.0

3.0

2.0

1.0

0.00.0 0.2 0.4 0.6 0.8 1.0

Fingering

Override withfingering

Override

Test 4, M=7.5

Test 5, M=7.5

Test 6, M=7.5

Test 7, M=22

Test 8, M=22

Test 9, M=22

Pore volumes injected

Fv

g’

=(q

µo)/

(4k

g∆

ρh2)

Figure 9.4: Transitions between displacement cate-gories.13

darcy.16 Here, a long-core assembly of reservoir rocksamples was mounted vertically and gas injected up-dip at reservoir conditions. Gravity-stable gasoodrequirements were fullled, as given by Eq. 9.1.Injected hydrocarbon gas also extracts compo-

nents into the owing stream from the residual crude(trapped behind the ood front by capillary forces)and from bypassed zones, and thereby contributesto the total recovery. Recovery by vaporization af-ter gas breakthrough is demonstrated by Hustad andHolt6 who also calculated the IFT by parachor for-mulation in their system of separator gas and NorthSea crude at reservoir condition to be in the range of0.5 to 4 mN/m. They conrmed these calculationsby later experimental determination of IFT betweenequilibrated gas and reservoir crude at reservoir tem-perature and pressure.The contribution to the total oil recovery by va-

porization is demonstrated in Fig. 9.6 which showsoil production versus pore volumes hydrocarbon gasinjected following downdip waterooding of the core.6

Separator gas pre-equilibrated with a North Sea crudeoil at reservoir pressure and temperature before startof the updip gas injection is represented by the lowerproduction curve. Vaporization was negligible in thisexperiment. The upper production curve is from in-jection of dry separator gas in which production af-ter gas breakthrough is dominated by vaporization.Gas breakthrough occurred in both cases at about0.5 pore volumes gas injected. The core was a one-piece 1.2 m long water-wet Bentheimer sandstone andthe reservoir crude exhibited a spreading behavior ona water surface.For the dry gas experiment, the production after

breakthrough is explained by lm ow and vaporiza-tion - the last mechanism being the dominant contrib-utor to the total recovery.6 This was also observed

9.0

8.5

8.0

7.5

7.0

6.5

6.0-100 -50 0 50 100 150

Gas

brea

kthr

ough

time,

year

Average gradient of horizontal permeability, md/m

Figure 9.5: Gas breakthrough time versus averagekh-gradient.15

0.16

0.12

0.08

0.04

0.000.0 0.4 0.8 1.2 1.6

Pore volumes gas injected

Stan

dard

pore

volu

mes

prod

uced

Equil. GasDry Gas

Figure 9.6: Oil production histories from gas injectionexperiments after waterooding.6

through the change in color of the euent in the sep-arator at the core outlet. The lm-ow mechanismis supported by ndings of Dumoré and Schols17 whoconcluded that lm ow may play a role in gasoodsin a highly permeable water-wet Bentheimer sand-stone.Nitrogen also has the ability to vaporize interme-

diates at very high pressures. Higher recovery e-ciencies result from increases in pressure from 215 to315 bar, as reported by Glasø7 for long-core nitrogengasoods. The higher recovery at higher pressure isexplained by more favorable vaporization of interme-diates at higher pressure.Niko et al.9 reported nitrogen gasood experi-

ments at 450 bar pressure in a long core contain-ing reservoir crude from the Statfjord eld. Theyconcluded that miscibility was not developed. How-ever, the displacement eciency was considerably im-proved by favorable mass transfer across the displace-ment front. They observed little oil production after

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9.1. DISPLACEMENT MECHANISMS 211

breakthrough. This is an important observation thatdiers from the result of th hydrocarbon gasood re-ported by Hustad and Holt,6 who observed produc-tion of light oil after gas breakthrough.A comparison of the ability of nitrogen to vaporize

intermediates from reservoir crude oil with that ofhydrocarbon gas at similar conditions can be madefrom the nitrogen coreood reported by Niko et al.9

and hydrocarbon gasood reported by Haugen et al.16

Miscibility was not developed and little oil productionafter breakthrough was observed in the 450 bar nitro-gen coreood. Haugen et al.16 reported miscibility ina pressure range from 317 to 352 bar, depending onthe richness of the injected hydrocarbon gas.Statfjord reservoir oil was used in both experiments

above. Miscibility was developed through a multiple-contact vaporization process in the hydrocarbon gasinjection experiment.16 This was not the case for thenitrogen experiment,9 and demonstrates the betterability of hydrocarbon gas to develop miscibility ascompared to nitrogen at reservoir conditions repre-sentative for North Sea gasooding operations.Vaporization of intermediate hydrocarbon fraction

reduces the interfacial tension (IFT) between the in-jected gas and reservoir crude oil in the ood frontand with a corresponding reduction of the gas-oil cap-illary pressure. For one particular system of sand-stone rock and hydrocarbon uids this behavior isillustrated by Delclaud et al.18 Fig. 9.7 shows gas-oilcapillary pressure curves adjusted for IFT between oiland gas in the range of 22.5 to 0.6 mN/m. The varia-

100

10-1

10-2

10-3

0 20 40 60 80 100

Cap

illar

ypr

essu

re,

bar

Gas saturation, %PV

0.6 mN/m

3.2 mN/m

22.5 mN/m

IFT

Figure 9.7: Capillary pressure curve for various IFT's,after Delclaud et al.18

tion in IFT was obtained by adjusting the equilibrium

pressure of the C1-nC5 system at constant tempera-ture, and was computed by parachors. It is, however,uncertain whether this reduction of capillary pressureby reduced IFT is representative for North Sea reser-voir uids.The lower the IFT, the lower is the viscous force

required to mobilize discrete ganglia of oil and coa-lesce them to form a continuous phase of oil, either asa lm or a mobile bank. The viscous force requiredcan be calculated after having dened wetting charac-teristics, made necessary assumptions, and obtainedrelevant petrophysical and uid data.19

Ypma2 dened a dimensionless capillarity/gravitynumber which incorporates the gas/oil IFT,

Ncg =σ (φ J/k)1/2

∆ρ g H, (9.4)

where φ is porosity, k is absolute permeability, Jis average Leverett J-function, and H is reservoirlayer thickness in stratied reservoirs or total reser-voir thickness.Ncg is related to a dimensionless recovery expres-

sion,(E − Emin)/(Emax − Emin),

where E, Emin and Emax are oil recoveries after in-jection of one movable pore volume, as shown inFig. 9.8. The gure shows that at Ncg = 0.01,

1.0

0.8

0.6

0.4

0.2

0.110-3 10-2 10-1 100 101

Gravity-dominated

Capillary-dominated

N c g

(E-E

min)/

(Em

ax-E

min)

Figure 9.8: Eect of gas/oil capillary pressure ongravity drainage in dipping layer, after Ypma.2

the capillary forces start to aect the recovery pro-cess and for Ncg > 0.1, the capillary forces can notbe neglected. For high-permeability reservoirs withlow IFT at the displacement front of a gravity-stablegasood, Ypma2 states that capillary pressure willnot signicantly aect the recovery of oil. How-ever, it is uncertain how low IFT must be in a high-permeability reservoir in order to fulll this state-ment.Hustad and Holt6 found that variations in gas-oil

capillary pressure have little eect on history match-ing recoveries, as shown in Fig. 9.9. The matchingwas, however, sensitive to oil-water capillary pres-sure. This was particularly true in matching waterproduction. The IFTs were in the range of 0.5 to

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212 CHAPTER 9. GASFLOODING

100

80

60

40

200 250 300 350

Berea A

Bentheimer

Pressure, bar

Oil

reco

very

,%

Figure 9.9: Recovery of oil as a function of pressure.7

4 mN/m at the displacement front of their gravity-stable gasoods in a 2.7 darcy waterwet Bentheimersandstone. The gasoods reported in Fig. 9.6 werecarried out at a capillarity/gravity number of about0.01, which supports the ndings of Ypma.2

Low residual oil saturations in gravity-stable ni-trogen displacements are reported by Glasø7 in core-oods of one-piece 50 cm long Bentheimer and Bereacores containing North Sea crude oil under high tem-perature and pressure. In these vertical coreoods,nitrogen was injected updip at pressures of 215, 260,and 315 bar. Higher recovery eciency with increas-ing pressure is reported. Recovery eciency was alsostudied at dierent mobile water saturations. Recov-eries of oil in these coreoods are reported to be inthe range of 50 to 80% of oil in place, with betterrecovery at higher water saturation. Recovery washigher in the 1.5 darcy Bentheimer as compared tothe 0.3 darcy Berea core.Fig. 9.9 shows the inuence of permeability, or

more precisely of the average pore diameter of sand-stone rock, on local displacement eciency for thetwo cores.7 A 1.5 darcy Bentheimer core with an av-erage pore diameter of about 25 microns and a 0.3darcy Berea core with an average pore diameter ofabout 8 microns were used in nitrogen gravity-stablecoreoods at comparable ooding conditions. Therecovery of oil from the Berea core is signicantlylower as compared to the Bentheimer core at the 215bar immiscible gasood. It is believed that partic-ularly the capillary forces are aecting the recoveryeciency for the Berea core, and that this recoveryprocess is within the capillary dominated regime de-scribed by Ypma,2 Fig. 9.8.

9.1.4 Miscible Processes

Miscible ooding is one of the most promising en-hanced oil recovery methods due to its potential forrecovering all of the oil ushed by solvent. However,only in rare instances eld performance come close tothe high recovery potentially possible from this pro-cess due to low volumetric sweep eciency.

There are four major miscible processes used: (1)vaporizing gas drive, (2) carbon dioxide ooding, (3)condensing gas drive, and (4) rst-contact miscibleooding.The most common injection gases for vaporizing

gas drive are lean gas, nitrogen, and ue gas. In acondensing gas drive, hydrocarbon gas, rich in inter-mediate components, is used. In a rst contact mis-cible process, a slug of ethane or a light hydrocarbonmixture is used.

Vaporizing Gas Drive

When a lean gas, ue gas or nitrogen is injected intoa reservoir at high pressure, hydrocarbons vaporizefrom the oil into the gas phase and concentrate atthe oil-gas front. The phase relationship between theinjected gas and the reservoir oil determines the pres-sure required for miscible displacement, and a highconcentration of C2C6 in the reservoir uid makesthis recovery process favorable. The miscible frontwill move through the reservoir, displacing oil andwater in the front as shown in Fig. 9.10.

Lean gas OilBank

Residual oil

Injected water

from waterflood

Miscible zone formed by gasbecoming enriched with C2-C6

Connate water

Figure 9.10: Schematic representation of high pres-sure lean gas miscible ooding.20

A vaporizing gas drive miscible process displacesnearly all of the oil in the area contacted, but thefraction of the reservoir contacted may be low due tooil/gas/water instabilities caused by the injected gasand reservoir inhomogeneities, resulting in a low oilrecovery.Two conditions must be met for the vaporizing gas

drive process to operate: (1) high pressure must existat the gas-oil interface, and (2) the reservoir oil mustcontain a high concentration of C2C6.Use of nitrogen or ue gas as injection gas seems

to be increasingly applied in secondary or tertiary oilrecovery processes. The phenomena of miscibility be-tween N2 and a hydrocarbon multicomponent systemis complex and only a few papers report result fromhigh pressure N2 gasooding of reservoir uids. Inmost cases, the ooding pressure required to achievemiscibility with lean gas, nitrogen or ue gas exceeds300 bar.

Carbon Dioxide Flooding

The mechanisms of multicontact displacement withcarbon dioxide is similar to the high pressure lean gas

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9.1. DISPLACEMENT MECHANISMS 213

process. The miscible zone between CO2 and oil, likethat of the lean gas process, originates from trans-fer of components from oil to CO2 through multiplecontacts between the leading edge of CO2 and theoil. The greatest dierence between the two processesis that in the lean gas process only C2C6 hydro-carbon components are vaporized from the oil, whileCO2 also can extract heavier components up to C30.21

CO2 has the following characteristics: (1) it promotesswelling, (2) it reduces oil viscosity, (3) it increases oildensities, (4) it is soluble in water, and (5) it can va-porize and extract portions of the oil.The solubility of CO2 in hydrocarbon oil causes the

oil to swell. CO2 expands oil to a greater extent thandoes methane. The swelling depends on the amountof methane in the oil. Because CO2 does not displaceall of the methane when it contacts a reservoir uid,the more methane there is in an oil, the lessis theswelling of the oil. A large reduction of viscosity of acrude oil will also occur as it becomes saturated withCO2 at increasing pressure. The oil density will in-crease when saturated with CO2, but the density ofwater will decrease. As CO2 is injected into the reser-voir, the dierence in the densities of oil and waterwill decrease, which in turn will reduce the chancesfor gravity segregation. The surface tension of oil andwater will also be reduced, resulting in a more eec-tive displacement.When CO2 is injected into a porous medium, it

will rst saturate the crude oil in the front portion ofthe system. Light components (C1-C4) are vaporizedfrom the oil into the gas phase and an equilibriumgas is developed. The injection of additional CO2 willextract heavier hydrocarbons, C5 through C30, whichwill form a transition zone separating the injectedCO2 from the reservoir uid. The multiple contactingof CO2 and oil will lead to miscibility. The distinctadvantage of CO2 as a miscible displacement uid isthat it achieves miscibility at pressures of only 100 to300 bars.

Condensing Gas Drive

The condensing gas drive, also called enriched gasdrive, is similar to vaporizing gas drive. Its pur-pose is to achieve miscibility through multiple con-tacts of enriched gas and oil. When the gas comesin contact with the reservoir oil, C2C6 componentsare extracted from the gas phase into the oil phase.After multiple contacts, the reservoir oil around theinjection well is so highly enriched with C2C6 thatit becomes miscible. Further injection results in gasdisplacing the oil. A suciently large volume of en-riched gas must be injected for the miscible front tobe maintained over a large part of the pattern area.An enriched slug of 10 to 20% of the reservoir porevolume is generally used. This is followed by less ex-pensive lean gas or water. Fig. 9.11 is a schematicrepresentation of the enriched gas process. The con-densing gas drive process requires a lower pressurethan the lean gas process. Typically, a of pressure

Lean gas OilBank

Residual oil

Injected water

from waterflood

Miscible zone formed by gasbecoming enriched with C2-C6

Connate water

Enrichedgas slug

Figure 9.11: Schematic representation of enriched gasprocess.20

150 to 400 bars is needed for this process, dependingon the composition of the injection gas, the reser-voir temperature and the API gravity of the reservoiruid.

First Contact Processes

In shallow reservoirs or reservoirs with low gravityoil, neither of the processes above can be used. If agas driven miscible displacement is desired, a miscibleslug must be created ahead of the gas. The slug mustbe completely miscible with the reservoir uid at itsleading edge and also be completely miscible with theinjected gas at its trailing edge. Slug material maybe propane or liqueed petroleum gas (LPG). In themiscible slug process, the miscibility of the interme-diate zone is created by the injected slug material.If this slug is bypassed or lost for any reason, thereis no way to replace the supply. The volume of slugmaterial to be injected must therefore be sucient tolast for the entire sweep process.

Field Tests

Lean Gas. At least 11 projects using lean gas havebeen reported in the literature. They typically havebeen large-scale oods involving thousands of sur-face acres.2233 Oil gravity was typically larger than40 API. The majority of these projects are consid-ered successful22,23,26,32,33 with high ultimate recov-ery, i.e. in the range of 50% of OOIP. They have beenconducted in highly stratied carbonate and sand-stone reservoirs as well as in less heterogeneous reser-voirs.An important dierence between vaporizing gas

drive eld ooding and the condensing gas drive andrst-contact miscible eld ooding is that for the rstprocess, solvent is being continuously injected. Thisis probably the most important reason for the rela-tive success of vaporizing gas drives. Additionally,the overall viscosity ratio of oil and driving gas hasgenerally been more favorable in vaporizing gas driveprojects than in other hydrocarbon-miscible oods.This is due to the fact that higher API gravity oilsare required to achieve miscibility in a vaporizing gasdrive ooding.

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214 CHAPTER 9. GASFLOODING

Nitrogen Gas. Gasooding using N2 and ue gasare recovery methods which use nonexpensive, nonhy-drocarbon gases to displace oil from deep reservoirs.In the Ryekman eld, that oil recover at reservoirpressure was found to be slightly higher for CO2 thanfor N2.34 This fact, in addition to considerations ofN2 and CO2 costs, compressibility, availability, andcorrosion properties producted the operator to selectN2 as the injection gas.

From thirty N2 gas injection eld projects reportedin the literature,20,34,3646 seven were multicontactmiscible displacement of reservoir oil. These are deepreservoir elds (deeper than 8000 ft) and contain lightoils with API gravity > 41.

Carbon Dioxide. The majority of the CO2 oodswere designed to investigate tertiary recovery e-ciency. Alternate injection of water and CO2 wastried in about half the projects and continuous injec-tion of large CO2 slugs driven by water was tried inthe rest. An equal number of sandstone and carbon-ate formations were tested. Oil gravity generally hasbeen in the range of 30 to 50 API with viscosities lessthan 2 cp (2 mPa·s). Operating problems were gen-erally more severe than in waterooding, with corro-sion problems, leaks and precipitation of a heavy hy-drocarbons. Incremental recovery ranged from 3.5%to 18% OOIP. Several papers concerning practices inCO2 oods have been published.4754

Enriched Gas. The condensing gas drive projectshave predominantly been secondary recovery oods.Oil gravities range from 30 to 50 API, with oilviscosities less than 2 cp (2 mPa·s). The slugsizes have ranged from 2% to more than 50% ofHCPV and were higher than 10% in the major-ity of projects. There has been a considerablerange in the observed/estimated incremental oil re-covery for secondary and tertiary condensing gasdrive projects,33,35,5565 ranging from 3% to 30% ofOOIP.

Slug. The majority of the eld tests were conductedduring the period from 1950 to 1970. Most of thesetests were secondary recovery processes in sandstonesand in reservoirs that are essentially horizontal. Sol-vent slug sizes were primarily in the 1 to 12% HCPVrange. Oil gravities ranged from 30 to 51 API witha majority between 36 and 42 API

There are a few published papers which compareperformance of secondary recovery rst-contact mis-cible ooding projects with anticipated wateroodperformance.21,66,67 The miscible oods were foundto recover from 8 to 35% more oil than primary pro-duction followed by waterooding.

Potential For Miscible Gas Processes in NorthSea Reservoirs

After 35 years of research, there is considerableknowledge concerning the mechanisms of miscibilityand uid ow in reservoir rock. Even so, advancesare needed in a number of important areas, such as:(1) improved understanding of CO2 ooding misci-bility, (2) displacement eciency of gas injection, (3)aspects within the slug processes, both in secondaryand tertiary recovery.The gases used were hydrocarbon gas, carbon diox-

ide, and nitrogen. Hydrocarbon gas is generallyeasiest available. Its composition ranges from puremethane to enriched gas. However, since there is al-ready a market for hydrocarbon gas, its cost/benetas injection gas has to be evaluated.Carbon dioxide is a very suitable injection gas with

many favorable properties. However, its density ex-ceeds that of the oil in some reservoirs, and it is cor-rosive and not always available.When hydrocarbon gas and carbon dioxide not are

available in required quantities, nitrogen is used asan alternative for deep reservoirs containing 35 to 45API gravity oil. In fact, nitrogen has been consideredas the best prospective injection gas for improved oilrecovery in light oil, waterooded and highly perme-able sandstone reservoirs in the North Sea. However,there has been a change in attitude towards the use ofCO2. A growing awareness of the emission of green-house gases, more emphasis is put on the exploitationof point sources of CO2 that could be injected into thereservoirs. Several aspects of utilazing exhaust gasesfor increased oil recovery puposes have also gainedconsiderable interest.As for all recovery methods in which the oil is mis-

cibly displaced by gas, the microscopic displacementeciency is assumed to be very high. The successof a miscible displacement process in a reservoir ismost likely due to an unfavorable viscosity ratios be-tween the reservoir oil and the injected gas. Thus,gas injection should be combined with close controllof the mobility of the reservoir uid. This can beachieved by alternating between gas and water injec-tion, WAG. Polymers and foams can also be used toincrease the volume swept by the gas.

9.1.5 Immiscible Gas Injection

Classication

There are three basic types of gas injection methodsthat have proved useful: dispersed gas injection, cre-stal (attic) gas injection and gasooding.68

1. Dispersed gas injection involves re-injecting gasat a number of locations in a eld and is mostbenecial when applied to gently dipping reser-voirs. It can yield about half as much oil as isproduced by solution gas drive.69

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9.1. DISPLACEMENT MECHANISMS 215

2. Crestal gas injection consists of injecting gas ei-ther into an existing gas cap, or into an oil zone inorder to create a secondary gas cap. It is ideallysuited for high relief reservoirs with high verticalpermeabilities, 200 md or more. It can yield to-tal recoveries from about 50% of oil in place fora reservoir of 15-degree dip to about 70% for areservoir of 25-degree dip. This represents an im-provement of about 10% over dipping reservoirswithout gas injection.

3. Gasooding consists of injecting gas in some pat-tern arrangement of wells to sweep oil toward theproducing wells.

Evaluation Methods

Schiltus70 proposed the material balance method.This is essentially a zero dimensional (tank) model.The method relates volumes to pressures. Later,Buckley and Leverett71 proposed a one dimensionalmodel for frontal displacement. The model is de-scribed in Sec. 4. Terwilliger et al.72 investigatedexperimentally the eect of gravity and comparedthe results with the predictions from the Buckley-Leverett theory. In the case of gravity drainage, theyfound that the recovery depends on the productionrate. A low rate is more ecient than a high rate.The Buckley-Leverett theory predicts microscopic

displacement eciency only. The overall displace-ment eciency may be found as a product of themicroscopic, horizontal and vertical displacement ef-ciencies.Dietz73 proposed a two-dimensional, analytical

model for displacement of oil by gas in a reservoirwith dip. The gas-oil contact may become unstable.Under such conditions, gas will override the oil caus-ing an early breakthrough in the downdip producingwells. Dietz derived a criterion for stable displace-ment of oil by gas. The criterion depends on the dipangle and the injection rate. The theory is discussedin Sec. 4.1.10.The above theories assume that there is no trans-

fer of mass between the liquid and gas phase duringthe displacement. In such a case, there is no dif-ference between water-oil displacements and gas-oildisplacements, except for an unfavorable mobility ra-tio in the latter case. The Buckley-Leverett theoryhas been extended to gas injection into an undersat-urated reservoir,74 see also Sec. 4. Then, part of thegas will be dissolved in the reservoir liquid. The re-covery could increase substantially as a consequenceof liquid swelling, improved viscosity ratio etc. Theeect of injecting a high pressure gas75 (condensing,vaporizing or combination drive) can be analyzed bya graphical technique similar to the one introducedby Welge.76

Pressure Maintenance

The objective of conducting a gas injection projectdepends on the type of reservoir:

Gas-Condensate Reservoir. In a gas-condensatereservoir, the liquid content is a valuable part of thehydrocarbons in place. As a result of pressure de-cline, a large fraction of this liquid may condense inthe reservoir. The impact of retrograde condensationcan be made less severe if some dry gas is re-injected.Ideally, the pressure should be kept constant and noliquid should accumulate in the reservoir. In addition,the injected gas will help expel the reservoir gas. Anecient miscible displacement process may be the re-sult. The injected gas can be recovered at a laterstage. Hence, the strategy is to send the oil to themarket rst and the gas afterwards.

Undersaturated Reservoir. If an undersatu-rated reservoir is produced by expansion of rock andliquids, then a gas phase would evolve as a conse-quence of the pressure decline after the bubblepointhas been passed. The free gas could severely decreasethe liquid transport capabilities of the rock. Hence,it is desirable to keep the reservoir pressure above thebubblepoint as long as possible. The reservoir pres-sure could be kept at a constant level by injectingwater and/or gas. In the case of gas injection it isdesired that the injected gas, which is not in equilib-rium with the reservoir liquid, will diuse into liquidbody. Then, reservoir liquid would swell as gas goesinto solution.77 Some liquid would be produced asconsequence of liquid swelling. The diusion mecha-nism is probably too slow for the swelling process tobe signicant.

Saturated Reservoir. In an oil reservoir with alarge gas cap and an active aquifer, the gas cap maybe produced concurrently with the oil zone. In sucha case the aquifer would push the oil zone into theshrinking gas zone. As a consequence, unrecoverableoil will not only occur in the oil zone, but also inthe portion of gas cap which is invaded by oil. Onestrategy to avoid this is to inject gas in the gas zoneto maintain the pressure, while the oil is produced.

North Sea Considerations

In North Sea reservoirs, gas injection is carried outeither for the purpose of maintaining pressure, or forstorage of the gas for future use.Most of the oil reservoirs of the North Sea contain

light oils, and have relatively high pressure and tem-perature. The mobility contrast between the oil andthe injected gas is therefore small. In many cases hy-drocarbon gas and oil will exhibit miscible or close tomiscible conditions.The reservoirs of the North Sea are generally het-

erogeneous with layers of high-permeable sands. Thepermeability is often very high, so that gravity over-ride of the gas would be a problem. Therefore, gravityassisted gas injection seems to be the method whichis most promising for North Sea conditions. Residualoil saturations as low as 10% have been reported in

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216 CHAPTER 9. GASFLOODING

gravity-stable immiscible core oods with cores fromthe Oseberg eld.4 Previously in this chapter, a dis-cussion on the factors aecting local displacement ef-ciency is given. In principle these factors are thesame for miscible and immiscible ooding.As an alternative to hydrocarbon gas for injection

there is a considerable interest in nitrogen. Becauseof the high pressure required for nitrogen to developmiscibility, it may be used in highly permeable reser-voirs for gravity-stable displacement.Bath1 describes the conditions for gravity-stable

conditions. For good recovery, high oil mobility andhigh vertical permeability in addition to a dippingreservoir are required.

9.2 Flow of Gas Condensate

9.2.1 Introduction

The aim of dry gas recycling and water injection isto maintain the pressure throughout the reservoir ator above the dewpoint pressure in order to preventliquid drop-out in the reservoir and, if possible, alsoin the near-well region. Liquid dropout will causereduced productivity of the production wells and lossof the heaviest components of the reservoir uid.Dry gas recycling is regarded as a miscible process

and consequently it should be possible to remove allthe wet gas from pores contacted by the advancingdry gas. For such displacements, the rock relativepermeability curves are linear and the end points forwet and dry gas are equal (krgw = krgd = 1). Inaddition, it is of great importance that the shape ofthe reservoir suppresses override or bypass.The endpoint mobility ratio for a water-gas dis-

placement is usually much less than unity, desity dif-ference between the uids is large. The water dis-placement of gas is therefore an extremely stable pro-cess. Unfortunately, water traps a signicant amountof gas. Because the residual gas saturation is believedto be independent of the pressure, the surface volumeof trapped gas is directly proportional to the pressure.Once the reservoir has been ooded with water, ad-

ditional recovery of trapped gas and condensate canbe achieved by pressure blowdown. The reductionin pressure allows the trapped gas to expand, coa-lesce and migrate to the crest of the structure whereit can be produced. The blowdown of a waterdrivereservoir is a process which is hard to control and dif-cult to describe quantitatively. Below the dew pointpressure, three phases (water, gas and condensate)coexist. Three-phase relative permeability functionsare therefore reuired.

9.2.2 Fluid Properties

A proper uid analysis for the two processes constantcomposition expansion (CCE) and constant volumedepletion (CVD), should include the measurements

of density, viscosity, and interfacial tension as func-tions of pressure and composition. Interfacial ten-sions should preferably be measured by laser-lightsurface scattering.78 The liquid viscosities can bemeasured by a Ruska Rolling Ball viscosimeter, whilegas viscosity can be calculated from correlations.79

Densities can be determined by a high-pressure An-ton Paar densitometer.Fig. 9.12 schematically shows the results from a

25

20

15

10

5

00 10 20 30 40 50

Revaporization

Retrogradconden-sation

Liq

uid

drop

out

volu

me,

%

Pressure, MPa

pd

Figure 9.12: Liquid dropout of gas condensate duringconstant volume depletion (CDV).

CDV experiment of a moderately rich gas-condensateuid system. At the dewpoint pressure, pd, liquidstarts dropping out of the uid. This condensationwith decreasing pressure is called retrograde conden-sation. At a certain pressure, a maximum amountof liquid is obtained. Further depletion causes revap-orization of condensed liquid. But, as can be seenfrom Fig. 9.12, this process is less pressure depen-dent. Thus, if condensation is permitted within themain part of the reservoir, most of this liquid will belost.

9.2.3 Relative Permeability

Only a few cases are reported in the literature of mea-surements of relative permeability or ow of gas con-densate in the retrograde area.Relative permeability of gas-condensate systems is

more complex than for ordinary gas/oil systems sincemass transport from one phase to another easily takesplace. In addition to viscous ow along the directionof the pressure gradient, segregation of condensedliquids may play an important role. Therefore, themechanisms of vertical ow are of great importance,specically in highly permeable, waterwet reservoirsat connate water saturation.Relative permeability to oil depends on the capil-

lary forces when the value of interfacial tension, σ,is small. This is specically the case in near-criticaluid systems such as rich gas condensates and volatileoils as well as in miscible displacement.79

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9.2. FLOW OF GAS CONDENSATE 217

Bardon and Longeron79 investigated the eect ofvery low interfacial tension on relative permeabil-ity. They used binary hydrocarbon mixtures withliquid and vapor phases in equilibrium at experimen-tal conditions. The level of interfacial tension wascontrolled by varying the equilibrium pressure andthus the composition of the mixtures. In each case,the two phases were separated after phase equilibriumhad been achieved. The porous medium was satu-rated with the liquid phase which then was displacedby the vapor phase with a very small pressure dropacross the core to maintain uid equilibrium. Theexperimental velocities were suciently low to securelaminar ow, so that the Darcy equation could beapplied.The relative permeabilities were calculated by the

graphical technique of Jones et al.,80 and shapes ofthe curve varied considerably depending on the phys-ical properties of the phases. Numerical simulationwas applied since only one or two experimental pointscould be measured close to the critical point.A classication made with respect to σ showed that

the major change in relative permeabilities seemed tooccur in the vicinity of σ = 0.05 mN/m. Above thisvalue, the relative permeability curves were similar tothose for ordinary gas/oil systems. The curves to beused in numerical simulation can therefore be handledwith reasonable accuracy by traditional techniques.Fig. 9.13 shows the gas-condenstae relative perme-

ability curves for the condensed liquid and its corre-

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Condenesate saturation

Rel

ativ

epe

rmea

bilit

y

kr at low pkr at p max. liq.kr at pd

Figure 9.13: Relative permeability for condensate andequilibrium gas at dierent pressure.

sponding equilibrium gas, at three dierent pressures:(1) dewpoint pressure, pd; (2) pressure of maximumliquid dropout; (3) at low pressure far from the ret-rograde region.For very low values of interfacial tension, especially

close to the critical point, prediction is dicult sincethere are large changes in relative permeabilities forrelatively small changes in interfacial tension. Forσ < 0.001 mN/m, the relative permeabilities can berepresented by straight lines with unity slope. Note

that the above values of interfacial tension are calcu-lated and not measured.Gravier et al.81 determined the gas-condensate rel-

ative permeability on whole cores at reservoir con-ditions. Data were obtained for eight rock samplesfrom an Abu Dhabi reservoir (carbonate). The resultsshowed that reduction of permeability to gas with in-creasing condensate saturation was more pronouncedthan for standard gas-oil systems. These observa-tions agree with the drastic drop in well productivityobserved during production of gas-condensate elds.The experiments were not conducted at near-criticalconditions and are therefore not necessarily contra-dictory to the results above.Critical condensate saturations for ow were mea-

sured for each core sample at reservoir conditions.It was impossible to correlate the critical conden-sate saturation with petrophysical rock characteris-tics. The critical saturation was found to be high inall experiments, with an average of 35%. The experi-ments were performed in cores mounted horizontally.This fact can easily explain the observed results.Asar and Handy82 performed measurements of the

inuence of interfacial tension on gas-liquid relativepermeability in a gas-condensate system. They foundthat the curves of the individual relative permeabil-ities (krg and krl) vs. gas saturation, approachedstraight lines as IFT approached zero. The rela-tive liquid permeability decreased more rapidly thanthe relative permeability to gas with increasing IFT.Residual gas and liquid saturations were higher withhigher IFT. The gas saturation at which the gas andliquid relative permeability curves intersected, washigher as the IFT decreased, indicating a decreasein the oilwet character of the system. The level ofkrl and krg at which these two curves intersected washigher for lower IFT values.Saturation history eects at intermediate IFT's

were investigated and appeared not to be signicant.Relative permeability results obtained at the higherIFT level (σ = 0.83 mN/m) approached results ob-tained for a nitrogen/kerosene ood, even though thenitrogen/kerosene IFT was much higher. The rela-tive gas and liquid permeabilities for gas-condensatereservoirs appear to correspond to those for normalgas/oil systems except near-critical conditions.Saeidi and Handy83 reported an investigation of

ow and phase behavior of gas condensate andvolatile oils in porous media. Gas and liquid per-meabilities were measured at high pressure. Sim-ilar measurements were made on bubblepoint sys-tems of dierent volatility close to atmospheric pres-sure. Comparison between these two systems indi-cated that gas-condensate relative permeability val-ues were lower in the region of low to intermediate liq-uid saturation than for the bubblepoint systems. Fur-thermore, gas-condensate relative permeability ratiosindicated a lower critical liquid saturation for owthan that of the bubblepoint systems.Laboratory phase behavior studies on hydrocarbon

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218 CHAPTER 9. GASFLOODING

binary mixtures owing through a porous mediumwere performed to test the validity of PVT relation-ships obtained from static, steady-state PVT cells inprediction of condensate reservoir uid behavior un-der actual ow conditions. The results indicated thatat lower laboratory pressure-depletion rates, althoughmuch greater than eld rates, equilibrium was main-tained during transient ow. At higher pressure de-pletion rates, nonequilibrium behavior was observed,particularly during the revaporization period.Dumoré and Schols17 pointed out that the resid-

ual oil saturation of very permeable sandstones inpresence of connate water, could be extremely low,4% in gravity stable gas injection processes. Theyalso showed that low oil saturation could be obtainedin gravity-drainage experiments after long drainagetimes in sand columns. These low values of residualoil saturation obtained by lm ow were independentof whether or not the oil phase spreads on water inpresence of gas.The superiority of the gravity-drainage process was

clearly recognized by King and Stiles84 in their eldstudy of the East Texas Hawkins reservoir, with 87%recovery. More recently, Hagoort85 observed that theliquid relative permeability was the key factor in thegravity-drainage process.Delclaud et al.18 have presented a series of

unsteady-state gas displacement experiments in ho-mogeneous and fairly permeable core samples. Thedisplacements were performed with an analysis oftransient phenomena by measurement of pressure andsaturation (x-ray attenuation) along the entire lengthof the core. From these local measurements, it waspointed out that the capillary pressure-saturation re-lationship, recorded during gasooding, was identicalto the one determined by the restored-state method.They also investigated the gas-oil relative perme-

abilities for a high permeability oil reservoir appli-cation by unsteady-state displacement tests. Thegas-oil relative permeabilities were unchanged whenthe interfacial tension between the phases varied inthe range 0.6 to 30 mN/m. For the type of porousmedium used (very permeable sandstone), very lowresidual oil saturations were found, 3.5 to 6%, corre-sponding to kro = 10−5, which was a practical lowerlimit for measurable ow.Several papers8690 presented results from mea-

surements of relative permeability for a gas-condensate system by two techniques (1) pressure de-pletion, and (2) dynamic displacement of retrogradeliquid at dierent pressures.The measurements were performed on a six-

component model system exhibiting typical behaviorof condensates. The designed system had a maximumliquid dropout of 30.1% at 18.2 MPa and 41C, de-termined by constant-volume depletion (CVD). Theinterfacial tension was measured by laser-light surfacescattering at dierent pressures.78

The relative permeability of the gas-condensatesystem was studied by pressure depletion through-

out the retrograde region and displacement of con-densed liquid by injection of equilibrium gas at dif-ferent pressure in the retrograde region. None of theseare standard experiments, and the equipment and ex-perimental procedures had to be specically designedand developed.In the relative permeability experiments, eects of

gravitational segregation were observed. The numer-ical simulations clearly showed that eects of gravitywere important on the recovery of liquids.Pressure depletion resulted in losses of the heaviest

components during both horizontal and vertical de-pletion. Due to gravitational segregation, the losseswere less for vertical than horizontal experiments.When immobile water was present, the recovery ofliquid components was higher than in absence of wa-ter. This was due to a smaller critical hydrocarbonliquid saturation to ow. The relative permeabilitycurves of the depletion experiments were similar tocurves of ordinary gas-oil systems. This implies thatstraight-line relative permeability curves should notbe used in evaluation of production by pressure de-pletion of gas-condensate systems similar to those inthis work.The dynamic displacement experiments showed

that relative permeability strongly depends on in-terfacial tension between gas and condensed liquid.The relative permeabilities were signicantly changedfrom almost straight lines at low IFT (σ < 0.1mN/m), to low-valued, highly curved permeabili-ties at higher interfacial tensions (σ > 0.1 mN/m).Therefore, eects of interfacial tension on relative per-meability must be taken into consideration in evalu-ation of gas-condensate reservoirs.

9.3 Tertiary Gasooding

9.3.1 Introduction

Flowing water is physically incapable of displacingall the original oil from a reservoir rock because capil-lary and surface forces acting in the uid-rock systemcounteract the pressure gradient caused by the water.A residual oil saturation, Sorw, which usually is in therange 20 to 40% of the pore space, will therefore beretained.Oil may also be present in large areas of water-

ooded reservoirs at saturations well above Sorw.This is due to macroscopic processes such as nonuni-form ow and sweep in parts of the reservoir less ac-cessible for water.Even after a successful waterood, large quantities

of oil will therefore remain in the reservoir, often morethan 50% of the original oil in place. This residual oilis the target for tertiary gasooding.Gas injection may recover additional oil because

lower residual oil saturations are usually obtainedwhen gas is introduced into a waterooded porousmedium. Injected gas may also ow in other path-ways and thus displace oil not already contacted by

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9.3. TERTIARY GASFLOODING 219

water. A tertiary gasood therefore recovers bothsecondary and tertiary oil, and in practice it is notpossible to distinguish between these two categories.The displacement mechanisms and physical condi-tions involved in the recovery of these two types ofoil will to a large extent be dierent. The discus-sion below will only be related to the recovery of oilretained in a porous medium swept by water.Fig. 9.14 illustrates idealized saturation proles

Inje

ctio

nga

s

Water saturation

Gas saturation

Oil saturationSorw

Sorg

Siw

0 1

Phas

esa

tura

tion

Dimensionless distance from injection point

Figure 9.14: Idealized saturation proles during a sta-bilized tertiary gas-ooding process.

during a stabilized (e.g. by gravity) tertiary gas-ooding process. When gas is injected, residual oilis mobilized either by a miscible or immiscible mech-anism. An oil bank is formed by the mobilized oil,displacing a considerable fraction of the mobile waterin front of it. Behind the oil bank, tails in the waterand oil saturations are seen. The water saturationapproaches the irreducible saturation, Siw, whereasthe oil saturation actually may be reduced below theresidual oil saturation after gasooding, Sorg, due tomass transfer of oil components into the owing gasphase.The shape of the oil saturation prole behind the

oil bank is therefore due to the combined eect of af-terow of oil and mass transfer into the gas. Threephases are owing in this region, for an immisciblesystem, and the three-phase relative permeability ofoil is decisive for the formation of the oil bank. Thepresence of a high water saturation may inuenceboth the ow of oil and the interphase mass trans-fer.

9.3.2 Some Characteristics of TertiaryOil Recovery

Stagnant Oil

The presence of mobile water may have a signicantinuence on the recovery of tertiary oil. Sometimeswater is also injected together with gas, giving higherwater saturations behind the oil bank than illustratedin Fig. 9.14. Tertiary gasooding and tertiary WAGinjection may thus have common features related tothe presence of mobile water.Analyses of experimental data indicate the pres-

ence of three oil-phase fractions during ooding pro-cesses at high water saturations; owing, dendriticand trapped oil.91 The owing fraction is directlymobilized oil by gas injection, whereas the dendritic

or lagging oil is assumed to be held in dead end ordendritic structures, which do not lie in the paths ofoil ow, but are nevertheless in contact with the ow-ing gas phase. The dendritic oil fraction can thereforedirectly exchange mass with the owing gas. Thetrapped or isolated oil fraction is shielded from thegas by water barriers, which may reduce the rate ofmass transfer.92 Both the dendritic and isolated frac-tions contribute to the stagnant oil phase saturation.In practise, it may be very dicult to distinguish be-tween dendritic and isolated oil, although the latterby many is considered to be the major part of thestagnant oil phase9195

The amount of trapped oil correlates well with thedierence in oil saturations between drainage and im-bibition for a given value of the oil relative perme-ability.9193 This is illustrated by the quantity a inFig. 9.15a, which shows typical oil relative perme-ability curves for primary drainage and imbibition ina strong waterwet porous medium. The total satura-tion of oil at a water saturation, Sw, is given by thequantity b. The fraction of oil phase trapped is givenby the ratio a/b, and is shown a function of watersaturation in Fig. 9.15b. Fig. 9.15c shows drainageand imbibition oil relative permeability curves for aoilwet porous medium showing no hysteresis eect,and consequently no oil should be trapped by mobilewater as illustrated in Fig. 9.15d.The oil trapping functions of Fig. 9.15 is consistent

with the experimental results,93 which always gaveclose to 100% recovery for miscible displacement ofthe wetting phase. The presence of large quantities ofnonwetting phase only delayed the recovery. The re-covery of nonwetting phase by miscible displacementwas, however, greatly reduced by a owing saturationof the wetting phase.Other experimental data conrm this interpreta-

tion,9498 although some trapping of oil has been de-ned at high water saturations also in oilwet porousmedia.98 For systems with mixed wettability, hys-teresis in both water and oil relative permeabilitiesmay occur. Some oil phase trapping may thus beexpected, but less than in waterwet porous media.The stagnant oil fraction may also be recovered

during a tertiary gas ood. Mass will be transferredbetween oil and owing gas by molecular diusion.This may cause oil swelling followed by mobilization,and immobilized oil may be recovered by evaporationinto the owing gas. Because diusion is time depen-dent, short experimental times in laboratory experi-ments may give very misleading results for the nalmicroscopic displacement eciency obtainable by gasinjection.92,94,95,99

The recovery of trapped oil by diusion of injectedgas (CO2) through the water barrier, and the subse-quent swelling of the oil phase, has been illustrated byblindpore micromodel experiments.99 As the trappedoil swells, the water barrier is displaced into the ow-ing gas. This causes the barrier to thin until it breaksat a critical waterlm thickness. The isolated oil

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220 CHAPTER 9. GASFLOODING

Siw0 100%

Oil

rela

tive

perm

eabi

lity

a

b

Water saturation

(a)

0

Frac

tion

ofoi

lph

ase

trap

ped

Siw 100%

Water saturation

(b)

0

Oil

rela

tive

perm

eabi

lity

Siw 100%

Water saturation

(c)

0Fr

actio

nof

oil

phas

etr

appe

dSiw 100%

Water saturation

(d)

Figure 9.15: Typical relative permeability curves for oil in waterwet (a) and oilwet (c) porous media, andcorresponding oil trapping functions (b and d).

phase can then be mobilized by owing gas.Various approaches to model recovery of stagnant

oil have been suggested. One is to allow only thedendritic fraction to exchange mass with the owingphase.100 A more general model allows trapped oilrecovery both by swelling and extraction.92 The rela-tive importance of swelling and extraction will dependon type of gas injected, the oil in question, pressure,and temperature.

Miscible - Immiscible Displacement

In miscible tertiary injection processes, all oil presentin the ow channels of the gas can be recovered. Theonly residual oil is found in dendritic structures andparts of the pore space isolated from the gas. Most ofthis oil can be recovered by the mechanisms discussed.For an immiscible process, the oil present in the

ow channels of the gas is generally not recovered byconvection. Some can be recovered by mass transferinto the owing gas phase, similar to the behaviorof dendritic and isolated oil. The amount of recov-erable oil depends on the ability of the gas to ex-tract oil components, and the quantity of gas injected.The ow behavior during immiscible displacementmust be described by permeability functions. Sincethree phases are owing simultaneously behind the oilbank, Fig. 9.14, three-phase relative permeabilities of

oil are needed.

Diusion Process

Fig. 9.16 illustrates eld-scale gas injection, showing

Water

Oilfilm

Sand

Gas

Low permeability zoneGas

Oil

Figure 9.16: Stagnant oil in a waterooded reservoirboth on microscopic and macroscopic scale.

zones unswept by gas on microscopic and macroscopicscale. The macroscopic stagnant oil may be recoveredin diusion times of 104 to 105 s (CO2 injection).99

Laboratory and eld-scale diusion length and times

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9.3. TERTIARY GASFLOODING 221

are related by

tfield = tlab

(xfield

xlab

)2

. (9.5)

Diusion into the low-permeability zone of 1 m lengthshould therefore require diusion times of 108 to 109 s,or 3 to 30 years. Water blocking of oil on length scalesof less than 0.5 m may therefore be unimportant foreld CO2 injection.92 The presence of water nev-ertheless reduces diusional uxes and lengths, andconsequently the amount of oil recoverable by suchmechanisms. For less water-soluble gases, diusionaluxes will be drastically reduced by the presence ofwater.

9.3.3 Three-Phase Relative Perme-ability of Oil

The importance of the three-phase relative permeabil-ity of oil on recovery during tertiary gas injection pro-cesses has been demonstrated both through presentedreservoir simulations and modelling of laboratory ex-periments.6,9, 101 The simulation studies presentedby Roberts and Peacock101 used two types of corre-lations given by Stone102,103 to calculate three-phaseoil relative permeabilities. Use of one correlation in-dicated that a protable tertiary recovery was withinreach. Use of the other excluded tertiary gasoodingbecause much less oil was mobilized.Standard correlations for reservoir core samples

may result in three-phase oil relative permeabilitiestoo pessimistic or too optimistic compared with mea-sured values.6,9 At present, no general rules existfor how to construct correct three-phase curves fromtwo-phase relative permeability data.The shape of the oil relative permeability curves re-

ect the mechanism of oil ow in three-phase mode.The very low residual oil saturations that have beenobtained during laboratory experiments of tertiarygas injection, as well as secondary injections withconnate water, have been attributed to lm ow ofoil under the inuence of gravity in waterwet porousmedia.8,17,104109 Experimental results indicate thatwhen the conditions required for oil to spread onthe water are satised, trapped oil blobs, after con-tact with gas, spontaneously spread to form a lm.This gives conditions for very ecient oil recovery. Ifenough oil is present, isolated oil blobs may reconnectby lm ow and form a continuous oil phase which canevolve into an advancing oil bank. If the advancinggas leaves small zones locally unswept, oil will be by-passed and become isolated behind the displacementfront. However, due to the spreading characteristicsof the system, water lms that separate oil from thegas may break and the oil will be connected to thecontinuous oil lm covering the water surface. By-passed oil can then ow towards the production endof the system. Mechanism of reconnection of discon-tinuous oil, oil bank formation, and after drainage by

lm ow are described in detail for ow in micromod-els and capillary tubes.104,106,108

The conditions for oil to spread on water is a posi-tive spreading coecient, ςow, given by

ςow = σwg − σog − σwo, (9.6)

where σwg, σog and σwo are the interfacial tension be-tween water and gas, oil and gas and water and oil,respectively, at conditions of thermodynamic equilib-rium.110 A positive spreading coecient indicatesthat less energy is associated with the oil phase beingspread as a continuous lm between water and gas,than having a water-gas interface and an oil drop.Examples of oil lm formation are frequently seen ascolor patterns when oil spill spreads on a wet street,or on the sea.Similarly, a water spreading (on oil) coecient may

be dened by:

ςwo = σog − σwg − σwo. (9.7)

A negative spreading coecient for water indicatesthat a water lm separating oil and gas is unstable.Typical values for the interfacial tensions σwg, σogand σwo are 65 (25), 20 (1), and 40 (15) mN/m re-spectively, where the gures within parentheses arevalid for reservoir conditions, and the other gures aretypical for low pressure laboratory experiments withrened oil. By Eqs. 9.6 and 9.7, an oil spreading co-ecients, ςow, of 5 (9) mN/m, and a water spreadingcoecients, ςwo, of -85 (-39) mN/m may be calcu-lated. High curvature surfaces may modify Eqs. 9.6and 9.7 somewhat. The values nevertheless indicatethat favorable spreading characteristics can usuallybe expected at tertiary gasooding conditions.Waterwet porous media and spreading oil systems

seem to give conditions for high oil recovery,104,106,109

although some examples of good recovery are also re-ported in nonspreading oil systems.17

It is generally agreed that waterwet porous me-dia with spreading oil oer conditions for ecientoil recovery by the lm-ow mechanism, and thatnonspreading oil systems are unfavorable for oil re-covery.104,109 The eect of oilwet solid surfaces forspreading oil systems is, however, not clear. As thewetting characteristics of oil reservoirs often are dif-ferent from waterwet, the eect of rock wettabilityon tertiary oil recovery from real systems should beinvestigated further.Besides the possibility of lm ow of oil in water-

wet porous media, the existence of connate water ina waterwet porous medium improves the relative per-meability to oil, by excluding oil from the smallestpores. The higher the connate water saturation, themore streamlined the pore structure for oil ow willbe. In such streamlined structures, the resistance tooil ow is greatly reduced.85,111

As rock wettability and spreading characteristicsseem to be important for oil recovery, these factorsshould also inuence the position of oil isoperms in aternary diagram. From the discussions above, it could

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222 CHAPTER 9. GASFLOODING

be expected that a system at, or close to, irreduciblewater saturation should be optimal for oil recovery.Experimental results indicate, however, that a highowing water saturation is not necessarily detrimen-tal to ecient oil recovery, and the process path seemsto be important for three-phase relative permeabili-ties.112 Various methods for measurements of threephase permeabilities are discussed by Oak.112 Deter-mination of oil relative permeabilities from tertiarygasooding experiments, using model assumptions, isdemonstrated by Foulser et al.105

A clear understanding of the factors inuencing thepermeability of oil in tertiary gas injection mode isnot yet established. Film ow of oil under the inu-ence of gravity seems to be a key factor for low resid-ual oil saturation in laboratory experiments. Mea-surements of three-phase oil relative permeabilitiesshould therefore be made at conditions reecting thesame type of ow mechanisms that will prevail in theoil reservoir.

