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Sciences,occupythecharacteristic,conventionalDarcysimulationandthesimulationofthresholdpressureofourpetroleumisnoBelhajpetroleumJournalofPetroleumScienceandEngineering78(2011)239–242ContentslistsavailableatScienceDirectJournalofPetroleumScience.else1999).Inthelowpermeabilityporousmedia,thefluidflowchannelisverynarrowandtheliquid–solidinteractionissignificant(Huang,1997).Fluidboundarylayerintheporesurfaceresultsinincreasedfiltrationresistanceandtheseepagevelocityisnotlinearwiththepressuregradientanylonger(DengandLiu,2001).Onlywhenthepressuregradientexceedsacertainvalue,theseepagevelocitywillhaveapseudo-linearrelationshipwiththepressuregradient(Luetal.,2002;Xiongetal.,2009).Underthiscircumstance,thepermeabilityisnotaconstant,whichinfactisaparametervaryingwithpressurecantakethenon-Darcybehaviorintoconsideration.Besides,arenovatednonlinearseepagemathematicalmodelisproposedinthispaperandanidealgeologicalmodelofdiamondinvertedninespotswellpatternisalsoestablishedwiththefieldandexperimentaldataforfurtherstudy.Atlast,thesimulationresulthasbeencomparedwiththeonessimulatedunderDarcy'slaw,ThresholdPressureGradientmodelandtherealproductiondata.Thenumericalmodelandthesimulationsoftwareshowveryconsistentresultsinallranges,whichisofgreatsignificanceforthepetroleumindustry.gradientduetocharacteristicsoflowpermeabarethreezonesinthelowpermeabilityreservoirsnonlinearseepagezoneandpseudo-linearseepagepressuregradientinazoneissmallerthanthe⁎Correspondingauthor.Tel.:+861069213294,+8E-mailaddress:xuqingyan007@163.com(Q.Xu).0920-4105/$–seefrontmatter©2011ElsevierB.V.Alldoi:10.1016/j.petrol.2011.05.012disciplineandnon-fromDarcy'slawat(AlvaroandFaruk,couldtakethenonlinearflowintoconsiderationandmakepredictionsforthelowpermeabilityreservoirs.Forthispurpose,thenewnonlinearnumericalsimulationsoftwarehasbeendevelopedwhichpetroleumdisciplines,demonstratingdepartureslowfluidvelocitiesinlowpermeabilityreservoirs1.IntroductionDarcy'slaw,thefoundationformuchflowinporousmedia,isaclassicexampleHowever,boththehydrologistsandthenosecretthattheempiricalDarcy'slawdescribefluidflowinporousmedia(bodyofexperimentalevidence,fromagreementwiththeproductiondata.Thereforetakingthenonlinearseepageintoaccountismoresuitabletoreflectthepercolationmechanismanddevelopmentperformanceoflowpermeabilityreservoirs.©2011ElsevierB.V.Allrightsreserved.understandingoffluidofascientificparadigm.engineersfinditislongeracceptedtofullyetal.,2003).Thereisagradient,thezoneiscalleddeadoilzone;whenthepressuregradientinazoneissmallerthanthecriticalpressuregradientbutbiggerthantherealthresholdpressuregradient,thezoneiscallednonlinearseepagezone;whenthepressuregradientinazoneisbiggerthanthecriticalpressuregradient,thezoneiscalledpseudo-linearseepagezone(Xiongetal.,2009).Butuptonow,thereisnocommercialnumericalsoftwarewhichilityreservoirs.There:deadoilzone,zone.Whentherealthresholdpressure2.Theestablishment2.1.NonlinearflowTypicalnonliincludethreeparts:andthepseudo-lin613651045892(mobile).rightsreserved.andthenonlinearsimulationresultshowsexcellentgradientarealsoconducted.Thesimulationresultshowsthat:inthelowpermeabilityreservoirs,nonlinearseepagedominatesinalargescaleofformationdiamondinvertedninespotswellpatterncondition,thenonlinearsimulation,ThemodelandalgorithmofanewnumericalpermeabilityreservoirsQingyanXu⁎,XianguiLiu,ZhengmingYang,JunWangInstituteofPorousFlowandFluidMechanics,GraduateUniversityofChineseAcademyofabstractarticleinfoArticlehistory:Received20August2010Accepted1May2011Availableonline11May2011Keywords:lowpermeabilityreservoirsnonlinearseepagenumericalsimulationLowpermeabilityreservesurgenttoresearchintensivelypermeabilityreservoirsdeviatesseepage.Accordingtothisdeveloped.Unlikemostmatureinfluencecausedbythenonlinearmathematicalmodelisproposed.idealmodelofdiamondinvertedjournalhomepage:wwwsimulationsoftwareforlowLangfang065007,Chinaalargeproportionintheremainingreservesallovertheworld.Soitislawoftheseepageinthelowpermeabilityreservoirs.FluidflowinthelowfromtheclassicDarcy'slawandinsteadconformstotheoneofnonlinearanonlinearseepagenumericalsimulationsoftwarehasbeencommercialnumericalsimulationsoftwaresatthepresentstage,theseepageisspeciallyconsidered.Inthispaper,anonlinearseepageInaddition,onthebasisofpracticalfieldandlaboratoryexperimentdata,anninespotwellpatternisalsoestablished.UnderthesamereservoirandEngineeringvier.com/locate/petrolofnonlinearflownumericalmodelmodelnearflowcurvesoflowpermeabilityreservoirnon-flowsection,nonlinearflowbendsectionearflowbendsection.InordertodescribetheFig.2showsthat,whenthepressuregradientissmall,theseepagecurveisconcave;whenthepressuregradientislarge,theseepageislinear.Attheminimumpressuregradient,thebiggestporesandthroatsparticipateintheflow;withtheincreaseofthepressuregradient,moreandmoreporesandthroatstakepartintheflowandtheseepagecurvepresentsaconcavecurvedsection;afterthewiththevariationoftheinletpressure.Besides,theflowtimeand(oil,gas,andwater)inthereservoir(2)waterandoilcannotbeFig.1.Typicalcurvesofdifferentflowregimes.Table1Datasetsofreservoirparameters.Porosity0.107Permeability5×10−16m2Netthickness9.4mOildensity0.76×103kg/m3Oilviscosity1.74×10−3PasWellarrayspacing100m×600mNonlinearseepageparametera=2.0,b=1.5×10−5(1/Pa)240Q.Xuetal./JournalofPetroleumScienceandEngineering78(2011)239–242nonlinearflowcurveoflowpermeabilityreservoirinmathematicalsense,experimentalcorrectionisusedonthebasisofclassicalDarcylaw(Yangetal.,2007).Hencetheequationofthemotioninthelowpermeabilityreservoircanbeformulatedasfollows:Vi=−MKKriμi∇Pið1ÞM=1−1a+b∇Pijjð2ÞVg=−KKrgμg∇Pgð3ÞInformula(2),aandbarebothfunctionsofpermeabilityandcanbeobtainedbylaboratoryexperiments.bisequivalenttothereciprocalofpseudothresholdpressuregradient.aisafactorthataffectthenonlinearconcavecurve(a≥0)andisalsoadimensionlessparameter.Whena=0,theformulasaboveturntopseudothresholdpressuregradientmodel;whenbisinfinite,reflectinginfinitesimalpseudothresholdpressuregradient,theinteractionbetweenfluidandsolidisweakandMapproacheszero,whichmaketheformulasabovebecomeDarcylinearflowmodel.Insummary,whena=0,appearingaspseudothresholdpressuregradientmodel,thenonlinearflowcurveisastraightlineandthelineintersectsthexaxisat1/b;when0bab1,thecurveintersectsthexaxisat(1−a)/bandthevalueisminimumstartingpressuregradient;whena≥1,thecurvegetsthroughthecoordinateoriginandtheminimumthresholdpressuregradientiszero.Whenaandbaregivendifferentvalues,accordingEqs.