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ContentslistsavailableatScienceDirectJournalofPetroleumScienceandEngineeringjournalhomepage:www.elsevier.com/locate/petrolImprovingmultiscalemixedfiniteelementmethodforflowsimulationinhighlyheterogeneousreservoirusingadaptivityNaZhanga,BichengYanb,QianSuna,YuheWanga,⁎aDepartmentofPetroleumEngineering,TexasA&MUniversityatQatar,EducationCity,Doha,QatarbDepartmentofPetroleumEngineering,TexasA&MUniversity,CollegeStation,TX,USAARTICLEINFOKeywords:ReservoirsimulationAdaptivityMultiscalemixedfiniteelementmethodHeterogeneityMultiplescaleABSTRACTWepresentanadaptiveMultiscaleMixedFiniteElementMethod(MsMFEM)formodelingmultiphaseflowinhighlyheterogeneousreservoir.Inthisframework,fractionalflowmodelisusedtoapproximatepressureandvelocitysolutiononcoarsescale,whileresolutioninfinescaleishonoredbythebasisfunctions,whicharecalculatedbylocalproblemsonfinescale.Theadaptivecomputationintheflowproblemispermittedbytwodifferentbasisfunctions.Inthenumericalexamples,ouradaptiveMsMFEMisappliedtosolvetwodimensionalandthreedimensionalreservoirsimulationproblemswithhighlyheterogeneousporosityandpermeabilityfields.Thepreliminarynumericalsimulationresultspresentedshowasignificantspeedupincomparisonwiththereferencenumericalmethod,whichencourageandisbeneficialforfurtherinvestigationoftheproposedmethodforreservoirnumericalsimulation.1.IntroductionAlotofimportantproblemsinengineeringandsciencefieldsarefundamentallymultiscaleproblems,suchasheterogeneousporousmediaandcompositematerials.Directnumericalsimulationforsuchproblemsisnotrealistic,becauseittakestremendouscomputermemoryandcomputationaltime,whichisevenbeyondthecapacityoftoday’scomputerresources.Consideringthedifficultiesinnumeri-callyresolvingallscales,itisoftensufficienttosimulatemultiscaleproblemsbyconservingacertainaccuracy.Therefore,upscalingormultiscalemethodsaretypicallyadoptedforsuchheterogeneoussystems(Karimi-FardandFiroozabadi,2003;Durlofsky,1991;Efendievetal.,2004;WeinanandEngquist,2003).Upscalingmethodsismainlyreliedontheideaofformingcoarsescalesolutionswithaprescribedanalyticalapproach.Ingeologicalmodelingarea,numericalupscalingmethodsbasedondetailedgeolo-gicalmodelshasreceivedmuchattentionrecently.Thoseupscalingmethodsevaluatespermeabilityfieldinporousmedia,andbasicallytwomainproceduresareinvolved.Fluidflowthroughporousmediaonfinescaleisfirstlymodeled.Secondly,finescaleinformationisincorporatedintocoarsescaleflowproperties,e.g.permeability.Herehowtorealizethesecondprocedureisvital,andthosemethodshavegainedsuccess(JuanesandPatzek,2005;Wuetal.,1973;Zhangetal.,2016;Yanetal.,2016).However,whencomplexfluidflowprocessesinhighlyheterogeneoussystemsarestudiedbycoarsemodelswithsimplifiedsettings,itisverydifficultorevenimpossibletoobtainaprioriestimatesoftheerrors.Ontheotherhand,inmultiscalemethodsfinescaleinformationisdynamicallyabsorbedthroughoutthesimulationloop.Besides,thosecoarsescaleequationsaregenerallyformulatedandsolvednumericallybutnotanalytically.Inliteraturesalotofmultiscalemethodshavebeenproposed,e.g.,HeterogeneousMultiscaleMethod(HMM)(WeinanandEngquist,2003),MultiscaleFiniteElementMethod(MsFEM)(HouandWu,1997)andvariationalmultiscalemethod(ChenandHou,2002).Multiscalemethodshavealsobeenpromisinginreservoirsimulationarea,sinceitgainstheadvantagesingriddiscretizationforreservoirheterogeneityandflowphysicsforcomplextransportprocesses.MsFEM(Aarnesetal.,2008)constructsthemultiscalefiniteelementbasisfunctions,whichareadaptivetolocalpropertyofthedifferentialoperator,anditsolvesellipticpartialdifferentialequationswithmultiscalesolutions.Further,oversamplingtechnologyisadoptedtolowertheimpactofboundaryconditionsofcoarseelements.However,velocityfieldonfinescaleisnotensuredtobelocallyconservativeinthismethod.Further,Aarnes(Aarnes,2004)proposesMsFEMusingglobalinformation,andtheseapproachesextractimportantinformationaboutnonlocalmultiscalebehaviorinfluidmechanicalprocessesbysurrogatemodels.Efendievetal.,(2006)alsoappliedMsFEMusingglobalinformation.Specificallyintheirworksolutionofsinglephaseflowisusedtoaccuratelyupscaletwophaseflowinheterogeneousporousmedia.DespiteMsFEMhonorsfinescalehttp://dx.doi.org/10.1016/j.petrol.2017.04.012Received3December2016;Receivedinrevisedform12March2017;Accepted7April2017⁎Correspondingauthor.E-mailaddress:yuhe.wang@qatar.tamu.edu(Y.Wang).JournalofPetroleumScienceandEngineering154(2017)382–388Availableonline08April20170920-4105/©2017ElsevierB.V.Allrightsreserved.MARKinformationinellipticproblems,thesesolutionsarenotnecessarilytobelocallymassconservative.ToremediatethisMultiscaleMixedFiniteElementMethod(MsMFEM)isintroduced(ChenandHou,2002),anditguaranteeslocalmassconservationoncoarsescale.Theydemon-stratedthateveninsinglephasetracerflow,accurateflowsimulationrequiresalocallyconservativevelocityfieldonfinescale.What’smore,MsMFEMisappropriatetosolveproblemswithcomplexflowmechan-ismsandirregularphysicaldomains.Forexample,itsimulatessinglephaseflowinvuggyporousmediathroughconstructingdifferentflowsystemsoncoarseandfinescales(Gulbransenetal.,2010).Moreover,inordertofurtherimprovetheefficiencyofmultiscalemethods,somecriterionforadaptivityhadbeenproposed.Aarnes(2004)proposedanadaptiveapproachforMsMFEMbasedontheabsolutedifferenceofthetotalmobilityinacoarseelement.Adaptivemultiscalefinitevolumemethodisdevelopedformulti-phaseflow(Jennyetal.,2003,2006).Thismethodisbasedonthefinitevolumemethodology,butitisabitmorecomplextoconstructafine-scalevelocityfieldfortheMsFVMthanitisfortheMsMFEM(Kippeetal.,2008).Concertedeffortshasbeentaketobuildhighlydetailedreservoirmodelsthatmaximallycharacterizemultiscaleheterogeneityofgeolo-gicalfeatures.Insuchmodels,alocallyconservativemultiscaleschemecapableofreconstructingflowfieldswithhighfidelitybringshugecomputationalgains(Jennyetal.,2006).MsMFEMcanhandlesuchlargemodels,anditsmultiscaleframeworkallowsadaptivenumericalschemesforfluidtransportequations.ThisworkreportsourMsMFEMthatallowsdynamicallyadaptivecomputationoffluidtransportequationsinhighlyheterogeneousporousmedia.Inourworkreconstructionofasmallportionofbasisfunctionsataparticulartimestepcancomputethedetailedflowfield.SeveraltypicalnumericalsimulationcasesdemonstratethatouradaptiveMsMFEMprovideswell-matchedresultswithreferencefinescalesimulation.Moreover,itscomputationalcostislinearwithproblemsize.SinceMsMFEMisquiteappropriateforparallelization,wealsofurtherreducethecomputationaltimeonamulticorework-stationusinghighlevelparallelconstructs.Thepaperisorganizedasfollows.InSection2,themathematicalformulationisintroduced.Section3providestheadaptiveMsMFEMformultiphaseflowthroughporousmedia.Numericalstudiesaddres-singaccuracyandefficiencyarepresentedinSection4.Finally,theworkisconcludedinSection5.2.MathematicalmodelTwophaseoil-waterimmiscibleflowinheterogeneousporousmediaobeystheDarcy’slaw,andfluidsareassumedtobeincompres-sibleforsimplification.Furthertheeffectofgravityandcapillarypressureisignored.Inthemodelweassumenoflowboundaryconditionsandneglectbodyforces.Therefore,thegoverningequationsfortheproblemcanbeexpressedbyfractionalflowmodel,asshowninEqs.(1)and(2),Kvλpvq=−∇,∇·=(1)ϕStfvq∂∂+∇·()=www(2)Wherevvv=+wo,isthetotalvelocity;pisthepressure;Kisapermeabilitytensor;λλλ=+woisthetotalmobility;qqq=+woisthetotalvolumetricflux;ϕisporosity;Swiswatersaturation;fλλ=/wwisthewaterfractionalflow.Basedonthemathematicalmodel,wefurtherstudyEq.(1)onbothcoarseandfinescales.Eq.(2)issolvedseparatelybyusingthesequentialfullyimplicitmethod.3.AdaptiveMsMFEMmethodInthissection,theadaptiveMsMFEMisintroducedtosolvethepressureEq.(1).AsmentionedinSection1,MsMFEMiscapableofprovidingamassconservativevelocityfieldatthecoarsegridscale.Thecorrespondingvelocityvariablescanthenbeusedtosolvethesatura-tionequationatthesubgridscale.Thenotationstobeusedinthepaperareprovidedasfollows.LetdΩ⊂(=2,3)dbethereservoirdomainandnbetheoutwardpointingunitnormalon∂Ω.Moreover,L(Ω)2denotesthespaceofsquareintegrablefunctionsdefinedinthedomain.ThefunctionspaceisgivenbyEq.(3),HvvLvLvn(Ω)={|∈((Ω)),∇·∈(Ω),and·=0on∂Ω}divd022(3)3.1.FinescalediscretizationandhybridsystemThepressureequationonthecoarsescaleissolvedbyapplyingamixedfiniteelementformulation.TheninEq.(1),wefindpvLH(,)∈(Ω)×(Ω)∼∼div020suchthat,∫∫Kuλvdpud·()Ω−∇·Ω=0∼∼Ω−1Ω(4)∫∫lvdqld∇·Ω=Ω∼ΩΩ(5)AlltestfunctionssatisfyluLH(,)∈(Ω)×(Ω)div020.ToderiveaspatialdiscretizationofEq.(1),letvbethevectoroftheoutwardfluxesorderedbycellandbethevectorofcellpressures.ThelocalequationscanthenbeassembledtoformahybridsystemasinEq.(6),⎡⎣⎢⎤⎦⎥⎡⎣⎢⎤⎦⎥⎡⎣⎢⎤⎦⎥BCC0vp0q−=T(6)Here,bB={}ij,cC={}ik,qq={}karerespectivelydefinedbyEqs.(7)–(9),∫Kbψλψd=·()ΩijijΩ−1(7)∫cϕψd=∇·ΩikkiΩ(8)∫qϕd=qΩkkΩ(9)3.2.MultiscalebasisfunctionsforvelocityfieldLetΤ={Ω}hibeacoarsegridpartitionofΩbyacollectionofpolyhedralelements,andΩ=Ω∪Γ∪ΩijiijjrepresentsaneighborhoodcontainingtwoneighboringgridblocksΩiandΩj.Withoutlossofgenerality,theMsMFEMisdescribedfora3Dgridsystem.Fig.1isaschematicofa3Dgridsystem,whichshowscoarsegridblockΩijandtheinterfaceΓij.ThecoarsegridblockΩijisenlargedtobetterdepictthefineelements.IntheMsMFEM,thefirststepistocomputethefluxesacrosstheinterfacesegments,whichlieinsideΩij.ThebasisfunctionassociatedwiththeinterfaceΓ=∂Ω∩∂ΩijijisconstructedbysolvingKψλϕ=−∇ijij(10)⎧⎨⎩ψwxxwxx∇·