9.3.4 Laboratory Experiments

Very low residual oil saturations have frequently beenobtained by immiscible gas displacement of modeloils under the inuence of gravity. Some examples ofgravity-drainage experiments in tertiary mode, or sec-ondary mode at connate water saturations, are givenin Table 9.1, which also gives some results from othertype of experiments. The centrifuge experiments85

showed better oil recoveries for increasing connatewater saturations. A good match between oil rela-tive permeability values obtained by centrifugationand by ooding may be obtained.111 Low oil satura-tions are easily obtained only by the former method.The reason for this may be the extremely low owrateof oil under inuence of moderate forces, as indicatedin Table 9.1 for the drainage experiments. The tableshows that low residual oil saturations are also ob-tainable during measurements of three-phase relativepermeabilities by the steady-state method, and some-times at relatively high water saturations.112 Low oilsaturations are seen for glass beads, sand packs, aswell as consolidated materials. Most of the exper-iments refer to (strongly) waterwet conditions, andspreading oil.Miscible gasooding at tertiary conditions can give

very high displacement eciencies if sucient contacttime between owing gas and stationary oil is allowed.This is illustrated by Grogan and Pinczewski99 forboth oilwet and waterwet sandstones, and carbonatecores, where the residual oil saturations approachedzero when the contact time during CO2 ooding in-creased.Table 9.2 summarizes some results from immisci-

ble, high-pressure tertiary gas injection experimentsin vertical cores using reservoir oils. The multicon-tact miscibility experiment is added for comparison.Hydrocarbon gases were injected in all experimentsbut one (N2), and in three of the experiments, equi-librium gases were used.

As seen from Table 9.2, the miscible tertiary dis-placement was very ecient. The tertiary N2 injec-tion gave a 13% residual oil saturation. This wasconsidered pessimistic for a reservoir ood becauseof experimental artifacts. There was, however, al-most no afterdrainage of oil as degassing made theoil rather viscous. Injection of equilibrium HC gasinto reservoir sandstone, both at high and low in-jection rate at irreducible water saturation (25.2%),gave notably better nal recoveries than tertiary in-jection into Bentheimer sandstone at an intermediaterate. At 1.7 PV injected, however, the oil saturationswere approximately 12% and 20% for the low andhigh rate ows, respectively. Some oil was still pro-duced at the end of the equilibrium gasood in theBentheimer rock which was terminated after 1.7 PVgas injected. Dry gas injection into Bentheimer sand-stone demonstrated the importance of mass transferof stationary oil into the owing gas phase. After1.7 PV dry gas injected into the 122 cm Bentheimercore, the oil saturation was 13%. The water satu-ration in the short Bentheimer core was more than50% higher than in the long core due to end eects.This apparently caused no negative inuence on theoil recovery. Slow afterdrainage of oil following the oilbank was demonstrated in all the experiments withequilibrium gas.

9.3.5 Field Experience

A large number of tertiary gas injection projectshave been realized, both as pilot projects and full-eld oods. The earliest projects were slug injec-tions of liqueed petroleum gas followed by cheapergas and/or water. A selection of some of theseprojects demonstrated increase in oil production of11% OOIP,11 on an average.Gravity-stabilized ue gas injection, and immisci-

ble CO2 injection, have proved to be very eectivefor water ooded parts of the Hawkens Field Unitand the Weeks Island Field.116,117 Residual oil satu-rations of 35% and 22% after water ooding were re-duced to respectively 12% and 2%, by gas injection.An average recovery of 12% of OOIP was obtainedfor nine tertiary CO2 projects.50,97,118123 The CO2

projects varied from small pilots to full-scale injec-tion. The indicated improved recoveries were eitherbased on postood evaluations, or estimates usuallyupdated during the production period. Many ter-tiary recovery projects combine gas injection withwater injection. This is done either to improve thesweep eciency, or to reduce the need for injectiongas. An average recovery of 14% OOIP was obtainedfor the seven CO2 projects.97,116,124128 The tertiaryWAG projects were generally made for reservoirs withthicker oil zones and larger well distances.The experience gained through this limited selec-

tion of eld projects indicate that tertiary gas injec-tion can recover substantial amounts of wateroodresidual oil. Gravity-assisted injection has proved tobe very eective in dipping reservoirs. Pure gas in-

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9.3. TERTIARY GASFLOODING 223

Table 9.1: Examples of laboratory oil recoveries from immiscible gas displacements of model oils

Porous medium Wettability Sorg [%] CommentsBentheimer sandst.17 water 3 Capillary pressure experiment170 darcy sand17 water ' 9 329 hr drainage time

So ' 1% in upper parts170 darcy sand17 water ' 12 576 hr, not oil-spreading system

So ' 4% in upper partsSandstone and dolomite110 water 512 Centrifuge drainageSand pack107 water 0.62 Low-rate drainageSand pack107 water 39 High-rate drainageBerea sandstone107 water ' 825 Prod. time 10002500 hrChlashach sandst. water 1829 1731% connate waterChlashach sandst.8 water 8 2000 hr drainage time

So ' 4% in upper partBerea sandstone111 water < 10 Three-phase rel. perm. measurem.Sand pack112 water 24 Three-phase rel. perm. measurem.Berea sandstone109 water 12 1330 hr drainage timeBerea sandstone109 oil 14 End eects eliminatedGlass beads109 water 2 Oil-spreading systemGlass beads109 water 17 Nonspreading oil systemGlass beads109 oil 13 Oil-spreading system

Table 9.2: Results from high-pressure gas injection experiments in vertical sandstone cores

Porous Core Wetta- Injection Duration Residual Inj.gasmedium length bility velocity [days] oil sat.

[cm] [cm/d] [%]Reservoir9 41+100 water 8-30 6 13.0 N2

Reservoir113 165 water 4.8 225 5.5 Equil.HCReservoir113 154 water 50.0 16 7.4 Equil. HCBoise114 126 weakly w. 15.0 7 0.7 Misc. HCBentheimer6 123 weakly w. 23.0 9 19.0 Equil. HCBentheimer6 122 weakly w. 23.0 17 10.0 HCBentheimer115 60 weakly w. 14.0 7 12.0 HC

jection is eective in thin, horizontal reservoirs withshort well distances, whereas WAG seems to improverecovery for larger well distances and thicker oil zones.

9.3.6 Tertiary Gas Injection in NorthSea Oil Reservoirs

Many North Sea oil reservoirs are today being wa-terooded, and large eorts are often made to obtainhigh volumetric sweep eciency. Although very ef-fective oil recovery often results, most of the OOIPwill be left in the reservoirs at the end of the wa-teroods. The potential of tertiary gravity injectionis therefore large, both through improved microscopicdisplacement eciency, and possibly by better sweep.In addition comes the recovery of attic oil, oil situatedin the upper regions of dipping reservoirs, above theuppermost perforation of the production wells. Theability of tertiary gas injection to mobilize residualoil has already been demonstrated in one North Seaoil reservoir.129

A few aspects related to tertiary gas injection inNorth Sea oil reservoirs will be discussed below, basedon an example of a gravity stabilized injection into adipping reservoir, Fig. 9.17, with characteristics given

α

Gas

Fv

Fg

Oil

Gas-oil contact

Figure 9.17: Schematic illustration of a gravity-assisted gas injection process.

in Table 9.3. (A detailed description of this type ofprocesses is given by Ypma2).The critical rate for gravity-stable gas injection is

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224 CHAPTER 9. GASFLOODING

Table 9.3: Reservoir and uid characteristics for hypothetical, tertiary, gas-injection process.

Reservoir permeability : 9.87 · 10−13 m2 (1 darcy)Reservoir dip angle : 10

Interwell distance : 1000 mOil zone thickness : 50 mOil and gas rel.perm. : 0.5 and 0.5Oil and gas viscosity : 3 · 10−4 Pa·s (0.3 cp) and 3 · 10−5 Pa·sDensity dierence oil/gas : 400 kg/m3

given by73

(q/A)crit = ucrit =k (ρo − ρg) g sinα(

µokro− µgkrg

) . (9.8)

Use of Eq. 9.8 gives a critical Darcy velocity of 0.1m/d or a front velocity of 0.4 m/d (porosity 25%).The water saturation is reduced from 65% to 25%

in front of the oil bank. The indicated injection veloc-ity corresponds to 2.7 years of water production be-fore the oil bank arrives, if the production wells canhandle the large amount of water produced (30 m3

water/day per meter interwell distance between wa-ter producers). For a real reservoir, heterogeneitiesare likely to reduce the critical velocity considerably.When the oil bank arrives, the danger of exten-

sive gas and water coning will limit the oil produc-tion rate. Use of horizontal wells may improve therate.101 The low productivity of vertical wells duringproduction of tertiary oil can also be optimized bythe positioning of production completions below theoil/water contact.130

Nevertheless, long project lifetimes seems to be as-sociated with tertiary gas injection, starting with aperiod of many years with only water production, andfollowed by a long period of moderate oil production.This may be in direct conict with oshore opera-tions original designed for limited project time scalesand high oil production.As illustrated in Fig. 9.17, the oil behind the front is

subjected to both gravity and viscous forces, (Fg andFv), and the nal owpath is determined by both themagnitude of these and the permeability anisotropy.Draining oil may eventually reach the lower boundaryof the reservoir, and ow at a relatively high ratedowndip.Drainage velocities of the order of one cm per day

are estimated for Berea sandstone and 3 cp reservoiroil.104 Higher permeabilities and low oil viscositieswill enhance drainage. The possibility for high lo-cal oil saturation due to heterogeneities can possiblybe benecial to oil recovery. Mass transfer from oilto owing gas (HC) may improve the oil recovery byextraction of intermediate components giving a richcondensate when ashed.6 The danger of immobi-lization of draining oil by compositional eects must,however, be kept in mind.The eld experience of tertiary gasooding proves

that substantial amounts of residual oil may be re-covered. As discussed, the physical conditions inmany North Sea oil reservoirs seem favorable for thismethod. Laboratory experiments demonstrate thatoil banks are formed at reservoir rates and condi-tions. Long time scales involved in tertiary projectsare probably the largest restriction for taking themethod into use.

9.4 Gravity-Stable Displace-ment

As discussed earlier, the main drawback of gas in-jection is the poor sweep eciency due to unstabledisplacement of the oil phase by the less viscous in-jected gas. The mobility of an accumulated oil bankis reduced by gas ngering, segregation and grav-ity override. In horizontal core displacements, gashas been observed to breakthrough very early afterstart of gas injection.4,131 Gravity-stable displace-ment means displacement vertically downwards at arate suciently low so that buoyancy prevents frontinstabilities. This type of displacement is one methodfor preventing viscous ngering and gravity tonguing.The eciency of gravity-stable displacement by gas

compared to waterooding in highly permeable ho-mogeneous reservoirs with considerable dip angle isexemplied by the Oseberg eld case,4 where theresidual oil saturation after waterooding is expectedto be about 0.27, while residual oil saturation aftergravity-stable gas displacement is about 0.1, based onexperimental results. Fortunately, many of the NorthSea oil reservoirs have a considerable dip angle, whichcan ensure a stable gas-oil front. A schematic illus-tration of a gravity-stable gas injection is given inFig 9.17. The oil is displaced downwards towards pro-duction wells located near the water-oil contact. Thegravity forces will not only contribute to stabilizationof the gas front, but also to continuing drainage of oilbehind the front until an equilibrium between gravity,capillary, and viscous forces is obtained.If the gas-oil mobility ratio is favorable, and pro-

vided gravity does not inuence the displacementby segregating the two phases, the gas displace-ment mechanisms are regarded to be less complicated.However, usually the mobility ratio is unfavorableand ngering or gravity tonguing may dominate thefrontal movement and lead to early breakthrough and

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9.4. GRAVITY-STABLE DISPLACEMENT 225

low oil recovery. For forced immiscible gas injection,a theoretical maximum velocity for stable front move-ment has been presented by Blackwell et al.,132 andwas earlier in this chapter described by Eq. 9.8 forNorth Sea reservoir conditions.This equation indicates instability of the displace-

ment front at frontal advancement rates in the rangeof 1 m/d, as illustrated by the example given in Ta-ble 9.3. Dietz73 and Hill133 have dened a criticalvelocity for vertical miscible displacement, which fora given vertical column and neglecting variation inrelative permeability between gas and oil (sinα = 1,kro = krg), can be described by

uc = k

(∆ρ

∆µ

)g. (9.9)

A more restrictive criterion for the displacement rateis the upper limit for stable displacement as expressedby Dumoré134 and Blackwell et al.,132

ust = k

(dρ

)min

g, (9.10)

where (dρ/dµ)min is the minimum of the derivative ofgas-oil mixture density with respect to gas-oil mixtureviscosity.The ratio of stable to critical velocity can then be

expressed by the viscosity ratio, M = µo/µg, for therelations between gas and oil viscosities and densitiesreported by Dumoré,134

ustuc

=1− (1/M)

lnM. (9.11)

For hydrocarbon or nitrogen gases,M 1 at reser-voir conditions of North Sea reservoirs. This impliesthat ust is always less than uc. If the frontal velocityis above the stable rate, part of the transition zonewill not be at stable conditions. If u lies betweenust and uc, a growing unstable tongue is developed.If miscibility is achieved, the transition zone nger-ing will cause more rapid mixing of uids than whatwould be obtained by dispersion only. The eective-ness of gravity segregation in improving displacementeciency decreases rapidly at rates above the criticalrate. Above the critical velocity, the front will beunstable and gravity tonguing will dominate. In lowpermeability reservoirs with low dip angle, the crit-ical rate is often found to be too low for practicalapplications.In a typical North Sea eld situation with well-

spacing of about 1 km, the transition zone will beshort compared to the total length of the ow path.This relatively short extent of the transition zoneshould imply a higher eld recovery scaled from thelaboratory results. However, dispersion and large-scale heterogeneities both have negative eects on theeld-scale recovery.The importance of reservoir description is obvious

from the permeability dependence of the critical ve-locity. If the permeability is decreasing downwards

from the injection well, severe instabilities can be cre-ated as the front velocity exceeds the critical velocity.If the permeability variations are unknown, the eld-scale breakthrough time of the injected gas can bevery unpredictable.For miscible gas injection into a layered reservoir,

the transverse dispersion of the injected gas to com-municating layers causes slug dilution and loss of mis-cibility and also crossow of gas and the mobilized oilbank between layers. In isotropic ow regimes in ho-mogeneous reservoirs, an unfavorable mobility ratiowill accelerate the growth of the gravity tongue andthereby reduce the oil recovery at gas breakthrough.Simulation studies have shown that reduced verticalpermeability retards the gas segregation.135 Thesestudies also show that reduced communication be-tween layers leads to higher recovery compared tohomogeneous reservoirs in all cases except where thehigh-permeability layer is located at the top of thereservoir model.Theoretical treatment of vertical sweep in homoge-

neous, isotropic cross-sections has been presented byDietz73 and Hawthorne.136 Both have derived equa-tions for immiscible gas drive characterized by a singlegravity tongue, showing that the vertical sweep is pri-marily dependent on the viscous/gravity ratio and themobility ratio. The equations show that when the vis-cous/gravity ratio becomes large, the vertical sweepdepends primarily on the mobility ratio. Ypma2 haslater developed a two-dimensional analytical modelfor gas-oil gravity drainage in a homogeneous, dip-ping reservoir by combining one-dimensional verticalBuckley-Leverett drainage theory85 with the Dietz73

segregated ow approach. Examples given by Ypma2

show that miscible-like recovery eciencies can be ob-tained by immiscible gas injection into highly perme-able dipping reservoirs containing light oil.The importance of buoyancy on gas displacement

is best illustrated by the gravity-drainage process,which is characterized by negligible viscous forcesor viscous forces acting perpendicular to the gravityforces. Then, a stationary saturation distribution isthe result of an equilibrium between gravity and cap-illary forces. The displacement is in this case com-pletely gravity dominated, and all movement of theoil phase will be due to gravity forces.An analytical model for oil recovery by vertical

gravity drainage has been presented by Richardsonet al.137,138 based on the theory of Cardwell andParsons,139

(∆z)So =kv ∆ρ g∆t

φ µo

(dkrodSo

)So

, (9.12)

where (∆z)So is the vertical distance the saturationSo has travelled in time ∆t. If the oil relative perme-ability can be represented by a Corey type of equation(kro = Sno ), then the saturation distribution can eas-ily be calculated as a function of the reservoir heightat any time before gas breakthrough. As an exam-ple, we will use a vertical column of 200 m thick-

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226 CHAPTER 9. GASFLOODING

ness with kv of 1 µm2, porosity 25%, and oil viscosity0.5·10−3 Pa·s. After 1 year, the oil saturation at 20 mbelow the gas-oil contact is estimated at 34%, whileat 200 m the saturation still is 60%. After 10 years,the oil saturations at the same depths are 18% and34%, respectively. Even in thick pay-zone reservoirslike in the North Sea, the oil recovery by gravity seg-regation alone may be possible within practical timeframes.Terwilliger et al.72 presented experimental data

on saturation distribution versus time and height.They performed forced gravity-drainage experimentsat constant injection rate, and showed that low ratesyielded three times higher oil recovery compared tohigh-rate experiments. These results are in agree-ment with expected ow behavior when oil is dis-placed at rates above the critical stability rate as dis-cussed earlier.Several experimental studies have shown that very

low residual oil saturation can be achieved by gravitydrainage.17,85,131,140,141 A possible pore-level expla-nation of such ecient oil recovery processes may befound in the uid distribution of the dierent pores.All the oil entering pores during a drainage process iscontained in the connected larger pores. These poresare generally larger than the pores containing the con-nate water. By this argument, all oil initially presentis in principle movable and the residual oil saturationmay approach zero. The key requirement for achieve-ment of very low residual oil saturations is the main-tainance of connectivity in the oil phase during theoil displacement.Waterood residual oil saturation can be eciently

mobilized by tertiary gasooding if conditions for pro-duction through thin oil lms are achieved.104,142

The presence of connate water may change the wet-tability conditions for the oil phase. The oil recoveryis found to be higher for positive spreading systems.This is attributed to ow of oil through a continuousthin lm which separates the oil and water phases.The oil spreading coecient has been dened earlierby Eq. 9.6.For systems of negative spreading, the absence of

oil lms and thus a direct gas-water contact, con-tributes to reduced oil production.142 The stabilityof thin oil lms is described by the disjoining pres-sure108,143 arising from molecular interactions. Theseinteractions can be due to Van der Waal forces, elec-trostatic forces, and structural/steric eects. Dullienet al.142 showed that very little oil was recovered bytertiary gas injection, for systems of negative spread-ing coecient. Recovery of residual oil after water-ood by gravity-drainage is improved if oil spreads onwater in the presence of gas.Although very low residual oil saturation is associ-

ated with gravity-stable displacement, it may not bepossible to achieve these saturations within practicaland economical time scales.Immiscible, gravity-stabile gasooding involves

complicated component exchanges. At the gas front,

the gas can donate lighter components to the oil. Thisleads to swelling of the oil phase which becomes moremobile due to lower oil viscosity and higher satura-tion. Behind the gas front, the gas extracts compo-nents from the residual oil. Several papers have dis-cussed experimental and theoretical studies of com-positional eects for immiscible displacement relatedto oil elds in the North Sea.144146 The displacementeciency is a function of the displacement rate, thegas-oil phase behavior, wettability, and rock proper-ties.Early eld experience showed that oil moved

downwards during production,72 and later eld re-ports have shown very high oil recovery by gravitydrainage.84,117,147

High recoveries by gravity-stable gas injection havebeen proved both for secondary and tertiary pro-cesses.11 Reduced interfacial tension has been foundto increase both the drainage rate and the oil recov-ery.148 Although research eorts on fundamentals ofgravity-stable displacement has been extensive, theknowledge of the pore level displacement mechanismson one hand, and on the other hand the eld-scale ef-fect of heterogeneity, etc., is insucient and shouldbe topics of further research.

9.5 Water-Alternate-GasInjection (WAG)

9.5.1 Introduction

Low gas viscosity, and often low density, are perhapsthe two most limiting gas properties with respect tosweep eciency during gravity-unstable oods. Onesolution to this problem is to take advantage of thereduced gas permeability in the presence of mobilewater.149 A favorable mobility ratio between injectedgas and oil may thus be obtained, even for mediumviscous reservoir oils,11,150 µoil < 2 · 10−3 Pa·s.It has been suggested to inject water and gas at a

ratio giving equal velocity of the injected uids.149,150

If gas is overinjected, a zone of high gas saturationmay accumulate behind the oil bank and cause owinstabilities. Too high water velocity will result incorrespondingly high water saturations which may besuboptimal for oil recovery. Estimation methods forequal water and gas velocity ooding are given byStalkup.11 Simulation results indicate that equal ve-locity injection provides optimum stabilization of vis-cous ngering at high total injection rates.151 Forlower injection rates, optimal recovery may be ob-tained at lower water/gas ratios.In eld projects, the water/gas ratio may rather

be determined by the availability of gas, or by therelative costs of water and gas. In dipping reser-voirs, gravity stabilized water/gas injection can beperformed in a manner giving two fronts.150 If thecritical velocity of the gas/oil front is higher than thecritical velocity for a subsequent water/gas front, wa-ter will not mix into the oil. As water lls parts of

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9.5. WATER-ALTERNATE-GAS INJECTION (WAG) 227

the reservoir behind the gas/oil front, the need forinjection gas is reduced.

9.5.2 Alternating Injection

Originally, the use of water for mobility control wasproposed as a continuous process.149 In eld oper-ations, water and gas are injected as alternate slugsrather than simultaneously, because better injectivityis obtained for each uid when injected as one phase.Further, gas and water can otherwise segregate in thewellbore, and alternating injection is also more con-venient operationally.11

For a WAG process, there exist two points of viewregarding the ow mechanism. One mechanism con-siders the uids to move through the reservoir as dis-crete slugs. This is supported by laboratory exper-iments,152 where it may be demonstrated that thesaturations of water and gas increase and decreasethroughout the core due to cyclic injection. In orderto model these saturations correctly, relative perme-ability curves with hysteresis eects must be used.The other proposed mechanism is that gas ngersinto the water phase, and the discrete slug identityis lost.11 Other mixing processes will also dispersethe uids, although not necessarily completely on amicroscopic scale. The saturation histories during aWAG process can be very important for immiscibledisplacement, since three-phase oil relative perme-abilities may depend on the ow path.112

Reservoir simulations indicate that continuous wa-ter/gas injection may yield improved oil recoverycompared to WAG.153 The cyclic nature of the WAGprocess may also give negative implications on theprole control for injection into layered reservoirs.154

Contrary to alternate injection, simultaneous injec-tion of water and gas can improve the prole controlrelative to continuous gas injection.

9.5.3 Gravity Segregation

Fig. 9.18a illustrates a water/gas injection process in-uenced by gravity. Near the injection well, waterand gas ow as dispersed phases, but gravity segre-gation will reduce the height of the zone of dispersedow until it eventually disappears. Instead, a zone ofowing gas without mobile water appears at the topof the reservoir, and a zone with owing water with-out gas is formed in the lower part. Analyses of theprocess in Fig. 9.18,155,156 have demonstrated thatthe extent of the zone of dispersed ow depends onthe dimensionless viscous-gravity ratio, F

′′

vg, denedby

F′′

vg =C qt

∆ρkvA

(krwµw

+krgµg

)

=C uh

∆ρkv`

(krwµw

+krgµg

) , (9.13)

Oil

Water and gas

Water

Gas

Water

Gas

(a)

Water

Gas

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0

Normalized distance

Nor

mal

ized

heig

ht

1.251.00.750.50.25

(b)

Figure 9.18: Water/gas injection into a vertical crosssection.

where qt is the total volume injected, u is Darcy ve-locity, ∆ρ is the density dierence between water andgas, kv is vertical permeability, kri relative permeabil-ity, and µi viscosity of gas or water. The constant Cis 0.102 s2/m when SI units are used, A is the hori-zontal area between the wells, h the thickness of theoil zone, and ` the distance between injection andproduction wells.

Fig. 9.18b indicates zone boundaries for dispersedow at stationary conditions for dierent values ofF′′

vg. The total sweep is improved when F′′

vg increases.Improved F

′′

vg may be obtained by increasing the in-jection velocity, decreasing the interwell distance, andchoosing (if possible) a dense injection gas. The wa-ter/gas mobility should be reduced as much as possi-ble. Reservoirs with thick oil zones and low verticalpermeabilities are favorable for water/gas injection.

The F′′

vg dened by Eq. 9.13 may be identied inFig. 9.18b as the ratio of the length of the dispersedow zone to the total length between injector andproducer. For F

′′

vg greater than one, the dispersedzone is not fully segregated before the producing wellis reached. In order to estimate the oil recovery dueto water/gas injection, it may be assumed that theresidual oil saturation reaches the same value, Sorg,wherever gas ows.156 The water-ooded part of thereservoir reaches the residual oil saturation for wa-terooding, Sorw. With these assumptions, oil re-covery is predicted to increase as long as the sweep

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228 CHAPTER 9. GASFLOODING

eciency of the gas increases. In a practical applica-tion, residual saturations will usually not be reachedfor a limited amount of uid injected, and much lessfavorable recoveries will be obtained. The recover-ies obtained for immiscible processes will strongly de-pend on the ow conditions (permeability functions),and on the amount of immobilized oil that can be re-covered by mass transfer. Also for miscible ooding,large amounts of mobile water may delay the recoveryof oil.Heterogeneous porous media may strongly inu-

ence the sweep eciency. Stratied layers can re-sult in improved recovery if the permeability increaseswith depth. The opposite behavior will be the resultif the highest permeability layers are on top. The ex-istence of thin, relatively low-permeability layers be-tween high-permeability strata may improve the re-covery. If these thin layers become impermeable, re-covery decreases below that of a homogeneous reser-voir.155,156

The presence of shales in an otherwise homoge-neous and isotropic reservoir will improve oil recovery.The improvement is largest for the lowest water/gasratios.151 The shales may act as baes, preventingthe gas from rising to the top of the reservoir, and theeective permeability in the vertical direction may begreatly reduced compared to the horizontal perme-ability. Two-dimensional simulations for a homoge-neous reservoir with the eective kv/kh ratio may givealmost identical oil production as a three-dimensionalshale model.Fig. 9.19 shows simulated zone boundaries for two

0 180 360 540 720 900

0

-25

-56

-87

Hei

ght

(m)

Distance (m)

Gas

Oil

Water

Water and Gas

(a) High gas mobility

0 180 360 540 720 900

0

-25

-56

-87

Hei

ght

(m)

Distance (m)

Gas

Oil

Water

WaterandGas

(b) Low gas mobility

Figure 9.19: Simulation of WAG injection into a ver-tical section.

cases of WAG injection into a homogeneous cross sec-tion. The only dierence between the simulationsare the gas relative permeabilities used, giving highgas mobility in Fig. 9.19a and low gas mobility inFig. 9.19b. The shape of the zone boundaries inFigs. 9.18 and 9.19 are somewhat dierent, but qual-itative agreement is seen. The oil recovery during the

simulation shown in Fig. 9.19a was less than during acorresponding waterood. The process in Fig. 9.19brecovered more oil than water injection. A simulatedinjection into the same homogeneous cross section,but dipping, gave increased oil recovery when waterand gas was injected from the top.

9.5.4 High Water Saturations

High water saturation in the dispersed ow zone mayinuence the oil recovery both for miscible and im-miscible WAG injection. For miscible processes, oilmay be isolated behind water barriers. The problemis largest in waterwet porous media. Isolated oil maybe recovered by breakdown of the barriers, swellingof oil, and mass transfer into the owing gas. Forimmiscible displacement, residual oil may be isolatedor in contact with owing gas. The ability of oil toow is represented by the relative permeabilities inthe three-phase space. Stagnant oil may partly berecovered by evaporation into owing gas.Some aspects related to oil recovery in the presence

of high water saturation are discussed in Sec. 9.3.

9.5.5 Three-Phase Relative Perme-ability

Figs. 9.18 and 9.19 illustrate that simultaneous owwater, oil, and gas may occur in signicant parts ofthe reservoir. The values of the three-phase relativepermeabilities may not only determine the amountand prole of oil recovery, but also the boundaries forthe zone of dispersed ow.Experimental determination of three-phase perme-

abilities is generally complicated and laborious. Thenumber of investigations found in the literature istherefore limited, and there is no general agreementon the eect of the presence of a third mobile phase.The shape of the isoperms varies between most

studies.153,157162 The oil permeabilities are gener-ally functions of the saturation of all three phasespresent. Often, lower residual oil saturations are ob-tained in the three-phase space compared to residualoil saturation after water and gasooding. The waterrelative permeabilities are usually only functions ofwater saturation, and equal to the two-phase perme-abilities.Less general interpretations can be made for

gas. Three-phase relative permeabilities may dependstrongly on the process path as well as the uidsused.112,162 It is therefore important that this typeof data is measured at conditions relevant for the pro-cess it shall represent.The importance of process path on nal oil satura-

tions for secondary and tertiary continuous water/gasinjection into Bentheimer sandstone is illustrated inFig. 9.20.115 Fig 9.20a is for a secondary WAG pro-cess, and the residual oil saturation is seen to stronglydepend on the water/gas ratio. For the tertiary in-jection in Fig. 9.20b, residual oil saturations are al-most independent of the water/gas ratio. It is seen

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9.6. FIELD EXAMPLES 229

1 2

3

4

w o

g

Sorw

Sorg

4 (0.25/5.3)3 (1.7/3.8)2 (2.9/2.5)1 (4.9/0.65)

Siw

(a)

Sorw

Sorg

g

ow

(0.1/5.7)(0.5/5.4)(2.7/2.6)(4.9/0.7)

Siw

(b)

Figure 9.20: Residual oil saturations in Bentheimersandstone for a system of model uids. Saturationhistories and ow rates of water/gas [ml/min] are in-dicated.115

that the recovery obtained by the secondary injec-tion with high water/gas ratio is almost identical tothe tertiary recovery. The secondary injection withlow water/gas ratio gives almost identical residual oilsaturation after gas injection.The importance of three-phase water and gas rela-

tive permeabilities is demonstrated in Fig. 9.19. Lowtotal water/gas mobility reduces the speed of segre-gation. The zone of dispersed ow of water and gastherefore increases.

9.5.6 WAG Injection in the North Sea

The possible use of WAG processes in North Sea oilreservoir will be discussed for a hypothetical reservoirusing the F

′′

vg as dened in Eq. 9.13, together withFig. 9.18. Reservoir and process characteristics aresummarized in Table 9.4.From Table 9.4, values of F

′′

vg between 0.073 and1.7 are obtained. The ratio u/(krw/µw + krg/µg) inEq. 9.13 is estimated using Darcy's law for linear ow.The highest F

′′

vg is obtained for CO2 injection, maxi-

mum pressure drop across the reservoir, and the low-est ratio between vertical and horizontal permeability.A good vertical sweep eciency is thus within reachfor optimal conditions. Reduced injectivity for radialow near the injector, reservoir heterogeneities, andalternating rather than continuous injection may re-duce the sweep eciency signicantly. For HC gasand N2, the F

′′

vg will be less than 0.75 at optimal con-ditions.

The results obtained by the simplied calculationsindicate that large variations in F

′′

vg can be expected,and also possibly outside the range indicated. Thecorresponding sweep eciency will vary from high atoptimal conditions to very poor at the other end ofthe scale.

WAG injection may prove benecial to oil recov-ery even if segregated ow dominates. This has beendemonstrated by simulation of various injection pro-cesses (water, gas, WAG) into a dipping North Seaoil reservoir, where WAG gave the best recoveries.163

This was mainly due to better vertical sweep e-ciency, and less to lower residual saturations after gas-ooding. The heterogeneities in the system, showingincreased permeability upwards in each isolated zone,proved to be favorable for WAG injection.

The simulations indicate that WAG injection intodipping reservoirs with increasing permeability up-wards and with long interwell distances, has processcharacteristics very dierent fromWAG injection intohorizontal reservoirs. Both kind of reservoirs can befound on the Norwegian Continental Shelf.

9.6 Field Examples

The following eld examples cover gasooding and oilrecovery of the upper Statfjord reservoir in the Stat-fjord eld. The Statfjord eld comprises two mainsandstone reservoirs, Brent and Statfjord, with unsat-urated oil. Oil production commenced in 1979 basedon waterooding of the Brent reservoir and reinjec-tion of the associated gas into the Statfjord reservoir.Laboratory tests had shown that the reservoir uidproperties were favorable for a high-pressure misciblegasood in the Statfjord formation. The eciencyof gasooding in a heterogeneous reservoir was, how-ever, uncertain.

After ten years of gas injection in the Upper Stat-fjord reservoir, 44% of STOOIP or 152 mill Sm3 havebeen produced, and the average GOR increased from154 to about 450 Sm3/Sm3. History has proven gasinjection to be an ecient oil recovery method, andit is predicted that continued gas injection will raisethe ultimate oil recovery to above 55% in the UpperStatfjord. A schematic cross section of the reservoirsand current production strategy is shown in Fig. 9.21.The Statfjord eld geology and development strategyare described in several papers.16,164,165

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230 CHAPTER 9. GASFLOODING

Table 9.4: Reservoir and ow characteristics for a hypothetical WAG process

Reservoir permeability : 9.87 · 10−13 m2 (1 darcy)Interwell distance : 1000 mOil zone thickness : 50 mVertical permeability : 10% or 50% of horizontalDensity dierence water/gas : 700 kg/m3 (HC, N2) or 300 (CO2)Maximum pressure drop between wells : 100 bar or 50 barViscosity water and gas : 0.3 · 10−3 Pa·s and 0.03 · 10−3 Pa·s

1000 2000 3000 4000

2500

3000

StatfjordGI

StatfjordOP

U. BrentOP

L. BrentWIU. Brent

WI L. BrentOP

Statfjord

Dunlin

U. BrentBrent OWC

Statfjord OWCL. Brent

West EastMeters

Met

ers

Subs

ea

Figure 9.21: Statfjord eld schematic cross section.

Geology

The oil accumulation in the Statfjord formation istrapped along the crest of a long north-south faultblock, dipping to the west with 6 to 8 degrees. Theformation consists of an overall coarsening-upwardsequence of interbedded sandstones, siltstones andshales, and is split into three layers named Nansen(top), Eiriksson and Raude. A widespread shale atbase of the Eiriksson layer acts as a pressure barrierover a signicant area of the eld and subdivides theStatfjord formation in two reservoirs, the Upper Stat-fjord and the Lower Statfjord reservoir. Gasoodingis performed primarily in the Upper Statfjord reser-voir.

The Eiriksson unit is characterized as a uvial bedsequence with upward improving reservoir properties.Shales are of limited lateral extent and act as verticalow restrictions rather than barriers. Average thick-ness is 45 m, net to gross ratio 0.75 and permeability200 to 1000 md. The Nansen member is a 7.5 mthick, clean, marine sheet sand with excellent reser-voir properties, e.g. permeability of 2000 to 5000 md.Fig. 9.22 shows a typical representation of the UpperStatfjord reservoir.

Fluid Properties

The initial reservoir pressure is 404 bar, and the bub-blepoint pressure 198 bar. Oil viscosity is low, about0.36 cp, and the viscosity of the injected gas 0.032cp. Gravity ratio of gas to oil at reservoir conditionsis about 0.4. Fluid in the reservoir shows negligiblevariation in PVT parameters and molecular compo-sition with depth and area.The injection gas composition varies with time

and the mixture of Brent and Statfjord gas produc-tion. Slim-tube experiments, equation-of-state calcu-lations, and swelling tests indicate a minimum mis-cibility pressure in the range of 317 to 352 bar, de-pendent on the molecular composition of the injectedgas. Enriched, contacting gas is obtained through va-porization of C2 to C6 components o the reservoiroil. Methane content of the injection gas is approxi-mately 76 mol%. First-contact miscibility is achievedat enriched gas compositions with less than 72 mol%methane.

Core Floods

Special core analyses have been performed to measuremicroscopic displacement eciencies of the gasood-ing. A study performed on cores from well B-39 indi-cated 0-3% residual oil when oil was displaced by gasabove the minimum miscibility pressure. Even be-

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9.6. FIELD EXAMPLES 231

Formation

0 GR 100

-0.15 NPHI 0.45

1.95 RHOB 2.950.1 Permeability 10000

MeterTDV/MSL

2725

2750

2775

2800

Dunlin

Nansen

Eriksson

Raude

Figure 9.22: Upper Statfjord formation.

low the minimum miscibility pressure, coreoods in-dicated a residual oil saturation of 15% compared to32% if ooded by water. The experiments were per-formed at reservoir pressure and temperature undergravity-stable conditions. The Statfjord formation iswaterwet, and mobility ratio of gas to oil is about 11.

Well Pattern

To optimize the gasood, the gas injection wells wereplaced stratigraphically low in a line pattern along thecrest of the eld. The oil production wells were placeddowndip, in a line midway between the gas injectorsand the OWC, but stratigraphically high in the highlypermeable Nansen and upper Eiriksson sands. Hori-zontal distance between injection wells and producersvaries from 700 to 900 m. The oil producers penetratetop of the reservoir approximately 75 m below thatof the gas injectors. Until Jan. 1991, a total of 20oil producers and 7 gas injectors have been drilledand completed in the Upper Statfjord reservoir, andgas breakthrough has been experienced in most wells.New wells will be added to recover oil in the unsweptwedge zone between the gas cap and the water leg.

Reservoir Performance

Injection of gas has induced a secondary gas cap in theUpper Statfjord reservoir. Cased hole logs (CNT),applied with time-lapse technique, are used as the

main tool both to monitor the movement of the gasfront, and to calculate the gas saturation and residualoil.166 Also, RFT pressures sampled in wells drilledthrough this area conrm the existence of the gas cap.The overall reservoir performance shows an ecient,gravity-dominant, drainage process:

• signicant volumes of oil are produced• the CNL measurements show that lm ow ofoil takes place in gas-invaded areas and leads tovery low values of residual oil (an average of 5%is assumed in the reservoir simulations)

• cusping and gravity overriding of gas occur athigh withdrawal rates, but in most wells the highoil recovery is obtained by continuing the produc-tion after gas breakthrough

• potential for recovery after breakthrough is con-siderably reduced when the vertical communica-tion between the layers is poor

• the gravity drainage is most eective when thevertical permeability is high

• the shale layers present in wells act as local owbarriers, and are in many cases successfully usedto isolate high GOR zones and to add new lowGOR zones for production

• to date 22% of the injected gas has been repro-duced as free gas in the producers, 78% remainsin the reservoir.

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232 CHAPTER 9. GASFLOODING

1979 1981 1983 1985 1987 1989 1991

50

40

30

20

10

0

20

15

10

5

0

500

400

300

200

100

0

Wat

eran

doi

lra

teSm

3 /dx

103

Gas

inje

ctio

nra

teSm

3 /dx

106

GO

RSm

3/Sm

3

Oil productionGas injectionWater productionGOR

Figure 9.23: Statfjord reservoir performance.

Field measurements prove that exchange of com-ponents has taken place in the vicinity of the displac-ing front. Several reservoir uid analyses were per-formed on uids samples before gas breakthrough inthe wells. The PVT analyses from these wells showedvariations in physical properties and molecular com-positions one to two months prior to the rise in thegas oil ratios. It is dicult, however, to give a fullexplanation of these observations in terms of phasebehavior models for miscible oods.Average initial well rates spanned from 10 000 to

50 000 bbl/d. Gas breakthrough occurred after about3.5 years in the rst wells, and mean oil recovery perwell exceeds 20 million barrels. Although large indi-vidual well dierences are observed, about 40% of theproduction has taken place after gas breakthrough.Management of wells after gas breakthrough dependson the overall situation. In the Statfjord eld, thegas-handling capacity of the facilities has requiredconsiderable planning with focus on the Upper Stat-fjord reservoir withdrawal and injection. Therefore,wells with high gas-oil ratios have been periodicallyon production and shut in. Fig. 9.23 summarizes theproduction history of the Statfjord reservoir.

Nomenclature

A = area, m2

E = recovery eciencyF = force, N= ratio

g = acceleration of gravity, m/s2

H = layer thickness, mh = height, mJ = Leverett J-functionk = permeability, mdL = length, m` = length, m

M = viscosity ratioN = dimensionless numberp = pressure, Paq = volumetric owrate, m3/s

S = saturation, fractiont = time, su = Darcy velocity, m/sx = diusion length, mα = dip angle

∆z = vertical height change, m∆ρ = oil-gas density dierence, kg/m3

∆µ = oil-gas viscosity dierence, Pa·sφ = porosity, fractionµ = viscosity, Pa·sρ = density, kg/m3

σ = interfacial tension (IFT), mN/mς = spreading coecient, mN/m

Subscripts

c = critical or capillarycrit = criticald = dewpoint or dryg = gas or gravityh = horizontali = initial, irreducible or phase il = liquid

max = maximumo = oilr = residual or relativest = stablev = viscous or verticalw = water or wet

1, 2 = phase label

Superscripts′= modied

˜= average

Operators

∆ = dierence

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REFERENCES 233

References

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[150] Blackwell, R.J. et al. Recovery of Oil byDisplacements with Water Solvent Mixtures,Trans., AIME (1960) 219, 293300.

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238 CHAPTER 9. GASFLOODING

[151] Christie, M., Muggeridge, A.H., and Barley,J.J.: 3D Simulation of Viscous Fingering andWAG Schemes, paper SPE 21238 presented atthe 1919 SPE/DOE Symposium on ReservoirSimulation, Anaheim, Feb. 1720.

[152] Morel, D. and Latil, M.: Basic Study ofSweep Eciency Improvement by Water Al-ternate Gas Injection, paper presented at the1987 European Symposium on IOR, Hamburg,Oct. 2729.

[153] Warner, H.R.: An Evaluation of Miscible CO2

Flooding in Waterooded Sandstone Reser-voirs, JPT (Oct. 1977) 133947.

[154] Gorell, S.B.: Implications of Water-Alternate-Gas Injection for Prole Control and Injectiv-ity, paper SPE/DOE 20210 presented at the1990 SPE/DOE Symposium on EOR, Tulsa,April 2225.

[155] Stone, H.L.: Vertical Conformance in an Alter-nating Water-Miscible Gas Flood, paper SPE11130 presented at the 1982 SPE Annual Tech-nical Conference and Exhibition, Dallas, Sept.2629.

[156] Jenkins, M.K.: An Analytical Model forWater/Gas Miscible Displacements, paperSPE/DOE 12632 presented at the 1984 SPESymposium on EOR, Tulsa, April 1518.

[157] Maloney, D.R., Mahmood, S.M., and Honar-poor, M.M.: The Eects of Viscous Forces onThree-Phase Relative Permeability, NIPER-392, DE89 000737, National Institute forPetroleum and Energy Research, Bartlesville(1989).

[158] Baker, L.E.: Three-Phase Relative Permeabil-ity Correlations, paper SPE/DOE 17369 pre-sented at the 1988 SPE/DOE Symposium onEOR, Tulsa, April 1720.

[159] Grader, A.S. and O'Meara, D.J.: DynamicDisplacement Measurements of Three-PhaseRelative Permeabilities Using Three Immisci-ble Liquids, paper SPE/DOE 18293 presentedat the 1988 SPE Annual Technical Conferenceand Exhibition, Houston, Oct. 25.

[160] Van Spronsen, E.: Three-Phase Relative Per-meability Measurements Using the CentrifugeMethod, paper SPE/DOE 10688 presented atthe 1982 SPE Symposium on EOR, Tulsa, April47.

[161] Maini, B.B. and Kokal, S.: Measurementsand Correlations of Three-Phase Relative Per-meability at Elevated Temperatures and Pres-sures, paper SPE 19677 presented at the 1989SPE Annual Technical Conference and Exhibi-tion, San Antonio, Oct. 811.

[162] Dria, D.E., Pope, G.A., and Sepehrnoori,K.: Three-Phase Gas/Oil Brine Relative Per-meabilities Measured Under Carbon DioxideFlooding Conditions, paper SPE/DOE 20184presented at the 1990 SPE/DOE Symposiumon EOR, Tulsa, April 2225.

[163] Olsen, G., Skauge, A., and Stensen, J.Å.: Eval-uation of the Potential Application of WAGProcesses in a North Sea Reservoir, paper pre-sented at the 1991 European Symposium onIOR, Stavanger, May 2123.

[164] Roberts, J.D., Mathieson, A.S., and Hampson,J.M.: Statfjord Field, Geology of the Norwe-gian Oil and Gas Fields, Norwegian PetroleumSoc., Graham & Trotman, London (1987) 319340.

[165] Buza, J.W. and Unneberg, A.: Statfjord Ge-ology, Reservoir Performance, Oil & Gas J.(July 21, 1986) 6872.

[166] Flølo, L.H.: Calculating Injection Gas Satu-ration from Cased Hole Compensated NeutronLogs in an Oil Reservoir, paper presented atthe 1990 Annual Logging Symposium, Lafay-atte, June 2427.

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Chapter 10

Surfactant Flooding

10.1 Displacement Mechanisms

10.1.1 Introduction

The aim of surfactant ooding is to recover thecapillary-trapped residual oil after waterooding.These microscopic oil droplets usually constitutemore than half the residual oil. By the injection ofsurfactant solution, the residual oil can be mobilizedthrough a strong reduction in the interfacial tension(IFT) between oil and water. A large increase inmicroscopic recovery depends on ecient surfactantsthat can reduce IFT by a factor of about 104, whichcorresponds to a value close to 1 µN/m.The fact that oil recovery could be related to IFT

was rst mentioned by Uren and Fahmy1 in 1927.However, serious work with surfactant ooding didnot start until the sixties. Most of this work was car-ried out in USA, aimed at completely wateroodedreservoirs. Usually a small, highly concentrated sur-factant slug was injected, followed by a large polymerdrive.2,3 A large number of eld tests were carriedout, but the results were often disappointing, show-ing a much lower recovery than predicted from lab-oratory tests. Some of the larger eld tests (90 to400 acres) have been technical successful, recovering25 to 30% of the residual oil with a volume ratio of9 to 27 Sm3 of oil per ton surfactant.4 In spite ofthis, the process seemed to be far from economical.This is to a large extent caused by the long time in-terval between heavy front end investments in costlysurfactants, and the production of the extra oil.As a result and because of the recent drop in oil

price, the U.S. activity within surfactant ooding hasdecreased dramatically. For the activity within sur-factant ooding in the North Sea, the attitude seemsto be more optimistic. One important reason is thepossibility to inject the surfactant before the reser-voir is completely waterooded, thereby improvingthe process economy by earlier production of the ex-tra oil, restricting us to a time window for the appli-cation of surfactant ooding.A recent study5 estimates the IOR potential for

surfactant ooding on the Norwegian continentalshelf to be approximately 100 million Sm3. Requir-ing an eciency corresponding to a volume ratio of 40between produced extra oil and injected surfactant,reduces the potential to about 70 million Sm3. This

is still a considerable amount of oil, correspondingto the total recoverable reserves in elds like Ula orDraugen. Because of the time window, the potentialdrops 50% if we wait for ten more years.5

10.1.2 Process Description

After the surfactant solution has been injected, thetrapped oil droplets or ganglions are mobilized dueto a reduction in interfacial tension between oil andwater. The coalescence of these drops leads to a lo-cal increase in oil saturation. An oil bank will startto ow and mobilize (incorporate) any residual oil infront. Behind the oil bank, the surfactant now pre-vents the mobilized oil from being retrapped. Theultimate residual oil saturation will therefore be de-termined by the interfacial tension between oil andsurfactant solution behind the oil bank.Due to the high cost of surfactant, only a small

portion of the pore volume can be injected. The sur-factant slug therefore has to be displaced by water,usually containing polymer to increase the viscosity,thereby avoiding ngering and breakdown of the slug.Polymer can also be added to the surfactant slug toprevent ngering into the oil bank.In the following, the main aspects of surfactant

ooding is discussed.

Capillary Desaturation

In order to reduce the residual oil saturation in thewaterooded zones, the pressure drop across trappedoil has to overcome the capillary forces that trapsthe oil. This is what happens when IFT between oiland water is reduced by surfactants. A large numberof studies have shown how the residual oil satura-tion correlates with the capillary number, the dimen-sionless ratio between the viscous and the capillaryforces,69

Nc =uµ

γ, (10.1)

where u is the Darcy velocity, µ is the viscosity of dis-placing uid, and γ is the interfacial tension betweenoil and surfactant solution.Using Darcy's law and taking into consideration

the eect of residual saturation, porosity, gravity, andwettability, have resulted in a large number of alter-native denitions.69 Typical plots of residual satu-

239

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240 CHAPTER 10. SURFACTANT FLOODING

ration as a function of Nc are shown in Fig. 10.1aand 10.1b. The Capillary Desaturation Curve (CDC)varies with pore-size distribution and wettability.

30

20

10

0

10-7 10-6 10-5 10-4 10-3 10-2

Nc

Non

wet

ting

resi

dual

satu

ratio

n

Wide pore-size distribution(e.g. carbonates)

Typical sandstone

Well-sorted sand

(a) Eect of pore-size distribution on the CDC.

30

20

10

010-7 10-6 10-5 10-4 10-3 10-2

Res

idua

lno

nwet

ting

orw

ettin

gsa

tura

tion

(%)

NcNonwetting

total

Normal rangewaterfloods

Nonwettingcritical Nonwetting

phase

Wettingcritical

Wettingphase

Snwr

Swr

(b) Eect of wettability on the residual saturation of wettingand nonwetting phase.

Figure 10.1: Characteristic capillary desaturationcurves.9

As pore-size distribution becomes more narrow, theoil saturation starts to drop at a higher Nc, but zeroresidual saturation is obtained at a lower Nc. Thatis, the CDC becomes steeper. The CDC for the dis-placement of the wetting phase is shifted to the rightof the CDC for the displacement of the nonwettingphase by approximately two orders of magnitude. Asurfactant ood therefore should perform best in awaterwet reservoir. If the capillary number is highduring the ood, which is hopefully the case, a reser-voir rock with a narrow pore-size distribution will givethe lowest residual oil saturation.CDC's for North Sea reservoir rock from the Brent

formations have been measured and compared withCDC's for Berea cores.10 Using the above denitionof Nc, ranges in values for the critical Nc (start of de-saturation) were from 5·10−7 to 5·10−5. The total Nc(zero residual oil saturation) ranged from 2 · 10−4 to5 · 10−3. The Ness formation had the most favorableCDC, with a critical Nc of 5 · 10−7 and a total Nc of2 · 10−4. Since the capillary number during a surfac-tant ood will be around 10−4, these data illustratethe importance of characterizing the reservoir withrespect to CDC's.

