Reservoir Characterization Using Expert Knowledge, Data and Statistics

Prepared with assistance from:

Peter Corvi, Head of Effective PropertiesKes Heffer, Head of Reservoir DescriptionPeter King, Senior Reservoir Engineer,

Effective PropertiesStephen Tyson, Computing ConsultantGeorges Verly, Senior Geostatistician,

Effective PropertiesBP Research Sunbury-on-Thames, England

Christine Ehlig-Economides, ReservoirDynamics Program Leader

Isabelle Le Nir, GeologistShuki Ronen, Geophysics Integration

Program LeaderPhil Schultz, Department Head Etudes et Productions SchlumbergerReservoir Characterization DepartmentClamart, France

Patrick Corbett, Research AssociateJonathan Lewis, Conoco Lecturer in GeologyGillian Pickup, Research AssociatePhillip Ringrose, Senior Research AssociateHeriot-Watt UniversityDepartment of Petroleum EngineeringEdinburgh, Scotland

Dominique Guérillot, Sr. Research Engineer, Reservoir Engineering Division

Lucien Montadert, Director of Exploration and Production

Christian Ravenne, Principal Research Associate,Geology and Chemistry Division

Institut Français du Pétrole Rueil-Malmaison, France

Helge Haldorsen, Assistant Director, InternationalField Developments

Norsk Hydro A.S. Oslo, Norway

Thomas Hewett, Professor of Petroleum Engineering

Stanford University California, USA

Accurately simulating field performance requires knowing properties

like porosity and permeability throughout the reservoir. Yet wells provid-

ing the majority of data may occupy only a billionth of the total reser-

voir volume. Transforming this paucity of data into a geologic model for

simulating fluid flow remains an industry priority.

January 1992

The simplest representation of a reservoir isa homogeneous tank with uniform proper-ties throughout. Characterizing such a sim-plified model requires merely enough datato define its volume and basic petrophysicalproperties. This information can then beused to help forecast reservoir behavior andcompare possible production scenarios.

But efficient exploitation of reservesrequires a more sophisticated approach.Reservoirs have been created by complexsedimentary and diagenetic processes, andmodified by a history of tectonic change.Rather than resembling simple tanks, reser-voirs are heterogeneous structures at everyscale. The need for accurate characteriza-tion is vital for oil company economic plan-ning (next page, top).

In the past, geologists have used expertknowledge to interpolate between wells andproduce basic reservoir models. However,during the last ten years, this effort has beentransformed by statistical modeling offeringinsight into the effects of heterogeneity.

This article details how reservoir descrip-tions are developed and is divided intothree stages (page 27):•Defining the reservoir’s large-scale struc-

ture using deterministic data•Defining the small-scale structure using

statistical techniques—geostatistics •Rescaling the detailed geologic model to

be suitable input for a fluid-flow simulator.

Cray is a mark of Cray Research Inc. MicroVAX is a markof Digital Equipment Corp.1. Weber KJ: “How Heterogeneity Affects Oil Recovery,”

in Lake LW and Carroll HB Jr (eds): Reservoir Charac-terization. Orlando, Florida, USA: Academic PressInc. (1986): 487-544.

Defining the Large-Scale StructureThe first stage aims to construct as detailed ageologic description of the reservoir as mea-surements will allow. The available informa-tion includes a structural interpretation fromseismic data, static data from cores and logsand dynamic production data from testing.To these, the geologist adds expert knowl-edge on geologic structures, obtained fromfield experience and outcrop studies (see“Gathering Outcrop Data,” page 28).

The first step is to recognize depositionalunits and then, if possible, correlate thembetween wells. It helps to know the generalshape and order of deposition of the units,and ideally the relationships between width,thickness and length for each unit.1 Well-to-well correlation is made using traditionalhard-copy maps and logs or, today, interac-tive computer workstations. Workstationsallowing simultaneous visualization of seis-mic, log and other data enable easier recog-nition of subtle correlations.

In most cases, this large-scale characteri-zation results in a conventional layer-cakemodel—a stack of subhomogeneous layers.There is no doubt that many reservoirsshould be characterized by more complexarrangements of depositional units—such asthe “jigsaw” and “labyrinth” models (nextpage, bottom).2 The ability to use thesemore complex models routinely remains anindustry goal (see “Trends in Reservoir Man-agement,” page 8).

25

2. Weber KJ and van Geuns LC: “Framework For Con-structing Clastic Reservoir Simulation Models,” Jour-nal of Petroleum Technology 42 (October 1990):1248-1253, 1296-1297.

nHow heterogeneity affects productivity. In this example developed at BP Research,Sunbury-on-Thames, England, two different models have been interpolated between apair of hypothetical wells. The lithologies are identical: only their spatial correlationhas been altered. The graph shows how the pore volume recovered versus pore volumeinjected varies for the two models. After one pore volume has been injected, the differ-ence in recovery between the two models is about 0.1 of a pore volume.

26

nDefining reservoir type. Current large-scale characterization tends to result in a sim-ple layer-cake model. However, efforts are being made to use more complex modelslike the jigsaw and labyrinth models shown.

Heterogeneity case A

Hypothetical well data

Heterogeneity case B

12345Shale Barrier

Lithology

Pore volume (PV) injected

PV

re

co

vere

d

0.5

0.4

0.3

0.2

0.1

00 0.2 0.4 0.6 0.8 1.0

Case BCase A

Predicted oil recovery

Layer cake Jigsaw Labyrinth

Tidal channel

Barrier footTransgressive deposit Barrier barBarrier bar Crevasse splay

Distributary channel fill

Defining the Small-Scale Structure UsingGeostatistical ModelingOnce the shape of the reservoir and itslarge-scale structure have been described,the next stage focuses on defining hetero-geneity within each depositional unit. Thisrequires more detailed interwell measure-ments than are currently available. Boreholeseismic data typically cannot span wells.Although surface and cross-well seismicdata do reach everywhere, there is insuffi-cient knowledge relating seismic data tophysical rock properties, and explorationseismic data have insufficient resolution. Inaddition, well testing data often lack direc-tional information. Therefore nondetermin-istic, or geostatistical, methods are required.3

Large-scale reservoir models typicallycomprise a few thousand grid blocks, eachmeasuring about 100 m square and morethan 1-m thick [about 1000 ft2 by 3 ft] Butmodels that take into account small-scaleheterogeneity use smaller grid blocks—1million or more are often needed. The sheerquantity of input data required to fill somany blocks also favors geostatistics.

Geostatistical techniques designed to pro-vide missing data can be classified in twoways: by the allowable variation of theproperty at a given point and by the waydata are organized. There are two classes ofproperty variation, continuous and discrete.Continuous models are suited to propertieslike permeability, porosity, residual satura-tion and seismic velocity that can take anyvalue. Discrete models are suited to geo-logic properties, like lithology that can berepresented by one of a few possibilities.4

Data organization has developed alongtwo paths depending on how the modelsare built up. One is grid based. All theproperties are represented as numbers on agrid, which is then used for fluid-flow simu-lation. The second is object based. Reser-voir features such as shales or sands aregenerated in space and a grid then superim-posed on them.

