A compressible viscous three-phase model for porous media flow based on the theory of mixtures

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Advances in Water Resources

journalhomepage:www.elsevier.com/locate/advwatres

A compressible viscous three-phase model for porous media flow based on the theory of mixtures

Yangyang Qiao

, Steinar Evje

University of Stavanger, Stavanger 4036, Norway

a r t i c le i n f o

Keywords:

Multiphase flow in porous media Three-phase flow

Viscous coupling Mixture theory Compressible model Water alternating gas (WAG) Waterflooding

a b s t r a ct

Inthispaperwefocusonageneralmodeltodescribecompressibleandimmisciblethree-phaseflowinporous media.TheunderlyingideaistoreplaceDarcy’slawbymoregeneralmomentumbalanceequations.Inparticular, wewanttoaccountforviscouscouplingeffectsbyintroducingfluid-fluidinteractionterms.In[Qiao,etal.

(2018)AdvWaterResour112:170–188]awater-oilmodelbasedonthetheoryofmixtureswasexplored.It wasdemonstratedhowtheinclusionofviscouscouplingeffectscouldallowthemodeltobettercaptureflow regimeswhichinvolveacombinationofco-currentandcounter-currentflow.Inthisworkweextendthemodel indifferentaspects:(i)accountforthreephases(water,oil,gas)insteadoftwo;(ii)dealwithboththecompressible andincompressiblecase;(iii)includeviscoustermsthatrepresentfrictionalforceswithinthefluid(Brinkman type).Amainobjectiveofthisworkistoexplorethisthree-phasemodel,whichappearstobemorerealisticthan standardformulation,inthecontextofpetroleumrelatedapplications.Wefirstprovidedevelopmentofstable numericalschemesinaone-dimensionalsettingwhichcanbeusedtoexplorethegeneralizedwater-oil-gasmodel, bothforthecompressibleandincompressiblecase.Then,severalnumericalexampleswithwaterfloodingina gasreservoirandwateralternatinggas(WAG)experimentsinanoilreservoirareinvestigated.Differencesand similaritiesbetweenthecompressibleandincompressiblemodelarehighlighted,andthefluid-fluidinteraction effectisillustratedbycomparisonofresultsfromthegeneralizedmodelandaconventionalmodelformulation.

1. Introduction Generally

Theprocessesofmultiphase flowin porousmediaoccurinmany subsurfacesystemsandhavefoundmanyapplicationsofpracticalin- terest,suchashydrology,petroleum engineering,geothermalenergy developmentandcarbonstorage(Bakhshianetal.,2019;Bakhshianand Hosseini,2019;Wu,2016).Theimmisciblethree-phaseflowisalways encounteredinwaterfloodingforoilreservoirswithgascap,inimmis- cibleCO₂storageindepletedoilandgasreservoirs,andsteamfloods andwater-alternating-gas(WAG)processes(BentsenandTrivedi,2012;

Juanes,2008).Darcy’slawwasoriginallydevelopedforsingle-phase flow(Darcy,1856).Conventionalmodelingofmultiphaseflowisnor- mallybasedonDarcy’sextendedlaw(Rose,2000)byincorporationof relativepermeabilities(Muskatetal.,1937).However,recentexperi- mentalobservationsindicatethattheflowmode(co-currentorcounter- current)canhaveastrongimpactontheflowingphasemobilities.That istosay,therelativepermeabilitiesarenotonlyfunctionofsaturation butarealsorelatedtotheeffectofhowthefluidsflowrelativelytoeach other(BentsenandManai,1992;BourbiauxandKalaydjian,1990).

∗Correspondingauthor.

E-mailaddress:steinar.evje@uis.no(Y.Qiao).

Viscouscoupling

Viscouscoupling(i.e.,fluid-fluidinteraction)wasfirstlymentioned byYuster(1951)byusingtheoreticalanalysistoderivethatrelativeper- meabilityisafunctionofbothsaturationandviscosityratio.Inaddition, capillarynumberwasalsoproposedtobeafactoraffectingrelativeper- meabilities(Ehrlich, 1993;AvraamandPayatakes,1995).Ingeneral, momentumtransferduetodifferencesininterstitialvelocitiesinduces accelerationofthesloweranddecelerationofthefastermovingfluid whenthefluidsaremovingco-currently.Decelerationofbothfluidve- locitieswilloccuriftheyaremovingcounter-currently(Ayodele,2006;

BentsenandManai,1993;DullienandDong,1996;Lietal.,2004).

Inordertoextendthesingle-phaseDarcy’slawtomultiphaseflow, delaCruzandSpanos(1983)derivedtheoreticallyDarcy’sempirical extendedlawbyapplyingthemethodofvolumeaveragingtoStokes equation.InKalaydjian(1987,1990),Kalaydjiandevelopedflowequa- tionsusingtheconceptsofirreversiblethermodynamics(Katchalskyand Curran, 1975) from amacroscopic understandingof two-phase flow in porous media.Inaddition,some researcherstried togaininsight into howtwo immisciblephases flow through a porousmedium by using simpleanalogous modelssuch as tubular flow (Yuster, 1951;

Bacrietal.,1990).InLangaasandPapatzacos(2001)LangaasandPa-

https://doi.org/10.1016/j.advwatres.2020.103599

Received15May2019;Receivedinrevisedform21April2020;Accepted22April2020 Availableonline28April2020

0309-1708/© 2020TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

patzacosusedtheLatticeBoltzmann(LB)approachtoinvestigateef- fectsof viscouscouplingandconcludedthatcounter-current relative permeabilitiescausedpartlybyviscouscouplingarealwayslessthan thecorrespondingco-currentcurvesunderdifferentlevelsofcapillary forces.Usingthesamemethod,Lietal.(2005)showedthattheirmodel wasabletocapturemainexperimentaleffectscausedbyviscouscou- pling.Theyalsomentionedthattheinterfacialareabetweenthefluids isakeyvariableforrelativepermeabilityfunctionsfortwoimmisci- blefluidsflowinporousmedia.Ageneralizedmodelwasdevelopedin Qiaoetal.(2018)fortwo-phaseflowwithviscouscouplingeffect.Nu- mericalinvestigationsshowedabetteragreementwiththeexperimental tests(BourbiauxandKalaydjian,1990)comparedtotheconventional modeling.TheauthorsinBentsenandTrivedi(2012)constructedmod- ifiedtransportequationsforbothco-currentandcounter-currentthree- phaseflowthroughaverticalincompressiblemodelbasedonpartition concepts.Theirequationsareusedtoestimatetheamountofmodeler- rorbecauseofafailuretoaccountfortheeffectofinterfacialcoupling whichhastwotypes:viscouscouplingandcapillarycoupling.Moreover, SherafatiandJessen(2017)investigatedtheeffectofmobilitychanges duetoflow reversalsfrom co-currenttocounter-currentflow onthe displacementofWAGinjectionprocesses.

