SPE9234 SPE Societu of PetroIelm EngIneers of AIME ASYSTEMATICAPPROACHTOTHERELATIVEPERMEABILITYPROBLEM INRESERVOIRSIMULATION byNelsonN.Molina,INTERCOMPResourceDevelopmentandEng.,Inc. @Copyright'1980, American Institute of Mining,Metallurgical, andPetroleum Engineers,Inc. This paper was presented at the 55th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, held in Dallas, Texas, September 21-24, 1980. The material is subject to correction by the author.Permission to copy is restricted to an abstract of not more than 300 words. Write: 6200 N.Central Expwy., Dallas, Texas 75206. ABSTRACT Directexperimentaldeterminationofthree-phase relativepermeability dataisextremelydifficult.Forthis reason,numericalmodelsdescribingmultiphaseflowin petroleumreservoirsdeterminethree-phaserelativeper-meabilityfunctionsusingtwosetsofthemoreeasily measuredtwo-phasedata,i.e.,water-oilandgas-oildis-placementsystems.However,becauseofthewiderange ofscatteringofthelaboratory-measuredtwo-phaserela-tivepermeabilitydata,itisusuallyverydifficultto determinearepresentativeaveragefunctionwhetherby reservoirlayers(differentlithologies)orreservoirregions (severalfaciesorreservoirunits).Itfollowsthatour greatestneedtodayintheareaof reservoirsimulation isa quick,relativelyinexpensiveproceduretomatchreservoir performancedatabasedonadjustmentsofrepresentative relativepermeabilityfunctionsassignedonapermeability distributionbasis.A systematicapproachtothisproblem isdiscussed inthis study. Fluidsaturation andrelativepermeability values cor-respondingtothebeginning,intersection,andendpointsof thedisplacementprocesswerecorrelatedasalogarithmic functionof airpermeabilitytodetermineinitjalsaturation functionsforeachrocktype(regionswithidenticalcapil-larypressureandrelativepermeabilityproperties).This procedureprovedtobeusefulforwater-oil and gas-oil sys-temsofcarbonateandsandstonereservoirs.Theso determinedsaturationfunctionswerecurvefittedusing exponentialequationsasstartingpointsduringthehistory matchphase.Subsequently,theshapesoftherelative permeabilitycurveswereadjustedbychangingtheex-ponentsofthefittingequationsinordertomatchWOR and/or GORproductionhistory.Thisprocedureassumesno massdiffusionandisothermalreservoirconditions.This approachhasbeensuccessfullyappliedtomatchreservoir productionhistoryof differentoilfieldsoftheworld,even whenadversemobilityratioconditionswerepresentin stratified reservoirs. Thefollowingareincluded:typicalfluidsaturation andrelativepermeabilitycorrelationsasalogarithmic functionofairpermeability,relativepermeabilityfitting References and illustrations at end of paper. equations,andapplicationstomatchtheobservedperfor-manceof selected case histories. INTRODUCTION Theinterpretationoflaboratory-determinedrelative permeabilitycharacteristicstoobtainanaveragerelative permeabilitycurveapplyingtoaparticularformationpre-sentsaconsiderableproblemforthepracticingreservoir engineer.Therearemanypossiblemethodsofaveraging relativepermeabilitydata.However,ofmoreimportance thantheaveragingmethodutilizedisthequestionof determiningjusthowwellthecurvesusedactuallyrepre-sent the formationunderconsideration. Theidealrelativepermeabilitytestshouldinvolve fresh,preserved,wholecoresamplestakenusingnative crudeoilasdrillingfluidandselectedpressure-coring techniques.Furthermore,thetestshouldbeperformed at reservoirconditionsusingliveoilcontainingdissolvedgas, andthedirectionofthesaturationchangeinthelabora-tory-determinedrelativepermeabilitypropertiesshould correspondtothatinthereservoir.Inactuality,these ideal samples and conditionsare seldom,if ever,obtained. Usually,mostoftheexperimentalworkonrelative per me abilitiesinvolvessmallsamplesofcoreorplugs obtainedfromreservoirrockswhoseoriginalwettability preferencehasalreadybeenalteredbythedrillingfluids, andasuitablelaboratoryprocedurethateliminatesinlet