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257015,WaterfingeringEnhancedoilrecoveryPermeabilitycontrolNumericalsimulatorMathematicalmodeldisplacementleadstoagreatdifferenceinoilrecoverybetweenlowandhighlypermeableoilformations.Therefore,theideaforalleviationinpermeabilitydifferencebypermeabilitycontrolinawaterfloodingoilofwellborehasntroleintheTang,2007;YouofPCA(Kangninlaboratory2000;Bolouriand).inperme-moresignificantnumericalsimu-lationtechnology(Juetal.,2007;Dahaghietal.,2011;Satoetal.,2012)hasbeenregardedasoneofthemaintechniquespredictingfluidflowinthereservoirs.However,simulationtechnologyforpredictingoilandgas*Correspondingauthor.SchoolofEnergyResources,ChinaUniversityofGeosciences(Beijing),XueyuanRoadNo.29,HaidianDistrict,Beijing,100083,China.E-mailaddress:jubs2936@163.com(B.Ju).ContentslistsavailableatScienceDirectJournalofPetroleumScienceandEngineeringjournalhomepage:www.elsevier.com/locate/petrolhttps://doi.org/10.1016/j.petrol.2017.08.056Received3June2016;Receivedinrevisedform28April2017;Accepted27August2017Availableonline21September20170920-4105/©2017PublishedbyElsevierB.V.JournalofPetroleumScienceandEngineering159(2017)360–375Permeabilityheterogeneityofoilformationshasanimportantimpactonoilrecovery.Duringtheprocessofoilfielddevelopment,theinherentheterogeneityofoilreservoirsmaycausethedifferencesinfluidmobility.Thismobilitydifferencesinthereservoircanleadtowaterfingeringphenomenonhorizontallyandverticallyinoneoillayerduringwaterflooding(Juetal.,2007).Itmayalsoleadtowaterpreferentialflowingintohighlypermeablelayersinamultilayerreservoir.Thesephenomenacanacceleratewaterbreakthroughandleadtoalowwatersweptvolumeduringoilproductionbywaterfloodinginoilfields.Whenanoilfieldentershighwater-cutperiod,thedisproportionationinoilTheresearchonpermeabilitycontrolinthevicinityseveraldecadesofhistoryanditmayplayanimportadevelopmentofdifferentstagesofanoilfield(Xiongandetal.,2009).Mostpublishedworksfocusedonpreparationetal.,2015)anditseffectonpermeabilityreductio(Nguyenetal.,2006;Seright,2009;McCooletal.,Ghoodjani,2011;Touchetal.,2013;Parketal.,2015Morerecently,thetechnologyresearchandapplicationabilitycontrolbetweeninjectorandproducerhasmadeprogress(Heetal.,2013;Chenetal.,2015).Reservoir1.Introductionreservoirhasbeenregardedasoneoftheeffectivemethodsforenhancedoilrecovery(Yaoetal.,2012).multilayerreservoiroftenleadstoalowsweepefficiencyinastrongheterogeneousoilreservoirdependingonwaterfloodingforenhancedoilrecovery(EOR).EORbymeansofpermeabilitycontrolduringwaterfloodingperiodsinheterogeneousoilformationshasbeencorroboratedbypreviousexperimentsinthelaboratoryandfieldtrialinmatureoilfields.However,thequantitativerelationshipsbetweenEORandpermeabilitycontrolinoilreservoirsarenotclearlyunderstoodcurrently.Inthispaper,themechanismsofpermeabilitycontrolinoilreservoirswereanalyzedbytheoreticalapproaches.Then,athreedimensionalmulti-phaseflowmathematicalmodelconsideringmigrationofsolidpermeabilitycontrolagents(PCA)waspresented.Anumericalsimulatorforpredictingmulti-phaseflowofpermeabilitycontrolinoilreservoirswasdevelopedinFortrancodes.Inaddition,thepermeabilityreductionandwater-cutchangeinducedbyPCAcalculatedbythesimulatorarevalidatedbyexperiments.Furthermore,atwoandathreedimensionalgeologicalmodelswithdifferenttypesofpermeabilitycontrolagentsinjectedintothehighlypermeablelayerforEORwerestudiedbyusingthenewsimulatordevelopedinthiswork.Thesimulationresultsdemonstratethepermeabilitycontrolprocesseswithelapsedtimeaswellasoilproductionperformanceinfluencedbypermeabilitycontrol.ItalsorevealsamechanismofpermeabilitycontrolforEOR,andsuggestsoilrecoveryimprovedby4.9–8.8%inthe2Dsimulationand6.1%inthe3Dsimulationexample.ItvalidatesthefeasibilityofnumericalsimulationandimportanceofpermeabilitycontroltechnologyforEOR.ARTICLEINFOKeywords:ABSTRACTWaterfingeringphenomenoninanoillayerorwaterpreferentialinjectionintohighlypermeablelayersinaAnumericalsimulatordevelopedformodelingenhancedoilrecoveryBinshanJua,b,c,*,YongYangd,ThompsonBrantsona,JieaSchoolofEnergyResources,ChinaUniversityofGeosciences,Beijing100083,ChinabKeyLaboratoryofMarineReservoirEvolutionandHydrocarbonEnrichmentMechanism,MinistrycKeyLaboratoryStrategyEvaluationforShaleGas,MinistryofLandandResources,ChinaUniversitydSinopecShengliOilfieldCompany,ExplorationandDevelopmentResearchInstitute,Dongying,permeabilitycontrolforChiaofEducation,ChinaUniversityofGeosciences,Beijing,ChinaofGeosciences,Beijing100083,ChinaChinaHowever,thereisnoPCAmassconservationequationmodelingPCAB.Juetal.JournalofPetroleumScienceandEngineering159(2017)360–375migrationinpreviousauthor'smathematicalmodel(Jiangetal.,2003).Theydefinedapermeabilityreductioncoefficienttoapproximatelydescribepermeabilitychangeintheirnumericalsimulation.Ourpreviouspaper(Juetal.,2007)focusedonfinesreleaseandmigrationproblemleadingtomorepermeabilityheterogeneityandlessoilrecoveryintheproductionofoilfrompoorlyconsolidatedsandstonereservoirs.Conversely,thiscurrentpaperfocusesonmodelingelimina-tionofpermeabilityheterogeneityandEORbyPCAinjectionintooillayerswithhighpermeability.Themainworksofthisresearchare:Firstly,thepermeabilitycontrolmechanismofreservoirsisanalyzedbyinteractionsinvolvingPCAadsorbedontoporesurfaceofthereservoirrock,pluggingporethroats,swellingeffectaswellasnon-NewtonianeffectofpolymertypePCA.Secondly,amathematicalmodelofthreedimensionalthreephaseflowswithpermeabilitycontrolbyPCAinjec-tionwasestablished.Thirdly,thenumericalsimulatorusedforpredictingoilproductionperformancesbyPCAwasdeveloped.Andfinally,permeabilitycontrolexamplesarestudiednumericallyonthesimulator.2.MathematicalmodelComparedwithconventionaltheoryofmultiphaseflowinporousmedia,theflowcomplexityliesinfluidflowaccompanyingwithPCAtransportinporousmedia.Therearefollowingfourmajorphenomena:(1)themigrationofsolidPCAcarriedbyfluidflow,(2)PCAadsorbedontheporesurfaceorcapturedatporethroats,(3)PCAreleasedfromtheporesurfaceundercertainconditions,(4)aswellingeffectofPCAandaneffectofnon-Newtonianpolymerfluid.Thepreviousmathematicalmodelsofpermeabilitycontrolflowaremainlydividedintotwocate-gories:macroscopicmathematicalmodelandmicroscopicnetworkmathematicalmodel(GruesbeckandCollins,1982;KhilarandFogler,1983;SharmaandYortsos,1986;OhenandCivan,1990;LiuandCivan,1994;JuandFan,2009;Juetal.,2013).ThemacroscopicmodelisbasedoncontinuumtheoryofporousmediafortransportmodelingofPCA(fines)suchas:GruesbekandCollin'smodel(GruesbeckandCollins,1982),KhilarandFogler'smodel(KhilarandFogler,1983),OhenandCivan's(OhenandCivan,1990)modelandJu'smodel(JuandFan,2009;Juetal.,2013).Fromthemicroscopicview,microscopicnetworkmodelisconductedtodescribefluidflowandPCAmigrationinmicroscopicporousmediasuchasSharmaandYortsosmodel(Sharma,andYortsos,1986)andRegeandFoglermodel(RegeandFogler,1987).Forthelimitationofthemicroscopicmodelwithastrongdependenceonprob-abilityanditsnumericalsolvingprocessconsumingtremendousamountofCPUtime,themodelinthispaperisbasedonthetheoriesofparticlemigrationandmacroscopiccontinuumporousmedia.Thedetailsofmodelassumptionsarestatedasfollowingsection.2.1.MathematicalmodelingassumptionsThemathematicalmodelwasdevelopedunderthefollowingassumptions:(1)Multiphaseflowsinporousmediaarethree-dimensionalandnon-isothermal.(2)Blackoilsystem.(3)Rockisacompressibleporousmedia.productionperformancebyusingpermeabilitycontrolforEORisatinitialstage.Therefore,itisurgenttodevelopanumericalreservoirsimulatortopredictoilproductionperformancewithtreatmentbyvarioustypesofPCA.Permeabilitycontrolisobtainedbythechangesinporosityandpermeabilityofreservoirrock,whichisrealizedbytheinteractionbe-tweenPCAandformationrock.Therefore,themathematicalmodelin-cludesmultiphaseflowequations,PCAmassconservationequation,andtheequationsofvariationofporosityandpermeabilityofthereservoir.361(4)ReservoirfluidiscompressibleNewtonianornon-Newtonianfluids.(5)Considergravityandcapillaryforces.(6)Thewettabilityofpermeabilitycontrolisconsidered.(7)ConsiderswellingeffectofPCA.2.2.ThreedimensionalthreephaseblackoilsystemflowmodelForthree-dimensionalthree-phaseblackoilreservoirsystem,theflowbehaviorsaregovernedbythreepartialdifferentialequations(Hanetal.,1993).WaterphaseflowequationdivC18K⋅KrwBw⋅μwgradΦwC19þqw¼η∂∂tðϕ⋅Sw=BwÞ(1)Oilphaseflowequation:divC18K⋅KroBo⋅μogradΦoC19þqo¼η∂∂tðϕ⋅So=BoÞ(2)Gasphaseflowequation:divC18K⋅KrgBg⋅μggradΦgþK⋅Kro⋅RsBo⋅⋅μogradΦoC19þqg¼η∂∂tC0ϕ⋅SgC14BgþϕSoRs=BoC1(3)Auxiliaryequations:Φo¼poþγo⋅z(4)Φw¼pwþγw⋅z¼poþpcwoþγw⋅z(5)Φg¼pgþγg⋅z¼poþpcgoþγg⋅z(6)SoþSwþSg¼1(7)Where:tistime,s;pispressure,Pa;Sissaturation;Kisformationpermeability,m2;Krisrelativepermeability;μisviscosity,Pa⋅s;Rsissolutiongas-oilratio;ziselevationfromreferencelevel,m;ϕisforma-tionporosity;pciscapillarypressure,Pa;γisspecificgravityoffluids;qissource(Injectionrate)andsink(productionrate)terms,m3/s;ηisunitconversionfactordependingonthesystemofunits;divisdivergenceoperator;gradispotentialgradient;subscript:w-water;o–oil;g-gas.2.3.FlowequationofPCAinfluidsInpercolationprocess,themovementofPCAiscontrolledbycon-vectionanddiffusionwhilesadsorptionreactionalsooccurs.Therefore,thetransportofPCAinporousmediacanbedescribedbyconvection-diffusion-adsorptionequation