Complexmultiphaseflowinporousmediaanduseofthetheoryofmixtures

Motivated by petroleum related applications variousattempts to solvethethree-phaseporousmediaflowmodelhavebeenreporteddur- ingthepastdecade(FallsandSchulte,1992;GuzmánandFayers,1997a;

1997b;JuanesandPatzek,2004).Aninterestinginvestigationwascar- riedoutinLieandJuanes(2005)whereafront-trackingalgorithmwas proposedforconstructing veryaccuratesolutionstoone-dimensional problems(forexampleWAGtesttherein).Thiswasexploredinthecon- textof streamline simulationwhich decouples thethree-dimensional problemintoasetofone-dimensionalproblemsalongstreamlines.This workislimitedtothree-phaseimmiscible,incompressibleflowandalso gravityandcapillaritywereignored.Differentnumericalmethodshave beenimplementedtosimulatethree-phaseflowinporousmedia.Afinite volumemethodwasusedinLeeetal.(2008)forsolvingcompressible, immiscibleflowwithgravityinheterogeneousformationsbyusingthe blackoilformulation.Ahybrid-upwindingschemeforphasefluxwas proposed inLee andEfendiev (2016)fora finitedifferenceapproxi- mationtosolvethethreephasetransportequationsinthepresenceof viscousandbuoyancyforces.Afiniteelementmethodwasappliedto simulatefluidinjectionandimbibitionprocessesinadeformableporous media(Gajoetal.,2017).Moreover,(DongandRivière,2016)applied asemi-implicitmethodwithdiscontinuousGalerkin(DG)discretization tosolvetheincompressiblethree-phaseflowintwodimensions.Addi- tionalphysicaleffectsarealsodiscussedandexploredforthree-phase porousmediaflow,suchashysteresiseffectsofrelativepermeabilities (Ranaee etal., 2019) andellipticregions(JuanesandPatzek, 2004;

Juanes,2008;LeeandEfendiev,2016).InJuanes(2008)Juanespre- sentedanonequilibriummodelofincompressiblethree-phaseflowin porousmedia.Thenonequilibriumeffectsbyintroducingapairofef- fectivewaterandgassaturationsintotheformulationshavetheability tosmearsaturationfrontsfromnumericalsimulations.

Thetheory ofmixturesoffersageneralframeworkfordeveloping modelsforcomplexmultiphaseflowsystems(Rajagopal,2007).More lately,biomedicalapplicationshavebeenadriverforthedevelopment ofvariousmodelsrelyingonthisapproach.Forexample,thestudyhow cancercellsareabletobreakloosefromaprimarytumorinvolvesasolid matrix(theso-calledextracellularmatrix),differenttypeofcells(can- cercells,stromalcells,immunecells),andinterstitialfluid(Evje,2017;

EvjeandWaldeland,2019).Arecentexample ofthis isdescribed in WaldelandandEvje(2018b);Urdaletal.(2019)where,respectively,a cell-fluidtwo-phasemodelandacell-fibroblast-fluidthree-phasemodel aredevelopedtoshedlightontheexperimentallyobservedtumorcell behaviorreported in Shieh etal. (2011). Themodel thatis derived

relies on replacingDarcy’slaw bymore generalmomentumbalance equationswhichincorporateboththecell-matrixresistanceforceand the cell-fibroblastinteraction. Thelatter is understood asa ”viscous coupling” effectcausedbyamechanicalcouplingthatcan occurbe- tweentumorcellsandfibroblastsandhasbeenreportedinexperimental studies(Labernadie,2017).Anotherexamplehowgeneralizedmomen- tumequationscanbeusedtocapturenon-standardmultiphasebehav- iorinthecontextofaggressive tumorcellsis exploredinWaldeland andEvje(2018a).InPolachecketal.(2011)twocompetingmigration mechanismswereobserved,oneintheupstreamdirectionandanother inthedownstreamdirection.Theuseofgeneralizedmomentumequa- tionsallowedustoaccountforboththisfluid-stressgeneratedupstream migrationandachemotactivemigrationinthedirectionofincreasing concentrationofchemicalconcentrations(WaldelandandEvje,2018a).

Theaimofthiswork

Theobjective ofthis paperistoinvestigate amixturetheory ap- proach tosimulatethreeimmisciblefluidsflowingina1Dreservoir.

Weshallconsiderboththecasewithcompressibleandincompressible fluids.Themodelwhichisintroducedisquitegeneralsinceitcanau- tomaticallycaptureflowthatinvolvesacombinationofco-currentand counter-currentflow.Thecurrentworkrepresentsextensionofprevious workintwoways:

• Extendtheincompressibletwo-phasemodelthatwasexploredin Qiaoetal.(2018);Andersenetal.(2019)toincludethreephases.

• Extend the compressible two-phase model studied in Qiaoetal.(2019a)toincludethreephases.

Inaddition,themodelswestudyinthecurrentworkaremoregen- eralthanthosestudiedinQiaoetal.(2018);Andersenetal.(2019)since we considerStokeslike momentumequationswhich involveviscous termsthataccountforinternalfrictionduetoviscosity.Inparticular, appropriatenumericalschemesareintroducedtoinvestigatecompress- ibleandincompressiblethree-phaseflowscenariosthataremotivated byinjection-productionflowscenarios.

Mainobservationsfrom ournumericalexperiments withtwoand three-phaseflowscenarioswheretheflowdynamicsaregeneratedby injectionof waterorgasin thecenterofthedomainandproduction offluidsattheleftandrightboundaryare:(i)Thesimulationcasesin- volvecompetitionbetweenpressuredrivenco-currentflowandcounter- currentgravitydrivenflow;(ii)Boththeincompressibleandcompress- iblediscreteversionofthemodelappeartohavegoodstabilityprop- erties.Thenumericalexperimentsindicatethatthenumericalschemes canbeusefulasatooltodeepentheinsightintotherelationbetweenthe incompressibleandcompressibleversionofthemodel.Themodeland itsdiscreteapproximatecounterpartsappeartobeagoodstartingpoint forextendingtomorecomplexflowsystems,asmentionedabove,that involvecompetitionbetweendifferentdistinct,non-standardtransport mechanisms.

The rest of this paper is organized as follows. In Section 2 we briefly describe the mixture flux approach in a three-phase setting.

In Section 3we summarize the generalizedthree-phase porous me- dia model, botha compressibleandanincompressible versionofit.

Section 4is devotedtonumerical simulationstodemonstratethree- phase dynamics andverifybasic featuresof the numericalschemes.

Thedetailsofthecompressibleandincompressibleschemearegiven inAppendixA–AppendixD.

2. Mixturetheoryframework

2.1. ConventionalmodelbasedonDarcy’slaw

We firstly describe the traditional formulation of incompressible multiphaseflowmodelwithoutsourceterms.Transportequationsfor

incompressibleandimmisciblephasesoil(o),water(𝑤)andgas(g)in porousmediaarenormallygivenby:

𝜕_𝑡(𝜙𝑠_𝑖)+∇⋅𝐔_𝑖=𝑄_𝑖, (2.1)

𝐔_𝑖=𝜙𝑠_𝑖𝐮_𝑖, (𝑖=𝑤,𝑜,𝑔), (2.2)

where𝜙isporosity,s_i isphasesaturation,Q_iisthesourceterm,and U_iandu_iaretheDarcyvelocityandinterstitialvelocityofeachphase 𝑖=𝑜,𝑤,𝑔,respectively.Forsimplicitytheirreducible(immobile)phase saturation(s_ir)isnotconsideredintheequationsbyassumingitisequal to0.Hence,thenormalizedphasesaturation(= ^{𝑠𝑖−𝑠𝑖𝑟}

1−𝑠𝑤𝑟−𝑠𝑜𝑟−𝑠𝑔𝑟)equalsthe phasesaturationvalues_i. Thetraditionalmacroscopicformulationof Darcy’slawthatrelatesthevolumetricfluxofaphasetothepressure gradientofthatphaseisgivenby:

𝐔_𝑖=−𝐾𝑘𝑟𝑖

𝜇_𝑖 (∇𝑝_𝑖−𝜌_𝑖𝐠), (𝑖=𝑤,𝑜,𝑔), (2.3) whereKistheabsolutepermeabilityofporousmedia,p_iisphasepres- sure,gistheaccelerationofgravityandk_ri,𝜌iand𝜇iarephaserelative permeability,densityandviscosity,respectively.