andoutletendeffectswhichmanifestthemselvesinsatu-rationgradientsmustbeselected.Moreover,thevery processofcuttingthecoresandbringingthemtothe surfacechangestheoriginalfluidsaturationsbeforethey canbemeasured.Properhandling,transporting,and storage ofthecorestobeanalyzed are necessaryto obtain goodrelativepermeabilitydata.Inwater-wetrocksthe interstitialwatersaturationatthestart ofthetest should closelyapproximatethereservoirconnatewatersatura-tion.. Asarule,toavoidcorepluggingproblems,refined oilratherthanreservoircrudeisusedinrelativepermea-bility tests. ItshOUldbenotedthatinherentsamplingproblems suchas:arealcoverage,corerecovery,andselectionof samplestocoverreservoirporosity,permeability,and lithologyrangeswerenotmentionedinthepreceding paragraphs.Ifsignificantrockpropertyvariationsare identified inareservoir, it can be subdivided intomeaning-ful rock types.Sinceflowproperties are afunctionof pore geometry,eachlithologicalunitusuallyhaspeculiarrela-tivepermeabilitycharacteristics.Unfortunately,relative permeabilitydataforeachunitarenotalwaysavailable; consequently,itisverydifficultornearlyimpossibleto assignrelativepermeabilitypropertiesonareservoirunit basisunlessareservoircorrelatingparameterisdeter-mined.Forinstance,reservoirstratificationandmobility ratiohaveprofoundeffectsonfractionalflowcharac-teristicsof awaterflood andshouldbeaccountedforwhen determiningtherelativepermeabilitycurvestobeusedin asimulationstudy.Similarly,the gas-oil relative permea-bilitycharacteristics can vary greatlywithdifferentreser-voir rock types. Ontheotherhand,theuseofcoredatainreservoir engineeringisalwaysinterpretativesincedatainthe interwellareaareincompleteandmustbeeitherinter-polatedorextrapolated.Thus,thedistributionofthe formationpropertiesinthereservoirusuallyhasalarge degree of uncertainty. Therelativepermeabilityproblemisevenmore com-plex,takingintoconsiderationthenon-systematicerrors associated withthe laboratorymeasurements,the inconsis-tenciesof laboratoryprocedureswhentwoormoresetsof dataareavailable,and,moreimportant,thefactthatthe laboratory-determinedcoreflowpropertiesareusually basedontwo-phasedisplacementtestsratherthanthree-phasesystems. Asaresultoftheabovementionedfactors,the experimentallydetermined,two-phaserelativepermea-bilitydatausuallypresentawiderangeofscattering. Consequently,it isverydifficulttodeterminearepresen-tativeaveragefunctiononanybasis,withareservoirunit basis beingthemost difficult. Tothespeculativeworkerwhotriestoarriveatthe solutionbyuseofimaginationandintuitionandthento testhishypothesis by experiment or observation (i.e.,trial-and-errorproceduresduringthehistorymatchphaseofa reservoirsimulationstudy),therelativepermeabilitydata may appearenormouslyoverwhelming,apparently inconsis-tent, difficulttocorrelateto anyreservoirparameter,and sometimesirrelevant,e.g.,data offracturedcores.Trial-and-errorprocedurescouldbeveryexpensiveandsome-timesfrustratingexperimentsduringthehistorymatch process.Furthermore,they couldprovide results whichare "needed"butaredifficulttointerpretandreconcilewith themeasuredlaboratorydataandreservoirrockproper-ties. Sincereservoirsimulationhasemergedasapowerful managementtool,resultsproducedbythismethodcould havedeepimplicationsinthefuturelifeofthereservoir. Confidenceinthepredictionsoffuturereservoirperfor-mancefollowingahistorymatchdependstoalarge degree ontheengineer'sconfidenceinthevaluesofthematching reservoir parameters. Thispaperdescribesaproceduretoassignrelative permeabilitycharacteristicstoeachreservoirunitona permeabilitydistributionbasis,aswellasatechniqueto changethe shapeof the relative permeabilitycurves during the historymatch phase. Fluidsaturation andrelativepermeability values cor-respondingto thebeginning,intersection,andendpointsof thedisplacementprocesswerecorrelatedasalogarithmic functionof absolute permeability