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CompositelinearflowmodelheterogeneousshalereseJieZenga,b,XiangzengWangc,JianchunaUniversityofRegina,PetroleumSystemsEngineering,bStateKeyLaboratoryofOilandGasReservoirGeologycYanchangPetroleumGroup,Xi'an,Shaanxi,710614,PRarticleinfoArticlehistory:Received31August2016Receivedinrevisedform1January2017Accepted2January2017Availableonline9January2017Keywords:inearly-middlerixblocksize,sec-Optimalfractureandcomparedthismodelisaofshalereservoirs.byverticallysub-Allrightsreserved.*Correspondingauthor.PetroleumSystemsEngineering,FacultyofEngineeringandAppliedScience,UniversityofRegina,Regina,Saskatchewan,S4S0A2,Canada.**Correspondingauthor.StateKeyLaboratoryofOilandGasReservoirGeologyandExploitation,SouthwestPetroleumUniversity,8XinduAvenue,Xindu,Chengdu,Sichuan,China.Zeng).ContentslistsavailableatScienceDirectJournalofNaturalGasScienceandEngineeringJournalofNaturalGasScienceandEngineering38(2017)527e548E-mailaddresses:guojianchun@vip.163.com(J.Guo),fanhua.zeng@uregina.ca(F.partiallypenetratingcases,anderrorsofAzari'smethodbecomeparticularlynoticeabletimeresponse.Theinfluenceofotherparameters,suchasmatrixpermeability,matondaryfracturepermeabilityandhydraulicfractureconductivity,arealsodiscussed.patternisselectedbasedoncumulativeproduction.Besides,fielddataareanalyzedgraphicallywithmodelingsolutions,andreliableresultsareobtained.Asnumericalandsemianalyticalmethodsrequireextensivecomputingprocessing,practicalalternativetopredictwell-testingresultsandselectoptimalwellpatternReservoirheterogeneitiesinverticaldirectioncanbefurtheraddedtoourmodeldividingthereservoirintomoreparts.©2017ElsevierB.V.contributestoproductivity.SolutionsfromAzari's(1990,1991)work,wheretheeffectoffractureheightismerelytreatedasaskinfactorareinvestigatedaswell.ResultsshowthatourmodelismoreaccurateinShaleheterogeneousreservoirMultifracturedhorizontalwellTransientrateandpressurebehaviorPartiallypenetratingfracturehttp://dx.doi.org/10.1016/j.jngse.2017.01.0051875-5100/©2017ElsevierB.V.Allrightsreserved.formulti-fracturedhorizontalwellsinrvoirGuob,**,FanhuaZenga,*Regina,SKS4S0A2,CanadaandExploitation,SouthwestPetroleumUniversity,Chengdu610500,ChinaChinaabstractMulti-stagefracturingiscurrentlykeytechniquetodevelopshalereservoirs.Differentanalyticalmodelshavebeenproposedtofastinvestigatepost-fracturingpressure-andrate-transientbehaviors,andhence,estimatekeyparametersthataffectwellperformance.However,previousanalyticalmodelsgenerallyneglectreservoirheterogeneityortypicalseepagecharactersofshale,suchasadsorption/desorption,gasslippage,anddiffusioneffects.Thispaperpresentsananalyticalmodelforpressure-andrate-transientanalysisofmulti-stagefracturedshalereservoir,consideringheterogeneity,typicalseepagecharactersand,specifically,fluidsflowfromupper/lowerreservoirwhenverticalfracturespartiallypenetratetheformation.Thismodelissimilartofive-region-flowmodel,butsubdividesthereservoirintosevenparts,namely,twoupper/lower-reservoirregions,twoouter-reservoirregions,twoinner-reservoirregions,andhy-draulicfractureregion,whicharealltransientdualporositymediaexceptthehydraulicfracture.Asreservoirheterogeneityalongthehorizontalwellboreisincluded,thefracturedistributioncanbevarious.Fractureinterferenceissimulatedbylocatingano-flowboundarybetweentwoadjacentfrac-tures.Theactuallocationsoftheseno-flowboundariesofaspecificheterogeneousreservoiraredeter-minedbasedonthepressurevaluewhichvarieswithtimeandspace.Thus,thetwosidesofthisboundaryhasminimumpressuredifference,satisfyingtheno-flowassumption.Adsorption/desorption,gasslippageanddiffusioneffectsareincludedforrigorousmodelingofflowinshale.Thismodelisvalidatedbycomparingwithcommercialwelltestingsoftware,obtainingagoodmatchinmostflowregimes.Log-logdimensionlesspressure,pressurederivativeandproductiontypecurvesaregeneratedtoconductsensitivityanalysis.Resultssuggestthatlargerdesorptioncoefficientcausessmallerpressureanditsderivativevalueasalargerproportionofgasisdesorbedinformationandjournalhomepage:www.elsevier.com/locate/jngseJ.Zengetal./JournalofNaturalGasScience5281.IntroductionShalereservoirs,whichhaverecentlygainedextensiveinterests,bringmanyuniqueengineeringchallengesthatmaynotcommonlyexistinconventionalreservoirs.Thenaturalgaswithinshalereservoircanbestoredinthematrixnanoporespace,naturalfracturesystemsandadsorbedintheshaleorganicmatter(ScheinandMack,2007).Currently,multi-stagefracturedhorizontalwells(MFHWs),whichsignificantlyincreasethecontactareaoffracturesurfaceandpayzone,areprovedtobehighlyefficientinshalegas/oilextraction.Inpublishedliteratures,anumberofanalyticallybasedsolutionshavebeenproposedtofastpredictpost-fracperformanceandevaluatecriticalfactorsthatdominatewellbehaviorsincenumer-icalmodelsaretime-consumingandneeddetaileddescriptionoffractureandreservoirpropertieswhichareusuallynotavailableorcomplete.Brownetal.(2011)proposedanewtrilinearflowmodelwithhydraulicfractureregion,innerstimulatedregion(dualporosity)andouterreservoirregiontofacilitateinvestigatingpressurebehaviorofMFHWsinshales.Theyalsostatedthattran-sientdual-porositymodelismoreapplicableforshalereservoircomparedwithpseudodual-porosityidealization.Basedontrilinearflowmodel,Ozkanetal.(2011)madecomparisonofMFHWsperformanceintightsandandshalereservoirs.Theyconcludedthatincreasingnaturalfracturedensityisaneffectivemethodtoenhanceproductivityofshale.Later,StalgorovaandMatter(2013)extendedBrownetal.(2011)modelintoafive-regionmodeltosimulatethefracturewithastimulatedsur-roundingarea.Dengetal.(2015)modifiedfive-regionmodeltodealwithnon-uniformdistributedfracturescases,however,thelocationofimpermeableboundarybetweentwoadjacentfracturesishardtodetermineinheterogeneousreservoir.Tooptimizefrac-turingtreatmentsizeandwellspacing,NobakhtandClarkson(2012)analyzedhowthefracturesinonewellaffecttheadjacentwellproductioninazipper-shapefracturepattern.Clarksonetal.(2013)investigatedthestresssensitivityofmatrixpermeabilityandfractureconductivityandfoundthattheyhavesignificantimpactonwellproduction,especiallyforoverpressuredshaleplay.Recently,Ozcanetal.(2014),andWangetal.(2015)combinedlinearflowwithfractaltheoryandanalyticallystudiedthepressurebehaviorofmultifracturedunconventionalwells,theyprovidedalternativemethodtosimulatewellperformanceinfracturedreservoir.Inviewofshale'smultipleflowmechanisms,Zhaoetal.(2013),Wang(2013),Liuetal.(2014),Zengetal.(2015),Zhangetal.(2016),andZhaoetal.(2016)developedanalyticalandsemi-analyticalmodelsaccountingfordesorption,diffusion,andviscousflow.InWang's(2013)andLiuetal.(2014)models,thestressdependentnaturalfracturepermeabilityanddifferentanglesbetweenfractureandhorizontalwellborearealsoincluded.Andbyaddingdiffusionanddesorptioneffects,Tianetal.(2014)improvedEzulikeandDehghanpour's(2013)quadrilinearflowmodel.RenandGuo(2015)presentedamoregeneralsemi-analyticalmodel,incorporatingvariousfracturelength,diversefractureintervals,differentfracture-wellboreangles,fractureasymmetryandpartiallypenetratingfracture.Insummary,thesemodelsareversatileenoughtomodelpre-dominantflowregimesforshalebutneglectoneorseveralofthefollowingtypicalcharacteristicsandpossiblesituationsinshalereservoir:(1)dualporosityfeaturecausedbynaturalfracturesandartificiallycreatedfracturenetworks,(2)gasslippageinthenanopores,(3)diffusionduetoconcentrationgradientinthema-trixpore,(4)desorptionfromthesurfaceoforganicmatter,(5)variousfracturespacing,(6)partiallypenetratingfracture,andspecifically(7)reservoirheterogeneity.1.1.FracturenetworksinshalereservoirNaturalfractures,includingdiscretemacrofracturesandcontinuousmicrofractures(Tian,2014),aregenerallyobservedinshalegasplayswhichconsistofthinlayersofeasilybrokenandfinegrainedrock.Theexistenceofthesefracturesisacriticalfactorthatenhanceswellproductioninsuchlow-permeabilityformationswithnanometertomicrometerpores.Normally,afterfracturingoperation,someofthepre-existingnaturalfracturescanbereop-ened,conductivewithproppantorintersectedbyhydraulicfrac-tures,thus,formingcomplexfracturenetworkwhichcanbedetectedthroughmicro-seismicmapping(Barreeetal.,2002;Mayerhoferetal.,2010).Fromanalyticalpointofview,dual-porositymodels,triple-porosityidealizationsandevenfractionaldiffusionmodelsareusedtosimulatethefluidtransportinfrac-turedunconventionalreservoir.Although,discretefracturenetworkmethodiscapabletoaccountfordetailedpropertiesofindividualfracture,itiscomputationallyexpensiveowingtotherequirementofthoroughcharacterizationdata(Ozcanetal.,2014).Dualporositymodels(Barenblattetal.,1960;WarrenandRoot,1963;Kazemietal.,1969;deSwaanO,1976)involveapplyingeffectivemean-characteristicstodescribefracturedformationaredevelopedbasedonuniformfractureandmatrixpropertyassumptionwhichisappropriateforcontinuous-fracturesystems.Although,dualporositymodelsmaynotalwaysbeauthenticinactualreservoircondition,theyarepracticalforsimulatingfrac-turedreservoirsonaccountofcomputer'srun-timeefficiencyandminimumdatarequirement.Asanalternativeapproach,triple-porositymodel(AbdassahandErshaghi,1986)wasproposedtoconsidertwogroupsofdifferentmatrixblocksinadditiontothefracture.Later,severalinvestigators,suchasAl-GhamdiandErshaghi(1996),Liuetal.(2003),Wuetal.(2004),Al-Ghamdietal.(2010),andDehghanpourandShirdel(2011),extendedthetripleporositymodel.Bydividingthematrixvolumeintotwosub-domains,EzulikeandDehghanpour(2013)developedquadrilinearflowmodeltoconductsimultaneousmatrixtomicrofractureandmatrixtohydraulicfracturedepletion.Andfractaltheoryprovidesanoptionalwayofsophisticatedanalyticalstudyoffracturedres-ervoirs(Cossioetal.,2012;Ozcanetal.,2014).Infact,dualporosityandtripleporositymodelsarebothspecialcasesofmultiporositymodel(Baietal.,1993),andthefundamentalformulasofourmodel,similarwithErtekinetal.(1986)andOzkanetal.(2010)works,arebasedondeSwaanO's(1976)transientdualporosityassumptioncombinedwithlinearflowmodel.1.2.Multi-mechanismflowofshalegasGastransfermechanisminshalereservoirsseemstobesignif-icantlydifferentcomparedwithconventionalgasformationsbecauseofthenanoporestructureandadsorption/desorptionef-fect.Therefore,itiscrucialtounderstandhowthefluidsarestoredandtransportedinthenanoporenetworkbeforedevelopingthegas-bearingformation.Although,athoroughunderstandingofgastransferinshalereservoirhasnotbeenobtained,thefollowingcommonlyacceptedmechanismsaresummarized,asshowninFig.1.1.2.1.Slipflow,KnudsendiffusionandDarcyflowGenerally,thenanoporesizeofshaleorganicmatrix(averagesizelessthan4nm-5nm(Kangetal.,2011))issmallerthanthegasmolecularfreepath,thus,apartfromgasmolecules'collision,thecollisionbetweengasmoleculesandnanoporewallsisessentialaswellandshouldnotbeignored(Wuetal.,2016).Inthissituation,slipflow(randommolecularflow)willoccurinnanopores,whereandEngineering38(2017)527e548theassumptionofcontinuumDarcyflowisnolongervalidandbecomes1.Therefore,Klinkenberg's(1941)andBrownetal.(1946)flowmodifiedDarcy'sequationisrequired.Moreover,asgasisverycompressiblefluid,apressuregradient,initiatedbywellproduc-tion,canimmediatelygenerateagasconcentrationgradient(SigalandQin,2008),whichdrivesthediffusionwithinnanopores(Ertekinetal.,1986).Ordinarily,Knudsendiffusiontheoryisfrequentlyutilizedtomodeldiffusioneffects(Javadpouretal.,2007,Javadpour,2009).AndthecorrespondingdiffusionvelocityinnanoporesofasimplifiedsphericalmatrixblockcanbeexpressedbythefollowingformgovernedbyFick'slaw(Ertekinetal.,1986):vdm¼C0MgDmrmgvCmvr(1)WherevdmisthevelocityofKnudsendiffusion,Mgisthegasmo-lecularweight,Cmismolarconcentration,rmgismatrixgasdensity,andDmisthediffusivitycoefficientandtheapproximatevaluecanbecalculatedfromEquation(2)proposedbyErtekinetal.