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well-driven,overallproceduretotheaccuracyofthecoarsemodelsisassessed.Applicationoftheoverallupscalingmethodologyisshowntocoarsenfine-scale,geocellulardescriptionsintoincludenumeroussimulationsforhistorymatching,theinvestigationofdifferentwellconfigurationsforperformanceprediction,andtheassessmentofuncer-JournalofPetroleumScienceandEngineering*Correspondingauthor.Tel.:+15104189473;fax:+1650725providesignificantlymoreaccuratecoarsemodelsthanthosegeneratedusingstandardprocedures.D2005ElsevierB.V.Allrightsreserved.Keywords:Near-wellupscaling;Two-phaseflow;Highmobilityratio;Effectivefluxboundaryconditions;Upscaledrelativepermeabilities;Reservoirsimulation;Horizontalwells1.IntroductionThefieldofupscalingismotivatedbytheneedtocoarse-scalemodelssuitableforfastandaccuratereservoirsimulations.EfficientflowcalculationsareessentialasatypicalreservoirengineeringstudymaydisplacementsinheterogeneousreservoirsM.Hui*,L.J.DurlofskyDepartmentofPetroleumEngineering,GreenEarthSciencesBuilding,StanfordUniversity,Stanford,CA94305-2220,USAReceived26May2005;receivedinrevisedform23July2005;accepted25July2005AbstractHighmobilityratiosareoftenencounteredinimproved-oil-recoveryprocessesbecausethedisplacedoilcanbemuchmoreviscousthantheinjectedwaterorgas.Inthiswork,anewtwo-phaseupscalingapproachformodelinghigh-mobility-ratiodisplacementsisdevelopedandapplied.Forthenear-wellregion,aspecializedupscalingprocedure,whichincludessingle-phaseandtwo-phaseupscalingcomponents,ispresented.Inthisupscaling,localregionsaroundeachwellareconsideredandthecoarse-scalewellindices,wellblocktransmissibilities,andrelativepermeabilitiesaredeterminedsuchthatthefine-andcoarse-scaleflowratesareinagreement.Awayfromwells,theupscaledrelativepermeabilitiesforeachcoarseblockarecomputedbyimposingeffectivefluxboundaryconditions,whichhavebeenshowntoprovidebetteraccuracythanstandardprocedures.Theperformanceofthesetechniquesisdemonstratedbyconsideringmultiplerealizationsofsynthetic3Dmodelswithvaryingcorrelationstructuresanddegreesofspatialvariability,aswellasdifferentfluidmobilitycontrastsandproductionscenarios(involvinghorizontalwellsandfive-spotpatterns).Byquantifyingtheupscalingerrorsusingwell-definedmetrics,thecontributionofeachcomponentoftheAccuratecoarsemodelingof0920-4105/$-seefrontmatterD2005ElsevierB.V.Allrightsreserved.doi:10.1016/j.petrol.2005.07.0042099.E-mailaddresses:mhhui@pangea.stanford.edu(M.Hui),lou@pangea.stanford.edu(L.J.Durlofsky).high-mobility-ratio49(2005)37–56www.elsevier.com/locate/petrolrealizations.taintiesusingmultiplegeostatisticalDespitecontinualimprovementsincomputationalefficiency(e.g.,parallelcomputing),upscalingisSciencestillessential.Thisisdueinparttothefactthatgeologicalmodelscontinuetogrowinsize.Numerouspastresearchershaveinvestigatedmanyaspectsoftheupscalingproblemandproposedawidevarietyofupscalingmethods(seereviewsbyFarmer,2002;Durlofsky,2003).Inthiswork,wedevelopandapplyanupscalingtechniquespecificallydesignedforhigh-mobility-ratio,two-phaseflowprocesses(e.g.,waterflooding,immisciblegasinjection)thataredrivenbywells.Thissubjectishighlypertinenttothepetroleumindustryforanumberofreasons.Waterfloodingpersistsasthesinglemostusedimproved-oil-recoveryprocessbecausewater/brineiscommonlyandcheaplyavail-able.Inaddition,immisciblegasinjectioninsomecasesoffersanattractivemeansofutilizingproducedgas.Theviscosityofthedisplacedoilisoftenconsid-erablygreaterthanthatoftheinjectedfluid,inwhichcasethedisplacementproceedsatahighmobilityratio.Inthecontextofreservoirsimulation,anaccuratewellmodelisneededtonumericallyrelatethewelltothegridblockinwhichitiscompleted(i.e.,thewell-block).Asaresult,thecoarse-scalesimulationmodelwillnotprovideaccurateflowresultsunlessthenear-wellregionisupscaledappropriately