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inmethodoleumKeywords:FracturedreservoirsDiscretefracturenetwork(DFN)modelOilandwatertwophaseflowinreservthepaper,implicitflowequationsofoilandwaterphaseinfracturedreservoirsareestablishedtoconsiderthenonlinearityproperties.AndthenthecontrolvolumefiniteelementmethodbasedontheedistributionandenormousSinceandShahvali,2016).Someresearchers(Longetal.,1982;Beggmodelorequivalentporousmedia(EPM)forfield-scalestudyontensormulti-scaleedreservoirs,Bacaetal.(1984)fracturesex-un-structuredmeshes,whichiscalledthediscretefracturemodelbriumwasestablishedbetweenthefluidatthefracturenodalContentslistsavailableatScienceDirectJournalofPetroleumScienceJournalofPetroleumScienceandEngineering146(2016)1211–1225http://dx.doi.org/10.1016/j.petrol.2016.08.024fracturedreservoirs,whichcanbeusedeasilytoremovethelocalpointsandthematrixnodalpointsadjacenttothefracture.Sub-sequently,someresearchers(KimandMilind,2000;Karimi-FardandFiroozabadi,2003;Huangetal.,2011)representedbyKimandMilind(2000)appliedthefiniteelementmethodtosolvetheDFM0920-4105/&2016ElsevierB.V.Allrightsreserved.nCorrespondingauthor.E-mailaddress:437677340@qq.com(R.-h.Zhang).etal.,1989;Durlofsky,1991)extendedanequivalentcontinuum(DFM).Theyemployedthefiniteelementmethodtoobtainthenumericalsingle-phaseflowsolutionandthecross-flowequili-twoporosity/permeabilitymodelsforsimulationofmulti-phasefluidflowininter-connectedfracturenetworks,whicharesolvedbythefinitedifferencemethod.However,Yaoetal.(2010)pointedoutthattheexistenceoflarge-scalefracturesleadstodis-connectedfracturenetworks,thusthemodelsmentionedabovecannotbeappliedtothistypeofreservoir.Moreover,theevalua-tionoffluidexchangebetweenthematrixandthefractureddo-mainistheothercomplextaskforthesemodels(Li,2008;Wangtheequivalentcontinuummodelisappropriateforfinitenumberofshortfractures;theREVandpermeabilitywouldbehardtoobtainforstrongheterogeneousreservoirs.Tosimulatetheflowindiscretelarge-scalefracturYoung(1981),NoorishadandMchran(1982)andproposedone-dimensionalelementstorepresentplicitlywhilethereservoirisdispersedbytwo-dimensionalsearchers(WarrenandRoot,1963;KazemiandMerrill,1976;DeanandLo,1988;StalgorovaandBabadagli,2012)proposedcontinuaodstodeterminetheequivalentproperties,suchaspermeabilitytensorineachREV.However,Khoeietal.(2016)pointedoutthatreservoirswith1.IntroductionDuetotheinherentcomplexfracturheterogeneity,thenumericalsimulationnaturallyfracturedreservoirshavebeenoilworkers(ZhangandYin,2014).unstructuredgridsystemandN-Riterationareusedtoobtainthenumericalsolution.Moreover,theaccuracyofthissimulatorisprovedtobereasonablebyfielddata;Sensitivityanalysisshowsthattheexistencesofcomplexmulti-scalefracturenetworkshavegreatimpactsonwaterflooding.Comparedwiththehomogeneousreservoir,naturalfracture(microandlargescale)networkscanchangethedi-rectionandvelocityofthewaterfront,whichfinallyleadstothemoreseriouswaterbreakthrough,higherwatercutandlessrecovery.Theresearchandthenumericalresultsobtainedinthispapercanprovidetheoreticalguidanceforefficientandscaledevelopmentofnaturalfracturedreservoirs.