2.2. Ageneralizedmultiphaseflowmodelbasedonmixturetheory

Forourinvestigations,themassbalanceequationswithsourceterms inthecaseofcompressiblewater-oil-gastransportcanbegivenby:

(𝜙𝑛_𝑤)_𝑡+∇⋅(𝜙𝑛_𝑤𝑢_𝑤)=−𝑛_𝑤𝑄_𝑝+𝜌_𝑤𝑄_𝐼𝑤, 𝑛_𝑤=𝑠_𝑤𝜌_𝑤 (𝜙𝑛_𝑜)_𝑡+∇⋅(𝜙𝑛_𝑜𝑢_𝑜)=−𝑛_𝑜𝑄_𝑝, 𝑛_𝑜=𝑠_𝑜𝜌_𝑜

(𝜙𝑛𝑔)_𝑡+∇⋅(𝜙𝑛𝑔𝑢𝑔)=−𝑛𝑔𝑄𝑝+𝜌𝑔𝑄𝐼𝑔, 𝑛𝑔=𝑠𝑔𝜌𝑔 (2.4) whereu_i, (𝑖=𝑤,𝑜,𝑔) representstheinterstitialvelocityof phasei in theporousmedia.Inaddition,Q_pistheproductionrateand𝑄_𝐼𝑤,Q_Ig representtheinjectionrateofwaterandgas,respectively.

Thestartingpointfordevelopingour modelthatcanaccount for moredetailedphysicalmechanismsforwater-oil-gasporousmediaflow thanconventionalmodeling,isthetheoryofmixtures.Thisisatheory basedonbalancelawsandconservationprinciples,whichiswellknown incontinuummechanics(Bowen,1976;RajagopalandTao,1995;Byrne andPreziosi,2003; AmbrosiandPreziosi,2002; PreziosiandFarina, 2002),andhasbeenwidelyappliedtothebiologicaltumor-growthsys- temswhichcanbecharacterizedasamixtureofinteractingcontinua.

Neglecting inertialeffects (accelerationeffects),as isusual when dealingwithcreepingflowinporousmaterials,themechanicalstress balanceisgivenbyAmbrosiandPreziosi(2002):

0=∇⋅(𝑠_𝑖𝜎_𝑖)+𝑚_𝑖+𝐺_𝑖, (𝑖=𝑤,𝑜,𝑔), (2.5) where𝜎ireferstotheCauchystresstensor,m_irepresentstheinteraction forcesexertedontheconstituentsbyotherconstituentsofthemixture, and𝐺_𝑖=𝑠_𝑖𝜌_𝑖𝑔istheexternalbodyforceduetogravity.Thestandard expressionforthestressterms𝜎i,isgivenby

𝜎_𝑖=−𝑝_𝑖𝛿+𝜏_𝑖, (𝑖=𝑤,𝑜,𝑔), (2.6)

where𝛿istheunitarytensorand 𝜏_𝑖=2𝜇_𝑖𝑒_𝑖, 𝑒_𝑖=1

2(∇𝑢_𝑖+∇𝑢^𝑇_𝑖), (𝑖=𝑤,𝑜,𝑔). (2.7) Theviscouspart𝜏i reflectsthatthewater,oilandgasbehaveasavis- cousfluid.Accordingtogeneralprinciplesofthetheoryofmixtures,the interactionforcesm_ibetweentheconstituentsappearingin(2.5)maybe describedasinPreziosiandFarina(2002);AmbrosiandPreziosi(2002); ByrneandPreziosi(2003):

𝑚_𝑜=𝑝_𝑜∇𝑠_𝑜+𝐹_𝑤𝑜−𝐹_𝑜𝑔+𝑀_𝑜𝑚, 𝑚_𝑤=𝑝_𝑤∇𝑠_𝑤−𝐹_𝑤𝑜−𝐹_𝑤𝑔+𝑀_𝑤𝑚,

𝑚_𝑔=𝑝_𝑔∇𝑠_𝑔+𝐹_𝑤𝑔+𝐹_𝑜𝑔+𝑀_𝑔𝑚, (2.8)

whereF_ij(𝑖,𝑗=𝑜,𝑤,𝑔),denotestheforce(drag)thattheiphaseexerts on thejphase.Thejphaseexerts anequalandoppositeforce −𝐹_𝑖𝑗. Similarly,M_om,𝑀_𝑤𝑚andM_gmrepresentinteractionforces(dragforces) betweenfluidandporewalls(solidmatrix),respectively,foroil,water andgas.Thetermp_i∇s_i isrelatedtointerfacialforceexertedbyother phaseson phasei, arisingfrommathematical derivationofaveraged equations(DrewandSegel,1971).Toclosethesystemwemustspecify thedragforce term𝐹_𝑤𝑜,𝐹_𝑤𝑔,andF_og andthestresses 𝜎i (𝑖=𝑜,𝑤,𝑔) andinteractionforcetermsM_imbetweenfluid(𝑖=𝑤,𝑜,𝑔)andmatrix.

Dragforce representstheinteractionbetweenonephaseandanother phaseandismodelledasRajagopal(2007);PreziosiandFarina(2002); AmbrosiandPreziosi(2002):

𝐹_𝑤𝑜 =̂𝑘_𝑤𝑜(𝑢_𝑤−𝑢_𝑜), 𝐹_𝑤𝑔 =̂𝑘_𝑤𝑔(𝑢_𝑤−𝑢_𝑔),

𝐹_𝑜𝑔=̂𝑘_𝑜𝑔(𝑢_𝑜−𝑢_𝑔), (2.9)

wherê𝑘_𝑖𝑗(𝑖,𝑗=𝑜,𝑤,𝑔),remainstobedetermined.Typically,̂𝑘_𝑖𝑗∼𝑠_𝑖𝑠_𝑗 toreflectthatthisforcetermwillvanishwhenoneofthephasesvan- ishes.Similarly,theinteractionforcebetweenfluidandporewall(ma- trix,whichisstagnant)isnaturallyexpressedthenas(Rajagopaland Tao,1995; Rajagopal,2007;PreziosiandFarina,2002;Ambrosi and Preziosi,2002):

𝑀𝑖𝑚=−̂𝑘_𝑖𝑢𝑖, (𝑖=𝑜,𝑤,𝑔). (2.10)

Thecoefficients ̂𝑘_𝑖𝑗 and̂𝑘_𝑖(dimensionPa· s/m²),thatcharacterizethe magnitudeofinteractionterms,canbechosensuchthatthemodelre- coverstheclassicalporousmediamodelbasedonDarcy’slaw.Atthe sametimetheapproachusedherewillopenfordevelopmentofreser- voirmodelswheremoredetailedphysicscanbetakenintoaccount.