todetermineinitial satu-rationfunctionsforeach rocktype.Thisprocedure proved tobeusefulforwater-oil andgas-oil systemsof carbonate andsandstonereservoirs.However,thecorrelationsand historymatchapplicationsdescribedinthispaperare limitedtocarbonatereservoirs.The so determined satura-tionfunctionswerecurvefittedusingexponentialequa-tionsasstartingpointsduringthehistorymatchphase. Thiswasaccomplishedutilizinga"modelinginsidethe model"techniquewhichincorporatestheadvantagesof an analyticalsolutionforeachreservoirunit.Thecompre-hensivelybroadandversatile,two-phaserelativepermea-bilitymodelsusedtofittheendpointsareincludedinthis paper.Subsequently,duringthehistorymatchphase,the shapes of the relativepermeabilityfunctionswere adjusted bychangingtheexponents of thefittingequationsinorder tomatchWORand/orGORproductionperformance.The resultisaquick,versatile,inexpensiveprocedurewhich convergestothedesiredsolutionfromthefirstrelative permeabilityadjustmentandwhichhonorsthelaboratory-determined endpoints. Thisprocedureassumesthattheendpointsremain constantduringthehistorymatchphase,andthisimplies nomass diffusion and isothermal reservoir conditions. THEORY Therelativepermeabilitytoafluidisdefinedasthe ratioofeffectivepermeabilityatagivensaturationof thatfluidtothe absolutepermeabilityat 100%saturation. Asarule,the relativepermeabilityisbasedinthespecific airpermeability(ka)thathasbeencorrectedforslippage (Klinkenberg effect). Eachporoussystemhasuniquerelativepermeability characteristicswhichmustbemeasuredexperimentally. Directexperimentaldeterminationofthree-phaserelative permeabilitypropertiesisextremelydifficultandinvolves rathercomplextechniquestodeterminethefluidsatura-tiondistributionalongthelengthofthecore.Forthis reason,themoreeasilymeasuredtwo-phaserelativeper-meability characteristics are experimentally determined. Inwater-oilsystems,themaximumrelativeper mea-bilitiestooilandwaterthatcannaturallyoccurduring displacementarecalledendpoints.Inthispaper,the common relative permeability valueatwhichoil andwater areequalisreferredasthe"intersectionpoint"ofthe displacementprocess.Similardefinitionsapplytogas-oil systemsiftotalliquid(criticalwaterandoil)andgas replace the displaced and displacing phases ofthewater-oil system.Sincenumericalmodelsdescribingmultiphase flowinpetroleumreservoirsdonotdistinguishbetween critical andirreduciblesaturations,theterm"critical" will beusedinthetwo-phaserelativepermeabilitymodel sectiontoindicatethechangefrompendulartofunicular saturation. Takingintoconsiderationthateachlithologicalunit, i.e., layer orreservoirregion,hasdifferentrockproperties thatcontrolporegeometryand,consequently,different relativepermeabilitycharacteristics,itwillbeidealto assignflowpropertiesonareservoirunitbasis,utilizing laboratory-obtained relativepermeability data,despitethe problemsassociatedwithsuchmeasurements.Usually,the mostreliableexperimentallydeterminedrelativepermea-bilityvaluesaretheso-calledendpointsofthedisplace-mentprocess.Unfortunately,relativepermeabilitydata arenotalwaysavailableforeachrocktype,andthereis littlereasontobelievethatasingle,averagerelative permeabilitycurvewillberepresentativeofthebehavior of each layerorthewholereservoir.Theproblemcouldbe approached if ameaningfulcorrelatingparameterisdeter-. mined forthe reservoir. Therelativepermeabilitiesaredependentonthe saturations of eachfluid.Additionally,theflowproperties arethecompositeeffectofreservoirwettability,pore geometry,fluiddistribution, and saturationhistory. Thewatercontentataspecificpointwithinthe reservoirisrelatedtothestructural positionof that point, to the poregeometry, and to thewettability of the rock. Somerockpropertieswhichaffectresidualoilsatu-ration(S)inawaterdisplacementprocessareper mea-orw bilitystratification,arealheterogeneities,andreservoir wettability.It