Grid-Based ModelingOne of the key geostatistical tools used ingrid-based modeling is kriging—named aftera pioneer of rock-property estimation, D. G.Krige, who made a series of empirical stud-ies in the South African goldfields.5 Krigingis designed to honor measured data andproduce spatially correlated heterogeneity.The heart of the technique is a two-pointstatistical function called a variogram thatdescribes the increasing difference (ordecreasing correlation) between sample val-ues as separation between them increases.

Oilfield Review

Stage 2: Defining small-scale structure

Stage 1: Defining large-scale structure

Core plugs Whole core Well logs Well testing

Borehole geophysics Surface seismicsOutcrop studies

Geologist’s expert knowledge

Kriging minimizes the error between theinterpolated value and the actual (butunknown) value of the property.6

For a given property, a variogram is con-structed from pairs of data generally mea-sured in wells. In the oilfield, criticism ofkriging centers on the low density of welldata when compared to mining. To over-come this problem, variograms can alterna-tively be constructed from data measuredon outcrops or in mature fields where bettersampling is available.

Although variograms are commonly usedto infer the spatial continuity of a singlevariable, the same approach can be used tostudy the cross-continuity of several differ-ent variables—for example, the porosity atone location can be compared to seismictransit time. Once constructed, this cross-correlation can be used in a multivariateregression known as cokriging. In this way,a fieldwide map of porosity can be com-puted using not only porosity data, but alsothe more abundant seismic data.7

Kriging and cokriging deal with quantita-tive, or hard, data. A third method calledsoft kriging combines expert information,also called soft data, with the quantitativedata. The expert data are encoded in theform of inequalities or probability distribu-tions. For example, in mapping a gas/oilcontact (GOC), if the contact is not reachedin a certain well, soft kriging uses theinequality: GOC is greater than well totaldepth. At any given point where there is nowell, an expert may define the probability offinding the GOC within a certain depthinterval. Soft kriging will use this probability.8

(continued on page 30)

nBuilding a reservoir model in three stages with measured data, expert knowledgeand statistics.

27January 1992

3. Haldorsen HH and Damsleth E: “Stochastic model-ing,” Journal of Petroleum Technology 42 (April1990): 404 -412. Haldorsen HH and Damsleth E:“Challenges in Reservoir Characterization Research,”presented at Advances in Reservoir Technology, Char-acterization, Modelling & Management, organized bythe Petroleum Science and Technology Institute, Edin-burgh, Scotland, February 21-22, 1991.

4. Journel AG and Alabert GF: “New Method For Reser-voir Mapping,” Journal of Petroleum Technology 42(February 1990): 212-218.

5. Krige DG: “A Statistical Analysis of Some of the Bore-hole Values of the Orange Free State Goldfield,” Jour-nal of the Chemical, Metallurgical and Mining Societyof South Africa 53 (1952): 47-70.

6. Journel AG: “Geostatistics for Reservoir Characteriza-tion,” paper SPE 20750, presented at the 65th SPEAnnual Technical Conference and Exhibition, NewOrleans, Louisiana, USA, September 23-26, 1990.

7. Doyen PM: “Porosity From Seismic Data: A Geostatis-tical Approach,” Geophysics 53 (October 1988):1263-1275.

8. Kostov C and Journel AG: “Coding and ExtrapolatingExpert Information for Reservoir Description,” in LakeLW and Carroll HB (eds): Reservoir Characterization.Orlando, Florida, USA: Academic Press Inc. (1986):249-264.

Stage 3: Scaling up

30,000 grid blocks

inflo

w fa

ce

outfl

ow fa

ce

Fluid-flow simulation

1 million grid blocks

nPhotomosaic of the Tensleep Sandstone in the Big Horn basin, Wyoming, USA. ThisPennsylvanian-age sandstone is being studied because of its similarity to the Rotliegen-des Sandstone, an important gas reservoir unit found in the Southern Basin of the NorthSea. Like most outcrops, it is irregularly shaped. Therefore, to obtain the mosaic withconstant scale and no distortion, overlapping photographs were taken from points in aplane parallel to a major depositional surface on the outcrop. With a helicopter posi-tioned in this plane, laser range finders were used to maintain a fixed distance fromthe outcrop to ensure constant scale. This was double-checked by positioning scalebars at regular intervals on the outcrop. The mosaic was prepared during studies byJon Lewis and Kjell Rosvoll at Imperial College, University of London, England andHeriot-Watt University, Edinburgh, Scotland with the support of Conoco (UK) Ltd.

Every reservoir is unique. Yet recurring deposi-

tional conditions create families of reservoirs

that are broadly similar. If reservoir rock outcrops

at the surface, it presents a golden opportunity to

gather data which can be used to help character-

ize related subsurface formations.1

Outcrop studies allow sampling at scales that

match interwell spacing. Small-scale permeabil-

ity can also be measured and compared with

depositional characteristics. Then the statistical

and depositional information measured at the

outcrop can be incorporated into the stochastic

modeling of the subsurface analog.2

Generally, there are two kinds of outcrop

study.3 Bed or bed-set scale genetic models

(GEMs) concentrate on reservoir heterogeneity as

a function of the sedimentary process. To date,

most outcrop studies have been of this type. The

second type, a field analog model (FAM), looks at

heterogeneity caused not only by depositional

processes but also by diagenesis. FAMs present

two difficulties. First, it is hard, given the paucity

of reservoir data, to decide which diagenetic pro-

cesses are relevant. Second, finding outcrops

that have undergone specific diagenetic pro-

cesses can be difficult.

Although the study of outcrops is a traditional

skill learned by most geologists, in the past they

have concentrated on such goals as describing

the sedimentological environment and the

geometries of the major depositional units. For

reservoir characterization, a much more detailed

and quantitative picture of the heterogeneity

within depositional sequences is required.

Therefore, in addition to the geologist’s tradi-

tional mapping skills, new analytical techniques

have been devised. Analyzing photographs of the

outcrop helps clarify the lithofacies distribution.4

Clearly, more than one photograph is needed,

and this technique can succeed only if all the

photographs are at the same scale and taken at

the same distance from the rock. A helicopter-

mounted camera with a laser range finder is one

way of achieving this. Successive photographs

are taken with a 30 to 40% overlap and then com-

bined to construct a photomosaic of the whole

Gathering Outcrop Data

28

outcrop (above). The mosaic can be digitized to

quantify lithofacies patterns. It can also be used

to plan and execute small-scale petrophysical

measurements on the outcrop.

