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UniversityArticlehistory:Received3January2012Accepted2June2012Availableonline21June2012Keywords:fractaldimensionfracturenetworkpermeabilityfracturedensityfracturelengthfractureorientationmultivariableregressionartificialneuralnetworksrepresentedbydiscretefractures.ThisapproachismoreusefulinandcharacterizefracturedreservoirproperlyandsimulatetheirContentslistsavailableatSciVerseScienceDirectJournalofPetroleumScienceJournalofPetroleumScienceandEngineering92–93(2012)110–123complexgeometrywhichpreventstheirdirectfeedingintotheE-mailaddress:tayfun@ualberta.ca(T.Babadagli).capturingthecomplexityofthenetworksbutthewholesystemisperformancesbyimprovingconventionaldual-porositysimula-tors,modelingofsuchsystemshasremainedadifficulttaskmainlyduetocomplexfracturenetworkgeometry(Bourbiauxetal.,1998,1999).Inotherwords,fracturenetworkshavea0920-4105/$-seefrontmatter&2012ElsevierB.V.Allrightsreserved.http://dx.doi.org/10.1016/j.petrol.2012.06.007nCorrespondingauthor.representingthecomplexityoffracturestructure.Analternativeistousesingleporosityoptioninwhichthemodelcouldbetomuchgreaterdistancesalongthesefracturesthanitcouldflowthroughmatrix(Murphyetal.,2004).DespiterecenteffortstoTwoapproacheshavebeenproposedtomodeltransportinfracturedreservoirs:dual-porosityanddiscretefracturenetworks(DFN).Theformerismorecapabletocapturetheinteractionprocessbetweentwodifferentmediathatconstitutethereser-voir,i.e.,matrixandfracture.But,thisapproachisstilllimitedinInthiscase,anaccuraterepresentationofpermeabilitydistribu-tion(orfracturenetworkpermeabilityifitisstronglyfracturedominated–lowmatrixpermeabilitysystem)isacrucialtask.Insubsurfacereservoirs,thepresenceofnaturalfracturescontrolsthefluidflowbecausefluidcanflowmorequickly1.Introductionrepresentedasasingleporositymediumandthepermeabilitydistributiondescribesthecontrastbetweenfractureandmatrix.abstractMappingfracturenetworksandestimatingtheirpropertiessuchasporosityandpermeabilitytobeusedasinputdatainsimulationstudiesaretwocriticalstepsinmodelingofnaturallyfracturedreservoirs.Thedatatoachievethesetwotasksarealmostalwaysinsufficientandmostlylimitedtowellscalemeasurements,seismicmaps,andoutcropstudies.Anyquantitativeinformationaboutfracturenetworksobtainedthroughthesesourceswouldmakeaccuratepreparationofstaticmodelspossible.Hence,itisessentialtouselimitedquantitativedataeffectivelyinfracturenetworkstudiesforaccurateestimationofreservoirperformanceinanysubsurfacemodelingstudy.Thisstudyfocusesononeofthecriticalproblemsnamelypracticalestimationofeffectivefracturenetworkpermeability(EFNP).First,aninvestigationontherelationshipbetweenstatisticalandfractalparametersoffracturenetworkgeometryandfracturenetworkpermeabilitywaspresented.Then,twelvestatisticalandfractalfracturenetworkpropertiesofrandomlygenerated2Dfracturepatternsweretestedagainstpermeabilityandcorrelationswereobtained.ApproximatelyhalfofthepropertiesshowedastrongrelationshipwiththeEFNP.Correlationsobtainedthroughmultivariableregressionanalysisweretestedonrandomlygenerateddifferentfracturenetworktypesandnaturalfracturepatterns.Inthisexercise,allfracturenetworkcharacteristics(density,length,orientation,connectivity,andaperture)wereconsidered.Finally,thecapabilityofartificialneuralnetworks(ANN)wasusedtocapturecomplexandnonlinearrelationshipsbetweenstatistical-fractalparametersandequivalentpermeabilityof2Dfracturenet-workstofurtherimproveempiricalEFNPpredictionmodels.Toachievethis,abackpropagation(BP)neuralnetworkwithonehiddenlayerinthemiddlewasused.Aftertrainingthisnetwork,itwasusedtopredicttheEFNP.TheANNwasobservedtobeamorepowerfulapproachthanthemultivariableregressionanalysisinhandlingthenon-linearityandcomplexityoftheproblem.Thisstudyshowedthatcertainfractalcharacteristicsoffracturenetworkcouldbeusedinfracturenetworkmappingandpreparationofpermeabilitydata.Thecorrelationsobtainedandtestedcouldalsobeusefultocalculateequivalentfracturenetworkpermeabilitytensorin2Dpracticallyandefficiently.Theproposedmethodcouldbepotentiallyextendedandfurtherdevelopedtobeapplicableon3Dmodelsaswell.&2012ElsevierB.V.Allrightsreserved.articleinfoEstimationofequivalentfracturenetworandstatisticalnetworkpropertiesAlirezaJafari,TayfunBabadaglinDepartmentofCivilandEnvironmentalEngineering,SchoolofMiningandPetroleum,journalhomepage:www.elsevierkpermeabilityusingfractalofAlberta,3-112MarkinCNRL-NREF,Edmonton,AB,CanadaT6G2W2.com/locate/petrolandEngineeringA.Jafari,T.Babadagli/JournalofPetroleumScienceandEngineering92–93(2012)110–123111reservoirsimulators(Bourbiauxetal.,1998).Thus,thisgeometry(withitsdefinedproperties)mustbescaledupandtheresultsfromthisprocessforinstanceequivalentpermeabilitycouldbethenfedintosimulatorsandrunningadualporosity,ordualpermeabilitysimulation(PetrelManual,2007).Fractureswithinasimulationgridblockcausethatblocktoactasaneffectivemediumwhosepropertiescanberepresentedbyequivalentvaluesforinstanceequivalentfracturenetworkpermeabilitytensor(Longetal.,1982).BarenblattandZheltov(1960)andBarenblattetal.(1960)studiedsuchsystemsasadouble-porositymedium.Inthisapproach,eachmedium,i.e.porousmatrixandthefracturenetwork,isoverlappedandmutuallycommunicating(Bogdanovetal.,2003).Later,WarrenandRoot(1963)modeledfracturedporousmediaasanidealizedtwoequivalentfractureandmatrixmediaconsistedofidenticalrectangularporousparallelepipedswhichareseparatedbyanorthogonalnetworkoffractures.Inthismodelfluidflowoccursinthefracturenetworkandmatrixblocksfeedthesenetworks.Becauseofminimizingcomputationworksforfieldscalesimulation,thisrepresentationoffracturereservoirsistypicallyusedinmostindustrialsimulators,butoneofthemostdifficultpartintheuseofWarrenandRootmodelistofindequivalentfracturenetworkpermeabilities(Bourbiauxetal.,1998).deDreuzyetal.(2001)studiedtheeffectofthefractureparameterssuchaslengthandaperturedistributionsonthepermeabilityof2Dfracturenetworks.Theyalsoinvestigatedthepermeabilityscalingandthescaledependenceoftheflowpatternstructure.JafariandBabadagli(2008,2009)showedthatnetworkpropertiessuchasfracturedensityandlengtharethemostcriticalparametersontheequivalentfracturenetworkspermeabilitysincethesenetworkpropertiescontrolthepercola-tionpropertiesofthefracturesystem(i.e.whetherornotthefracturesspantwooppositesidesofthedomain).Iffact,theirstudysuggeststhatforequivalentnetworkpermeabilitypur-poses,oneshouldpayattentiontonetworkproperties(fracturelengthanddensity)ratherthansinglefractureproperties(fractureaperture).Thereareseveralmethodstocalculateequivalentfracturenetworkpermeability.Oda(1985)introducedamethodforcalculatingequivalentpermeabilitytensorofafracturedreservoirusingthegeometryoffracturenetwork.Themethoddoesnotrequireaflowsimulatortoobtainthepermeabilitytensoranditdoesnotaccountforthefractureconnectivity;andthus,itislimitedtowell-connectedfracturenetworks.Inotherword,itwouldunderestimateequivalentfracturepermeabilitywhenfracturedensityislow.Bourbiauxetal.