isconvenienttopointoutthatthelabora-tory-determinedSat100%water-cutanduniformly orw distributedfluidsaturationsisdifferentfromthefield Swhichisleftbehindinthemorepermeablewater-orw sweptzoneswhentheproducingwater-oil ratioreachesits economiclimitandwhichalsocouldberelatedtoother importantfactorssuchasgravitysegregation,totalwater throughput,mobilityratio (M),etc. It isinterestingtoobservetherelationbetweenpore geometryandreservoirwettabilitywithrelativepermea-bility,connatewatersaturation,andSorw'Poregeometry isafunctionofthereservoirrockcharacteristicsand affectssuchreservoirpropertiesaspermeability,porosity, tortuosity,irreduciblewatersaturation,andsurfacearea. Therockpropertywhosevariationisthe mostimportant in influencing reservoir performance isthepermeabilityvalue sinceitwillaffecttheratiooftheviscoustothe gravita-tionalandcapillaryforces.Fromtheseconsiderations, selectionoftheabsolutepermeabilityvalueasapossible correlating parameter can beinferred. Ifacceptabletrendsoftheendpointsandresidual fluidsaturationsareobtainedasafunctionofpermea-bility,thenrelativepermeabilitycharacteristicscouldbe assignedonareservoir unit basis.Statistical criteria could beusedindetermininghowwellanequationfitsagiven setofdata.Ifacceptabletrendsorcorrelationsare obtained,theirphysicalsignificanceshouldbeevaluated andthemostmeaningfulcorrelationselected,considering theproblemsassociatedwiththelaboratorymeasurements andtheactualreservoirconditions.Forinstance,since connatewatersaturationdependsonreservoirfluiddistri-bution,abetter definitionofthisparameterforeachrock typeshouldcomefromcapillarypressureand/orwelllog data.The"intersectionpoint"relativepermeabilityand fluidsaturationcorrelationsshouldbeinterpretedonlyas indicationsofthecurvatureoftheflowpropertiessubject to change duringthe historymatch phase. Stone1,2developedaprobabilitymodeltoestimate three-phaserelativepermeabilitydatafromthelabora-tory-measuredtwo-phasedata.Themodelcombinesthe channelflowtheoryinporousmediawithprobability conceptstoobtainasimpleresultfordeterminingthe relativepermeabilitytooilinthepresenceofwaterand gasflow.Themodelaccountsforhysteresiseffectswhen waterandgassaturationsarechanginginthesamedirec-tionofthetwosetsofdata.Theuseofthechannelflow theoryimpliesthatwaterrelativepermeabilityandwater-oilcapillarypressureinthethree-phasesystemarefunc-tionsofwatersaturation alone,irrespective of therelative saturationsofoilandgas.Moreover,theyarethesame functioninthethree-phasesystemasinthetwo-phase water-oilsystem.Similarly,thegasphaserelativeper-meabilityandgas-oilcapillarypressurearethesame functionsof gassaturationinthethree-phasesystemasin thetwo-phasegas-oilsystem.ThesecondStone'smodel suggeststhefollowingequationasameansofestimating three-phase oil relative permeability (kro) data: Theprobabilitymodelissuchthatitwillsatisfythese assumptionsandyieldthecorrect oil relativepermeability inthethree-phasesystemonlyiftherelativepermeability attheendpointsisequaltoone;otherwise,itwillonly approximatethetwo-phasedata. Consideringthatthegas-oildataaremeasuredinthe presenceofirreduciblewatersaturation,Dietrichand Bondor3 suggestedthefollowingmodificationtoStone's equation: (2) Thismodeltendstopredictincorrectkvalues(greater thanunity)forsmall kvalues,i.e.,kro :-;0.3. rocwrocw Nolen 4 asreferencedbyDietrich 5 suggestedadif-ferentnormalizationof equation (1)whichremains bounded as kapproacheszero andhastheform: rocw k= k + kro+ k- k+ k3 rorocwkrocw rw.krocw rgrwrg() Thismodel givesareasonableapproximationtothethree-phaseoilrelativepermeability.Numericalmodelsavail-abletothepetroleumindustryuseathree-phaseoil relativepermeabilitymodelgivenbyequation(1),(2)or (3).InthemodifiedversionsoftheStone'smodel,anoil-watersystematcriticalwatersaturationandanoil-gas systematzerogassaturationarephysicallyidentical. Consequently,theend-pointrelativepermeabilitytooilis