A Mixed Finite Volume Element Method for Flow Calc

AMixedFiniteVolumeElementMethodforFlowCalculationsinPorousMediaJimE.JonesInstituteforComputerApplicationsinScienceandEngineeringNASALangleyResearchCenterSUMMARYAkeyingredientinthesimulationofowinporousmediaistheaccuratede-terminationofthevelocitiesthatdrivetheow.Thelargescaleirregularitiesofthegeology,suchasfaults,fractures,andlayerssuggesttheuseofirregulargridsinthesimulation.Workhasbeendoneinapplyingthenitevolumeelement(FVE)methodologyasdevelopedbyMcCormickinconjunctionwithmixedmethodswhichweredevelopedbyRaviartandThomas.Theresultingmixednitevolumeelementdiscretizationschemehasthepotentialtogeneratemoreaccuratesolutionsthatstan-dardapproaches.ThefocusofthispaperisonamultilevelalgorithmforsolvingthediscretemixedFVEequations.Thealgorithmusesastandardcellcenterednitedierenceschemeasthe‘coarse’levelandthemoreaccuratemixedFVEschemeasthe‘ne’level.Thealgorithmappearstohavepotentialasafastsolverforlargesizesimulationsofowinporousmedia.TheMixedFiniteVolumeElementDiscretizationInthisrstsection,webrieyintroducethemixednitevolumeelement(FVE)dis-cretizationtechnique.Wewillnotdwelltoomuchonthedetailsofthediscretizationitselfasourfocushereisonsolvingthediscretesetofequationsthatthediscretizationproduces;adetaileddescriptionofthediscretizationcanbefoundin[7].WebeginbyconsideringthefollowingpartialdierentialequationdenedonadomaininR2:(rA(x)r(x)=f(x)x2;r(x)=g(x)x2@:(1)HereweassumethediusioncoecientAisdiagonal,butvaluesofthecoecientsmayjumpordersofmagnitudeatmaterialinterfaces.Inthecontextofreservoirsimulation,thisisthepressureequationforincompressiblesingle-phaseowwhereisthepressureinthereservoir,andtheboundaryconditionspeciestheuxon@.Asoneofourgoalsforthenewdiscretizationisaccurateapproximationsofowvelocities,wewillbeginbyreformulatingthisequationasarstordersystemofequationswherevelocityappearsexplicitlyintheequations.Thisisdonebyintroducingtheowvelocityvariablesviathedenition,vAr;(2)andthenrewritingthepartialdierentialequationin1as,rv=f:(3)Againinthecontextofreservoirsimulation,Denition2isDarcy’sLawandEqua-tion3isthemassconservationlaw.Inreservoirsimulation,thissameapproachoftreatingowvelocityexplicitlyhasbeenusedinmixednite-elementmethodswithconsiderablesuccess[5],[6],[13].Equations2and3alongwiththeboundaryconditionfromequation1representtherstordersystemthatwediscretizeusingthemixedFVEmethod.Becauseoftheirregularityofreservoirgeology,faults,layers,etc.,uniformrectangulargridsarenotadequateinmodelingtheow.ThemixedFVEdiscretizationwasdevelopedforalogicallyrectangulargridofirregularquadrilater-als.Aexampleofsuchagridisshowningure1.Todiscretizethissystem,wefollowthenitevolumeelement(FVE)principlesdevelopedin[3],[8],[9].ThetwomajorcomponentsofanyFVEdiscretizationschemeareachoiceofcontrolvolumestointegratethecontinuousequationoverandachoiceofniteelementspacesfortheunknowns.Importantindevelopingthediscretizationforgeneralquadrilateralsisthemap-pingrelatingageneralquadrilateraltoareferenceone.ConsiderthequadrilateralPwithvertices(x00;y00);(x10;y10);(x01;y01);and(x11;y11)showningure2.Letthereferencequadrilateral^Pbetheunitsquare.Thenthereisauniquebilinearmappingof^PontoPgivenby,x(^x;^y)=x00+(x10x00)^x+(x01x00)^y+(x11x10x01+x00)^x^yy(^x;^y)=y00+(y10y00)^x+(y01y00)^y+(y11y10y01+y00)^x^yIfPisconvex,thenthismappinghasinverse.Werestrictourselvestoconvexquadrilaterals,soforeach(x;y)2Pwehaveanassociatedpoint(^x;^y)2^P.Showningure2areseveralvectorsthatwillbeusefullaterindescribingthecomponentsofourdiscretizationtechnique.Foreach(x;y)2Pwedenefourvectors.X(x;y)istheimagethetheunitvector(1;0)in^P;Y(x;y)istheimagethetheunitvector(0;1)in^P;x(x;y)isaunitvectororthogonaltoY(x;y);y(x;y)isaunitvectororthogonaltoX(x;y):FortheniteelementspacesweusethelowestorderRaviart-Thomaselementsonthequadrilateralelements,see[2],[14]and[11].Theycanbedenedasfollows.Thecharacteristicfunctionsofthequadrilateralsprovideabasisfortheniteelementspacefor.Thebasisfunctionsforv,arebestseenbyassociatingdegreesoffreedom2withnormalcomponentsonedgesofquadrilaterals.Atypicalbasisfunctionfortheniteelementspaceforvhassupportontwoadjacentquadrilateralsandhasaconstantnormalcomponentontheedgesharedbythequadrilateralsanditsnormalcomponentiszeroonotheredges.Themagnitudeofthebasisfunctionissuchthattheuxonthecommonedgeisone,Zedgevds=1:Theseconditionsalonedonotuniquelydeterminethebasisfunction;thefollowingadditionalconditionontheniteelementspaceisneeded.WithinanyquadrilateralP,vxkYkvarieslinearlywith^x;constantwith^y;vykXkvarieslinearlywith^y;constantwith^x:Atypicalbasisfunctionisrepresentedinthegure3.Wenotethatthebasisfunc-tionshavecontinuousnormalcomponentsacrossgridinterfaces.Withthiswecanguaranteethatourcomputedowvelocitywillalsohavecontinuousnormalcompo-nentacrossgridedges.Thetruephysicalsolutionalsohasthisproperty,continuousnormalcomponentofvelocities,butnoteverynumericalschemeforapproximatingitdoes,aspointedoutin[12].Wenowneedtochoosethecontrolvolumes.Thequadrilateralsusedtodescribethegridarethenaturalc