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ArabKeywords:TPFAMPFAFEMPermeabilitytensorMultiphaseflowposesseriouschallengestoaccurateandefficientnumericalsimulation.Themathematicalmodeldescribingthemultiphaseflowbehaviorequations;reservoirashighlyheteroge-geometryofthegonalandunstruc-cultiesfacedbytheFluxApproximationintheindustrialequationwithanellipticnature)pressureequation.However,robustandefficientasTPFAmaybe,itisonlyvalidontheso-calledK-orthogonalgrids*Correspondingauthor.E-mailaddresses:wezhang@pi.ac.ae(W.Zhang),malkobaisi@pi.ac.ae(M.AlKobaisi).ContentslistsavailableatScienceDirectJournalofPetroleumScienceandEngineeringjournalhomepage:www.elsevier.com/locate/petrolhttp://dx.doi.org/10.1016/j.petrol.2017.06.003Received22December2016;Receivedinrevisedform16April2017;Accepted1June2017Availableonline3June20170920-4105/©2017ElsevierB.V.Allrightsreserved.JournalofPetroleumScienceandEngineering156(2017)282–2981.IntroductionInjectingwaterintooilreservoirstomaintainreservoirpressureandtopushtheoiltotheproductionwellsistheprimarymethodforimprovingoilrecoveryaroundtheworld.Todesignanoptimalwaterfloodingschemeortoidentifybypassedoilzones,reservoirsimulationisanindispensabletoolforthispurpose.Despitedecadesofintensiveresearch,thesimultaneousflowofwaterandoilinthereservoirsstillconsistsofnonlinearlycoupledflowandtransportpropertiessuchaspermeabilityisoftenrepresentedneousandanisotropictensors.Moreover,thecomplexreservoiroftenentailstheuseofgeneralnon-orthoturedgrids.Allthesefactorscontributetothediffidiscretizationmethod.Traditionally,theTwo-Point(TPFA)method(AzizandSettari,1979)iswidelyusedcodestodiscretizetheelliptic(oramoregeneralparabolicEfficientandaccuratesimulationofmultiphaseflowandtransportinporousmediaplaysakeyroleinoptimizingrecoveryofhydrocarbonresources.Two-PointFluxApproximation(TPFA)iswidelyusedintheindustryfordiscretizingthepressureequationbecauseofitsrobustnessandefficiencydespitethefactthatitintroducesO(1)fluxerrorsonnon-K-orthogonalgrids.Multi-PointFluxApproximation(MPFA)providesaconsistentdiscretiza-tionoftheflowequationsatthecostoflargerfluxstencilswhichleadstoadenser,asymmetricsystemoflinearequationsrequiringmorecomputermemoryandworkforlinearsolvers.ThisdrawbackofMPFAbecomesmorepronouncedforgenerallyunstructuredtriangulargridsandisevenworsein3Dapplications.Inthiswork,wedevelopatwo-stepfinitevolumemethod(TSFVM)thatisconsistentfortheanisotropicpressureequationontriangularandtetrahedralgridswithimprovedefficiencyascomparedtoMPFAmethods.Duringeachtimestep,thepressureequationissolvedimplicitlyandsaturationisthenupdatedexplicitly(IMPES).Todiscretizethepressureequationwithfullpermeabilitytensors,Galerkinfiniteelementmethod(FEM)isemployedinthefirststeptocomputepressureatgridvertices.Pressuresatgridverticesarethenutilizedinthesecondsteptoderiveatwo-pointfluxstencilwithaconstantforeachgridface.PressureatcellcentersandfluxacrossgridfacescanthenbesolvedusingaTPFAapproach.Finally,thefluxfieldatthenewtimelevelisusedtoupdatesaturationsexplicitly.AlthoughduringeachtimestepourTSFVMneedstosolvetwosystemsoflinearequations,resultsofthenumericalexperimentsclearlydemonstratethatitisstilllessexpensivethantheclassicalMPFA-Omethodwhileprovidinghigh-qualitynumericalsolutionsmatchingthatofMPFA-O.Thegainedefficiencyisespeciallysignif-icantfor3Dtetrahedralgrid.Asthescaleoftheproblemincreases,MPFAbecomeslessappealingbecauseofitslargefluxstencilsthatresultinmuchdensermatriceswhileourTSFVMstillremainscomputationallytractablethankstotheflexibilityofFEMinthefirststepandthetwo-pointfluxstencilinthesecondstep.Ontheotherhand,thefirststepoftheTSFVMisreminiscentoftheControlVolumeFiniteElementMethod(CVFEM)butwithimportantdifferences.UnlikeCVFEM,ourTSFVMdoesnotneedtoconstructadualcontrolvolumemesharoundgridverticesonthebasisoftheprimarymeshandthesaturationunknownsarenotassociatedwithnode-centeredcontrolvolumesbutarepiece-wiseconstantongridcellsinwhichreservoirpropertiesaredistributed.Thisen-suresthatasaturationvaluedoesnotstraddleacrossmaterialdiscontinuitieswhichCVFEMsufferfrom.ARTICLEINFOABSTRACTAtwo-stepfinitevolumemethodforthesimulationinheterogeneousandanisotropicreservoirsWenjuanZhang,MohammedAlKobaisi*DepartmentofPetroleumEngineering,ThePetroleumInstitute,AbuDhabi,P.O.Box2533,UnitedofmultiphasefluidflowEmiratesW.Zhang,M.AlKobaisiJournalofPetroleumScienceandEngineering156(2017)282–298(Aavatsmark,2002).IfthegridisnotK-orthogonal,TPFAisnolongerconsistent:differentnumericalresultswillbeobtaineddependingontheorientationofthegridandtheresultswillnotconvergetothecorrectsolution.TheO(1)errorinfluidvelocityincurredbyTPFAwillthencauseerrorsinthesubsequenttransportcomputations.TheshortcomingsofTPFAwererealizedearlybyresearchersandaclassofconsistentdiscretizationmethodscalledMulti-PointFluxApproximation(MPFA)havebeendevelopedoverthelasttwodecades,see(Aavatsmarketal.,1996a,1996b;EdwardsandRogers,1998;Aavatsmarketal.,1998a,1998b,1998c,2008;NordbottenandEigestad,2005).MPFAmethodswithfullpressurecontinuitywithintheinteractionregionaredevelopedmorerecentlyin(Chenetal.,2008;EdwardsandZheng,2010a,2010b;FriisandEdwards,2011).TheclassofMPFAmethodshasbeendemonstratedtobecapableofreducingthegridorientationeffectsignificantlyandcapturingthetensorialinfluenceofthepermeabilitytensorongeneralnon-orthogonalgridsaccurately.Fig.1.CellmoleculeofMPFAfortriangulargrids.ConsideringthedeficiencyofTPFAandincreasedaccuracyofMPFA,itmaybesurprisingthatTPFAisstillthepredominantdiscretizationmethodinmostcommercialreservoirsimulatorsandMPFAisrarelyusedbytheindustry.ThereluctanceonthepartoftheindustrytoembraceMFPAmethodsasanext-generationreservoirsimulationtechnologycanbepartlyascribedtothefactthatMPFAisgenerallymorecomplexandlessefficientthanthesimpleTPFAmethod.WhileTPFAusespressureattwocellstoapproximatethefluxacrossonecellinterface,thefluxstencilofaninterfacebyusingMPFAismuchlarger.Forexample,onstructuredquadrilateralgrids,pressureatsixcellsareneededbymostMFPAmethodstoapproximatefluxacrossoneinterface.AlargerfluxstencilmeansadenserJacobianmatrixandmorecomputermemoryisneeded.ThecellmoleculeofTPFAinvolves5pointson