3. Asummaryofthegeneralthree-fluidmodelforporousmedia flow

3.1. Thecompressiblecase

Weareinterestedinstudyinga1-Dmodelforthreecompressibleim- misciblefluidsmovinginaporousmedia.Aftercombining(2.4)-(2.10) themodeltakesthefollowingform:

(𝜙𝑛_𝑤)_𝑡+(𝜙𝑛_𝑤𝑢_𝑤)_𝑥=−𝑛_𝑤𝑄_𝑝+𝜌_𝑤𝑄_𝐼𝑤, 𝑛_𝑤=𝑠_𝑤𝜌_𝑤, (𝜙𝑛_𝑜)_𝑡+(𝜙𝑛_𝑜𝑢_𝑜)_𝑥=−𝑛_𝑜𝑄_𝑝, 𝑛_𝑜=𝑠_𝑜𝜌_𝑜, (𝜙𝑛_𝑔)_𝑡+(𝜙𝑛_𝑔𝑢_𝑔)_𝑥=−𝑛_𝑔𝑄_𝑝+𝜌_𝑔𝑄_𝐼𝑔, 𝑛_𝑔=𝑠_𝑔𝜌_𝑔

𝑠_𝑤(𝑃_𝑤)_𝑥=−̂𝑘_𝑤𝑢_𝑤−̂𝑘_𝑤𝑜(𝑢_𝑤−𝑢_𝑜)−̂𝑘_𝑤𝑔(𝑢_𝑤−𝑢_𝑔) +𝑛_𝑤𝑔+𝜀_𝑤(𝑛_𝑤𝑢_𝑤𝑥)_𝑥,

𝑠𝑜(𝑃𝑜)_𝑥=−̂𝑘_𝑜𝑢𝑜−̂𝑘_𝑤𝑜(𝑢𝑜−𝑢𝑤)−̂𝑘_𝑜𝑔(𝑢𝑜−𝑢𝑔) +𝑛_𝑜𝑔+𝜀_𝑜(𝑛_𝑜𝑢_𝑜𝑥)_𝑥,

𝑠_𝑔(𝑃_𝑔)_𝑥=−̂𝑘_𝑔𝑢_𝑔−̂𝑘_𝑤𝑔(𝑢_𝑔−𝑢_𝑤)−̂𝑘_𝑜𝑔(𝑢_𝑔−𝑢_𝑜) +𝑛_𝑔𝑔+𝜀_𝑔(𝑛_𝑔𝑢_𝑔𝑥)_𝑥,

Δ𝑃_𝑜𝑤(𝑠_𝑤)=𝑃_𝑜−𝑃_𝑤, Δ𝑃_𝑔𝑜(𝑠_𝑔)=𝑃_𝑔−𝑃_𝑜 (3.11) withcapillarypressureΔ𝑃_𝑜𝑤definedasthepressuredifferencebetween theoilandwaterandcapillarypressureΔP_go definedasthepressure differencebetweenthegasandoil.Wemaychoosetousethefollowing expressionsforcapillaryforce

Δ𝑃_𝑜𝑤=𝑃_𝑜−𝑃_𝑤=Δ𝑃_𝑜𝑤(𝑠_𝑤)=−𝑃_𝑐^∗₁ln(𝛿1+𝑠_𝑤 𝑎1

) and 𝛿1,𝑎1>0, Δ𝑃_𝑔𝑜=𝑃_𝑔−𝑃_𝑜=Δ𝑃_𝑔𝑜(𝑠_𝑔)=𝑃_𝑐^∗₂𝑠^𝑎_𝑔² and 𝑎2>0 (3.12) withnon-negativeconstants 𝑃_𝑐𝑖^∗ representinginterfacialtension.This allowsustomimiccapillarypressurefunctionsthatprevisouslyhave beenproposedforthree-phasereservoirflow(ChenandEwing,1997;

OddandDavid,2010).Inaddition,wehavethefundamentalrelation thatthethreephasesfilltheporespace

𝑠_𝑜+𝑠_𝑤+𝑠_𝑔=1. (3.13)

Theabovemodelmustbecombinedwithappropriateclosurerelations for𝜌_𝑖=𝜌_𝑖(𝑃_𝑖).Werepresentthethreephasesbylinearpressure-density relationsoftheform

𝜌_𝑤−̃𝜌_𝑤0=𝑃_𝑤

𝐶_𝑤, 𝜌_𝑜−̃𝜌_𝑜0= 𝑃_𝑜

𝐶_𝑜, 𝜌_𝑔= 𝑃_𝑔

𝐶_𝑔 (3.14)

where𝐶𝑤,C_oandC_grepresenttheinverseofthecompressibilityofwa- ter,oilandgas,respectively.

WerefertoAppendixBforasemi-discreteapproximationof(3.11)as wellasafullydiscretescheme.

Remark3.1.Wemayalsostudyahigherdimensionalcase(e.g.,2D) wherethe modelconsists of three massbalanceequations for three phases(water,oilandgas)andsixmomentumequations(eachphase hastwodirectionssuchasxandy).Theschemehasbeentestedin2D fortwophasesandshowssimilarpropertiesasin1D.

3.2. Theincompressiblecase

3.2.1. Viscousflow

Wemaylet𝐶_𝑤,𝐶_𝑜,𝐶_𝑔 gotoinfinityin(3.14).Thenweobtainthe incompressibleversionofthemodel(3.11).WerefertoAppendixCfor asemi-discreteaswellasafullydiscreteschemeforthisincompressible case.

3.2.2. Inviscidflow

Moreover,inordertorelatethisincompressibleversiontotheclas- sicalDarcy-basedformulationweignoretheviscositytermsinthemo- mentumequations by setting 𝜀_𝑖=0(𝑖=𝑤,𝑜,𝑔)in (3.11)_4,5,6. Solving momentumequationswithrespecttointerstitialphasevelocitiesu_i,the Darcyvelocitiesoffluidphaseareexpressedasfollowsbasedon(2.2): 𝑈𝑤 =𝜙𝑠𝑤𝑢𝑤=−𝜆𝑤𝑤(𝑃𝑤𝑥−𝜌𝑤𝑔)−𝜆𝑤𝑜(𝑃𝑜𝑥−𝜌𝑜𝑔)−𝜆𝑤𝑔(𝑃𝑔𝑥−𝜌𝑔𝑔),

𝑈_𝑜 =𝜙𝑠_𝑜𝑢_𝑜=−𝜆_𝑤𝑜(𝑃_𝑤𝑥−𝜌_𝑤𝑔)−𝜆_𝑜𝑜(𝑃_𝑜𝑥−𝜌_𝑜𝑔)−𝜆_𝑜𝑔(𝑃_𝑔𝑥−𝜌_𝑔𝑔), 𝑈_𝑔 =𝜙𝑠_𝑔𝑢_𝑔=−𝜆_𝑤𝑔(𝑃_𝑤𝑥−𝜌_𝑤𝑔)−𝜆_𝑜𝑔(𝑃_𝑜𝑥−𝜌_𝑜𝑔)−𝜆_𝑔𝑔(𝑃_𝑔𝑥−𝜌_𝑔𝑔), (3.15) andthefollowingrelationsaredefined:

𝜆𝑤𝑤 = 𝜙𝑠²_𝑤

𝑅 (𝑅𝑜𝑅𝑔−̂𝑘²_𝑜𝑔), 𝜆𝑤𝑜=𝜆𝑜𝑤=𝜙𝑠_𝑤𝑠_𝑜

𝑅 (̂𝑘_𝑤𝑜𝑅𝑔+̂𝑘_𝑜𝑔̂𝑘_𝑤𝑔), 𝜆_𝑜𝑜 = 𝜙𝑠²_𝑜

𝑅 (𝑅_𝑤𝑅_𝑔−̂𝑘²_𝑤𝑔), 𝜆_𝑤𝑔=𝜆_𝑔𝑤= 𝜙𝑠_𝑤𝑠_𝑔

𝑅 (̂𝑘_𝑤𝑔𝑅_𝑜+̂𝑘_𝑜𝑔̂𝑘_𝑤𝑜), 𝜆_𝑔𝑔 = 𝜙𝑠²_𝑔

𝑅 (𝑅_𝑤𝑅_𝑜−̂𝑘²_𝑤𝑜), 𝜆_𝑜𝑔=𝜆_𝑔𝑜=𝜙𝑠_𝑜𝑠_𝑔

𝑅 (̂𝑘_𝑜𝑔𝑅_𝑤+̂𝑘_𝑤𝑔̂𝑘_𝑤𝑜), (3.16) where

𝑅_𝑤= ̂𝑘_𝑤+̂𝑘_𝑤𝑔+̂𝑘_𝑤𝑜, 𝑅𝑜= ̂𝑘_𝑜+̂𝑘_𝑤𝑜+̂𝑘_𝑜𝑔, 𝑅_𝑔 = ̂𝑘_𝑔+̂𝑘_𝑤𝑔+̂𝑘_𝑜𝑔,

𝑅= ̂𝑘_𝑤̂𝑘_𝑜̂𝑘_𝑔+(̂𝑘_𝑤+̂𝑘_𝑜+̂𝑘_𝑔)(̂𝑘_𝑤𝑔̂𝑘_𝑤𝑜+̂𝑘_𝑜𝑔̂𝑘_𝑤𝑜+̂𝑘_𝑤𝑔̂𝑘_𝑜𝑔)