Gamma ray logging of outcrops is used to help

describe the composition of the lithofacies. A

standard truck-mounted sonde can be lowered

down the face of an outcrop and gamma ray mea-

surements continuously recorded. Although

washouts are known to affect borehole gamma

ray logging, tests show that the sonde can be as

much as 0.6 m [2 ft] away from the rock before

the reading is compromised.5

An alternative technique employs a lightweight,

portable gamma ray spectrometer to measure

either total radiation or individual radioelement

concentrations. Total radiation measurements

are typically made at 0.65-m [2.1-ft] intervals.

Selected intervals may be logged in greater

detail if required. Because the tool measures an

area of the rock face with a diameter of about 0.3

m [1 ft], readings taken for thin beds will be

slightly influenced by adjacent strata.

One of the principal pitfalls in reservoir charac-

terization stems from the use of unrepresentative

permeability data measured from cores and core

plugs taken from only a few wells. Densely sam-

pled outcrop permeability data are of key impor-

tance. These can be gathered by performing labo-

ratory-based flow tests on samples gathered in

the field. However, a comparatively recent devel-

opment has seen the introduction of portable

electronic minipermeameters—also known as

probe permeameters (next page, below).6 These

evaluate permeability—from 0.5 millidarcies

(md) to 15 darcies—by injecting nitrogen into a

prepared location on the outcrop. Thousands of

nondestructive measurements can be made in

situ and the results compared to traditional labo-

ratory tests for quality control. This permits

examination of permeability changes on a very

small-scale—on the order of millimeters.

Through such measurements, permeability has

been shown to vary dramatically within a deposi-

tional unit.7

A typical study, carried out by the University of

Texas, Bureau of Economic Geology, Austin,

Texas, USA, centers on a Ferron Sandstone out-

crop in central Utah, USA. Ferron is a wave-mod-

ified, deltaic sandstone with permeability distri-

bution strongly related to lithofacies type and

grain size. It is believed to be an analog for Gulf

Coast deltaic reservoirs, which account for 64%

of Texas Gulf Coast gas production.8

First, the depositional framework of the out-

crop was established using color photomosaics to

map the distribution and interrelations of the

sandstone components. This involves determin-

ing lithofacies architecture, delineating perme-

able zones and their continuity, identifying flow

barriers and baffles and establishing permeabil-

ity trends.

On the basis of this framework, more than 4000

minipermeameter readings were made, all on

Oilfield Review

1. Weber KJ and van Geuns LC: “Framework for Construct-ing Clastic Reservoir Simulation Models,” Journal ofPetroleum Technology (October 1990): 1248-1253, 1296-1297.

2. North CP: “Inter-Well Geological Modelling of ContinentalSediments: Lessons From the Outcrop,” presented atAdvances in Reservoir Technology, Characterisation,Modelling & Management, organized by the PetroleumScience and Technology Institute, Edinburgh, Scotland,February 21-22, 1991.

3. Lewis JJM: “A Methodology for the Development of Spa-tial Reservoir Parameter Databases at Outcrop,” pre-sented at Minipermeametry in Reservoir Studies, orga-nized by the Petroleum Science and Technology Institute,Edinburgh, Scotland, June 27, 1991.

4. A lithofacies is a mappable subdivision of a stratigraphicunit distinguished by its lithology.

5. Jordan DW, Slatt RM, D’Agostino A and Gillespie RH:“Outcrop Gamma Ray Logging: Truck-Mounted andHand-Held Scintillometer Methods Are Useful for Explo-ration, Development, and Training Purposes,” paper SPE22747, presented at the 66th SPE Annual Technical Con-ference and Exhibition, Dallas, Texas, USA, October 6-9,1991.

6. Lewis JJM: “Outcrop-Derived Quantitative Models of Per-meability Heterogeneity for Genetically Different SandBodies,” paper SPE 18153, presented at the 63rd SPEAnnual Technical Conference and Exhibition, Houston,Texas, USA, October 2-5, 1988.Jensen JL and Corbett PWM: “A Stochastic Model forComparing Probe Permeameter and Core Plug Measure-ments,” paper 3RC-24, presented at the 3rd InternationalReservoir Characterization Technical Conference of theNational Institute for Petroleum and Energy Research andthe US Department of Energy, Tulsa, Oklahoma, USA,November 3-5, 1991.

7. Kittridge MG, Lake LW, Lucia FJ and Fogg GE:“Outcrop/Subsurface Comparisons of Heterogeneity inthe San Andres Formation,” SPE Formation Evaluation 5(September 1990): 233-240.

8. Tyler N, Barton MD and Finley RJ: “Outcrop Characteriza-tion of Flow Unit and Seal Properties and Geometries,Ferron Sandstone, Utah,” paper SPE 22670, presented atthe 66th SPE Annual Technical Conference and Exhibition,Dallas, Texas, USA, October 6-9, 1991.

Variogram analysis confirmed that the hetero-

geneity of the permeability was structured in a

way that depended on the rock type sampled and

the sampling scale. For instance, at sampling

intervals of 0.6 m, the range over which perme-

ability shows correlation was 5.5 m [18 ft] for

tough-cross-bedded sandstone and 1.2 m [4 ft]

for contorted beds. Planar cross-bedded strata

exhibited a range of 3.6 m [12 ft] for a 0.6 m

sampling interval and 0.6 m for the 0.1 m inter-

val. According to the University of Texas

researchers, this suggests that the permeability

structure may be fractal.

This is just one example of work underway.

Over the past five years, there has been an explo-

sion of outcrop studies—a hundred or more liter-

ature references (above, right). And many pro-

jects are afoot to bring these studies together and

create outcrop data bases. Key information

needed to condition geostatistical modeling will

become readily available.

vertical exposures of rock to minimize the effects

of weathering. Two sampling strategies were

employed. First, more than 100 vertical strips

were mapped every 30 to 60 m [100 to 200 ft] on

three channel complexes. Permeability measure-

ments were taken at 0.3-m spacing on each strip.

Second, two sampling grids were constructed

to examine the small-scale spatial variability of

permeability within lithofacies. One was 12-m

[40-ft] square with 0.6-m subdivisions. The sec-

ond, inside the first, measured 1.8-m [6-ft]

square with 0.1-m [0.3 ft] subdivisions.

The study successfully identified patterns of

heterogeneity. Delta-front sandstones were found

to be vertically heterogeneous but laterally con-

tinuous over a typical well spacing. However, the

sand belts of the distribution system—composed

of amalgamated, lateral accretion point-bar

sandstones—were found to have both lateral and

vertical heterogeneity.