=(),∈Ω−(),∈Ωijiijj(11)ψn·=0,on∂Ωijij(12)ψnv·=,onΓijijijij(13)Wherenistheoutwardunitnormalon∂Ω;ijnijistheunitnormaltoΓijorientedfromΩitoΩj;wx()iisasourcedistributionfunction,whichistoproduceaflowwithunitaveragefromΩitoΩj.ForallΩisuchthat,∫qdΩ≠0Ω(14)Wechoosewx()itobe,N.Zhangetal.JournalofPetroleumScienceandEngineering154(2017)382–388383∫wxqxqξd()=()()ξiΩi(15)Toguaranteeaconservativeapproximationonthefinescale,thesimplestchoiceiswx()=1/Ωiiforblockgridswithoutsource/sink.Thelocalboundaryconditionvijshouldhonorheterogeneousstructures.ThuswedefinevijaccordingtoEq.(16),∫vvvs=()dsijijij0Γ0ij(16)WhereKvnλn=·()·ijijij0.Ifthereservoirishomogeneous,simplyv=1/Γijij.3.3.BasisfunctionsforpressurefieldFunctionsthatareconstantoneachcoarsegridblockwithnosource/sinkareusedtoapproximatethepressurefield.Thus,abasisfunctionϕV∈iisassignedforeachΩi,suchthat⎧⎨⎩ϕxxx()=1,∈Ω0,∉Ωiii(17)Itshouldbenotedthatinthelowest-orderRaviart-Thomasmethodthistypeofapproximationspaceforpressurefieldisalsoapplied.3.4.WellmodelWellistypicallyformulatedbyPeaceman’smodel(Peaceman,1983),asshowninEq.(18),qλWpp=−(−)wew(18)WhereWisthewellindex;peisthecellpressure;pwisthebottom-holepressure.CombinedwithEq.(18),Eq.(6)canthenbeextendtoformalinearsystem,shownasEq.(19),⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥⎡⎣⎢⎢⎤⎦⎥⎥⎡⎣⎢⎢⎤⎦⎥⎥B0C0BCCC0vqp00q−−=wwTwTw(19)HereBwisablock-diagonalmatrixwithNwblocks,forwhichblockBkisann×kkdiagonalmatrixwithentriesλWB{}=()iikik−1,whereNwiswellnumber,nkisthenumberofcellsperforatedbywellk.Forthebasisfunctionsassociatedwiththewell-blockinterfaces,itisassumedthatwellblockΩiisperforatedbywellkwithboundaryTk.ThebasisfunctionψikassociatedwiththeblockΩiisconstructedbysolvingKψλϕ=−∇ikik(20)⎧⎨⎩ψwxx∇·=−(),∈Ω0,otherwiseikii(21)Withψ·n=0ikonT∂Ω/ikandψikisconstantonT∂Ω∩ik.3.5.CoarsescalehybridsystemTogeneratethemultiscaleequations,thebasisfunctionsforvelocityfieldneedstobesplitintotwoparts,ψψψ=−ijijHjiH(22)⎧⎨⎩ψEψEEE()=(),∈Ω/Ω0,∉ΩijHijijji(23)⎪⎪⎧⎨⎩ψEψEEE()=−(),∈Ω0,∉ΩjiHijjj(24)ThenallthebasisfunctionsψijHareassembledascolumnsinamatrixΨ.Similarly,allψikarearrangedascolumnsinamatrixΨw.Inaddition,allbasiswellratesisarrangedinamatrixRw.LetΙistheprolongationfromblockstocells.Ifblocknumbericontainscellnumberj,thenΙ=1ij,otherwiseΙ=0ij.Thecoarsegridsystemcanthenbeobtainedintheform(Eq.(25)),⎡⎣⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥⎡⎣⎢⎢⎤⎦⎥⎥BBCBBCCC0vqp00q−−=∼∼∼∼11c12c12c22cccccTwTwTw(25)BBBB=ΨΨ,=ΨΨ11cf12cwfwTT(26)BBRBR=ΨΨ+22cfwfwwwwTT(27)CCRC=(Ψ+)Ι∼fwfwwTT(28)CC=ΨΙ∼fT(29)qq=ΙΙcfT(30)WhenthenonlineariterationinEq.(25)isconverged,fluxescanbeobtainedonthefinegridbyvvq=Ψ+Ψfcwwc.Thesaturationissolvedonafine-scalegrid.Themultiscalemethodinthispaperisappliedtoapproximateofpressureandvelocity.Finitevolumemethodisusedtocomputethesaturationequation.Eachtimethepressureandvelocitysolutionsareobtained.Thenthevelocityisusedtoupdatethesaturation.3.6.AdaptivecomputationInthisworkflowandtransportequationsaretreatedseparatelyanddifferently.AteachtimestepaloopforsolvingthecoupledproblemsisincludedinourMsMFEMscheme.ThetransportproblemissolvedbyNewton