+̂𝑘_𝑔̂𝑘_𝑤𝑜(̂𝑘_𝑤+̂𝑘_𝑜)+̂𝑘_𝑤̂𝑘_𝑜𝑔(̂𝑘_𝑜+̂𝑘_𝑔)+̂𝑘_𝑜̂𝑘_𝑤𝑔(̂𝑘_𝑤+̂𝑘_𝑔). (3.17) Usingcapillarypressurerelations(3.12)itfollowsthat(3.15)take thefollowingequivalentform:

𝑈_𝑤 =−̂𝜆_𝑤𝑃_𝑤𝑥−(𝜆_𝑤𝑜+𝜆_𝑤𝑔)Δ𝑃_𝑜𝑤𝑥−𝜆_𝑤𝑔Δ𝑃_𝑔𝑜𝑥 +(𝜆𝑤𝑤𝜌𝑤+𝜆𝑤𝑜𝜌𝑜+𝜆𝑤𝑔𝜌𝑔)𝑔,

𝑈_𝑜 =−̂𝜆_𝑜𝑃_𝑤𝑥−(𝜆_𝑜𝑜+𝜆_𝑜𝑔)Δ𝑃_𝑜𝑤𝑥−𝜆_𝑜𝑔Δ𝑃_𝑔𝑜𝑥+ (𝜆_𝑤𝑜𝜌_𝑤+𝜆_𝑜𝑜𝜌_𝑜+𝜆_𝑜𝑔𝜌_𝑔)𝑔, 𝑈_𝑔 =−̂𝜆_𝑔𝑃_𝑤𝑥−(𝜆_𝑔𝑔+𝜆_𝑜𝑔)Δ𝑃_𝑜𝑤𝑥−𝜆_𝑔𝑔Δ𝑃_𝑔𝑜𝑥

+(𝜆_𝑤𝑔𝜌_𝑤+𝜆_𝑜𝑔𝜌_𝑜+𝜆_𝑔𝑔𝜌_𝑔)𝑔. (3.18)

Herewedefinethefollowingnotationforgeneralizedphasemobilities

̂𝜆_𝑖:

̂𝜆_𝑤=𝜆_𝑤𝑤+𝜆_𝑤𝑜+𝜆_𝑤𝑔,

̂𝜆_𝑜=𝜆_𝑜𝑜+𝜆_𝑤𝑜+𝜆_𝑜𝑔,

̂𝜆_𝑔=𝜆_𝑔𝑔+𝜆_𝑤𝑔+𝜆_𝑜𝑔.

(3.19)

Bysumming𝑈_𝑤,U_oandU_gin(3.18)andusingthenotationintroduced in(3.19),thetotalDarcyvelocitycanbeexpressedasfollows:

𝑈_𝑇=−̂𝜆_𝑇𝑃_𝑤𝑥−(̂𝜆_𝑜+̂𝜆_𝑔)Δ𝑃_𝑜𝑤𝑥−̂𝜆_𝑔Δ𝑃_𝑔𝑜𝑥+(̂𝜆_𝑤𝜌_𝑤+̂𝜆_𝑜𝜌_𝑜+̂𝜆_𝑔𝜌_𝑔)𝑔 (3.20)

wherewehaveused

̂𝜆_𝑇= ̂𝜆_𝑤+̂𝜆_𝑜+̂𝜆_𝑔. (3.21)

Therefore,thewaterpressuregradientcanbederivedfrom(3.20): 𝑃_𝑤𝑥=− 1

̂𝜆_𝑇𝑈_𝑇−(𝑓̂_𝑜+𝑓̂_𝑔)Δ𝑃_𝑜𝑤𝑥−𝑓̂_𝑔Δ𝑃_𝑔𝑜𝑥+(𝑓̂_𝑤𝜌_𝑤+𝑓̂_𝑜𝜌_𝑜+𝑓̂_𝑔𝜌_𝑔)𝑔 (3.22)

withgeneralizedfractionalflowfunction:

𝑓̂_𝑖= ̂𝜆_𝑖∕̂𝜆_𝑇, (𝑖=𝑤,𝑜,𝑔). (3.23)

Inserting(3.22)into(3.18)weget:

𝑈_𝑤=𝑓̂_𝑤𝑈_𝑇+(𝑊_𝑜+𝑊_𝑔)Δ𝑃_𝑜𝑤𝑥+𝑊_𝑔Δ𝑃_𝑔𝑜𝑥−(𝑊_𝑤𝜌_𝑤+𝑊_𝑜𝜌_𝑜+𝑊_𝑔𝜌_𝑔)𝑔, 𝑈_𝑜=𝑓̂_𝑜𝑈_𝑇+(𝑂_𝑜+𝑂_𝑔)Δ𝑃_𝑜𝑤𝑥+𝑂_𝑔Δ𝑃_𝑔𝑜𝑥−(𝑂_𝑤𝜌_𝑤+𝑂_𝑜𝜌_𝑜+𝑂_𝑔𝜌_𝑔)𝑔, 𝑈_𝑔 =𝑓̂_𝑔𝑈_𝑇+(𝐺_𝑜+𝐺_𝑔)Δ𝑃_𝑜𝑤𝑥+𝐺_𝑔Δ𝑃_𝑔𝑜𝑥−(𝐺_𝑤𝜌_𝑤+𝐺_𝑜𝜌_𝑜+𝐺_𝑔𝜌_𝑔)𝑔, (3.24) where

𝑊_𝑖= ̂𝜆_𝑤𝑓̂_𝑖−𝜆_𝑤𝑖, 𝑂_𝑖= ̂𝜆_𝑜𝑓̂_𝑖−𝜆_𝑜𝑖,

𝐺_𝑖= ̂𝜆_𝑔𝑓̂_𝑖−𝜆_𝑔𝑖, (𝑖=𝑤,𝑜,𝑔). (3.25)

Itshouldbenotedthat𝑊_𝑖+𝑂_𝑖+𝐺_𝑖=0(𝑖=𝑤,𝑜,𝑔)inlightof(3.16), (3.21),and(3.23).

4. Numericalexamples

Wemainlyfocusonareservoirmodelwherethereareoneinjection well atthecenterandtwoproductionwellsdistributedattwosides.

Theinjectionrateisequaltothetotalproductionrateandtheratesof twoproductionwellsarealsosame(SeeFig.1).Inaddition,reservoir inclination𝜃isalsoaccountedforinthemodel.

Fig.1. Reservoirmodelwithinjectionandproduction.

Fig.2. Left:Capillarypressurebetweenwater andoil.Right:Capillarypressurebetweenoil andgas.Wereferto(3.12)fortheirexpressions andTable1fortheinputparameters.