January 1992

Gas supply (nitrogen cylinder) R1 R2

Four way valve

Injection pressuremeasurement system

Flow units

Gas injection probe

Pressure transducer (0-20psi)

Roc

k

Four way valve

Four way valve

nMinipermeame-ter in detail. Theprobe is applied tothe cleaned sur-face of the rockwith enough pres-sure to ensure agood seal. Nitro-gen is injected intothe rock and thepressure and flowrate measured.These values,along with thearea of the probe,are then used tocalculate perme-ability. To measurethe range of per-meabilities foundat the outcrop, fourdifferent flow ele-ments are required.

nGoing underground—outcrops are notalways on the surface. A three-dimen-sional (3D) study of permeability varia-tion was undertaken in a subsurface sil-ica sand mine in Morvern, Scotland. Some8000 permeability measurements havebeen collected over a 1-km [0.6-mile]square section of the mine. The work wascarried out by Jon Lewis and Ben Lowdenat Imperial College, University of London,England, and supported by Den NorskeStats Oljeselskap A.S. (Statoil).

nThe Sierpinski Gasket. This self-simi-lar fractal structure was devised by thePolish mathematician W. Sierpinskiabout 90 years ago. It is formed bydividing the largest triangle intosmaller triangles with sides half aslong as the original. In three of theresulting four triangles the process isrepeated. The process is repeated atprogressively finer scales. At everyscale the same patterns can be found.Random fractals generated by a simi-lar process, but with added stochasticvariations, produce remarkably realis-tic simulations of geologic variability.

These kriging methods yield smooth inter-polations, but do not describe small-scaleheterogeneity. In a process called stochasticmodeling that superimposes correlatednoise onto smooth interpolations, more real-istic pictures emerge. A probability distribu-tion determines how this noise is generated.A number of pictures, or realizations, willusually be created, each with different noisesampled from the same distribution (top,right). By analyzing many realizations, theextent to which geologic uncertainty affectsreservoir performance can be studied (mid-dle, right).

An example is sequential Gaussian simu-lation (SGS). The data are constructed gridblock by grid block. At the first selectedblock, the smooth interpolated value is cal-culated by kriging using the available mea-sured data. The interpolated value and itsvariance, also calculated by kriging, definea Gaussian distribution function from whicha noise value is randomly drawn and addedto the interpolated value. For the next point,selected at random, the process is repeated,using as a base this newly derived datapoint together with the measured data. Asthe grid blocks are filled, all previously cal-culated values contribute to computing thenext in the sequence.

The technique depends on the reservoirproperty being normally distributed, so aproperty like permeability, which has askewed distribution, requires transformationto normality. The reverse process has to beperformed once all the grid blocks are filled.

An increasingly popular method for gen-erating noise for stochastic modeling usesthe concept of fractals—a statistical tech-nique that produces remarkably realisticimitations of nature. Fractal objects exhibitsimilar variations at all scales of observa-tion. Every attempt to divide fractal objectsinto smaller regions results in ever moresimilarly-structured detail (right).9 This sim-plifies stochastic modeling. The variogram isdefined from a single number—the fractaldimension, calculated from measured datain the reservoir or outcrops. And becausefractals are self-similar, the variance of thenoise need be determined only at a singlescale. Fractal modeling has been used topredict the production performance of sev-

0

20

40

60

80

0

20

40

60

80

Realization A

Realization B

Dep

th, f

t

Above 100 md

100–0 md

10–1 md

1.0–0.1 md

Below 0.1 md

0 200 400 500 600 800 1000 1200 1400Distance, ft

nTwo stochasticrealizations of reser-voir permeabilityshowing how differ-ent noise added tothe same krigedinterpolation givesquite different real-izations. In both thehorizontal and ver-tical directions,realization A hasrelatively high con-tinuity whereasrealization B hasrelatively low conti-nuity. Continuity isgauged by measur-ing the size of groupsof contiguous blockswith permeability of100 millidarcies(md) or more.

30 Oilfield Review

After Fogg GE, Lucia FJ and Senger RK: “Stochastic Simulation of Interwell-Scale Het-erogeneity for Improved Prediction of Sweep Efficiency in a Carbonate Reservoir,” inLake LW, Carroll HB Jr and Wesson TC (eds): Reservoir Characterization II. San Diego,California, USA: Academic Press Inc. (1991): 355-381.

Distance

Measured data

Simulation 1Kriged interpolationActual dataSimulation 2

Valu

e

nImitating reality’s variability. The kriging technique yields asmooth interpolation, ignoring nature’s true small-scale het-erogeneity. By adding noise to a kriged interpolation, stochas-tic modeling produces more lifelike realizations.

eral large fields. Recent studies have alsoapplied fractal models to investigate the per-formance of miscible gas floods.10

Discrete data require a special interpola-tion technique called indicator kriging. Thiscan be combined with stochastic modelingto form a process called sequential indicatorsimulation (SIS). Take a model where eithersand or shale must be chosen—there is nomiddle ground. At the same time, a randomelement must be introduced to obtain multi-ple realizations. Consider building a modelwhere sand is 1 and shale 0. Indicator krig-ing will assign every grid block some valuebetween 0 and 1—giving an indication ofthe likely lithology. For example, a value of0.7 indicates a 70% chance of sand. In thestochastic stage, this percentage is then usedto weight the random choice of sand orshale for the grid block.11

Object-Based ModelingIn object-based modeling, a picture of thereservoir model is built from genericunits—for example, sand bodies or shalebarriers, each unit having uniform internalproperties. These models are built by start-ing with a uniform background matrix andadding units with a contrasting property:either a high permeability background towhich shale barriers are added, or a lowpermeability background to which sandbodies are added.

When starting with a high-permeabilitybackground, the vertical distribution of

nStochastic shales in object-based modeling. The location ofeach stochastic shale is random and independent of othershales. The units are drawn randomly from a size distributionfunction and placed at random locations until the required shaledensity has been achieved.

31January 1992

9. Hewett TA: “Fractal Distributions of Reservoir Het-erogeneity and Their Influence on Fluid Transport,”paper SPE 15386, presented at the 61st SPE AnnualTechnical Conference and Exhibition, New Orleans,Louisiana, USA, October 5-8, 1986.Hewett TA and Behrens RA: “Conditional Simulationof Reservoir Heterogeneity With Fractals,” paper SPE18326, presented at the 63rd SPE Annual TechnicalConference and Exhibition, Houston, Texas, USA,October 2-5, 1988.“Fractals and Rocks,” The Technical Review 36, no.1 (January 1988): 32-36.Crane SD and Tubman KM: “Reservoir Variabilityand Modeling With Fractals,” paper SPE 20606, pre-sented at the 65th SPE Annual Technical Conferenceand Exhibition, New Orleans, Louisiana, USA,September 23-26, 1990.