(Azizetal.,1999).Inourproposedupscalingapproach,thenear-wellandinter-wellregionsofthereservoiraregivendifferenttreatmentsinaccordancewiththedis-tinctlydifferentflowregimesthatprevail.Inthenear-wellregion,theflowisradialandthepressurefielddoesnothaveaconstantgradient.Asaresult,upscalingapproachesthatassumealinear,con-stant-gradientpressurefield(e.g.,constantpressure–noflowboundaryconditions)arenotappropriatenearthewell.Amongtheapproachesthatdoapplyacon-stantpressuregradient,alargenumberprovideupscaledabsolutepermeability,designatedk*(seereviewbyRenardanddeMarsily,1997).Inthiswork,wewillrefertotheseupscalingtechniquesasstandardk*(ork*only)approaches.Ithasbeenestablishedthattheuseofsuchk*tocalculatethewellindicesinthecoarse-scalemodelcanleadtolargeinaccuraciesinflowresults(MascarenhasandDur-lofsky,2000).Clearly,theproperupscalingofthenear-wellregionrequiresatreatmentspeciallydesignedtoaccountfortheradialnatureoftheflow.M.Hui,L.J.Durlofsky/JournalofPetroleum38Onepossibleapproachforanaccuratenear-welltreatmentistheuseoflocalgridrefinementaroundthewell.Thismethodmaybeexpensive,however,asmoreblocksareincludedinthesimulationandsmalltimestepsmayberequiredduetotheintroductionofsmallblocksinthehigh-flowregion.Inthepast,avarietyofanalyticalnear-wellupscalingmethodshavebeenproposed(Lin,1995;Soeriawinataetal.,1997).Thesemethodsareeffectiveandefficientinsomecasesbuttheyrelyonsimplifyingassumptions.Thus,forhighlyheterogeneousreservoirs,theseapproachesarelikelytobelesseffectivethannumer-icaltechniques.Thefirstnumericalnear-wellupscalingtechniquewasproposedbyDing(1995).Heappliedaglobalfine-scalesolutionofasingle-phase,steady-state,incompressible,well-drivenflowproblemtodeter-minetheupscaledwellindex(WI*)andwellblocktransmissibilities(Tw*).Durlofskyetal.(2000)extendedthemethodto3Dcasesinvolvingverticalwellsandemployedasmaller(extendedlocal)fine-scaledomaintocomputetheupscaledparameters.Theirnear-wellmethodperformedbetterthanstan-dardapproachesinmanycaseseventhoughtheyusedarelativelysmalldomainforthecalculationofnear-wellparameters.Inasimilarbutindependentwork,Muggeridgeetal.(2002)alsoextendedDing’smethodandconsideredareducedcomputationaldomain.Theyfoundthenear-well,single-phaseupscalingapproachtobequiteeffectiveforavarietyof2Dand3Dproblemsinvolvingpartiallypenetratingandnon-verticalwells.MascarenhasandDurlofsky(2000)furtherextendedthenear-wellmethodologybyintroducinganefficientGauss2Newtonoptimizationproceduretoforceagreementbetweenthecoarse-andfine-scaleflowrates.Theyobservedasignificantimprovementoverthestandardk*approachina3D,three-phaseflowprobleminvolvingahorizontalproducer.Recently,Zhangetal.(2005)presentedafullyglobalupscalingtechniquethatcalculatesWI*andtransmissibilitiesandshowedthatitcouldprovideaccurateresultsforhighlyheterogeneousreservoirs.Whilethesingle-phaseupscalingtechniquesdescribedaboveoftengivesatisfactoryresults,theireffectivenessdeteriorateswithextremelevelsofcoar-sening(twoordersofmagnitudeormore)andifmultiphaseeffectsaredominant,forinstanceinahigh-mobility-ratio,two-phasedisplacement.InsuchandEngineering49(2005)37–56cases,two-phaseupscalingtechniquesmayalsobeneeded.Inthecontextofnear-wellupscaling,thisposedatwo-phaseoptimizationmethodforthenear-wellregionbuttheirmethodwasonlyappliedtoScienceimpliesthat,apartfromtheupscaledsingle-phaseflowparameters(WI*andTw*),two-phasepara-meters(e.g.,pseudofunctions)mayalsoberequired.Numeroustwo-phaseupscalingapproachesinvolvingdynamicpseudofunctionshavebeenproposedtodate(seereviewsbyBarkerandDupouy,1999;Darmanetal.,2002).Therearetwogeneraltypesofdynamicpseudofunctions—thosebasedondirectlyupscalingDarcy’slaw(KyteandBerry,1975)—andthosethatuseanaveragetotalmobilityformulation(Stone,1991).Ofthelargebodyofworkaddressingthecalculationofpseudofunctions,relativelylittlecon-siderationhasbeengiventoflowinthenear-wellregion.Instead,mostofthepreviousapproachesareformulatedbasedonlinearflowassumptionsandarethusnotsuitableforuseinthenear-wellregion.Previously,EmanuelandCook(1974)andWoodsandKhurana(1977)computedpseudofunctionsbasedontheupscalingofDarcy’slawinthenear-wellregioninordertoreducethedimensionalityoftheflowproblem.Theseeffortsweredirectedataccountingfornumericaldispersioninthecoarsegridandconsideredrelativelysimplereservoirdescriptions(homogeneousorpurelylayered).Assuch,theywerenotspecificallydesignedtocapturetheeffectsofsubgridheterogeneity.Toourknowledge,therehasnotyetbeenanin-depthinvestigationintotheupscalingofnear-well,two-phaseflowparametersforhigh-mobility-ratiodisplacementsinheterogeneousreservoirs.Weiniti-allymodifiedtwoofthepopulardynamicpseudoiza-tionmethods(KyteandBerry,1975;Stone,1991)toaccountforwell-drivenflowbutfoundthattheydonotprovidesatisfactoryresultsforthehigh-mobility-ratiocasesofinterest(seeSection3.2).Instead,weproposeanewupscalingtechniquethatcomprisestwocomponents:anear-well,single-phase(NW1P)pro-cedureandanear-well,two-phase(NW2P)procedure.Theupscalingcalculationsareperformedonanextendedlocalwellmodelextractedfromtheglobalsimulationgrid.TheNW1PprocedureisbasedontheworkofMascarenhasandDurlofsky(2000).Awell-driven,single-phaseflowproblemisimposedandthemismatchinsteady-stateflowratesbetweenthefineandcoarsemodelsisminimizedbyoptimizingWI*andTw*.TheNW2PproceduredeterminesupscaledM.Hui,L.J.Durlofsky/JournalofPetroleumrelativepermeabilities(krj*)byminimizingthediffer-encesinboththeoilandwaterflowratesbetweenthecompletelylayeredmodelsandwasnotusedincon-junctionwithanynear-well,single-phaseupscalingtechnique,asisperformedinourapproach.Thetwo-phaseflowphenomenainregionsawayfromwellsmustalsobeupscaledinsomecasestoachieveglobalaccuracy.Forthisupscaling,weapplyeffectivefluxboundaryconditions(EFBCs)forthecalculationofthekrj*foreachcoarseblock.Theseboundaryconditions,introducedbyWallstrometal.(2002a),accountapproximatelyfortheeffectsoftheglobalflowfieldintwo-phaseupscalingcalculationsbyappropriatelyattenuatingthefluxthroughhigh-permeabilityfeatures.TheeffectivenessofEFBCshasbeendemonstratedinavarietyofimmiscibledisplacements(Wallstrometal.,2002b;Chen,2005).Huietal.(2005)systematicallyinvestigatedtheuseofEFBCkrj*infirst-contactmiscibledisplace-mentsandobtainedreasonableaccuracyforpartiallylayeredpermeabilityfields.Thispaperproceedsasfollows.Wefirstdescribethethreeindividualcomponentsofourupscalingapproach(NW1P,NW2P,EFBCkrj*)inSection2.WethenpresenttheresultsfromtheapplicationofourapproachtoalargenumberofexamplesinSection3.Here,weinvestigatemultiplerealizationsof3Dsyntheticfieldswithvaryingcorrelationstructuresanddegreesofspatialvariability,aswellasdifferentfluidmobilitycontrastsandwellconfigurations.Wealsoquantifytheupscalingerrorsinordertogaugethecontributionofeachupscalingcomponenttowardtheoverallaccuracyofthecoarsemodel.ConclusionsarepresentedinSection4.2.MethodologyInthiswork,wepresentanewtwo-phaseupscal-ingapproachthatentailsdifferenttreatmentsforthenear-wellandinter-wellregionsofthereservoir.Forfineandcoarsemodelsforawell-driven,two-phaseflowproblem.Inthepast,someinvestigatorshaveadoptedanoptimizationapproachinthecontextoftwo-phaseupscaling(Tan,1995),butwithoutafocusonthenear-wellregion.Johnsonetal.