Fig.3. Waterfractionalflowfunction𝑓_𝑤(𝑠_𝑤,𝑠_𝑜)(definedin(4.27))witheffectsofmodelinclination𝜃andtotalflowdirectionofU_T.(A):𝑠𝑖𝑛𝜃=0,𝑈_𝑇=𝑄_𝑝∕2;(B):

𝑠𝑖𝑛𝜃=1,𝑈_𝑇=−𝑄_𝑝∕2;(C):𝑠𝑖𝑛𝜃=1,𝑈_𝑇 =𝑄_𝑝∕2.

Interactionterms

Themodel(3.11)_4,5,6shouldbearmedwithappropriatefunctional correlationsforfluid-rockresistanceforcê𝑘_𝑤,̂𝑘_𝑜,̂𝑘_𝑔andfluid-fluiddrag forcê𝑘_𝑤𝑜,̂𝑘_𝑤𝑔,̂𝑘_𝑜𝑔.Hereweusetheinteractiontermssuggestedinthe recentworks(Standnesetal.,2017;Qiaoetal.,2018;Andersenetal., 2019):

̂𝑘_𝑤 ∶=𝐼_𝑤𝑠^𝛼_𝑤𝜇_𝑤

𝐾𝜙, ̂𝑘_𝑜∶=𝐼_𝑜𝑠^𝛽_𝑜𝜇_𝑜

𝐾𝜙, ̂𝑘_𝑔∶=𝐼_𝑔𝑠^𝛾_𝑔𝜇_𝑔 𝐾𝜙,

̂𝑘_𝑤𝑜 ∶=𝐼_𝑤𝑜𝑠_𝑤𝑠_𝑜𝜇_𝑤𝜇_𝑜

𝐾 𝜙, ̂𝑘_𝑤𝑔∶=𝐼_𝑤𝑔𝑠_𝑤𝑠_𝑔𝜇_𝑤𝜇_𝑔

𝐾 𝜙, ̂𝑘_𝑜𝑔∶=𝐼_𝑜𝑔𝑠_𝑜𝑠_𝑔𝜇_𝑜𝜇_𝑔 𝐾 𝜙.

(4.26)

Alltheinteractiontermŝ𝑘_𝑖and̂𝑘_𝑖𝑗havedimensionPa· s/m².Theparam- eters𝛼,𝛽and𝛾aredimensionlessexponentswhereas𝐼_𝑤,I_oandI_gare dimensionlessfrictioncoefficientscharacterizingthestrengthoffluid- solidinteraction.Finally,𝐼_𝑤𝑜,𝐼_𝑤𝑔andI_ogarecoefficientscharacterizing thestrengthofthefluid-fluiddragforcewithdimension(Pa· s)⁻¹.

Inputdata

TheinputparametersusedinthesimulationsarelistedinTable1. We use 101 grid cells for a 100-meter reservoir layer. We refer to AppendixDforaconvergencetest.Themagnitudeoftheinteraction coefficients𝐼_𝑤𝑜,𝐼_𝑤𝑔,andI_ogarechosenasinQiaoetal.(2018)where weappliedageneralizedtwo-phasemodeltomatchtheexperimentally measuredrelativepermeabilitycurvesandobtainedvaluesfortheinput parameterssuchas𝐼_𝑤𝑜whosemagnitudeisaroundseveralthousands.

Inorder toavoid toomanycomplicatingeffectsatthesametimein thesubsequentdiscussion,wehavesettheviscositytermstozero,i.e., 𝜀_𝑤=𝜀_𝑜=𝜀_𝑔=0.

We use the similar capillary pressure relations as Qiaoetal.(2019b)forwaterandoilandLewisandPao(2002)foroil andgas(seeFig.2).Theexpressionofaneffectivewaterfractionalflow function𝑓𝑤(𝑠𝑤,𝑠𝑜)intheconventionalwater-oil-gasmodel(assuming nocapillarypressure,i.e.,Δ𝑃_𝑜𝑤=Δ𝑃_𝑔𝑜=0)is

𝑓_𝑤(𝑠_𝑤,𝑠_𝑜)∶=^def 𝑈_𝑤 𝑈𝑇 =

̂𝜆_𝑤

̂𝜆_𝑇𝑈_𝑇−(𝑊_𝑤𝜌_𝑤+𝑊_𝑜𝜌_𝑜+𝑊_𝑔𝜌_𝑔)𝑔sin𝜃

𝑈𝑇 (4.27)

Fig.4. Resultsofthehorizontalcompressiblethree-phasemodelduringa400-daywaterfloodingperiod.Thesourcetermeffectscanbeseenclearlyinallplots whereproductionwellsarelocatedat10mand90mandinjectionwellat50m.(A)Waterpressureplotshowsastrongpressuregradientregionattheearlystage (before130days).(B)Watervelocityprofile.Itcanbeseenthatwater—-frontreachestheproductionwellafteraround100days.(C)Normalizedwatersaturation showsthatthewaterfrontisfastwhereastheotherphases(oilandgas)areproducedslowly(takesalmost300days).(D)Oilpressureprofilegivesasimilarresult aswaterpressure.(E)Oilvelocitybehaviorissimilartowatervelocity.(F)Normalizedoilsaturationplotillustratesthatoilisdisplacedquiteslowly.(G)Thegas pressuregradientisverylowinthegas-displacedregionattheearlystageduetothehighmobilityofgas.(H)Thereisnogasadvancingfrontsincegasflowseasily.

(I)Gasisdisplacedfastlyandalotofgasisrecoveredbefore130days.

Table1

Referenceinputparametersinthesimulations.

Parameter Dimensional value Parameter Dimensional value

L 100 m 𝐼 _𝑤 2.5

𝜙 0.25 I _o 1.8

̃𝜌_𝑤0 1 g/cm ³ I g 1.1

̃𝜌_𝑜0 0.8 g/cm ³ 𝐼 _𝑤𝑜 3000 (Pa · s) ⁻¹

̃𝜌_𝑔0 0.018 g/cm ³ 𝐼 _𝑤𝑔 3000 (Pa · s) ⁻¹

𝑠 _𝑤𝑟 0 I og 3000 (Pa · s) ⁻¹

s or 0 𝛼 0.01

s gr 0 𝛽 0.01

𝜇_𝑤 1 cP 𝛾 0.01

𝜇o 1.5 cP 𝑃 _𝑐^∗₁ 4 ^∗10 ⁴Pa

𝜇g 0.015 cP a 1 2

K 1000 mD 𝛿1 0.08

𝑘 ^max_𝑟𝑤 0.4 𝑃 _𝑐2^∗ 10 ⁵Pa

𝑘 ^max_𝑟𝑜 0.5556 a 2 2

𝑘 ^max_𝑟𝑔 0.9091 𝐶 _𝑤 10 ⁶m ²/s ² 𝑄 _𝐼𝑤 0.125 m ³/day C o 5 ^∗10 ⁵m ²/s ² 𝑄 _𝐼𝑔 0.125 m ³/day C g 10 ⁵m ²/s ² 𝑄 _𝑝 0.0625 m ³/day 𝜀 _𝑤 0.0 cP

N x 101 𝜀 o 0.0 cP

A 1 m ² 𝜀 g 0.0 cP

𝑃 _𝑤𝐿 10 ⁶Pa x I 50 m

△t 1570 s x P(1,2) 10 (1)&90 (2)m

wherewehaveused(3.24)and(3.25)where𝑈_𝑇=∫0^𝑥(𝑄_𝐼−𝑄_𝑝)𝑑𝑥.Sim- ilarly,f_oandf_gcanalsobeexpressedinthesamemanner.Inorderto illustratethephaseflowfraction𝑓_𝑤(seeFig.3)werepresentU_Tbya

referencetotalvelocity𝑈_𝑇∈ [−^𝑄𝑝

2 ,+^𝑄𝑝

2].WerefertoTable1forother inputdatathatareused.