&2016ElsevierB.V.Allrightsreserved.andstrongdynamicanalysisofchallengesforthe1960s,somere-dominantproblem,suchasthemasstransferbetweenshortfracturesandthematrix.Theequivalentpropertiescombiningthefractureandmatrixsystemarecomputedforeachrepresentativeequivalentvolume(REV)inwhichthereservoirandfluidproper-tiesareuniform.Afterthat,Boe(1994),Chenetal.(2003),Zhouetal.(2008)proposedtheup-scalingandpseudo-functionmeth-NumericalsimulationControlvolumefiniteelementmethodNumericalsimulationofwaterfloodingbasedoncontrolvolumefiniteelementRui-hanZhanga,n,Lie-huiZhanga,Jian-xinLuoa,Zhi-dongaStateKeyLaboratoryofOilandGasReservoirGeologyandExploitation,SouthwestPetrbProductionPlantNo.1ofXinjiangOilfield,Karamay,Xinjiang834000,ChinaarticleinfoArticlehistory:Received25September2015Receivedinrevisedform9May2016Accepted22August2016Availableonline24August2016abstractAnumericalmethodisemploypaper.Owingtothemulti-scalesasignificantroleforasuccessfulpaperproposesacombinationusedtosimulatetheflowjournalhomepage:www.elsevnaturalfracturedreservoirsYanga,b,Ming-yangXubUniversity,Chengdu,Sichuan610500,Chinaedtoinvestigatethefluidflowingdynamicsinfracturedreservoirsinthisandstrongheterogeneity,descriptionoffracturenetworksystemsplaysfracturedreservoirsimulationmodel.Comparingtopreviousworks,thisofdualcontinuumanddiscretefracturenetwork(DFN)model,whichisoirswithmulti-scalefracturesandprovedfeasible.Inthefirstpartofier.com/locate/petrolandEngineeringexamples.R.-h.Zhangetal./JournalofPetroleumScienceandEngineering146(2016)1211–12251212formulti-phaseflowinlarge-scalefracturedreservoirs.However,theexistenceoflarge-scalefracturesinthedomainimposesthediscontinuitiesinthefieldvariablessuchasthesaturationandfluidpressureacrossthefractures(Karimi-FardandFiroozabadi,2003;NickandMatthäi.,2011a,2011b;Abushaikhaetal.,2015).RabbaniandWarner(1994),HelmigandHuber(1998),Mon-teagudoandFiroozabadi(2004)pointedoutthat,whetherthetraditionalfiniteelementmethod(FEM)orupstreamimprove-mentsupwindfiniteelementmethodcanhardlyguaranteethelocalconservationofmass,whichispronetonumericaldispersionformultiphaseflownumericalsimulation.Bastianetal.(2000)proposedthefinitevolumemethod,whichbasedontheflowdi-rectionupstreamweightedcriteriaandmetthelocalconservationofmass.However,itneedstospendalotoftimetodealwiththeconnectionbetweenfinitevolumegridsandhadpoorefficiencyandaccuracy.Therefore,Durlofsky(1993),Geiger(2003)andGeigeretal.(2004)combinedthefiniteelementmethodandfinitevolumemethod,usedthefiniteelementmethodtodealwiththepressureequationandfinitevolumemethodtodealwiththeconvectionequation.However,thisIMPESformulationsuffersfrominstabilityandtimesteplimitationespeciallyforthemodelNomenclatureBooilformationfactor,dimensionlessBwwaterformationfactor,dimensionlessCocompressibility,MPaC01heffectivethickness,mkpermeability,mDkrrelativepermeability,dimensionlessppressure,MPaqminter-porosityfluxratefrommatrixtonaturalfracturesystem,m3/sqscttotalwellproductionrate,m3/sSwsaturation,dimensionlessTreservoirtemperature,Kttime,dxdistanceonxdirection,mydistanceonydirection,mzdistanceonzdirection,mcontainingsmallelements(Matthäietal.,2010;NickandMatthäi.,2011a,2011b).Chenetal.