2-Dquadrilateralgrids,structuredornot,while9pointsareinvolvedinthecellmoleculeofMPFAonstructuredquadrilateralgrids.ThecellmoleculecanbeevenlargerfortriangulargridsasisshowninFig.1.13points(denotedbythereddots)areinvolvedinthecellmoleculewhileTPFAonlyuses4points.Thesituationisevenworsefor3-Dgrids:27pointsareneededtoconstructacellmoleculebyMPFAfora3-Dhexahedralgrid.Incontrast,thecellmoleculeofTPFAforhexahedralgridscontainonly7points.Inadditiontotheincreasedusageofcomputermemory,unlikeTPFAwhoseJacobianmatrixisalwayssymmetric,theJacobianmatrixofMPFAisgenerallynotsymmetric,makingitmoredifficultformanylinearsolverstosolvethediscretizedsystemofequationsefficiently.Consideringtheincreasingpopularityofgenerallyunstructuredtriangularand283tetrahedralgridsinreservoirsimulationtoaccommodatecomplexreservoircharacterizations,consistentnumericaldiscretizationschemesthataremoreflexiblethanMPFAareneeded.In(ZhangandALKobaisi,2016)wedevelopedaTwo-StepFiniteVolumeMethod(TSFVM)todiscretizethepressureequationforsingle-phaseflowonnon-K-orthogonalgrids.Thefluxacrossacellinterfaceisreducedtoastricttwo-pointstencilplusaconstantinatwo-stepprocess.Inthefirststep,theControlVolumeFiniteElementMethod(CVFEM)isusedtocomputepressureatallthegridverticeswhicharethenusedinthesecondsteptoconstructtwo-pointfluxstencilsforcellinterfaces.Oncethefluxstencilsaredetermined,pressureatcellcenterscanbecomputedinthesamewayastheusualTPFAfinitevolumemethod.ExtensivenumericalexperimentsdemonstrateditsrobustnessandaccuracywhichiscomparabletoMPFAmethods.Inthisworkwemodifyandextendtheschemetotwo-phaseflowandtransportproblemsontriangulargridsin2Dandtetrahedralgridsin3D.Thefirststepisstilltocomputepressureatgridvertices.However,unliketheoriginalformulationwhereCVFEMisused,hereweusetheGalerkinfiniteelementmethod(FEM).TheadvantageofusingFEMinsteadofCVFEMisthatFEMdoesnotrequiretheconstructionofdualcontrolvolumesaroundgridverticesoftheprimarymesh.Uponcomputingpressureatgridvertices,atwo-pointfluxstencilwithaconstantcanbeconstructedforeachgridface,resultingina4-pointcellmoleculein2Dtriangulargridsand5-pointcellmoleculein3Dtetrahedralgrids.ResultsofthenumericaltestsdemonstrateexcellentagreementbetweenourTSFVMandtheclassicalMPFA-Omethodforfullabsolutepermeabilitytensors.Althoughduringeachtimestep,ourTSFVMneedstosolvetwosystemsoflinearequations,oneforpressureatgridverticesandtheotherforpressureatcellcenters,ourresultsshowthatTSFVMisstilllessexpensivethanMPFA-Omethod.ThisisbecausethesizeofthematrixinthefirststepisgenerallysmallerthanitforMPFAforgeneralunstructuredtriangularandtetrahedralgrids.Furthermore,theresultantmatrixissparse,symmetricandpositivedefiniteandthereforethesystemoflinearequationscanbesolvedefficientlybymanysolverswhileinthesecondstep,thecoefficientmatrixhasastructuresimilartothatofTPFA.Insummary,duringeachtimestep,MPFA-Osolvesonesystemoflinearequationsthatisdenserandmoredifficulttosolve,andourTSFVMsolvestwosystemsoflinearequationsthataresparserandeasiertosolve.Ontheotherhand,comparedtoCVFEM,ourTSFVMdefinessaturationunknownsonthesamemeshasarematerialpropertiessuchasabsolutepermeabilityandporosity,thusavoidingtheartificialsmearingeffectofCVFEMforhighlyheterogeneousmedia(Abushaikhaetal.,2015).Moreover,CVFEMneedstoconstructadualgridfromtheprimarymesh,whichcanbeproblematicforcomplexscenariosandplaceshighre-quirementsonstorageandCPUfor3Dtetrahedralgrids(Geigeretal.,2004).Thedifficultyofdual-gridconstructionisavoidedcompletelyinourTSFVM.Therestofthepaperisorganizedasfollows.InSection2,themathematicalmodelisdescribed.Section3givestheformulationofourTSFVM,followedbyresultsofnumericalexperimentsinSection4.Sec-tion5concludesthispaper.2.MathematicalmodelImmiscibleoil-watertwo-phaseflowinoilreservoirscanbedescribedbythefollowingmathematicalmodel(Chenetal.,2006a):∂ðϕSlρlÞ∂t¼C0∇C15ðρlvlÞþρlql;l¼o;w(1)whereϕisporosity;Slandρlaresaturationanddensityofphasel,respectively;vldenotestheDarcyvelocityofphaselandqlisvolumetricsourceofphasel.VelocityvlisrelatedtothephasepressuregradientbyDarcy'slaw:vl¼C0λlK∇pl;l¼o;w(2)W.Zhang,M.AlKobaisiJournalofPetroleumScienceandEngineering156(2017)282–298whereλlisphasemobilityandisdefinedbyλl¼krl/μl.krlandμlarerelativepermeabilityandviscosityofphasel,respectively.Kistheab-solutepermeabilitytensor.Herewehaveneglectedthegravityforce.Inaddition,saturationandpressureofthetwophasesareconstrainedbythefollowingtwoequations:SwþSo¼1;poC0pw¼pcowðSwÞ(3)whereoilisassumedtobethenon-wettingphaseandwateristhewettingphase.pcowiscapillarypressurebetweenthetwoimmisciblephases.Sincethefocusofthisworkistoprovideaconsistentdiscretizationmethodoftheflowequationsonnon-K-orthogonalgrids,weneglectthecapillaryforcesforsimplicityandpressureofthesystemisp¼po¼pw.Ifbothcompressibilityofthefluidandtheporousmediaarenegli-gible,themathematicalmodelcanbesimplifiedtothefollowingcoupledsystemofequations:∇C15ðC0λtK∇pÞ¼qt(4)ϕ∂Sw∂tþ∇C15ðfwvtÞ¼qw(5)wherevtistotalvelocityandvt¼vwþvo;qtistotalfluidsourceterm;λtistotalmobilityandλt¼λwþλo.fwisthefractionalflowofthewaterphaseandisdefinedasfw¼λw/λt.Equation(4)isusuallycalledtheflowFig.2.Fluxapproximationforgridfacei-jsharedbytwotriangularelements.equationandequation(5)iscalledthetransportequation.Toclosethemathematicalmode,boundaryandinitialconditionsneedtobesupplied.3.Two-stepfinitevolumemethod(TSFVM)Equations(4)and(5)arecoupledtogetherbythemobilitytermλtandtotalvelocityvt.Tosolvethemathematicalmodelnumerically,differentsolutionstrategiessuchasfullyimplicitandsequentialsplittingarecommonlyusedintheliterature.Inthiswork,weemployasequentialsplittingstrategytodecouplethenonlinearlycoupledflowandtransportequations.Specifically,theimprovedImplicit-PressureandExplicit-Saturation(IMPES)method(Chenetal.,2006b)isused.Duringeachtimestep,mobilitytermiscalculatedusingwatersaturationfromthelasttimestepandthensubstitutedintoequation(4)tosolveforthepressureandtotalvelocityatthenewtimelevelimplicitly.Totalvelocityatthenewtimelevelisthenheldconstantandsubstitutedintoequation(5)toupdatewatersaturationexplicitly.LocaltimesteppingforthesaturationequationisdictatedbytheCFLconditionandisthereforemuchsmallerthanthetimestepforthepressureequation.Weremarkherethatthesequentialfullyimplicitstrategyintroducedin(Jennyetal.,2006)canbe284equallyappliedtoourTSFVMtorelievetherestrictionontheallowabletimestep.Forsequentialfullyimplicitformulation,duringeachtimestep,watersaturationfromthelastiterationisusedtocomputethemobilityfieldinequation(4)whichisthensolvedforpressureandfluxatthenewiterationlevel.Theupdatedfluxfieldisthenusedtoupdatethesaturationfield.Theloopcanbeiteratedonuntilthesaturationsolu-tionconverges.Finally,wenotethatunlikeCVFEMwhosesaturationunknownsareassociatedwithdualcontrolvolumesthatareformedaroundgridverticesandcancontaincontrastingpropertiesinthesamecontrolvol-ume,ourTSFVMassociatesonesaturationunknowntoeachcellofthemeshwithoutusinganydualcontrolvolumes.3.1.Step1:computingpressureatgridverticesInthisstep,astandardGalerkinfiniteelementmethodisusedtocomputepressureatgridvertices.ThedomainΩispartitionedintoatriangularmesh.Pressureisinterpolatedoverthedomainbylinearbasisfunctions:pnþ1ðxÞ¼Xmi¼1pnþ1iNiðxÞ(6)wherepiispressureatnode(vertex)iandsuperscriptnþ1denotethenewtimelevel.misthenumberofnodesinthemesh.Ni(x)isthelinearbasisfunctionofnodeidefinedas:NiC0xjC1¼C261;i¼j0;otherwise(7)wherexjisthecoordinatevectorofnodej.TheGalerkinweakformofequation(4)withno-flowboundaryconditionsisgivenby:∫ΩλtK∇pC15∇Nidx¼∫ΩqtNidx(8)forallbasisfunctionsNi.Substitutingequation(6)into(8),weobtainasystemoflinearequationsforpressureatnodes:Anodepnþ1node¼qnode(9)wherepnþ1nodeisthevectorofpressureunknownsatallgridnodes.TheelementsofmatrixAnodeandright-handsideqnodearegivenby:Aði;jÞ¼∫ΩλtK∇NjC15∇Nidx;qðiÞ¼∫ΩqtNidx:(10)herethepermeabilitytensorKandsource/sinktermsqtaredefinedontriangularelements.Moreover,sincethesaturationvariableisalsopiece-wiseconstantoneachtriangularelement,thetotalmobilityλtisalsodefinedoneachelement,whichisanimportantdifferencebetweenourmethodandtheonein(Geigeretal.,2004)wheresaturationunknownsareassociatedwithcontrolvolumesformedaroundgridvertices.3.2.Step2:computingfluxfieldandupdatingsaturationInthesecondstep,weuseacell-centeredfinitevolumemethodtocomputepressureatcellcentersandacontinuousfluxfieldthroughcellfaces.Thefluxfieldisthenusedtoupdatesaturationsexplicitly.Inte-gratingequation(4)overatriangularelementΩeandapplyingthedivergencetheorem,wehave:∫∂ΩeC0λntK∇pnþ1dS¼∫Ωeqtdx(11)wheredSistheorienteddifferentialareaelementwiththedirectionpointingoutward.∂Ωedenotestheboundarysurfaceoftheelement.Fig.2showstwotriangularelements(theircentroidsaredenotedbyx1andx2,respectively;x1andx2willalsobeusedtodenotetherespectivetriangularelement)thatshareacommongridfaceσijwhosenodesarexiandxj.Acrucialpointhereistoobtainacontinuoustotalfluxfieldacrossgridfacesdefinedas:Fnþ1ij¼∫σijC0λntK∇pnþ1dS(12)Thetotalmobilityλntisevaluatedusingsingle-pointupstreamweighting.Theproblemisthenreducedtofindingacontinuousapproximationofthestaticpartoffluxusingpressureatcellcenters:Fnþ1sij¼∫σijC0K∇pnþ1dS(13)Fornotationconvenience,wewilldropthesuperscriptnþ1fornow.Theabovefluxs
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