(1998)proposedamethodwhichcalculatestheequivalentnetworkpermeabilityfromincompressiblesteady-stateflowthroughtheactual3Dfracturenetworkineachdirectionbyapplyingapressuredropbetweenthetwosidesoftheparallelepipednetworkwithaspecificboundarycondition.Priortothisapproach,Longetal.(1985)andCacasetal.(1990)developed3DfractureflowmodelsandMassonnatandManisse(1994)thenintroduced3Dfractureflowmodelswhichtakeintoaccountthematrixpermeability.Inasubsequentstudy,Loughetal.(1996)developeda2Dfractureflowmodeltakingintoaccountthecontributionof3Dmatrixflows.Odling(1992a,1992b)introduceda2Dmodelbyconsideringthematrixperme-ability.Thecommonpointintheseeffortswasintroductionofanewfracturenetworkdiscretization.Minetal.(2004)usedstochasticrepresentativeelementaryvolume(REV)approachtocalculateequivalentfracturepermeabilitytensoroffracturedrocks.Theycalculatedtheequivalentpermeabilityvaluesusingthe2DdistinctelementcodeUDEC(Itasca,2000).Nakashimaetal.(2000,2001)usedtheboundaryelementmethodtoestimatetheeffectivepermeabilityofnaturallyfracturedreservoirs.Lateron,Teimoorietal.(2003)usedthesamenumericalmethodtoimprovethecomputationofeffectivepermeabilitytensorinnaturallyfracturedreservoirs.Allthesemethodsusednumericaltechniques.Inhighlyfracturedandcomplexreservoirswhichneedmorenodesforcomputations,thisapproachmightrequireheavycomputationaleffortsandbetimeconsuming.Sincetheclassicaltheoryofpercolationwasfirstintroduced,ithasbeenthefocusofagreatnumberofstudiestosolvenetworkrelatedproblemsindifferentareasofscienceandengineering.Thebuildingcomponentsofthistheory,i.e.scalingequationsanduniversalconstantsandexponents,arewelldefinedforsystematiclattices(Kirkpatrick,1971;Stauffer,1979;Sahimi,1993;BerkowitzandEwing,1998).Firstattemptsonmodelingfracturenetworksusingthepercolationtheorywereonlatticenetworks(StaufferandAharony,1994).However,percolationoffracturenetworksneededanotherclassofpercolationcalledcontinuumpercolationwithregardtothenatureoffracturedistributionwithintherocks(Mourzenkoetal.,2005).Khamforoushetal.(2008)showedthenecessityofthisapproachasboththeindividualfractureproper-tiesandalsotheirinterconnectivitypropertiesareaffectingthetransportpropertiesoffracturednetworks.Mourzenkoetal.(2004,2005)studied2Dfractureplaneswithapower-lawsizedistributionuniformlylocatedina3Dspacemodel.Then,bytriangulatingeachfractureandsolvingflowequations,theycalculatedpermeability.Khamforoushetal.(2008)alsostudiedthepercolationthresholdandpermeabilityofanisotropic3Dfracturenetworks.Inthisstudy,weproposeanotherapproachtopracticallyestimateequivalentfracturenetworkpermeability.Themethodisbasedonusingstatisticalandfractalpropertiesofthenetworks.Ithasbeenknownformorethantwodecadesthatnaturalfracturepatternsexhibitfractalcharacteristics(BartonandLarsen,1985;LaPointe,1988;BartonandHsieh,1989).Thefractaldimensionshowsthetendencyofanobjecttospreadorfillinthespacewhereitislocatedandexpressedbynon-integerdimensionvalues(BerkowitzandHadad,1997).Ithasalsobeenshownthatthefractalpropertiesoffracturenetworkhavesomeimplicationsonconductivitypropertyofthatnetwork(LaPointe,1988).Fractureconnectivity,length,density,aperture,andorientationaretheparameterswhichcontrolthepermeabilityoffracturenetwork.Forinstance,Babadagli(2001)speculatedthatfracturesthatarealignedperpendiculartothedirectionofflowmighthaveanegativeeffectonpermeability.ItwasalsoshownbyRossenetal.