(2006)andYang(2010)constructedthevirtualfinitevolumemesh,whichavoidedthecouplingproblembetweentwosetsofmeshes,andanewfiniteelement-finitevo-lumemethodisestablished.However,modelsaboveallassumedthematrixsystemisasingleporosityorhomogeneouswithdis-cretefracturenetwork,whichneglectedthenaturalmulti-scalefractures.Leeetal.(2001)andKhoei(2016)proposedanapproachtocombinethediscretefracturedescriptionwithanEPM.JiangandRami(2015)proposedamulti-continuummediawithdiscretefracturesforsinglephaseshalegasreservoir,whichhasnotyetbeenappliedtotwophaseflowinfracturedreservoirs.Thesecombinedmodelscanbeusedefficientlytosimulatetheflowthroughporousmediacontainingmultiplelengthscales.Inaddi-tion,Peaceman(1978)pointedoutthatduetothedimensionsofanygridblockcontainingawellaremuchlargerthanthewellboreradiusofthatwell,thepressurecalculatedforawellblockweredifferentgreatlyfromthebottom-holeflowing(BHF)pressureofthemodeledwell.Somepreviousworks(Geigeretal.,2004)ne-glectedthenumericalwellmodelwhichwouldreducetheaccu-racyofsimulatoronwellproductionandpressureperformance.Viewofthis,thispaperestablishesthefullyimplicitseepageequationsconsideringthenonlinearitypropertieswithpressure2.Mathematicalmodel2.1.ModelassumptionAsFig.1shows,thenaturalfracturenetworkoflowperme-abilityfracturedreservoirisverycomplex.Naturalfracturesmainlyrefertotheporesformedbythetectonicmovementanddissolution.TheresearchofLi(2006)showsthatwhenthedepth41000m,usuallydevelopshighangleandverticalfractures.Inchanges,andusesthecontrolvolumefiniteelementmethodtoobtainnumericalsolution.Inbrief,theworksofthispapercanbelistasfollows:First,basedontheprincipleofmaterialbalanceandflowequations,deducedtheoil-watertwophaseimmiscibility,compressiblefluidflowequation;second,usedthecontrolvolumefiniteelementmethodtoestablishthenumericalformatandsolveditbytheN-Riterationmethod;Finally,weprogrammedtheCVMnumericalsimulatortosimulatethewater-floodingperfor-manceoffracturedreservoirs,thesimulatorhasbeenprovenbySubscriptfmicro-scalefracturesystemFlarge-scalefracturellocalcoordinatesystemalongthefracturedirectionmmatrixsystemscstandardstatettotalGeeksymbolsαinter-porositycoefficient,1/m2λmobility,mD/mPaC1s;μoviscosity,mPaC1sρooildensity,g/cm3Φporosity,fractionthispaper,basedonthedualcontinuumanddiscretefracturemodel,thefracturedreservoiriscompositeofthreesystems:matrixsystem,micro-scalefracturesystemanddiscretelarge-scalefracturesystem.Tomakethismathematicalmodelmoretractableandeasytounderstand,thefollowingassumptionsanddescriptionsareapplied:(1)Flowisisothermalandinoil-watertwophases,fracturedre-servoirinthispaperisdevelopedunderpressureabovethebubblepointpressuretoavoidthespilloverofdissolvedgas;(2)Naturalfracturedreservoirissimplifiedasdual-porositydual-permeabilitycontinuummedia,inwhichtherearetwosetofreservoirpropertiesformatrixandfracture(micro-scale)systemrespectivelyasFig.1(c);(3)AsshowninFig.1(b),reservoirissimplifiedtothetwo-dimensionalplaneofthemiddledepthofreservoir,andthelarge-scalefractureisexpressedbyone-dimensionalsolidbasedonthediscretefracturemodel.