10. Perez G and Chopra AK: “Evaluation of Fractal Mod-els to Describe Reservoir Heterogeneity and Perfor-mance,” paper SPE 22694, presented at the 66th SPEAnnual Technical Conference and Exhibition, Dal-las, Texas, USA, October 6-9, 1991.Payne DV, Edwards KA and Emanuel AS: “Examplesof Reservoir Simulation Studies Utilizing Geostatisti-cal Models of Reservoir Heterogeneity,” in Lake LW,Carroll HB Jr and Wesson TC (eds): Reservoir Char-acterization II. San Diego, California, USA: Aca-demic Press Inc. (1991): 497-523.

11. Journel AG and Alabert FG: “Focusing on SpatialConnectivity of Extreme-Valued Attributes: Stochas-tic Indicator Models of Reservoir Heterogeneities,”paper SPE 18324, presented at the 63rd SPE AnnualTechnical Conference and Exhibition, Houston,Texas, USA, October 2-5, 1988.

12. Haldorsen HH and Chang DM: “Notes on StochasticShales; From Outcrop to Simulation Model,” in LakeLW and Carroll HB Jr (eds): Reservoir Characteriza-tion. Orlando, Florida, USA: Academic Press Inc.(1986): 445-486.

13. Geehan GW, Lawton TF, Sakurai S, Klob H, CliftonTR, Inman KF and Nitzberg KE: “Geologic Predic-tion of Shale Continuity, Prudhoe Bay Field,” in LakeLW and Carroll HB Jr (eds): Reservoir Characteriza-tion. Orlando, Florida, USA: Academic Press Inc.(1986): 63-82.

shales may be inferred from cores and logs,particularly gamma ray logs. But unless thewell spacing is extremely dense, nothing isrevealed about the shales’ lateral dimen-sions. Some shales correlate from well towell, but most do not, and their lateral extentmust be generated statistically.

These stochastic shales are drawn at ran-dom from a size distribution and placedrandomly until the precalculated shale den-sity has been achieved. Shale density is esti-mated from the cumulative feet of shalemeasured in wells compared with the grosspay, and then assumed to represent thereservoir volume under study. To ensure thatthe realizations honor the deterministicdata, shales observed in wells are placed inthe model first, their lateral extent still deter-mined randomly. Subsequent, randomlyplaced shales that intersect a well arerejected (below).12

To generate these models, the key statis-tics specified by the geologist are shalelength and width, together with some guid-ance as to their interdependence. Tradition-ally, the geologist describes shales qualita-tively as wide, extensive or lenticular—toovague for building realistic stochastic mod-els. Instead, ranges of length and width arerequired, along with gross pay thickness andaverage shale density. The depositionalenvironment has great bearing on these val-ues and the primary sources of this informa-tion are outcrop studies.13

nBuilding an object-based model using SIRCH, a software pack-age developed at BP Exploration. In this model, two differenttypes of fluvial channel belt have been generated in a three-stage process. First, one type of channel belt is generated (yel-low). Then a second set (purple) is added to the first. This secondtype has associated overbank deposits (red)—which are thinner,poorer quality sands that nevertheless improve connectivity.Finally, the combined channel belts are faulted.

Each channel belt is created using about 50 properties. Someare constants, some are picked at random from input distribu-tions and some are calculated from other properties. Propertiesthat determine channel shape include width, thickness, reachlength and angle, azimuth and depth. Flow properties, like per-meability and porosity, are assigned either by lithology or byusing a random sample. All well data are honored. Net-to-grossratio is used to control the quantity of sand generated.

14. Channels are the depositing rivers; channel belts arethe resulting sandstone bodies.

In theory, a similar method could beapplied to generate sandstone bodies in alow-permeability background. However, BPResearch, Sunbury-on-Thames, England, hasdeveloped SIRCH—system for integratedreservoir characterization—to generate real-istic reservoir heterogeneities using a libraryof depositional shapes. Each shape has a setof attributes defined by sedimentology andbased on observations of depositional sys-tems. These attributes determine the sizeand position in the reservoir model of eachshape and its relations with other shapes.

For example, channel belts are con-structed as continuous links of a specifiedshape.14 As a belt is constructed, input val-ues for the attributes are repeatedly sampledfrom the library and the generated belts maybe conditioned to honor well data. Themodel can also be faulted, structured toaccount for regional dip and clipped backusing the mapped top and bottom of thereservoir to give it a volume indicated bystructural maps (left).

SIRCH is still stochastic, so several real-izations are required to estimate reservoircharacteristics. The realizations can be usedto estimate the volume of sandstone withina specified interval, reservoir connectivityand hydrocarbons in place for a givenradius around a well. This process hasproved valuable in indicating sensitivity ofresults to the input parameters. Influentialinputs can then be targeted for extra study.

Individually, all these techniques paint aportion of the total picture. In most cases,gaining a full view of the reservoir requires ahybrid (see “The HERESIM Approach toReservoir Characterization,” next page).Object-based, discrete modeling may beused to describe the large-scale hetero-geneities in the reservoir—the sedimento-logical units—while different continuousmodels may describe the spatial variationsof properties within each unit. For example,for fluvial environments, BP Research isplanning to use SIRCH to generate chan-nels, SIS for the facies within them and SGSfor permeability within the facies.

(continued on page 36)

32 Oilfield Review

The HERESIM Approach to Reservoir Characterization

An example of a software package that combines

traditional geologic analysis with geostatistical

techniques is HERESIM, developed jointly by the

Institute Français du Pétrole, Rueil-Malmaison,

France, and the Centre de Géostatistiques, Ecole

Nationale Supérieure des Mines de Paris, Foun-

tainebleau, France. This allows sedimentological

and geostatistical structural analysis to be car-

ried out on interactive workstations, followed by

geostatistical conditional simulations using

small-scale grid blocks. Each simulation creates

an image of the reservoir geology, consistent

with well data. Petrophysical data, like perme-

ability and porosity, are then mapped between

wells using deterministic or stochastic

algorithms. Finally, the petrophysical information

is scaled up for fluid-flow simulation (right).1

First, a sedimentological study is carried out to

subdivide the area under investigation, which can

include formations bordering the reservoir as

well as the reservoir itself. The largest divisions

are depositional units that include sets of geneti-

cally linked strata, bounded by major sedimen-

tary discontinuities or sudden variations in the

sedimentological environment. Distinction of

units is usually achieved by correlating features

between wells. The next step is to identify litho-

facies within each depositional unit, collect sta-

tistical information about them at the wells and

then generate stochastic realizations of lithofa-

cies between wells.

Statistical information obtained from well data,

seismic interpretation and geologic knowledge are:

• Proportion curves—the percentage occurrence

of each lithofacies in a depositional unit (right)

• Experimental variograms—to quantify the spa-

tial continuity of each lithofacies in the reser-

voir (far right).2

With this information and a form of indicator krig-

ing employing a Gaussian random function,

nHorizontal proportion curve. Proportioncurves provide information on the relativefrequency of each lithofacies in a deposi-tional unit. Here, four lithofacies have beenproportioned using data from 20 wells.

33

nOrganization and use of HERESIM, a geostatistical reservoircharacterization package.