Fractional Flow

Pope11 showed how fractional-ow theory could beapplied to surfactant ooding. The treatment islimited to two-phase ow, no surfactant in the oilphase, continuous injection of surfactant and no n-gering or dispersion. Fig. 10.2 shows saturation pro-les for both a secondary and a tertiary process to-

1 1

0 01 1

Sorw Sorc Sorw Sorc

fw fw

(Sw2,fw2)(Sw2,fw2)

Sw Sw

SwrSwr-Ds

Lowtension

Lowtension

(Sw3,fw3) (Sw3,fw3)

Secondary Tertiary

1

1

1

10 0

Sw Sw

1-Sorc

1-Sorc

XD XD

Sw2 Sw2

Swr

1-Sorw

Lowtensionwater

Lowtensionwater

Connatewater

Connatewater

(a) Fractional flow diagram

(b) Saturation profiles before oil bank BT

Figure 10.2: Fractional-ow curves and saturationproles in a secondary and tertiary surfactant ood.11

gether with the appropriate fractional-ow curves.The fractional-ow curve for the surfactant ood ter-minates at Sorc, residual oil saturation after surfac-tant ooding, determined by the CDC. The viscosityof the surfactant solution is increased by the addi-tion of polymers. This can be observed by a smallertendency for the surfactant fractional-ow curve tobend downwards, compared to the water fractional-ow curve. The saturation of water in the oil bank,Sw2, is given by the intersection between the water-ood fractional-ow curve and a tangent to the sur-factant fractional-ow curve, drawn from the point(−Ds, 0). Ds is the adsorption of surfactant (porevolumes of solution) and is given by

Ds =(1− φ)ρr asφ ρs Cs

, (10.2)

where φ is the porosity, ρr is the density of rock, ρs isthe density of surfactant, as is surfactant adsorptionper unit mass of rock, and Cs is surfactant concen-tration per unit volume of solution.The velocity of the surfactant front is proportional

to the slope of this tangent, and will therefore de-crease with increasing Ds. From Eq. 10.2, this is seento correspond to increasing adsorption and decreas-ing concentration. The adsorption level is therefore ofgreat importance since it determines the time neededto produce the extra oil. Adsorption is also important

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10.1. DISPLACEMENT MECHANISMS 241

regarding the economy of surfactant ooding, and isdiscussed in detail later in this chapter.The secondary process is predicted to be identi-

cal to the tertiary process except for the rst shock,where the saturation changes from initial water satu-ration to oil-bank saturation, while in the tertiaryprocess the change is from saturation after water-ooding (1−Sorw) to oil-bank saturation. The veloc-ities of these shocks are proportional to the slopes ofthe lines between the corresponding changes in watersaturation on the waterood fractional-ow curve, asindicated in Fig. 10.2. This treatment therefore pre-dicts the same surfactant breakthrough time, residualsaturation and mobility ratio for the tertiary and thesecondary case.

Relative Permeability And Mobility Ratio

When the residual oil saturation decreases, oneshould expect an increase in the relative permeabilityto water, simply because the water is blocked by lessoil. In a surfactant ood, the mobility of the injectedsolution therefore will increase as interfacial tensionand residual oil decreases. This has been conrmedby experiments where relative permeability has beendetermined as a function of Nc.12 Based on theseobservation it is often assumed that the endpoint rel-ative permeability to water increases linearly from itsvalue at Sorw to 1.0 as the residual oil saturation de-creases to zero.9 For a typical waterwet rock, the end-point water permeability could increase by a factor ofve as residual saturation approaches zero. Assumingthe same viscosity for water and surfactant solution,as would be the case for low surfactant concentra-tion, this corresponds to the increase in mobility ofthe displacing uid.The total mobility of oil and water ahead of the

surfactant is determined by the oil-bank saturation.It is usually less than the total mobility ahead of theinitial waterfront. Measurements of oil-bank mobilityhave shown that it could be more than 10 times lowerthan the mobility of oil at initial water saturation.13

With the mobility of the displacing phase increas-ing and the mobility of the displaced phase decreasingby quite large amounts, one should expect a large in-crease in mobility ratio going from waterooding tosurfactant ooding. In unfavorable cases, it couldincrease more than ten times. Mobility control insurfactant ooding, either by the use of polymer orby adjusting the viscosity of the surfactant solution,therefore seems to be of crucial importance.

Phase Behavior And Recovery

The interfacial tension of a surfactant/oil/brine sys-tem is closely related to the phase behavior. Fora given type of oil and surfactant, the brine salin-ity strongly aects the phase behavior. In addition,the phase behavior will change with temperature,pressure, surfactant type and composition, surfactantconcentration, and water oil ratio. Phase behavior for

surfactant ooding is discussed in detail in Sec. 10.3.However, some of the important features aecting dis-placement mechanism are also briey discussed here.Fig. 10.3 illustrates how phase behavior and inter-

facial tension are aected by salinity.14

10-1

10-2

10-3

10-4

γ m oγ m w

Inte

rfac

ial

tens

ion,

mN

/m

20

16

12

8

4

00 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

Salinity, % NaCl

σ oor

σ w

TypeII(-)

TypeIII

TypeII(+)

σo σw

Figure 10.3: Eect of salinity on the solubilizationparameters for oil (σo) and water (σw), together withthe corresponding interfacial tensions between mi-croemulsion and oil (γmo) and between microemul-sion and water (γmw).14

At low salinity, the surfactant stays in the waterphase where it forms a microemulsion by solubilizingsome of the oil (Type II()). As salinity is increased,more oil is solubilized into the microemulsion. Thisis accompanied by a drop in the interfacial tensionbetween oil and microemulsion that can be relatedto the amount of oil solubilized, by the Chun Huhrelation15

γ =0.3

σ2, (10.3)

where γ is the interfacial tension, and σ is the sol-ubilization parameter; volume of oil solubilized pervolume surfactant.Further increase in the salinity results in the for-

mation of a water phase, giving rise to three phases(Type III) with a second interface. The interfacialtension between microemulsion and water depends onthe amount of water solubilized into the microemul-sion. Since the increase in salinity leads to an increasein oil content and a decrease in water content of themicroemulsion, the interfacial tension towards waterwill increase and towards oil will decrease as salinityincreases. This is illustrated in Fig. 10.3. Above acertain salinity, all the oil will be solubilized, and anew two-phase system is formed (Type II(+)), with

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242 CHAPTER 10. SURFACTANT FLOODING

water in equilibrium with a microemulsion consist-ing of all the oil and surfactant and some solubilizedwater.Core-ood experiments have shown that highest re-

covery is obtained at the salinity where the two in-terfacial tensions cross, Fig. 10.3. This salinity valueis therefore referred to as the optimal salinity. Healyand Read14 explained this observation by suggestingthat the highest interfacial tension would determinethe recovery (controlling IFT). Even though interfa-cial tension between oil and microemulsion is lowerat salinities higher than optimum, corresponding toa more ecient displacement of residual oil, the rearof the microemulsion slug would be trapped by thehigher interfacial tension between microemulsion andwater, resulting in slug breakdown.

Phase Gradients

Changes in phase behavior along the direction of owcan result in large variations in the performance ofa surfactant ood. This was shown by Nelson etal.16 who varied the salinity to create phase gradi-ents in core oods. Using two dierent salinities forthe waterood brine, surfactant solution and poly-mer solution, altogether 8 dierent core oods wereconducted. The highest salinity gives a three-phasesystem close to a type II(+) system, while the lowestsalinity results in a type II() system. The resultingrecoveries and retention are shown in Table 10.1.From Table 10.1, low retention and low residual

oil recovery correspond to a negative salinity gradi-ent, with the salinity decreasing from the wateroodbrine to the polymer solution. To explain this, a di-agram was constructed showing the phase behavioras a function of both the concentration of surfactantand salt. From Fig. 10.4, the salinity where the phasebehavior changes from type III, the shaded region,to type II(+), the region above, decreases rapidlyas surfactant concentration decreases, resulting in adecreasing width of type III region with decreasingsurfactant concentration for this surfactant system.The white zone within the shaded region correspondsto the optimal salinity. Solubilization parameters atoptimum are given for three dierent concentrations(numbers in circles), and seem to increase with in-creasing surfactant concentration. Realizing that thesurfactant slug will be diluted both at the front andat the rear, as it is transported through the core, al-lows for the construction of idealized compositionalpaths for each of the eight experiments in Table 10.1.Here the symbol → denotes the phase environ-

ment path from the midpoint of the slug towards thefront, while ← denotes the phase environment pathin the opposite direction. Looking at ood number3, which performed best, we observe the following,moving from the front to the rear of the slug:1. Due to the dilution of surfactant at the high

salinity level, a type II(+) system is formed at thefront. Since type II(+) corresponds to a water-in-oilemulsion, the surfactant now has the ability to mo-

2.5

2.0

1.5

1.0

0.5

00 2.0 4.0 6.0 8.0

2 3 5

6

8

3

4

1 4 65

7

Percent surfactant in the system

Perc

ent

sodi

umch

lori

dein

the

brin

e

6

7

10

Figure 10.4: Idealized ow paths of the oods in Ta-ble 10.1 illustrated by the salinity requirement dia-gram.16

bilize residual oil by swelling, which is an eectivemechanism that does not require low interfacial ten-sion. This is, however, only an advantage at the startof the ood, or, if retrapping of mobilized oil occursbehind the oil bank. Otherwise, the residual oil ismobilized by the oil bank.2. Towards the rear of the slug, both the concentra-

tion of salt and surfactant decrease, corresponding toa compositional line starting at the upper right andending at the lower left. The surfactant system willpass through optimum (indicated by the white zonein the middle of the type III region), and a large partof the slug will be in the type III region, ensuring lowinterfacial tension and low residual saturation3. At the rear of the slug, the phase environment is

type II(), below the shaded region. The surfactantnow is in the oil-in-water phase which is miscibly dis-placed by polymer solution. Since no surfactant re-tention due to phase trapping occurs, type II() atthe rear of the surfactant slug helps to transport thesurfactant through the core.Looking again at Table 10.1, we observe that all

the displacements having high salinity in the poly-mer slug is accompanied by 100% retention of thesurfactant. This is a result of the trapping of theoleic phase containing all the surfactant at the highinterfacial tension associated with a type II(+) sys-tem.For this surfactant system we see that a negative

salinity gradient gives the best eciency. In fact, ithas been observed that ood 3 described above per-forms better than a ood at constant optimal salin-ity. The reason is that dilution will transfer the phasetype from type III (at optimum) in the middle of theslug, to type II(+) at the front and rear. However,

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10.1. DISPLACEMENT MECHANISMS 243

Table 10.1: Residual oil saturations and surfactant retention for dierent phase environments of wateroodbrine, surfactant slug and polymer drive16

Flood Waterood Surfactant Polymer Residual Retention ‡

number brine slug drive oil saturation1 II() II() II() 29.1∗ 522 II(+)/III II(+)/III II(+)/III 25.2∗ 100∗

3 II(+)/III II(+)/III II() 2.0∗∗ 61∗

4 II() II() II(+)/III 17.6∗ 100∗

5 II() II(+)/III II(+)/III 25.0 1006 II(+)/III II() II() 5.6∗∗ 59∗∗

7 II() II(+)/III II() 7.9∗ 73∗

8 II(+)/III II() II(+)/III 136.7∗∗ 100∗

∗ Average of duplicates ‡ % of injected surfactant∗∗ Average of triplicates

this need not be true for other surfactant systems thatcan have dierent salinity requirement diagrams. Amore general requirement, therefore, is that the phasetype should change from type II(+) in front, via typeIII to type II() at the rear of the slug. However, asalinity requirement diagram determined for a surfac-tant designed for North Sea conditions (ethoxylatedsulfonate) is similar to the diagram in Fig. 10.4.17

Here, also the advantage of a negative salinity gradi-ent was demonstrated both in core oods and simu-lations with a modied version of UTCHEM.17

Volumetric Sweep Eciency

To eciently displace the oil bank towards the pro-duction wells, the mobility ratio should be as low aspossible. In addition to preventing ngering of thesurfactant slug into the oil bank, low mobility ra-tio also reduces large-scale dispersion caused by per-meability contrasts, gravity segregation, and the wellpattern. A low mobility slug improves the volumetricsweep eciency by forcing more of the injected uidsinto low-permeable layers and into areas far from theline between the injector and producer. The spread incapillary number will be less, giving a lower residualoil saturation in the low-permeable zones and a higherresidual saturation in the high-permeable zones.

Simulation studies of surfactant oods in layeredreservoirs indicate that the mobility ratio is of vitalimportance to oil recovery.18 In these studies, theimportance of utilizing a salinity gradient was em-phasized. Injecting a given amount of surfactant asa large or small slug made little dierence in perfor-mance, and the eect of heterogeneities on large-scaledispersion of the slug is diminished by viscous cross-ow and transverse dispersion.

10.1.3 Surfactant Flooding in NorthSea Reservoirs

The use of surfactants to improve the recovery inNorth Sea reservoirs has been discussed for more thanten years. The main candidates have been the Fortieseld19,20 on the English side, and the Oseberg eld21

and Gullfaks eld22 on the Norwegian side. Reservoirconditions require new ways of designing a surfactantood.20 Many of these conditions are related to theoshore location of the reservoirs, leading to largewell spacing, use of seawater as injection uid, andlimited storing capacity of chemicals.

Low Surfactant Concentration

Using a large slug with low surfactant concentration,less than 1 wt%, seems attractive for several reasons:(1) supply and handling of the chemicals will be eas-ier; (2) a large slug is less sensitive to dispersion andbreakdown, which is believed to be important withlarge well spacing and often highly stratied reser-voirs; (3) with a large, low-viscosity slug, a polymerdrive solution to prevent ngering into the rear of theslug could be omitted. Such a solution is also attrac-tive for economic reasons, since smaller investmentsare needed early in the project.Perhaps the most important requirement for such

a system is low adsorption of the surfactant. Other-wise, the velocity of the surfactant front will be sloweddown too much (high value of Ds), resulting in a longproduction period, and low oil-bank saturation. Evenif high recovery is achieved, the production cost willbe too high.Designing a system having optimum conditions at

seawater salinity leads to the formation of a typeIII-phase environment where the microemulsion is atrather low saturation. This could cause severe trap-ping of the microemulsion phase, leading to muchhigher surfactant retention. Since seawater will beused both for the surfactant slug and the preceding

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244 CHAPTER 10. SURFACTANT FLOODING

waterood, the advantages of the salinity gradientseem dicult to obtain. Looking again at the salinityrequirement diagram in Fig. 10.4, we see that lower-ing the surfactant concentration, which will happenat the rear of the slug, could lead to formation of atype II(+) system. The result will be an even higherdegree of phase trapping. Type II() design, withhigher interfacial tensions, might be necessary in or-der to avoid phase trapping.

Avoid the Use of Polymer?

Avoiding the use of polymer for mobility control byusing a large, low-concentration slug, has several ad-vantages. Injectivity is higher, and therefore pro-duction is faster, without polymer. Also, polymerscompatible with North Sea reservoir conditions, aredicult to nd. One candidate is scleroglucan, butit cannot yet be produced in large quantities. Xan-than is a promising candidate for temperatures below80C.A low mobility ratio between the surfactant slug

and the oil bank and water owing ahead, would re-quire the use of polymer. Even though the water-ood mobility ratio at North Sea conditions usuallyis well below unity, it could increase to rather highvalues during a surfactant ood, as discussed previ-ously. The main eects would be ngering of the sur-factant slug into the oil bank and a low volumetricsweep eciency. Using polymer to reduce mobilityand push the surfactant slug has shown signicantimproved recovery in simulation studies.18,20

Temperature Variation

Temperature gradients resulting from the injection ofcold seawater into the reservoir will aect the phasebehavior of the surfactant. An interesting questionis whether this could result in a favorable phase gra-dient. Unfortunately, the contrary seems true. Theoverall trend for the sulfonates, which are the mostprobable surfactant candidates, is a change from typeII(+) towards type II() as temperature increases, re-sulting in an unfavorable phase gradient.2326 Thepossibility of phase trapping is thereby further in-creased. It has also been observed that increasingthe pressure generally creates a change in phase be-havior towards type II().23,25,26 The eect of thepressure gradient during a ood therefore will be theopposite of the temperature gradient, but is expectedto be much smaller.Simulation studies, taking into account the eect of

temperature on adsorption, solubilization (IFT) andviscosity, in addition to phase behavior, showed thatrecovery increased from 15 to 23% of OOIP abovewaterood recovery when the temperature gradientwas accounted for, and polymer was used both in andbehind the surfactant slug.20 The corresponding re-covery increase without polymer was 7 and 14% ofOOIP. However, a favorable phase gradient was as-sumed, which is not likely to occur for sulfonates.

Using an alcohol that will create an optimal phasechange by chromatographic separation from the sur-factant slug, has been suggested as a remedy.27

Core Floods

At least two surfactant systems have been identiedwith promising behavior in core oods at North Seareservoir conditions. A branched ethoxylated alcoholhas been especially designed for the Gullfaks eld.28

Injecting 0.5 PV of a 2% surfactant in seawater, re-duced the residual oil saturation from 44% (after wa-terooding) to 5%, corresponding to an eciency ofabout 40 Sm3 oil per ton injected surfactant. Thescreening of a large number of commercial surfactantsfor a pilot injection in the Oseberg eld has resultedin another surfactant system showing promising re-sults in Berea core displacements at reservoir condi-tions.29 Here, an ethoxylated sulfonate is mixed withalkyl-aryl-sulfonate, alpha-olen sulfonate and an al-cohol. Injecting 0.5 PV of a 2% solution of surfactantin seawater recovered 90% of the residual oil after wa-terooding. This system also seemed to perform wellin the phase-trapping danger zone of low temperatureand low concentration.

Low Tension Polymer Flood

Recent studies have shown that injecting biopolymertogether with certain surfactants at low concentra-tion, 0.3 wt%, can recover oil very eciently in coreoods at reservoir conditions.30,31 It is clear thatthe polymer (scleroglucan or xanthan) improves theperformance of the surfactant by reducing adsorptionlevels, in some cases down to below 0.1 mg surfac-tant per gram rock. It also results in the loweringof the interfacial tension, and gives low interfacialtension over wider ranges of salinity and surfactantconcentration. This system therefore is more robust,reducing the possibility of phase trapping. It also hasthe advantage of good mobility control. However, itis not clear by which mechanism the polymer aectsthe surfactant behavior. In the most favorable coreoods, 80% of the waterood residual oil was pro-duced and 350 Sm3 of oil was produced per ton ofsurfactant injected.31 Field-scale simulations usingBPOPE (a modied version of UTCHEM) gave aneciency about four times lower, with a recovery ofup to 37.5% of OOIP. In these simulations, tempera-ture gradients were neglected.

Timing

The optimal time for the start of a surfactant oodhas been discussed for the Oseberg eld.21 With amodied black oil simulator, it was shown that for ahomogeneous reservoir the optimal injection time wasafter some water had been injected. Injecting ear-lier did not lead to earlier production, since the frontof the oil bank would catch up with the waterfrontand therefore be retarded by the production from the

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10.2. RELATIVE PERMEABILITY 245

waterood. However, for a heterogeneous case, thesurfactant should be injected as early as possible.

10.2 Relative Permeability

Few publications3235 are found on relative perme-ability of micellar solutions. The reason is probablythat surfactant ooding has been tried in only a lim-ited number of oil reservoirs, and most of the researchon relative permeability has therefore been on water-oil-gas systems. Also, relative permeability of micel-lar systems is a complex subject, mostly because ofthe complicated phase behavior, see Sec. 10.3.There are, in addition to the publications on micel-

lar systems, also some publications on low interfacialtension oil-water systems.10,3640 These are relevantpapers in the sense that they discuss the eect of lowinterfacial tension, or high capillary number, on rela-tive permeability.

10.2.1 Two-Phase Relative Permeabil-ity

Two-phase relative permeability curves are commonlyrepresented in tabular form or by some saturation-dependent function. The latter is preferable for micel-lar systems because of the complexity of the relativepermeability function. A most common expressionfor micellar systems is

kr = k0r · (S?)e. (10.4)

Early two-phase relative permeability models allowedthe exponent e to approach unity as interfacial ten-sion decreased. This is reasonable since it is gen-erally accepted that the relative permeability curvesare straight lines at low interfacial tensions. Coats41

used the following expression for relative permeabilityof oil as a function of interfacial tension in an oil-gassystem,

kro = (γ

γ0)

1n (S?o )nog + [1− (

γ

γ0)

1n ]S?o , (10.5)

where kro will approach So as γ approaches zero.Pope42 suggested e to be unity in Eq. 10.4, for mi-cellar systems. Later, Lake9 suggested that k0

r ande should be measured at both high and low capil-lary number, and to use a linear interpolation func-tion to calculate the relative permeability at the ac-tual capillary number. This method to representtwo-phase relative permeability at varying capillarynumber is probably the most common at the present.The change in residual saturation with capillary num-ber is taken care of through the CDC, discussed inSec. 10.2.4.Actual relative permeability measurements have

shown that Eq. 10.4 is not suciently exible. Solv-ing with respect to the exponent e,

e =ln(kr/k

0r)

ln(S?), (10.6)

shows that if e is a constant, a plot of the right-handside of Eq. 10.6 as a function of saturation shouldgive a horizontal line. Fig. 10.5 shows how e actu-ally varies with saturation. In order to reproduce the

2.0

1.6

1.2

0.8

0.4

0.00.0 0.2 0.4 0.6 0.8 1.0

e

S

Figure 10.5: The exponent e as a function of normal-ized saturation.35

shape of the observed relative permeability curves,the exponent e could be expressed as a polynomial ofthe form

e = a0 + a1 · S? + a2 · (S?)2. (10.7)

10.2.2 Three-Phase Relative Perme-ability

Information on measured three-phase relative perme-ability of micellar systems is very limited. Delshadet al.32 measured three-phase relative permeabil-ity using the steady-state technique. Their conclu-sion was that three-phase relative permeability of oil,water, and microemulsion is function of the respec-tive phase saturation only. Fig. 10.6 shows measuredtwo- and three-phase relative permeability from their

∆∆

∆∆

∆∆

∆ ∆∆

1.25

1.00

0.75

0.50

0.25

0.000.0 0.2 0.4 0.6 0.8 1.0

Microemulsion saturation

Mic

roem

ulsi

onre

lativ

epe

rmea

bilit

y

Figure 10.6: Two- and three-phase relative perme-ability of microemulsion.32

work. Skauge et al.33 have also measured three-phase relative permeability using the unsteady-statetechnique. They concluded that the three-phase rel-ative permeabilities of oil, water and microemulsion

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246 CHAPTER 10. SURFACTANT FLOODING

are probably not only dependent on the respectivephase saturations. Kvanvik et al.34 measured three-phase relative permeability using the unsteady-statetechnique. They concluded that the water and mi-croemulsion relative permeabilities are functions ofall three phase saturations. Fig. 10.7 shows the mi-croemulsion isoperm.34 The relationship between rel-

+ +

++

+

+

Ο Ο

Ο

ΟΟ Ο

x x

x

x

xx

∆∆ ∆

SwSo

Sm

krm=0.0krm=0.1krm=0.2krm=0.3krm=0.4krm=0.5krm=0.6

0.8

0.6

0.4

0.2

Figure 10.7: Three-phase relative permeability isop-erms of microemulsion.34

ative permeabilities and the saturations is subject forfurther research and at the present, any three-phaserelative permeability model should be used with care.A major dierence between micellar systems and

gas-oil-water systems is the change in microemulsioncomposition that may take place during a ood.43

When the eective salinity increases, the microemul-sion phase changes from almost pure water to almostpure oil. This dramatic change almost excludes theuse of any two-phase relative permeability curves ina three-phase correlation scheme.One of the rst three-phase relative permeabil-

ity models was suggested by Lake.9 In the absenceof measured data, he proposed to set the II() mi-croemulsion relative permeability equal to that of wa-ter, and the II(+) microemulsion relative permeabil-ity equal to that of oil. In the three-phase region, k0

r

and e in Eq. 10.4 are estimated by interpolation, forexample by

k0r = k0

rw + F (k0ro − k0

rw), (10.8)

e = ew + F (eo − ew), (10.9)

where

F =So(1− Sw)

Sw + So. (10.10)

Where measured two-phase relative permeabilitiesare known, a better approach is to use the actualtwo-phase relative permeability data of the II(+)- andII()-phase microemulsion or the III phase as inputto the traditional three-phase models, see Sec. 3.4.3.

The saturation-weighted interpolation model is usedas an example,

krm =(Sw − Swr)kII(+)

rm + (So − Sor)kII(−)rm

(Sw − Swr) + (So − Sor)(10.11)

Another approach is to assume that each three-phase microemulsion relative permeability is a func-tion of its own saturation only.32 Then two-phase mi-croemulsion relative permeability could be measuredusing the III-phase microemulsion and applied in thethree-phase region directly. This method will, how-ever, generate discontinuities at the two-phase bor-ders if the two-phase II(+) and II() microemulsionrelative permeability is dierent from that of the IIIphase.Kvanvik et al.34 tested several three-phase models

and concluded that the Stones44,45 models gave bet-ter prediction than the Lake model9 or the Delshad32

assumption that each relative permeability is a func-tion of its phase saturation only.

10.2.3 Hysteresis

Two-phase relative permeability hysteresis of micellarsystems has been observed by several authors.3335

Delshad et al.,32 however, did not report this eect.In oil-gas-water systems, hysteresis in the nonwet-ting phase has been explained as a capillary trap-ping phenomenon. Skauge et al.33 observed hystere-sis in excess-water relative permeability, and Kvanviket al.34 for microemulsion and excess-water.Eikje et al.35 observed hysteresis for all three

phases in a series of unsteady-state experiments. Theinterfacial tension between microemulsion and wa-ter and microemulsion and oil was 0.08 mN/m and0.005mN/m, respectively. Figs. 10.8 and 10.9 showthe two-phase relative permeability curves of mi-croemulsion and excess-water. The sequence is

Figure 10.8: Two-phase relative permeability of mi-croemulsion.35

primary drainage, primary imbibition and secondarydrainage, where water is assumed to be the wettingphase. The core material was burned Berea sand-stone. The relative permeability exhibits strong hys-teresis. Unlike the hysteresis behavior reported at

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10.2. RELATIVE PERMEABILITY 247

Figure 10.9: Two-Phase relative permeability of wa-ter.35

high interfacial tension, Eikje et al.35 report a stronghysteresis also for the imbibition relative permeabil-ity.The fact that both phases show strong hysteresis

in both drainage and imbibition relative permeabilitymay be explained by the viscous instability that ispresent during these low-tension, unsteady-state ex-periments. Delshad et al.32 conducted all the rela-tive permeability measurements at steady-state con-ditions, thereby avoiding instability.Several authors have discussed the eect of viscous

instability.4648 Pavones46 instability number is

Ns =(µd − µD)2

µdγφ, (10.12)

and a displacement is said to be stable if Ns < 10−4.The eect of instability on the measured relative per-meability curves is clearly illustrated by Peters andKhataniar.47 Eq. 10.12 shows that microemulsionsystems are generally far less stable than oil-water-gas systems because of the low interfacial tension. Alltwo-phase relative permeability measurements whereoil or water displaced microemulsion by Eikje et al.35

were conducted at unstable conditions according toEq. 10.12, and the degree of instability thereforechanges from drainage to imbibition. It is beyondthe scope of this chapter to discuss in detail whichimplications the presence of instability during rela-tive permeability measurements would have for thepractical use of these data. But it may seem obviousthat the use must be limited to simulation of caseswith the same degree of (in)stability as those presentduring the measurements. The restrictions on theusage may be alleviated by including the instabilitynumber in the relative permeability expressions.35

Modeling of Hysteresis

None of the common hysteresis models are able tohandle the hysteresis in relative permeability that ispresented in Figs. 10.8 and 10.9. Carlson49 assumesno hysteresis for wetting phase, while Killough50 as-sumes no hysteresis for the imbibition relative per-meability. A new model has been suggested35 that

takes into consideration all the hysteresis loops inFigs. 10.8 and 10.9. The model uses Land's equa-tion51 for calculating residual nonwetting phase satu-ration. Empirical correlations and interpolation rou-tines are used to calculate relative permeability atany saturation. A more detailed description of themodels is found in the Sec. 3.4.4.

10.2.4 Desaturation Curves

The capillary desaturation curve (CDC) describes therelationship between the capillary number and resid-ual uid saturation. Discussion on capillary numbersand desaturation curves is found in Sec. 10.1.2.Eciency of a surfactant ooding, as predicted by

numerical simulation, will rely upon the CDC used.Since the desaturation curve depends on wettabilityand pore-size distribution, it should be measured forevery distinct rock type. The dierence in CDC's forNorth Sea formations is clearly illustrated by Garneset al.10 At a capillary number of 0.0001, the relativeresidual oil saturation Sor/Sorw is 0.85 in the Tarbertformation and 0.5 in the Ore formation.The CDC's are most commonly modelled using log-

linear expressions of the form

Sr = b0 + b1 logNc. (10.13)

As illustrated by several authors,37,39,52 the shapeof the CDC's may deviate from the linear form, andEq. 10.13 will in those cases have a limited applica-bility. In order to model these curves, slightly moreexible expressions are suggested.Because of the close relationship between CDC and

relative permeability, the latter should be measuredat several capillary numbers within the decline partof the CDC. But since the relative permeability mea-surements are time consuming and expensive, theyare normally carried out only at two capillary num-bers. One measurement is made at a capillary num-ber less or equal to the critical capillary number andthe other at the highest expected capillary numberor at the capillary number where the residual satu-ration is at its minimum. The endpoint relative per-meability will normally change as the capillary num-ber changes.32,37,39,40,53 For a two-phase system, theendpoint relative permeability of a phase is thereforenormally modelled as a function of the residual satu-ration of the other phase.The change in relative permeability with capillary

number will aect the modelling of hysteresis de-scribed in Sec. 3.4.4. A complete hysteresis loopmust be established at high and low capillary num-bers. Land's trapping constant for these two situ-ations must be dened, as it is assumed that theresidual saturation of the nonwetting phase will bedierent at high and low capillary numbers, for thesame historical maximum nonwetting phase satura-tion. Hysteresis is therefore not only a function ofsaturation history, but also of capillary number.During a microemulsion displacement process, the

capillary number and the saturation paths will change

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248 CHAPTER 10. SURFACTANT FLOODING

in time and space, leading to a continuous changein the relative permeability curves, caused both byhysteresis and changes in capillary number.

10.3 Phase Behavior

10.3.1 Introduction

The oil-recovery eectiveness of a chemical ood is re-lated to the phase behavior of the brine-oil-surfactantsystem. The important phenomenon in such systemsis the ultra-low interfacial tensions (IFT) that can oc-cur. Low interfacial tensions are needed to reduce thecapillary forces that keep residual oil trapped. Phasebehavior of these systems has been extensively stud-ied for the past 20 years.A surfactant molecule is amphiphilic; the molecule

consists of a hydrophilic head group and a lipophilicchain, Fig. 10.10. The hydrophilic part can be ionic

S - O NaO

O

+-

Figure 10.10: Anionic surfactant.

or nonionic depending on the polar head group ofthe surfactant. An aqueous solution of surfactantswill contain micelles, above a certain critical micelleconcentration, CMC.54,55 In the nonpolar interior ofsuch micelles, organic molecules can be solubilized.The dual nature of the surfactants makes them re-

side at the interface between the aqueous and organicphases, lowering the interfacial tension.For chemical ood processes, anionic surfactants

are studied because they have a tendency of loweradsorptions to the reservoir roch than nonionic andcationic surfactants. This section will therefore con-centrate on phase behavior of anionic systems. Com-mercial surfactants will often consist of mixtures ofanionic and nonionic surfactants. The eect of non-ionic material present will therefore be noted.A three-component system of water, oil, and surfac-

tant can form single- or multiphase microemulsions.A microemulsion is dened as a translucent and ther-modynamic stable solution consisting of oil, water(electrolyte), and one or more amphiphilic compo-nents, surfactant or alcohol.56 In Fig. 10.11, a waterexternal II() phase is shown in a ternary diagram.

Ternary Diagram

Actual surfactant systems in oil recovery will nor-mally produce from one to three phases. For simplesystems containing only three components; oil, purewater, and a single component surfactant; the phasebehavior may be fully described by ternary diagrams.The three corners in the diagram will each represent100% of one of the three components. Every possible

PM

E

Water Oil

Surfactant

Binodal curve

Tie line

I-phase

T

Figure 10.11: II(), water external phase behavior.

mixture or phase composition of the surfactant sys-tem will be represented by points within the diagram.The surfactant system may exhibit phase behav-

ior of II(), III, or II(+) type, depending on the oiland surfactant type. But once the components aregiven and the pressure and temperature are held con-stant, the phase behavior may be fully represented bya binodal curve and tie lines on a ternary diagram.The binodal curve describes the boundary betweenthe single I-phase and the multiphase region(s). Foran overall composition within the multiphase region,both the composition of the microemulsion phase andthe excess phase(s) will lie on the binodal curve.Their positions on the binodal curve are described bya tie line, a straight line passing through the overallcomposition point.The II()-case is illustrated in Fig. 10.11. An over-

all composition given by point T is separated into anexcess oil phase E and a microemulsion phase M .Point P denotes the Plait point. When the overall

composition approaches P , both the microemulsionand the excess oil phase will approach the same com-position.

10.3.2 Parameters

Salinity

Two-phase microemulsions are oil or water external.In the middle phase of a type III microemulsion, it isassumed that the structure is bicontinuous.57 Bothoil and water are solubilized in the middle phase.When the solubilization of oil and water are equal, anoptimum phase behavior is reached. There is an in-verse relationship between solubilization and interfa-cial tension at the optimum condition (σw = σo = σ)according to Huh's equation,15 cfr. Eq. 10.3,

γσ2 = 0.3, (10.14)

where γ is the interfacial tension, σ the solubilizationparameter at optimum dened by the volume of oilor water in the middle phase divided by the volume

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10.3. PHASE BEHAVIOR 249

of surfactant. The constant 0.3 has been veried byseveral authors.23,58,59

The phase behavior or the phase transitions fromII() to III to II(+) are strongly dependent on thesalinity of the water, Fig. 10.12. At low salinity, a

0.432

0.692

0.985

0.996

1.038

1.060

1.081

1.124

1.157

1.192

1.284

1.427

1.729

Increasing salinity, meq/ml

Stat

ic P

hase

Beh

avio

ur

V

olum

fra

ctio

n

excess water

0.0

0.5

1.0

excess oil

microemulsion

Figure 10.12: Phase behavior of 2 wt% AS41 andoctane in brine containing NaCl and CalCl2 at a ratio4:1.

lower II() microemulsion phase is formed. Whensalinity is increased, the solubility in the aqueousphase decreases. An inversion of the water exter-nal microemulsion to an oil external microemulsionappears due to a screening of the microemulsions inwater. At intermediate salinity, a type III microemul-sion is formed. This phase solubilizes both water andoil. At optimal salinity, the solubilization of oil andwater is equal, Fig. 10.13. It has also been shown

1.51.00.50.0

0

10

20

30

40

50

Total Cl-concentration, meq/ml

Solu

biliz

atio

n fa

ctor

s

σoσw

Figure 10.13: Solubilization parameters of oil andbrine as functions of salinity at T = 295 K. 2wt%AS41, octane, and brine containing NaCl and CaCl2at a ratio of 4:1.

that the solubilization is highest and the IFT lowestwhen the width of the III-phase region is at a mini-mum.60,61

Divalent Cations

The formation water of oshore reservoirs typicallycontains 4 to 10% salt. The main cations are Na+

and Ca2+, but often a signicant amount of Ba2+ ispresent. Also, the injection water will be seawater.The eect of the cations on phase behavior will dif-fer. Divalent ions will have a larger eect than pureNaCl.59,62,63 The eect increases with higher valenceand larger molecular weight of the cation.64 With amixture of dierent cations, the ion composition of anexcess water phase will dier from the overall salin-ity due to the competitive interaction between thecations and the anionic surfactant.The strong preference for divalent cations to mono-

valent, makes the phase behavior dependent uponsurfactant concentration. At higher surfactant con-centration, less divalent cations are available for thelast surfactant added and the eective salinity de-creases, Fig. 10.14.

43210

0

2

4

6

8

Wt% (aq) AS 41

Wt%

NaC

l(aq

)

II(-) phase

III phase

II(+) phase

Figure 10.14: Salinity requirement diagram for AS41octane and brine containing NaCl and CaCl2 at aratio 4:1.

To be suitable for EOR applications, a surfactantmust have low injectivity, low adsorption, low cost,and a suitable phase behavior. Optimal phase behav-ior means low interfacial tension with respect to resid-ual oil and high solubilization. The surfactant systemmust be made of a minimum number of compoundsand not form gels and high viscous liquid crystals ormacroemulsions.The surfactants for North Sea elds should have

optimal salinity close to seawater salinity. Dierentsurfactant structures and mixtures therefore have tobe studied. Skauge and coworkers have found a suit-able system29 through a large number of phase tests,core oods and simulations.

Surfactant Structure

The surfactant molecular weight and structure willinuence the phase behavior.24,58,59,61,62,65 Whena hydrophilic surfactant forms a lower phase mi-croemulsion, the surfactant must be made morelipophilic to reach optimum salinity. This can bearchived by increasing the alkylchain length, mini-mize alkylchain branching or reduce the polarity ofthe head group.

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250 CHAPTER 10. SURFACTANT FLOODING

When the alkyl chain length increases, the widthof the three-phase region decreases and the optimalsolubilization of oil increases.56,58,60,66 In surfactantood design, a compromise often must be made be-tween the desire of having highest possible solubiliza-tion and the need for a robust system, fairly insensi-tive to phase environment changes in the reservoir.Formation of unwanted high viscous liquid crys-

tal phases can be reduced by altering the surfac-tant structure. Linear alkylchains have the best per-formance regarding solubility, but also highest ten-dency of making gel phases.60,62 Aromats like ben-zene and toluene are often part of the hydrocarbonchain. It has been found that for ethoxylated alkyl-sulfonates branching with a benzene molecule doesnot reduce the liquid crystalline phases.24 Reservoiroil has aromatic compounds and the phase behaviormay improve with respect to reservoir oil using thesegroups.56

Surfactants studied for EOR applications havebeen:(1) Lipophilic: petroleum fractions, olens, alkyl-

benzens, alkanes, alkylaryls(2) Hydrophilic: sulfonates, sulfates, carboxylates,

phosphates, ethoxylated alcohols.The most promising surfactants for oshore North

Sea reservoirs are ethoxylated (EO) anionic sul-fonates.24,59,6668 The reason is that the ethoxyla-tion improves the salt tolerance. Increasing the num-bers of EO-groups makes the surfactant more hy-drophilic.24,29,58,59,65,66 Optimum salinity can befound near seawater salinity, which is not the casefor e.g. petroleum sulfonates.Ethoxylated components are rather expensive

chemicals. It is therefore desirable to mix them withcheaper chemicals, e.g. petroleum sulfonates.29 Thissystem will have a high salt tolerance and a favorablephase behavior due to the ethoxylated fraction.

Cosurfactants and Cosolvents

For a surfactant system to be soluble and avoid gel-structure at reservoir conditions and low injectiontemperature, cosolvents like short-chained (C3-C5)alcohols are added to the system.Alcohols lower the solubilization at optimum

and also change the optimum.29,69,70 The di-rection of phase change will depend on the hy-drophilic/lipophilic balance of the alcohol comparedwith the surfactant. When the alcohol is most water-soluble, the transition II(+) to II() occurs, and thedrop in optimal solubilization is signicant. An oil-soluble alcohol will move the phase behavior in theII(+) direction, and the lowering in solubilizationwill decrease as the molar volume of the alcohol ap-proaches that of the oil.Due to its eect on optimal salinity, alcohol may

also be used to control phase behavior. By addingan appropriate alcohol to the surfactant system inFig. 10.14 (salinity requirement diagram), the alter-

ation in optimum salinity with surfactant concentra-tion may be less or even go in the opposite direction.Liquid crystals or gels may disappear, adding al-

cohols to the phases.Mixing one hydrophilic and one lipophilic surfac-

tant enables us to obtain optimal salinity at the in-jection water salinity. When using a two-surfactantsystem, the surfactants should have an optimal salin-ity as close as possible. This minimizes the possibil-ity of chromatographic separation during the ood.29

Using one hydrophobic and one hydrophilic surfac-tant, the hydrophilic can make the system more sol-uble by forming mixed micelles.71 Fully isomerizeda-olens have been reported to give good results withno alcohol added concerning solubility in high salinewater and no gel-forming.61 When the system wasonly partially isomerized, some alcohol was neededto get clear and low-viscous phases. A class of one-component surfactants, new ethoxylated sulfonates,28

have been developed, designed for reservoirs in theNorth Sea. These surfactants do not need alcoholsor a cosurfactant for compatibility with seawater orfor having optimal salinity at seawater concentration.They contain some nonsulfonated components whichinuence on the phase behavior.

Temperature

The reservoirs in the North Sea are usually water-ooded and injection of cold water will give temper-ature gradients. Eorts must be made to keep thesurfactant system stable, or at least to avoid unfavor-able phase behavior, with respect to the temperaturegradients.Anionic surfactants usually become more water/-

brine soluble when the temperature increases. Theoptimal salinity increases and the observed transi-tions are II(+) to II().23,60 Nonionic surfactantsshow an opposite and larger temperature eect. Athigh temperatures, the nonionic surfactants may beabove the cloud point and either phase separate orbecome oil soluble.The general temperature trend found for anionic

surfactants, may be reversed by adding nonionic hy-drophilic groups as hydroxy- or ethoxy groups to themolecule and cause favourable phase behavior gradi-ents wih respect to the temperature distribution inthe reservoir, see Sec 10.1.2.Ethoxylated sulfonates may show anormal temper-

ature dependence, like having two optimum temper-atures. This is caused by nonsulfonated material inthe blend.23

Phase behavior is improved by increasing the tem-perature, liquid crystal and macroemulsion phasesmelt at higher temperatures.59,60,62

The width of the III-phase region is usually broad-ened, while the optimum solubilization decreases withincrease in temperature.70

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10.3. PHASE BEHAVIOR 251

Pressure and Oil Composition

In deep North Sea reservoirs, there will be high pres-sures (300 to 600 bars), and the oil will have largeand varying solution gas-oil ratios. Both the pressureand the composition of the reservoir oil will inuencethe phase behavior.The composition of reservoir oil varies from

methane gas to heavy resins that may contain aro-matic and polar compounds. The resins and aromat-ics in the oil may be important for the phase be-havior.72 The dierent oil structures may fraction-ate between the excess phases and the microemulsionphases.Phase behavior as a function of hydrocarbon

number of n-alkanes has been extensively stud-ied.23,56,7378 Increasing the carbon number in-creases the optimal salinity and leads to a phase tran-sition from II(+) to III to II() due to less compati-bility between the surfactant and the oil. Increasingthe oil-chain length generally broadens the width ofthe III-phase region and decreases the solubility dueto steric hindrance.7678

The general trend in optimal solubilization andsalinity is shown in Fig. 10.15. However, for lowcarbon number a maximum in solubility and a mini-

4321

0

10

20

30

40

50

Optimal salinity % NaCl

Opt

imal

sol

ubili

zatio

n

Vo

/Vs=

Vw

/Vs

n-C14

n-C12

n-C10n-C8

n-C6

n-C5

Figure 10.15: Variation in optimum solubilizationand salinity for n-alkanes. Replotted from Fig. 10,Ref. 70.

mum in salinity may be observed. The existence andlocation of such a maximum and minimum depend onthe surfactant structure.The presence of disolved gas in live crude oil lowers

the average molecular weight of the oil. The eect onthe phase behavior is similar to decreasing the carbonnumber of n-alkanes.For microemulsion systems, there is not much pub-

lished material concerning pressure eects on phasebehavior. In one study,79 gas was added to crude oil,and only small eects were observed. It may then bedicult to separate the pressure eect from composi-tional eects. Most published results indicate thatpressurizing dead oil favors II() phases.23,25,26,72

This is caused by changes in oil density.Recent results with live model oil (44.5 mole% n-

decane and 55.5 mole% methane) and live reservoiroil show an increasing optimal salinity with increasing

pressure.25,26 The phase behavior of live model oil ex-hibited two middle phases. This is also sporadicallyobserved for live reservoir oil. It is dicult to obtainoil-rich middle phases because a sudden change fromIII to II(+) occurs before reaching optimum. An in-crease in salinity does not enhance the solubilizationof oil in the middle phase.

Polymers

Polymers have been used in surfactant ooding as amobility buer behind the surfactant slug. Interac-tions between surfactants and polymers were mini-mized to avoid changes in phase behavior. Recentresults have shown that some polymers positively af-fect the surfactant phase behavior, lowering interfa-cial tension and increasing oil recovery.31

Summary

Based on the most commonly observed eects of theparameters aecting phase behavior, a table for driv-ing the phase transition in the direction

II()→ III→ II(+)

is summarized in Table 10.2, together with the gen-eral eect upon optimum solubilization indicated byarrows.

Table 10.2: Normal parameter change for II() → III→ II(+)-phase change.

Parameter change Opt.solub.

increase salinity -increase alkyl chain length of surfactant ↑decrease surfactant concentration∗ ↓decrease temperature ↑decrease pressure -increase alcohol concentration ↓decrease number of EO-groups ↓

*) In the presence of divalent cations.

10.3.3 Modelling and Simulations

A chemical simulator can be used to optimize surfac-tant slug composition and injection procedure, pro-vided the simulator has a an adequate phase-behaviordescription, taking into account all pertinent param-eters.Among the available chemical simulators,

UTCHEM80 and SCORPIO81 should be men-tioned. SCORPIO applies measured pseudoplanes,and its applicability in the North Sea reservoirs hasrecently been signicantly improved by Alvestad etal.,82 by a more direct use of measured data.Most of the discussion below, however, is based

on the phase-behavior description in UTCHEM. Itis modular and seems exible in handling phase en-vironment changes. The modularity simplied any

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252 CHAPTER 10. SURFACTANT FLOODING

inclusion of new phenomena and parameter descrip-tion. The phase-behavior description in UTCHEM isbased on submodels described by equations. It is rel-atively easy to incorporate new formulations, exten-sions or changes to the existing models. A possiblepractical advantage is that it requires less measuredinput data. A fairly good temporary match to thephase behavior can be done at an early stage basedon few data.

10.3.4 Representation of Simple Sur-factant Systems by Ternary Di-agrams

As described above, phase behavior of simple systemscontaining only three components: oil, pure water,and a single-component surfactant may be fully rep-resented by ternary diagrams. At constant tempera-ture and pressure, a given surfactant system will becompletely described by a binodal curve and tie linesin a xed ternary diagram.

II()-Phase Behavior

The II() case is illustrated in Fig. 10.11 with a Plaitpoint and tie lines. Normally, the Plait point for aII() system is located close to the oil corner. For allwater/oil ratios of practical importance in a surfac-tant ood of a reservoir, sucient accuracy will beachieved by placing the Plait point at the oil corner.This implies that the tie lines will intercept at the oilcorner, and that the excess oil phase is assumed to be100% oil. This will simplify the calculations of phasevolumes and phase compositions.The validity of placing the Plait point at the oil cor-

ner is demonstrated by the line IME on Fig. 10.16.Any dilution of an injected aqueous surfactant so-

excess oilphase

II(-) micro-emulsion

V 2

V 3

T

Water Oil

Surfactant

a

b

M

EC3M

C3T

C 1T

C 1M

I

Figure 10.16: II()-phase behavior.

lution (point I) by oil or water, will produce a mi-croemulsion composition far from the Plait point andbelow the line IME.Regarding a II() system with an overall composi-

tion given by CiT , (i = 1, 3) and phase volume frac-tions ϕo (excess oil phase) and ϕm (microemulsion).

From a geometrical evaluation of the ternary diagramin Fig. 10.16, the microemulsion volume fraction, ϕm,is expressed by the relationships:

ϕm =C3T

C3M=

b

a+ b. (10.15)

By this equation, the binodal curve may be deter-mined from a set of phase-behavior measurements.The binodal curve may be represented by an equa-tion with adjusted constants, or be in tabular form.Once the binodal curve is known, the microemul-

sion composition may be easily calculated for anyoverall system composition within the two-phasearea. The composition is given by the intersectionpoint between the binodal curve and the tie line. Thetie line, ETM, passing through the oil corner, is de-termined by the ratio

C3T

C1T=C3M

C1M. (10.16)

II(+)-Phase Behavior

The ternary diagram for the II(+)-phase behavior issimilar. But contrary to the II() case, where theactual part of the binodal curve is limited by the sur-factant concentration of the injected slug (line IMTEin Fig. 10.16), the actual microemulsion compositionsin a II(+) ood may cover all, or most of the binodalcurve. If the aqueous surfactant solution in Fig. 10.17(point I) is diluted with oil, it will enter the two-phase

T

Water Oil

Surfactant

ab

M 2

E 2

IP

M 1

E 1

excess waterphase

II(+) micro-emulsion

V 3C 3 MC 2 M

V 1C 3 1C 2 1

Figure 10.17: II(+) ternary diagram.

area in point E1. In this case, point E1 is locatedbelow the Plait point P , and the initial two-phaseequilibrium upon dilution with oil, is represented byan excess water phase at point E1, and a II(+) mi-croemulsion at point M1. Note that the volume ofthe microemulsion at point M1 will be zero.If the Plait point had been ignored (placed at the

water corner), the predicted phase behavior would bea II(+) microemulsion at point E1 in equilibrium withan excess water phase (of zero volume) located at thewater corner.

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10.3. PHASE BEHAVIOR 253

In a surfactant ood of a reservoir, a signicantpart of the surfactant slug will have a low content ofoil, especially with a II(+)-phase behavior at the frontof the slug where the oil is eciently displaced bythe oil external microemulsion. In simulations of sur-factant ooding, where II(+)-phase behavior occurs,the location of the Plait point should be included toachieve optimal results.The determination of the Plait point involves more

laboratory work, as also the phase compositions ofthe equilibrium phases must be known. For the over-all composition given by point T in Fig. 10.17, themicroemulsion volume fraction, ϕm is given by;

ϕm =C3T − C3W

C3M − C3W=

b

a+ b. (10.17)

The tie line passing through T is given by

C3T − C3W

C2T − C2W=C3M − C3W

C2M − C2W. (10.18)

In addition to the binodal curve, the tie line mustalso be represented by a table or correlation describ-ing the relation between the microemulsion and theexcess phase composition. The calculation of phasecomposition and volumes will also require some kindof iterative method.

III-Phase Behavior

For a III-phase system containing only three compo-nents, the microemulsion composition is given by theinvariant point I in Fig. 10.18. Any overall system

excess waterphase

III-phase micro-emulsion

V 3

V 1

W O

Surfactant

e

I

f

V 2excess oilphase

T

Figure 10.18: III-phase behavior.

composition within the triangle WOI, will producemicroemulsions of identical compositions, lying on theinvariant point. Assuming that the excess phases are100% oil and water, the microemulsion volume frac-tion for point T is given by

ϕm =C3M

C3T=

e

e+ f. (10.19)

If the composition of the invariant point is known,Eq. 10.19 gives the volumes of the microemulsion

phase. The volume of the excess phases are givenby mass balance for oil and water.The II-phase lobes are treated in the same manner

as real II() or II(+) phases.

10.3.5 Multicomponent Systems

Surfactant systems that could be used in a reservoirwill contain additional components making the phasebehavior more complex. Most important is that allreservoirs will contain some kind of electrolyte in thewater. There will be a competitive interaction be-tween the dierent cations present in both seawaterand formation water, with surfactant and also themineral surface.83

Most of the currently available commercial surfac-tants need some kind of cosolvent to achieve solubilityin high salinity water. Normally a short-chained al-cohol (C3-C5) is used. The alcohol will inuence thephase behavior, and also partition between all thephases present. However, the number of surfactantsthat do not need alcohol as a cosolvent is increasing.Let us consider a surfactant system containing sea-

water, a given surfactant/alcohol ratio, and an oil.By placing the surfactant + alcohol in the surfactantcorner and seawater in the aqueous corner, a phase-behavior map may be constructed in a ternary dia-gram. A typical ternary diagram with seawater, butwithout alcohol, is sketched in Fig. 10.19. With alco-hol, the phase behavior becomes even more complex.

Seawater Oil

Surfactant

I-phase

II(-)

II(+)-lobe

II(+)

II(-)-lobe

III-phase

II(-)-lobe

II(-)

Figure 10.19: Schematic phase behavior with seawa-ter (no alcohol).

