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Figure5—FracturelengthdistributioninFigure4.

wheren(l)dlisthenumberofnaturalfracturestaneously.havingalengthintherange[l,l?dl],?isacoef-Injectionrate60bpm

ficientofproportionality,andaisanexponentvary-Lengthofastage300ftLayerheight100ftingbetween1and3.AsshowninFigure1,the

Minimumhorizontalstress4450psiprobabilitytointroducelargefracturesismuch

FluidLeak-offCoefficient0.00001ft/min0.5smallerthanthatofhavingsmallfractures.TheViscosity1cprelativeabundanceofsmallversuslargefracturesisYoung’sModulus6.53?106psiinverselyproportionaltothemagnitudeoftheex-Poisson’sRatio0.2No.ofPerforations60ponenta(Figure3).Foragivensetofnatural

DensityofSlurry1.2g/cm3fractures,orientationsareroughlythesame.Inthe

DiameterofPerforations16mm

Barnettshale,thedominanttrendofnaturalfrac-DiameterofWellbore0.1m

turesiswest-northwest,andothersetstrendnorth-Proppant40/70MeshProppantdiameter0.0124inchsouth(Gale,etal.,2007).Toinvestigatethesensi-So0psitivityofthecreatedgeometryoffracturenetworks,

To900psi

wegeneratedasinglesetofverticalnaturalfrac-?0.6

tureswithastrikeorientationof45oina1000ft?1000ftsquarereservoirofsize(Figure4).Fracturespacingwasvariedfromrandom(Figure4a)to

regular(Figure4b).Bothtwopatternswerecreatedusingexponentofpowerlawa?2(Figure5).Inourcurrentmodel,itisassumedthatonceahydraulicfracturedivertedalongthepre-existingpathofanaturalfracture,thehydraulicfracturemustpropagatetotheendofthenaturalfracturebeforekickingintothematrix.Besides,naturalfracturesarepotentialdistributedinthereservoir.Deformationofnaturalfracturesisnotcalculateduntilintersectedbyhydraulicfracturesinthemodel.Inotherwords,thenaturalfracturescannotaffectthehydraulicfracturepropagation,beforehydraulicfracturesintersectingnaturalfractures.Thecomputationalefficiencyofthemodelisnotincreasedafterincludingnaturalfractures.

Table1—Inputparametersformultiplefracturespropagatingsimul-

Sensitivityanalysisofcomplexfracturenetworks

Aseriesofsimulationswasperformedusingourfracturepropagationmodeltoillustratetheimpactofperforationclusterspacinganddifferentialstress(DS?SHmax-Shmin)onfracturegeometryandinjectionpressureinthenaturallyfracturedreservoirs.Theparameters(Table1)areprescribedbasedonthepublisheddataforshalereservoirs.Fractureheightisconstantandequaltothethicknessofthereservoir

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Figure6—Effectivefracturelengthdistributionandtotaleffectivefracturelengthforfourcaseswithdifferentperforationclusterspacing(slickwater,60bpm,DS?100psi,relativeanglebetweenNFandHF?45o,a?2,NFspacing?55ft).

formation.Hence,fracturesurfaceareaisdependentontheeffectivelengthofhydraulicfractures.Theeffectivefracturelengthisdeterminedbyproppanttransportinthecomplexfracturenetworkswhichisrelativetoproppantdiameter.Cipollaetal.(2011)statedthattheeffectivelengthisfracturelengthwithapertureof2.5timesgreaterthantheaverageproppantdiameter,whichcanallowproppanttoenterafracture.

Theeffectsofperforationclusterspacing

Perforationclusterspacingisaveryimportantfactorthatcanbeoptimizedtomaximumoil/gasproduction.Theclusterspacingaffectsoil/gasproduction,intwoways-pressureinterferencefromproductionandfractureinteractionbetweenmultiplefractures(YuandSepehrnoori,2014;Yu,etal.,2014).Wesolelyfocusonstudyingthefractureinteraction.Thestressshadoweffectsofsimultaneousmultiplefracturescanresultinunevenfracturegrowth(Olson,2008;RousselandSharma,2011;WuandOlson,2013).Furthermore,naturalfracturesalsoaffectfracturegrowthandmightgeneratecomplexfracturenetworks(OlsonandDahi-Taleghani,2009;Dahi-TaleghaniandOlson,2009;Kresseetal.,2013;WuandOlson,2014(a)).Thegoalofstimulationdesignsistoexposeasmuchfracturesurfaceareaaspossibleattheleastamountofcost,becauseitisbelievedthatdrainageareaisdirectlyrelatedtothesurfacearea.

