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AmethodforevaluationofwaterfloodingperformanceinfracturedreservoirsShaohuaGua,b,n,1,YuetianLiua,ZhangxinChenb,CuiyuMaaaMOEKeyLaboratoryofPetroleumEngineering,ChinaUniversityofPetroleum,Beijing102249,ChinabDepartmentofChemicalandPetroleumEngineering,UniversityofCalgary,Calgary,Alberta,CanadaT2N1N4articleinfoArticlehistory:Received19December2013Accepted3June2014Availableonline17June2014Keywords:WaterfloodingFracturedreservoirDual-porosityImbibitionRelativepermeabilityabstractAmathematicalmodelisdevelopedforevaluationofwaterfloodingperformanceinahighlyfracturedreservoir.Themodeltransformsadual-porositymediumintoanequivalentsingleporositymediumbyusingapseudorelativepermeabilitymethodtonormalizetherelativepermeability.Thisapproachallowsbothfracturesandmatrixtohavepermeability,porosity,endpointsaturation,andendpointrelativepermeabilitybythemselves.ImbibitionisalsotakenintoaccountbymodifyingChen'sequation.Someeffects,includingimbibitionandrecoveryratesareinvestigated.Theinvestigationshowsthatimbibitioncandeterminethepotentialofafracturedreservoirandalowrecoveryratecanimprovethewaterfloodingsituationintermsofretardingwaterbreakthroughandcontrollingtheriseofwatercut.Anewchartcomposedbywatercutvs.recoverycurvesisprotractedtoestimatetheultimatewater-floodingrecoveryrate.Thewaterfloodingperformanceoftworeservoirsisevaluated.Comparedwithnumericalsimulationmethod,theerrorofthesetwocasesarenotmorethan2%,whichprovedthatthismethodisreliable.Bothlabtestdataandfielddataareappliedtoafurtherdiscussionofthecharacteristicsofwaterfloodingperformanceinfracturedreservoirs.Oncomparisonwiththeclassicalmethod,suchasTong'smethodandtheX-plotmethod,thereasonwhythenewmethodismoresuitabletofracturedreservoirsisaddressedbyatheoreticalanalysis.Anappropriateapplicationofthismethodcanhelpthereservoirengineertooptimizethereservoirmanagementwithlowcostsandhighefficiency.&2014ElsevierB.V.Allrightsreserved.1.IntroductionExperiencesfromoilrecoveryaroundtheglobehaveshowndistinctwaterfloodingperformanceinfracturedreservoirsthaninconventionalreservoirs.Inmostcases,therecoveryusuallybeginswithahighproductionrateinanearlystageandthendeclinesdramaticallyoncewaterbreaksthroughduetoarapidriseinwatercut,especiallyinsomehighyieldwells.Moreover,thegeologicalcomplexityisalsoabarrierforaccurateestimationofthewaterfloodingperformanceandthepotentialofafracturedreservoir.Furthermore,aseveryoneknows,itissignificanttoperformreservoirmanagementandinvestmentdecision.Forinterpretationofwaterfloodingperformanceinfracturedreservoirs,manyresearchpapershavebeenpublished.Currentlyusedmethodscanbeclassifiedastwocategories:reservoirsimulationandareservoirperformanceanalysis.Thereservoirsimulationmethodsconsistofnumericalsimulationandphysicalsimulation.Modelsofdual-porosity(Barenblattetal.,1960)andshapefactors(WarrenandRoot,1963;Kazemietal.,1976)arewidelyusedinnumericalsimulationofthefracturedreservoirs.Butoneofthemainproblemsisthatthesemodelsareover-simplifiedtomeetthedemandofcomputing.Anotherproblemisthathistorymatchingisasubjectiveprocess.Thatis,variousresultsmaybeobtainedonthebasisofthesamedata.Becauseofmoretunableparametersinadual-porositymodel,moreprobablechoicesmaybemadebyreservoirengineers.Somenewtechnol-ogies,suchasadiscretefracturenetwork(DFN)modelandunstructuredgrids(HoteitandFiroozabadi,2008a,2008b;Huangetal.,2011),cancharacterizeafracturenetworkmoreaccurately.However,technicallimitationoninformationcollectionofin-situfracturesandenormousamountofcomputingareimpedimentstotheirapplication.Actually,thephysicalsimulation(Yuetianetal.,2013)providesanobjectivewaytopresentthewaterfloodingperformanceinfracturedreservoirs,buthighcostsandlowefficiencyarebottleneckproblems.Comparedwiththereservoirsimulationmethods,thereservoirperformanceanalysismethodsareeasy,fastandcheaptools,whicharecomposedofanalyticalmodels,empiricalmodelsandsemi-empiricalmodels.ButthesetypesofmethodsneedmorefielddataContentslistsavailableatScienceDirectjournalhomepage:www.elsevier.com/locate/petrolJournalofPetroleumScienceandEngineeringhttp://dx.doi.org/10.1016/j.petrol.2014.06.0020920-4105/&2014ElsevierB.V.Allrightsreserved.nCorrespondingauthor.Tel.:þ861089732260.E-mailaddress:cc0012@126.com(S.Gu).1VisitingscholarofUniversityofCalgary.JournalofPetroleumScienceandEngineering120(2014)130–140andrecoveryexperiencetodevelop,andthepredictingresultsalsoneedmorecheckswithfieldproduction.ThetheoryofBuckleyandLeverett(1942)andtheWelge(1952)equationwerefirstproposedtoexplainthephenomenaoftwo-phaseflowinreservoirs.Accord-ingtoexperimentsofEfros(1958),arelationshipbetweenoilcut,oilviscosityandoutflowendwatersaturationinaprocessofwater–oildisplacementwasobtained.Timmerman(1971)foundarela-tionshipbetweencumulativeoilproductionandanoil–waterratio(i.e.,(1C0fw)/fw)byfielddata,whichwasfromawaterfloodingreservoirinIllinois.Tong(1978,1988)studiedstatisticaldatafrommorethan20waterfloodingreservoirsaroundtheglobeanddrewachartforengineerstoevaluatethewaterfloodingperformance.Chen(1985)deducedsomewaterdisplacementcurve(WDC)meth-odsbyusingthetheoryofBuckley–Leverett,theWelgeequationandtherelationfoundbyEfros,andtheresultswereconsistentwithTong'ssurvey.Asmoreadvancesinthetechnologyofreservoirwaterfloodingevaluationaremade,moretypesofreservoirshavebeenputintoconsiderationbyresearchers.El-khatib(2001,2012)appliedtheBuckley–Leverettdisplacementtheorytostudywaterfloodinginnon-communicatingstratifiedreservoirsandininclinedcommunicatingstratifiedreservoirs.Yang(2009)proposedanewdiagnosticanalysismethodforwaterfloodingperformanceinconventionalreservoirs.Infact,manylessonsandmuchexperiencehavealreadybeenlearnedfromhundredsoffracturedreservoirs(AllanandSun,2003;SunandSloan,2003)duringpastmanyyears(Dangetal.,2011).Manyresearchershavepublishedmanymathematicalmodelstointerpretmulti-phaseflowinfracturedmedium,suchastheDeSwaan(1978)model,theKazemianalyticalmodel(1992)andtheCivan(1998)model.However,theexistingproblemsofevaluatingwaterfloodingperformanceinfracturedreservoirshavenotbeenfiguredoutproperly.Oneofthecriticalproblemsishowtodealwithoil–waterflowinadual-porositymedium.Anotherissueishowtodetecttheinfluenceofimbibitiononthein-situflowandtheperformanceofoilwells.Thispaperaimstosolvetheabovementionedproblems.First,amodelisproposedforwater–oilflowinamatrix-fracturemediumbyusingthemethodofpseudorelativepermeabilitycurves.ThenChen'smodel(1982)ismodifiedforcalculationofthewaterbreakthroughtimeandwatersaturationatthebreakthroughtime.Achartiscom-posedforwater-floodingevaluationbyestimationoftheultimaterecoveryfactor.Thenthewaterfloodingperformanceintwofracturedreservoirsisevaluated.Comparedwiththeclassicalmethod,suchasTong'schartandX-plotmethod(1978),someanalysesareconductedandinfluentialfactorsarediscussed.2.Mathematicalmodel2.1.AssumptionsanddefinitionsAwellgroupconsistsofoneinjectorandoneproducerinahighlyfracturedreservoir,andtheKazemimodelingconcept(1976)isused,asshowninFig.1.Theadditionalassumptionsaregivenasfollows:theflowislinear,isothermal,andincom-pressible,anditobeysDarcy'slaw;inadual-porositymodel,fractureandmatrixhaveitsownirreduciblewatersaturation,permeability,porosityandrelativitypermeability;thewater–oildisplacementinthiscaseisnon-piston-like;finally,thereservoiriswater-wetandtheimbibitioneffectistakenintoaccount.2.2.PseudorelativepermeabilityHearn(1971)usedthepseudorelativepermeabilitymethodtosimulateastratifiedreservoirbywaterflooding,whichmeansthatthereservoirisdividedintomanylayers.BabadagliandErshaghi(1993)introducedthismethodintothedualporosityconceptandproposedtheeffectivefracturerelativepermeability(EFRP)methodtoreducethemodeltoasingleporosityfracturenetworkmodel.Inthestratifiedreservoir,eachlayerhasitsownthickness,porosity,initialwatersaturation,andresidualoilsaturation.Similarly,inafracturedreservoir,eitherfracturesormatrixhasNomenclatureAcoefficient,dimensionlessBcoefficient,dimensionlessbfractureaperture[L],mfwwatercut,dimensionlessf/wfthederivativeofwatercutoffracture,dimensionlesshformationthickness[L],mkf,kffconventional/intrinsicfracturepermeability[L]2,μmkmmatrixpermeability[L]2,μmkTtotalpermeability[L]2,μmkrof,krom,kroToilrelativepermeabilityinfracture/matrix/total,dimensionlesskrwf,krwm,krwTwaterrelativepermeabilityinfracture/matrix/total,dimensionlessLlength[L],mP1–P27coefficient,dimensionlessQocumulativeoil,dimensionlessqimbimbibitionrate,dimensionlessqwf,qwm,qwTfracture/matrix/totalflowrate[L]2[T]C01,m2/sRrecoveryfactorofOOIP,dimensionlessR0ultimaterecoveryfactor,dimensionlessRnrecoveryinnormalizedrange,dimensionlessRf,Rm,RTfracture/matrix/totalrecoveryfactorofOOIP,dimensionlessRf',Rm',RT'fracture/matrix/totalultimaterecoveryfactor,dimensionlessSwf,Swm,SwTwatersaturationoffracture/matrix/total,dimen-sionlessSof,Som,SoToilsaturationinfracture/matrix/total,dimen-sionlessSorf,Sorm,SorTresidualoilsaturationinfracture/matrix/total,dimensionlessSwif,Swim,SwiTinitialwatersaturationinfracture/matrix/total,dimensionlessSnwef,Snwem,SnweTfracture/matrix/totalwatersaturationatout-flowendinnormalizedrange,dimensionlessSAwf,SAwm,SAwTaveragewatersaturationinfracture/matrix/total,dimensionlessSnAwTfractureaveragewatersaturationinnormalizedrange,dimensionlessSnAwBTwatersaturationatbreakthroughtimeinnormalizedrange,dimensionlessttime[T],stBwaterbreakthroughtime[T],sVwf,Vwm,VwTfracture/matrix/totalwatervolume[L]3,m3Wrecoveryrate[L][T]C01,m/sXlength[L],mμo,μwoilviscosity[M][L]C01[T],Pasϕf,ϕmfracture/matrixporosity,dimensionlessλimbibitionindex,dimensionlessS.Guetal./JournalofPetroleumScienceandEngineering120(2014)130–140131itsownproperties,sotheycanberegardedastwodifferent“layers”.Thepseudorelativepermeabilitymethodisintroducedtotransformadual-porositymediumintoanequivalentsingleporositymedium,asdisplayedinFig.2.Thisprocesscansimplifythecalculationbyreducingthenumberofequationsandpara-meters.Actually,theendpointsofsaturationsofbothmatrixandfracturesarenotthesamevalue.Therefore,themovablesatura-tionranges(fromSorto1C0Swi)ofthetwomedia,aretotallydifferentfromeachother.Intheprocessofcalculationofpseudorelativepermeability,normalizationisanecessaryprocedureforeliminatingtheeffectoftheendpoints.Thenormalizationprocessaimstotransformvariousoriginalsaturationrangestothenormal-izedrangefromzerotoone,whichenablestheendpointsofmatrixandfracturestobethesamevalue,asdemonstratedinFig.2(a)and(b).Theequationisgivenasfollows:Snw¼SwC0Swi1C0SwiC0Sorð1ÞBythenormalizationprocess,allsaturationsaretransformedtothenormalizedrange,andthentheprocessforpseudorelativepermeabilitybegins.TherelativepermeabilitycanbetestedandcalculatedbytheWelge–JBNmethod(Johnsonetal.,1959)andthesaturationusedincalculationisthewatersaturationattheoutflowendSwe;therefore,thewaterrelativepermeabilitycanbewrittenaskrw(Swe).Thepseudorelativepermeabilityofwaterinthenormalizedrangeis(seederivationinAppendixA)krwTðSnweTÞ¼ðkff=kmÞUðϕf=ϕmÞUkrwfðSnwefÞþkrwmðSnwemÞðkff=kmÞUðϕf=ϕmÞþ1ð2ÞSimilarly,thepseudorelativepermeabilityofoilinthenormal-izedrangeiskroTðSnweTÞ¼ðkff=kmÞUðϕf=ϕmÞUkrofðSnwefÞþkromðSnwemÞðkff=kmÞUðϕf=ϕmÞþ1ð3ÞThefracturerelativepermeabilitycurvesseemX-shaped,asdisplayedinFig.2(c).Theycanbewrittenasfollows:krwfðSnwefÞ¼Snwefð4ÞkrofðSnwefÞ¼1C0Snwefð5ÞFig.1.Modelofwaterfloodinginwater-wetfracturedmediaandimbibitionprocess.Fig.2.Modelofwaterfloodinginwater-wetfracturedmedium:(a)matrixrelativepermeabilitycurvesinoriginalrange;(b)matrixrelativepermeabilitycurvesinnormalizedrange;(c)fracturerelativepermeabilitycurvesinnormalizedrangeand(d)totalrelativepermeabilitycurveinnormalizedrange.S.Guetal./JournalofPetroleumScienceandEngineering120(2014)130–140132Thematrixrelativepermeabilitycurvesinthenormalizedrange(Fig.2(b))aretransformedfromrelativepermeabilitycurveintheoriginalrange(Fig.2(a)).Heretheoil/waterrelativepermeabilityratiocanbepresentedinapolynomialfittingform,andwhythepolynomialformbecomesthechoicewillbeexplainedinSection4kromðSnwemÞ¼p1þp2USnwemþp3USn2wemþp4USn3wemþp5USn4wemp6USn5wemþp7USn6wemþp8USn7wemþp9USn8wemð6ÞkrwmðSnwemÞ¼p10þp11USnwemþp12USn2wemþp13USn3wemþp14USn4wemþp15USn5wemþp16USn6wemþp17USn7wemþp18USn8wemð7ÞAccordingtoEqs.