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IntroductionInreservoirengineeringorCO2geologicalstorage,aproperwellmodelingisrequiredtosimulateac-curatelymultiphaseflow.Anefficientmethodmusttakeintoaccountimportantwellparameterssuchasthesingularpressuredistributioninthewellvicinityandthelargedifferenceofscalesbetweenthewell-boreradiusandthereservoirdimension.Currentmethodsarebasedonsingle-phasenear-wellanalyticalsolutionsandwellindexesdevelopedinthepapersofPeaceman(1978,1983).Despitetheassump-tionsof2Duniformrectangulargridsinhomogeneousmedia,Peaceman’sapproachiswidelyusedinreservoirsimulation.However,thisapproachcannotbeusedinordertotakeaccuratelyintoaccountheterogeneities.Thisisinparticularthecasefordeviatedwellsandstratifiedheterogeneities.Forthisreasonaspecificmodelisneeded.Thisworkaddressesthe3Dnumericalsimulationofmultiphaseflowinnear-wellregions.Thenear-wellmodelproposedinthispaperisbasedon3Dmeshesthatarerefinedaroundthewellandontheuseofaccuratefinitevolumeschemes.Thefirststepofthediscretizationistocreatearadialmeshthatisexponentiallyrefineddowntothewellboundary.ThisradiallocalrefinementimpliestobuildamatchingmeshbetweentheradialgridandthereservoirCPGgridusingeitherhexahedraorbothtetrahedraandpyramidsasseeninFigure1.Thistransitionzoneenablestocouplethenear-wellsimulationandaglobalreservoirsimulation.(a)exponentiallyre-(b)unstructuredmeshwithonly(c)hybridmeshwithhexahedra,nedradialmeshhexahedratetrahedraandpyramidsFigure1Near-wellmeshesDuetothecomplexgeometry,multiphaseflowsimulationswithsuchkindofmeshesrequiremultipointfluxapproximationschemes(MPFA).Inthisframework,variousMPFAmethodsareused,namelytheO,L,GandGradCellschemes.TheOandLmethodsarewidelyusedintheoilindustryforthediscretizationofDarcyfluxesinmultiphaseporousmediaflowmodels,seeforexampleAavatsmark(2002,2007);Aavatsmarketal.(2008);Age´lasandMasson(2008);Edwards(2002);EdwardsandRogers(1998)andreferencestherein.TheGandGradCellschemesarenewermethods,detailedinAge´lasetal.(2010a,b)andbrieydescribedinthenextsection.TheGschemeisanextensionoftheLscheme.Ituses,astheOandLschemes,afluxformulationbasedontheconstructionofasubcellgradientaroundeachvertexsatisfyingfluxandpotentialcontinuityconditions.TheconstructionoftheGradCellschemeratherstartsfromanonsymmetricdiscretevariationalformulationbasedontwocellwiseconstantgradients.Thefirstgradientisconsistentandthesecondgradientsatisfiesaweakconvergenceproperty.BoththeO,L,GandtheGradCellschemesarecompactinthesensethattheyhaveasparsestencil,typicallyatmost27pointsfortopologicallyCartesianmeshes.Unfortunatelytheirnonsymmetryleadstoaconditionalcoercivitydepending,forgeneralpolyhedralmeshes,bothonthedistorsionofthemeshandonthepermeabilitytensor.TheSUSHIscheme(SchemeUsingStabilizationandHybridInterfaces)introducedinEymardetal.(2009)isasymmetricversionoftheGradCellschemeusingonlytheconsistentgradientinthediscretevariationalformulation.Itleadstoanunconditionallycoercivescheme,butthisisattheexpenseoftheECMORXII–12thEuropeanConferenceontheMathematicsofOilRecovery6–9September2010,Oxford,UK
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