Initialconditions

For the waterflooding case, we assume the reservoir initially is mostlyfilledwithgas(90%)andsomeoil(10%):

𝑠_𝑔(𝑥,𝑡=0)=0.9, 𝑠_𝑜(𝑥,𝑡=0)=0.1. (4.28) FortheWAGinjectioncase,thereservoirisassumedinitiallyfilledwith oil(90%)andsomeextrawater(10%):

𝑠_𝑜(𝑥,𝑡=0)=0.9, 𝑠_𝑤(𝑥,𝑡=0)=0.1. (4.29) Forthecompressiblecase,areferencepressure𝑃𝑤𝐿attheleftboundary ofthelayerisgivenatinitialstate,

𝑃𝑤𝐿(𝑥=0,𝑡=0)=10⁶Pa. (4.30)

Boundaryconditions

Weassumeaclosedboundaryforbothcompressibleandincompress- iblemodels,whichmeansthat

𝑢_𝑖(𝑥=0,𝑡)=0, 𝑢_𝑖(𝑥=𝐿,𝑡)=0, 𝑖=𝑤,𝑜,𝑔. (4.31) Fortheincompressiblecase,wegiveareferencepressure𝑃_𝑤𝐿attheleft boundaryofthelayer,

𝑃_𝑤𝐿(𝑥=0,𝑡)=10⁶Pa. (4.32)

Sourceterms

ForWAGexperiments,gasandwaterareinjectedatdifferenttime periodsduringthewholeoilrecoveryprocess.WeassumethatQ_I(x)and Q_p(x)taketheform

𝑄_{𝐼𝑤,𝐼𝑔}(𝑥)= 𝑄_{𝐼𝑤,𝐼𝑔} 𝜎

{1, if|𝑥−𝑥_𝐼|≤𝜎∕2;

0, otherwise. , 𝑄_𝑝(𝑥)= 𝑄_𝑝

𝜎

{1, if|𝑥−𝑥_𝑝,𝑖|≤𝜎∕2;

0, otherwise. (4.33)

where(𝑖=1,2)and𝑄_{𝐼𝑤,𝐼𝑔}=0.125m³∕dayand𝑄_𝑝=0.0625m³∕day.The widthofthesmallregionassociatedwiththeinjectorandproduceris 𝜎.Inthenumericalscheme𝜎=Δ𝑥.

4.1. Waterfloodinginagasreservoir

Wefirsttesttheproposedcompressiblethree-phasemodelappliedto agasreservoirdevelopment.Inthisexample,waterisinjectedat50m intoagasreservoirlayeroflength100mwithalittleproportionofoil (10%).Twocases,respectively,forthehorizontal(Fig.4)andvertical reservoir(Fig.5)areshownbelow.

Theresultsofthehorizontalcompressiblethree-phasemodelwith waterinjectionforatotalperiodof400daysareshowninFig.4where pressures (first column), velocities (middle column) and saturations (rightcolumn)aresymmetricwiththeinjectionwelllocatedatthecen- terofreservoirlayer.Thegasismostlyrecoveredduringthefirst130 days,see(I),whereasoilrecoverytakesplaceovermorethan300days, see(F),duetoitslowermobilitythangas.Itisalsoobservedthatat earlystagegaspressurealongthereservoirlayerhaslessgradientthan boththewater’sandtheoil’s(seefirstcolumninFig.4).Theinjected waterdisplacesbothoilandgasinthereservoirneartheinjectionwell regionwhereahighpressuregradientisnecessaryforbothwaterand oiltoflow,seepanel(A)and(B),becauseoftheirlowmobilities.After waterhasarrivedtheproductionwellsataround100days(seeC),wa- terandoilpressuresdropowingtothefactthatwaterthencanfindan easyflowpathtotheproductionwells.

InFig.5,weshowtheresults(phasepressures,velocitiesandsat- urations)ofacompressibleverticalthree-phasemodelwitha400-day waterfloodingdisplacement.Waterisinjectedtodisplaceoilandgasat bothsidesofthereservoirlayer.Itquicklyfillsthebottompart,then starts accumulating,see panel(C).Correspondingly,gas is displaced faster inthelowerpartthanin theupperpartbecausethereservoir layerisvertical.Gravitysegregationisseeninthelowerpartwheregas

Fig.5. Resultsoftheverticalcompressiblethree-phasemodelduringa400-daywaterfloodingperiod.Thesourcetermeffectsareidentifiedinthevelocityand saturationplotswhereproductionwellsarelocatedat10mand90mandinjectionwellat50m.(A)Waterpressureplotindicatesthatalotofwaterflowstoward thebottomandbythatgreatlyincreasesthepressureinthatregion.(B)Duetothestronggravityeffectwaterflowsfastertowardsthebottomoflayercompared thewaterdisplacementintheupperlayer.(C)Normalizedwatersaturationshowsthatwaterflowsfastlytothebottomwhereitisaccumulatedbeforeitbegins toefficientlydisplacetheupperpartofthelayer.(D)Oilpressurefollowsthesimilarbehavioraswater.(E)Waterdisplacestheoiltowardsbothsidesfromthe center.However,atearlytimesomeoilintheupperpartofthelayerwillmovedownwardlyduetogravity.Later,thewaterfrontwilldisplaceoilupwardly.(F) Theoiladvancingfrontbehavessimilarasthewaterfront.(G)Gaspressurebehavessimilartothewaterpressure.(H)Atanearlystagegasisdisplacedtowardsthe productionwellfromthecenter.Afterthewaterfronthasreachedthebottomproductionwellthewholebottompartofgas(50mto100m)startsmovingupwards.

(I)Gasisrecoveredslowlyintheupperpartwhereasgasrecoveryinthelowerpartconsistsoftwostages:initially,gasisdisplacedbywatertothebottomproduction well.Then,gasinthelowerzonestartsflowingupwardly.

Fig.6. Comparisonbetweenthecompressibleandincompressiblemodelwithverticalthree-phaseflow.(A,D)Phasevelocity𝑢_𝑤andu_gforwaterandgas,respectively.

(B,E)Pressure𝑃_𝑤andP_gforwaterandgas,respectively.Thecompressiblemodelaccountsforthefactthatgasissignificantlycompressedandstoresenergywhich isremovedfromthesystemasgasisproduced.Thisgivesrisetolowerpressureprofilesforthecompressiblecaseascomparedtotheincompressiblecase.Thisgives risetoalowerpressurelevelforthecompressiblemodelascomparedtotheincompressible.(C,F)Saturation𝑠_𝑤ands_gforwaterandgas,respectively.

issqueezedupwardly,see(H)and(I).Incontrasttowhatisshownin Fig.4G,gaspressuredistributionshowsasimilarbehavioraswaterand oil(higheratbottomandlowerattop),seefirstcolumninFig.5.We refertothefiguretextformoredetails.

4.1.1. Comparisonofthecompressibleandincompressiblemodels WecontinuethediscussionofthecaseshowninFig.5.Inparticular, wewanttocomparethebehaviorofthecompressibleandincompress- iblemodel.Constantdensityvalues 𝜌_𝑤=1000kg∕m³,𝜌_𝑜=800kg∕m³ and𝜌_𝑔=18kg/m³areusedintheincompressiblemodel.