(6)and(7),thetotalrelativepermeabilitycanbewrittenaskrwTðSnweTÞkroTðSnweTÞ¼1ðp19þp20USnweTþp21USn2weTþp22USn3weTþp23USn4weTþp24USn5weTþp25USn6weTþp26USn7weTþp27USn8weTÞð8ÞwherethecoefficientsP19–P27canbedeterminedbyfitting.ThetotalrelativepermeabilitycanbereferredtoasFig.2(d).Thetotalwatersaturationis(seederivationinAppendixA)SwT¼ðϕf=ϕmÞUSwfþSwmðϕf=ϕmÞþ1ð9ÞSimilarly,thetotaloilsaturationSoT,thetotalresidualoilsaturationSorTandthetotalinitialwaterSwiTsaturationareasfollows,respectively:SoT¼ðϕf=ϕmÞUSofþSomðϕf=ϕmÞþ1ð10ÞSorT¼ðϕf=ϕmÞUSorfþSormðϕf=ϕmÞþ1ð11ÞSwiT¼ðϕf=ϕmÞUSwifþSwimðϕf=ϕmÞþ1:ð12Þ2.3.FlowinfracturemediumwithimbibitionAnotherkeyprobleminthiscaseishowtodetecttheeffectofexchangebetweenfractureandmatrix.Productionfromthematrixblockscanbeassociatedwithvariousphysicalmechanismsincludingoilexpansion,capillaryimbibition,gravityimbibition,diffusionandviscousdisplacement.Inwaterfloodingreservoir,oilexpansionisnotsignificantrole.Diffusionisnotanobviousphenomenon.Whenfracturepermeabilityisfarhigherthanmatrixpermeability,viscousdisplacementisnegligibleaswell.Andthemainmechanisminproductionfrommatrixtofractureisimbibition.Aronofskyetal.(1958)proposedanempiricalmodelofimbibitioncorrelatedwiththeoilrecoveryfactorandultimaterecoveryfactor,whichisRm¼R0mð1C0eC0λtÞð13Þwhereλisanimbibitionindex,whichdeterminesoftheconver-genceratetotheultimaterecoveryfactor.Infact,itillustratesthemagnitudeofimbibition,andtheunitis[1/s].Althoughλisanempiricalparameter,itincludesphysicalmeaning.Kazemietal.(1992)useanequationtocharacterizethisparameter.Andλcanalsobeobtainedbyspontaneousimbibitiontestorhistorymatching.ByusingtheDuhamelprinciple,ChenandLiu(1982)deducedanewdynamicimbibitionmodelbyusingtheAronofskymodelwithrespecttodynamicwatersaturationinthefracturesystem.TerezandFiroozabadi(1999)usedthesamemodelintheirresearchtointerprettheexperimentalresult.However,Chen'sequationhasanerrorleadingtoanobviouscalculationmistake,whichwillbediscussedinSection4.Thusthemodelneedscorrection,andChen'smodelcanbemodifiedas(seederivationinAppendixB)qimbðx;y;z;tBÞ¼ð1C0SwimÞϕmR0mλSwfðx;y;z;tBÞC0λZtB0Swfðx;y;z;τÞeC0λðtBC0τÞdτC20C21ð14ÞSupposethatthereisahorizontal,linear,water-wet,naturallyfracturedoil-bearingformationoflengthL,asFig.3.TheinitialwatersaturationdistributionsofthematrixandfractureareSwm(x,0)¼SwimandSwf(x,0)¼0,respectively.Waterhasbeeninjectedintotheinletend(x¼0)sincet¼0.Thedimensionlessparametersareintroduced,suchasx¼x=L,t¼λt,WðtÞ¼WðtÞ=lλandqimb¼qimb=λ.Theequationsofdimensionlessflowinthefracturedporousmediumcanbewrittenasfollows:WðtÞf0wfðSwfÞ∂Swf∂tþϕf∂Swf∂tþð1C0SwimÞϕmR0mSwfC0Rt0Swfðx;τÞeC0ðtC0τÞdτhi¼0Swfðx;0Þ¼SwifðxÞSwfð0;tÞ¼18>>>:ϕmdSwmdtþqimb¼0qimb¼ð1C0SwimÞϕmR0mRt0Swfðx;τÞeC0ðtC0τÞdτC0SwfhiSwmðx;0Þ¼SwimðxÞ8>>>:ð15Þwherethederivativeofwatercutf0wfðSwfÞcanbewrittenasf0wfðSwfÞ¼μo=μw½ððμo=μwÞC01ÞSwfþ1C1382ð16ÞAprogramiscraftedtosolvethetwo-phaseflowequationfornumericalanalysis.ThedataappliedinnumericalcalculationcanbereferredinTable1.Somedynamicparameterscanbedeter-minedthroughthiscomputing,includingthefracturewatersaturationSwfandth
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