(1986)inft2/D:Dm¼31:54ffiffiffiffiffiffiffiMgpk0:67m(2)Wherekmistheabsolutepermeabilityofmatrixnanopore.Simultaneously,thepressureforcesresultinDarcyflow,com-Fig.1.SchematicofgasJ.Zengetal./JournalofNaturalGasSciencebinedwithslipflow,themodifiedDarcyvelocityiswrittenas:vsm¼C0kammmgvpmvr(3)WherevsmismodifiedDarcyvelocity,kamisthemeasuredmatrixgaspermeabilitywithslippageeffects,mmgisgasviscosityinthematrixandpmismatrixpressure.Totakeslippageeffectintoac-count,Klinkenberg's(1941)equationisappliedtodescribeactualgaspermeability:kam¼km1þbkpm;ave!(4)Wherebkisgasslippagefactor,andpm;aveisaveragematrixpres-sure.Klinkenberg(1941)gavetheexpressionofslippagefactor.Then,theaboveequationcanalsobewritteninthefollowingform:kam¼kmF(5)AndF¼1þ4claverp(6)Wherecisacollisionproportionalityconstantwhichisnormallyequalto1(Zhangetal.,2015),laveisthegasmolecularmeanfreepath,andrpisthenanoporeradius.lave¼ffiffiffip2rmmg;avepm;aveffiffiffiffiffiffiffiRTMgs(7)Wheremmg;aveismatrixaveragegasviscosity,andTistemperatureinK.Javadpour(2009)alsousedthefollowingformofcbasedonBrownetal.(1946):c¼2atC01(8)Javadpour(2009)pointedoutthatathereisthetangentialmomentumaccommodationcoefficientwithavaluerangingfrom0to1,determinedbythesmoothnessofnanoporewall.Normally,experimentalmeasurementisneededtodetermineatforspecificshalesystemsJavadpour(2009).Asnoexperimentaldataareavailabletoobtainat,here,atisassumedtoequal1,andthen,cinmatrixnanopores.andEngineering38(2017)527e548529slippagefactorshavethesameexpressionwherecequals1andeitherofthemcanbeusedtocalculatecorrectionfactorforslippage.SubstitutingEquation(7)intoEquation(6):F¼1þC188pRTMgC190:5cmmg:avepm;averp(9)NotethepressureunitinEquation(9)isinPa.Next,thetotalflowvelocitythroughthenanoporeinthematrixisassumedtobethecombinationofparalleldiffusionvelocityandmodifiedDarcyvelocityandisgivenby:vtm¼C0kammmgvpmvrC0MgDmrmgvCmvr(10)Thetotalvelocityvtm,asshowninEquation(10),hasquiteanalogousformtothatproposedbyErtekinetal.(1986),however,theymerelyusedFick'slawtocharacterizeslipvelocity.Similartoconventionalformations,Darcyflowissuitableforgasflowinmacrofractures(secondaryfractures),whichwillbediscussedlater.Forgastransferfromshalematrixtofracture,wefollowthederivationofErtekinetal.(1986)andOzkanetal.(2010)asshowninAppendixA.1.2.2.Adsorption/desorptionandsurfacediffusionDuetothelargesurfaceareaoforganicnanoporousmaterialsandtheiraffinitytomethane(Yuetal.,2016),acertainpartofgaswithinshaleformationisadsorbedintheorganic-richrocks.Oncethewellproduces,reservoirpressurestartsdecreasing,andthegasmoleculesonthesurfaceoforganicmatterdesorbtoreducepres-suredecline.Asgasdesorptioncarrieson,agasconcentrationgradientformsfromorganicbulktoitssurfaceandcontrolsthesurfacediffusionofkerogen(Javadpour,2009).Traditionally,theclassicLangmuirisotherm(Langmuir,1916)hasbeenappliedtorepresentgassinglelayerdesorptionforshaleformations.Here,theequivalentvolumeconcentrationconceptisusedtodescribeLangmuirtheory,asshownbelow:VEsc¼VLpmpLþpm(11)partiallypenetratingfractureandreservoirheterogeneity.2.MethodologiesInthispaper,deSwaanO's(1976)transientdual-porosityidealizationcombinedwithlinearflowassumption,asshowninFig.2,isappliedtomodelfluidsflowwithinfracturedformation.Thus,hydrocarbonswithinthesphericalmatrixblockwouldfirsttransfertothefracturenetworks(secondaryfractures),andthen,flowtohydraulicfracturethroughconnectedsecondaryfractures.Theentirereservoirisconsideredtobetransientdual-porositybecauseshalegasreservoirgenerallyhasextremelylowperme-abilitywhichconduceslong-timetransientflowregimestopro-ducegas(Brownetal.,2011;KimandLee.2015).ThesphericalmatrixelementassumptionhasbeenmadeasKucu
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