Freq

uenc

y

Relative distance

100%

0-1000 0 1000 2000

Lithofacies1 2 3 4

▲ ▲▲▲▲▲▲▲ ▲▲▲ ▲▲▲▲ ▲ ▲ ▲Wells▲▲

Sill

Range

Variogram

Distance

Valu

e

nVariograms quantify the spatial continu-ity of each lithofacies in the reservoir. Thesill equals the variance of data; while therange indicates the separation beyondwhich two points are uncorrelated.

1. Eschard R, Doligez B, Rahon D, Ravenne C and Leloch G:“A New Approach for Reservoirs Description and Simula-tion Using Geostatistical Methods,” presented atAdvances in Reservoir Technology, Characterization,Modelling & Management, organized by the PetroleumScience and Technology Institute, Edinburgh, Scotland,February 21-22, 1991.

2. Matheron G, Beucher H, de Fouquet C, Galli A, Guérillot Dand Ravenne C: “Conditional Simulation of the Geometryof Fluvio-Deltaic Reservoirs,” paper SPE 16753, pre-sented at the 62nd SPE Annual Technical Conference andExhibition, Dallas, Texas, USA, September 27-30, 1987.

January 1992

Geostatisticalanalysis• proportioncurves

• variograms

Simulation of lithofacies

Simulation ofpetrophysicaldata

Scaling up ofpetrophysicaldata

Connectivity studiesOptimization of well spacingSwept volume estimation

Fluid-flow modelingEnhanced oil recoverystudies

Stratigraphic studies

Reinterpretation of the results interms of sequential stratigraphy

numerous realizations of lithofacies distribution

can be readily obtained for each unit. However,

the algorithm works only in rectangular formats,

so it is necessary to first geometrically transform

each unit into a rectangle. After stochastic real-

izations have been generated, the real shape of

the unit is restored. HERESIM offers two types of

transform, depending on whether the prevailing

environment since deposition has been domi-

nated by erosion or differential subsidence—

nA step in HERESIM processing: creating rectangles to allow generation of stochastic realizations. Before the stochastic realizationcan be generated, the depositional units have to be transformed into rectangles. HERESIM has two ways of doing this. One assumesthat the unit has been eroded and that all the correlation lines within the unit are parallel. To create a rectangle, these correlationsare extrapolated across the eroded sections of the unit. Then the rest of the structure can be stochastically generated and the result-ing realization re-eroded to recreate the original structure. The alternative technique assumes that differential subsidence hasoccurred, causing the correlation lines to diverge. To create a rectangle, the effects of the subsidence are reversed. Then the stochas-tic realization is created and resubsided.

34 Oilfield Review

Fill in eroded strata to create rectangle, then perform stochasticrealization.

Erode stochastic realization to recreate structure.

Erosion transformation

Reverse effect of subsidence to create rectangle,then performstochastic realization.

Subside stochastic realization to recreate structure.

Subsidence transformation

3. Guérillot D, Rudkiewicz JL, Ravenne C, Renard G andGalli A: “An Integrated Model for Computer Aided Reser-voir Description: From Outcrop Study to Fluid Flow Simu-lations,” presented at the 5th European Symposium onImproved Oil Recovery, Budapest, Hungary, April 25-27,1989. Reprinted in Revue de l’Institut Français du Pétrole45, no. 1 (January-February 1990): 71-77.

nStochastic realization generated using HERESIM.

although some forms of fracturing can also be

accommodated (previous page).

Having selected the most likely realizations of

lithofacies, geologists and reservoir engineers

assign porosity and permeability values.

HERESIM assumes that petrophysical data are

strongly related to lithofacies. For each lithofa-

cies, either constant data values can be assigned

or a Monte Carlo-type distribution employed.

The geologic model describes a reservoir at a

decametric scale (on average 50-m [165 ft] square

or smaller with a thickness of 0.5 m [1.5 ft]). As a

result, simulated grids are huge (millions of

cells), too big for fluid-flow simulation (above).

Scaling up in HERESIM is performed using a fast

algorithm that calculates absolute permeabili-

January 1992

ties.3 First, average permeability is estimated

from the root of the product of the arithmetic mean

and harmonic mean of the grid-block permeabili-

ties. The quality of this value can be judged by

computing the log of the difference between the

two means. The larger the difference, the less

satisfactory the simple average. Grid blocks that

fail this test are subjected to a more accurate and

time-consuming fluid-flow simulation.

35

Rescaling the Geologic Model For Fluid-Flow SimulationGeologic modeling can describe a reservoirwith millions of grid blocks. For the com-puter, this mainly presents problems of stor-age. But the next major operation is fluid-flow simulation, which involves complexnumerical calculation—a monumentallylarge task given a million grid blocks (see“Simulating Fluid Flow,” page 38).

Today, fluid-flow simulators can copewith up to about 50,000 grid blocks—the

36

nModeling the North Sea Rannoch forma-tion. This reservoir sandstone is character-ized by hummocky cross-stratified bed-forms. The Department of PetroleumEngineering at Heriot-Watt Universitydetermined the flow performance of thisdepositional structure using the two deter-ministic simulations shown here. Thesetwo images represent a slice of rock mea-suring 2.4 m [7.9 ft] and 14.4 m [47.2 ft].

In the first (top), saturation is at initialconditions. Low permeability, rippledcrests have the lowest oil saturations(green). The second (bottom) shows satu-ration distribution after 0.2 pore volumesof water have been injected at a lowadvance rate of 0.25 m/d. Saturation isreduced most rapidly in the low perme-ability, rippled crests (dark blue).

Total pore volume injected

Total pore volume injected

0Rec

over

y of

orig

inal

oil-

in-p

lace

0

40%

80%

100%50%

Horizontal Flood

Detailed geologic modelArithmetic average and rock curves

Vertical Flood

00 50% 100%

Harmonic average androck curves

Detailed geologic model

Rec

over

y of

orig

inal

oil-

in-p

lace

40%

80%

nRannoch formation’s flooding character-istics. Using densely-sampled permeabil-ity data, capillary pressures measuredfrom appropriate rock types and knowl-edge of the depositional structure to con-struct a detailed geologic model, theDepartment of Petroleum Engineering atHeriot-Watt University simulated both hor-izontal and vertical flooding. The resultsusing the detailed geologic model are sig-nificantly different from those derivedusing traditional techniques employingaverage permeabilities and rock curves—particularly for the vertical flood.

greater the number of grid blocks, the moreexpensive the fluid-flow simulation is incomputation. So before the simulation isrun, the geologic model, with its high-reso-lution petrophysical data must be enshrinedin a smaller number of larger grid blocks—aprocess called scaling up. Reorganizing somuch data may introduce error, maskingheterogeneities in the geologic model thatdirectly affect simulated production.