identical inbothsystems. Assumingnomassdiffusionandisothermalreservoir conditions,thecriticalfluidsaturationsandtheend-point relativepermeabilitiesshouldremainconstantduringthe historymatchphase.Mo;ganandGordon6 illustratedthat withineachreservoirrocktype,therelative permeabilities aresimilar,varyingonlyslightlyforratherlargechanges inairpermeability.Eventhoughthereservoirpermea-bilitywillchangeduringthehistorymatchphase,theflow propertiesassumedtoremainconstantshouldmaintain theirrelativevalueandphysicalsignificanceforeach reservoirunit.Anyadjustmentinthe shape of therelative permeabilitytomatchreservoirperformanceshouldac-countforfluiddistributionwithinthereservoir,whichin turnisdependentonthesaturationhistoryandonthe wettingcharacteristicsoftherock.Thiscouldbeaccom-plishedbyexponentialiterationofthealreadycorrelated two-phaserelativepermeabilitydata,if convenient analyt-icalexpressionsaredeterminedtocurvefittheabove mentionedendpointsandtheircorrespondingfluidsatura-tions.Thisinturnincitesthedevelopmentofauniversal two-phasepermeabilitymodeltofitthelabora-tory-determined relative permeability data. Theadvantagesofusinganalyticalexpressionsfor theexperimentallydeterminedrelativepermeabilitydata canbefurthervisualizedtakingintoconsiderationthe followingfactsaffectinggas-oilandwater-oildisplace-ment systems:themostsignificantpart of theoilproduc-tionoccursnearthecriticalgassaturation;thehistory matchprovidesanestimateofthegasrelativepermea-bilityonlyforsmallgassaturations;andforunfavorable mobilityratiodisplacement,thefractionalflowcurveis verysensitivetothewaterrelativepermeabilitycurve near critical water saturation. TWO-PHASERELATIVEPERMEABILITYMODELS Relativepermeabilitymathematicalmodelsincite theanalysttothoroughlyconsideralltheflowproperty variablesandcharacteristicsofthebasicnatureofthe two-phaseflow. Typicalsetsoflaboratory-determinedwater-oiland gas-oil relativepermeabilitycurves are showninFigures1 and2 respectively. Water-Oil System Inattemptingtocurvefittheend-pointrelative permeabilitiesandtheircorrespondingfluidsaturations,if thenormalizedwatersaturationisdefinedastheratioof thedisplacingfluidsaturationtothemaximumdisplacing fluidsaturation,therelativepermeabilitytowatercanbe expressed asfollows: k= kSw- Swc ()nw rwrwro1-S-S wcorw (4) where: S0 1 " 10100 10100 ka-md 1000 1000 oCENTRAL REGIONCN.W. REGIONoN.W. REGION.8R-160 TPAODATA Fig.7 - BatiRamanFieldGas-OilRelativePermeability Correlations-EndofDisplacement. .>-z .. z 3.20 3.10 z o

or.. lc 3.00 o .. N

Z

RUN No.3 2.90nw:2.90RUN No." CUMULATIVEWATER PRODUCTION-bbi Fig.8 - ExponentialIterationofWaterRelativePermeability Equation-BR-17HistoryMatch-Layer1. 1.00,----------,,----------, .. .. :::;ii ... .. :I G:o.e.. Go.. > ;::: ... ..J .. G:O.O.'-:------==--::IL:--.......::::!l:::-------:-' 0.01.0 WATE"SATURATION Fig.9 - BR-17Water-OilRelativePermeabilityCurves-Layer1(KUni t). 1.00',----------.-----------, Kro. 0.0L------"'----......::::::.-=!!:::-""------l 0.01.0 WATERSATURATION Fig.10- BR-17Water-OilRelativePermeabilityCurves-Layer2(G2Unit). >-

::; iii ... "'2 a: 1.0',--------r------------, 0.5 "'> >=... ..J iI1I-""'O"J:::.llN AUG 1,71 Fig.15- Br-17WaterfloodTest- WaterProductionHistoryMatch. I.O,----,----,----,------,------,-----,--------r------,rr-----, .. .8 .7

. .5 B .4 '" i .3 .2 AUGI,71 OCTI,71oDeSERVED OATA -ALLOCATED PRODUCTION6CALCULATED DATAJAN1,72"PRI,nJULI,72OCT 1,72"ANl t 73"PRI,73 Fig.16- BR-17WaterfloodTest- Water-cutHistoryMatch. JULI,735EPI,73 N TOPGARZAN rOPEl .5-z '" z 0 CoX '" Z 0 i= Q: :> !C(U) U)

Q'" N oJ 2 Q: 0 z 3.20 3.00 2.80 2.60 2.40 2.20 Fig.17- 8atiRamanField- GeneralizedCrossSectionofGarzan Formation- CentralArea. RUNNo.2 n=2.19 2.00 RUNNO.3 1.80 20003000 CUMULATIVEGASPRODUCTION-MMcf Fig.18- ExponentialIterationofGasRelativePermeability Equation- N.AmericanOilField. 6000