The ternary diagram may be used to predict phasetype for a multicomponent system, as long as theoverall system composition lies on the same plane.But the phase compositions can no longer be repre-sented by simple binodal curves and tie lines. Addi-tional modelling tools are needed to calculate phasevolumes and compositions.It is also important to distinguish between real

II-phase behavior and II-phase lobes such as inFig. 10.19. The microemulsion in the II(+) and II()

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254 CHAPTER 10. SURFACTANT FLOODING

lobes can be regarded as type III-phase microemul-sions undersaturated with oil or water. They willtherefore generally have lower viscosities and less ten-dency of creating macroemulsions than real II phases.The microemulsion and excess phase compositions

will lie outside the ternary plane due to partitioningof cations and alcohol. Due to dierent phase veloc-ities in the reservoir and ion exchange with the min-eral surface, the overall system composition will soonmove away from the plane, and the ternary diagrammay predict wrong phase type.

Pseudocomponent Representation

Let us consider a three-phase, multicomponent sys-tem with a middle-phase microemulsion in equilib-rium with an upper-excess oil phase, and a lower-excess water phase. The two excess phases may be an-alyzed for all possible components and their compo-sition determined. Let the composition of the excesswater and oil phase be denoted as the aqueous, W ∗,and oleic, O∗, pseudocomponents. The microemul-sion composition is given by mass balance and is de-noted M∗. Furthermore, the maximum amounts ofW ∗ and O∗ are extracted from M∗, and the resid-ual is denoted S∗, the surfactant pseudocomponent.The extracted volumes are regarded as solubilizedamounts of W ∗ and O∗ in the microemulsion.A new ternary diagram where the three corners are

represented by the pseudocomponents W ∗, O∗, andS∗ can then be constructed. The microemulsion,M∗,will lie on the same plane as it is simply a mixture ofthe three pseudocomponents.Any mixture of the three pseudocomponents with

an overall composition inside the triangle W ∗O∗M∗,will produce equilibrium phases that are identical toW ∗, O∗, and M∗ in composition. This is evident,since any of the overall compositions may be obtainedby mixing the three equilibrium phases in dierentratios. The important conclusion is that any three-phase system in equilibrium may be represented in asimple way on a ternary diagram by the use of pseu-docomponents.If the pseudoternary diagram should be used in

phase-behavior modelling, then also the equilibriumphases in the II-phase lobes must lie on the sameplane. That is, the distribution of component be-tween the three pseudocomponents must be indepen-dent of phase type. This last assumption may at bestbe a good approximation, as it is most probable thatchanges in the micellar structure will have some inu-ence on the thermodynamic distribution equilibriumof ions and alcohol between surfactant, water and oil.Work by Biais et al.84 suggests that the depen-

dency upon phase type for distribution of alcohol isminor and can be ignored. The distribution of ions isless studied regarding pseudoplanes. Foulser et al.85

have reported some work with NaCl and Na2SO4.They measured phase compositions for a II(+) lobeon a pseudoplane determined from a three-phase so-lution. For the NaCl system, both the microemulsion

and the excess water phase stayed close to the plane,while for the Na2SO4 system the deviation was some-what larger, about 10% deviation in the excess wa-ter phase salinity. The measured alcohol content forboth the II(+) and II() lobes conrms the assump-tion above.The larger deviation with Na2SO4 is most proba-

bly due to the increased valence of the anion, SO2−4 .

However, the major part of anions in seawater andformation water are monovalent Cl−. For divalentcations, Ca++, Mg++, etc., no reported measure-ments on pseudoplane compositions are found.The conclusion is that the pseudoternary plane can

be used in surfactant phase-behavior description inthe same way as for the simple three-component sys-tem described above.

10.3.6 Modelling

In order to describe phase behavior by ternary dia-grams, the following must be determined(1) The binodal curve itself, and its variation when

the phase type is changed.(2) Which components that partition between two

or all three of the main components (water, oil andsurfactant) and the distribution of them.(3) The eect of any of the components on the

phase type and binodal curve.

Binodal Curve

The binodal curve can be described by the Handequations for both the microemulsion and the excessphase,86 here shown for the II() case,

C3M

C2M= A ·

(C3M

C1M

)B, (10.20)

C3O

C2O= Ae ·

(C3O

C1O

)F. (10.21)

If the Plait point is placed at the aqueous corner,Eq. 10.21 is ignored. The Hand parameters A and B,are measured in the laboratory.The chemical simulator UTCHEM uses a symmet-

ric form of Hand Eq. 10.20 for all phase types to de-scribe the binodal curve,87 i.e. B = −1, which makesthe curve symmetric around the line C1 = C2. Theintersection point between this line and the binodalcurve gives the maximum height of the binodal curve.At this point, C3MAX denotes the maximum amountof surfactant in the microemulsion along the binodalcurve.The relationship between C3MAX and A is found by

substituting for C3M in Eq. 10.20 by C3MAX, settingC2M = C1M and using C1M + C2M + C3M = 1,

A =

(2 · C3MAX1− C3MAX

)1−B

. (10.22)

UTCHEM needs input of C3MAX at zero eectivesalinity, CSE = 0, at optimum CSE , (CSEopt) and

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10.3. PHASE BEHAVIOR 255

twice CSEopt. The Hand parameter A is calculatedfor these salinities and interpolated for the actualsalinity of the system.There is no experimental reason for the choice of a

symmetric binodal curve, but the calculation of phasevolumes and compositions are somewhat simplied.The shape parameter B in the Hand equation could

be seen as a parameter describing the interactionforces between the micelles. If there were no such in-teraction forces between the micelles, for a II() mi-croemulsion, the solubility of oil, C2M/C3M , wouldbe independent of the surfactant concentration. Thiswould require B = 0 in Eq. 10.20, and the binodalcurve would be represented by a straight line throughthe aqueous corner.Due to the existence of such interaction forces, and

the fact that they become stronger when the sur-factant concentration is increased (closer packing ofmicelles), the solubility C2M/C3M will also increase.This means that B should be slightly less than zero.Recent measurements88 indicate a typical value forB ranging from -0.15 for II(), to -0.2 for II(+) mi-croemulsions.Fig. 10.20 shows the lower part of a II()

pseudoplane with measured points and calculated

98.27 % water (0.5 % NaCl)1.73 % IBA

99.71 % octane0.29 % IBA

93.73 % RL30116.27 % IBA

1.00.80.60.20.00.00

0.05

0.10

0.15

0.20

0.4

Figure 10.20: II() binodal curve on a pseudoternarydiagram. Measured points and calculated curves withB = −0.16 and B = −1 (symmetrical).88

binodal curve with B = −0.16 and B = −1. The sig-nicance of the shape factor B is even more obviouswhen the interfacial tension is calculated, Fig. 10.21.The binodal curve modelling in UTCHEM has been

extended to also include asymmetrical cases.88

Oil

The solubility of oil in water is very low and can be ne-glected. In surfactant environment, the oil is usuallytreated as a single component. Dierent oils will havedierent phase behavior, but once the oil is given, itis assumed that it will have no inuence on the phasebehavior. This assumption holds for pure oils with anarrow distribution in molecular size. Real reservoiroils may behave dierently.Austad et al.26 measured phase behavior for a one-

component surfactant with live crude oil from a NorthSea reservoir at dierent pressures, temperatures andsalinities. In the III-phase region, they observed two

0.200.150.100.050.00.001

.01

.1

1

10

100B = -1.0B = -0.16measured

Surfactant concentration, ml/ml water

Inte

rfac

ial t

ensi

on,

mN

/m

Figure 10.21: Interfacial tension vs. surfactant con-centration in a II() microemulsion. Experimentalpoints from Lohne.88

microemulsions, an upper and a middle phase in equi-librium with an excess water phase. When the salin-ity was increased, the volume of the upper phase in-creased while the middle phase decreased until II(+)-phase behavior was reached.A reasonable explanation for this existence of two

microemulsion phases in equilibrium with an excesswater phase, is that the composite oil separates sothat the oil in the upper phase has a lower molecu-lar weight forming a II(+)-type microemulsion, andthe oil in the middle phase has a higher molecularweight forming a III-type microemulsion. This kindof separation is probably enhanced by the very broaddistribution in live crude-oil composition. Simplied,it can be taken as a mixture of two main oils, pres-surized gas and dead crude oil, with a large dierencein molecular weight and other physical properties.At present, there exist no model to handle this sep-

aration of live crude oil.

Water and Salts

As a basis, the amount of water solubilized in pureoil is neglected. For alcohol-containing systems, somesalt-free water is solved by the alcohol in the oilphase.84 However, this will only have consequencesat very high alcohol concentrations, which is of nointerest in surfactant ooding.Several authors have reported an increase in the

excess water salinity of surfactant system with NaClas the only inorganic salt.84,85 Biais et al.84 sug-gest a model for this phenomenon that takes into ac-count both hydration of the polar head of the surfac-tant, and dissociation of counterions from the anionicsurfactant. The volume of hydrated water made thesmallest contribution to the increase in excess salin-ity. The dissociation of the surfactant is described byan equilibrium constant for the chemical equation

NaS Ki

S− + Na+, (10.23)

where NaS and S− are the associated and dissociatedsurfactant. An additional equilibrium constant wasintroduced to deal with the ratio of chemical potentialof salt ([Na+]·[Cl−]) between the microemulsion and

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256 CHAPTER 10. SURFACTANT FLOODING

the excess water phase. Another approach is to useelectrostatic considerations.85,89 The micelles havean anionic surface in equilibrium with a diuse ioniclayer where the concentration of Na+ decreases andCl− increases with increasing distance from the sur-face. This is illustrated in Fig. 10.22. The increase inbulk or excess salinity is due to the repulsion of Cl−

Ioni

c -

conc

entr

atio

n, m

eq/m

l

Distance from the micelle surface

Bulk solution orexcess phase

increase in excesssalinity

Initial water salinity

excesssalinity

CNa

AD

CCl

Figure 10.22: Distribution of Na+ and Cl− close toan anionic micellar surface.90

from the micelle surface.An estimation based on data from Foulser et al.85

gives a 3.6% rise in excess salinity with 2% surfac-tant in water. This agrees well with an average in-crease of 3% for 1.5% surfactant in water reported byProvoust.87 The increase depends on the molecularsize and surfactant structure.For the pseudocomponent model, it seems appro-

priate to indicate this eect as a fraction of salt-freewater in the surfactant pseudocomponent.In water containing divalent cations, the salinity

increase will be less due to a closer packing of ions inthe diuse layer. Thus, in seawater, this eect maybe neglected.

Ion Exchange

The presence of divalent cations as Mg++, Ca++,Sr++, or Ba++ increases the eective salinity dra-matically. The eect increases with increasing atomicweight (decreasing hydrated radius).64 Hirasaki83,89

investigated the ion exchange between Na+, Ca++

and surfactant by the electrostatic Donnan model andmeasurements. He found that a mass action typeequation could describe the ion exchange

C29S

C6S= KS · C3 ·

C29O

C6O, (10.24)

where C9O and C9S are the concentrations of mono-valent cations (Na+) in free form and associated withsurfactant; C6O and C6S are similarly the concen-trations of divalent cations (Ca++); C3 is the totalconcentration of surfactant. All units are convertedto [meq/ml water]; KS is an experimental constant.The concentration C3 is equal to the sum of C9S

and C6S . It has no practical consequences whether

the associated ions are assumed to be at the micellesurface or in a diuse ionic layer.A similar equation is used for the ion exchange with

the mineral surface in the reservoir.An important aspect is that the ratio between

Ca++ and Na+ concentration is much higher in themicroemulsion than it will be in an excess waterphase. Then, if phase separation occurs in the reser-voir due to dierent phase velocities, the overall salin-ity composition will change.Due to the competitive association between dier-

ent cations and the surfactant and the mineral sur-face, and the dierent eect on phase behavior, theion exchange model is extended to handle an arbi-trary number of mono-, di- and tri-valent cations.88

The model requires the presence of one monovalentcation (usually Na+), serving as a reference for the ionexchange. The general mass action equation is

CZ9SC6S(i)

= KS(i) · C(Z−1)3 · CZ9O

C6O(i), (10.25)

where Z is the valence of the cation C6(i). With Ncations present in addition to Na+, there will be Nsuch equations. In the presence of a mineral surface,another set of N similar equations must be solved.The mass equation describes the ion exchange very

well, provided the change in ionic strength and activ-ity not to large.

Alcohol Distribution

The distribution of alcohol between the three pseu-docomponents is reported by many authors, and iswell described by the accepted model of Biais et al.84

The model assumes the alcohol to be monomericin the aqueous and surfactant pseudocomponents,while in the oleic phase, self-association of the al-cohol molecules takes place. The amount of mono-,di-, tri-, . . ., and n-meric alcohol is described by theself-association constant K, which is assumed to beidentical for all levels of association. The fraction ofmonomeric alcohol in oil is

ϕoa1 =ϕoa

1 +K · ϕoa. (10.26)

The distribution of alcohol between water and oil,and surfactant and oil, is described by the partition-ing constants Kw and Km,

Kw =ϕwaϕoa1

=ϕwa · (1 +K · ϕoa)

ϕoa, (10.27)

Km =ϕsaϕoa1

=ϕsa · (1 +K · ϕoa)

ϕoa, (10.28)

where ϕsa is the volume fraction of alcohol in the mem-brane part of the micelle,

ϕsa =V sa

V sa + a · Vs, (10.29)

where a is the ratio between the mean molecularlength of alcohol and surfactant.

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10.4. LOSS OF CHEMICALS 257

Phase Calculation Procedure

The calculation procedure for a given overall compo-sition is summarized by

1. The pseudocomponents are determined by calcu-lating the partitioning of alcohol and cations.

2. The eect of alcohol and cations is taken care ofby calculating an eective salinity (the equiva-lent amount of NaCl to achieve the same phasebehavior) as a function of associated amount ofalcohol and cations with the surfactant.

3. The Hand parameter A is calculated as a func-tion of eective salinity and alcohol associatedwith the surfactant. The phase-type boundariesare input parameters in units of eective salin-ity. The shape parameter B is determined by theinterpreted phase type.

4. Phase volumes and compositions are calculated.

5. Calculation of physical properties such as phaseviscosities and interfacial tension, based on thephase compositions (and phase type).

10.4 Loss of Chemicals

10.4.1 Introduction

In Sec. 2.2.2 various retention mechanisms and funda-mental aspects related to adsorption of surfactants atthe solid/liquid interface have been discussed. In thepresent section, experimental methods and problemsin obtaining reliable retention data will be handled.Nearly 90% of the surfactants injected are believed

to be retarded by the formation when passing throughthe reservoir rock. Thus, only a small amount is ac-tive in lowering the interfacial tension between oil andwater. It is therefore extremely important to be ableto quantify the amount of surfactant needed for asuccessful chemical ood.In the later years, more advanced surfactant sys-

tems have been synthesized in order to meet thevarious reservoir conditions.62,67,91,92 Injection wa-ter (salinity/hardness), cation exchange ability of thereservoir rock, and temperature are very importantparameters in selecting the correct chemical formula-tion.In the petroleum literature, retention of surfactants

has very often been termed adsorption. However,the large amount of papers dealing with the subjecthas revealed that several retention mechanisms (ad-sorption, precipitation, and phase trapping) are ac-tive depending on the experimental conditions. Inthis section, adsorption is related to the retentionmechanism involving adsorption at the solid/liquidinterface, and in the following discussion the term re-tention will be used if the retention mechanism is notclearly stated.

10.4.2 Methods to Measure the Lossof Surfactants

In general, two methods have been applied in theliterature to study the retention of surfactants, i.e.static and dynamic experiments. Each of the meth-ods will be shortly commented below.

Static Retention

Static retention studies usually involve determinationof retention isotherms using crushed reservoir mate-rial or model reservoir minerals; 24 hours have oftenbeen suggested to be an appropriate equilibrium time.For single-component surfactant systems, the shapeof the isotherm, retention vs. surfactant equilibriumconcentration, will give information about the reten-tion mechanism.93 However, for commercial multi-component systems it is often dicult to relate theretention prole to a specic mechanism.94 The draw-back using this method is that it is nearly impossibleto determine the eect of residual oil on the retention,and furthermore, the pore geometry, which obviouslyis important in determining the access of the surfac-tants to the mineral surfaces, is lost.95

Static retention studies using cores can, accordingto Smidt et al.,96 be done by equilibrating the corewith surfactant solution and then ush it with iso-propyl alcohol followed by quantication of the sur-factants removed. This method raises several ques-tions. When is retention equilibrium established,and will the alcohol remove all the surfactant ini-tially present in the core? The reproduced amountof surfactant must also be corrected for the surfac-tant present in the pore volumes.

Dynamic Retention

Dynamic retention studies are generally conducted inthe following ways: (1) determination of the euentsurfactant concentration ratio, Cs/Cs0, at dierentcumulative injection volumes; (2) circulation of about20 pore volumes (PV) of solution through the core for24 hours;97 (3) extraction of the surfactant from thecore with an ecient solvent. During the equilibriumtime, one should use a realistic injection rate and besure that equilibrium is established.98

In the rst method, the retention can be calculatedfrom the plot of Cs/Cs0 vs. injected volume using anonretarding tracer, Fig. 10.23. The experiment isstopped when the euent surfactant concentration isequal to the initial concentration, i.e. Cs/Cs0 = 1.The retention process can in many cases be rathertime consuming.95 Depending on the ooding rate,the time the surfactants spend inside the core maybe too short to cause signicant change in the eu-ent concentration. In order to be sure that retentionequilibrium has been obtained, the surfactant solu-tion can be circulated through the core for severalweeks. Nonequilibrium adsorption will be handled inmore detail later. Concerning the other two meth-

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258 CHAPTER 10. SURFACTANT FLOODING

3210PV

Cs

/Cs

o

0.0

0.2

0.4

0.6

0.8

1.0

SurfactantTracer

Figure 10.23: Schematic euent prole of a nonad-sorbing tracer and a surfactant.

ods, the question is again: At what time is the ad-sorption equilibrium established? In the dynamic ex-periments, the eects of oil and pore geometry willbe realistic compared to real ow of surfactants inporous media.

10.4.3 Type of Surfactants

Petroleum and Alkyl/Aryl Sulfonates

A large amount of work has been published deal-ing with petroleum sulfonates. The reason may bethat they are the least expensive surfactants avail-able. However, petroleum sulfonates have been shownto have a rather low salt tolerance, especially formultivalent cations. Even though the surfactantsare quite soluble in the injection water, petroleumsulfonates will increase cation exchange between thebrine and the rock, causing increased concentrationof multivalent cations in solution. Maximum in theretention isotherm is very often observed, and So-masundaran et al.93 have described the abstractionof surfactants as a combined adsorption and pre-cipitation/resolubilization process. It has, however,been found that the hardness tolerance of alkyl sul-fonates99,100 and sulfates101,102 is enhanced in thepresence of an electrolyte solution. Thus, in generalthe hardness tolerance can be increased by increasingthe surfactant concentration above the critical micelleconcentration, CMC, or by adding a monovalent elec-trolyte, NaCl.93

Bae and Petric98 have shown that the dynamic re-tention of petroleum sulfonates in Berea cores de-pends on the ow rate and the surfactant concentra-tion. Static experiments also showed that the equilib-rium time varied with the concentration, and the sul-fonate retention isotherm went through a maximum,conrming a mixed retention mechanism. It shouldalso be noted that petroleum sulfonates may inter-act with multivalent cations to form a high-viscosity,sticky precipitate which is capable to partially orcompletely block the pores of a porous medium.103

Increased oil recovery may be observed, which wasaccounted for by the pore blockage phenomenon,that diverted the ood to previously unswept pores.Such a surfactant system is acting more like a gel-

blockage rather than a chemical slug moving throughthe porous medium.So far, retention behavior has been discussed with-

out oil present in the system. In the presence of oil,phase trapping, in addition to adsorption and pre-cipitation, can occur. The optimum condition for oilrecovery has been determined to take place when thesystem is hydrophilic/lipophilic balanced, HLB.104

A surfactant-rich middle phase is formed containingequal amounts of water and oil termed the III state. Asalinity requirement diagram proposed by Nelson16 isan important tool to keep the surfactant ooding sys-tem at the optimum. Surfactant retention/adsorptionfrom the excess water phase in equilibrium with themiddle phase is found to be at the minimum at theoptimum for oil recovery.105 However, due to min-eral dissolution and cation exchange on the clays,the HLB can be changed and the phase behavior isusually moved toward a water-in-oil microemulsion,termed II(+) state.106108 Divalent cations are shownto strongly inuence the microemulsion phase behav-ior through the formation of divalent cation sulfonatespecies, and increased retention can be related to for-mation of an upper-phase microemulsion.107 The sur-factant is trapped in the residual oil, and the chemicalood becomes more or less an ordinary water ood.The literature contains many conicting reports of

the eects of the oil/water ratio on surfactant loss.Meyers and Salter109 have summarized most of the re-sults, and the experiments conducted by the authorssuggest that both for static and dynamic retentionstudies the retention of surfactants is independent ofthe brine/oil ratio.Novosad110 has proposed a procedure to dierenti-

ate between surfactant retained in porous media be-cause of adsorption and surfactant retained becauseof unfavorable phase behavior. The method consistsof ooding the core consecutively with dierent or-ganic solvents. The method does not dierentiate be-tween surfactant adsorbed and precipitated by mul-tivalent cations.

Salt-Tolerant Surfactants

It is quite obvious from the discussion above thatmany of the problems related to the retentionof petroleum sulfonates can be avoided by usinghardness-tolerant surfactant systems; α-olen sul-fonates62 and ethoxylated olen sulfonates67 are be-lieved to have very high salinity (including divalentcation) tolerance. In the later years, alkoxylatedalkyl/aryl anionic surfactants, mostly sulfonates,have received increasing attention.24,95,111116 Thegeneral chemical formula can be represented by

R1 − (O− R2)x − SO−3 ,

where R1 is hydrophobic alkyl/aryl group, O−R2 isalkoxy group (usually O-CH2 CH2), and x is degreeof alkoxylation (ethoxylation).The surfactants can be used at seawater salinities

without any precipitation. In a system without oil

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10.4. LOSS OF CHEMICALS 259

present, the surfactant loss can be related only to ad-sorption at the solid/liquid interface.95,113,114 Theethoxy-groups, EO, are active in initiating the ad-sorption at the reservoir minerals by forming hydro-gen bonds between the ether oxygens of the surfac-tant and hydroxyl groups of the solid.113,114 Thus,contrary to the petroleum sulfonates, it is interestingto note that EO-sulfonates adsorb onto the net nega-tively charged quartz surface.114 At very low surfac-tant concentrations, well below the CMC, the adsorp-tion increases as the number of EO-groups increases.However, above the CMC the adsorption decreases asthe number of EO-groups increases.113,114 It is alsoreported that in some cases surfactant aggregates areformed as temperature or salinity is scanned, and theaggregates may be trapped in the porous media.111

Nonionic ethoxylated surfactants are also rathersalt tolerant. The adsorption at the rock surface israther high, and furthermore, small temperature gra-dients may cause phase trapping.117,118

Surfactant Mixtures/Cosurfactants

In order to overcome the poor hardness tolerance ofsurfactants, it has been observed that surfactant mix-tures may in many cases improve the cation tolerance.It is well known that the addition of ethoxylated non-ionic surfactants to alkyl/aryl sulfonates or sulfateswill decrease the precipitation boundary in the pres-ence of multivalent cations.119122 The eects havebeen explained in terms of mixed micelle formationbased on regular solution theory taking into consid-eration the counterion exchange at the micelles.Static adsorption studies of mixed surfactant sys-

tems on model reservoir minerals have shown thatsynergistic interaction between the surfactants takesplace during the adsorption process.114,123125 Inthe case of a nonionic/anionic surfactant system, itis found that the adsorption of the ionic surfactantonto kaolinite was enhanced by the presence of thenonionic surfactant and vice versa in the premicel-lar region. Decrease in the adsorption level of theanionic surfactant with increase in the nonionic sur-factant was observed above the CMC.125 Commer-cial products of ethoxylated sulfonates usually con-tain 10 to 40 mole% of the corresponding nonionicsurfactant. The total plateau adsorption of the sur-factant mixture on kaolinite and quartz increases asthe amount of nonionic surfactant increases.114 Dueto the dierent anities of the surfactants towardthe reservoir mineral in such systems, it is very likelythat chromatographic separation will occur in a dy-namic ooding process. This is one of the most seri-ous problems in using surfactant mixtures, especiallyin oshore reservoirs where the distance between theinjector and the producer is large.Very often, low molecular weight alcohols have

been used as cosurfactant in the chemical mixture.Beside a distribution of the alcohol among the variousliquid phases, coadsorption of alcohol and surfactanttakes place at the solid/liquid interface. An adsorp-

tion model for the coadsorption of alcohol and sur-factant has been obtained by using the lyotropic liq-uid crystalline D-phase as a model.126130 The struc-ture of the crystalline D-phase is characterized by X-ray, and it is described as amphiphilic double layersand intervening water layers. It is observed that theplateau adsorption of the surfactant decreases as theconcentration of alcohol increases.

10.4.4 Chromatograpy Eects

Using a multicomponent surfactant slug, the chro-matographic eects of the reservoir may change thecomposition of the surfactants in solution, which hasinuence on the interfacial tension between the reser-voir uids. Poor oil recovery can be the result. Theessential feature of surfactants, being able to ag-gregate in aqueous solution to form mixed micelles,makes it dicult to predict the chromatographic be-havior of these compounds. Theoretically, the mixedmicelles in the aqueous phase, which contain repre-sentatives of all the surfactant molecules in solution,are in equilibrium with surfactant monomers, surfac-tants adsorbed on the rock surface, surfactants in aneventual middle phase and surfactants trapped in theoil phase. Depending on the surfactant concentra-tion, dierent proportions of the surfactant moleculeswill appear in the micellar, monomer, and adsorbedstates.131

Based on experimental data of two isomericallypure anionic surfactants, Trogus et al.132 showedthat for continuous injection of the surfactant slug,high and low concentration slugs gave dierent break-through curves of the two surfactants. Later, Har-well et al.133 reported a slug model describing thechromatographic movement of surfactant mixturesthrough porous media. The model was based onthe results of Trogus et al.,132 the so-called pseu-dophase separation model, stating that the micellescould be treated as a separate equilibrium phase,134

and the adsorption plateau model, stating that onlymonomers are adsorbed. Harwell et al.135 veriedexperimentally that the pseudophase model and theplateau adsorption model predicted the surfactantchromatographic movement quite well using surfac-tant slug concentrations well above the CMC, whilebelow the CMC the predicted euent history was lessaccurate. However, it predicted the general trendthat the concentration waves of the less adsorbedspecies is rapidly moving at low concentrations. Veryrecently, Mannhardt and Novosad136 have publishedan adsorption model for the ow of binary surfactantmixtures in porous media, using the ideal mixed mi-celle theory and the surface excess concept. They alsoconcluded that above the CMC the micelle formationmay dominate the adsorption process.Miller et al.92 have suggested a dual surfactant

system consisting of an expensive EO-sulfonate anda cheaper secondary alkane sulfonate for enhancedoil recovery at high salinities. According to the au-thors, preliminary experiments suggest that in the

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260 CHAPTER 10. SURFACTANT FLOODING

dual surfactant system, the two surfactants are aboutequally strongly adsorbed on the rock, and they be-lieved that interactions between surfactants in suchmixtures tend to equalize the adsorption of the com-ponents. However, no experimental results were pre-sented to support the suggestion.It is encouraging to note that a dual surfactant sys-

tem developed by Exxon for the Loudon eld whichconsisted of a mixture of alkyl propoxy/ethoxy sul-fonates, appears to have an optimal salinity that isnot a strong function of surfactant concentration ordivalent ion concentration.116 Furthermore, the chro-matographic separation of components is expected tobe minimal with negligible impact on surfactant per-formance.Austad et al.137 have recently studied chromatog-

raphy eects of commercial products of ethoxylatedsulfonates containing 12 mole% of the correspondingnonionic surfactant. Both of the surfactant groupswere strongly polydisperse in the ethoxylation degree,Fig. 10.24. It is interesting to note that the ana-

0.0 0.2 0.4 0.6 0.8 1.0

x 102 minutes

x10

2vo

lts 5.0

4.0

3.0

2.0

1.0

Nonionic

10E

O9E

O8E

O7E

O6E

O5E

O

4EO

3EO

Sulfonate

Figure 10.24: HPLC chromatogram of the commer-cial 6EO-sulfonate mixture (12 mole% nonionic and88 mole% sulfonate) showing the polydisperse natureof the anionic surfactant.137

lytical chromatogram in Fig. 10.24 is based on ionpair interaction, and the relative retention times canbe regarded as a relative cation anity of the vari-ous EO-sulfonate isomers. The core oods were con-ducted by circulating the surfactant solution througha prepuried and oil-containing reservoir sandstonecore which contained about 20% detrital clay miner-als. At the experimental conditions, seawater salini-ties and 70 C, the nonionic surfactant was trappedin the oil phase and the anionic surfactant remainedin the aqueous phase. The surfactant concentra-tion was kept above the CMC during the oodingprocess. In the case of the precleaned core, an in-crease in adsorption of the low molecular weight EO-sulfonate isomers relative to the high molecular iso-mers was observed. No signicant preferable adsorp-tion of the EO-sulfonate isomers was detected for the

oil-containing core. Static adsorption onto kaolinitebelow the plateau showed a change in the selectiveadsorption of the dierent isomers, resulting in an in-creased adsorption of the high EO-number sulfonatesat very low equilibrium concentrations, Fig. 10.25.It should also be noted that the adsorption ratio

1098765432

5

10

15

20

Number of EO-groups

Nor

mal

ized

pea

k ar

ea, %

+

Sample 1Sample 2Standard

Figure 10.25: Normalized percentage area of the dif-ferent EO-sulfonates in the equilibrium solution as afunction of the number of EO-groups for the static ad-sorption onto kaolinite at 70 C and seawater salinity.The standard is the initial 6EO-sulfonate prole.137

between the nonionic and the anionic surfactantschanges all the time during the adsorption process inthe precleaned core. The change in the adsorption ra-tio has been related to the nonequilibrium adsorptionprocess and dierent anities of the two surfactantstoward the various reservoir minerals. This will bediscussed in the next section.

10.4.5 Nonequilibrium Adsorption

Experimental studies have shown that surfactant re-tention depends both on the owrate and on the sur-factant concentration.98 This means that the reten-tion is taking place under nonequilibrium conditions,and that the retention process is time dependent.The higher the concentration, the longer the equi-librium time required. These observations are basedon petroleum sulfonates, and it is known that pre-cipitation of the surfactants due to cation exchangebetween the brine and the rock may also take place,in addition to adsorption.It is also obvious that the mechanism of the

nonequilibrium adsorption of surfactants in porousmedia must be related to the geological propertiessuch as tortuosity and clay content of the core. Thetime it takes to achieve adsorption equilibrium willdepend on the owrate and the accessibility of thesurface. In sandstones, most of the surface area is re-lated to the clays. Authigenic kaolinite typically has aspecic surface area of 10 m2/g, while the specic sur-face area of a clay-free, medium-grained sandstone isin the order of 0.1 m2/g. Hence, in a medium-grainedsandstone containing 5 to 10 wt% kaolinite, approxi-mately 90% of the surface area is related to the kaoli-nite. A large part of the surface area in sandstones

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10.4. LOSS OF CHEMICALS 261

containing kaolinite, or other clays, will thus be as-sociated with the micropores in the sandstone. Theaccessibility of these pores will aect the time it takesto achieve equilibrium saturation. Of special interestis the surface area associated with the dead-end pores,since adsorption equilibrium in these pores will prob-ably depend on the diusion-controlled transport ofthe surfactant.Austad et al.95 have studied the kinetics of the

nonequilibrium adsorption of EO-surfactant systemsonto clay-containing reservoir cores by circulating thesurfactant solution through precleaned cores for sev-eral weeks. The decrease in the surfactant concentra-tion was monitored during the adsorption process. Atypical prole of the surfactant concentration vs. in-jected volume (PV) is shown in Fig. 10.26. The fourregions of Fig. 10.26 may be related to the followingprocesses:

1. Dilution with brine from the core and fast sur-factant adsorption.

2. Adsorption where there is a surfactant concen-tration gradient between the mixing bottle andthe solution inside the core.

3. Diusion-controlled adsorption where the surfac-tant concentration in the mixing bottle is nearlyequal to the solution inside the core.

4. A constant concentration indicating that naladsorption equilibrium has been obtained.

6005004003002001000

Injected volume, PV

Log

Cs

+ 3

1

2

34

0.4

0.6

0.8

1.0

1.2

1.4

Figure 10.26: Total surfactant concentration ver-sus injected volume (PV) at 80 C using the 6EO-surfactant system which contained 12 mole% nonionicsurfactant.95

The position of the break in the curve between re-gion 2 and 3 is depends on the ooding rate. Theslope of the curve in region 3, which has been calleddiusion-controlled adsorption, will depend on thenature of the micropores in the clay minerals. It hasbeen veried that the diusion-controlled adsorptionrate constant for cores containing mostly authigenickaolinite is larger than for cores containing detritalkaolinite. This is in line with the mineralogical prop-erties of the two clays, stating that the authigenic

kaolinite has a larger microporosity and more con-nected pores relative to the detrital kaolinite, which isbelieved to contain a large number of dead-end poresand a lower porosity.The rate of the diusion-controlled adsorption can

be described by a simple rst-order, kinetic ratepseudo-equation in the available adsorption area.Thus, a mechanism in line with these observationssuggests that only surfactant monomers are diusedthrough the throats of the micropores. It is also ob-served that the ratio between the anionic sulfonateand the nonionic alcohol in the ooding solutionchanges during the whole adsorption process, and thechange can be related to the relative anities towardsthe minerals of the cores (relatively, nonionic surfac-tants have greater anity towards the easily accessi-ble quartz surface and the opposite is true for the lessaccessible kaolinite surface), Fig. 10.27.

80060040020000

Ani

onic

/Non

ioni

c

Injected volume, PV (1 day = 59.48 PV)

2

3

1

7.0

7.5

8.0

8.5

9.0

Figure 10.27: The change in the concentration ra-tio between the anionic and the nonionic surfactantsduring the adsorption process.95

The eects of residual oil on the diusion-controlledadsorption have been tested by comparing the adsorp-tion rate of a precleaned core and an oil-containingcore, Sor = 0.31. 138 The two cores were sampled atthe same depth and they showed identical mineralcomposition, about 20% detrital clay (illite and kaoli-nite). The surfactant system, a 6EO-sulfonate systemcontaining 12 mole% nonionic surfactant, showed anoil-in-water microemulsion, II() state, in the pres-ence of oil, but most of the nonionic surfactant wastrapped in the oil phase. The results of the experi-ments conducted at 70C may be summarized as fol-lows:138

- Both the cleaned core and the core with residualoil and reservoir wettability showed two adsorp-tion regimes of surfactants, one fast adsorptionand one slow adsorption.

- A temporary delay in the adsorption processwas observed between the fast and the slowdiusion-controlled adsorption regime. The de-lay was 5 times longer for the oil-containing corethan for the cleaned core.

- The amount of surfactants adsorbed both inthe fast and the slow process is higher for the

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262 CHAPTER 10. SURFACTANT FLOODING

cleaned core than for the oil-containing core,Table 10.3.

- The quantities adsorbed in the diusion-controlled adsorption process correspond to 50and 25% of the total adsorption for the cleanedand the oil-containing core, respectively.

- The rate of the diusion-controlled adsorptionprocess is 2.3 times faster for the cleaned corerelative to the oil-containing core.

- The adsorption rate in the diusion-controlledregime was independent of the surfactant con-centration, Cs > CMC, and the area availablefor adsorption.

- A mechanism for the diusion-controlled ad-sorption process, which is in line with the ex-perimental results, suggests that only surfac-tant monomers diuse into the micropores. Therate-determining step is believed to be the dif-fusion of surfactants through the pore throats.

Table 10.3: Surfactant adsorption values at dierentoil saturations.

Γfast Γeq

mole/m2 mole/m2

So = 0.0 1.5 · 10−6 3.0 · 10−6

So = 0.31 1.0 · 10−6 1.4 · 10−6

10.4.6 Sacricial Chemicals

Various precautions acting with dierent mechanismshave been studied in order to reduce the retentionof surfactants, and the literature has been reviewedby Surkalo and Pouska.139 In most cases, a preushusing dierent chemicals is suggested, i.e. injectionof:

- Sodium chloride solution in order to reducehardness.

- Alkaline additives, sodium hydroxide, carbon-ates, phosphates, and silicates, to both decreasehardness and to render the rock more negativelycharged.140,141

- Sacricial chemicals, i.e. lignosulfonates142,143

and polybasic carboxylic acids144 that will ad-sorb and block the active sites of the rock.

Alkaline additives are mainly used in fresh water,and it cannot be used in hard water like seawater be-cause of precipitation. Surfactant loss can be reducedsignicantly (> 50% reduction) by pretreatment witha lignosulfonate preush.142 However, no signicantreduction is obtained when lignosulfonate is incorpo-rated with the chemical slug. It is also found that lig-nosulfonate causes dissolution of minerals to a muchgreater extent than brine or petroleum sulfonate, pro-ducing undesirable divalent cations.Supposing that adsorption is the only mechanism

by which surfactants are retained, a potential sacri-cial agent should show similar retention mechanisms

as the surfactant in order to be able to block theactive adsorption sites on the reservoir rock. Fur-thermore, the adsorption onto the active sites shouldbe nearly irreversible. Finally, the sacricial agentshould not aect the brine tolerance and the opti-mum phase behavior of the surfactant system.Beside direct electrostatic interaction between the

rock and the charged head group of the surfactant,the EO-groups of ethoxylated sulfonates and alcoholsare believed to play an important role in the adsorp-tion onto the mineral oxides.113,114 Hydrogen bondsbetween nondissociated silanol groups of the reser-voir rock and the ether oxygens of the surfactants aresuggested to take place. Polyethylene glycol, PEG, isa class of sacricial chemicals that can compete withthe surfactants towards some of the active sites of therock material.145 Although PEG is patented as a sac-ricial adsorbate by Texaco, very little work on thismatter has been reported in the literature.146 Thereason may be that PEG is believed to be a ratherexpensive chemical, and that ethoxylated sulfonateshave quite recently been reported used in a real eldtest.116 Austad et al.147 have evaluated PEG asa sacricial chemical towards ethoxylated sulfonates,and the results are summarized below.

Static adsorption studies. PEG will act as a sac-ricial agent regarding adsorption onto quartz andkaolinite, and the decrease in adsorption is greaterfor quartz than for kaolinite, Fig. 10.28.

Concentration of PEG-4000, wt%

Ads

orpt

ion

ratio

Quartz

Kaolinite

0.6

0.7

0.8

0.9

1.0

1.00.80.60.40.20.0

Figure 10.28: The fractional decrease in the total ad-sorption of the 6EO-surfactant system on quartz andkaolinite as a function of the PEG-4000 concentra-tion.147

The sacricial eect of PEG is related to its molec-ular weight and to some extent the concentration, es-pecially for the kaolinite. PEG of dierent molecularweights (400 to 10000) was tested, and the PEG-4000showed the best properties.Changes in pH between 7.5 and 3.5 have little ef-

fects on the sacricial behavior of PEG towards thequartz surface, while the sacricial behavior towardsthe kaolinite surface appears to decrease as the pH islowered.

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10.5. TEMPERATURE EFFECTS 263

Dynamic adsorption studies. The experimentswere performed in two dierent ways: (1) PEG-4000was added after adsorption equilibrium of the surfac-tant was established; (2) PEG-4000 was added to thesurfactant solution from the beginning of the ood.The surfactant solution was circulated through thecore for several weeks. The results can be summa-rized as follows:A desorption of the surfactant corresponding to 30

to 35% is observed by adding PEG-4000 to the sur-factant solution after equilibrium adsorption has beenestablished.The decrease in the surfactant adsorption is only

temporary, and a slow readsorption to the initialvalue takes place. During this period, a pressurebuildup over the core was observed, and it was relatedto phase separation of PEG-4000 at the hydrophilicmineral oxide surfaces, Fig. 10.29.

Injected volume, PV, (1 day = 29.24 PV)

Ads

orpt

ion,

mol

e/m

2 x

10-6

PEG-4000

8006004002000

TotalAnionicNonionic

0.0

0.5

1.0

1.5

2.0

Figure 10.29: Dynamic adsorption of the ethoxylatedsurfactant mixture onto a Berea core as a function ofinjected volume. At 250 PV, 0.4 wt% PEG-4000 wasadded; temperature, 80C; brine, synthetic seawater;pH, 6.9 to 7.1. 147

A minimum in the surfactant adsorption vs. timeprole is observed if PEG-4000 is present from thestart. Afterwards, there appears to be a linear in-crease in the surfactant adsorption with time. Also,in this case a slow phase separation of the PEG wasnoticed as the pressure builds up over the core.Experiments using oil-containing reservoir cores,

showed that the desorption eects of PEG is stronglydepressed, and in this case no pressure buildup overthe cores was noticed.In conclusion then, the experiments suggest that

PEG is not a potential sacricial chemical for ethoxy-lated sulfonates applied at reservoir conditions.Minssieux91 has developed a method to reduce the

retention of surfactants to be used in hard water. Thebasic idea is to remobilize surfactants retained in therock by means of a proper additive capable to desorbthem. The surfactant system consisted of a mixtureof alkylaryl sulfonate and a nonionic ethoxylated al-cohol containing eight ethylene oxide groups. Thedesorbing agent, a very hydrophilic ethoxylated non-ionic surfactant (30 EO-groups), was added to theinjection water behind the micellar slug. The desor-

bent was able to remobilize both of the surfactantsof the slug signicantly, i.e. 37 to 47% of sulfonateand 33 to 46% of the cosurfactant. The drawback is,however, that the two-component surfactant systemwill probably chromatograph in the reservoir.Simultaneous injection of surfactant and polymer

at a xed concentration ratio, termed Low-TensionPolymer Flood, LTPF, has been suggested by BP toreduce the loss of surfactants.31 The surfactant re-tention was reported to be less than 0.18 mg/g rock.The breakthrough proles showed that the polymerwas slightly ahead of the surfactant, and in that waythe polymer may act as a sacricial agent. However,interactions between the polymer and the surfactantmay also be responsible for the low retention of sur-factant. The results are interesting and future workto understand the phenomena should be done by look-ing into the basic work on the subject which has beenreviewed by Saito148 and Goddard.149,150

10.5 Temperature Eects

Signicant temperature gradients will arise when lowtemperature seawater is injected into a high temper-ature reservoir and inuence properties of reservoirrock and uids.151 Heat transfer between the in-jected uid and the reservoir uids and rock causesthe temperature front to travel slower than the uidfront.9 This means that uids injected after a periodof cold water injection will move through a temper-ature gradient as they move into the reservoir. Theheat transfer is controlled by the heat capacity, con-ductivity, and volumes of uid and rock involved. Ifthe thermal conductivity is nite, uid injection ratewill also aect the ratio of the temperature- and uid-front velocities.This section is limited to the eect of temperature

gradients on properties of reservoir uids, excludingphenomena like thermally induced fractures and theirinuence on injectivity.Relevant processes for North Sea reservoirs are

standard waterooding, polymer ooding, surfactantooding, polymer-gel operations and single-well sur-factant tracer tests. The eect of temperature gradi-ents on the key parameters of these processes will bebriey discussed.

10.5.1 Standard Waterooding

Water viscosity decreases from 1 cp at 20 C to 0.28cp at 100 C. The initial injected water temperaturetherefore has a positive eect on the mobility ratio,when the front is close to the injection well, but anegative eect on injectivity.Precipitation of barium and strontium sulfate takes

place when sea- and formation water are mixed andmay aect both injectivity and productivity throughpermeability reduction. It occurs both at high andlow temperatures, but the rate of precipitation in-creases with temperature.152,153 Temperature and

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264 CHAPTER 10. SURFACTANT FLOODING

the degree of mixing increases with distance from theinjector and the problem is therefore normally mostsevere close to producers. Injection of scale inhibitorsin zones close to the producers is normally necessaryto prevent dramatic reductions in productivity.

10.5.2 Polymer Flooding

Polymers are used to achieve favorable mobility ratiosduring water- or surfactant ooding. It is thereforeessential that the viscosity of the polymer solutionis not reduced during the ooding period, which forNorth Sea reservoirs can be several years. Temper-ature will aect polymer viscosity both with respectto the change in state of energy and temperature-dependent chemical breakdown of the polymer chain.The high viscosity of the polymer solution may reducethe injectivity dramatically and lead to low injectionrates. But, as demonstrated by Sorbie et al.,154 theviscosity of the polymer solution is very temperaturedependent. Increased injectivity may therefore begained by preheating the injection uid.The long-term, temperature-dependent stability of

polymers has been studied for the promising polymersxanthan and scleroglucan.155,156 At unfavorable con-ditions, which may be present in real eld cases, thexanthan viscosity decay-time constant, τ , as denedby

µ?p = exp(−t/τ), (10.30)

can be in the order of days. At optimum conditions,however, the decay-time constant can be in the orderof years, illustrating the level of uncertainty in pre-dicting polymer viscosity. The decay-time constantdecreases by a factor of more than 100 when the tem-perature increases from 65 C to 120 C.155 The sametendency is displayed by scleroglucan,156 though thispolymer seems to be slightly more temperature resis-tant than xanthan.

10.5.3 Surfactant Flooding

Surfactant is injected in order to increase the micro-scopic displacement eciency by lowering the residualoil saturation which depends on phase behavior, in-terfacial tension, and surfactant retention. The mostdominant eect of temperature is on the phase behav-ior.20,23,25,26,110 It is generally accepted that anionicsurfactants, which are probably the most promisingfor North Sea conditions, at high or optimum salini-ties (II(+) or III) move towards II() as the tempera-ture increases. This behavior has been observed bothwith and without cosurfactant.At low salinity, the II()-case, the phase behavior is

more complex.26 Since surfactant systems normallyare designed to be near or at optimum at reservoirconditions, the above discussion implies that the sur-factant system may exist as II(+) in a zone close tothe well during injection.Surfactant retention is caused by precipitation,

phase trapping and adsorption. At optimum condi-tions, only the latter is considered as a major source of

surfactant retention.157 Both increasing and decreas-ing adsorption has been observed as temperature in-creases.20,110 When an upper-phase microemulsionis present, which may be the case at least in thecool, near-well area, the dominant surfactant reten-tion mechanism will be phase trapping.110 Low tem-perature will have a positive eect on solubility andphase viscosity. Both increase with decreasing tem-perature and enhance the initial capillary number andmobility ratio.

10.5.4 Polymer Gel

Polymer gel is normally used to block high-permeablezones. Key factors are gelation time and gel strength.The gelation time is a critical parameter with respectto planning. Overestimation may lead to formation ofgel in tubing, while underestimation may imply thatwells are set on production or injection before the gelplug is properly established. Kolnes et al.158 showthat the gel time can be expressed by

tg = exp(a+ b/T ), (10.31)

where T is the temperature (K) and b is approx-imately 5000 using Cr(III) as crosslinker. It istherefore necessary to take the temperature gradientaround the well into careful consideration when de-ciding which crosslinker concentration, injection rateand shut-in time to use. If the exact temperaturegradient is unknown, the operation may easily fail.

10.5.5 Single-Well Surfactant TracerTests

Because of the large costs involved in (future) NorthSea surfactant ooding projects, the eciency willprobably be evaluated by small-scale eld test. Theresidual oil saturation before and after the surfac-tant ood is measured by interpretation of the returnproles of injected partitioning and nonpartitioningtracers. The partitioning tracer is generated in thereservoir, normally through hydrolysis of an ester.159

Both the hydrolysis rate and the partitioning coe-cient are temperature dependent. Depending on themethod of interpretation, one or both of these param-eters are taken into consideration when calculatingthe residual oil saturation. Temperature changes dur-ing the tracer test will make the interpretation of thetracer return proles dicult and may lead to incor-rect residual oil saturation. Even if the temperatureprole is constant during the test, the interpretationmay be dicult because the prole is normally notknown.

10.5.6 Relative Permeability

Several authors have investigated the eect of temper-ature on the oil-water relative permeability.53,160162

The general, observations are that residual oil satu-ration decreases, residual water saturation increases,

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10.6. FIELD EXAMPLES 265

relative permeability to oil increases, and relativepermeability to water decreases as temperature in-creases. It is also observed that the eect of temper-ature on interfacial tension is moderate and cannotexplain the observed changes in residual saturationsand relative permeability. Possible explanations arechanges in wettability and viscous instability.

10.6 Field Examples

10.6.1 Introduction

Surfactant ooding has been proven on both labora-tory and pilot scales as an EOR process having a highpotential of oil recovery. This is due to the capabil-ity to reduce oil saturation in the oodable reservoirportions to a much lower level as compared to water-ooding. However, it is an expensive process, and anyapplication of surfactants to increase the oil recoveryin a specic eld normally requires high oil prices.

10.6.2 Oshore Application

Operations in an oshore environment put severe con-straints on surfactant ooding projects. Limited stor-age capacity on platforms create problems for thedaily injection of large quantities of surfactant. Forinstance, injection at 3000 m3/d of a 2% surfactantsolution, implies the handling of around 60 tons ofchemicals per day. In most cases, the storage re-strictions will limit the surfactant injection to a pilottest.22,163,164

The long lead time normally associated with o-shore installations therefore requires early planningof any surfactant ooding project. This may be re-garded as a major disadvantage compared to conven-tional water or gas injection.22,164

The high cost of eld development and production,together with the severe constraint on early plans,increase the nancial risks associated with a tech-nically promising but expensive surfactant oodingproject. Therefore, there is a need to carefully de-velop a strategy for surfactant ooding in each indi-vidual case.22,164

10.6.3 Surfactant Injection Strategy

Theoretically, the ultimate residual oil saturation tosurfactant ooding is the same in secondary and ter-tiary operations, but the reservoir may respond dif-ferently.For secondary operations, two factors introduce a

delay in the production of additional oil:9

1. The connate water, which is not saturated withsurfactant, has to be miscibly displaced by the aque-ous surfactant slug.2. Adsorption of surfactant onto the rock will re-

duce the velocity of the surfactant by the followingfactor,

a =1− φφ× ρr ×

Γ

Cs, (10.32)

where a is the velocity reduction factor due to surfac-tant adsorption, φ is porosity, ρr is density of rock,Γ is adsorption (weight of surfactant/weight of rock),and Cs is surfactant concentration. Thus, the veloc-ity reduction increases with Γ and decreases with Cs.Typical values of these parameters for the Brent

formation are: φ= 0.33, ρr = 2650 kg/m3, Γ = 0.0005kg/kg, Cs = 25 kg/m3.For this set of parameters, Eq. 10.32 yields a veloc-

ity reduction of 0.11, which means that the velocityof the surfactant in the porous medium is divided by1.11, compared to the ideal case without adsorption.Schematically, 0.11 pore volume of water is requiredto drive secondary oil ahead of the surfactant ood.Taking into account the additional 0.10 pore vol-

ume of connate water present in the Brent forma-tion, the production of surfactant oil will be associ-ated with the production of 0.10 + 0.11 = 0.21 porevolume of water.For tertiary operations, adsorption plays the same

role. In addition, dilution of surfactant in water maybe more active, due to the larger water saturationpresent and its high mobility. Such a dilution canreduce the mobilization of oil in the case of low, con-centrated surfactant solutions. Propagation of themicellar solution will be governed by permeabilitycontrasts, possibly without smoothing gravity eects.This normally necessitates the injection of chase wa-ter thickened by a polymer behind the micellar slug,in order to improve the volumetric sweep eciency.A signicant time lag will be expected between thestart of the surfactant injection and the breakthroughof the surfactant oil bank. Simplied calculations in-dicate a time lag corresponding to 0.2 to 0.3 of thepore volume.A comparison of secondary and tertiary injection

strategies indicate that a better performance can beexpected from surfactant ooding during secondaryoperations.