(Juetal.,2003).Thethree-dimensionalmassconservationequationforanykindofPCAis:∂∂xC18uxwCsiC0ϕswDsxi∂Csi∂xC19þ∂∂yC18uywCsiC0ϕswDsyi∂Csi∂yC19þ∂∂zC18uzwCsiC0ϕswDszi∂Csi∂xC19þϕsw∂Csi∂tþRasiþRrsiþQsi¼0(8)Initialconditions:Csi¼Csi0(9)WhereCsiismassconcentrationofPCA,kg/m3;Csi0isinitialmasscon-centrationofPCA,kg/m3;RasiisadsorptionrateofPCAontoporesur-faces,kg/(m3⋅s);RrsiisreactiontermofPCA,kg/(m3⋅s);QsiisPCAchangeinducedbysourceandsinkterms,kg/(m3⋅s);Dsxi,Dsyi,Dsziiscientofparticlesonporesurface,m⋅kg;χwj-VelocitycoefficientofB.Juetal.JournalofPetroleumScienceandEngineering159(2017)360–375trappedparticlesinporethroat,m2⋅kgC01;vw-flowvelocityofwaterphase,m⋅sC01;vcwj-Criticalphaseflowvelocityinducingtheparticlecomponentjreleasefromtheporesurface,m⋅sC01;Cs-massconcentrationoftotalsaltions,kg⋅mC03;Cscj-Criticalconcentrationoftotalsaltionsinducingthemensionsrespectively,m/s.WhenparticlesarenotBrownian(particlesizesisusuallygreaterthan1μm),thecoefficientis0.Rlijinequation(10)isexpressedasRlij¼RrclijþRrhlijþRdlijþRclij(11)Where:Rclijisparticlereleaseratefromporewallbycolloidalforces(Rrhlijisparticlereleaseratefromporewallbyhydrodynamicforces(Rdlijisparticledepositionrateonporesurfaces;andRrclijisparticlecapturerateatporethroats.)2.5.VolumechangerateofwaterwettingPCAForparticulatereleasecausedbyhydrodynamics,acriticalflowratemustbereached(vwC21vcwj);whenCsC21Ccsj,colloidalforcecausesparti-clesrelease.Accordingtomechanicalconditionandtheprincipleofparticlereleaseandtrappingprincipleofparticlesadsorbedontoporesurfaceandporethroat,thevolumechangerateofwaterwettingparti-clesinaunitrockvolumeis∂Λwj∂t¼C0RrcwwjþRrhwwjþRdwwjþRcwwjC1C14ρwj¼αhwwjΛwjC0vwC0vcwjC1þαcwwjΛwjC0CsC0CscjC1C0βwjvwCwwjC0χwjvwCwwj(12)Thefourpartsoftherighthandsideoftheequationarewaterwettingparticlechangeratescausedbyhydraulicerosionrelease,colloidalforcerelease,surfaceadsorptionandtrapping.whereΛwj-Volumeofwaterwettingparticlesreleasedfromunitvolumeofrock,m3/m3;ρwj-Densityofwaterwettingparticles,kg/m3;αhwwj-Particlereleaseratecoefficientcausedbyhydrodynamicforces,1/m;αcwwj-Particlereleaseratecoeffi-cientcausedbycolloidalforce,m3⋅kgC01⋅sC01;βwj-Adsorptionratecoeffi-2C01comprehensivediffusioncoefficientPCAinthreedimensionscausedbydynamicandmoleculardiffusion,m2⋅sC01;uxw,uywanduzwisflowvelocityinthreespacedimensions,m/s.2.4.MassconservationequationforconventionalblockingtypeofPCASincesolidparticulatePCAhasaffinityforwettability,theparticulatePCAisdividedintoanoilwettingPCA,waterwettingPCA,andaneutralPCA(LiuandCivan,1994).However,becausetheparticlesizesarenon-uniformaccordingtoparticlesizes,itwasdividedintoncomponents(Juetal.,2003).Foranyradiusrjparticlecomponent,massconservationequationisgivenby:∂∂xC18uxlClijC0ϕSlDxlij∂Clij∂xC19þ∂∂yC18uylClijC0ϕSlDylij∂Clij∂yC19þ∂∂zC18uzlClijC0ϕSlDzlij∂Clij∂zC19þϕSl∂Clij∂tþRlijþQlij¼0(10)Wheresubscript:l¼o(oil);w(water);g(gas);i¼o(oilwet),w(waterwet),i(neutralwet),j¼1,2,3……n,groupedbyradiusofparticlecomponent.Clij-themassconcentrationofthejthcomponentofthewetting-typeiinphasel,kg/m3.Rlij-thechangerateofparticlesmasscausedbyreleasefromthesurfaceofporewallorporethroat,kg/(m3⋅s),Qlij-Thechangerateofparticlemasscausedbysourceandsinkterm,kg/(m3⋅s).Dxlij,Dylij,Dzlij-Diffusioncoefficientofanycomponentsinthreespacedi-mensions,m2⋅sC01.uxl,uylanduzl-Flowvelocityoflinthreespacedi-362particlecomponentjreleasefromporesurface,kg⋅mC03.2.6.VolumechangerateofoilwettingPCAThefactorscontrollingoilwetPCAmigrationaretheshearofoilphaseflow,adsorptiononporesurfaceandthetrappingatporethroats.OilwettingPCAvolumechangerateinaunitvolumeofrockcanbeexpressedas∂Λoj∂t¼C0RrhoojþRdoojþRcoojC1C14ρoj¼αhoojΛojðvoC0vcoiÞC0βojvoCoojC0χojvoCooj(13)ThreepartsoftherightsideoftheequationarevolumechangeratesofoilwettingPCAparticlecausedbyhydrodynamic,surfaceadsorption(deposition)andcapture.WhereΛoj-Volumeofoilwetparticlesreleasedfromunitvolumeofrock,m3/m3;ρoj-Densityofoilwettingparticle,kg/m3.Themeaningoftheothersymbolsissimilartothatofthecorre-spondingsymbolicmeaninginequation(12).2.7.VolumechangerateofneutralwettingparticlePCANeutralparticlescanexistinoilandwaterphase,butthefactorsthatcontrolthemigrationoftheparticlesarehydrodynamic,adsorptionandtrappingeffect.Thevolumechangerateofneutralparticlesperunitvolumeisexpressedas:∂Λij∂t¼∂Λoij∂tþ∂Λwij∂t¼RrhoijC14ρoijþRrhwilC14ρwijþRdoijC14ρoijþRdwijC14ρwijþRcoij=ρoijþRcwijC14ρwij¼αhoijΛoijðvoC0vcoiÞþαhwijΛwijðvwC0vcwiÞC0βojvoCoijC0βwjvwCwijC0χoijvoCoijC0χwijvwCwij(14)Where:Λij-Volumeofneutralparticlesreleasedfromunitvolumeofrock,m3/m3.Themeaningoftheothersymbolsissimilartothatofthecor-respondingsymbolicmeaninginequation(12).2.8.ThechangesinporosityandpermeabilitycausedbyPCAThe“bridging”ofPCAinporethroatsandPCAadsorptiononporesurfacescancausethechangesinporosityandpermeability.Theinstantaneousporosityisexpressedby:ϕ¼ϕ0C0XΔϕ(15)WherePΔϕdenotesthechangeofporosityinducedbyreleaseandretentionofparticlesinporousmedia,anditisexpressedby:XΔϕ¼Xnj¼1C2C0ΛwjþΛojþΛijC1C0C0Λwj0þΛoj0þΛij0C1C3(16)ThechangeinpermeabilityinthisstudyiscausedbyPCA.ModifyingtheKozenyequation,takingintoaccountthefinespluggingeffect,andaccordingtothestudyofLiuandCivan(1993)andJuetal.(2003),theexpressionforinstantaneouspermeabilitychangedbyreleaseandretentionofparticlesisgivenby:K¼K0C2ð1C0fÞkfþfϕ=ϕ0C3n(17)Where:Koandϕoareinitialpermeabilityandporosity,kandφareinstantaneouslocalpermeabilityandporosityofporousmedia;kf-Constantflowcoefficientofseepageparticlesallowedbythepluggedpore.Therangeofindexnvaluesisfrom0to1.0.Aflowefficiencyfactorfisdefinedasthefractionoftheoriginalcross-sectionalareaopenforflow.Theparticleslargerthanporethroatradiusmayblockattheporethroats,andmanyparticlesmayalsoproducea“bridging”plugginginaporethroat.Therateofvolumechangeofalltrappedparticlesintheporethroatsis∂Λcj∂t¼∂Λcwwj∂tþ∂Λcooj∂tþ∂Λcwij∂tþ∂Λcoij∂t¼χwjvwCwwjþχojvoCoojþχwijvwCwijþχoijvoCoij:(18)where:Λcj-Volumeofjparticlestrappedinporethroatsinunitvolumeofrock,m3/m3;Λcwwj-Volumeofjwaterwettingparticlestrappedinporethroatsinunitvolumeofrock,m3/m3;Λcooj-Volumeofjoilwettingparticles
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