(1)and(2),seepagecurvesareshowninFig.2.Fig.2.Differentseepagecurves.solublewitheachother(3)thegascanbedissolvedintheoilphase(4)thedissolvedgasandthefreegascanexchangefreely(5)oilandwaterreachthebalanceinstantaneously(6)theseepageinthereservoirisisothermal(7)theseepageofoilandwaterabidesbythenonlinearseepagelawand(8)thecompressibilityofrockandfluidisconsidered.2.3.MathematicalmodelBysubstitutingtheequationofmotionintotheequationofcontinuityofeachphase,themathematicalmodeloflowpermeabilityreservoircanbeobtained.flowratecanalsobemeasured,andthentheflowvelocitycanbeobtained.Throughmeasurementofthepressuregradientandthefluidseepagevelocity,scatterdiagramofflowvelocityversuspressuregradientcanbeobtained.Thusbasedontheabovemathematicalmodel,bymeansofnonlinearregression,thevalueaandbcanbeachieved.2.2.FundamentalassumptionFundamentalassumptionofthemodel:(1)therearethreephasespressuregradienthasreachedthecriticalvalue,thenumberoftheporesandthroatsinvolvedintheflowisstableandthecurveturnsintostraightline.Inthelaboratoryexperiment,forsomekindoffluidandspecifiedcores,theinletandoutletpressuredifferenceandthelengthofthecorecanbemeasured.ThuswecangetdifferentpressuregradientFig.3.Relativepermeabilitycurve.241Q.Xuetal./JournalofPetroleumScienceandEngineering78(2011)239–242Oilcomponentequation:∇⋅kkroBoμo1−1a+b∇Po−ρ0gDðÞjjC18C19∇Po−ρogDðÞC20C21+qov=∂∂tϕS0BoC18C19ð4ÞWatercomponentequation:∇⋅kkrwBwμw1−1a+b∇Pw−ρwgDðÞjjC18C19∇Pw−ρwgDðÞC20C21+qwv=∂∂tϕSwBwC18C19ð5ÞGascomponentequation:∇⋅kkrgBgμg∇Pg−ρggDC16C17"#+∇⋅RsokkroBoμo∇Po−ρogDðÞC20C21+qgv=∂∂tϕSgBg+RsoSoBo!"#ð6Þ3.ExampleandanalysisUtilizingthedevelopedsoftware,aconceptmodelofdiamondinvertedninespotswellpatternisestablishedabstractedfromarealoilfieldandsubsequentlysimulated.ThepetrophysicalpropertyandfluidcharacteristicsareshowninTable1.Agridsystemof61×41×1hasbeenusedasillustratedinFig.4.TheadoptedrelativepermeabilitycurveispresentedasFig.3.AsillustratedinFig.1,thenonlinearbendingsectionisbetween0andFig.4.Griddivisionand0.1MPa/m.Hydraulicallyfracturingisconsideredforboththeproductionwellsandinjectionwellsandthefracturepenetrationratioissetto1/3.Duringthesimulation,withthebottomholeflowingpressureasconstraintcondition,theinjectionwellstakethevalueof30MPaandtheproductionwellstakethevalueof12MPa.Thesimulationtimeis10yearsandthetimestepis30days.Basedonthisconceptmodel,thereservoirdevelopmentispredictedundernon-linearflowmodel.Thepressuregradientdistributionofthereservoirafter10yearssimulationbythenewsoftwareisdisplayedinFig.5.AsshowninFig.5,quasi-linearseepageonlyoccursinthevicinityofthewellboresandpressuregradientinmostregionsissmall(i.e.,locatinginthenonlinearseepagezone).Nonlinearseepagedominatesinalargescaleofformationinthelowpermeabilityreservoirs,whichwellsdistribution.Fig.5.Schematicofpressuregradientdistributionafter10years.Krirelativepermeabilityforeachphase(i=o,w,g)(Dimen-ϕporosity(dimensionless)242Q.Xuetal./JournalofPetroleumScienceandEngineering78(2011)239–242consequentlyverifiestheaccuracyandprecisionofthemodelandthesoftware.Besides,theperformanceissimulatedunderDarcyflowmodelandthresholdpressuregradientmodel.Finally,thecomparisonismadebetweenthecalculatedresultsandtherealdataasdepictedinFigs.6and7.Fig.6depictstheoilrecoveryunderthreemodelsandtherealproductiondata.Withthesamecondition,thesimulatedresultunderDarcy'slawisthebiggestwhichisnotdifficulttounderstandbecauseDarcy'slawdoesnottakethenonlinearseepageresistanceintoaccountandtosomeextentimprovestheflowcapacityofthepayzone;theresultofnonlinearflowisslightlybiggerthantheresultofthresholdpressuregradientmodelandisgenerallyconsistentwiththeactualproductiondata.It'sjustbecausethatadoptingthethresholdpressuregradientmodelisequivalenttoincreasethethresholdvalueoftheinitialflow,exaggeratetheformationflowresistanceandreducetheseepagerangeoftheoilpool.Fig.7showsthecomparisonofwatercutunderdifferentcircumstancesincludingtherealproductiondata.ItisveryclearthatthewatercutofDarcymodelrisesfastestandthewatercutofnonlinearmodelincreasesfasterthanthethresholdpressuregradientmodel.Theabovefactssuggestthatthewaterfloodingofthelowpermeabilityisslow-effectandthespeedofpressurepropagationislow.Inthiscase,itisdifficulttoformeffectivedrivingsystem,whichisinaccordwiththedynamiccharacteristicsofdevelopedlowpermeabilityoilfield.4.ConclusionsLotsofcriticismhasbeenreportedagainsttheuseofDarcy'sequationtorepresentallrespectsofporousmediaflow.Thereisalsoabodyofexperimentalevidenceandpracticaldatarevealingde-parturesfromDarcy'slawatlowfluidvelocitiesinlowpermeabilityreservoir.SomepetroleumengineersproposethethresholdpressureFig.6.Recoveryversustimeunderdifferentcircumstances.gradientmodel,holdingthatonlywhenthepressuregradientexceedsacertainvalue(i.e.thresholdpressuregradient)thefluidjustbeginstoflow.Nevertheless,thehistorymatchedreservoirsimulationbasedonthismodelusuallygivesincorrectpredictionsoffuturefieldperformanceandtheexistenceofthresholdpressuregradientisalsocontroversial.Inordertodescribethenon-Darcybehaviorbetter,anonlinearseepagemathematicalmodelisproposedandthenonlinearseepagenumericalsimulationsoftwareisalsodeveloped.Basedonthenewsoftware,anidealmodelisestablishedtoperformthenonlinearsimulation,conventionalDarcysimulationandthresholdpressuregradientsimulation.Thecomparisonamongthethreesimulationresultsmanifeststheproposedmodelshowsexcellentconsistencyinallrangesandreflectsthepercolationmechanismanddevelopmentperformanceoflowpermeabilityreservoirsbetter.Biformationvolumefactorforeachphase(i=o,w,g)(Dimensionless)Sisaturationforeachphase(i=o,w,g)(Dimensionless)ρidensityforeachphase(i=o,w,g)(M/L3),kg/m3Rsodissolvedgas–oilratio(dimensionless)Dverticaldepth(L),mqivfluidvolumetricflowrateperunitofrockforinjectionorproductionundersurfacestandardconditions(L3/T),m3/sAcknowledgmentsTheauthorswouldliketoexpressgreatgratitudetotheNationalNatureScienceFoundationofChinafortheirsupport.AndthanksalsotoDr.YuRongzeandDr.HeYingfortheirgeneroushelp,cooperationandencouragement.ReferencesAlvaro,P.,Faruk,C.,1999.ModificationofDarcy'slawforthethresholdpressuresionless)∇Pipressuregradientforeachphase(i=o,w,g)(M/L2T2),Pa/mμiviscosityforeachphase(i=o,w,g)(M/LT),Pa⋅sMnonlinearflowcorrectionfactor(Dimensionless)anonlinearflowparameter(dimensionless)bNonlinearflowparameter(L2T2/M),1/PaNomenclaturesViseepagevelocityforeachphase(i=o,w,g)(L/T),m/sKabsolutepermeability(L2),m2Fig.7.Watercutversustimeunderdifferentcircumstances.gradient.J.Petrol.Sci.Eng.22(4),237–240.Belhaj,H.A.,Agha,K.R.,Butt,S.D.,Islam,M.R.,2003.Acomprehensivenumericalsimulationmodelfornon-Darcyflowincludingviscous,inertialandconvectivecontributions,SPEPaper85678Presentedatthe27thAnnualSPEInternationalTechnicalConferenceandExhibition,Abuja,Nigeria,August4–6.Deng,Y.E.,Liu,C.Q.,2001.Mathematicalmodelofnonlinearflowlawinlowpermeabilityporousmediaanditsapplication.ActaPetroleiSin.22(4),72–76(inChinese).Huang,Y.Z.,1997.Nonlinearpercolationfeatureinlowpermeabilityreservoir.Spec.OilGasReservoir4(1),9–14(inChinese).Lu,C.Y.,Wang,J.,Sun,Z.G.,2002.Anexperimentalstudyonstartingpressuregradientoffluidsflowinlowpermeabilitysandstoneporousmedia.Pet.Explor.Dev.29(2),86–89(inChinese).Xiong,W.,Lei,Q.,Liu,X.G.,Gao,S.S.,Hu,Z.M.,Xue,H.,2009.Pseudothresholdpressuregradienttoflowforlowpermeabilityreservoirs.Pet.Explor.Dev.36(2),232–236(inChinese).Yang,Q.L.,Yang,Z.M.,Wang,Y.F.,Ji,H.X.,2007.Studyonflowtheoryinultra-lowpermeabilityoilreservoir.Drill.Prod.Technol.30(6),52–54(inChinese).
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