Fig.6showsacomparison between thecompressibleandincom- pressiblemodelafter30 and120days.(A)shows thatatearlystage theinjectedwaterinthecompressiblemodelpreferstodisplacegasin thelowerpart(highpositivevalue)sincewaterleadstohigherpressure atthebottomsuchthatthegasiscompressedthere.Withcompressed gasproducedatthebottomandgasexpandingintheupperpart,gas willonly slowlymigratetowardstheupperpartresultingin compa- rablylowervelocity(negative)inthecompressiblemodel.Thevelocity differenceshownin(D)fitswellwiththesaturationdifferenceafter120 days.Attheearlytime(30days)thesaturationdifferencesarenotdis- tinct,see(C).However,afteralongtime(120days)thedifferencesare moresignificant,especially,inthewaterdisplacingpart,see(F).This isduetotheincreasingphasepressuredifferencebetweencompressible andincompressiblemodel,see(B)and(E).Theremovalofcompressed gasfromthegasreservoiras(almostincompressible)waterisinjected clearlygeneratesadditionalspaceforthewatertofillwhichgivesrise toalowerpressure.

4.2. Thecompressiblethree-phasemodelwithaWAGexperiment

InWAGprocesses,theinjectedwaterwillmigratetowardsthebot- tomoftheformationwhiletheinjectedgaswillflowupwardly.There- fore,counter-currentflowoccursintheverticaldirectionofthereser- voir duetothegravity segregationof water, oilandgas. Significant differencesintermsofsaturationdistributionandproducingGOR(gas- oil-ratio)havebeenreportedbetweenaconventionalmodelandmodels thatbettercanaccountforthemixofdifferentflowregimes(co-current andcounter-current).Forexample,inSherafatiandJessen(2017)an explicit representation of flow transitions between co-current and counter-currentflowwasusedtoimprovethedesignofWAGinjection processes.

Inthispart,weconductawateralternatinggas(WAG)injectionina 1Dreservoir(250mD)layerwhichinitiallycontains90%oiland10%

water.Thewaterandgasinjectionwellislocatedat50mandtwopro- ductionwellsaresetat10mand90m.Gasisinjectedforthefirst10 daysfollowedbythewaterinjectionthenext10days.Fluidscanbepro- ducedinbothproductionwells.ThewholeWAGexperimentcontinues withaninjectioncirculationofwaterandgas(eachfor10days).

Fig.7showstheresultforaWAGinjectionprocessproducedbythe compressiblethree-phaseverticalmodelwheregravitysegregationhas asignificanteffect.Fromthesimulationweseethatpressureincreases withtime(firstcolumninFig.7).Moreover,pressurevaluesatthelower partofthelayerarelargerthanattheupperpart.Duetothedensity difference,waterdisplacesoilfasterinthebottompart,see(B)and(C).

Inaddition,gasflowsquicklytowardstheupperpartofthereservoir

Fig.7.Resultsoftheverticalcompressiblethree-phasemodelfora400-dayWAGinjectionprocess.Thesourcetermeffectsarevisibleinthevelocityandsaturation plotswhereproductionwellsarelocatedat10mand90mandinjectionwellat50m.(A)Ahighpressureregioninthelayercenterduetothewaterorgasinjection andgravityeffect.(B)Wateradvancingfrontimpliesthatwaterflowsfastertowardsthebottomoflayercomparedthewaterdisplacementintheupperlayerdue togravitysegregation.(C)Waterpreferstoflowtowardsthebottomoflayerwheretheedgeregion(90m-100m)isalsosweptbywater.(D)Oilpressurefollows similarbehavioraswaterpressure.(E)Theupperpartofoilisrecoveredfasterthanthelowerpart.(F)Duetothelargedensitydifferencebetweenoilandgas,the upperpartoilisrecoveredveryquickly,evenfortheedgeregion(0m-10m).(G)Gaspressure.(H)Gasadvancingfrontisfastintheupperpartoflayerbecauseof thestronggravitysegregation.(I)Gasreachesthebottomproductionwellwhereasalotofgasisaccumulatedinthetopregion.

layer,seethesaturationplots.Intheupperpartoilisrecoveredfaster thaninthelowerpartbecauseofthelargerdensitydifferencebetween gasandoilthantheonebetweenwaterandoil,seethesecondcolumnin Fig.7.Wealsoobservethatgasreachesthebottomproductionwellbut doesnotmovefurther.Thiscanbeexplainedbythefactthatgravity segregationeffectovercomesthecapillarity.However,alotofgasis accumulatedintheupperedgeregion(0m-10m)duetothebuoyancy force,see(I).

4.3. Comparisonofcompressibleandincompressiblethree-phasemodels withWAGexperiments

Inthispart,wecomputesolutionsfromincompressiblethree-phase modelswithsameWAGinjectionprocessandcomparetherelevantre- sultswiththosefromthecompressiblethree-phasemodel.Constantden- sityvalues𝜌_𝑤=1000kg∕m³,𝜌_𝑜=800kg∕m³and𝜌_𝑔=18kg/m³areused intheincompressiblemodel

Fig.8showsacomparison between thecompressibleandincom- pressiblemodeloftheverticalthree-phasereservoirwithaWAGpro- cess.Similartowhatwasobservedin Fig.6,differencesareseenfor

phasevelocity,pressureandsaturation.Withincreasingtime,thisdif- ferencewillbeenhanced,especiallyforthepressure.Thisismainlydue tothegascompressibility.See(B)and(E)andthefiguretextformore explanation.Becauseofthedensitydifferencewaterpreferstoflowto- wardsthebottomofthelayerwhereas gasmovesfastertowards the upperpartoflayer,see(C)and(F).

4.3.1. Effectoffluid-fluidinteractions

Here wewanttoillustratetheimpactfrom fluid-fluidinteraction termsonthecompressiblemodelwithaWAGprocess.Twosituations arecompared below:onewith𝐼_𝑤𝑜=𝐼_𝑤𝑔=𝐼_𝑜𝑔=0(𝑃𝑎⋅𝑠)⁻¹ andone with𝐼𝑤𝑜=𝐼𝑤𝑔=𝐼𝑜𝑔=5000(𝑃𝑎⋅𝑠)⁻¹.

Fig.9comparestheresultsforthehorizontalmodelforaWAGpro- cesswithandwithoutfluid-fluidinteractioneffectat60and120days.

In(B)and(E),weobservethatduetothefluid-fluidinteraction,pres- sureiselevatedcomparedwiththecasewithnofluid-fluidinteraction.

Thewatervelocity(A)andsaturationprofiles(C)showthatwaterto alessextentdisplacesoilandinsteadflowsthroughtheoriginalwa- terchannelswhenfluid-fluidinteractionisincluded.Thedifferencein thewatersaturationprofilesbetween(C)and(F)isenhancedwithtime

Fig.8. Comparisonbetweenthecompressibleandincompressiblemodeloftheverticalthree-phasereservoirwithaWAGprocess.Resultsareshownafter60and 120days.(A)Gravitysegregationresultsinafastadvancingfrontofgasintheupperpartoflayerandafastadvancingfrontofwaterinthelowerpartoflayer.

(B)Phasepressureinthecompressiblemodelishighersincethecompressedgaswantstoexpandwhenitmovestoaregionwithlowerpressurebutcannotexpand duetotheconstrainedspaceforgas.(C)Gaspreferstomovetowardstheupperpartoflayerandwaterpreferstoflowtowardsthelowerpart.(D)At120days,gas reachestheupperproductionwellandwaterarrivesatthebottomwell.(E)Phasepressureinthecompressiblemodelincreaseswithtimecomparedwith(B).(F) Thedifferencebetweenthetwomodelsisenhancedwithtime.

Fig.9. ComparisonforthehorizontalcompressiblemodelforaWAGprocesswithandwithoutfluid-fluidinteractioneffects.Thesituationafter60and120days areplotted.(A)Phasevelocityat60days.(B)Phasepressureat60days.(C)Normalizedsaturationat60days.(D)Phasevelocityat120days.(E)Phasepressureat 120days.(F)Normalizedsaturationat120days.