The key to achieving scale up is derivinga single set of reservoir properties—calledpseudos—for the larger blocks so that thefluid-flow simulation after scale up is asclose as possible to what it would havebeen with the small-scale data.

Over the years, a variety of methods hasbeen developed to do this. These includecalculations based on simple averagingprocedures like the oft-quoted Kyte andBerry method and more complex simula-tions of parts of the reservoir using small-scale grid blocks, measuring about 1 msquare and 0.5 m thick.15

The impact of small-scale effects on two-phase flow performance is revealed in astudy of the North Sea Rannoch formation,carried out at Heriot-Watt University, Edin-burgh, Scotland. Rannoch subfacies arestrongly laminated and rippled. The studyutilized newly acquired minipermeameterdata at a millimetric scale, much denserthan that conventionally obtained fromcore analysis.

These data were combined with rock cap-illary pressure curves to define pseudos foreach lamina subfacies. A geologic model forhummocky cross-stratification combined thesubfacies at the bedform scale (left).16 Thesimulated flow performance of this modelwas then compared with simpler flow simu-lations calculated using arithmetic and har-monic average permeabilities for horizontaland vertical directions, respectively.

The flow behavior of the detailed modelwas anisotropic, whereas the simplerapproach yielded more isotropic behaviorand significantly higher recoveries for verti-cal displacement (right). So, small-scaleheterogeneity affects reservoir producibility,and in many cases, variations in capillarypressure with rock type and small-scalesedimentary structure cannot be ignored.

A predominant theme in scaling up is theidentification of a number of discrete scalesof heterogeneity. To preserve the effects of

heterogeneity at all scales, a series of scale-up operations can be performed, each deal-ing with heterogeneity larger than the previ-ous operation. At each stage in the process,the size of the scale up should be as large aspossible without introducing the next levelof heterogeneity. If a reservoir is divided intodepositional units, lithofacies, beds andsmall-scale heterogeneity within a bed, thescale-up process will have four stages. Forexample, lithofacies become the basic build-ing blocks at the depositional unit scale.17

Oilfield Review

Fluid density (ρ)

0

0

0

the range of REVρPorosity (φ)

the range of REVφPermeability (k )

the range of REVk

the range of REV for all properties

Volume

Microscopic Macroscopic Megascopic

Nor

mal

ized

ave

rage

val

ue

nHow sample volume affectsproperty averages. At small sam-ple volumes, property averagesfluctuate because of microscopicheterogeneity. As sample volumeis increased, heterogeneities willaverage out and properties stabi-lize—this gives the representativeelementary volume (REV). This sta-bility continues until larger-scaleheterogeneity begins to takeeffect. Different properties havedifferent volumes where hetero-geneity is stable. The ideal samplevolume should fall within rangesof stability for all key properties.

Heterogeneity can be assessed by measur-ing the average value of a reservoir propertywithin a growing volume. When the volumeis very small, the average value will fluctuatebecause of microscopic heterogeneities. Butas the averaging volume increases, more ofthese heterogeneities will be encompassedand the average value will stabilize. Furthergrowth may result in capturing larger-scaleheterogeneity and the value will again fluc-tuate (right).18 Since this behavior variesamong properties, the size of the smallestgrid block should be selected so that all keyproperties are stable. 19

No matter how many scales are used,existing methods of calculating pseudoshave their drawbacks. The elementarymethods, using averaged properties, arequick but not accurate enough. The simula-tion methods are computationally expen-sive. And if sensitivity studies are requiredusing a number of stochastic realizations,the time and cost of scaling up can be pro-hibitive. Recently, BP Research developed anew algorithm—real-space renormaliza-tion—that promises speedy yet accurate cal-culation of pseudos.

The method starts with a group of eightgrid blocks (2×2×2) with permeabilities dis-tributed to represent the original data. A newrelative permeability is calculated andadopted for a new single block the size ofthe eight original blocks—it becomes thestarting point for the next renormalizationwhen the process is repeated with sevenother similarly generated blocks. The newblock’s effective relative permeability is cal-culated using a semianalytical technique thatdepends on an analogy between Darcy’s lawand Ohm’s law—block permeabilities aremodeled by an equivalent resistor network.Through a series of transformations, theresistor network is reduced to a single resis-

16. Corbett PWM and Jensen JL: “An Application ofSmall Scale Permeability Measurements—Predictionof Flow Performance in a Rannoch Facies, LowerBrent Group, North Sea,” presented at Miniperme-ametry in Reservoir Studies, organized by thePetroleum Science and Technology Institute, Edin-burgh, Scotland, June 27, 1991.

17. Lasseter TJ, Waggoner JR and Lake LW: “ReservoirHeterogeneities and Their Influence on UltimateRecovery,” in Lake LW and Carroll HB Jr (eds):Reservoir Characterization. Orlando, Florida, USA:Academic Press Inc. (1986): 545-559.

18. Any volume for which the property is stable is calleda representative elementary volume (REV).

19. Haldorsen HH: “Simulator Parameter Assignmentand the Problem of Scale in Reservoir Engineering,”in Lake LW and Carroll HB Jr (eds): Reservoir Char-acterization. Orlando, Florida, USA: Academic PressInc. (1986): 293-340.Kossack CA, Aasen JO and Opdal ST: “Scaling UpHeterogeneities With Pseudofunctions,” SPE Forma-tion Evaluation 5 (September 1990): 226-232.Norris RJ and Lewis JJM: “The Geological Modelingof Effective Permeability in Complex HeterolithicFacies,” paper SPE 22692, presented at the 66th SPEAnnual Technical Conference and Exhibition, Dal-las, Texas, USA, October 6-9, 1991.

37January 1992

15. Kyte JR and Berry DW: “New Pseudo Functions toControl Numerical Dispersion,” Society ofPetroleum Engineers Journal 15 (August 1975): 269-276.Tompang R and Kelkar BG: “Prediction of Water-flood Performance in Stratified Reservoirs,” paperSPE 17289, presented at the SPE Permian Basin Oiland Gas Recovery Conference, Midland, Texas,USA, March 10-11, 1988.

1. Mattax CC and Dalton RL: “Reservoir Simulation,” Jour-nal of Petroleum Technology 42 (June 1990): 692-695.Breitenbach EA: “Reservoir Simulation: State of the Art,”Journal of Petroleum Technology 43 (September 1991):1033-1036.

2. Cheshire IM and Pollard RK: “Advanced Numerical Tech-niques for Reservoir Simulation and Their Use on Vectorand Parallel Processors,” in Edwards SF and King PR(eds): Mathematics in Oil Production. Oxford SciencePublications, Oxford, England: Clarendon Press (1988):253-268.

3. Por GJ, Boerrigter P, Maas JG and de Vries A: “A Frac-tured Reservoir Simulator Capable of Modeling Block-Block Interaction,” paper SPE 19807, presented at the64th SPE Annual Technical Conference and Exhibition,San Antonio, Texas, USA, October 8-11, 1990.