(4)Crudeoilflowsfrommatrixintofracturesystem(microorlarge-scale),andthenintoproductionwells.Theinter-porosityflowbetweenmatrixandfracturesystemisapplytoFick'sfirstlaw;injectionandR.-h.Zhangetal./JournalofPetroleumScienceandEngineering146(2016)1211–122512132.2.MathematicalmodelInthispaper,weassumeaslightlycompressiblefluidinfrac-turedreservoirandneglectthecompressibilityofrock.Thusthematerialbalanceforthetwophasesleadstotheequations:()()ρϕρ∇=∂∂()σσσσσvSt1wheres¼w,o.Auxiliaryequationsasfollows:+=SS1wo=−pppwocBasedonEq.(1),wededucetheseepagegoverningequationsformatrixandmicro-scalefracturesystemrespectively.Formatrixsystem:()()()μαμϕϕ∇⋅∇−−=∂∂−∂∂()⎛⎝⎜⎜⎞⎠⎟⎟kkBpkkBppBtSBt//2mromooommroooomofmommwmom()()μαμ∇⋅∇−−−−⎡⎣⎢⎢⎤⎦⎥⎥kkBppkkBpppwwmrwmwomcmmrwwomcmofFig.1.Diagramofwaterfloodinginnaturalfracturedreservoir.(a)adiagramofonemodeldispersedbyunstructuredgrids;(c)dualcontinuummediamodelformatrix()ϕ=∂∂()SBt/3wmmwm00.20.40.60.811.21.41.620406080PressurePbformationfactorviscosityFig.2.NonlinearitypropertiesFormicro-scalefracturesystem:()()()μαμϕϕ∇⋅∇+−+=∂∂−∂∂()⎛⎝⎜⎜⎞⎠⎟⎟kkBpkkBppqBtSBt//4frofofofofmroooomofofoffwfof()()()μαμϕ∇⋅∇−+−−+=∂∂()⎡⎣⎢⎢⎤⎦⎥⎥kkBppkkBpppqSBt/5frwfwfwfofcfmrwwwomcmofwwfmwfAlocalcoordinatealongthelarge-scalefractureisestablishedtoobtaintheseepageequationsforlarge-scalefracturesystem:()()μϕϕ∂∂∂∂=∂∂−∂∂()⎛⎝⎜⎜⎞⎠⎟⎟lkkBplBtSBt//6FrofofofofFofFwFof()()μϕ∂∂∂−∂=∂∂()⎛⎝⎜⎜⎞⎠⎟⎟lkkBpplSBt/7FrwfwfwfwfcfwFFwfInthissection,wetakeintoaccountthestronglynonlinearityofoilpropertieswithpressurechanges,whichisalwayssimplifiedtobeunderoriginalreservoirconditionsoraveragepressureforanalyticalsolution.AsFig.2shows,theformationfactorBandthewell(I)andoneproductionwell(P)model;(b)asimplifieddiscretefracturemicro-scalefracturesystem.oviscosityμovarywithpressurechanges.Forsimplicity,weassumethatthecapillarypressurecurveisafunctionthatdependsupon00.511.522.500.20.40.60.81PCswmwithpressurechanges.thesaturationvalues(MonteagudoandFiroozabadi,2007).()=−()=()pBSilnm,f,F8iiicwMoreover,MonteagudoandFiroozabadi(2004),Khoeietal.(2016)solvedthediscontinuityofsaturationbetweenlarge-scalefracturesandreservoirbasedonthecross-equilibriumconcept.Thus,thesaturationoflarge-scalefractureSwFissubstitutedbySwfas:()==()SSdSdtdSdSdSdt9BBwFwf/wFwFwfwfFfInthispaper,thefullyimplicitcalculationformatisestablishedfornonlinearityseepageequations.TherefortheunknownsδPCFandδSwFinequationareexpressedbyδSwffromchainrule:δδδδ==()pdpdSSdSdSS10ScFcFwfwFwFwfwfwf2.3.Discretizationandnumericalsolution2.3.1.DiscretizationThenumericalsolutionofreservoirsimulationisachievedbymeansofnumericaldiscretization,thus,thecontinuousphysicalrelationsinthewholeflowfieldareexpressedasthefinitenum-beroftheunitsofacertainvolumeandtimescale.AsFig.3shows,inthispaper,thewholereservoirregionisdiscretizedbyun-structuredDelaunaytriangulation;the1-Dthicklineisdecom-theparametersofthetransverseparametersareconstantandthewidthdiametereisafactoroftheone-dimensionalintegral.Hence,theintegrationoverthewholedomainΩdcanbewrittenas∫∫∫ΩΩΩ=+×()ΩΩΩeCEQdCEQdCEQd11mfdmfFlowpotentialvariables(pom,pof,pc)andsaturationvariables(Swm,Swf)ofmatrixormicro-scalefracturesystemareapproxi-matedinsideeachelement(Delaunaytriangle)bylinearapprox-imations:∑∑()=()()=()()∼∼==pxyNxypSxyNxyS,,,,12liklllikllww=(++)=()Nabxcylijk12A,,13llllWherethetermAistheareaofthetriangle.Forasimplifiedlarge-scalefracturelineelement,thecorre-spondingshapefunctionis:()()()=−−==()Nxxxxij/1,2;1,214ljijfInthispaper,theGalerkinmethodisusedtoestablishtheweakintegralformofseepageequationsystems.Thusthepressureorsaturationdivergencecanbetransformedintothefluxfi,whichisbasicbary(c)R.-h.Zhangetal./JournalofPetroleumScienceandEngineering146(2016)1211–12251214posedintriangularelementsthatarefacesofthetriangulationsurroundingthereservoir-largescalefractureinterface.Further-more,unlikethetraditionalfiniteelementmethod,weestablishthevirtualfinitevolumearoundeachvertextoobtaintheCVFEnumericalcalculationformat.Indetail,thecomputationaldomainΩdcomprisestwosubdomainsΩmandΩf.Forsimplicity,wewillassumethefracturesofwidtheareexpressedbyone-dimensionalsolidstotwodimensionalreservoirs.Inthispaper,CEQisusedtoexpressthecontrolequation.Basedonthediscretefracturemodel,(a)Fig.3.ExtractofDelaunaytriangulationconformingtofracturedreservoir:(a)afractures;(b)avirtualcontrolvolumecell(dashlines)formedbyconnectingthehomogeneous.Thefractureedge(bluethickline)issharedbytwoneighboredtriangles;partoftheboundaryoftheCVcellaroundi.continuousacrossthesurfaces(edges)ofthecontrolvolumeVi.ThespecificderivationisshowninAppendixA.()()∑=−−()=++++⎧⎨⎪⎪⎩⎪⎪fTppTppfortrangleforline15llnlnnnij,ki1i1ij1ji1Thetimeelementmatrixisproposedbyfiniteelementmethod,throughlumpingmethodtoenhancethestabilityoftheresultie(b)G6G1G2G4G5G3ijlijMilM1Cnn1S(c)Delaunaytriangulationformatrix-microscalefracturesystemandlarge-scalecenters(�G1,G2…G6�)oftrianglesaroundthenodeiandtheCVcellislocallyfluxdirectionofCVcell,thesegmentsC1MijandC1MilwithoutwardnormalnareR.-h.Zhangetal./JournalofPetroleumScienceandEngineering146(2016)1211–12251215(Dalen,1979).Theexplicit-Eulerintegrationoftimetermcanbeexpressedas:ϕϕϕϕϕ∂∂=Δ−∂∂=Δ−()++⎡⎣⎢⎤⎦⎥⎡⎣⎢⎢⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟⎤⎦⎥⎥⎡⎣⎢⎤⎦⎥⎡⎣⎢⎢⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟⎤⎦⎥⎥tBtBBtSBtSBSBNNNN16nnnnoiTo1owoiTwo1wo2.3.2.NumericalsolutionWeextractrespectivelyamatrix-microscalefractureelement(triangle)andalargescalefractureelement(thickline)forflowanalysis,basedonthedualcontinuummodel,eachtriangleele-mentcontainstwosetsofpropertieswhichrespectivelyre-presentsmatrixandmicroscalefracturesystem.Thesolutionforthewholeregionisachievedbysuperposingtheelementstiffnessmatrixonebyone.ItwillbeverydifficulttodecouplethematrixandfracturesystemequationsandsolveindependentlywhenconsideringthestrongnonlinearityofPDEs.Thus,afullyimplicitsimultaneoussolutionmethodisusedtoovercomethenumericaldispersionandthesolutionisstablepositively.BasedontheN-Riteration,thepre
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