ThetotaleffectivefracturelengthforfourcaseswithdifferentperforationclusterspacingisasshowninFigure6forastagewithalengthof300ft.Changingthteperforationclusternumberfrom3to6changesclusterspacingfrom150ftto60ft.Perviousworkonstressshadows(WuandOlson,2013)suggeststhatwhenfractureshaveaspacingequaltoorlessthantheirheight,fracturewidthwillbehinderedintheinteriorfracturesofamulti-clusterstage.ThewidestspacingcaseinFigure4(S/H?1.5)wouldconsequentlybeconsideredamildinteractioncase,andthe6fracturecasewouldbestrong(S/H?0.6).Withdecreasingperforationclusterspacing,thevariabilityofgrowthbetweenthefracturesofagivenstageincreases.Inthestronginteractioncase,theexteriorfractureshavelongereffectivelengthsthantheinteriorones.Thelargerthenumberoffractures,themoreuneveneffectivefracturelengthdistribution.Theresultsalsoshowthatincreasingthenumberofperforationclustersinfixedlengthstage

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Figure7—Widthprofileoffracturenetworks(leftside)andthechangeofthemaximumshearstresswithhypotheticalmicroseismiceventpatterns(rightside)attheendofinjectionforfourcaseswithdifferentperforationclusterspacing(slickwater,60bpm,DS?100psi,relativeanglebetweenNFandHF?45o,a?2,NFspacing?55ft,‘*’:microseismicactivity).

doesnotnecessarilyresultinincreasedtotalfracturelength.Fortherangeofinputexamined,themaximumtotalfracturelengthwasforthecaseof4fractures.Thegreatestpenetrationawayfromthewellbore,however,wasachievedfortheclosestspacedcasewhichhadthestrongestmechanicalinteraction.Only2ofthe6fracturesgrew,allowingthemtopenetratefurthestawayfromthewellbore.Figure7showsfracturewidthprofilesattheendofinjectionforeachcase.Thegreylinesrepresentnaturalfracturesandlineswithdifferentcolorsrepresentthehydraulicfracturewidthdistribution.Fracturewidthprofilesofthefourcasesillustratethatnearthewellbore,fracturesurfaceareaforcaseswithsmallclusterspacing(e.g.thecasewith6fractures)ismuchlargerthanthatforcaseswithlargeclusterspacing(e.g.thecasewith3fractures).Awayfromthewellbore,however,thecaseswithsmallclusterspacinghavesomewhatlesssurfaceareaandsmallerwidth.Thismightimplyahorizontalwellwithsmallclusterspacingcouldhavehighproductionratesattheearlytime,butdeclinerapidlyduringthelatetime(Khan,2013).

Theeffectsoftheremotedifferentialstress

Remotedifferentialstressalsohasagreatimpactoninjectionpressureandfracturecomplexity.Figure2indicatesthatthecrossingconditionisaffectedbythein-situstressesratio.Generally,theratioisgreater

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Figure8—Variationsofthenetinjectionpressureattheheelofhorizontalwellboreforthreecaseswithdifferentdifferentialstresses(clusterspacing?100ft,slickwater,60bpm,relativeanglebetweenNFandHF?45o,a?2,NFspacing?55ft).

Figure9—Effectiveandtotalfracturelengthoffourfracturesforthreecaseswithdifferentdifferentialstresses(clusterspacing?100ft,slickwater,60bpm,relativeanglebetweenNFandHF?45o,a?2,NFspacing?55ft).

than1andlessthan2.Whentherelativeangleofthehydraulicandnaturalfractureis45oandfrictioncoefficientis0.6,itisevidentfromthefigurethatthehydraulicfracturescannotcrossthenaturalfracturesforanydifferentialstresses.Furthermore,non-zerotensilestrengthofrockmakesthehydraulicfracturesevenhardtocrossthenaturalfractures.Therefore,inthiscasehydraulicfractureswillnotcrossnaturalfracturesforthreedistinctdifferentialstresses(0psi,100psi,300psi).Whenahydraulicfracturegrowsalongamisaligneddirection,additionalstresseswillactonthehydraulicfracture.Thelargerdifferentialstressimpliesthehigheradditionalstresses.Figure8showsnetinjectionpressureatthewellborewithtimeandillustratesthatthenetpressureincreaseswithincreasingdifferentialstress.Inaddition,thenetpressureofthecaseswithdifferentdifferentialstressesexhibitsdistictmagnitudesofpressurechangeasintersectingwithnaturalfractures,emphasizingthatnetpressuretrendmightbeabletoaidincharacter-izingcomplexfracturegeometry.

TotalfracturelengthandeffectivelengthareshowninFigure9,illustratingthatfracturegrowthis