Simulating Fluid Flow

Today, most simulators solve fluid-flow problems

by segmenting a portion of the reservoir into a

series of grid blocks—either two- or three-dimen-

sional. The fluid phases in each block are mod-

eled with finite difference equations similar to

the conventional volumetric material balance

equations. Darcy’s law is then used to describe

fluid flow between grid blocks.

Early efforts in reservoir simulation, in the late

1940s, began with simple material balance cal-

culations. Throughout the 1960s, black oil simu-

lations dominated the scene. These describe

nonvolatile oil as a single component containing

gas in solution as a second component. There are

three compressible, immiscible phases: gas,

water and oil. Recovery methods simulated

included pressure depletion and some forms of

pressure maintenance.1

The advent of enhanced recovery techniques

such as chemical flooding, steamflooding and in-

situ combustion, ushered in more sophisticated

simulators. Most describe the hydrocarbon in

terms of a number of components and solve ther-

modynamic equilibrium equations to determine

the distribution of components between the liquid

and vapor phases in the reservoir. 2

A common strategy for fluid-flow simulation is

to use history matching to adjust the geologic

38

model. Adjustments should be consistent with

measured data and improve upon postulated vari-

ations in reservoir properties that cannot be or

have not been measured. The results of the simu-

lation can be extremely sensitive to variations in

the reservoir description. Models for reservoir

management are usually routinely updated as

new production data or reservoir information are

acquired. This allows for reexamination of cur-

rent and future production scenarios. But history

matching always results in nonunique combina-

tions of variables. And it could be said that the only

fully simulated reservoir is a fully depleted one.

Simulator technology is by no means mature.

Some current developments center on improving

the ability to cope with different production sce-

narios, for example, naturally fractured forma-

tions and horizontal wells.

During production by gravity drainage in natu-

rally fractured reservoirs, oil drains from matrix

blocks into the fracture system. From there it can

pass either into other blocks below or along frac-

tures. A fractured reservoir is modeled using two

interacting continua, one for the matrix and one

for the fractures—the so-called dual porosity

concept. The matrix contains most of the oil and

the fractures most of the conductivity. So trans-

missibility between matrix blocks is usually

ignored. But to correctly model ultimate recovery,

the capillary contacts between matrix blocks

should be considered.

Work at Koninklijke/Shell Exploratie en Produk-

tie Laboratorium, Rijswijk, The Netherlands, has

shown that a vertical stack of oil-saturated matrix

blocks surrounded by gas does not drain indepen-

dently.3 Oil from one block passing into the frac-

ture system will be absorbed by the underlying

block, slowing production. Therefore, a flow sim-

ulation that ignores this effect will overestimate

initial production rates. To combat this, Shell has

devised a simulator that allows flow in all direc-

tions between neighboring blocks.

Fluid-flow simulation for horizontal wells is

currently being approached in at least two ways.

The productivity of a horizontal well can be

approximated, using numerical solutions of sim-

plified equations, and the approximation used in

a conventional reservoir simulator. Alternatively,

for single wells in which short-term performance

is being studied, fine-scale, black oil simulations

have been used. In this approach, pseudos can

be derived for a coarser, field-sized simulation.

Oilfield Review

tance, equivalent to the new block’s effectiverelative permeability.

The effect of each stage of renormaliza-tion is to produce larger blocks whose per-meability approaches that of the whole.Results of renormalization have been com-pared with direct numerical simulation andthe maximum error was found to be only7%.20 The main difference was in computa-tional effort. Renormalization speeds are upto two orders of magnitude faster than thoseof direct numerical simulation. The largestproblem yet tackled totalled 540 milliongrid blocks.

Renormalization has drawbacks: it is diffi-cult to represent contorted flow paths withsmall cells and the estimation of effectivepermeability suffers. This is seen in veryshaly reservoirs, where there are high con-trasts between neighboring permeabilities.

This problem may be resolved by anothermethod that short-circuits the need for scaleup and requires a different kind of reservoircharacterization. This is the streamtubeapproach, championed by Chevron OilfieldResearch Co., La Habra, California, USA,among others.21 The plan view of the reser-voir is simulated as a network of tubes thatcarry the flow between injection and pro-duction wells (right). The technique takesinto account well placement, areal hetero-geneity and the relative flow rates of wells.The key to the technique’s success is a pre-liminary flow simulation on two-dimen-sional vertical slices between wells. Theobserved scaling behavior is then mappedonto individual streamtubes and their contri-butions summed to get a three-dimensionalprediction of reservoir performance.

Because of the relatively greater arith-metic complexity of fluid-flow simulationcompared with model building, computers

may always be able to generate more com-plex geologic models than can be simu-lated. Scaling up will therefore remain a keyelement in reservoir characterization.

The fluid-flow simulation bottleneck willhave to be addressed, either by developingnew, more rapid processing hardware likeparallel processing22 or by smarter algo-rithms, or probably both. Then, the capabil-ity for cost-effective and repeated stochasticsimulation will be realized. The impact ofthis capability on reservoir management,and ultimately on improved recovery,remains to be felt. —CF

39January 1992

nCutting simulation time using streamtubes, a network of conduits conveying fluidfrom injection to production wells. This field-scale model incorporates five solvent injec-tion wells (pink) and seven producers (yellow). Using streamtubes, simulating the injec-tion of 2.5 pore volumes of solvent took just 30 seconds on a MicroVAX 3200 worksta-tion. Using conventional fluid-flow simulation for a coarse grid and a Cray XMP-48, thecalculation required 630 seconds. The two calculations predicted similar solvent break-through. (After Hewett and Behrens, reference 21.)

20. King PR: “The Use of Renormalization for Calculat-ing Effective Permeability,” Transport in PorousMedia 4 (July 1989): 37-58.King PR, Muggeridge AH and Price WG: “Renor-malization Calculations of Immiscible Flow,” Sub-mitted to Transport in Porous Media, August 1991.

21. Emanuel AS, Alameda GK, Behrens, RA and HewettTA: “Reservoir Performance Prediction MethodsBased on Fractal Geostatistics,” SPE Reservoir Engi-neering 4 (August 1989): 311-318.Hewett TA and Behrens RA: “Scaling Laws in Reser-voir Simulation and Their Use in Hybrid Finite Dif-ference/Streamtube Approach to Simulating the theEffects of Permeability Heterogeneity,” in Lake LW,Carroll HB Jr and Wesson TC (eds): Reservoir Char-acterization II. San Diego, California, USA: Aca-demic Press Inc. (1991): 402-441.

22. Mayer DF: “Application of Reservoir SimulationModels to a New Parallel Computing System,” paperSPE 19121, presented at the SPE Petroleum Com-puter Conference, San Antonio, Texas, USA, June26-28, 1990.