-Raphsonmethodbasedontheupdatedpressureandvelocityfield.AsimplifiedcomputationalflowchartispresentedinFig.2.Thesubscriptsnandn+1representtheprevioustimestepandcurrenttimestep.InFig.2,S,v,parerespectivelysaturation,velocityandpressure.TheoperatorsB,AandCondrespectivelyrepresenttheΓΩijijFig.1.Structuredorthogonalgrid.Thedualvolumeisenlargedtoshowtheunderlyingfinegrid.N.Zhangetal.JournalofPetroleumScienceandEngineering154(2017)382–388384computationofbasisfunctions,thecomputationoftheflowequationandtheadaptivecondition.ThemostexpensivepartofMsMFEMschemeisgenerallythereconstructionofmultiscalebasisfunctionsofvelocityfield.Therefore,toachievehighercomputationalefficiency,itisfavorabletoupdatetheminthescenariothatthisisabsolutelynecessary.ThefollowingschemeisemployedtoadapttheflowcomputationsofMsMFEM.Ifthecondition(Eq.(31))isnotsatisfiedinacoarseelement,thenthemultiscalebasisfunctionsofthiselementhavetobeupdated.Ingeneralthesmallerεis,themorebasisfunctionsareupdatedateachtimestep.Throughnumeroustestswefindthatifthechangeofmobilityratiobetweentwoconsecutiveiterationsislessthan18%,itisnotnecessarytorecomputethebasisfunction.Touseadaptivecomputation,wecanavoidcomputingthetotalvelocitybasisfunctionsandsavecomputationaltimesignificantly.ελλε11+0isadefinedvalue.Fig.2.FlowchartofMsMFEM.Fig.3.Permeabilityfieldsandwatersaturationprofilesat0.65PVI:(a)permeabilityfield(logarithm);(b)finescalesolution(64×64);(c)MsMFEM(8×8);(d)upscaledpermeabilityfield(logarithm);(e)Upascalingmethod(8×8).Table1Saturationfielderrors(L2norm).UpscalingfactorMsMFEMUpscaling2×20.001980.00824×40.00720.0608×80.01770.110N.Zhangetal.JournalofPetroleumScienceandEngineering154(2017)382–3883854.NumericalexamplesInthissection,theadaptiveMsMFEMisappliedtothreedifferentcases.Case1considersaheterogeneousquarter-fivespotproblemin2D.InCase2and3,weapplythismethodtosimulatefluidflowinhighlyheterogeneousandlargescalereservoirs.4.1.Case1:2Dheterogeneousquarter-fivespotsTheadaptiveMsMFEMandanupscalingmethodareappliedtomodelincompressibleandimmiscibletwophaseflow.Theporosityandpermeabilityfieldareconstructedona64by64gridsystem.Thisisaquarter-fivespotproblem.Asourceandasinkareintroducedincells(1,1)and(64,64),respectively.Itisassumedthatthereservoirisinitiallyfullysaturatedbyoil.ThephasemobilityisdefinedbyEq.(32),Sμαwoλ=,(=,)ααα2(32)L1errorsinsaturationvaluesgivenbyEq.(33)iscalculatedtoevaluatetheoverallresolutionoftheflowpattern,eSSS=−srefref11(33)Fig.3showsthewatersaturationprofilesafteraninjectionofwatercorrespondingto65%ofthetotalporevolumeinthereservoir.Thegridsareupscaledbyafactorof8ineachaxissothatthecoarsesystemconsistsof8×8gridblocks.Itindicatesthatthewatersaturationprofile(Fig.3(c))bytheadaptiveMsMFEMperfectlymatcheswiththereferencefinescalesolution(Fig.3(b)).Ontheotherhand,theupscalingmethodappliestheupscaledpermeabilityfield,anditswatersaturationprofile(Fig.3(e))failscapturingalmostallofthefinescale(a)Innjector(b)ProduccerFig.4.Case2:(a)horizontalpermeability(logarithm);(b)coarseelementswithinjectoran
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