10.6.4 Data Needed

Reduction of Residual Oil Saturation

The major unknown in the evaluation of surfactant oilrecovery is the reduction of the residual oil saturation,compared to water ooding,

∆So = Sorw − Sorc, (10.33)

where Sorw is residual oil saturation after waterood-ing, and Sorc is residual oil saturation after a surfac-tant (chemical) ood.The volumetric sweep eciency can reasonably

be estimated (for secondary or tertiary operations)through the usual reservoir calculations and reser-voir modelling studies.9,11,21 Residual oil saturationafter waterooding should be checked during water-ood operations through logging or single-well tracertests.165,166

Residual oil saturation after a surfactant ood ispartly an operational parameter, depending on com-

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266 CHAPTER 10. SURFACTANT FLOODING

position and quality of the micellar solution. After anoptimal surfactant system has been designed in thelaboratory, the quality of the system must be checkedat eld conditions. A rst-stage eld test may be theinjection of a surfactant slug into a conned sand ina single well, combined with the measurement of Sorbefore and after the surfactant injection. This willgive an in-situ measurement of the eciency of thesurfactant system at eld conditions.167170

Retention of the surfactant is a crucial parame-ter governing the amount of chemical needed. Thisparameter is measured in the laboratory but shouldalso be checked in eld conditions. Small-sized pilottests as mentioned above, can provide eld retentiondata.116,171176

Reservoir Description

Information about geometry of fault blocks, in-placeoil and water volumes, and possible reservoir connec-tion through nonsealing faults is vital for the selectionof reservoir sections that are candidates for surfactantooding. Early reservoir description techniques thatnormally are used for optimizing waterood opera-tions, may be used for the evaluation of the surfactantoil potential. Implementing a tracer program duringthe water injection period can signicantly increasethe knowledge of the ow pattern in the reservoir.All the usual information acquired from drilling, welltesting, injection and production, should be consid-ered.9,116,171176

Laboratory Data

For the planning of a surfactant pilot test, labora-tory studies covering optimization of surfactant sys-tem, evaluation of surfactant retention in reservoirrock, and evaluation of recovery eciency of the op-timized surfactant system have to be carried out. Atoptimized conditions, the interfacial tension betweenthe microemulsion and oil or water should be in therange of 10−6 N/m. In addition, studies of thermalstability, slug-size optimization, and evaluation of theneed for mobility control buers must be performed.Detailed laboratory programs have to be designed ineach individual case.9,21,116,171176

Fig. 10.30 shows production and saturation prolesvs. pore volumes injected during water and surfactantcore-ood experiments. Half a pore volume of 2% sur-factant solution followed by 1.5 pore volume of seawa-ter brings the residual oil saturation down from 36%after waterooding to 8% after the surfactant ood-ing. During the course of the experiment, the oil cutincreased from zero at the end of the waterood to50% during the surfactant injection.Numerical simulation and history matching of the

core-ood experiment using a multicomponent chem-ical ooding simulator may yield important informa-tion about relative permeability, phase behavior, vis-cosity, pressure buildup, and surfactant retention dur-ing the core ood. This is vital information for the

100

0

100

0

1.0 2.0 3.0

1.0

0.5

0.0

water

Waterflood Surfactant Waterflood

water

Production

%W

atersaturation

Pore volume

Irreducible water saturation

oil oil

oil

Figure 10.30: Production and saturation during wa-ter and surfactant core ood.22

pilot planning and eld evaluation work. However,the scale-up procedures of laboratory-measured datato eld use is still a matter of discussion in the liter-ature.

Single-Well Injection Test (SWIT)

Injection and back production of a surfactant slug ina single well in a conned reservoir sand will give rel-evant in-situ measurements of the recovery eciencyof the surfactant system. In addition, careful datasampling in the back production period can give rele-vant in-situ measurements of retention, and potentialproduction problems due to back-production of mi-croemulsion can be detected. However, a small-scale,single-well test is not expected to yield any informa-tion about the volumetric sweep eciency.167,168

The data expected from a SWIT are Sorw, Sorc,reservoir temperature after waterooding, and injec-tion pressure. From these parameters, the eective-ness of the surfactant system can be determined. Bymodelling and history matching the test, realistic elddata for relative permeability, microemulsion viscos-ity etc. can be expected.

Risks

An economical risk evaluation of a large surfactantproject rst requires reduction of the technical uncer-tainties to an acceptable level. This implies to carryout a pilot test in order to acquire suciently accurateinformation about oil recovery (absolute and incre-mental), production prole, and to gain operationalexperience. The pilot will at least require two specif-ically designed wells, one injector and one producer,preferably in a conned area, and with interwell dis-tances reduced compared to regular waterood wellspacing.

10.6.5 Planning

In order to combine the need for reliable data withpossible reduction of acquisition costs and projecttiming for a possible surfactant ooding in the eld,the following steps are recommended:

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10.6. FIELD EXAMPLES 267

1. Perform a screening study in order to identifythe most promising elds in a given region for surfac-tant ooding.2. As early as possible, perform a single-well sur-

factant injection test in a conned area of the selectedreservoir to evaluate the technical eciency and theretention of the surfactant system.3. Based on the results from the single-well surfac-

tant injection test, design and implement a two-wellpilot surfactant test. The total duration of the pilotshould not exceed two years. The information fromthe pilot will be used to evaluate the feasibility of alarge-scale surfactant ooding project.4. Perform a feasibility study of a large-scale sur-

factant ood, including the results of the previousthree steps. The feasibility study will require:

• optimization of the surfactant composition in thelaboratory

• a specic reservoir engineering study with a nu-merical simulation model

• evaluation of the source and cost of chemicals• plans for surface facilities and storage• an economical risk analysis.

10.6.6 Forecasting

In the following, a eld performance forecast is carriedout based on limited reservoir data. This is usuallythe case in the early planning period of a surfactantooding project. These calculations should be a rstevaluation, preceding more sophisticated numericalsimulation work.The data used are typical of several elds, Ta-

ble 10.4, in the North Sea.

Table 10.4: Oil reserves (MSm3) for the four largestNorwegian North Sea oil elds.163

Field Oil In Recoverable RemainingPlace Oil Oil

Statfjord 1030 475 555Gullfaks 450 180 270Oseberg 450 200 250Snorre 450 110 340

The petrophysical data and uid properties, re-spectively, are given in Tables 10.5 and 10.6. Thefollowing reservoir data are used in the calculating:

Vp = 129.2× 106m3 φ = 0.33N = 87.8× 106Sm3 Γ = 0.5mg/g

Npw = 36.9× 106Sm3 ρr = 2.65ton/m3

Bo = 1.25Rm3/Sm3 EV = 0.7Sorw = 0.35 EMB = 0.7Swi = 0.15

Additional oil recovery by chemical ooding,

ERc = (Npc −Npw)/N,

Table 10.5: Reservoir data for the four largest Nor-wegian North Sea sandstone reservoirs.

Statf. Gullf. Oseb. SnorreDip (deg.) 7.0 12.0 8.0 -φ (%) 23.0 33.0 25.0 25.0T (C) 100.0 70.0 100.0 92.0krw(Sor) 0.13 0.20 0.35 0.40Sorw 0.30 0.35 0.28 0.28Swi 0.28 0.07 0.20 0.25µo (cp) 0.49 1.11 0.43 0.47µw (cp) 0.30 0.45 0.34 0.34M 0.20 0.49 0.44 0.54

can be calculated from,

ERc = EV × EDc × EMB ×Sorw

1− Swi. (10.34)

Volumetric sweep eciency, EV , is the volume ofoil contacted divided by the volume of target oil. EVis a function of slug size, retention, Γ, and reservoirheterogeneity based on Dykstra-Parsons coecient,VDP.9 Simulation studies indicate an ultimate oil re-covery by water ooding of 42% of OOIP, and EV isestimated at 0.70.EMB is mobility buer eciency reecting loss of

microscopical displacement eciency in the reservoir,depending on EV and the reservoir heterogeneity.EMB is subjected to a high degree of uncertainty andis estimated at 0.70, from simulation.The microscopic displacement eciency by chemi-

cal ooding, EDc, is dened by:

EDc =Sorw − Sorc

Sorw. (10.35)

Surfactant ooding at reservoir conditions in Brentcores indicates microscopic displacement eciencyaround 0.75, Fig 10.30.Chemicals are lost in the reservoir by adsorption

and other retention mechanisms like trapping in wa-ter, trapping in oil and trapping in dead-end pores.An useful way to estimate the amount of surfactantrequired is described by Lake.9 The overall surfactantretention can be correlated with the clay content ofthe reservoir sand as shown in Fig. 10.31. The Brentsands are usually clean with low clay content, and theadsorption Γ = 0.5 mg/g is estimated.The amount of surfactant needed (in tons) to sat-

isfy the retention in the waterooded part of thereservoir is

Vp

(1− φφ

)EV Γ ρr.

The results are given in Table 10.7. The additionaloil calculated from Eq. 10.34 varies from 2.9% forSorc = 0.30 to 17.3% for Sorc = 0.05, and the sur-factant eciency varies from 10.5 to 62.4 Sm3 oil/tonsurfactant.For optimal surfactant usage, the amount of sur-

factant injected should be large enough to contact

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268 CHAPTER 10. SURFACTANT FLOODING

Table 10.6: PVT-data for Brent uids

Fluid B, Rm3/Sm3 GOR, Sm3/Sm3 ρ ∗ µ (cp) c (1/bar)×10−4

Oil 1.25 95.5 0.76 0.94 1.595Water 0.987 - 1.02 0.41 0.44

* Measured at initial reservoir conditions, pi = 290 bar, Ti = 70CBubble point pressure: pb = 209.5 barWater Salinity: 38000 ppmRock compressibility: 0.44× 10−4/bar

0 0.05 0.10 0.15 0.20

1.0

0.5

0

Field data

Lab data

Lab data regression lineField

data regression line

Weight fraction clay

Act

ive

sulf

onat

ere

tent

ion

(mg/

g)

Figure 10.31: Overall surfactant retention correlatedwith reservoir clay.9

all of the pore volume, but small enough to preventexcessive production of the surfactant.9

Table 10.7: Additional oil calculated according toEq.10.34, and the corresponding surfactant eciencyfactor, Π

Sorc EDc ERc Π ∗

0.30 0.143 0.029 10.50.25 0.286 0.058 20.90.20 0.429 0.087 31.40.15 0.571 0.115 41.50.10 0.714 0.144 52.00.05 0.857 0.173 62.4

*: The surfactant eciency factor,

Sm3/ton surfactant

Nomenclature

A = parameter in Hand equationAe = Hand parameter for excess phasea = length of line on a ternary diagram, be-

tween overall and II phase microemulsioncomposition

= parameter, dimensionless= velocity reduction factor due to surfactant

adsorption, dimensionlessas = surfactant adsorption (g surfactant/g

rock)ai = free parameter, i = 0, 1, 2B = parameter in Hand equation

= formation-volume factor, Rm3/Sm3

b = length of line on a ternary diagram, be-tween overall and excess phase composi-tion

= parameter, Kbi = free parameter, i = 0, 1C = concentration, amount/volumeCi = concentration of component i, (i = 1,

aqueous; i = 2, oleic; i = 3, surfactant)CiO = concentration of component i in the oleic

phaseCiM = concentration of component i in the mi-

croemulsion phaseCiT = total concentration of component iCiW = concentration of component i in the aque-

ous phaseCSE = eective salinity, meq/ml waterC6O = divalent cation in free form, meq/ml waterC6S = divalent cation associated with surfactant,

meq/ml waterC9O = monovalent cation (NaCl) in free form,

meq/ml waterC9S = monovalent cation associated with surfac-

tant, meq/ml waterc = compressibility, 1/bar

Ds = surfactant adsorption in pore volumes ofinjected solution.

E = sweep eciencyEDc = microscopic displacement eciency by

chemical oodingEMB = mobility buer eciencyEV = volumetric sweep eciencyERc = additional oil recovery by chemical ood-

ing, fractione = length of line on a ternary diagram, be-

tween overall and III phase microemulsioncomposition

= exponent, parameterF = Hand parameter for excess phase= So(1− Sw)/(Sw + So)

f = length of line on a ternary diagram, be-tween overall and the water-oil baselinefor III phase system

fw2 = fractional ow of water at Sw2

fw3 = fractional ow of water at Sw3

K = self-association constant for alcohol in oilKi = equilibrium constant for the dissociation

of anionic surfactant in water, meq/mlKS = cation exchange constant (units depend

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REFERENCES 269

on Z)Kw = alcohol partitioning constant between wa-

ter and oilKm = alcohol partitioning constant between sur-

factant and oilkr = relative permeabilityM = mobility ratio, dimensionless

= composition of microemulsionN = dimensionless number

= OOIP, Sm3

Np = oil volume produced, Sm3

Npw = oil volume produced after waterooding,Sm3

Npc = oil volume produced after chemical ood-ing, Sm3

O = composition of excess oleic phaseR1 = hydrophobic alkyl/aryl groupS = saturation, dimensionless= composition of surfactant

Sw2 = saturation of water in waterood shockfront

Sw3 = saturation of water in surfactant oodshock front

T = temperature, K or Ct = time, d or su = Darcy velocity, m/sV = volume, m3 or ml

VDP = Dykstra-Parson coecient, dimensionlessVp = pore volume, m3

W = composition of excess water phaseXD = dimensionless lengthx = number of ethoxy groupsZ = ionic valenceα = constantΓ = adsorption (kg surfactant/kg rock)γ = interfacial tension, N/m

∆S = residual oil saturation reductionµ = viscosity, Pa·sΠ = surfactant eciency factor, Sm3/tonρ = density, kg/m3

σ = solubilization parameter (solubilized vol-ume/surfactant volume)

τ = viscosity decay constant, daysφ = porosity, fractionϕ = volume fraction

Subscripts

a = alcohola1 = monomeric alcoholc = chemical (surfactant) or connate or capil-

lary or criticalD = dimensionless or displacingd = displacede = excess (phase)g = gel or gasi = initial= component label

j = phase labelm = microemulsiono = oil

p = polymer or porer = residual or relative or rocks = surfactant or stabilityw = water0 = base value

Superscripts

0 = endpointo = oil pseudocomponents = surfactant pseudocomponentw = aqueous pseudocomponent∗ = pseudocomponent? = normalized

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[162] Hing, Y.L. and Mungan, N.: Eect of Tem-perature on WaterOil Relative Permeability inOilWet and Water-Wet Systems, paper SPE4505 presented at the 1973 SPE Annual FallMeeting, Las Vegas, Sept. 30Oct. 3.

[163] Thomassen, P.R. and Skontorp, O.: ModernReservoir Management a Contribution to Im-proved Recovery, Proc., 1987 European Sym-posium on IOR, Hamburg, Oct. 2729, 65774.

[164] Hawes,R.I. and Skontorp,O.: The Challengeof Improved Recovery from O-Shore Fields,Proc., 1989 European IOR-Symposium, Bu-dapest, April 2527, 5362.

[165] Causin, E., Rochon, J., and Marzorati, D.:Field Measurements of Remaining Oil Satura-tion,paper SPE/DOE 20260 presented at the1990 SPE/DOE Symposium on EOR, Tulsa,April 2225.

[166] Briggs, P.J., Grist, D.M., and Woodhouse,R.: Locating the Remaining Oil in Produc-ing Fields, presented at the 1990 InternationalONS Conference and Exhibition, Stavanger,Aug. 2831.

[167] Sheely, C.Q. Jr. and Baldwin, E. Jr.: SingleWell Test for Evaluating Chemical EnhancedOil Recovery Processes using Single Well TracerTests, paper SPE 8838 presented at the 1980SPE Annual Technical Conference and Exhibi-tion, Dallas.

[168] Holland, K.M. and Porter, L.T.: Single-WellEvaluation Program for Micellar/Polymer Re-covery, Main and West Pools, West CoyoteField, California, paper SPE 11990 presentedat the 1983 SPE Annual Technical Conferenceand Exhibition, San Fransisco, Oct. 58.

[169] Park, Y.O.J., Deans, H.A., and Tezduyar,T.E.: Thermal Eects on Single-Well Chem-ical Tracer Tests for Measuring Residual OilSaturation, paper SPE 19683 presented at the1989 SPE Annual Technical Conference and Ex-hibition, San Antonio, Oct.811.

[170] Tang, J.S. and Harker, B.: Mass BalanceMethod to Determine Residual Oil Saturationfrom Single Well Tracer Test Data, JCPT(March-April 1990) 11524.

[171] Bragg, J.R., Gale, W.W., and McElhannon,W.A. Jr.: Loudon Surfactant Flood Pilot Te

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276 CHAPTER 10. SURFACTANT FLOODING

st, paper SPE/DOE 10862 presented at the1982 SPE/DOE Symposium on EOR, Tulsa,April 47.

[172] Ware, J.W.: Salem Unit Micellar/PolymerProject, paper SPE 11985 presented at the1983 SPE Annual Technical Conference and Ex-hibition, San Francico, Oct. 58.

[173] Bourdarot, G., Sardin, M., and Putz, A.:Chateaurenard Field Test Recovery Mecha-nisms and Interpretation, paper SPE/DOE12685 presented at the 1984 SPE/DOE Sym-posium on EOR, Tulsa, April 2225.

[174] Farrell, H.H., King,D.W., and Sheely, C.Q.:Analysis of the Low-Tension Pilot at BigMuddy Field, Wyoming, paper SPE 12683 pre-sented at the 1984 SPE/DOE Symposium onEOR, Tulsa, April 2225.

[175] Reppert, T.R. et al.: Second Ripley SurfactantFlood Pilot Test, paper SPE/DOE 20219 pre-sented at the 1990 SPE/DOE Symposium onEOR, Tulsa, April 2225.

[176] Milton, H.W.: Evaluation Coring Results ata Micellar/Polymer Flood, paper SPE/DOE12684 presented at the 1984 SPE/DOE Sym-posium on EOR, Tulsa,April 22-25.

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Chapter 11

Sweep Improvements

11.1 Foams for Gasooding*

11.1.1 Introduction

The high mobility of gas in the reservoir frequentlyleads to poor conformance in gas ooding processes.Poor sweep eciency (vertical or horizontal) maybe observed on the reservoir scale. Localized prob-lems include gravity override, gas channeling in high-permeable streaks, and gas coning into productionwells. These problems can be more dramatic in gasoods than in wateroods because of the adverse gas-oil mobility ratio. The conventional practice of in-jecting alternate slugs of water and gas (WAG) is notalways eective, and may create other problems byretrapping mobilized oil.1,2

Placing a foam around problem wells, or deep intothe reservoir, can be an addition or an alternative toWAG. Because of its dispersed nature, foam can pro-foundly inuence uid ow patterns in the reservoir.A properly designed foam can reduce gas mobilityby several orders of magnitude. Notably, foam selec-tively reduces gas ow, leaving the relative permeabil-ity of the liquid phase essentially unchanged. Foam isalso selective to permeability in that gas mobility isreduced relatively more in higher-permeable regions(within a range of, arguable, 0.1 to 10 µm2), eec-tively smoothing out reservoir heterogeneities. Thesedesirable properties, discovered around 196036 areunique to foam and distinguish it from simple plug-ging agents such as polymer gels or cement.Foam has been applied to treat injection wells (pro-

ducers may also be treated), with the objectives ofblocking gas from owing into portions of the reser-voir having a higher eective gas permeability (thiefzones) and diverting it to unswept zones. In reser-voirs with no crossow and short interwell distances,near-wellbore gas diversion will often provide overallsweep correction,7 but in North Sea reservoirs, morein-depth treatments may be necessary. On the reser-voir scale, foam was originally thought of as a viscosi-fying agent for the gas,4 but the relative importanceof gas thickening versus blocking-and-diversion in thereservoir remains unclear and is discussed later.

*Acknowledgment: The author is indebted to Professor C.J. Radke of the University of California for many stimulatingdiscussions and patient review of this manuscript.

This section attempts to convey some currentknowledge on how foams inuence ow in reservoirmedia, with an emphasis on phenomenology. Pore-level processes, ow mechanisms, modelling, and eldexperience are summarized from a perspective of ap-plication to hydrocarbon-gas and nitrogen oods inNorth Sea reservoirs. A comprehensive literature re-view is not attempted since several recent publica-tions on the subject are available.710 In general, theliterature on foam in porous media abounds with in-consistencies and contradictions, partly due to thenonequilibrium nature of foam, and partly due toinsuciently controlled experiments. Further, mostprevious workers have been interested in applyingfoam to injection of steam and CO2. The specialproperties of these gases make direct comparisonswith hydrocarbon-gas or nitrogen foams dicult inmany cases.

11.1.2 Fundamentals

A foam is dened as gas dispersed in a continuousliquid phase, usually taken to mean high gas volumefraction (foam quality). Because expanding the sur-face area requires increasing the free energy of thesystem, foams of more than transient lifetime requirethe presence of a surface-active solute to stabilize thelarger surface area in the foam. Foam may be stabi-lized from collapsing by nonequilibrium surface ten-sion gradients (the Marangoni eect), by the presenceof gel-like surface layers, and by thin-lm forces.Foam conned inside the pore network of a reser-

voir rock has a fundamentally dierent morphologyfrom the complex three-dimensional structure of bulkfoam, Fig. 11.1. The conned foam is made up of in-dividual bubbles of gas separated by liquid sheaths orlamellae; this term is often used for the pore-bridgingliquid lm alone. It is this structure that gives thefoam its ability to reduce uid mobilities. Interac-tion between lamellae and pore walls dominates owbehavior11 and imparts on the conned foam manycharacteristics dierent from bulk foams. Bulk foamproperties, extensively studied,12,13 should thereforebe applied to conned foams only with caution. Stud-ies of thin liquid lms, a more scientic approachto characterizing foams,14 are more applicable, butsome of this work pertains to thinner lms than those

277

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278 CHAPTER 11. SWEEP IMPROVEMENTS

gasrock

oil

Figure 11.1: Typical morphology of foam in a porousmedium of dimensions comparable to a North Seasandstone.

in conned foams which are thought to be of order0.1 µm.15

11.1.3 Bubble Generation and Coales-cence

Pore-level processes in conned foams have been stud-ied by direct observation in etched-glass micromod-els and micromodels more or less representative ofreservoir rock.1619 Recent ndings based mostly onstudies in micromodels with sandstone patterns con-sidered quite relevant to North Sea reservoirs havebeen summarized.15 The principal generation pro-cesses of snap-o, leave-behind, and lamella divisionare illustrated on Fig. 11.2. Snap-o occurs at specicsites (pore throats of large aspect ratio) and is seen asessentially a mechanical process determined by geom-etry and liquid saturation. Each generation site is ca-pable of repeated snap-o events, each making a partof the gas phase discontinuous, and the lamellae maymove away from the generation site. Thus, a highow resistance is generated.* Snap-o is favored asgas ows from a lower to a higher permeability region.It occurs in homogeneous media only above a limit-ing velocity,18 which may vary between surfactants.20

The leave-behind mechanism is an important sourceof lamellae at lower velocities. Leave-behind lamellaedo not make the gas phase discontinuous, and eachgeneration site normally will create only one lamellawhich remains at the site. Some authors term foamscontaining mostly leave-behind lamellae weak foams(in contrast to strong foams resulting chiey fromsnap-o processes) due to their lesser eect on gasmobility reduction, and consider their lifetimes rela-

*There are actually three varieties of snap-o depending onthe pore body/pore throat diameter ratio and whether (local)liquid saturation is increasing or decreasing. Neck constrictionsnap-o is similar to the classic (Roof) trapping of oil dropletsin a waterood and is expected to dominate in transient foamgeneration (initial drainage) because it needs liquid saturationto be above a certain value. Pre-neck constriction and rectilin-ear snap-o dominate when both phases are owing at steadystate.15

(a)

(b)

(c)

(d)

(e)

Figure 11.2: Mechanisms of bubble generation andcoalescence in porous media, schematic after.15 (a)Leave-behind; arrow indicates lamella formed by con-vergence of advancing liquid ngers. (b) Snap-o; ar-rows indicate growing collar of liquid that snaps oas new gas bubble is formed. (c) Lamella division;arrows indicate movement. (d) Capillary-suction co-alescence; arrows indicate stretched state of thinninglamella. (e) Diusion coalescence; arrows indicateshrinking small bubble.

tively short. Division of moving lamellae can increasebubble densities signicantly because ow paths tendto be tortuous. Some confusion exists as to whethersnap-o or mobilization and division of leave-behindlamellae is the principal foam-generation mechanismin drainage experiments.7,18

Coalescence of bubbles into fewer, larger bubblesis the result of lamella destruction. Lamellae can begenerated by snap-o also in the absence of surfac-tant, but will coalesce instantly since the extendedsurface is not stabilized. It is known that thin sur-factant lms are unstable above a critical capillarypressure P ∗c .

21 This applies also to conned foams.In a study where capillary pressure was measuredat constant fractional ow of gas and surfactantsolution in beadpacks, a limiting value was foundabove which the owing pressure drop became un-

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11.1. FOAMS FOR GASFLOODING 279

stable, rapid coalescence was observed, and mobilityincreased sharply. This limit was identied with theaverage P ∗c for all bubbles in the owing foam.22 P ∗cis a function of the surfactant solution and was foundto vary with ow velocities. A theory which explainsthis velocity dependence has been presented.23

At constant saturation, P ∗c of static lamellae cor-responds to the maximum disjoining pressure π in athinning lm. For moving lamellae, P ∗c is lower due toadditional thinning as lamellae move into large porebodies. This capillary-suction coalescence is an im-portant mechanism for destruction of lamellae. It oc-curs at specic geometric sites in the porous medium.The Pc in a porous medium will reach the P ∗c of agiven foam at a critical water saturation S∗w, whichwill be lower in higher-permeability media. Note thata PcSw relationship measured in a foam should beused in calculating S∗w because of gas trapping andhysteresis eects which are dierent from gas/watersystems.24

Moving lamellae may be destroyed at pore branchpoints, instead of dividing, due to the increased cap-illary suction associated with creating a new Plateauborder.25 A general coalescence mechanism is gas dif-fusion from smaller to larger bubbles. This relativelyslow process may be dominant in the stagnant, ortrapped (see below) part of a conned foam.

11.1.4 Foam Flow

Applicability

Steady-state foam ow is readily achieved in labora-tory cores, sandpacks or beadpacks. The accompany-ing pressure gradients are, however, typically higherthan reservoir pressure gradients, even at reservoir ve-locities. In some cases, unstable foams have been pur-posely chosen for study to keep pressures down. Theapplicability of many data in the literature to reser-voir processes may therefore be questioned. Steadystate is commonly not reached in laboratory cores un-til many pore volumes of foam have passed throughthe core. The question has been raised26 whether theassumption of steady-state ow with foam presentin the reservoir is correct under all circumstances.Unsteady-state or transient foam ow has been littlestudied and the scaling relations involved have notbeen quantied. While the value of studying ow infoam-lled porous media at steady state is obvious,these reservations should be noted.

Experimental

The apparatus for foam ow studies is basically thatof a steady-state relative permeability experiment.Due to the complex nature of foam, however, as manyvariables as possible should be monitored. Direct vi-sual observation of pore-level events can only be re-alized in micromodels. On a core or pack, pressureproles and in-situ local saturations should be mea-sured to ensure uniform conditions.

There is no standard experimental procedure. Gasmay be injected alone or alternating with surfactant.Gas and liquid may be coinjected, mixing just up-stream from the core or at the core inlet. Foam maybe generated externally in a foam generator and in-jected. Injecting at xed rates is more common thanat xed pressure drop, which is dicult to realize ex-cept for injection of gas alone. The porous mediummay be initially saturated with gas, brine, surfac-tant, or oil, and may contain residual saturations ofother uids. A particular mode, intended to elimi-nate leave-behind lamellae, is to drive the core to anintermediate water saturation by coinjecting gas andbrine at some liquid volume fraction and then substi-tute surfactant solution for brine.27,28

Variations in the mode of experiment have mostprobably inuenced many published studies throughtheir eects on transient foam generation and dis-placement, and may also inuence steady-state prop-erties to the extent that these depend on the satura-tion and generation history.To minimize the uncertainties associated with scal-

ing for as many parameters as possible, experimentsshould be performed at realistic ow rates, eld pres-sure gradients, and preferably at reservoir tempera-ture and pressure. The importance of the latter re-quirement varies with the gas in question. In anycase, the pressure drop should be insignicant com-pared to the mean pressure in order to minimize bub-ble expansion eects.

Flow Mechanisms

Foam does not ow as a continuum uid. Its twoconstituent phases ow by separate mechanisms.29

The liquid phase (usually water) ows as a contin-uous phase and seems to follow its ordinary rela-tive permeability and Darcy's law, taking into ac-count the dierent saturations from those in nondis-persed gas/liquid ow.5 Gas ow mechanisms havebeen much discussed.9,11,16,29,30 The current under-standing is that the break-and-reform29 and bubble-train11 ow mechanisms are essentially the same.7,17

When trains of bubbles move through tortuous pathsin the pore network in what macroscopically appearsas steady-state ow, each train is in motion intermit-tently, bubbles are added to or leave the train, andlamellae in the train are continuously broken and re-formed. The identity of a single bubble or train isnot conserved for any macroscopic distance and thebubble train is a time-averaged quantity.Foam ow behavior is highly nonlinear. Early stud-

ies attributed most varieties of non-Newtonian rheol-ogy to conned foam and it was realized that a properdescription requires accounting for its texture, or thebubble size and bubble-size distribution.31 This isdiscussed further in the section on modelling.Foam mobility is not a unique function of satu-

ration. In steady-state foam ow, a constant wa-ter saturation somewhat above the Swi to gasoodwas obtained, independent of gas and liquid ow

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280 CHAPTER 11. SWEEP IMPROVEMENTS

rates.32 Saturation was uniform except for the inletsection where foam is being shaped.33 Three stud-ies have conrmed and extended these recent ndingsfor three dierent foams in 1-µm2 Bentheimer, Boise,and Berea sandstones.3436 A realistic range of veloc-ities (1 to 7 m/d) and gas fractional ow (0.7 to 0.99)was studied; note, however, the large pressure gradi-ents (10 to 600 bar/m). The pressure drop at xedow rates was found to be nearly independent of gasrate, and to increase with increasing liquid rate, withno hysteresis, Fig. 11.3. One study34 found a breakpoint.

10

100

1 10 100

N2 foam at 52 bar, 20 oC

in 1.3-µm2 Boise ss.

Pres

sure

gra

dien

t (ba

r/m

)

Gas velocity (m/d) at constant liquid velocity

vliq = 0.881 m/d

vliq = 0.185 m/d

vliq = 0.153 m/d

(a)

10

100

Pres

sure

gra

dien

t (ba

r/m

)

0.1 1Liquid velocity (m/d) at constant gas velocity

vgas = 6.26 m/d

vgas = 2.14 m/d

Common fit

∆p/∆x = 78 vliq0.89; R= 0.987

(b)

Figure 11.3: Dependence of pressure gradient insteady-state foam ow on gas and liquid velocity. Re-drawn from data of Perso.35

When pregenerated foam is injected, the porousmedium changes foam texture to an essentially in-variant steady-state value.36 This indicates that foamtexture also is controlled by P ∗c . The observed behav-ior can be interpreted in terms of P ∗c and pore-sizedistribution.35,37 Wetting water occupies the small-est pores, in which foam lamellae cannot be stablebecause the local capillary entry pressure would belarger than P ∗c . Foam ows as time-averaged bubbletrains in the larger pores (and possibly as free gas inthe very largest ones), while the medium-sized poreshold foam that is trapped at pore constrictions.More studies have been conducted at constant in-

jected fractional ow (also known as owing-foamquality by analogy to bulk foam quality). This maybe more representative of eld processes where pre-formed foam is injected, but does not allow separationof liquid- and gas-ow rate eects on foam proper-ties. A representative, recent set of data is shownin Fig. 11.4.38 An increasing or constant mobilitywith increasing foam velocity (liquid and gas rates

0

0.001

Foam

mob

ility

, mm

2 /cp

0.002

0.003

0.004

Foam velocity (m/d) at constant fractional flow

0.59

0.68

0.81

0.90

0 1 2 3 4

Gas fractional flow ƒg:

CO2 foam at 111 bar, 25 oC, in 0.30 µm2 Berea

Figure 11.4: Steady-state dependence of mobility ontotal, or foam, velocity at constant fractional ow.Redrawn from data of Ref. 38.

increased in the same proportion) was found. Thisgenerally agrees with the data on Fig. 11.3. A cor-responding decrease in foam mobility with foam ve-locity has been found by other investigators,38,39 butan opposite trend has also been reported.40 In somestudies, liquid saturation was found to change withow rates, so true steady state may not have beenachieved, perhaps due to end eects or weak foams.These error sources may also be present in other lit-erature data.

Trapping and Mobilization

At any given time, only part of a foam in steady-stateow actually conducts ow while the rest is trapped.Using gas-tracer techniques, trapped fractions from30 to as much as 99% have been measured.29,37,4043

The trapped fraction is a function of the bubble gen-eration/coalescence and ow parameters and has notbeen well characterized.Away from the owing steady state, it appears that

trapping in foams may be essentially 100%. In ex-periments where a porous medium saturated withfoaming-agent solution is ooded by gas at xedpressure, a persistent state of constant and uniformliquid saturation and very low gas mobility can bereached.6,4446 This so-called gas blockage, appar-ently a stagnant foam letting at most a trickle ofgas through, occurs when the pressure gradient fallsbelow the mobilization pressure,47 which is super-cially similar to a yield point. Gas blockage can beproduced also in sandstone cores at reservoir condi-tions,48 but is sensitive to the presence of oil except

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11.1. FOAMS FOR GASFLOODING 281

with certain foamers,45 and a (possibly surfactant-dependent) minimum porous-medium length seems tobe required.33,49 It follows from our discussion of gen-eration mechanisms that lamellae existing in steady-state ow may be dierent from those generated ini-tially. Perhaps blockage occurs when the initial lamel-lae are preserved.39 Although persistent for as longas months, at eld pressure gradients, gas blockageis essentially a transient state and thus dicult tomodel even by a population-balance simulator. It isof interest in using foam against gas coning in pro-duction wells,32,50 or to seal gas storage reservoirs,51

and may be important in parts of a reservoir underfoam-assisted gasood. Some investigators have cho-sen to use both gas blockage and steady-state owtests in product screening.52

Mobilization and trapping of foam has been an-alyzed within a bubble-train model and by apply-ing percolation theory.28,53,54 The results indicatethat mobilization pressures will be higher than typicalreservoir pressure gradients so that foams should notow except very close to the wellbore. Core experi-ments also have indicated this.20 These observationsimply that steady-state foam ow in the reservoir maybe a minor contribution to the overall mobility reduc-tion, which must instead be ascribed to blocking anddiversion. This conclusion is further supported bythe results of the xed-pressure experiments discussedabove. Field-wide propagation of foam has actuallynever been demonstrated in any eld test, despite de-signs for this.55,56 Foam observed in wells far awayfrom the injector may have been generated there bygas and surfactant.20 Even with a small slug injected,surfactant (not foam) was found to transport at least1.6 km from an injector.57

Eects of Fluid and Rock Parameters

Good foaming agents can be found among mostclasses of surfactants. Common products like ethoxy-lated alcohols, alkyl ethoxy sulfates (limited to tem-peratures below 60 to 70C), alkylaryl sulfonates andα-olen sulfonates have been chosen for reservoir testsand application as foamers. The search for an optimalfoamer for a given application has obviously receivedmuch attention.32,52,5860 Still, few generally validscreening criteria have been established. Some au-thors report successful product selection by bulk foamstability,61 while others45 nd no correlation betweenbulk and porous-medium foam properties. Very likelythe type of tests used (both bulk and porous-mediatests) are critical for the screening results.High temperature is generally detrimental to bulk

foam stability as well as to its porous-medium prop-erties. Through careful screening and product opti-mization, suitable foamers have been identied evenfor steamood temperatures approaching 300C.62

Foam properties may vary with pressure. The eectsof pressure itself and increased density may be morecritical for CO2 and steam than for N2 and hydrocar-bon gases, but the eects of increased mass transfer

and lowered interfacial tensions must be consideredfor all gases.

Up to a certain level, usually some multiple ofthe critical micelle concentration, increasing sur-factant concentration produces stronger gas mobil-ity reduction. Depending principally on their hy-drophilic/lipophilic balance, some surfactants producestronger foams in more concentrated brines, whileothers are impaired by high salt concentration.

The presence of oil almost invariably is detrimen-tal to mobility reduction by foam and has been muchstudied. Still, the mechanisms of oil/foam inter-action remain unclear. In some cases there is nomobility reduction until the oil saturation becomeslow enough.27 In other cases, residual saturationsare sucient to eliminate mobility reduction com-pletely.63,64 Oil apparently does not change the ba-sic ow mechanism.29 The type of surfactant and oilstrongly inuence mobility reduction,65 but even inmodel systems the interactions are complex.63 Mostaqueous foaming agents are not appreciably solubi-lized by oil. A wettability change brought about bythe combined action of oil and surfactant cannot beruled out with crude oils and rock, but is unlikely inmodel systems of pure hydrocarbons and glass beadswhich show the same oil sensitivity of foam.63,65 Not-ing the similarity of Roof snap-o of oil dropletsand gas bubbles, residual oil has been proposed toblock generation sites.27 Film elasticity was foundto correlate with foam oil-displacement eciency inone study,66 but was not a predictor in another.67

Surface-active compounds in crude oil may act asfoam inhibitors by adsorbing to the surfactant/gasinterface.68,69 Oil spreading on a foam lm (as ex-pressed by the spreading coecient) may be a nec-essary condition for destabilizing them,70 but appar-ently is not sucient.67,71,72 AOS foams producedmore steady-state mobility reduction in the presenceof (model) oils with negative entering coecients.73

The breakdown of both bulk and conned foams iscontrolled by breakdown of the pseudoemulsion lmsseparating oil droplets from gas bubbles.74,75 Sta-ble pseudoemulsion lms can be formed using cer-tain uorinated surfactants, which also were foundto create oil-resistant conned foams.76 Some inves-tigators have proposed general relationships on oilsensitivity from comparison of one or two uorosur-factants with several conventional ones.67,72,77 How-ever, in a study that found 4 of 8 uorosurfactantfoams to be oil-tolerant in gas-blockage tests (andnone of 36 conventional systems), neither spread-ing nor entering coecients could fully predict foamperformance.45 The higher cost of uorosurfactantprobably precludes their eld use except in special-ized small-volume treatments such as against gas con-ing in producing wells.50 Luckily, in steady-statefoam ow, strong mobility reduction in the pres-ence of oil is reported also for suitable conventionalsurfactants.65,77,78 It has been proposed that evenan intrinsically oil-sensitive foam can be dynamically

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282 CHAPTER 11. SWEEP IMPROVEMENTS

metastable and produce an appreciable pressure dropif the moving lamellae are in contact with oil for alimited time.75 It should also be noted that oil tol-erance may not be required or even desirable in allcases; it has been stated that foam is required toblock principally the oil-depleted regions while gasmay be allowed to ow more freely in regions of highoil saturation.58

The eect of rock wettability may be related to theeects of oil. Generally, mobility reduction by foamhas been found to be poorer or nonexistent in oil-wetporous media.64,71,79 One study found surfactant tochange an oil-wet core back to water-wet, allowingstrong foam to form, but most of the mobility re-duction was lost when oil also was present.80 Thesestudies must be interpreted with some caution be-cause the reagents commonly used to make porousmedia oil-wet also are strong foam inhibitors. Lit-tle is known about foam behavior in mixed-wet orintermediate-wet media.The eect of permeability has been mentioned.

During gas blockage in unconsolidated packs, a max-imum relative mobility reduction at k ∼ 7 µm2 wasmeasured.6 Other systems in the persistent gas-blocking state have shown a somewhat lower per-meability for maximum blockage.48,49 In steady-state ow, foam mobility reduction was found to bestronger in 0.305-µm2 than in 0.015-µm2 sandstone.38

Possibly as important as permeability in itself is theeect of permeability contrasts, which has been littlestudied.7

Adsorption to rock surfaces is an issue with foamsas in other surfactant applications, worsening processeconomics by increasing chemical needs.81,82 How-ever, since most foaming agents are Winsor II() sys-tems, losses are generally expected to be smaller thanfrom micellar ooding formulations. Losses of order0.15 mg AOS surfactant per gram rock have been re-ported for water-wet Berea cores, including systemswith North Sea crudes; a four-fold increase was seenin cores made articially oil-wet.81 A slug dilutionproblem similar to that in surfactant ooding exists,and suggestions to counteract this have been made.71

When injecting a low surfactant concentration foam,and/or injecting into a surfactant-free core, large vol-umes of foam may need to be injected to reach steadystate,26 or only a weak foam may be obtained,83 be-cause of surfactant adsorption.

11.1.5 Modelling

Models treating foam as a non-Newtonian pseudouidin a conventional simulator have not demonstratedany great predictive ability. A predictive foam simu-lator should be mechanistically based, and represent,at least in simplied form, the actual phenomena.The key to modelling foam ow is the foam tex-

ture, which can be quantied (ignoring polydisper-sity) as the average bubble density and calculatedby the bubble population balance.84 The populationbalance has the same form as the ordinary mass con-

servation equations. It can be solved by the sametechniques and incorporated into an existing simu-lator. Foam texture cannot be measured in rock,but it can be observed in transparent media such assmooth capillaries,11 micromodels, or viewing cellsat the core outlet.36 Work towards developing fullfeatured population-balance reservoir simulators hasbeen done.41,85 These models assume foam to alteronly the gas-phase relative permeability and viscos-ity, and describe its eect both by assigning foaman apparent viscosity increasing with the number ofowing bubbles, and a higher trapped-gas fraction.The trapped-gas fraction can be measured, but foamgeneration rates, the eect of oil, and the surfactant-dependent minimum velocity for onset of strong-foamgeneration at intermediate core saturations41 must beset by history matching core oods.Other investigators have aimed primarily at de-

scribing laboratory data.36,86 A model incorporat-ing kinetically based bubble-generation and coales-cence rate expressions, i.e., fewer adjustable parame-ters, has been described.36 For modelling eld tests, asimpler approach has been used instead of the popu-lation balance. Here radial foam ow is modeled witha constant average mobility between wells.87 A quitedierent approach to foam modelling is to representthe porous medium as a network and use relationsderived from percolation theory to describe selectedproperties.19,28 A comprehensive model has also beenpresented.88

11.1.6 Field Experience

The use of foam in steam oods is increasing becausefoam seems to be eective in diverting steam and pre-vent gravity tongues and channeling.7,89 As much as14% incremental recovery (of STOOIP) has been re-ported, and the cost per barrel incremental oil dueto foam ranges from $1 to $11. In some respectsviscous oils, short interwell distances, low pressuresthese reservoirs are very dierent from those in theNorth Sea, but similar formations are produced: typ-ically heterogeneous, high-permeable sandstones.The experience with foam in CO2 injection is more

mixed. Treatment to correct the prole of an injectorwas successful,90 but trials in West Texas carbonatereservoirs under CO2 ood have generally not givenany major sweep improvement.91 The reasons for thisare unclear, but the high in-situ density of CO2, re-ducing its tendency to form gravity tongues, could beone.Only a few tests of foam used with hydrocarbon

gases and nitrogen, of primary interest in North Seaenvironments, have been described. An early eldexperiment at Siggins56 was done by cycling air andwater in a shallow 10 to 300 md sandstone reservoirproducing 36 API crude. Foam reduced injectivityand stopped severe gas channeling to a well ≈100 maway. Foam was never detected in producers or obser-vation wells or producers within the test pilot area,but unexpectedly appeared in a capped well 450 m

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11.2. POLYMER FLOODING 283

away from the injector. * Another eld test showedfoam acting principally to block gas, propagating onlyshort distances from the wellbore: At Painter,92 alarge N2 ood was conducted in a thick, deep Juras-sic sandstone with uids similar to those in Statfjord.Foam was injected to divert gas and correct leaksin doubly completed wells. The treatment producedonly reduced injectivity, no positive eects and somefracturing. The authors discuss several reasons forthe outcome of the test; it is also possible that theaverage permeability of this reservoir, 7 · 10−3 µm2,is too low for foam to be eective. At Kaybob Southin Alberta, a 150-million-barrel eld, where light oilis produced from a complex carbonate at 180 bar and88C, a foam was eective in optimizing gas injectiv-ity in a tertiary rich-gas ood, and reducing the pro-duced GOR.93 Its eect on oil production was hardto quantify due to uncertain baseline estimates. Alsoin Alberta, a smaller reservoir, part of the large Pem-bina eld, produces 42 API oil from a carbonate at57C and 105 bar by reinjection of stripped eld gas.A gas injector was treated with foam to improve thesweep. Injectivity sank and oil production rose by30%, at constant GOR, and remained above baselinefor ve months.57 The Alberta tests are of particu-lar interest for comparison with North Sea conditionsbecause of the similar oils, depths and temperatures,and the relatively large well spacings. Volumetric ef-ciency and economic success approached the rangeof steam-foam treatments.

11.1.7 Application

In medium- to high-permeable North Sea sandstonereservoirs under gas ood, application of foam shouldbe considered if poor sweep or channeling occurs de-spite an optimized WAG strategy. The limited eldexperience shows that especially channeling in high-permeable zones or gravity tongues can be stopped ef-fectively by foam. It may be less ecient in reservoirsof low permeability and less permeability contrast.The obvious choice is to try rst in a single injec-tor where gas channeling is suspected. If successful,regular treatments and/or eld-wide foam injectionmay be implemented. Treating production wells willprobably be restricted to cases of severe coning.By analogy waterooding, using foam in gas in-

jection is thus more akin to using polymer-gel treat-ments in water injection than to a polymer ood.Compared to surfactant and polymer injection, theproblems of nding suitable products for North Seaconditions are expected to be much smaller. Chem-ical costs are lower, helped further by the fact thatmost of the foam is gas which already is being in-jected. However, the mechanistic and quantitativeunderstanding of foam is much poorer than that

*In a recent steam eld trial,20 foam in an observation wellwas concluded to have been generated from surfactant andsteam transported there separately, because the foam propa-gated only a short distance in cores at eld pressure gradients.This could also have been the case at Siggins.

of any other reservoir injectant. Foam is never apanacea that can compensate for a poorly plannedgas ood, and there have been nasty surprises as-cribed to insucient understanding of foam behaviorin the reservoir. The greatest risk is likely to be exces-sive loss of injectivity if strong blockage occurs closeto the well. However, in contrast to a failed polymer-gel treatment, this damage can readily be reversedby ushing with water to dilute the foam, or with anantifoam to destroy it completely. Reinjecting someproduced oil will kill an oil-sensitive foam instantly;several foam inhibitors exist that can be used to re-move oil-tolerant foams.The necessary R&D tasks that must be addressed

to allow a safe and ecient eld application of foamin a gasood in the North Sea are: (1) Measuringof eects of foam on uid ow in reservoir cores atrelevant conditions. (2) Quantitative description ofow mechanisms to permit scaling of core ood re-sults. (3) Understanding reservoir ow patterns andresidual oil distribution in the absence of foam. (4)Designing a eld process, based on (1), (2), and (3)to nd the optimum surfactant type, concentration,slug design, injection strategy, etc.Of these, point (1) is the universal need for better

reservoir description. Even after a good mechanis-tic understanding, perhaps even a working numericalmodel, has been obtained under point (2), specictests for each reservoir and foam will still be neededbecause many unclear points will remain in the chem-istry of foams and reservoir materials.

11.2 Polymer Flooding

11.2.1 Introduction

Polymer ooding is an EOR method where polymeris added to the injected water. This will increase theviscosity, η, of water and decrease the mobility, λ, tothe displacing phase. In reservoirs with unfavorablemobility ratio, M , a decrease will improve the vol-umetric sweep eciency. It was Muskat94 that rstpointed out that mobility control will aect water-ood performance. Pye95 and Sandiford96 in 1964established that small amount of polymer added tothe brine can reduce the mobility signicantly. Sincethen, a number of papers have described this topic.It should be noted that the irreducible oil satura-

tion in a polymer ood will be the same as in a water-ood, but in a polymer ood the oil can be recoveredover a shorter time period and at lower watercut. Intheory, polymer ood is therefore more an acceleratedthan an enhanced oil recovery method.Polymer ooding will be favorable in reservoirs

where the oil viscosity is high, or in reservoirs thatare heterogeneous, with oil-bearing layers at dierentpermeability. For North Sea reservoirs, the hetero-geneous reservoirs are the potential candidates forpolymer ood. (For more details about rheology,cf. Sec. 3.2.)

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284 CHAPTER 11. SWEEP IMPROVEMENTS

11.2.2 Mechanisms

Viscosity Increase

Adding high molecular weight, water-soluble poly-mers to the injection water will increase the viscos-ity. The polymer is non-Newtonian and pseudoplas-tic. This means that it has a low viscosity, close towater viscosity, in the injection zone and an increas-ing viscosity into the reservoir. A typical viscositycurve is plotted in Fig. 11.5.

100

50

20

10

5

2

1100 101 102 103 104

ηr

2400 ppm

1200 ppm

800 ppm

400 ppm

200 ppm100 ppm

25 ppm50 ppm

Figure 11.5: Viscosity vs. shear rate for dierent xan-than concentrations in 5 g/l NaCl at 30C.97

Permeability Reduction

In addition to viscosity increase, there is also a per-meability reduction. For some types of polymers thiseect can be signicant. The reduction in permeabil-ity is due to retention of polymer by adsorption ortrapping in pore channels.

11.2.3 Polymers

There are two sets of EOR polymers that have beenused in reservoirs, synthetic polymers and biopoly-mers. The major eld experience is with syntheticpolymers.

Synthetic Polymers

The most used polymer in eld operations is poly-acrylamide, PAM, or hydrolyzed polyacrylamide,HPAM (normal degree of hydrolysis is 30 to 35%).These are polymers where the monomeric unit is acry-lamide, Fig. 11.6. The polymer structure is randomlycoiled, and due to large molecules the viscosifying ef-fect in water is good. The polymer structure is sensi-tive to salinity and at seawater salinity the viscosify-ing eect is reduced. Because of an elastic behaviorthe polyacrylamides will easily be degraded by highshear rates in porous media. They are normally tem-perature stable up to about 60C at seawater salin-ity,99 but is stable at higher temperatures if the diva-lent concentration is reduced.Other synthetic EOR polymers are copolymers

consisting of dierent types of monomers, normally

CH2 CH2CH CH

NH2 O-

C=O C=O

K+ or Na+x-ν

ν

(a) Partially hydrolyzed polyacrylamide, degree of hydrolysisis given by ν/x.

CH2OH

CH2OHCH2

CH3

CH2OHCH2OH CH2OHO

O

O

O

OO

OO

OO

O O

OO OO OO O( )()N M

AcOCH2AcOCH2

CC

OO

Na+

O_

Na+

O_

CCNaO+ _

(b) Xanthan.

Figure 11.6: Molecular structures.98

sulfonate and amides. They have similar rheologi-cal properties as polyacrylamide. Some of them arereported to be more salt resistant and temperaturestable.100

Biopolymers

Two biopolymers are used for EOR purposes, xan-than and scleroglucan. They are formed from poly-merization of saccharide molecules in a fermentationprocesses. Both have a helical, rodlike structure andare extremely pseudoplastic. The viscosity is al-most insensitive to salinity. Temperature stabilityfor xanthan is reported in the range 70C to above90C,101,102 and above 105C for scleroglucan.103

The polymers and especially the biopolymers aresusceptible to bacterial attack in the low-temperatureregion in the reservoir. To prevent biological degra-dation, biocides like formaldehyde are used.104 Fromeld experience and also laboratory experiments itis found that formaldehyde in concentrations 500 to1000 ppm eectively prevent biodegradation of poly-mer.105108

11.2.4 North Sea Reservoir Condi-tions

The North Sea reservoir conditions put strong restric-tions on the polymers; high injection rates, high tem-peratures, large interwell distances and the use of sea-water with high salinity.

High Injection Rates

The typical injection rates are in order of 103 m3/day,which means shear rates in order of 104 s−1. Poly-mers with a viscoelastic behavior will be mechanicallydegraded by these high shear rates due to elongation.The viscosity will be reduced because of reduction

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11.2. POLYMER FLOODING 285

in molecular weight. In core ood experiments, themobility reduction factor RF will be an increasingfunction of velocity, with highest RF in the injectionzone.110 This is shown in Fig. 11.7.111 The synthetic

A

B

C

A - 2 . 106 mol wt.B - 5.5 . 106 mol wt.C - mol wt. not determined

80

60

40

20

00 20 40 60 80

Res

ista

nce

Fact

or

Velocity, ft/day

Figure 11.7: Resistance factor as function of velocityfor dierent polyacrylamides.111

polymers like polyacrylamides are most sensitive toshear degradation. The critical shear rate, where theviscoelastic eects start dominating, is given by theDeborah number,112 which depends on the polymerstructure and the porous media. Mechanical degra-dation in core ood experiments is found to occur atshear rates of about 103 s−1 for polyacrylamide.113

The shear degradation can be reduced by injectinglower molecular weight polymers at higher concentra-tion, but the shear-thinning eect is thereby reduced.All the synthetic polymers known today will thereforegive a signicant reduction in injectivity.The biopolymer xanthan will be shear stable up

to shear rates far above the typical reservoir rates,and the pseudoplastic behavior will give a much lowerinjectivity reduction.If the reservoir is fractured the injectivity problem

is reduced.

Temperatures

The reservoir temperature can vary from 70 to 130C.The temperature around the injector will be muchlower due to injection of cold water over a long time.Since the temperature front moves slower than thepolymer front, polymer injected late in the life-timeof the reservoir will also reach the original reservoirtemperature. The polymer must therefore be stableup to the original reservoir temperature.Many chemical reactions that can degrade the poly-

mer is enhanced by temperature and content of oxy-gen. It is reported that the oxygen level must be low(in the range of ppb) to prevent degradation at hightemperatures.101 Amounts of Fe(II) or Fe(III) canstart degradation at high temperatures,114 in addi-

tion to aggregation.115 The Fe content can be stabi-lized by citrate.The water viscosity is decreasing by increasing tem-

perature. For polymer, the relative viscosity is nor-mally weakly reduced by increased temperature, be-cause of reduction in intrinsic viscosity and Hugginsconstant.

Large Interwell Distances

Large interwell distances indicate that the polymermust be stable over a long time (years) at high tem-peratures. In a conventional polymer ood, it is im-portant that the polymer does not form aggregates orgel that increase the ow resistance.

11.2.5 Laboratory Experiments

Laboratory experiments are important in planning apolymer ood. Dierent types of polymers will beevaluated. Viscosity, compatibility to the brine, andtemperature stability are important factors. After ascreening, core-ood experiments have to be carriedout to give information about the injectivity and re-tention level. In-situ rheology of polymer can also bedetermined by core oods.

Core Floods

Injectivity. The injectivity of a polymer solutionis important, especially when the injection pressureis close to the fracture pressure. From core oods,the injectivity can be calculated from the mobilityreduction factor, RF, by measuring the dierentialpressure ∆p for water and polymer in a constant-rate experiment. The permeability reduction factor,RRF, is the ratio between the dierential pressurebefore and after a polymer ood if a large volume ofpolymer is injected, followed by water,

RF =∆pp∆piw

, (11.1)

RRF =∆pfw∆piw

. (11.2)

Since the polymer viscosity depends on the shear rate,the resistance factor will also have this dependency.This means that RF will increase into the reservoirs.For some polymers, however, that become elongatedat high shear rates, there has been reported an in-crease in RF by increasing shear rate.109

Simple analytical models can be found by usingDarcy's law for radial systems and a power law forspecic viscosity.116,117 Lake118 used the same radialmodel but assumed a power law in absolute viscosity.At high shear rates near the well, Lake's model willgive too low injectivity.Darcy's law for radial geometry is

∇p =1

kv(r)× η(r). (11.3)

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286 CHAPTER 11. SWEEP IMPROVEMENTS

The viscosity, η, is shear-rate dependent and followsa Carreau model.119 In the injection region, wherethe shear rates are high, this model gives a power lawin specic viscosity,

ηsp(γ) = Kγn−1. (11.4)

The shear rate is proportional to the radius r, andthe resistance factor RF(r) can be found as a functionof the position of the polymer front by integrating thepressure gradient from rw to r,

RF(r) = RRF(r)

×

1 +

1

nηsp(γ = 0)

(γwγc

)n [1−

(rwr

)n]ln

r

rw

.(11.5)

Skin eects can easily be incorporated into the modelby substituting the well radius, rw, by an eectiveradius, reff , given by the skin, exp(S) = rw/reff , andγc is the critical shear rate given by λ γc = 1.Eq. 11.5 shows that the mobility reduction is

strongly dependent on the shear thinning index, n.This model ts experimental data well.117

To get a good injectivity, it is important to mini-mize RF in the injection zone. This can be done byusing an extremely pseudoplastic (also at high shearrates) polymer.A good injectable polymer has a low-permeability

reduction, close to 1 at high injection rates, and astable mobility reduction. An increase in the mo-bility reduction over time indicates gradual pluggingof the pores and results in a loss of injectivity thatcan be quantied by calculating the pressure increaseover time. Test setups and maximum level of pres-sure increase in sand packs and linear cores have beenreported.109

From constant-pressure experiments, RF can becalculated as a function of rate.The permeability reduction is aected by the reten-

tion of polymer. A layer at the pore surface will re-duce the eective pore radius. Also, plugging of poresdue to aggregates will reduce the water permeability.The plugging eect can be reduced by preltration orby using nonaggregating polymers.120122

In a polymer ood, it is not possible to measurethe permeability to polymer, but it is normally setequal to the permeability to water after a polymerood. (In oil-saturated cores it has to be measuredat the same saturation.) The relative viscosity of thepolymer in the core can be calculated, and is foundas the ratio between the resistance factors,

ηr(q = const) =RF

RRF. (11.6)

The injection rate is transformed to an apparentshear rate in the porous medium. Dierent equa-tions have be reported,123126 but all are based on

the capillary-bundle model that easily can be calcu-lated from ow through capillary tubes and Mooney-Weissenberg equation,109

γapp =3n+ 1

4n

4αv√8kwφw

, (11.7)

where n is the shear thinning index, α is a materialconstant, dependent on both porous media and typeof polymer, normally in the range 1.7 to 15,123127

v is the Darcy velocity, kw is the permeability andφw is the porosity to water. The viscosity in theporous medium, often mentioned as apparent viscos-ity can then be compared with viscosities measuredby rheometers.The apparent viscosity experiments are not a

straight-forward process, especially at low shear rateswhere the dierential pressure is very low and faceplugging of the core easily occurs. This is impor-tant to have in mind when dierent conclusions fromreported results are compared. The most commonresults are listed below:The α-factor is found to increase by decreasing per-

meability and also increasing molecular weight.127

The Newtonian viscosity in porous media is lowerthan calculated from bulk results.123,127129

The shear-thinning index for polysaccharides isclose to bulk results.126,130

The same results can be found in radial cores wherethe pressure is measured at dierent distances fromthe injector.117 The ratio between the polymer pres-sure, pp, and water pressure after polymer, pfw, givesthe relative viscosity as function of shear rate,

ηr(v) =pp

pfw. (11.8)

This method requires accurate pressure measure-ments at several positions from the injector.

Retention. Retention or adsorption of polymer isanother important factor that has to be measuredin core oods. Static adsorption levels measured insand or clay are reported to be a factor of ten higherthan measured in core oods.131133 This is becausethe pore surface is not easily accessible to the highmolecular weight polymer molecule. Retention is in-creased by increased salinity, especially for polyacry-lamides, and the concentration dependency followsa Langmuir isotherm. Some work has been carriedout at dierent saturations.133136 Highest retentionwas measured at Sw = 1. At Sor in waterwet cores,there was only a minor reduction in retention com-pared with Sw = 1, but in oil/mixed wet cores theretention was lowered by a factor of ten,136 and is ex-plained by reduced ability of polymer to contact thesurface.The normal way of measuring retention is to in-

ject a xed volume of polymer and measure the pro-duced polymer concentration. The dierence is the ir-reversible retention. Another way is to inject a seriesof polymer slugs and calculate the produced amount

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11.3. POLYMER GELS 287

of polymer. The dierence between the rst and thelast slug will give the retention. The last experimen-tal setup will also give the inaccessible pore volumeto polymer, IPV.

In a polymer ood, one fraction of the pore volumeis inaccessible to polymer because of too small poresand a wall-exclusion eect.123,137,138 Typical valuesof IPV are in the range of 10 to 30% of the totalpore volume, dependent on the porous medium andpolymer. The retention and IPV give a chromato-graphic eect, and the breakthrough time of polymeris a function of these two parameters. Low retentiongives higher eective velocity of polymer than water,and vice versa.

11.3 Polymer Gels

11.3.1 Introduction

Waterooding, using seawater, has been the most fre-quently applied recovery technique in the North Seareservoirs. Since these reservoirs very often containconsiderable permeability contrasts and thief zones,early water breakthrough and rapidly increasing wa-tercut have been observed in many cases. A typicalexample is Gullfaks, a reservoir where the use of poly-mer gels recently has been suggested for improvingthe recovery.139

Polymer gels have been used to improve watersweep eciency, and for reducing watercut, since theearly seventies.140143 Successful treatments with thistechnique are based on the formation of a highly vis-cous gel through the reaction between a polymer anda crosslinking agent, in high-permeable zones or frac-tures of the reservoir. Water is thereby diverted intothe unswept, low-permeable zones. It is importantthat the gel is transported far from the well, to givethe largest increase in sweep eciency. This is of spe-cial importance in cases of high vertical permeability.The gel should also be stable at reservoir conditions,and the low-permeable zones should not be damagedby the gel.

The use of polymer-gel technology can be very costeective, with costs of less than $0.10 per barrel ofextra oil.144 Other advantages using this techniqueare low risk and quick response. Compared to otherinjectants for water diversion, polymer gels are par-ticularly attractive due to the low content of solids.An eective polymer gel can consist of 99.8 to 99.9%water.

Several crosslinkers and polymers have been sug-gested and used in polymer-gel treatments.145149

However, in most cases either polyacrylamide orxanthan are used as polymer, with chromium ascrosslinking agent. Most of the following discussionwill therefore be limited to these compounds.

11.3.2 Gelation in Bulk

The purpose of studying gelation in bulk, is to obtaina detailed description of how gelation is aected byimportant parameters encountered in the reservoir.Also, screening a large number of dierent compo-sitions is only practical through bulk studies. Themost frequently measured parameters are: (1) gela-tion time, indicating how far from the well the gelsolution can be transported; (2) gel strength, indicat-ing the degree of ow resistance; and (3) gel stability,indicating for how long the ow resistance will last.

Chemicals

Polymers.109,150,151 Polyacrylamide, usually inits partially hydrolyzed state, is the most widely usedpolymer. This is mainly due to low cost, formationof strong gels and resistance towards bacterial degra-dation. In addition, the properties of the polyacry-lamides can be specied through variations in molec-ular size and the degree of hydrolysis, which can be ofgreat importance regarding gelation. The major dis-advantage is the low compatibility with hard brines,making them unsuitable for seawater which proba-bly will be the solvent for North Sea application.However, recent studies have shown that polyacry-lamides with a low degree of hydrolysis can be com-patible with seawater in the presence of a chromiumcrosslinker.152

Xanthan has in recent years been regarded as anattractive polymer, mainly due to high salt toleranceand absence of shear degradation. High degree ofshear thinning makes xanthan easy to inject. Xan-than gels can also undergo shear thinning, and areable to redevelop its gel strength after shear forceshave been removed.153 Bacterial degradation is themain drawback using xanthan, and temperature sta-bility is not as good as for polyacrylamide.

Chromium Crosslinkers. There are three maintypes of chromium crosslinkers being used to gel poly-mers: (1) Cr3+ usually with chlorine or nitrate asanion; (2) dichromate Cr2O2−

7 with potassium orsodium as anion; and (3) Cr3(RCO2)3+

6 with R =ethyl giving chromium propionate,154 or R= methylgiving chromium acetate.152 The crosslinks formedare ligand bonds between chromium and carboxylategroups on the polymer.Cr3+ is the crosslinker that is most frequently stud-

ied. To understand how the polymers are crosslinked,it is of vital importance to know the solution chem-istry of Cr3+. In water, the chromium ion is bondedto six water molecules, thereby forming the hexaaquaion Cr(H2O)3+

6 . Due to the small radius of thechromium ion, the hexaaqua ion behaves like an acidloosing protons,155

Cr(H2O)3+6 →

Cr(H2O)6−n(OH)(3−n)+n + nH+.

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288 CHAPTER 11. SWEEP IMPROVEMENTS

Increase in pH will shift this equilibrium towardsthe right. This can be achieved by adding base, orby diluting the chromium solution. For n = 3 (maxi-mum value) a gellike precipitate of chromium hydrox-ide (Cr(OH)3) is formed, corresponding roughly to achromium concentration of 25 ppm and a pH of 6.For n = 1 or 2, the hexaaqua ion can polymerize

into oligomers (olation). The formation of chromiumdimers, trimers, tetramers etc. have been extensivelystudied,156 and is illustrated in Fig. 11.8 togetherwith the structure of the complexes.

Cr Cr

Cr

H2O

H2OH2O

OH

OH

OH

H2OOH

OH2OH2

OHOH2

OH2

6+

Cr(H2O)5OH2+

Trimer

Cr Cr

Cr

Cr

H2O

H2O

H2OH2O

H2O

OH

OHOH

H2OOH

OH2OH2

OH2

OH2

OHOH2

OH2

6+

Cr(H2O)5OH2+

Tetramer

Cr Cr

H2OH2O

H2OH2O

O

O

H

H OH2

OH2OH2

OH2

Dimer

4+

2Cr(H2O)5OH2+

Figure 11.8: Formation and structure of chromiumoligomers.156

The chromium oligomers have been shown to gelmuch faster than the monomer with both poly-acrylamide157 and xanthan.158 This, together withdata from reaction kinetics,159 led to the conclu-sion that the oligomers are the reactive species dur-ing crosslinking. However, a recent study by Lock-hart shows that in crosslinking with polyacrylamide,the monomeric form of chromium is the reactivespecies.160 This is explained by the fact that evenif the crosslinking with monomeric chromium is slow,it is faster than the formation of oligomers.In order to slow down the gelation process,

in order to obtain deeper penetration of the gelsolution, chromium in its sixth oxidation state(chromium (VI)) as dichromate (Cr2O2−

7 ) has beenused as crosslinker, together with a reducing agent.

Due to the slow reduction of chromium(VI) tochromium(III), gelation now becomes much slower.However, this system is complex, and appears to bedicult to control in a porous medium.145 Other dis-advantages are the high toxicity of the dichromate,high sensitivity towards H2S,145 and precipitation ofdichromate with barium in hard brines.154

A more recent proposal for slow gelling chromi-umbased crosslinkers is the chromium carboxylates(Cr3(RCO2)3+

6 ). They are reported by several au-thors145,152,154 to be less sensitive to pH and brinehardness than Cr3+ and Cr2O2−

7 . The structure ofthe acetate is shown in Fig. 11.9.161 At low pH (be-

H2O H2O

H2O

H2OH2O

Cr Cr

Cr

Cr CrCr

Ac

Ac Ac AcAc

Ac Ac Ac Ac

Ac

OHOH

OH OH

O

(a)

(b)

Figure 11.9: Chromium acetate structures.161 a)Cyclic structure found at pH below 4. b) Linear struc-ture found at pH 4-6.

low 4.5) it is a cyclic trimer with the acetate groupsas bidentate ligands. As pH increases, the acetatebonds are replaced by hydroxide bonds, leading to alinear structure. At a pH above 6, slow precipitationoccurs.The pH in most reservoirs (also in the North Sea)

ranges from 6 to 8, and seawater has a pH around 8.This seems to limit the applicability of the chromiumcrosslinkers, since they all will precipitate at a pHabove 6. A possible result is poor gelation betweenthe polymer and the precipitate (Cr(OH)3. The mostimportant drawback, however, is that the precipitatewill not be transported into the reservoir, preventinggel formation far from the well. To minimize pre-cipitation of chromium one therefore could use highchromium concentration and/or generate a low pHenvironment, either by a low pH preush162 or by us-ing a buer.163 The possibility of generating a low pHenvironment far from the well will to a large extentdepend on the buer capacity of the reservoir rock.The pH sensitivity of chromium acetate is reported

by Sydansk152 and Lockhart160 to be signicantlysmaller than for Cr3+, probably due to the slow pre-cipitation of chromium acetate. It is therefore pos-sible that this crosslinker will be the more suitable,at least for in-depth treatments. However, this willto a large extent depend on the gelation time. It

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11.3. POLYMER GELS 289

has also been shown that preformed gels from Cr3+

and polyacrylamide can be stable in the pH range of7 to 10.160 If they can be injected, as is the casefor the xanthan-based gels, they will be another pH-insensitive alternative.

Gelation Time

The most important parameters governing gelationtime for a given system are pH and temperature. Inaddition, concentrations and properties of the poly-mer and the crosslinker are important.

Eect of pH. Gelation time of Cr3+/polyacryl-amide and chromium acetate/polyacrylamide asa function of pH, has been studied by Lock-hart.160,163165 By adding buers to the gel solution,pH was kept constant during the entire gelation pro-cess. As can be seen from Fig. 11.10, the eect ofpH is largest when Cr3+ is used as crosslinker, where

106

105

104

103

102

101

2 4 6 8pH

Gel

atio

ntim

e,m

in

Chromiumacetate

Cr3+

Figure 11.10: Eect of pH on the gelation time ofCr3+ and chromium acetate with polyacrylamide inbuered solutions.160

the gelation time increases by a factor of 10 when pHdecreases with one unit. The same change in gela-tion time for the chromium-acetate system requiresa change in pH from 9 to 2. At a pH of 6, proba-bly close to the pH in the reservoir, we can see thatthe chromium acetate will gel about 500 times slowerthan the Cr3+ system. The same delay of the gelationprocess can be obtained using Cr3+, if pH is loweredto about 3.5. In fact, the use of buers to control thepH is suggested as a method of delaying the gelation,thereby obtaining in-depth treatment.163

The eect of pH has also been studied in unbueredsystems. For a Cr3+ -system, where xanthan was usedas polymer, the gelation time increased by a factor of2 as pH decreased from 4.5 to 3, showing a muchlower sensitivity than in the buered case.117,162 Achromium acetate system showed an increase in gela-tion time by a factor of approximately 5 as pH de-creased from 10.6 to 4.0 which is not too dierent

from the buered case.145 Here gelation was seen tooccur at a pH as high as 12.5.

Eect of Temperature. Few studies have fo-cused on the eect of temperature on the gela-tion time. One study has been conducted ona dichromate/polyacrylamide-system166 and anotheron a xanthan/Cr3+-system.162,167 In both thesestudies, the activation energy, Ea, of the gelationreaction was calculated, based on the gelation timemeasurements at dierent temperatures using anArrhenius-like relationship,

Ea = Rd(ln t)

d(1/T ), (11.9)

where R is the gas constant, T is absolute tempera-ture, and t is the time.For the xanthan/Cr3+-system, Fig. 11.11, Ea was

determined to be 96 kJ/mol. This corresponds to

4

3

23.0 3.1 3.2 3.3 3.4

Log

t,t

inse

cond

s

1000/T, 1/ oK

200 ppm400 ppm800 ppm800 ppm*

Figure 11.11: Arrhenius-type plot of gelation time.162

an increase in gelation time by a factor of 3 whentemperature is decreased from 50 to 40C. In a typi-cal North Sea reservoir, temperature can range from5C, which is the temperature of seawater, to 100C,which is the average reservoir temperature. In thistemperature range, the gelation time of the above de-scribed xanthan/Cr3+-system will vary by a factor of35000. This indeed shows that the temperature willbe an important parameter when polymer-gel tech-nology are to be used in the North Sea.The above mentioned dichromate/polyacrylamide

system had a lower Ea, ranging from 60 to 80 kJ/mol.These systems therefore are less sensitive to temper-ature. Measurements of gelation time at dierenttemperatures have also been carried out in connec-tion with other studies. Even if these data are notaccurate enough for calculating activation energies,they give an estimate of the temperature dependence.For the chromium acetate/polyacrylamide system,Ea can be calculated to be around 100 kJ/mol,,152

while the polyacrylamide/Cr3+-system gives Ea inthe range of 40 to 70 kJ/mol.163 This variation in

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290 CHAPTER 11. SWEEP IMPROVEMENTS

activation energies for the gelation does not only indi-cate that the temperature dependence is dierent fordierent systems, but also that the gelation mecha-nism is dierent. A possible explanation of the appar-ent dierence in Ea for the xanthan/Cr3+-system (96kJ/mol) and the polyacrylamide/Cr3+-system (40 to70 kJ/mol) is that the rate-determining step in theformer reaction is the formation of oligomers.

Eect of Polymer/Crosslinker Concentrationand Properties. Most of the studies performed toinvestigate the dependence of gelation time on theconcentration of polymer and crosslinker, show thatgelation time is inversely proportional to both poly-mer and crosslinker concentration.160,162,168,169 Thechemicals and the methods used for determining gela-tion time vary, however, and other relations have alsobeen proposed.159

Eect of the variation of polymer properties hasbeen studied for the polyacrylamides. Sydansk152 hasshown that the gelation time increases with decreas-ing molecular weight. The gelation time increases bya factor of about 5 as molecular weight decreases from270 000 to 180 000.The gelation time increases as the degree of hy-

drolysis of the polyacrylamide decreases, and alsothe sensitivity to changes in polymer concentrationseems to increase with increasing degree of hydroly-sis.169 For unhydrolyzed (< 0.1% hydrolyzed) poly-acrylamide, hydrolysis of the polymer becomes therate-determining step in the crosslinking reaction.Since this reaction is slow, this leads to very longgelation times even at high temperatures. Using lowmolecular weight polymers and chromium acetate tofurther slow down the process, gelation times longerthan 1 day at 124C were obtained, corresponding togelation times longer than 1 month at 70 to 80C.152

The properties of the various chromium crosslinkerswere discussed earlier in this chapter. In most cases,Cr3+ will gel much faster than chromium acetate,while the dichromate-redox system probably will liesomewhere in between. It should be stressed that thepH dependence is very dierent for the crosslinkersand the relative rates may change with pH.

Gel Strength

Gel strength is governed by the concentrations andproperties of polymer and crosslinker. For a owingsystem, shear rate will also be an important param-eter. Batycky et al.141 showed that gel strength in-creased with increasing polyacrylamide concentrationwhile an optimum concentration existed in crosslinker(dichromate). A yield strength of 600 Pa was mea-sured with 1500 ppm polymer and 1000 ppm dichro-mate, using a glass tube. For the same type of system,Liang et al.170 measured yield strength in the rangeof 80 to 200 Pa.Gels formed by xanthan are usually claimed to

be weaker than polyacrylamide gels.145 However, ifviscosity is measured during gelation at comparable

shear rates for systems containing similar chemicalconcentrations, the results can be quite similar. Thiscan be seen from two measurements using 2000 ppmxanthan/800 ppm CrCl3,162 and 2500 ppm polyacry-lamide/1060 ppm Na2CrO7.167 At a shear rate ofapproximately 10 s−1, both systems have a viscosityclose to 200 mPa·s. Also, both systems experience astrong increase in viscosity as shear rate is decreased,with the xanthan gel having a viscosity of 2 600 000mPa·s at a shear rate of 0.001 s−1, Ref. 162.Xanthan gels dier from polyacrylamide gels by the

ability to reform after having been broken down byshear.153 This reformability depends strongly on theshear intensity, and gel strength or chromium con-centration.170 Weak gels sheared at low intensity aremost likely to regain their original strength.

Gel Stability

Gels have to maintain their original strength andshape for several months in order to give an eectivetreatment. This seems to be most dicult at hightemperatures. The two most important processes re-garding stability are the rupture of internal polymerbonds and shrinking of the gel, which both can leadto loss of permeability reduction.If xanthan is used as polymer, cleavage of the poly-

mer backbone will occur at temperatures above 70 to80C.101,103 As mentioned, xanthan can also undergodegradation by microbes. It is not clear whether thepresence of chromium, or the fact that xanthan iscrosslinked, will have any inuence on the stabilitytowards microorganisms or high temperatures.Polyacrylamide gels are reported to shrink at high

content of divalent ions and high temperature. At127C the volume of a gel formed by polyacrylamideand Cr3+/dichromate was reduced to only 10% after 4days.171 After 2 days, the permeability reduction hadalmost disappeared, corresponding to 50% volume re-duction. This shrinking was believed to be caused byhydrolysis of the polyacrylamide at high temperature,followed by overcrosslinking resulting from the reac-tion between divalent ions and the hydrolyzed amide(carboxylate) groups.More recent work has shown that polyacrylamide

gels can be stable for several years at high tempera-ture and high content of divalent ions.152 In thesestudies, chromium acetate and unhydrolyzed poly-acrylamide (< 0.1% hydrolyzed) were used, and op-timal stability conditions were created by the carefulremoval of oxygen. The lack of overcrosslinking andshrinking for this system still remain unexplained.Other systems that are reported stable at high

salinity and high temperature are acrylamide copoly-mers crosslinked with a mixture of phenols andformaldehyde.149 Also scleroglucan, being stable asa polymer solution at temperatures above 100C,103

has shown good thermal stability at 90C as achromium gel.172

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11.3. POLYMER GELS 291

11.3.3 Gelation in Porous Media

As mentioned in Sec. 11.3.2, bulk measurements onlygive an indication of how a gel treatment will work.To get a better description, studies of gelation inporous media have to be performed. Important fac-tors are how the gel solution is transported throughthe porous medium, where the gel solution is trans-ported, how the gel aects permeability towards oiland water, and for how long the permeability can re-main reduced. Also the injection procedure will beimportant for the gelation process. In the following,we shall assume that the polymer and crosslinker areinjected together immediately after mixing. Alterna-tive injection procedures are to inject crosslinker andpolymer sequentially (as with the aluminum-citratesystems), or to inject the gel itself (as can be donewith xanthan/Cr3+- systems).

Transport of Gel Solution

Filtration Mechanism. The rst detailed stud-ies of gelation during ow through a porous medium(a linear sandpack) showed that the gelation timewere much shorter than gelation times measured inbulk.173 It was later suggested that in-situ gela-tion involved the formation of gel aggregates thatwere retained by the sandpack.174 Based on fur-ther measurements of in-situ gelation, where it wasseen that ow resistance only developed over a shortzone in the sandpack used, the ltration mechanismwas postulated.175 This model has been conrmedby later measurements of in-situ gelation,162,176178

and seems to be valid both for xanthan and poly-acrylamide. The ltration process can schematicallybe described as follows:

1. Crosslinking leads to the formation of aggregates(pregel clusters). The size of the aggregates in-creases as they combine.

2. As the polymer-gel solution ows, the clustersare ltered from solution. Filtration rate in-creases with increasing concentration and size ofthe aggregates.

3. After a given time, and at a specic position inthe sandpack, ltration accelerates and a zone ofhigh ow resistance is developed.

A numerical model, based on the ltration mecha-nism, predicts the location of the zone having highow resistance with reasonable accuracy.179 Themodel uses equations for chemical transport, reactionkinetics between polymer and crosslinker, distribu-tions of particle size for pregel aggregates, ltrationrate (based on two pore sizes) and the eect of ltra-tion on permeability.One would expect that in-situ gelation time would

increase as permeability increases, since the poresthat are to be plugged then becomes larger. Experi-mental results exist that indicate that this is the case,and that in-situ gelation time at high shear rate and

high permeability in fact can be higher than the bulkgelation time.162 However, other results indicate thatthe in-situ gelation time is independent of permeabil-ity.174

Retention of Cr3+ and xanthan, both alone and inmixture, has been measured in Berea sandstone.180

Due to presence of clay the retention is here muchhigher than in a sandpack. A retention of 60 to 65µg/g was measured for Cr3+. Since the chromiumconcentration used in gel treatments is low, this reten-tion will have a great impact on the rate of transportof chromium. A retention value of 40 µg/g, measuredfor xanthan, will not aect the transport of polymersignicantly. Injecting a 50 ppm solution of Cr3+

would require more than 10 pore volumes in orderto saturate the core. Hence, if the retention processis fast compared to the ow rate, the polymer frontwill travel about 10 times faster than the chromiumfront. Also, the average chromium concentration inthe porous medium will be much lower than the in-jected concentration. Depending on the clay content,the injected concentration therefore should be higherthan determined from bulk studies. This was con-rmed by the fact that a much higher ow resistancewas measured in in-situ gelation experiments whenthe chromium concentration was increased from 50to 150 ppm. In these experiments, the retention ofchromium and polymer increased, as suggested by theltration mechanism described previously.

Permeability Reduction and Stability

In a gelation process, we have two ways of permeabil-ity reduction. First, permeability is reduced as thegel solution ows, due to the previously describedltering of gel aggregates. This will mainly aectthe placement of the gel. Secondly, the permeabilityis reduced during shut-in of the well, when gelationproceeds in a bulk-like manner. This will aect thewater-diverting ability of the gel.Like viscosity, the permeability reduction during

ow can depend on shear rate and the composition ofthe gel solution. For a xanthan/Cr3+ system, mea-surements have shown that the permeability reduc-tion becomes signicantly smaller as shear rate in-creases.177,178 The same observation has been madevarying the temperature. Decreasing temperatureleads to strong reduction in apparent viscosity to-gether with an increase in in-situ gelation time.162

For polyacrylamide gels, shear does not seem to re-duce ow resistance.176 However, the number of datapoints is small.After a shut-in period exceeding the bulk gelation

time, a permeability reduction by a factor of 1000to 10 000 can be observed, both for polyacrylamideand xanthan gels.152,162 Such gels can resist pressuregradients of 100 bars/meter or more. It is often as-sumed that shrinking of the gel, due to overcrosslink-ing caused by e.g., too high chromium concentration,or high temperature combined with hard brines, will

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292 CHAPTER 11. SWEEP IMPROVEMENTS

result in a gel that does not plug the core. However,experimental results show that there need not be adirect correspondence between shrinking and perme-ability reduction.181 Xanthan/chromium (III) gels,that were shrinking by 70% as bulk samples, eec-tively plugged unconsolidated sandpacks. This is in-teresting since it seems to allow for the injection ofhigh chromium concentrations that can prevent theprecipitation of chromium.

Gel Placement in the Reservoir

Placement of the gel solution will to a large extentbe governed by the nature of the permeability con-trasts, which also are responsible for the water break-through. In general we may say that the larger thepermeability contrasts, the earlier the water break-through and the more ecient the gel placement. Inan optimal gel treatment the gel solution penetratesdeep into the high-permeable zones without aect-ing the low-permeable zones. However, using analyt-ical models,182 it can be shown that low-permeablezones will experience a signicant penetration of thegel solution. This penetration will increase as mobil-ity ratio between the gel solution and reservoir u-ids decreases, and will be larger in a radial geometrythan in a linear geometry. If no barriers can pre-vent crossow between the layers, penetration intothe low-permeable layer will increase farther if themobility ratio is lower than one, but decrease if it ishigher than one.183 In order to minimize the dam-age of low-permeability zones, the viscosity of the gelsolution therefore should be as low as possible. Forpolymer-gel systems this is best obtained with lowmolecular weight polymers.The need for zone isolation during place-

ment of polymer gels have been analyzed bySeright.182,184186 Studying the eect of geome-try, shear thinning, mobility ratio, polymer reten-tion, diusion/dispersion, crossow, variations in re-sistance factors with permeability, and the possibilityof higher reduction in water permeability than oil per-meability, the conclusion is that zone isolation wouldbe required in most cases. One could then ask: Iszone isolation sucient to prevent penetration intolow-permeable zones? The answer is of course yesif the layers are separated with impermeable barri-ers. However, if the mobility ratio is low and somedegree of communication exists between layers, cross-ow into the low-permeable zones probably will hap-pen at some distance from the well. This tendencywill increase after the gel has started to ltrate fromsolution, and it will therefore be important duringplacement of the gel that injection stops before thein-situ gelation time is reached.One would expect the retention of chromium to be

of importance during placement. As mentioned previ-ously, the ion exchange with the clay in the reservoiris expected to be high,180 Sec. 11.3.3, and since theclay content usually increases with decreasing perme-ability, the chromium concentration will rst reach

a value below the critical value for gelation in thelow-permeable, oil-saturated zones. Therefore, theamount of gel compared with the amount of injectedgel solution should be smaller in the low-permeablezone.

Simulation Studies

Some guidelines for polymer-gel treatment can be ob-tained from simulation studies.187190 However, themodels used have simplistic ways of representing thein-situ gelation process, and little new information isobtained. As for the gel placement studies describedabove, they conclude that gel treatment eciency in-creases with depth of treatment, use of zone isolationand decreasing vertical communication between lay-ers.

Field Experience

Experience from eld treatments using polymer gelsis growing. An overview based on 130 eld projectsduring the period 1980 to 1989 was given by Schurzet al.191 Minimum, maximum and median values ofthe most important parameters were given, and it isinteresting to see that both the initial oil content andwater-oil ratio are high at start-up, that the incre-mental oil recovery is low (only 2% of OOIP), butthat the eciency is high. The median value of 2.4bbl of oil per lb of polymer corresponds roughly to acost of one dollar per extra barrel of oil recovered.In the early eighties, Mobil used xanthan

crosslinked with Cr3+ in more than 200 wells.190 Bythe summer in 1983, incremental recovery was 375000bbl, corresponding to 0.5 bbl of oil per lb of polymer.More recent eld tests have shown higher e-

ciency.144,192 Using polyacrylamide crosslinked withchromium acetate, the average incremental recoveryfrom 29 wells, in naturally fractured reservoirs, was13 bbl of oil per lb of polymer injected. It is here im-portant to note that the treatments were much moreecient in the injection wells compared to produc-tion wells (by a factor of 40). Results also showed thattreatments in carbonate rock were more ecient thanthose in sandstone formations. It is not clear whetherthe high eciency mainly is due to the nature of thegel system or to the fact that the reservoirs were frac-tured. High eciency of gelation in fractured reser-voirs has also been reported elsewhere.182 This is notsurprising since a fractured reservoir represents thehighest degree of permeability contrast there is, onlya small volume of gel is required to ll the fracture.Little gel is expected to enter the matrix if a pregelledsolution is injected.

11.3.4 Summary and Conclusions

Field experience indicates that polymer-gel treat-ments seem to be most ecient in fractured reser-voirs. In the case of unfractured reservoirs with

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11.4. FIELD EXAMPLES 293

strong variations in permeability, zone isolation prob-ably will be necessary in order to reduce the penetra-tion of gel solution into the low-permeable zone. Thisis also the result if the gel solution has low viscosity.Both during placement of the gel solution, and dur-ing the later production, it is an advantage that thevertical communication between the high-permeableand low-permeable zones is as low as possible.The strength of the gels that can be made seems

to be more than sucient to prevent the water fromowing, even very close to the well. A much largerproblem with todayÕs technology is to make gels thatare stable, and that are formed suciently slowly athigh temperatures. This is of special importance re-garding North Sea application since the temperaturesare high, gelation rate increases rapidly with temper-ature, and large well spacing requires deep penetra-tion.Deep penetration requires that the various compo-

nents are transported through the reservoir withoutany signicant losses. Experimental results show thatthe retention of chromium can be high compared tothe concentrations used, and therefore will travel onlya small fraction of the distance travelled by the poly-mer. This retention will to a large extent depend onpH and in which form the chromium is injected, andthe eect on the gel treatment will be large.

11.4 Field Examples

11.4.1 Introduction

Polymer processes to improve oil recovery have beenapplied world-wide during the last decades. Sev-eral review articles150,193,194 conclude that polymerprojects very often are reported to be successful, bothtechnically and economically. The frequency of pos-itive reports has increased the last years due to abetter planning and understanding of the interactionbetween reservoir description and polymer injectionstrategies.We limit ourselves to consider a few cases where

the polymer slug size amounts to a signicant partof the reservoir pore volume. Such projects seem togive the best results regarding incremental oil produc-tion when done at an early stage of the eld develop-ment. The design of the optimum polymer treatmentis, however, strongly dependent on the knowledge ofthe reservoir. The production history may reveal situ-ations where eld treatments are actualized at a laterdevelopment stage.

11.4.2 North Sea Applications

The application of polymer processes to improve oilrecovery in North Sea reservoirs has been extensivelydiscussed during the last years.139,195,196 The discus-sions and evaluation program have been focused onthe Gullfaks eld.

The main reservoir unit in Gullfaks is the BrentGroup. The Etive and Rannoch formations are de-veloped in the early production phase. The highestpermeabilities are found in the Etive formation, giv-ing a strong permeability contrast with the lower per-meable Rannoch formation. Water injection has beenchosen for pressure maintenance and as drive mech-anism. The reservoir temperature is 70C and theoil viscosity at reservoir conditions is 1.1 mPa·s. Thelow oil viscosity implies that the mobility ratio to wa-ter is such that a high oil recovery is expected beforewater breakthrough. Possible benet from increasingviscosity in the injection uid to improve fractionalow of oil and mobility ratio is marginal.The strong permeability contrast between the Ran-

noch and Etive formations, and also permeabilitycontrasts within each of the formations, may lead toa nonecient sweep in the low-permeable parts of thereservoir. This implies that the oil from these partswill be produced late and partly at high watercuts.The addition of polymer to the injection uid will

build up resistance to ow during injection in thehigh-permeable layers. This is mainly due to the in-creased viscosity, but polymer adsorption, loweringthe permeability to water, may also give some ef-fect. The net result is that the injection uid willdivert into low-permeable layers, giving a more e-cient drive in these layers.Polymer ooding under these conditions is strongly

related to an improvement of volumetric sweep e-ciency, an earlier production of the mobile oil and pro-duction at lower watercuts. The eect of uid diver-sion and a stabilization of the front have been demon-strated in laboratory experiments,197 see Fig. 11.12.The upper frame shows the performance of a water

Figure 11.12: X-Ray tomography visualization ofwater- and polymer oods in a stratied core.197

ood at the water breakthrough point in the high-

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294 CHAPTER 11. SWEEP IMPROVEMENTS

permeable part of the core. The lower frame shows aviscous polymer ood. The water front is stabilizedand the polymer is contacting the whole core area.

Observations During Production

The Lower Brent is developed by fault block. Injec-tion wells are drilled structurally low and productionwells structurally high and close to the fault planes.Both injection and production wells are perforatedin the Rannoch formation. The present water over-ride situation is shown in Fig. 11.13. The eect ofpermeability contrasts has been more severe than an-

Etive

Sw < 0.25

0.25 < Sw < 0.55

Sw > 0.55

Injector Producer

Rannoch

Figure 11.13: Override of water and water saturationin a Lower Brent cross section of Gullfaks.139

ticipated. The water segregation into the Rannochis also hindered by low vertical permeability betweenEtive and Rannoch. The observations suggest thatthe sweep eciency of Rannoch is poor.This situation led to an extension of the evalua-

tion program to include other polymer processes. Theprocess should preferentially be evaluated for its po-tential of uid diversion.As mentioned, a polymer process will generate ow

resistance - due to viscous eects - in the thief zonesand improve areal contact and sweep eciency inlower permeable zones.The concept of Deep Penetration Gels (DPG) may,

however, be a more ecient way to improve the sweepeciency. The DPG will give ow resistance in thiefzones by permeability reduction, which can be morepermanent than the viscous eect. The eect of per-meability reduction can also be higher and reducethe necessary volume of the treatment. The processhas the potential that subsequent injected water e-ciently will divert into the previously unswept lowerRannoch.

The Oshore Operation

The oshore operation implies that large volumes ofpolymer have to be handled on a limited space. Thespace limitations make it necessary to nd technicalsolutions where the polymer product can be mixedand diluted directly into the injection water ow line.The salinity of the injection water (seawater) and thelarge shear forces in such a mixing process already put

strong constraints on the polymer product to be used.The high injection rates expose the polymer to ad-ditional shear forces when entering the sand matrix.The eect of shear degradation during these processeshas to be seriously considered, to be able to denethe polymer properties in the reservoir. The techni-cal and practical problems by handling the polymercombined with the size of the reservoirs might be thelimitation of the size of a polymer operation.

As example, the polymer quantity required to feedan injector with 5000 m3/d polymer solution will be2 to 10 tons, dependent on polymer type (referringto a dry product). Also, the production and regularsupply of a given product for such an operation is aproblem to be solved.

The reservoir description and production history isof signicant importance to dene the potential andeconomic basis for a eld project. The simulationstudies have to be based on detailed polymer proper-ties at relevant eld conditions. As mentioned earlier,the polymer should preferentially be resistant to shearforces, and have a good injectivity.198 The chemicaland thermal stability199,200 must be known, as wellas properties related to ow and interaction in thesandstone matrix.126,135

11.4.3 The Vorhop-Knesebeck Poly-mer Flood

In the process of developing eld technology for NorthSea applications, the successive scale-up of knowledgefrom the laboratory to full-eld processes has to beconsidered as a very important task.

Statoil is engaged in the Vorhop-Knesebeck poly-mer eld ooding project.109 The eld is locatedin North-Germany and is operated by Preussag.

H3a

845

14

12

10

48

8

1220

1280

1300 1260

1240

Figure 11.14: Structure map of block H3a, Vorhop-Knesebeck.109

Preussag and its partner BEB planned for this eldproject on the basis of the positive results from aresearch pilot at Eddesse-Nord.105 Compared to anoperation in the North Sea, the Vorhop-Knesebeckproject can be considered as a pilot.

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11.4. FIELD EXAMPLES 295

-50 -30 -10 10 0 5 10 50 500 1000 5000 22 24 26 28 30

SP GR SFLU ILD Permeability in md Porosity in %Depthin m1390

1395

1400

1405

1410

1415

Figure 11.15: Log and core data from well VK48.109

Field Information

The project is performed in the upper layer of theDogger-sandstone in block H3a. A structure map ofthe block is shown in Fig. 11.14. The block is todaycomprising the wells VK8, VK45, VK48 and VKH3a.The eld was developed in 1959 with well VK8, andwas produced the rst four years by decreasing thepressure to bubblepoint pressure. The next years theproduction mechanism was solution-gas drive at thebubblepoint pressure. This had to be abandoned in1985, and the well H3a was drilled for water injec-tion. Water injection started in 1987. The reservoirpressure increased very quickly and production couldsoon be started from all three producers in the struc-ture. The production history shows that the blockH3a Upper Layer is conned by sealing faults with asmall communication to the lower layer, as well as tothe block in the East.A summary of eld parameters is given in Ta-

ble 11.1.

Table 11.1: Reservoir parameters of polymer projectVorhop Knesebeck block H3a109

Depth 1250mArea 145 000m2

Thickness 8-14mPorosity 27%Pore volume 430 000m3

Init. water sat. 14%OOIP 370 000Rm3

Form. volume factor 1.04Rm3/Sm3

Form. temperature 56 COil viscosity 3.5mPa sOil density 860 kg/m3

Salinity form. water 21% TDSPermeability 1000md

Fig. 11.15 shows a log sequence of the upper layerin well VK48 together with results from core analy-

sis. The data show that this reservoir is divided intotwo zones. The upper zone has permeabilities in therange 1 to 5 darcies and is separated from the lowerzone (about 500 md) by a low-permeable clay layer.The formation water has a salinity of 21% TDS (To-tal Dissolved Solids), and brine from the treatmentplant is reinjected. The viscosity ratio between oiland injection water is 4, somewhat more unfavorablethan at Gullfaks, where the value is 2.5.Water breakthrough was observed in early 1988

both in well VK45 and well VK8, and the watercutsincreased fast. At the start of the polymer injectionprogram in January 1990, well VK45 produced onlywater and the watercut in well VK8 was approach-ing 90%. In well VK48, the water breakthrough wasobserved late in 1988. The watercut increased moreslowly than in the other producers and was at thestart of the polymer injection approximately 30%.

Polymer Program

The polymer injection program has been divided intotwo phases. Phase 1 started in January 1990 with in-jection in well VK3a. The injection rate is 120m3/dand the polymer concentration is approximately 500ppm. The polymer solution has a viscosity of 4 mPa·sat reservoir ow conditions rates. The injection isplanned for 22 months. The polymer selected forthe project is a nonaggregating xanthan broth prod-uct,198,202 produced at Statoil, Biosentrum.The dilution of the 3% fermentation broth to the in-

jection concentration of 0.04 to 0.05% is performed byfeeding the broth into the water ow line. The use ofin-line static and dynamic mixers assured a solutionwith excellent injectivity. The injection-pressure de-velopment during the rst year of injection is shown inFig. 11.16. The injection pressure increased fast therst days to a value expected from the enhancementof viscosity, and then leveled out. The stable injectionpressure has been maintained during the rst year ofinjection. The variations in injection pressure follow

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296 CHAPTER 11. SWEEP IMPROVEMENTS

+

+

+

++

++ +

+ +

Inj. rate

140

120

100

80

60

40

20

0JAN MAR MAY JUL SEP NOV

pwH VKH3a

pwf (595.4 m)VK45

Time

Pres

sure

,ba

r

Figure 11.16: Injector well-head pressure, pwH (+)and formation pressure pwf ( ) development duringthe rst year of polymer injection.109

the variations in reservoir pressure.Results from a simulation study to match the wa-

tercut development in the production wells are shownin Figs. 11.17 to 11.19. Water breakthrough in bothwell VK45, Fig. 11.17, and in well VK8, Fig. 11.18,was observed less than one year after injection

Measured watercutWaterfloodingpredictionPolymer prediction

Time

Wat

ercu

t

1.0

.8

.6

.4

.2

.086 89 92 95 98

Figure 11.17: Watercut, Well VK45.109

started. A clear response of the polymer oodingprocess was observed early in 1991. The watercut de-creased from 100% to 95% and the well produced oilfor the rst time in one year. The measured water-cut development in well VK8 indicates a stabilization.The measurements in well VK48, Fig. 11.19, showeven a slight decrease in watercut about 3 monthsafter the start of polymer injection. A break inthe polymer injection program during summer 1990gave production at somewhat higher watercuts afterrestarting the operation. A stabilization of watercutsis, however, again observed at these watercut levels.These results indicate a much earlier response of

the polymer injection than given by simulation, whichpredicted a lowering of water-cut in the middle of1991.The rst part of the polymer slug will have prefer-

ence for the parts of the reservoir already watered out.The improvement of mobility ratio will give a more ef-cient and earlier production of the rest of the mobile

1.0

.8

.6

.4

.2

.0

Time

Wat

ercu

t

Measured watercut

Waterflooding

prediction

Polymer prediction

86 89 92 95 98

Figure 11.18: Watercut, Well VK8.109

1.0

.8

.6

.4

.2

.0

Time

Wat

ercu

t

Measured watercut

Waterflooding

prediction

Polymer prediction

86 89 92 95 98

Figure 11.19: Watercut, Well VK48.109

oil. It is anticipated that the deeper, lower permeablezone in the reservoir, Fig. 11.19, has been less e-ciently contacted by water than the high-permeablezone. Water segregation may to some extent be hin-dered due to the low permeable layer between thezones.The increased viscosity will give a diversion of the

polymer into lower permeable layers. The viscosityof the diverted polymer solution will secure mobil-ity control and give a good recovery from the low-permeable layer, which can not eciently be sweptby water at economic water/oil ratios.The response of the polymer process in the eld is

so far very promising. The decision to start Phase 2of the injection program was scheduled be taken inthe middle of 1991.109 The outcome is not known.

Nomenclature

Ea = activation energy, J/Molfg = gas fractional owK = consistency parameterk = absolute permeability, µm2 or darcy or m2

M = mobility ration = shear thinning index= number, n ∈ 1, 2, 3

Pc = capillary pressure, Pap = pressure, bar or Paq = rate, m3/dR = gas constant, J/K mol

RF = Resistance Factor, mobility reduction fac-tor

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REFERENCES 297

RRF = Residual Resistance Factor, permeabilityreduction factor

r = radius, mS = saturation, fraction= skin factor

T = temperature, K or Ct = time, sv = velocity (Darcy), m/sx = distance, mα = α-factor (material constant), dimension-

lessγ = shear rate, s−1

η = viscosity, Pa·sλ = mobility, m2/Pa· s or relaxation time, sπ = disjoining pressure, Paφ = porosity

Subscripts

c = capillary or criticalf = owingg = gasH = headi = irreducible or 1,2 (water, polymer)p = polymerr = relative or residualw = water or well

Superscripts

f = nali = initial∗ = critical

Operators

∆ = dierence∇ = gradient

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[133] Zaitoun, A and Kohler, N.: Two phase FlowThrough Porous Media. Eect of Adsorbed

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302 CHAPTER 11. SWEEP IMPROVEMENTS

Layer, paper SPE 18085 presented at the 1988SPE Annual Technical Conference and Exibi-tion, Houston, Oct.25.

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[143] Routson, W.G., Neale, M., and Penton, J.R.:A New Blocking Agent for Waterood Chan-neling, paper SPE 3992 presented at the 1972SPE Annual Fall Meeting, San Antonio, Oct.811.

[144] Sydansk, R.D. and Moore, P.E.: Produc-tion Responces in Wyoming`s Big Horn BasinResulting from Application of Acrylamide-Polymer/ Cr(III)-Carboxylate Gels,paper pre-sented at the 1990 Symposium on EOR, Uni-versity of Wyoming, May 34.

[145] Sydansk, R.D.: A New Conformance-Improvement-Treatment Chromium(III) GelTechnology, paper SPE/DOE 17329 presentedat the 1988 SPE/DOE Symposium on EOR,Tulsa, April 1720.