(1982)pro-andEngineering49(2005)37–5639thenear-wellregion,thesingle-phaseupscalingfor-mulationproposedbyMascarenhasandDurlofskyworkofMascarenhasandDurlofsky(2000).Webeginbyconsideringthesingle-phasepressureequa-tionforanincompressiblefluid:jdkljpC19¼q;C18ð2Þwherelisthe(constant)fluidviscosity,kisthespatiallyvaryingpermeabilitytensor,pisthepres-sure,andqisthesource/sinktermassociatedwithwells(positiveforproducers).Forthenear-wellupscalingcalculations,thesizeofthefine-scalecom-putationaldomainaroundthewell(extractedfromtheglobalsimulationgrid)andtheboundaryconditionstoScienceandEngineering49(2005)37–56(2000)isextendedtoincludeanewnear-well,two-phaseupscalingprocedure,whichforcestheagree-mentofoilandwaterflowratesbetweenthecoarseandfinegridsforeachcoarsewellblock.Awayfromwells,effectivefluxboundaryconditionsorEFBCs(Wallstrometal.,2002a)areappliedtocomputethepurelylocalkrj*foreachcoarseblock.Thesethreeupscalingcomponentsarenowdescribed.MoredetailedexplanationscanbefoundinHui(2005).2.1.Near-well,single-phaseupscaling(NW1P)Weconsiderafine-scale,structured,Cartesian,3Dreservoirmodel.Thefine-scalepermeabilityishighlyvariableandisconsideredtobeadiagonaltensorkwithcomponentskx,ky,andkz.Thegridisassumedtobeorthogonalandtwo-pointfluxapproximationsareappliedonboththefineandcoarsegrids(i.e.,off-diagonaltermsink*areneglected).Inreservoirsimulation,awelliscoupledwiththesimulationgridblockinwhichitiscompletedviathewellindex(WI).TheWIforawellblocki,j,krelatesthewellrateqwtothewellbottomholepressurepw(evaluatedattheblockcenter)andgridblockpressurepi,j,k:qw¼WIi;j;kpi;j;kC0pwC0C1:ð1ÞThePeaceman(1983)equationiscommonlyusedtocomputethevalueofWIasafunctionofthewellradius,wellblockpermeabilities,andwellblockdimensions.Uponupscaling,awellthatiscompletedinaparticularlocationinthefine-scalemodelwillbelocatedinthecorrespondinggridblockinthecoarse-scalemodel.Astandardapproachistoemploythedimensionsandupscaledabsolutepermeability(k*)ofthecoarsewellblockinthePeacemanexpressiontocomputethecoarseWI.Ithasbeenamplydemon-strated(Ding,1995;Durlofskyetal.,2000)thatthisapproachcanleadtosignificantinaccuracies.Oneimportantproblemwiththisapproachistheassump-tionofalinearflowfieldinthecalculationofk*.Moreover,thePeacemanequationisderivedbasedonahomogeneouswellblockandisthereforeinappropri-ateforacoarsewellblockcontainingsignificantsub-gridheterogeneity.M.Hui,L.J.Durlofsky/JournalofPetroleum40Thenear-well,single-phaseupscaling(NW1P)procedureemployedinthisworkisbasedonthebeappliedonthedomainmustbeprescribed.Thesizeofthedomainorlocalwellmodel(LWM)canbequantifiedbytheparameterr,whichdefinesthelocalfine-scaleregionintermsofthenumberofringsofcoarseblocksaroundthecoarsewellblock.Attheupperlimit,theentireglobaldomaincanbeconsid-ered(Zhangetal.,2005),thoughthecomputationaleffortcanbecomecostly.Previousinvestigationsintotheimpactofdomainsizeinthenear-wellregion(MascarenhasandDurlofsky,2000;Muggeridgeetal.,2002)concludedthatr=1generallyprovidesreasonableresults.Inournear-wellupscalingmetho-dology,weemployaLWMwithr=1.Awell-drivenflowproblemisimposedbyspecifyingaconstantwellborepressure(pw)anddomainboundarypressure(pb)suchthatpwNpb.WeillustrateourextendedlocalapproachinFig.1,whichcaneitherrepresenttheWellatthecenterofLWMFlowfromonefaceofthecoarsewellblocktoaneighboringblockCoarsewellblockFinegridCoarsegridFig.1.Localwellmodelshowingthefinegrid,coarsegrid,andcoarsewellblock.Scienceareal-sectionofaverticalwellorthecross-sectionofahorizontalwell.IntheNW1Pprocedure,theincompressible,single-phase,well-drivenflowproblem(asgovernedbyEq.(2))isfirstsolvedatthefine-scalelevelfortheLWM,yieldingthepressureatthecenterofeachfineblock.Ifi,j,andkarethefine-gridindicesinthex-,y-,andz-directionsoftheLWMrespectively,thenthefine-gridflowratefromblocki,j,ktoblocki+1,j,k,whichwedesignateqi+1/2,j,k,isgivenby:qiþ1=2;j;k¼Tiþ1=2;j;kpi;j;kC0piþ1;j;kC0C1;ð3ÞwhereTi+1/2,j,kisthefine-gridtransmissibilityinthex-direction.DesignatingI,J,andKasthecoarse-gridindicesinthex-,y-,andz-directionsrespectively,wecancomputethetransmissibilitybetweencoarsewell-blocksI,J,KandneighboringblockI+1,J,Kasfollows:Tw4Iþ1=2;J;K¼hqiIþ1=2;J;KhpiI;J;KC0hpiIþ1;J;K;ð4ÞwherehpiI,J,Kisthebulk-volume-weighted
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