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NUMERICALANALYSISOFFINITEELEMENTMETHODSFORMISCIBLEDISPLACEMENTSINPOROUSMEDIASandraM.C.MaltaAbimaelF.D.LoulaLaborat orioNacionaldeComputa c~aoCient caRuaLauroM uller,455,Botafogo,RiodeJaneiro,RJCEP22290-160,Brazil.Finiteelementmethodsareusedtosolveanonlinearsystemofpartialdi erentialequationswhichmodelsincompressiblemiscibledisplacementofone uidbyanotherinporousmedia.Fromabackward nitedi erencediscretizationwede neasequentiallyimplicittime-steppingalgorithmwhichuncouplesthesystem.TheGalerkinmethodisemployedtoapproximatethepressure,andimprovementvelocity eldapproximationsarecalculatedviaapost-processingtechniquewhichinvolvestheconservationofthemassandDarcy’slaw.Astabilized niteelement(SUPG)methodisappliedtotheconvection-di usionequationdeliveringaccuratesolutions.Quasi-optimalordererrorestimatesareobtainedundersuitableregularityhypotheses.1.IntroductionWestudy niteelementapproximationsforasystemofnonlinearpartialdi erentialequationsgov-erningincompressiblemiscibledisplacementintwodimensionalporousmedia.ThemathematicalmodelconsistsofanellipticsystemcomingfromtheconservationofmassandDarcy’slawandadegenerateparabolicequationexpressingtheconservationoftheinjected uid(concentrationequation).Let beaboundeddomainintheplaneR2withsmoothboundary@ andT0a xednumber.Ourdi erentialsystem,underappropriatephysicalassumptions,isgivenby[1]divu=fin (0;T);(1:1)1u= K(x) (c)rpin (0;T);(1:2)withtheboundaryconditionu =0on@ (0;T);(1:3)and @c@t+div(cu) div(Drc)=^cfin (0;T);(1:4)withtheboundaryandinitialconditionsDrc =0on@ (0;T);(1:5)c(x;0)=c0(x)on ;(1:6)wherepanduarethepressureandDarcy’svelocityofthemixture, = (x)andK(x),theporosityandpermeabilityofthemedium,respectively, =( 1; 2)denotestheexteriornormalto@ ,fthesourceandsinktermsand^c=^c(x;t)istheinjectedconcentrationatinjectionwellsandtheresidentconcentrationatproductionwells.Thedi usion-dispersiontensorDwillbeconsideredasin[1],i.e.,D=D(u)= mI+jujf lE(u)+ tE?(u)g;(1:7)withE(u)=1juj2u u;E?(u)=I E(u);(1:8)where m, land tare,respectively,moleculardi usion,longitudinalandtransversedispersioncoe cients.Normallydispersionisphysicallymoreimportantthanthemoleculardi usion;also, lisusuallyconsiderablylargerthan t,andweshallmakethisassumptioninouranalysis.Sincep(x;t)isdetermineduptoanarbitraryadditiveconstant,wenormalizeitbyimposingtheconditionZ p(x;t)dx=0;t2(0;T):(1:9)Finally,wenotethatinEquation(1.2) = (c)isthelocalviscosityofthemixturewhichdependsontheconcentrationc.Suchdependenceisveryimportantintheemergenceofviscous ngering,2displacemente ciency,andultimateoilrecovery.Inthepetroleumengineeringliteraturethemostcommonlyusedformtorepresent is (c)= (0)h1 c+M14ci 4;c2[0;1];(1:10)whereM= (0)= (1)isthemobilityratio.Inspiteofbeingc(x;t)themostimportantvariableintothesolutionofsystem(1.1)-(1.6)wehavedemonstratedin[2,3,4,5]thatthevelocity eldu(x;t)hasstronglyin uenceontheaccuracyoftheconcentration.Thisfactismainlynotedforadversemobilityratiocases,i.e.,whenM1inEquation(1.4).WhenM=1,whichcorrespondstothetracerinjectionprocessesinoilreservoirs,wepresentedin[6]anumericalanalysisoftheuncoupledsystem(1.1)-(1.6)consideringanon-standardGalerkinmethodfortheconcentrationequationwiththevelocity eldu(x;t)givenbyapost-processingtechnique.Hereweshallextendthatproceduretothetwophaseincompressiblemiscibledisplacement,inotherwords,weshallassumeM1throughoutthispaper.Theorganizationofthepaperfollows.Insection2,afterintroducingthebasicnotationandas-sumptions,acontinuous-spaceapproximateproblemisde nedbyusingasequentiallyimplicittimediscretization.Insection3,consideringpressuregivenbytheusualGalerkinmethod,weproposedapost-processingtechniquetorecoveranaccuratevelocity eld.Next,thepost-processingap-proachiscombinedwithatechniqueofsubtractionofsingularitytotreatthecaseofpointsourcesandsinks.Insection4theellipticsubsystemisassumedtobesolvedandastabilized niteelementformulation,SUPG(StreamlineUpwindPetrov-Galerkin),isusedtoapproximatetheconcentra-tionequation.Errorestimateswillbeexhibiteddemonstratingquasi-optimalconvergenceratesinL1(L2)-norm.Section5summarizesour ndings.2.Preliminaries2.1NotationsandDe nitionsLetm 0beanarbitraryintegerand1 q 1,wedenotebyWm;q( )theusualSobolevspace3on containingfunctionsvhaving nitenormkvkm;q:= Xj j mZ j@ vjqdx 1q;where =( 1; 2;:::; n)isamulti-index,and@ :=@ 1x1@ 2x2:::@ nxn.Whenq=2thenWm;2( )correspondstotheHilbertspaceHm( )withnormk km;2=k km.ThecaseH0( )=L2( )hasitsnormdenotedbyk k0=k k,andtheinnerproduct(v;w):=Z wvdx:Wealsorequirespacethatincorporatetimedependence.LetXanormedlinearspaceconsistingofasetofunctionsde nedon ,suchasmentionedabove.Ifwisafunctionde nedon[0;T] wesayw2Lp([0;T];X),1 p 1,iffort2[0;T],w( ;t)2XandkwkX2Lp([0;T]),andwede nekwkLp([0;T];X):= ZT0kw(t)kpXdt 1p:2.2RegularityoftheDataInitiallyweimposeverystrongregularityhypothesesintheanalysisofsystem(1.1)-(1.3),someofthesewillbeweakenedinsubsection3.3.Therefore,thefollowingregularityassumptionswillbemadeonfunctionsK,fandonthesolutionfu;pgof(1.1)-(1.3):(1)ThereexistconstantsK andK suchthat,forallx2 ,0K K(x