[146] Vossoughi, S. and Putz, A.: Reversible In-Situ Gelation by the Change of pH Withinthe Rock, paper SPE 20997 presented at the1991 SPE International Symposium on OileldChemistry, Anaheim, Feb. 2022.

[147] Strom, E.T., Paul, J.M., Phelps, C.H., andSampath, K.: A New Biopolymer for High-Temperature Prole Control: Part 1 - Labo-ratory Testing, paper SPE 19633 presented atthe 1989 SPE Annual Technical Conference andExhibition, San Antonio, Oct. 811.

[148] Borchardt, J.K.: Chemicals Used in Oil-eld Operations, in Oil-Field Chemistry: En-hanced Recovery and Production Stimulation,ACS Symp. Ser., 396, p 354 (1989).

[149] Moradi-Araghi, A., Bjørnson, G., and Doe,P.H.: Thermally Stable Gels for Near-WellborePermeability Contrast Corrections, paper SPE18500 presented at the 1989 SPE InternationalSymposium on Oileld Chemistry, Houston,Feb. 810.

[150] Chang, H.L.: Polymer Flooding Technology -Yesterday, Today and Tomorrow, JPT (Aug.1978) 111328.

[151] Hawk, W.A. and Cooke, R.W.: Crosslinkingof Polymers in Oileld Conformance ControlApplications: Advantages and Disadvantages ofNatural vs. Synthetic Polymers. presented atthe 1986 American Chemical Society NationalMeeting, Anaheim, Sep. 712.

[152] Sydansk, R.D.: Acrylamide - Polymer /Chromium(III) Carboxylate Gels for near Well-bore Matrix Treatments, paper SPE/DOE20214 presented at the 1990 SPE Symposiumon EOR, Tulsa, April 2225.

[153] Avery, M.R., Burkholder, L.A., and Gruenen-felder, M.A.: Use of Xanthan Gels in ActualProle Modication Field Projects, paper SPE14114 presented at the 1986 International Meet-ing on Petroleum Engineering, Beijing, March1720.

[154] Mumallah, N.A.: Chromium(III) Propionate:A Crosslinking Agent for Water-Soluble Poly-mers in Hard Oileld Brines, SPERE (Feb.1988) 24350.

[155] Cotton, F.A. and Wilkinson, G.: Advanced In-organic Chemistry., 4th Ed., John Wiley, NewYork, (1980) 727.

[156] Stünzi, H., Spiccia, L., Rotzinger, F.P., andMarty, W.: Early Stages of the Hydrolysis ofChromium(III) in Aqueous Solution. 4) Stabil-ity Constants of the Hydrolytic Dimer, Trimerand Tetramer at 25C and I = 1.0 M., Inor-ganic Chemistry, 28, No. 1 (1989) 66.

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[160] Lockhart, T.P.: Chemical and StructuralStudies on Cr3+/Polyacrylamide Gels, paperSPE 20998 presented at the 1991 SPE Interna-tional Symposium on Oileld Chemistry, Ana-heim, Feb. 2022.

[161] Tackett, J.E.: Characterization of Chromi-um(III) Acetate in Aqueous Solution," AppliedSpectroscopy, (March/April 1989) 4909.

[162] Kolnes, J., Stavland, A., and Thorsen, S.: TheEect of Temperature on The Gelation Timeof Xanthan/ Chromium(III) Systems., paperSPE 21001 presented at the 1991 SPE Interna-tional Symposium on Oileld Chemistry, Ana-heim, Feb. 2022.

[163] Lockhart, T.P., Albininico, P., and Burrafato,G.: Slow-Gelling Cr+3/Polyacrylamide Solu-tions for Reservoir Prole Modication. Depen-dence of the Gelation Time on pH., J. Appl.Polym. Sci., submitted for publication (1990).

[164] Lockhart, T.P. and Burrafato, G.: A New MildChemical Method for the Degelation of Cr+3

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[165] Lockhart, T.P., Burrafato, G., and Bucci, S.: Cr+3 - Polyacryl- amide Gels for Prole Modi-cation. Crosslinking Structure and Chemistry.paper presented at the 1989 European Sympo-sium on IOR, Budapest, April 2527.

[166] Jordan, D.S., Green, D.W., Terry, R.E.,and Willhite, G.P.: The Eect of Temper-ature on Gelation Time for PolyacrylamideChromium(III) Systems. SPEJ (Aug. 1982)463.

[167] Thorsen, S.: A Laboratory Study of XanthanChromium(III) Gels, MSc Thesis, RogalandUniversity Center, Stavanger (1989).

[168] Terry, R.E. et al.: Correlation of GelationTimes for Polymer Solutions Used as Sweep Im-provement Agents, SPEJ (April 1981) 22935.

[169] Hunt, J.A., Young, T.S., Green, D.W.,and Willhite, G.P.: A Study of Cr(III)-Polyacrylamide Reaction Kinetics by Equilib-rium Dialysis, Am. Inst. Chem. Eng. J. (Feb.1989) 35, No. 2.

[170] Liang, J., Tsau, J.S., Hill, A.D., andSephenoori, K.: Laboratory Study of PolymerGel Behavior, paper SPE 18506 presented atthe 1989 SPE International Symposium on Oil-eld Chemistry, Houston, Feb. 810.

[171] DiGiacomo, P.M. and Schramm, C.M.: Mech-

anism of Polyacrylamide Gel Syneresis Deter-mined by C-13 NMR, paper SPE 11787 pre-sented at the 1983 International Symposiumon Oileld and Geothermal Chemistry, Denver,June 12.

[172] Nagra, S.S., Batycky, J.P., Nieman, R.E. andBodeux, J.P.: Stability of Waterood Divert-ing Agents at Elevated Temperatures in Reser-voir Brines, paper SPE 15548 presented at the1986 SPE Annual Technical Conference and Ex-hibition, New Orleans, Oct. 58.

[173] Huang, C-G., Green, D.W., and Willhite, G.P.:An Experimental Study of the In-Situ Gela-tion of Chromium(+3)- Polyacrylamide Poly-mer in Porous Media, paper SPE/DOE 12638presented at the 1984 SPE/DOE Symposiumon EOR, Tulsa, April 1518.

[174] Hubbard, S., Roberts, L.J., and Sorbie, K.S.:Experimental and Theoretical Investigation ofTime-Setting Polymer Gels In Porous Media,paper SPE/DOE 14959 presented at the 1986SPE/DOE Symposium on EOR, Tulsa, April2023.

[175] McCool, C.S., Green, D.W., and Willhite,G.P.: Permeability Reduction MechanismsInvolved in In-Situ Gelation of a Polyacry-lamide/Chromium(VI)/Thiourea System, pa-per SPE/DOE 17333 presented at the 1988Symposium on EOR, Tulsa, April 1720.

[176] Marty, L., Green, D.W., and Willhite, G.P.:The Eect of Flow Rate on the In-SituGelation of a Chrome/Redox/Polyacryl-amideSystem, paper SPE 18504 presented at the1989 SPE International Symposium on OileldChemistry, Anaheim, Feb. 810.

[177] Heiri, S., Green, D.W., and Willhite G.P.: In-Situ Gelation of Xanthan/Cr(III) Gel Systemin Porous media, paper SPE 19634 presentedat the 1989 SPE Annual Technical Conferenceand Exhibition, San Antonio, Oct. 811.

[178] Jousset, F., Green, D.W., Willhite G.P., andMcCool C.S.: Eect of High Shear Rate onIn-Situ Gelation of a Xanthan/Cr(III) System,paper SPE/DOE 20213 presented at the 1990SPE/DOE Symposium on EOR, Tulsa, April2225.

[179] Todd, B.J., Willhite, G.P., and Green, D.W.:A Mathematical Model of In-Situ Gelationof Polyacrylamide by a Redox Process, pa-per SPE/DOE 20215, presented at the 1990SPE/DOE Symposium on EOR, Tulsa, April2225.

[180] Garver, F.J., Sharma, M.M., and Pope G.A.:The Competition Beetween Xanthan Biopoly-mer and Resident Clays in Sandstones, paperSPE 19632, presented at the 1989 SPE AnnualTechnical Conference and Exhibition, San An-tonio, Oct. 811.

[181] Eggert Jr., R.W., Willhite, G.P., and Green,D.W.: Experimental Measurement of the Per-

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304 CHAPTER 11. SWEEP IMPROVEMENTS

sistence of Permeability Reduction in PorousMedia Treated withXanthan/Cr(III) Gel Sys-tems, paper SPE 19630 presented at the 1989SPE Annual Technical Conference and Exhibi-tion, San Antonio, Oct. 811.

[182] Seright, R.S.: Placement of Gels To ModifyInjection Proles, paper SPE/DOE 17332 pre-sented at the 1988 SPE/DOE Symposium onEOR, Tulsa, April 1720.

[183] Wright, R.J. and Dawe, R.A.: Fluid Displace-ment Eciency in Layered Porous Media - Mo-bility Ratio Inuence, Rev. de L`Inst. Franc.du Petrole, 38, 455-474 (1983).

[184] Seright, R.S.: Eect of Rheology on GelPlacement, paper SPE 18502 presented at the1989 SPE International Symposium on OileldChemistry, Houston, Feb. 811.

[185] Seright, R.S.: Impact of Dispersion on GelPlacement for Prole Control, paper SPE20127 presented at the 1990 SPE Permian BasinOil and Gas Recovery Conference, Midland,March 89.

[186] Liang, J., Lee, R.L., and Seright, R.S.: Place-ment of Gels in Production Wells, paper SPE20211 presented at the 1990 SPE Symposiumon EOR, Tulsa, April 2225.

[187] Hughes, D.S., Woods, C.L., and Crofts, H.J.:Numerical Simulation of Single Well PolymerGel Treatments in Heterogeneous Formations,paper SPE/DOE 20242 presented at the 1990SPE/DOE Symposium on EOR, Tulsa, April2225.

[188] Gao, H.W., Chang, M.M., Burcheld, T.E.,and Tham, M.K.: Studies of the Eects ofCrossow and Initiation Time of a PolymerGel Treatment on Oil Recovery in a WateroodUsing a Permeability Modication Simulator,paper SPE/DOE 20216 presented at the 1990SPE/DOE Symposium on EOR, Tulsa, April2225.

[189] Scott, T., Roberts, L.J., and Sharpe, S.R.:In-Situ Gel Calculations in Complex ReservoirSystems Using a New Chemical Flood Simula-tor, paper SPE 14234 presented at the 1985SPE Annual Technical Conference and Exhibi-tion, Las Vegas, Sep. 2225.

[190] Abdo, M.K., Chung, H.S., and Phelps, C.H.:Field Experience With Floodwater Diversionby Complexed Biopolymers, paper SPE/DOE12642 presented at the 1984 SPE/DOE Sympo-sium on EOR, Tulsa, April 1518.

[191] Schurz, G.F., Martin, F.D., Seright, R., andWeiss, W.W.: Polymer Augmented Water-ooding and Control of Reservoir Heterogene-ity, paper NMT 890029, presented at the 1989Centennial Symposium: Petroleum TechnologyInto the Second Century, New Mexico Tech.,Socorro, Oct. 1619.

[192] Aslam, S., Vossoughi, S., and Willhite, G.P.:Viscometric Measurement of Chromium(III)-

Polyacrylamide Gels by Weissenberg Rheogo-nimeter, paper SPE/DOE 12639 presentedat the 1984 SPE/DOE Symposium on EOR,Tulsa, April 1518.

[193] Needham, R.B., Doe, P.H.: Polymer FloodingReview, JPT (Des. 1987) 15037.

[194] Taber, J.J.: Environmental Improvements andbetter Economics in EOR Operations.In Situ14, No. 4, 345405 (1990).

[195] Thomassen, P.R. and Skontorp, O.: ModernReservoir Management. A Contribution to Im-proved Recovery, paper presented at the 1987European Symposium on EOR, Hamburg, Oct.2729.

[196] Skontorp, O.: The Challenge of Improved OilRecovery from the Gullfaks Field Oshore Nor-way, paper presented at the 1987 InternationalEnergy Agency Oil Recovery Symposium, Syd-ney, Sept. 30.

[197] Hove, A.O., Nilsen, V., and Leknes, J.: Visu-alization of Xanthan Flood Behaviour in CoreSamples by Means of X-Ray Tomography,SPERE (Nov. 1990), 47580.

[198] Lund, T., Børeng, R., Bjørnestad, E. Ø., andFoss, P.: Development and Testing of XanthanProducts for EOR-Applications in the NorthSea, paper presented at the 1989 EuropeanSymposium on IOR, Budapest, April 2325.

[199] Lund, T., Lecourtier, J., and Muller, G.: Prop-erties of Xanthan Solutions after Long TimeHeat Treatment at 90C, Polymer Degrada-tion and Stability, 27, 21125, 1990.

[200] Audibert, A., Lecourtier, J., and Lund, T.:Xanthan Injection in North Sea Fields - Lab-oratory Study, paper presented at the 1991European Symposium on IOR, Stavanger, May2123.

[201] Littmann, W., Kleinitz, W. Kleppe, G., andLund, T.: A Commercial Scale Xanthan Poly-mer Flood Project in a High Salinity, Low Vis-cosity Oil Reservoir in North Germany, paperpresented at the 1991 Symposium on IOR, Sta-vanger, May 2123.

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Chapter 12

Special Topics

Table 12.1: Representative dataset for Norwegianthin oil zones,3 see Appendix ??

ho=20 m hp=5 mkv/kh=1 λo=1 darcy/cp

re=500 m rw=0.1 m∆ρwo=150 kg/m3 ∆ρog=700 kg/m3

12.1 Thin Oil Zones

A thin oil zone sandwiched between an overlyinggas cap and an underlying aquifer poses particularproduction problems. Oil productivity may be low.Large volumes of free gas and/or water tend to beproduced together with the oil, except at uneconom-ically low production rates.Most oil elds will experience similar problems, at

least when approaching the end of their life. Withthin oil zones, however, these problems can be sosevere as to rule out protable production entirely,in particular in heavy-investment, oshore environ-ments.

12.1.1 Productivity

Textbooks in reservoir technology1 give the steady-state productivity index of a vertical well as

Jv =2π λoho

ln(re/rw), (12.1)

where λo = khkro/µo is the (horizontal) oil mobility,ho is the oil zone thickness, and rw and re are thewell and drainage radii, respectively. Assumptionsinclude the well to be completed over the full oil-zoneheight, negligible water and gas production, and zeroformation skin. For an oil production rate qo, thepressure drawdown, ∆p = pe − pw, becomes

∆p = Jvqo. (12.2)

Consider the dataset in Table 12.1, where ∆ρog and∆ρwo are the density dierences at reservoir condi-tions. Assume the reservoir to be at a depth of H =2500 m, and normally pressured. For a rough esti-mate of the maximum possible oil production rate,qomax, assume the pressure loss in the productiontubing to be dominated by the hydrostatic pressure

loss, conservatively set equal to the gravity gradientof reservoir oil. Then qomax ≈ 4650 Rm3/day from

g∆ρwoH = Jvqomax. (12.3)

This is a high rate, even by North Sea stan-dards. Lien et al.2 reported test results around1000 Sm3/day in the Troll eld. These high ratesdo indicate that productivity will probably not be alimiting factor when producing from thin oil zones inhigh-permeability, light-oil reservoirs, such as thosefound in the North Sea.

12.1.2 Production of Free Gas andWater

Attendant production of free gas and/or water is un-desirable for a number of reasons. We shall limit at-tention here mainly to loss of oil as residual, by inva-sion of the gas cap, which may be a severe problemin producing from thin oil zones.Assume for simplicity gas-oil and water-oil contacts

to be horizontal, as illustrated in Fig. 12.1. Originalcontacts are indicated by solid lines. After some pro-duction, the GOC and the WOC will have moved(dashed lines). With a perfect waterdrive, the WOC

Perforations

hg

ho

hw

WOC

GOC

Figure 12.1: Gas-oil and water-oil contacts.

would rise to replace any produced volume. The risevelocity, vwo, of the WOC becomes

vwo =qtot

Aφ(1− Siw − Sorw), (12.4)

where qtot is the total volumetric production rate atreservoir conditions, A reservoir area, φ porosity, Siwconnate water saturation, and Sorw residual oil satu-ration after waterooding.

305

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306 CHAPTER 12. SPECIAL TOPICS

Similarly, the rise velocity, vgo, of the GOC be-comes

vgo =qg

Aφ(1− Siw − Sgro), (12.5)

where qg is the volumetric rate of production of freegas at reservoir conditions, and Sgro is the residualgas saturation after oil ooding. The connate watersaturation, Siw, is by assumption the same in theoriginal oil and gas zones.A minimum volume AhoφSorw of oil is unrecover-

able by water displacement. The rising GOC causesadditional oil to be lost as residual. Let Sorwg bethe residual oil saturation after waterooding, withtrapped gas present. Then the rate qloss of loss of oilas residual in the invaded gas zone is

qloss = vgoAφSorwg =Sorwgqg

1− Siw − Sgro. (12.6)

Requiring the rate of loss to be smaller than the pro-duction rate itself, i.e., qloss/qo < 1, we obtain a crit-ical gas/oil ratio, RFc, expressed at reservoir condi-tions,

RFc =1− Siw − Sgro

Sorwg. (12.7)

Converting to surface conditions, Ekrann3 found crit-ical ratios in the 5001000 Sm3/Sm3 range for a setof Norwegian oil elds.The arguments above can be repeated for the oppo-

site case of a perfect (innite) gas-cap drive, in whichcase critical water/oil ratios in the 510 Sm3/Sm3

range were found for the same elds.3

To put the critical ratios into perspective, we needto compare with the natural high-rate, long-termgas/oil and water/oil ratios, RF lim and Fwolim. Re-ferring again to Fig. 12.1, we assume steady-state pro-duction, which would require a constant-pressure lat-eral boundary. Assume furthermore that the produc-tion rate is suciently high to allow neglect of gravityeects. If the reservoir is suciently thin, contacts farfrom the well will remain approximately horizontal.All three phases then have the same horizontal pres-sure gradient, and RF lim and Fwolim can be estimatedat

RF lim =hgλghoλo

, (12.8)

Fwolim =hwλwhoλo

. (12.9)

For the elds mentioned above,3 RF lim is betweenone and three orders of magnitude larger than RFc.Low gas viscosity is instrumental in producing thelarge RF lim's. Fwolim is a factor of 24 larger thanFwoc.With gas-cap drive and a depletable (nite) gas

cap, production of free gas would imply loss of driveenergy. A GOR-criterion to avoid intolerable loss ofdrive energy was also developed.3 Again, allowableGOR's turned out to be order(s)-of-magnitude lowerthan RF lim for the example reservoirs.

The development above is somewhat simplistic, ofcourse, and would have to be carried out in muchmore detail in an actual eld study. Nevertheless,the following general conclusions are suggested

• In producing from thin oil zones, attendant pro-duction of free gas is likely to pose severe prob-lems

• Water production is likely to be less of a problem.

These conclusions are based on comparisons withthe limiting gas/oil and water/oil ratios RF lim andFwolim, whose derivation relied on neglect of gravityeects, and on steady-state considerations. In thenext section these restrictions are removed.

12.1.3 Vertical Well Coning

The horizontal contacts of Fig. 12.1 are unrealistic.For gas to be produced, the GOC has to intersect thewell below the top of the perforations, which in turnrequires viscous forces to be large enough to pull thegas down against the forces of gravity. A cone-shapedgas-oil interface is formed, as illustrated in Fig. 12.2(dashed curve). The process is commonly referred toas gas coning. Water coning may take place simul-taneously. For convenience, we shall only discuss gasconing in what follows, assuming the reservoir to beclosed at the original water-oil contact.

GOC

hg

ho

AB

d

hp

re

Figure 12.2: Gas coning

Two concepts are in frequent use to characterizea well's tendency for gas coning: The time to break-through, tbtv, is the length of the initial period whenfree gas not yet has reached the perforations. Thecritical rate to coning, qcv, is the highest steady-stateoil rate without attendant free gas production.Steady-state solutions require special boundary

conditions such as the lateral drive boundary condi-tions indicated in Fig. 12.2. The GOC is xed at thelateral boundary, which is also taken to be a constant-pressure boundary. Below the critical rate, a stablegas-oil interface forms, whose apex is situated somedistance above the perforations, as illustrated in thegure. Hydrostatic pressure prevails in the gas cone.The pressure must be continuous across the interface.In particular,

Φ(A)− Φ(B) = ∆ρoggd, (12.10)

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12.1. THIN OIL ZONES 307

where Φ(A)−Φ(B) is the viscous pressure drop in theoil between points A and B, ∆ρog is the density dier-ence between oil and gas, and g is the acceleration ofgravity. The critical rate is the largest rate allowingEq. 12.10 to have a stable solution with d < ho − hp.In the classical Muskat approximation,46 one ne-

glects the inuence of the gas cone when comput-ing the viscous pressure drop Φ(A) − Φ(B), whichis computed as if the reservoir boundary were closedat the original GOC. More accurate results are alsoavailable,7,8 where Muskat's assumption is removed.Fig. 12.3 is reproduced from Høyland et al.7 The

-2

-1.5

-1

-0.5

0

0.5

-0.5 0 0.5 1 1.5

0.905

0.714

0.4760.238

0.048

log(

q D)

log(reD)

hp

ho =

Figure 12.3: Critical rate to coning, adapted fromHøyland et al.7

gure shows the dimensionless critical rate for gasconing, qcvD,

qcv = π∆ρgh2oλoqcvD, (12.11)

vs. the dimensionless drainage radius

reD =reho

(kvkh

)1/2.

The well is completed from the bottom of the reser-voir. The fractional well penetration, hp/ho, is a pa-rameter in the gure.Using the data of Table 12.1, reD = 25, and from

the above gure, qcvD ≈ 0.18. Consequently, qcv ≈132 Rm3/d. This is order(s)-of-magnitude lower thanrates dictated by productivity alone, as discussed pre-viously. A similar observation was made by Ekrann3

in his study of Norwegian elds. Critical rates werein the range of 3 to 830 Sm3/d, which is one to sev-eral orders of magnitude lower than the correspond-ing maximum rates. Thus, if produced at or nearthe maximum rate, we would expect the steady-stategas/oil ratio to approach the limiting value RF lim,with potentially disastrous consequences, such as lossof oil as residual or loss of drive energy.It remains to be discussed whether the dynamics

of coning may be of any help. If breakthrough oc-curs only after a substantial part of the oil has beenproduced, then steady-state considerations are of lessrelevance.The time to gas breakthrough for vertical wells

can be estimated using the Sobocinski and Cornelius9

method. The procedure is summarized in the follow-ing three expressions,

z =2π∆ρoggλoho(ho − hp)

q, (12.12a)

tbtvD =z

4

16 + 7z − 3z2

7− 2z, (12.12b)

tbtv =φµoho

∆ρoggkv

2

1 +Mαog

tbtvD, (12.12c)

where α = 0.6 for 1 < Mog < 10.Using Mog = 10, φ = 0.25, and the example data

of Table 12.1, we obtain tbtv ≈ 12 days for an oilproduction rate of 500 Rm3/d. In reality, the mobil-ity ratio Mog will be higher, and the time to break-through even less. A totally insignicant quantityof oil will therefore have been produced before gasbreakthrough. This observation should be generallytrue, as long as the production rate is strongly super-critical. Only a very low kv will delay breakthroughto a level of practical interest. If kv = 0.01kh, thentbtv ≈ 1200 days for the example, which is a verysignicant delay.

12.1.4 Production Techniques

The previous discussion strongly indicates that themajor challenge in producing from thin oil zones is toavoid excessive production of free gas. Apart from re-ducing the total production rate, which may often im-ply uneconomically low oil rates, several potentiallyuseful techniques exist.Gas reinjection implies processing of large quanti-

ties of gas. This is generally costly. A rough in situseparation of gas and oil can be obtained by dual pro-duction, as illustrated in Fig. 12.4, and conceivablyreduce processing cost.

GOCOil

Gas

Packer

Casing

Figure 12.4: Dual production tubing.

The critical rate to coning is very sensitive to theoil zone height ho. A technique sometimes referred toas reverse coning1013,15 seeks to exploit this fact bycompleting the well below the original WOC. This in-creases the distance to the GOC, allowing a larger to-tal liquid withdrawal rate without attendant produc-tion of free gas. Oil will cone downwards (reverse con-ing) to be produced at the completions. The method

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308 CHAPTER 12. SPECIAL TOPICS

has apparently had considerable success in the East-ern European countries.11 Allen12 claims that USexperience up to 1954 is inconclusive. A more recentstudy13 nds reverse coning to be benecial in theTroll eld, increasing ultimate recovery (for a givenshut-in criterion) by 10 to 15% over that obtainedby conventionally positioned completions. Very re-cently, Haug et al.14 found in the order of 45% in-creased recovery after 5 years of gas-free production,with reverse coning into horizontal wells, again in theTroll eld. Roberts and Hulbert15 nd reverse coningto be clearly benecial in their study of tertiary gasinjection.Periodic shut-in of wells has been advocated16 as a

means of reducing the average gas production. Thisstrategy is presently found useful in the Statfjordeld.17 A theoretical basis for the method does notseem to exist.Coning is caused by viscous pressure loss in the oil,

as indicated by Eq. 12.10. Any technique that signif-icantly reduces this pressure loss should also reduceconing tendencies. Fracturing would appear to be amethod with some potential. Fractures would pre-sumably have to extend more than a few meters intothe reservoir, since critical rates are not sensitive todetails of the near-well region.Other methods try to block o the gas. Foam has

been suggested18 to be a suitable substance for near-well treatment, since it has the ability to reduce thepermeability to gas by orders of magnitude more thanit reduces the permeability to oil.19 We suspect thattreatment extending only a few meters into the reser-voir may be useful for a limited time period at best,even if the integrity of the treatment itself remains in-tact: For a suciently thin reservoir, and neglectinggravity eects, the steady-state gas/oil ratio is givenby Eq. 12.8, regardless of near-well eects. To com-pensate for low gas permeability, the near-well regionmust then have a high gas saturation at steady state.Thus, the long-term eect of even a perfect near-welltreatment is suspected to be reduced oil productivity.Long-term improvement would seem to require the

treated zone to extend deep into the reservoir. Ad-vantage must taken of gravity eects. Horizontal bar-riers can be xed20 or exible19 (foam). They mustbe positioned in the oil zone between the perfora-tions and the original gas-oil contact. Their eective-ness increases with barrier radius and with barrierdistance from the GOC.21,22

12.1.5 Horizontal Well Performance

Horizontal wells have been drilled in increasing num-bers in recent years. The technology is rapidly ap-proaching proven status. The chief attraction ofhorizontal wells is their increased contact with thereservoir, allowing increased production rates or re-duced drawdown.In two dimensions, neglecting end eects, exact re-

sults have been developed for the critical rate to gasconing (or cresting ,23 as it is sometimes called in this

geometry). Such results exist both for gas- and water-drive situations,23,24 and also for the lateral drive ge-ometry23,25 considered here. The dimensionless crit-ical rate, qhcD, is dened by25

qhc = Lhoλo∆ρogg

(kvkh

)1/2

qhcD, (12.13)

and the dimensionless distance to the outer boundary,xeD, by

xeD =xeho

(kvkh

)1/2.

The relationship between qhcD and xeD is given by

qhcD = c1(xeD)c2 , (12.14)

where c1 = 0.9437 and c2 = − 0.9896, for wells com-pleted at the bottom of the oil zone.Plotting Eq. 12.14 onto Fig. 12.3, it is observed

that the critical rate decreases much faster with xeDfor horizontal wells than with the corresponding reDfor vertical wells. This is caused by the viscous pres-sure gradient away from the well being much larger(linear, as opposed to logarithmic variation in radialgeometry). The constant pressure condition imposedat the lateral boundary is unrealistic. The strongdependence on xeD (or reD), therefore, makes therelevance of the concept of critical rates somewhatuncertain.3,23

In particular, quantitative comparisons of verticaland horizontal wells are dicult, due to the dierentdrainage geometries, and due to the dierent sensi-tivity to changes in reD or xeD. Nevertheless, byapplying the Table 12.1 data for xeD = 25 we obtainqhc = 0.46L Rm2/d. For a 500 m long horizontal well,qhc = 230 Rm3/d, which is nearly twice the vertical-well result. Ekrann3 found 400 m horizontal wells tohave between 5 and 10 times the critical rate of verti-cal wells for typical Norwegian reservoirs. He utilizedChaperon's method,6 however, which has later beenobserved to yield a factor 2.5 to 3 too large criticalrates for horizontal wells.25 Correction for this errorbrings his results more in line with the present exam-ple. Thus, indications are that horizontal wells havesomewhat more favorable coning characteristics thanvertical wells.In addition to the well length, the superiority of

horizontal wells will depend on the vertical perme-ability. By Fig. 12.3 and the denition of reD, thecritical rate to coning will increase with decreasingkv for vertical wells. The critical rate of horizon-tal wells is slightly decreased as kv is reduced (cf.Eq. 12.14 and recall that qhc ∝ k

1/2v qhcD). In our

example, if kv = 0.01kh, then qcv = 276 Rm3/d andqhc = 220 Rm3/d. Thus, the critical rates are aboutequal.A similar pattern emerges with impermeable bar-

riers of limited lateral extent. Ekrann22 consideredhorizontal barriers in the oil zone, symmetrically po-sitioned with respect to the well, as illustrated inFig. 12.5. He developed methods to compute criti-

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12.2. FIELD EXAMPLES 309

re or xe

GOC

r or x

Barrier

Horizontal or vertical well

Figure 12.5: Horizontal barrier.

cal rates in such geometries, with wells completed atthe bottom of the oil zone. Table 12.2 gives the ratioof critical rates with and without the presence of abarrier, for vertical and horizontal (italics) wells. Abarrier increases the critical rate if this ratio is largerthan unity.

Table 12.2: Critical rates ratios for vertical and hori-zontal (italics) wells, from Ekrann22

Critical rate changereD/ rD or xDxeD 2 4 8 16

43.741.31

131.87 2.97 7.220.68 0.83 1.49

401.37 1.78 2.54 4.460.56 0.59 0.67 0.89

The table refers to a situation where the barrier ispositioned in the middle of the oil zone, as illustratedin the gure. The barrier radius (half-width) is r (x),and rD = r(kv/kh)1/2/ho and xD = x(kv/kh)1/2/ho.As can be seen, a barrier increases very signicantlythe critical rate to coning for vertical wells. Horizon-tal wells experience a decrease in critical rate, exceptfor very wide barriers.Finally, we use the method of Papatzacos et al.26 to

estimate the time to gas breakthrough for a horizontalwell. Papatzacos et al. considered a horizontal wellat the bottom of the oil zone in a laterally innitereservoir. With a dimensionless rate

qD =µoq

2πL√kvkh∆ρoggho

, (12.15)

the dimensionless time to breakthrough,

tbthD =kv∆ρogg

φµohotbth, (12.16)

is given by the correlation

ln(tbthD) = c0 + c1u+ c2u2 + c3u

3. (12.17)

Here u = ln(qD) and c0 = −2.9351, c1 = −1.0678,c2 = 0.088277, and c3 = −0.034931. Note that theexpression is valid for qD > 0.07.

In our example reservoir, with φ = 0.25, q =500Rm3/d, and L = 100m, we obtain tbth ≈ 60 days.This is a factor 5 larger than for the correspondingvertical well, but still too small to allow signicantquantities of oil to be produced prior to breakthrough.With kv = 0.01kh, tbth ≈ 139 days, a factor 8.6 lessthan for the vertical-well example. Thus, while a lowvertical permeability delays breakthrough with a hor-izontal well, the eect is much weaker than with verti-cal wells. Note, however, that for the isotropic case, ifL is increased from 100 to 500m and q is proportion-ally increased from 500 to 2500Rm3/d, we still havetbth = 60 days, while Eqs. 12.12c give tbtv = 1.1days.For such a high rate, a delay of 60 days in break-through time may be signicant.In summary, horizontal wells oer an attractive

remedy for coning problems in thin oil zones. Whiletheir superiority over vertical wells increases withlength, the dierence is signicantly lower, however,in reservoirs with very low vertical permeability, orwith impermeable horizontal barriers.

12.2 Field Examples

There are several reservoirs with a thin oil zone inthe North Sea, Table 12.3. In all cases discussedbelow, the oil zone is between a gas cap and a wa-ter zone. The oil production potential from thin oilzones is strongly dependent on reservoir parameterslike oil-zone thickness, anisotropy ratio, absolute andrelative permeability, as well as on production pa-rameters such as well length, well location, well ori-entation, and the allowable water and gas productionlevels. In addition, the overall development strategy,and in particular any independent production fromthe gas cap, will have inuence on the oil productionprole.28

Hence, the potential for exploiting thin oil zoneswill vary from case to case and the selected pro-duction strategy must be tailored to the actual set-ting. The eld examples below clearly illustrate thesepoints.For developing thin oil zones, horizontal wells usu-

ally oer a great advantage over vertical wells. In theTroll West Oil Province, the oil rate from the rst hor-izontal test well was more than 4 times higher thanwhat can be expected from a vertical well.29

A horizontal well now typically costs about 1.2to 1.5 times that of a vertical well (per unit lengthdrilled), and several hundred horizontal wells weredrilled world-wide in 1991.

Troll

The Troll eld29,31,33 contains oil rims sandwichedbetween a large gas cap and an active aquifer. Theoil-zone thickness is 22 to 26 m in the Troll WestOil Province and 12 to 15 m in the Troll West GasProvince,29 see Fig. 8.16. The oil is located in un-consolidated, high-quality sands with permeabilities

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310 CHAPTER 12. SPECIAL TOPICS

Table 12.3: North Sea thin oil-zone reservoirs.

Oil zone Dist. from Horizontal Well Oil Subcritical∗∗

FIELD thickness GOC to well permeability length∗ viscosity oil rate (1 yr)(m) (m) (md) (m) (cp) (std m3/d)

Troll oil prov. 22.0 19 11000 500 1.3 2600Troll gas prov. 12.0 11 6000 800 1.8 500†

Visund 39.0 33 105000 800 0.3 200†

Gamma nord 26.5 10 5005000 500 0.4 250†

Snøhvit 14.0 12 500-2500 800 0.6 400†

Midgard27 11.5 N/A 6000 N/A 0.4 N/A

* Horizontal section, ** Horizontal wells, † From simulation only,

ranging from 1000 to 12 000 md. The oil viscosity is1.3 to 1.8 cp.The oil production will be limited mainly by gas

coning, resulting in rapidly decreasing oil rates.In parts of the eld, a relict oil zone below the

initial oil-water contact exists. This results in an un-usually low relative permeability to water in the waterzone; 0.10 to 0.40. The low water mobility allows awell location very close to the water-oil contact. Infact, it has been shown that the theoretical optimumwell location with respect to oil recovery under certainreservoir conditions is below the water-oil contact,30

Fig. 12.6. Completion in the water zone relies upon

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

k=6500 md

k=13000 md

500

400

300

200Cum

ulat

ive

oil

prod

uctio

n,10

3st

dm

3

Meters below WOC

Figure 12.6: Cumulativ oil production vs. completiondepth.30

oil coning down into the completions through the wa-ter zone; the so-called inverse coning process.Two horizontal wells have been drilled in the Troll

West oil zone; one in the Troll West Oil Province, well31/2-T1, and one in the Troll West Gas Province, well31/5-T1, Fig. 8.16.The 500 m horizontal well in the Troll West Oil

Province was drilled and completed with a pre-packedslotted liner early in 1990. The drilling target was 4 mabove the oil/water contact. The 11 month long-termproduction test at an average liquid rate of 5000 stdm3/d conrmed the pre-test reservoir simulations

0 80 160 240 320

4.0

2.0

0

300

200

100

Days

Oil

rate

,10

3st

dm

3 /d

GO

R,

m3/

m3

Figure 12.7: Oil rate and GOR vs. time in Troll testwell, 31/2-T1.29

1.0

0.8

0.6

0.4

0.2

0.00 80 160 240 320

Days

SimulatedMeasured

Wat

ercu

t

Figure 12.8: Water cut vs. time in Troll test well,31/2-T1.29

showing that the initial oil rate from a Troll horizontalwell would be at least 4 times higher than what couldbe expected from a vertical well in the same area.29,33

The test also conrmed the prediction that the watercut stabilizes at a low (30 to 35%) level. Gas break-through occurred after 183 days. The test behaviorwas continuously history-matched in a detailed nu-merical simulation model. Figs. 12.7 and 12.8 showmeasured and simulated values for oil rate, GOR andwater cut.A typical feature of the Troll sediments is the occur-

rence of numerous calcite-cemented layers. These areof two classes; extensive, up to kilometer-wide sheetsfound at the boundaries of the geological zones andthe less extensive (1 to 100 m) calcite-cemented layerswithin the geological zones. In the horizontal section

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12.2. FIELD EXAMPLES 311

10m

250m

NW SE2750

2800

2850

2900

2950

3000

3050

3100

3150

3200

2750

2800

2850

2900

2950

3000

3050

3100

3150

3200

Dep

th(m

MSL

)

Depth

(mM

SL)

34/8-3A

34/8-3

GOC2904 mMSL

OWC2943 mMSL

Draupne

Heathe

rFM

Tarbert

FM. II

Ness

FM.

Etive

FM.

Ranno

chSST

Draupne FM.

Heather FM.

Tarbert II

Tarbert I

Ness FM.

Etive FM.

Rannoch SST

10 - 50 md

10 - 50 md

50 - 700 md

100 - 5600 md

50 - 150 md

Figure 12.11: Cross-section of the Visund eld.

1516

1548

15800 200 400

Observed calcites (strong well log response)Possible calcites (weak well log response)

GOC

OWC

6o

Length (m)

Dep

thM

SL(m

)

Figure 12.9: Observed calcites in Troll test well, 31/2-T1.29

of 31/2-T1, several dipping calcites were identied,Fig. 12.9. The eect of these on the 31/2-T1 testwell behavior has been found to be minor.29,31

The test also demonstrated that after a period ofsupercritical (relative to gas coning) production fol-lowed by a 10 days shut-in period, the well resumedthe original oil rate without any indications of hys-teresis eects (cf. Fig. 12.7, nal test period).

The second test well was completed in the 12 mthick oil zone in the Troll West Gas Province. The800 m long horizontal section was positioned approx-imately 1 m above the water-oil contact. The three-month long production test commenced early in Jan-uary 1991. A well production prognosis generatedprior to the drilling of the well is given in Fig. 12.10.The expected stabilized oil rate after one year is about500 std m3/d, which is 1/5 of that of well 31/2-T1.

In general, the high oil rates obtained in Troll areattributed to the clean, high-permeability sand andto the low mobility of water in the water zone.

3000

2500

2000

1500

1000

500

00 20 40 60 80 100 120 140 160 180

1.0

0.8

0.6

0.4

0.2

0.0

A

A

F

F

C

C

case A : reference case (k=6.5 darcy)case C : k=10 darcycase F : without residual oil below OWC

Days

Oil

rate

,st

dm

3/d

Wat

ercu

t

Figure 12.10: Predicted oil rate and water cut vs.time in Troll well, 31/5-T1.

Visund

In the thin oil zone of the Visund eld the reservoirproperties are quite dierent, leading to a consider-ably lower oil production potential than in the TrollField.The A-North compartment of the Visund Field has

an oil column thickness of 39 m. There seems, how-ever, to be no residual oil in the water zone to reducethe water mobility as in Troll, the permeability is gen-erally lower than in Troll (10 to 2000 md), and thereservoir consists of 8 to 9 dipping layers with severepermeability contrasts, Fig. 12.11. The oil viscosityis 0.3 cp. The production from Visund is foreseen tocommence in the second half of the 1990's.Although the oil zone is almost twice as thick as

in the Troll Oil Province, reservoir simulation studieshave shown that the gas will break through quicklyin the high-permeable Etive layer resulting in criticaloil rates after 1 year of only 200 Sm3/d. An oil pro-

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312 CHAPTER 12. SPECIAL TOPICS

0 0.5 1

2500

2600

2700

km

Dep

thm

MSL

Top Statfjord reservoirLayer 1

Layer 5 X 3

Layer 2Layer 3Layer 4Layer 6

SHALE

Well30/6-15

GOC

FWL

Lower Statfjord reservoir

10o

Layer

123456

Thicknessm

30.46.2

13.719.23.9

29.4

khmd

17505290235036601960540

kvmd

2.69030157010

Figure 12.13: 30/6 Gamma Nord eld, cross section, Upper Statfjord reservoir.32

1200

1000

800

600

400

200

00 365 730 1095 1460 1825

1.0

0.8

0.6

0.4

0.2

0.0

Days

Oil

Rat

e,Sm

3 /d

Well 6.0 m above OWCWell 6.0 m above OWC sub. crit

Well 19.5 m above OWCWell 19.5 m above OWC super crit.

Wat

ercu

t

Oil rate

Water cut

Figure 12.12: Visund eld, oil rate and water cutpredictions.

duction prognosis for a 750 m long horizontal well inVisund is shown in Fig. 12.12.

In Troll, high oil rates were obtained assuming noproduction of free gas (i.e., subcritical rates to gasconing). Avoiding free gas production oers severaladvantages. Energy is conserved in the reservoir, uidcontact movements are minimized, and the surface-gas handling capacity can be kept at a minimum.

In Visund, subcritical production rates are very lowand the eld is not considered economically viable asa stand-alone project. Fig. 12.12 shows that super-critical oil production might give considerably higheroil recovery than subcritical production. The con-densate drop-out from the gas, which amounts to acomparatively large volume in Visund, is not includedin the proles in Fig. 12.12.

30/6 Gamma Nord32

Supercritical rates are also planned for the horizon-tal well in the thin oil zone in 30/6 Gamma Nord.In fact, in addition to producing the thin oil zone,this horizontal well is designed to deliver gas for rein-jection into the Oseberg Field.32 The Gamma Nordhorizontal well is scheduled to be drilled in the rsthalf of 1991.The Gamma Nord structure is an easterly 9 to 10

dipping fault block. Permeabilities parallel to thebedding planes vary between 500 and 3500 md, per-meabilities perpendicular to the bedding planes arebetween 2 and 100 md, i.e., the kv/kh-ratio is verylow; 0.002 to 0.2. Fig. 12.13 shows a cross section ofthe eld. Oil viscosity is 0.37 cp.The distance between the free water level and the

gas-oil contact is 26.5 m, but most of the oil zone iswithin the oil-water transition zone.Figs. 12.14 and 12.15 illustrate the results from a

simulation study of the Gamma Nord horizontal wellpublished in 1988.32 The horizontal well section waslocated 10 m below GOC, and the initial oil rate was2000 Sm3/d. These assumptions ensured that themain target was achieved; suciently fast buildup tothe target gas rate of 1.5 std m3/d and reasonablylow water rates, Fig. 12.15.In Gamma Nord, the orientation of the well is very

important for the recovery due to the dipping layersand the high degree of permeability anisotropy. Withthe 500 m horizontal well crossing the bedding planes,the oil recovery was almost twice as high as with thewell positioned along the bedding planes (Fig. 12.14).According to the simulation study,32 the oil recoveryfrom a vertical well is only 30% of horizontal well

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REFERENCES 313

0.0 1.0 2.0 3.0

2000

1500

1000

500

0

West - East OrientedHorizontal WellNorth - South OrientedHorizontal WellVertical Well

std

m3 /d

Time(Year)

Figure 12.14: 30/6 Gamma Nord eld, oil productionrates.32

0 1 2 3

2.0

1.5

1.0

0.5

0.0

Liq

uid

Rat

e(1

03st

dm

3 /d)

Time (Years)

1.5

1.0

0.5

0

Gas

Rat

e(1

06st

dm

3 /d)

Oil Prod. Rate, supercriticalWater Prod. Rate , supercriticalGas Prod. Rate, supercriticalOil Prod. Rate, subcritical

Figure 12.15: 30/6 Gamma Nord eld, gas and liquidrates.32

crossing the beds.Fig. 12.15 also shows that subcritical (to gas) pro-

duction leads to oil rates below 200 Sm3/d in GammaNord after 1 year of production, compared to 400Sm3/d in the supercritical case.

Field Development Strategy

Since all the aforementioned elds contain large gascaps, the combination of gas-cap depletion and oilproduction becomes an important eld developmentissue. Generally, due to the possibly large globaluid contact movements and pressure drops causedby rapid depletion, gas production prior to, or si-multaneously with oil production is often expectedto reduce the nal oil recovery.28 However, for somecomplex elds like the Troll Field, the combinationof drive mechanisms arising from various gas produc-tion scenarios are so intricate, that the eect of gasproduction on oil recovery might be positive in someparts of the eld, and negative in others.28 The op-timum phasing of gas and oil development has to bedetermined in each case, preferably by means of full-eld reservoir simulation models.

Nomenclature

A = horizontal area of reservoir, m2

d = distance from original GOC to apex ofcone, m

F = ratio, Rm3/Rm3

FWL = free water levelGOC = gas-oil contact

g = acceleration of gravity, m/s2

h = height, mH = depth of reservoir, m ssJ = productivity index, Rm3/Pa sk = permeability, m2 or md or darcyL = length of vertical well, mM = mobility ratioq = volumetric production rate, Rm3/s

qloss = volumetric rate of loss of oil as residual,Rm3/s

qtot = volumetric total uid production rate,Rm3/s

r = barrier radius, mRF = gas/oil ratio, Rm3/Rm3

S = saturationt = time, sv = contact velocity, m/sx = barrier half width, mp = pressure, PaΦ = oil pressure potential, Paφ = porosityλ = horizontal mobility, kh kr/µ, m2/Pa sµ = viscosity, Pa sρ = density, kg/m3

Subscripts

bt = breakthroughc = critical, for coning (rates) or for loss of oil

(fractions)D = dimensionlesse = externalg = gash = horizontali = initial, connate

lim = limiting, when rate approaches innitymax = maximum

o = oilp = perforatedr = residual or relative (endpoint value)v = verticalw = water or well

References

[1] Joshi, S.D.: Horizontal Well Technology, Penn-Well Publishing Company, Tulsa (1991).

[2] Lien, S.C., Selnes, S., Havig, S.O., and Kyd-land, T.: The First Long-Term Horizontal-Well Test in the Troll Thin Oil Zone, JPT(Aug. 1991) 91417 and 97073.

[3] Ekrann, S.: Production from Thin Oil Zones,research report SPT T-6/87, Rogaland Re-search, Stavanger (Sept. 1987).

[4] Muskat, M. and Wycko, R.D.: An Approx-imate Theory of Water-coning in Oil Produc-tion, Trans., AIME (1935) 14, 14461.

[5] Chierici, G.L., Ciucci, G.M., and Pizzi, G.: A

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314 CHAPTER 12. SPECIAL TOPICS

Systematic Study of Gas and Water Coning ByPotentiometric Models, JPT (Aug. 1964) 92329.

[6] Chaperon, I.: Theoretical Study of Coning To-ward Horizontal and Vertical Wells in Anisotro-pic Formations: Subcritical and Critical Rates,paper SPE 15377 presented at the 1986 SPEAnnual Technical Conference and Exhibition,New Orleans, Oct. 58.

[7] Høyland, L.A., Papatzacos, P., and Skjæve-land, S.M.: Critical Rate for Water Coning:Correlation and Analytical Solution, SPERE(Nov. 1989) 495502.

[8] Wheatley, M.J.: An Approximate Theory ofOil/Water Coning, paper SPE 14210 presentedat the 1985 SPE Annual Technical Conferenceand Exhibition, Las Vegas, Sept. 2225.

[9] Sobocinski, D.P. and Cornelius, A.J.: A Corre-lation for Predicting Water Coning Time, JPT(May 1965) 594600.

[10] van Lookeren, J.: Oil Production From Reser-voirs With an Oil Layer Between Gas and Bot-tom Water in the Same Sand, JPT (March1965) 35457.

[11] Cottin, R.H. and Ombret, R.L: Application ofa Multiphase Coning Model to Optimize Com-pletion and Production of Thin Oil ColumnsLying Between Gas Cap and Water Zone, pa-per SPE 4632 presented at 1973 SPE AnnualMeeting, Las Vegas, Sept. 30Oct. 3.

[12] Allen, T.O.: Thin Oil Column Completion andProduction Practices, World Oil (April 1954)21825.

[13] Darley, J.C., Røssland, L., van Golf-Racht, T.,and Krakstad, O.: Reservoir Simulation of theTroll Field, paper presented at the 1984 O-shore Northern Sea, Stavanger.

[14] Haug, B.T., Ferguson, W.I., and Kydland, T.:Horizontal Wells in the Water Zone: The MostEective Way of Tapping Oil From Thin OilZones?, paper SPE 22929 presented at the1991 SPE Annual Technical Conference and Ex-hibition, Dallas, Oct. 69.

[15] Roberts, L.J. and Hulbert, G.: A Numericaland Analytical Study of O-Shore Oil Rim De-pletion During Tertiary Gas Injection, paperpresented at the 1991 European Symposium onIOR, Stavanger, May 2123.

[16] Chaney, P.E., Noble, M.D., Henson, W.L., andRice, T.D.: How to Perforate Your Well to Pre-vent Water and Gas Cusping, Oil and Gas J.(1956) 55, 108.

[17] Haugen, S.A. and Haaland, S.: Gas Floodingof the Statfjord Reservoir, Statfjord Field, pa-per presented at the 1991 European Symposiumon IOR, Stavanger, May 2123.

[18] Heuer, G.J., Flesenthal, M., and Jacocks, C.L.:Control of Gas-Oil Ratio in Producing Wells,"U.S. Patent No. 3,368,624 (1965).

[19] Hanssen, J.E. and Dalland, M.: Foam Barri-

ers for Thin Oil Rims: Gas Blockage at Reser-voir Conditions, paper presented at the 1991European Symposium on IOR, Stavanger, May2123.

[20] Karp, J.C., Lowe, D.K., and Marusov, N.:Horizontal Barriers for Controlling Water Con-ing, JPT (July 1962) 78390.

[21] Strickland, R.F.: An Analysis of Articial Bar-riers for Controlling Water Coning, MS thesis,Texas A&MUniversity, College Station, (1974).

[22] Ekrann, S.: On the Protection Against Con-ing Provided by Horizontal Barriers of LimitedLateral Extent, paper presented at the 1991European Symposium on IOR, Stavanger, May2123.

[23] Giger, F.M.: Analytic 2-D Models of WaterCresting Before Breakthrough for HorizontalWells, paper SPE 15378 presented at the 1986SPE Annual Technical Conference and Exhibi-tion, New Orleans, Oct. 58.

[24] Kidder, R.E.: Flow of Immiscible Fluids inPorous Media: Exact Solution of a Free Bound-ary Value Problem, J. Appl. Phys. (1956) 27,No. 8, 86769.

[25] Dikken, B.J.: Pressure Drop in HorizontalWells and Its Eect on Their Production Per-formance, paper SPE 19824 presented at the1989 SPE Annual Technical Conference and Ex-hibition, San Antonio, Oct. 811. Abridged ver-sion in JPT (Nov. 1990) 142633.

[26] Papatzacos, P., Herring, T.R., Martinsen, R.,and Skjæveland, S.M.: Cone BreakthroughTime for Horizontal Wells, SPERE (Aug.1991) 31118.

[27] Gilhuus, T: A Study of Possible Oil Productionfrom the Thin Oil Zone of the Midgard Field,paper presented at the 1988 NPD seminar Re-covery from Thin Oil Zones, Stavanger, April.

[28] Kydland, T., Haugan, P.M., Bousquet, G.,and Havig, S.O.: Application of Unconven-tional Techniques in Constructing an IntegratedReservoir Simulation Model for Troll Field,SPERE (Aug. 1988) 96776.

[29] Lien, S.C., Seines, K., Havig, S.O., and Kyd-land, T.: Experience From the Ongoing Long-Term Test With a Horizontal Well in the TrollOil Zone, paper SPE 20715 presented at the1990 SPE Annual Technical Conference and Ex-hibition, New Orleans, Sept. 2326.

[30] Haug, B.T.: Inverse Coning In HorizontalWells: Theory, Applications and Generaliza-tion, MS thesis, NTH, Trondheim (1990).

[31] Lien, S.C., Manner, M., and Haldorsen, H.H.:Horizontal Wells: Still Appealing in Forma-tions With Discontinous Vertical Permeabil-ity Barriers? paper SPE 20962 presented atthe 1990 SPE European Petroleum Engineer-ing Conference, The Hague, Oct. 2224.

[32] Fantoft, S., Krogh, P.K., and Pollen, S.: Evalu-ation of Oil Recovery From a Thin Oil Column

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REFERENCES 315

by a Horizontal Well in the Oseberg GammaNorth Reservoir, paper SPE 18341 presentedat the 1988 SPE European Petroleum Confer-ence, London, Oct. 1719.

[33] Hovland, S., Jones,C., and Whittle, T.: Plan-ning, Implementation, and Analysis of the FirstTroll Horizontal Well Test, paper SPE 20963presented at the 1990 SPE European PetroleumEngineering Conference, The Hague, Oct. 2224.

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316 CHAPTER 12. SPECIAL TOPICS

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Part IV

Appendices

317

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Appendix A

The SPOR Program

A.1 General Information

A.1.1 Purpose

The main objective of the SPOR Program (1985-91) has been to build up a Norwegian R&D exper-tise within Improved Oil Recovery (IOR) and reser-voir technology. SPOR was initiated at the requestof governmental authorities. A key purpose was todevelop highly competent and independent researchgroups, which could oer expert advice to the au-thorities (NPD) when carrying out their supervisingtasks, and assessing alternative development strate-gies for the Norwegian oshore reservoirs.Additional aims of the program were national coor-

dination, development of international contacts andcooperation, and to strengthen reservoir related edu-cation in Norway.

Subject Areas and Execution of theTechnical Part

The scope of the program was initially described inthe SPOR Program Outline. This comprehensive doc-ument from 1985 was written with the purpose ofguiding the rst project proposals and further techni-cal implementation of the R&D program. The outlinehas been a very useful tool throughout the programperiod, to concentrate the eort, focused on qualityand applicability, were important issues.Four subprograms of SPOR were established from

the start,

1 Optimization of Reservoir Data (SPOR-OPT)2 Gas Injection (SPOR-GAS)3 Water Injection (SPOR-WATER)4 Other Areas.

Responsibility for the three main areas of SPORwas given to three Norwegian research institutes.Thus they were encouraged to focus on further IOR-activity in a specic direction, avoiding overlappingareas of R&D in the future.The subject of reservoir description was recognized

as an important and integrated part of the subpro-grams in SPOR. In addition, a more general ap-proach to reservoir description has been the topicof SPOR-OPT, with the main objective to improve

reservoir data generation for dynamic reservoir simu-lation models. SPOR-OPT was assigned to the Insti-tute of Energy Technology (IFE) at Kjeller, and theactivity has mainly been concentrated on

- Quantitative Geological Data Base and IntegratedReservoir Description

- Homogenization Procedures- Tracertesting and Modelling.

The gas injection subprogram has aimed at criti-cal factors in connection with the use of hydrocar-bons and nitrogen gases to improve oil recovery in amiscible or nonmiscible manner. Responsibility forSPOR-GAS was given to the Continental Shelf andPetroleum Technology Research Institute (IKU) inTrondheim. The main themes of the gas injectionarea have been

- Mobility Control by Foams- WAG-Processes and Tertiary Gas Injection- Gravity Assisted Displacement of Oil- Gas Condensate Recovery- Reservoir Description and Reservoir Hetero-geneities' Impact on Fluid Flow.

SPOR-WATER was assigned to Rogaland Research(RR) in Stavanger. This subprogram has mainly beenfocused on the use of chemical additives such as sur-factants and polymers to improve oil recovery fromseawater injection. Since such methods are promisingbut somewhat exotic, there is a need for extensive re-serarch, and SPOR-WATER received the largest bud-get within the SPOR Program. The activity has beenconcentrated on the following main themes:

- Analytical Models of Displacement Mechanisms- Phase Behavior and Modelling of Flowing Pro-cesses

- Chemical Retention in the Reservoir- Reservoir Description on Micro and Larger Scales- Simulation of Model Reservoirs and PilotProjects.

The fourth subprogram included a smaller projecton the topic of production from thin oil zones, andthe preparation of the SPOR Monograph (1990-91).

319

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320 APPENDIX A. THE SPOR PROGRAM

Table A.4: SPOR Budget, million NOK

1985 -86 -87 -88 -89 -90 -91 TotalSPOR-OPT 1.8 4.6 4.2 3.8 3.6 3.0 1.1 22.1SPOR-GAS 2.5 4.5 5.3 5.2 5.8 5.1 3.3 31.7SPOR-WATER 2.7 5.0 6.7 7.7 6.6 5.3 3.8 37.8OTHER AREAS 0.7 - 0.4 - - 0.6 0.6 2.3ADMINISTRATION 0.6 0.9 1.1 1.0 1.0 1.0 1.1 6.7TOTAL 8.3 15.0 17.7 17.7 17.0 15.0 9.9 100.6

A.2 Statistics and Management

A.2.1 Budget

The total budget of the SPOR Program hasamounted to NOK 100 million in the seven-year pe-riod, all governmental funding from the Ministry ofPetroleum and Energy.

The yearly and total budgets of the subprogramsare shown in Table A.4.

Here the administration costs cover the Secretariatat NPD, seminars, workshops and other general pro-gram activities.

A.2.2 Organization

The following chart shows the SPOR organization:

SPORSECRETARIAT

InternationalAdvisors

RRSPOR-WATER

IKUSPOR-GAS

IFESPOR-OPT

AdvisoryGroup

SPORBOARD

MPE

A.2.3 SPOR Board and Advisors

SPOR has been accountable to the Ministry ofPetroleum and Energy (MPE), which appointed aSteering Committee (Board) with full nancial andprofessional responsibility for the program. The Min-istry was initially represented by a member of theBoard, and later by an observer. The following Boardmembers were appointed in 1990, to serve during thenal period of SPOR:

Director Farouk Al-Kasim NPD, ChairmanDirector Henrik Carlsen StatoilAdvisor Jan C. Høiland NTNFDirector Steinar Njå NPDDirector Rolf Prydz Norsk HydroDirector Arild Unneberg Saga Petroleum

To assist the program management in technicalmatters, a SPOR Advisory Group of experts fromoil companies and universities was established by theBoard. The Advisory Group has made visits to the in-stitutes twice a year on a regular basis, and made rec-ommendation to the Board on new projects, progressof research etc. The members of the Advisory Grouphave been:

Sector Manager Odd Skontorp Statoil,Chairman

Professor Jon Kleppe NTHSupervisor Peter A. Read StatoilDept. Manager Arne Skauge Norsk HydroManager Olav Skinnarland RestekProfessor Svein Skjæveland HSRProfessor Ron Steel U. BergenDirector Rolf Wiborg PPCoN

SPOR also established close relationships with theinternational R&D environment. Three internationalexperts have assisted the SPOR Board through theprogram period by advising on the research activityfrom an international point of view. SPOR's interna-tional advisors have been:

Dr. Robert I. Hawes EOR Program Manager,Winfrith, UK

Dr. Pierre Simandoux Director at IFP, FranceDr. Joseph J. Taber Director Emeritus,

New Mexico PetroleumRecovery Research Center, USA

A.2.4 Subprogram Management andSecretariat

At each of the three SPOR institutes, the subpro-gram was led by one appointed researcher, who hasbeen responsible for the total planning and reportingfrom the subject area. These persons have also beenthe direct contact between the SPOR managementand the respective institutes and project leaders.

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A.3. SPOR PUBLICATIONS 321

SPOR-OPT

At IFE, Jan Nitteberg led the subprogram duringthe rst years and was succeeded by Arne Råheimfrom 1987. The activity within SPOR-OPT, split on11 dierent projects, amounted to about 30 researchman-years.

SPOR-GAS

At IKU, SPOR-GAS has been supervised by severalprogram leaders during the SPOR-period. The fol-lowing persons have been responsible for the subpro-gram: Steinar Hagen (1985 to 86), Stein-Børre Torp(1987), Jan Ole Aasen (1988), Ole Torsæter (1989 to90), and nally Idar Akervoll in 1991. The researchhas been divided on 16 dierent projects covering thegas injection area, with a total of about 48 researchman-years.

SPOR-WATER

At RR, Jostein Kolnes has led the subprogramthroughout the SPOR period. The activity startedout with 16 dierent, and rather small projects in1985 to 86. Six of these early projects were performedat external institutes. From 1987 on, as a technicalknowledge base was established, the eort was con-centrated to 5 to 6 yearly projects. The rst doctor'sdegree nanced by SPOR was completed at the Uni-versity of Bergen in 1987 as part of a SPOR-WATERproject. The subprogram activity has amounted toabout 89 research man-years during the SPOR pe-riod.

SPOR Secretariat

The day-by-day administration of the total programhas been conducted by the SPOR Secretariat locatedat NPD's oces in Stavanger. Throughout the SPORperiod, Anna Inger Eide has been the program man-ager, with Marta Eliassen taking care of the SPORoce functions from 1987.

A.2.5 Institutes

The three Norwegian research institutes, being re-sponsible for the SPOR main subprograms are iden-tied below,

SPOR-OPT: Institutt forenergiteknikk

SPOR-GAS:

SPOR-WATER:

A large number of researchers, mainly from thethree institutes, but also from universities and otherR&D institutes in Norway, have contributed valuableproject work and results. Reservoir engineering stu-dents have also interacted by performing their mastertheses within SPOR projects, and SPOR has sup-ported several doctorate theses at universities.

A.2.6 Seminars and Meetings

According to the Ministry's Guidelines for SPOR, theresults from the program should be made generallyavailable to the Norwegian R&D environment andcompanies working on the Norwegian Shelf.The yearly SPOR Seminars have been arranged to

ensure dissemination of results by technical presen-tations and printed proceedings. An average of 150national and international participants have gatheredfor each of the two-day meetings in Stavanger, for in-formation and discussion of the program results.Throughout the program period, the need emerged

for smaller fora to discuss special technical topics.Several workshops arranged by SPOR at the initia-tive of the researchers have been successful in bringingthe Norwegian specialists from institutes, NPD andindustry together. This resulted in improved commu-nications between the researchers and the users of theR&D results.

A.3 SPOR Publications

A large number of technical reports have been pre-pared from the project work in SPOR. During therst years, most of the reports were done in Norwe-gian. The last few years, most reports are in En-glish, in order to reach a broader audience. Many ofthe SPOR publications are referred to in the dierentchapters of the monograph.

A.3.1 Technical Reports

During the SPOR period, 164 technical reports havebeen prepared based on the project work, by the in-stitutes. These reports are available directly from theinstitutes, or from the NPD Library. A separate com-plete list of reports is available from NPD. The num-ber of reports assigned to each subprogram of SPORis shown in Table A.5.

Table A.5: SPOR reports, international presentationsand publications

Techn.Rpts. InternationalSPOR-OPT 41 22SPOR-GAS 43 20

SPOR-WATER 79 56

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322 APPENDIX A. THE SPOR PROGRAM

A.3.2 International Presentations

Close to 100 technical papers have been presentedinternationally during the period 1985-91. Manymore are currently being prepared as presentationsor posters at international conferences, and as pub-lications in technical journals. The numbers in Ta-ble A.5 refer to papers from project work, completelyor partly nanced by SPOR. They do not include pa-pers from the SPOR Seminars.

A.3.3 Database of SPOR Projects

More detailed information on each of the SPORprojects, including reports and papers, is stored in theinternational Infoil-Sesame Database. This databasesystem gives online access to petroleum related re-search projects supported by EEC, UK and Norway.For additional information, please contact the Infoil-Sesame host centre in Norway:

SDS - Norwegian GovernmentComputer Centre LtdUlvenveien 89 BN-0581 OsloNorway

Telephone: (47) 2 95 63 00Telefax: (47) 2 64 84 07

A.4 New R & D Programs

The SPOR Program was concluded by the end of1991, after having achieved its overall goals. Stillthere is a great need to continue the R & D withinadvanced oil recovery methods, aiming at future im-plementation of such technology in elds on the Nor-wegian Shelf.From 1990, a new 5-year R & D program, spon-

sored by oil companies and NPD, was established.This program, called PROFIT, is concentrating onthe topics of reservoir characterization and near-wellow.The Norwegian authorities, being aware of the im-

portant IOR-topics not included in PROFIT, decidedto initiate a new govermental-sponsored R & D pro-gram from 1992. This program, RUTH, will concen-trate on gas-based and water-based advanced recov-ery methods. The program is also opened for partic-ipation by industry sponsors, with a total frame ofNOK 95 mill over 4 years.The administration of the two new R & D pro-

grams to follow SPOR is located in NPD's oces inStavanger. For additional information, please con-tact:

Norwegian Petroleum DirectorateP.O.Box 600N-4001 StavangerNorwayTelephone: (47) 4 87 60 00Telefax: (47) 4 55 15 71

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Appendix B

Key Parameters for Norwegian Sandstone

Reservoirs

B.5 Introduction

A summary is given of parameters for North Sea sand-stone reservoirs. Even though it may be impossibleto identify the key parameters for IOR-methods ingeneral, without detailed and specic reservoir knowl-edge, many existing or contemplated methods may bejudged as promising or unrealistic based on a smallset of general reservoir parameters.The two chalk elds Ekosk and Eldsk are in-

cluded at the bottom of the tables for the purposeof comparison.The parameters are presented in two tables. The

two-page Properties Table, Tables B-1a and B-1b,gives information about formation properties, uidproperties and composition, and reservoir tempera-ture and pressure. Initial data are reported, as arange or an average value for most of the parame-ters. In some instances, however, a typical or repre-sentative value has been selected, e.g. depth to uidcontacts.The Volume Table, Table B-2, contains reservoir

area and bulk volume; initial hydrocarbon volumesin place and recoverable reserves; production periodand method; cumulative produced volumes of oil andgas.The data in the Properties Table and the reser-

voir area and bulk volume have been reported to thismonograph by the operators of the elds. The restof the data in the Volume Table is taken from theresource data base of the Norwegian Petroleum Di-rectorate (NPD). (Note that since two dierent datasources have been used, there is not necessarily a fullconsistency between specied bulk volume and initialoil in place.) Updated reserves are reported in NPD'sannual reports.

B.6 Parameters

First are presented explanatory lists where the pa-rameter categories appear in the same order as in thetables.Special Conventions: The eld of an unknown pa-

rameter is left blank and a dash, , indicates irrele-vance.

B.6.1 Properties Table

Group/Form. - Abbreviations

IDS: Intra Dunlin SandL-BC: Lunde-BC

L-DEF: Lunde-DEFN-mela: Nordmela

Age = Geological age

Cr: CretaceousDa: DanianE: EoceneJ: Jurassic

Ma: MaastrichtianP: PaleoceneT: Triassic

prexM: MiddleL: Late

T = Trap type

an: anticlinalef: eroded fault blockh: horstrf: rotated faultsa: stratigraphicu: structural

D = depth to crest, m subseaowc = oil-water contact, m subsea

go/wc = gas-oil or gas-water contact, m subseaht = gross reservoir thickness, mhn = net reservoir thickness, mΘ = dip angle, degreeshg = gas column thickness, mho = oil column thickness, mφ = porosity, %kh = horizontal permeability, md (D if stated)

kv/kh = permeability ratioSoi = initial oil saturation; initial gas saturation

is reported for Frigg, Heimdal, Midgard,Smørbukk

Sorw = residual oil saturation if wateroodedSorg = residual oil saturation if gasoodedVclay = clay content, ratioρo = oil density, kg/m3

323

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324 APPENDIX B. KEY PARAMETERS FOR NORWEGIAN SANDSTONE RESERVOIRS

γg = gas gravity (air=1)ρc = condensate density, kg/m3

µo = oil viscosity, cpµg = gas viscosity, cpµc = condensate viscosity, cppb = bubblepoint pressure, barpd = dewpoint pressure, barRs = solution gas/oil ratio, Sm3/Sm3

CL = condensate content, Sm3/(MSm3 gas)Bo = FVF of oil, Rm3/Sm3

B−1g = expansion factor for gas, Sm3/Rm3

C1 = mole percent C1

C2-4 = mole percent C2-C4

C5-12 = mole percent C5-C12; the number re-ported is C5-C12+ (or C5+) for most of theelds; C5-C12 is reported for the Garn for-mation of Sm¿rbukk S¿r, Statfjord Nord,Sn¿hvit, and Ula.

CS = salinity, mg/lCDi = concentration of divalent ions, mg/lT = reservoir temperature, Cp = reservoir pressure, bar

B.6.2 Volume Table

A = area of reservoir, km2

Vb = gross rock volume, 106 Sm3

N = initial oil in place, 106 Sm3

NRsi = initial associated gas in place, 109 Sm3

GFi = initial free gas in place, 109 Sm3

R-oil = oil reserves, 106 Sm3

R-gas = gas reserves, 109 Sm3

R-ngl = NGL reserves, 106 tonsPeriod = estimated production period for elds de-

cided to be developedMeth. = Recovery method decided

g: gas injectionp: pressure depletionw: water injection

Np = cum. oil production July 1991, 106 Sm3

Gp = cum. gas production July 1991, 109 Sm3

B.6.3 Comments

Some eld-specic comments are given in this para-graph since the data reported in some cases cannoteasily be included in a simple table.The same symbols and units are used as in the

tables, and the units are therefore not repeated here.

Balder

Molefraction C5-9: 8.6; range of pb: 145165.

Brage

Statfjord ; molefraction C5-7: 14.78.

Fensfjord ; molefraction C5-7: 5.79.

Gullfaks

Brent ; pb: 217270; Rs: 87.5100.

Cook ; reported parameters in Properties Tableand A and Vb in Volume Table are for Phase II.

Gullfaks Sør

Brent ; owc: 33003470; goc: 33213220. Com-position in table is for oil. Gas composition is84,10,4.

Statfjord ; gas composition is 85,9,4.

Heidrun

Tilje and Åre; owc: 24152452; goc: 22922314.

Tilje; µo: 0.841.63; pb: 221245; Rs: 78114; Bo: 1.211.31; C1: 4146; C2-4: 711;(C5-12,CO2,N2): 4351.

Oseberg

Rs: 132158.

Smørbukk

This condensate eld only has oil in Tilje withRs = 440. A producing GOR of 1458 is reportedfrom the gas zone of Tilje. For Garn, Ile, Tofteare reported producing GOR's in the range 8401700.

Ile; Ile-1 go/wc: 4085.

Tilje; Bo: 2.422.64; B−1g : 238224.

Smørbukk Sør

Garn; pb: 330393; Rs: 295570; Bo: 1.953.00.

Tilje-2 ; Rs: 513362.

Snorre

Statfjord ; owc: 25952599; ρo: 710765; µo:0.770.84; pb: 90126; Rs: 5884; Bo: 1.191.27;C1: 2026; C2-4: 2022; C5-9: 2426; C10+: 2434.

Lunde-BC ; owc: 25742595; C5-9: 20; C10+: 25.

Lunde-DEF ; owc: 25612574; CS : 3400042000;CDi: 20002500; C5-9: 19; C10+: 21.

Ula

Varying contact owc: 3789, 3759, 3714, 3561,3544, 3491.

Frigg

Initial gas saturation is reported in the table inthe Soi column. Initial oil saturation in the oilrim was 73.9.

Midgard

Ile; Composition in the table is for gas. Oil com-position is 47,16,37. Initial saturation in the ta-ble is for gas. Initial oil saturation is 80. Oil rockvolume is reported as 47·106 m2.

Sleipner Vest

Tabulated values are from the southern part ofthe eld. The eld has varying gwc. Range ofcondensate content CL: 250470.

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B.7. FORMATIONS. 325

For the northern part, the following values havebeen reported: γg = 0.86; ρc = 637; µg = 0.041;pd = 431; CL: 250470; Bo = 1.688; B−1

g =271.6; C1: 77.29; C2-4: 13.51; C5-12: 4.52; p =447.5.

Troll

Sognefjord ; owc: 15491569; goc: 15431547; ρo:890910; µo: 1.31.8; Rs: 5569; Composition intable is for oil, the gas composition is 93,4,3.

B.7 Formations.

In this section a short description is given of thegroups and formations in the tables in order to qual-ify the average values of the parameters.13 When-ever reading or applying average values, the eect ofreservoir heterogeneities, such as permeability con-trasts and ow barriers, should be considered.

LUNDE FORMATION. The Lunde Formationconsists of channel and sheet sandstones (ne tomedium grained), deposited in braided and low sin-uosity river systems. Associated with the sandstonesare shales deposited in oodplains and in lacustrinebasins. The shales can represent ow barriers. Theow communication is moderate to good in the lowerparts of the formation, but variable in the upperparts. This is mainly due to increasing frequency ofshale as barriers.

STATFJORD FORMATION. The StatfjordFormation is stratigraphically divided into threeunits: Nansen, Eiriksson and Raude. The forma-tions consist of interbedded sandstones, siltstones andshales. Eiriksson and Raude represent uvial de-posits, while Nansen is deposited in a shallow ma-rine environment. The river systems in Eirikssonand Raude are partly braided and partly meanderingwith varying sinuosity. The Nansen unit has excellentreservoir characteristics. The horizontal permeabilityin Eiriksson and Raude is good, while the vertical owcapacity is to some extent limited by the shale lay-ers. The shales can represent continous pressure bar-riers over large areas of the elds. Some of the chan-nel sandstones have excellent reservoir characteristicsand thereby represent high-permeable zones. Diage-nesis is important throughout the formation and ispartly destroying the good primary porosity. Car-bonate cement with limited distribution can locallyrepresent ow barriers.

COOK FORMATION. The Cook Formation inthe Dunlin Group is deposited in a shallow marine,tidally inuenced environment. The reservoir unitsare heterolitic with interbedded negrained sand-stones, siltstones, and shales. This generally impliesan overall low permeability, and the vertical perme-ability is supposed to be low. The reservoir charac-

teristics are best in the uppermost parts of the for-mation. Shale beds, calcite cement and internal tex-tures are strongly reducing the vertical permeabilitythroughout the formation.

INTRA DUNLIN SAND. The Intra DunlinSand belongs to the Lower Jurassic Dunlin Group,and comprises three coarsening upward sequences,where only the uppermost has reservoir quality. Thegeological model is uncertain, both with respect todepositional environment and sedimentological pro-cesses, but mapped geometry and lithological vari-ations indicate an elongated rather than a sheetlikesandbody. The prevailing interpretation is a near-shore sand ridge/sand wave, probably deposited byenforced tidal currents. The reservoir ow character-istics vary from fairly good, with permeabilities thatmay attain values of 3 D, to extremely poor, and seemto be in accordance with the elongated geometry ofthe sandbody.

BRENT GROUP. The Brent Group is strati-graphically divided into the following formations(from base): Broom, Rannoch, Etive, Ness, and Tar-bert. In the Oseberg/Veslefrikk area, the OsebergFormation is developed below the Rannoch Forma-tion.The Broom Formation is not regarded as very pro-

ductive in the Viking Graben.The Oseberg Formation represents a fan delta com-

plex deposited at the edge of the Horda Platform intoa marine environment. The fan delta complex pre-dates the huge uvial Brent Delta and is geneticallynot related to the Brent Delta. The Oseberg For-mation consists of stacked coarse-grained sandstoneswith excellent reservoir characteristics. In parts ofthe basin, the sand layers are alternated with marineshales. Some of these shales are continous and act asow barriers. In parts of the formation, carbonate-cemented horizons are frequent. Some of these hori-zons represent omission surfaces and are widely dis-tributed over each fan. Other carbonate-cementedlayers have a more limited distribution, but still actas local barriers during production.The Rannoch Formation is interpreted as the

lower/middle part of a prograding deltafront and con-sists of interbedded shales, siltstones and negrainedmicaceous sandstones with increasing sand contentand permeability upwards. In some areas, the forma-tion is regarded as a nonreservoir unit.The Etive Formation represents the upper part of

the deltafront, consisting of sandstones with high per-meabilities in the lower parts and increasing occur-rence of siltstones and shale horizons in the upperparts. The most important problems in Rannoch andEtive are related to high permeable zones (in Etiveand partly in the upper parts of Rannoch), semiper-meable layers (in lower/middle parts of Rannoch)and barriers (coal/shales in the upper parts of Etiveand calcite-cemented zones in Rannoch).

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326 APPENDIX B. KEY PARAMETERS FOR NORWEGIAN SANDSTONE RESERVOIRS

Formation PropertiesField Group/ Age T D owc go/wc ht hn Θ hg ho φ kh kv

kh

Soi Sorw

Form. m ss m ss m ss m m m m % md % %

Balder Heimdal P au 1680 1760 - 116 95 0-2 - 80 33 1-10 D 0.1-0.5 85 15

BrageStatfj. EJ h 2300 2381 - 110 88 0 - 61 23 1 D 0.1-0.2 68 15Fensfj. LJ an 2080 2149 - 35 20 1-2 - 27 26 100 0.1-0.4 48 22

Draugen Rogn LJ an 1596 1638 - 0-46 0-46 - - 0-37 32 6-7 D 0.6-0.7 72-85 30-35

Brent MJ u 1740 1947 - 250-300 190 12 - 207 31 3.2 D 0.5-0.9 80 30Gullf. Cook EJ u 1750 2090 - 50-75 48 0-15 - 340 28 550 0.05-0.2 72 21

Statfj. EJ u 1860 2028 - 175 105 1-2 - 168 27 1 D 0.5 85 30

Brent MJ u 2850 300 135 17 0-470 5-250 20 5-250 0.1 70-95 32Gullf.-S Statfj. EJ u 3000 3365 3222 260 170 17 222 143 20 0.01-3 D 0.1 60-85 28

Lunde LT a 3250 3680 - >420 >50 17 - 15 0-20 0.1 40-70

Gyda Farsund LJ au 3700 4155 - 50 45 4 - 45 17 30 0.1-1 47-83 15

Fangst MJ h 2080 2478 2283 70 63 7 203 195 28 0.7-20 D 0.9 86-97 25-35Heidrun Tilje EJ h 2100 2452 2292 120 90 6 203 130 25 0.07-2 D 0.5 51-75 12-25

Åre EJ h 2240 2415 2314 205 103 3 63 130 27 0.1-10 D 0.1 60-71

Oseberg Brent J ef 2120 2711 2497 186 111 8 377 214 23 0.5-3.5 D 0.1-1 60-89 27

Garn J u 3952 - 4077 40 36 6 125 - 11 >10 76 -

Smørb.Ile J u 4025 - 4550 60-70 2-45 6 525 - 12 >10 57-75 -Tofte J u 4143 - 4305 80 47 6 163 - 12 >10 64 -Tilje J u 4208 140-185 7-102 6 14 >10 61-84 -

Garn J u 3799 3982 - 87.5 70.6 <3 - 182.5 15 2-850 0.6 64-76 24Smørb. Ile J u 3925 - 4039 71 21 <3 114 - 14 6 60 25-Sør Tilje-3 J u 4125 - 4242 65 <3 117 - 12 1.7 60 17

Tilje-2 J u 4190 4242 - 52 <3 - 52 17 60 55 17

Statfj. EJ u 2300 2595 - 90 40 9 - 120 24 1000 0.3 86 17Snorre L-DEF LT u 2360 2561 - 570 140 8 - 90 24 380 0.25 76 22

L-BC LT u 2340 2574 - 180 45 8 - 50 24 125 0.25 71 22

Statfj.Brent MJ u 2360 2586 - 155 115 7 - 226 28 2300 0.50 84 30Statfj. EJ u 2575 2806 - 125 63 7 - 231 21 1000 0.10 64 18

Statfj-NVolgian LJ u 2600 2718 - 46 37 10 - 118 25 785 0.30 83 35Brent MJ u 2620 2718 - 140 105 10 - 98 25 1000 0.4 80 35

Ula Ula LJ u 3345 3561 - 110 100 10 - 90 17 300 0.15 82 30

Brent MJ u 2755 2906 - 125 71 1-3 - 151 18 200-700 72 30Veslef. IDS EJ u 3000 3079 - 52 12 1-3 - 79 20 10-600 70 30

Statfj. EJ u 3170 3208 3194 >200 >100 1-3 24 14 16 50 76

Frigg Frigg E a 1785 1957 1948 55 52 0 50 9 28 0.5-4 D 0.001-1 91

Heimdal Heimdal P an 2020 2150 2146 0-126 0-101 0-126 4 26 1000 0.01-0.8 86.7 29

Garn MJ h 2300 - 2490 50 49 <10 200 29 5000 0.5 94 -Midgard Ile MJ h 2300 2500 2490 65 62 <10 200 12 27 5000 0.2-0.3 92 25

Tilje EJ h 2350 - 2490 220 110 <10 200 25 1000 0.01-0.1 76 -

Sleip.-V Hugin MJ au 3450 - 203 70-158 10 88-160 - 22 10-450 0.4-0.5 - -

Sleip.-Ø Heimdal P u 2260 - 2417 100 95 0 157 - 27 400 0.1 - -

SnøhvitStø LJ u 2300 2419 2405 70-95 62-85 2 105 14 16 250 0.10 93N-mela MJ u 2300 2419 2405 60-105 2 105 14 14 10 0.10 93

Troll Sognefj. LJ rf 1300 1559 1547 230 1-4 230 0-26 27 103-104 0.25-0.9 30-95 20-40

EkoskEkosk Da u 2900 3200 - 170 120 3-7 - 170 33 1-100 0.025-0.10 88 38-63Tor Cr u 3030 3250 - 85 60 3-7 - 85 30 1-100 0.025-0.10 82 38

Eldf.-AEkosk Da u 2683 - 45 45 7-9 - 45 35 1.0 0.005-0.1 80 25-40Tor Ma u 2896 - 30 30 7-9 - 30 35 2.5 0.005-0.1 85 25-40

Eldf.-BEkosk Da u 2807 2940 - 76 76 5-7 - 76 33 1.0 0.1 70 25-40Tor Ma u - 91 91 5-7 - 91 35 3 0.1 80 25-40

Table B-1a: Properties Table formation, uids, composition, pressure, and temperature

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B.7. FORMATIONS. 327

contd. Hydrocarbon Properties HC-comp. Water Res.

Sorg Vclay ρo γg ρc µo µg µc pb pd Rs CL Bo B−1g C1 C2-4 C5-12 CS CDi T p

% % cp cp cp bar bar % % % mg/l mg/l C bar

0-25 914 0.677 - 3.0 0.016 - 155 - 53 - 1.15 120 33.9 5.9 77 177

835 - 0.7 - 90 - 60 - 1.22 - 20.1 17.5 50210 5400 98 2447 843 0.766 - 0.58 0.024 - 168 - 93 - 1.29 164 36.5 17.5 41710 2700 87 215

- < 10 824 1.010 - 0.68 - 59 - 52 - 1.19 - 16 23 60 37500 400 71 165

5-25 882 0.705 - 1.12 - 244 - 94 - 1.25 - 45.4 6.2 47.0 41300 1970 72 31020-40 844 0.821 - 0.43 - 217 - 116 - 1.41 - 43.6 12.9 42.1 80 319

15 838 0.810 - 0.40 - 270 - 171 - 1.46 - 49.6 13.6 35.7 80 320

30 10 860 0.670 790 0.35 0.035 383 418 200 240 1.60 286 58 10 30 41200 1560 125 45030 5 860 0.670 790 0.35 0.035 386 463 200 310 1.60 278 56 12 29 38800 1210 125 470

15 865 - 0.41 - 381 - 180 - 1.60 - 58 10 31 129 510

- 0-30 822 1.030 - 0.28 - 207 - 327 - 1.69 - 34.3 25.4 37.7 273000 33000 154 595

5-15 0.8 882 0.660 0.75 0.021 241 245 117 108 1.34 219 44.3 12.7 43.0 27000 Low 85 25220-50 15 900 0.660 1.24 0.020 232 224 79 85 1.21 219 43.8 9.0 47 85 251

9 922 0.660 2.29 0.020 201 60 1.16 219 40.8 3.4 55.8 32000 Low 85 251

10 8 850 0.68 728 0.43 0.023 - 281 281 145 310 1.43 222 46.3 13.8 38.9 37800 1450 100 281

- - 0.905 766 - 0.13 - 447 - 2.32 238 73 13.9 9.2 143 467- - - - 246 74.2 14.2 7.7 155 476- - 0.926 771 - 0.12 - 382 - 2.32 238 70.8 16.5 8.5 150 473

774 0.946 0.09 410 440 2.53 231 66.9 16.8 11.6 152 474

832 0.961 - 0.14 0.055 - 362 - 433 - 2.48 - 56 17 14 50000 140 40320 - 0.992 768 - 0.052 - 386 1356 - 234 70 17 9 48000 140 41420 - 1.146 788 - - 1247 - 230 145 43620 820 1.100 - 0.070 - 323 - 438 - - 55 25 15 64350 145 436

10 6 - - - - 34000 2000 90 383- 10 690 - 0.42 179 - 133 - 1.40 - 37 23 40 93 383- 10 700 - 0.48 155 - 105 - 1.33 - 32 22 45 34000 2000 93 383

- 13 824 0.771 - 0.31 - 276 - 190 - 1.58 - 49 17.7 32.6 14800 540 92 3835 17 840 0.848 - 0.29 0.032 - 196 - 155 - 1.54 - 39.5 19.5 40.3 14000 1250 99 404

- 842 0.844 - 0.74 - 131 - 93 - 1.30 - 27.6 21.1 30.0 22000 600 98 398- 846 1.154 - 0.71 - 109 - 66 - 1.24 - 23.5 19.5 30.4 22000 600 98 398

- 0-15 689 1.293 - 0.37 0.015 - 166 - 164 - 1.35 - 28.1 20.6 28.4 200000 34000 143 491

- 10 840 0.950 - 0.43 - 195 - 125 - 1.46 - 37.0 20.1 42.9 19800 250 122 321- 12 830 0.970 - 0.34 - 200 - 140 - 1.49 - 36.0 20.8 43.2 29500 990 128 346

10 838 0.838 - 0.17 - 332 320 - 2.1 54.8 17.1 28.1 43727 1663 133 355

- 835 0.581 797 4.83 0.021 61 5 1.15 194 95.5 3.7 0.07 60000 61 198

- 9 0.712 710 0.022 210 156 211 86.3 10.3 3.4 60000 76 218

- - 0.762 - 0.022 - 251 - 190 - 214 82 13.5 3.0 87000 90 25110 663 0.762 0.36 0.022 251 251 159 190 1.48 214 82 13.5 3.0 87000 90 251- - 0.762 - 0.022 - 251 - 190 - 214 82 13.5 3.0 87000

- 10 0.84 649 - 0.047 - - 356 - 360 1.54 271 70.8 15.2 4.1 70000 120 442

- 5-10 - 1.456 653 - 0.009 1.2 - 235 1729 336 2.19 213 71.0 21.4 6.2 36986 2820 93 245

- 866 0.749 750 0.59 0.023 263 259 149 112 1.44 229 81.4 8.8 2.28 93000 5210 93 267- 866 0.749 750 0.59 0.023 263 259 149 112 1.44 229 81.4 8.8 2.28 93000 5210 93 267

20-60 900 0.610 750 1.60 0.002 - 158 158 62 33 1.18 151 36 7 40 50000 68 158

< 1 838 0.700 - 0.13 0.040 - 383 - 273 - 1.78 - 58 15 27 50000 4400 131 497< 1 838 0.700 - 0.13 0.040 - 383 - 273 - 1.78 - 58 15 27 75000 3300 131 497

< 5 830 0.830 - 0.11 0.048 - 340 - 471 921 2.42 266 54.7 16.9 28.4 50000 850 126 472< 5 830 0.840 - 0.11 0.049 479 933 2.45 268 126< 5 842 0.880 - 0.10 0.061 - 400 458 1113 2.29 270 63.3 14.1 22.6 47000 1050 126 476< 5 842 0.880 - 0.10 0.061 - - 458 1113 2.29 270 - -

Table B-1b: Properties Table contd. formation, uids, composition, pressure, and temperature

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328 APPENDIX B. KEY PARAMETERS FOR NORWEGIAN SANDSTONE RESERVOIRS

Size Resources Prod.plan Cum.Prod.Field Group/ A Vb N NRsi GFi R-oil R-gas R-ngl Period Meth. Np Gp

Form. km2 Mm3 MSm3 GSm3 GSm3 MSm3 GSm3 Mtons years MSm3 GSm3

Balder Heimdal 30 310 131 7 0 35

BrageStatfj. 7.7 470

149.2 9.8 0 46.2 1.7 1 1994-2010 wFensfj. 47 1260

Draugen Rogn 60 956 155 14 0 68 3 1993-2010 w

Brent 35 2777 438 41 0 190Gullf. Cook 15 789 71 11 0 15 16.5 2.3 1986-2006 w 54 3

Statfj. 10 651 52 10 0 25

Gullf.Brent 6200

80.9 15.1 77 22.3 56.1 3.0-Sør

Statfj. 88 1220Lunde

Gyda Farsund 20 1000 75 23 0 30.5 2.9 2.4 1990-2010 w 3 0.4

Fangst 30 1311 153 15.3 14.7 62.7Heidrun Tilje 22 2329 172 14.3 23.7 23.1 37.8 1995-2011 w

Åre 14 1046 10 1 1.5

Oseberg Brent 115 7662 460.4 67.2 64.0 226 70 6 1988-2017 wg 44

Garn 592

Smørb.Ile

905794

80 125 20 65Tofte 1534Tilje 7589

Garn 1200Smørb. Ile 1268

89.4 30.5 9.8 31 24-Sør Tilje-3 1618

Tilje-2 601

Statfj. 30 2045341 39.2 0 106 6.7 3.2

Snorre L-DEF 50 3730 1992-2011 wL-BC 30 3195

Statfj.Brent 80 6513 795.5 152

0396.4

59 18 1979-2009w

339 18Statfj. 35 4964 295 48 130 g

Statfj.-NVolgian

13157

66.8 5.1 0 30.9 2.5 1994-2015 wBrent 280

Ula Ula 16 1800 130.7 15.8 0 69.2 4.7 3.5 1986-2009 w 25 1.7

Brent 22 109792 12 36.4 3.1 1.3 1989-2008 w 4

Veslef. IDS 278Statfj. 12 229

Frigg Frigg 104 5030 1.2 235 0.7 180 1977-1995 p 173

Heimdal Heimdal 1616 9.4 60 5.7 35.6 1985-1997 p 18

GarnMidgard Ile 53 3000 15 80

Tilje

Sleip.-V Hugin 80 6972 0 0 189 27 135 9 1996-2013 p

Sleip.-Ø Heimdal 59 1446 0 0 88 19 51 10 1993-2002 p

SnøhvitStø

80 5850 65.2 6.5 76 5.7N-mela

Troll Sognefj. 710 60000 670 48 1812 41 1288 30 1996→ p

EkoskEkosk 49 8300

1080 304.4 0 320 154 14.9 1971-2048wgp

161 73Tor 49 4150 w

Eldf.-AEkosk 11 488

Tor 11 325366 95 0 74.6 55.3 5.2 1979-2025 p 47 18

Eldf.-BEkosk 8 577

Tor 8 691

Table B-2: Volume Table reservoir size; resources; production period, volumes, and method

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B.7. FORMATIONS. 329

The Ness Formation represents the uvial parts ofthe delta with straight to low sinuos channels. In thenorthern part of the Viking Graben, the formationis inuenced by shallow marine processes implyingmore continous sandbodies. Production is compli-cated by zones with very good reservoir characteris-tics, interbedded with poor quality layers. The coallayers are mostly interpreted to be widely distributedand may therefore act as barriers to ow. The hori-zontal communication is moderate to good, with re-duced vertical communication.The Tarbert Formation represents the nal event

of the Brent delta system and shows large variationsin the dierent elds. The Tarbert Formation has ingeneral a very high sand content, but some interbed-ded shale barriers distributed over large areas are rec-ognized. The formation has overall good reservoircharacteristics. The vertical permeability contrastsare larger than the horizontal. The vertical ow ca-pacity is limited by the coal and shale layers.

HUGIN FORMATION. The Hugin Formationis of early Bathonian to early Oxfordian age, repre-senting a marine transgression over the underlyingdelta complex of the Sleipner Formation. The thick-ness and content of the Hugin Formation is highlydependent on the structural setting. The forma-tion is dominated by coastal, near-shore marine sand-stones, with the depositional environment rangingfrom coastal marsh and backbarrier, through barrierand oshore bars to oshore sand and mud sheets.The reservoir characteristics are generally good, butchange depending upon type of deposition. Thenet/gross-values vary between 80% and 95% for themost productive zones.

FENSFJORD FORMATION. The FensfjordFormation consists of oshore siltstones and ne-grained, well sorted, clean sandstones of lowershoreface environment. A cyclisity of coarsening up-wards sequences probably reects a variation in de-positional environment, from transitional/inner shelfto lower shoreface, caused by relative sea level uctu-ations.Consequently, there are permeability contrasts ver-

tically which have great inuence on reservoir behav-ior, whereas only gradual changes are expected in thehorizontal continuation of these laterally persistentsands.

SOGNEFJORD FORMATION. The Sogne-fjord Formation comprises, together with the underly-ing Middle Heather Formation, six depositional cyclescontrolled by minor sea level uctuations during anoverall transgressive period. Each cycle commenceswith a micaceous, poorly sorted upward coarsen-ing progradational sequence, and is terminated by atransgressive component including excellent reservoirsands. These are cleaner, have good sorting sands

and good reservoir characteristics, with permeabili-ties ranging from 1 to 10 D. The reservoir characteris-tics in the progradational sequences are rather poor,with permeabilities in the order of 100 to 200 md.Laterally, continuous and extensive calcite cementedhorizons are believed to occur related to the interfacesbetween most of the depositional cycles, and also re-lated to the interfaces between some of the prograda-tional and transgressive components. It is expectedthat these horizons will act as eective barriers foruid ow in the Troll Reservoir.

VOLGIAN SAND (=Intra Draupne Sand).The Volgian Sand is deposited as submarine fans.The association comprises an upper proximal part,consisting of nearly 100% sand with very good reser-voir characteristics. The lower distal part of the fansconsists of alternating sands and shales, where theshales can be widely distributed. Some carbonate-cemented intervals with limited lateral distributionare recognized in the sandstones.

ULA FORMATION AND THE FARSUNDFORMATION. The Ula Formation and the Gydasandstone member of the Farsund Formation were de-posited in marine environment on a storm-dominatedshallow marine shelf. The sandstones generally con-sist of several coarsening upwards units, representingeither regressive events caused by relative sea levelchanges, or storm generated progradational blanketsands.The dierent units can be correlated within most

of the individual elds. Although almost the entireinterval consists of sandstones, the reservoir qualityvaries from very good to extremely poor. This is dueboth to variation in grainsize and clay content, anddiagenetic eects (i.e. Qz and carbonate cementation)caused by great burial depth.

HEIMDAL FORMATION. The Heimdal For-mation is of Paleocene age and comprises alterna-tions of two dominating facies, massive sandstonesand ning upward sequences of bouma divisions, re-spectively. The facies association dened by thesetwo facies suggests a submarine fan depositional envi-ronment, with deposits representing both inll chan-nels and suprafan areas. The reservoir propertiesare good, in spite of zones of low permeability as-sociated with argillaceous and clayey divisions of thebouma sequences. Sandstone permeabilities often ex-ceed 1 D. Concretions and discontinuous layers of car-bonate cement occur, but do not aect the gas pro-duction in the Heimdal Field.

FRIGG FORMATION. The Frigg Formation ofEocene age is a part of a Paleocene-Lower Eocene sub-marine fan complex. The lower part of the formationis mainly sandy, but also contains heterolithic faciesand breccias with strong synsedimentary deformation

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330 APPENDIX B. KEY PARAMETERS FOR NORWEGIAN SANDSTONE RESERVOIRS

due to rapid sedimentation related to turbiditic de-position.The pay zone (Frigg Formation Upper Member)

consists of clean, unconsolidated sandstones withgood reservoir characteristics. Average permeabili-ties range from 0.9 to 3 D. The sandstones are orga-nized in amalgamated beds with locally strong ero-sional contacts, related to deposition by grainows inthe channelized, proximal part of a submarine fan.Some shale layers also occur. These layers are local

and have limited lateral extent. However, they causerelatively strong dynamic pressure barriers that lo-cally and temporarily have prevented the rise of waterlevel in the reservoirs during production.

BÅT GROUP. The Båt Group comprises four for-mations: Åre (base), Tilje, Tofte, which is only rec-ognized on the western part of the Halten Terrace,and Ror (top).The Åre Formation is interpreted as coastal plain

to delta plain deposits with swamps and channels passupwards into marginal marine facies. It consists of al-ternating sandstones and claystones interbedded withcoals and coaly claystones. Especially in the lowerpart the claystones act as vertical and lateral barri-ers.The Tilje Formation consists of very ne to coarse-

grained sandstones interbedded with shales and silt-stones. A near-shore marine to intertidal depositionalenvironment is typical of the formation. The forma-tion is divided into reservoir zones with signicantvariation in reservoir properties separated by shalebarriers.The Tofte Formation is interpreted as fan deltas

and consists of moderately to poorly sorted coarse-grained sandstones. In the Smørbukk Field, the for-mation is part of the reservoir, with permeabilities >10 md.The dominant lithology in The Ror Formation is

mudstones. Towards the top, interbedded silty andsandy coarsening upwards sequences are common.The formation was deposited mainly below wave ba-sis in open shelf environments. The formation acts asa barrier between the Tilje Formation and the FangstGroup.

FANGST GROUP. The Ile Formation is inter-preted to represent various tidal inuenced delta orcoastline settings. The formation consists of neto medium and occasionally coarse-grained sand-stones interbedded with thinly laminated siltstoneand shales. Mica-rich intervals are common. Thereservoir properties are generally good. Especiallywithin the lower part barriers may locally reduce thepossibility for vertical uid ow.The basal part of The Not Formation consists of

claystones deposited in lagoons or sheltered bays. To-ward the top, the formation consists of claystoneswhich coarsen upwards into bioturbated ne-grainedsandstones. This part of the formation consists of

prograding deltaic or coastal front sediments. Gener-ally, the formation acts as a barrier between the Ileand Garn Formation.The Garn Formation consists of medium to coarse-

grained, moderately to well-sorted sandstones. Mica-rich zones are present, and the sandstone is occasion-ally carbonate cemented. The formation may repre-sent progradation of braided delta lobes. Delta topand delta front facies with active uvial and wave in-uenced processes are recognized. The permeabilityis generally good ( > 10 md ).

ROGN FORMATION. The Rogn Formationshows a coarsening upward sequence from siltstonesand shales to sandstones which constitute the bulk ofthe unit. The sandstones are interpreted as shallowmarine bar deposits. The permeability ranges fromgood (1001000 md ) in the lower part to very good( > 1 D ) in the upper part of the reservoir.

STØ FORMATION. The sands in the formationwere deposited in prograding coastal regimes, and avariety of linear clastic coast lithofacies are repre-sented. Moderately to well-sorted and mineralogi-cally mature sandstones are dominant. Thin unitsof shale and siltstone are clear markers. Phosphaticlag conglomerates occur in some wells, especially inupper parts of the unit.

NORDMELA FORMATION. The formationwas deposited in tidal at to ood plain environ-ments. Individual sandstone sequences represent es-tuarine and tidal channels which dissected this low-lying area. The formation consists of interbedded silt-stones, sandstones, shales, and claystones with minorcoals. Sandstones become more common towards thetop.

SHETLAND GROUP. The Shetland Group in-cludes the formations of the former Chalk Group.The group consists of pelagic limestones (chalk) andcalcareous shales. The main reservoirs are situatedin the pure chalks in the upper part of the group(Tor and Ekosk Formations). Sedimentologicallythe chalks can be separated into two main facies,the open marine pelagic chalk and reworked subma-rine chalk facies. Generally, the reworked chalk fa-cies have the best reservoir characteristics. Superim-posed on the primary reservoir characteristics diage-nesis and fracturing play an important role for theporosity and permeability development in the chalkreservoirs. The overpressure is also an important fac-tor in conserving the anomalously high porosity inthe chalk.Tor Formation. Overall, the Tor Formation has

the most extensive and productive reservoirs in theShetland Group. The formation is characterized byvery pure chalks. The depositional environment ofthe Tor Formation is a mixing of pelagic chalk andsubmarine debris ows and turbidities. In the upper

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B.7. FORMATIONS. 331

part of the formation, mass ows are very frequentand create stacked submarine fans of highly porouschalk. Another typical feature is the developmentof synsedimentary faults creating small grabens lledwith resedimented chalk, also highly porous. Theporosity in the Tor Formation ranges from 2040%,in extreme cases up to 50%. The matrix permeabil-ity, on the other hand, is low and typically between24 md, but may locally exceed 10 md. Eective per-meability caused by fracturing may reach 150 md.Ekosk Formation. The deposition of pelagic

chalk and mass ows continue into the Ekosk For-mation. In the lower part the sediments are moreargillaceous and tight, and can act as a ow bar-rier between the Tor and Ekosk Formations. Thebest reservoir zones in the Ekosk Formation are sit-uated in stacked mass-ow deposits containing re-worked material from the Tor Formation. In otherparts, slumped chalk plays an important role to im-prove the reservoir quality. Even the pelagic chalksare an important reservoir facies in the Ekosk For-mation, but in this facies the permeability is generallylow. The porosity range in the Ekosk Formation is1845%, and the matrix permeability is not dramat-ically dierent from the values in the Tor Formation(0.510 md).

References

[1] Geology of the Norwegian Oil and Gas Fields,A.M. Spencer, C.J. Campbell, S.H. Hanslien,P.H.H. Nelson, E. Nys¾ther, and E.G. Or-maasen (eds.), Graham and Trotman, London(1987).

[2] A revised Triassic and Jurassic lithostrati-graphic nomenclature for the Norwegian NorthSea, J. Vollset and A.G. Doré (eds.), NPD-bulletin No. 3, Norwegian Petroleum Direc-torate, Stavanger (1984).

[3] A lithostratigraphic scheme for the Mesozoicand Cenozoic succession oshore mid- andnorthern Norway, A. Dalland, D. Worsley, andK. Ofstad (eds.), NPD-bulletin No. 4, Norwe-gian Petroleum Directorate, Stavanger (1988).