(2000)thatasfracturelengthanddensityincrease,theconnectivityofafracturenetworkincrease.Thepermeabilityofthefracturenetworkincreaseswithanincreaseinfractureapertureanddensity(Zhangetal.,1996).Hence,havingapracticaltechni-quetoestimateequivalentfracturenetworkpermeabilityusingstatisticalandfractalcharacteristicsoffracturenetworkpropertieswillfacilitatethepreparationofstaticmodels.Inthisstudy,wefirstfocusedondevelopingrelationshipbetweenstatisticalandfractalpropertiesof2Dfracturenetworksandequivalentfracturenetworkpermeability(EFNP).Thenweintroducedcorrelationstocalculatepermeabilitytensor.Also,becauseofthepossiblecomplexityintheserelationships,anANNwasusedtopredicttheEFNP.Finally,bothmethodswerevalidatedagainstsyntheticandnaturalfracturenetworks.Thisisanefficientandpracticalapproachbutyetcapableoftakingintoaccountthewholecomplexityoffracturenetworks.Thewayitisusedinthepracticalapplicationsisthefirsttoobtainthestatisticalandfractalcharacteristicsoffracturenetworksindifferentpartsofthereservoirandthen,toestimateequivalentpermeabilityforeachgridblocktofeedintotheconventionalsimulators.2.Building2DsyntheticfracturenetworkmodelsWebeginwithdevelopinganalgorithmthatgeneratessyn-thetic2Dfracturemodelinsideasquaredomain.Thevaryingparametersarefracturelength,density,andorientation.Thefractureconnectivityisimplicitlyconsideredasitisafunctionofvaryinglength,densityandorientation.Inotherwords,non-parallelfractureswithhigherdensityandlongerlengthswouldyieldabetterconnectivityandthishasbeentakenintoaccountbyvaryingfracturedensity,lengthandorientation.Ineachmodel,fractureseedsaredistributedaccordingtoauniformdistributionandeachfractureisrepresentedasalineinthedomain.Therangeofeachparameterinthisalgorithmisasfollows:(1)Fracturelength(inmeters):(a)constant:20,40,60,and80and(b)variablewithanormaldistributionandameanvalueof20,40,60,and80.(2)Density(#offractures/domain):50,100,150,200and250(domainis100C2100m2),and;(3)Orientation:twofracturesetsinthedomainwiththedirec-tionsof(a)N–SandE–Wand(b)NW–SEandNE–SW.(4)Aperture:insteadofapertureweusedconductivity(JafariandBabadagli,2008).TheconductivityisrelatedtoaperturesincetheproductoftheintrinsicfracturepermeabilityandtheceptshavelengthpercolatingEachdatasethas20differentcombinationsofthelengthanddensity(Table1).Tendifferentrealizationsusingdifferentran-domnumberseedsforeachcombinationweretestedtoincludetheeffectofrandomness.Foreachcase,thisamountsto200realizationsandthereforeforallfourdifferentorientationcases,atotalof800realizationswereperformed.PermeabilityvaluesinthistablewereobtainedusingacommercialsoftwarenameFRACA(seeSection4).Theywereconsideredastheactualortargetvaluesandusedtoevaluatetheaccuracyoftheproposedmethodsthroughoutthisstudy.Figs.1and2showtwodifferentrepresentationsofthe2Dpatternswithdifferentorientationsandfixedfracturelength.Inasense,eachofthesemodelscouldbeconsideredasrepresentationofone2Dgridcellinthedual-porositysimulatorsorthewholenetworkofthefieldthatcouldlaterbedividedintosubgrids.the1000mD.mfractureconductivity.FD:Fractaldimension.inYConnectivityindexMaximumtouchwithXscanninglineMaximumtouchwithYscanninglineFD(box-counting)Permeability(mD)Lines20Fig.1.Fracturenetworkwithtwofracturesets,N–SandE–W,andconstantlengthinthe100C2100m2domain.A.Jafari,T.Babadagli/JournalofPetroleumScienceandEngineering92–93(2012)110–123112202001.7681.8111.5591.7631.7791.933202501.8421.8491.6501.5671.6921.74440501.4251.3921.6581.6501.7991.833401001.7401.6161.6291.7701.7301.790401501.8471.7281.6231.6941.8421.777402001.9001.7791.6421.6901.8321.784402501.9291.8191.5791.7481.7981.73960501.6401.3831.4061.5141.6571.741601001.8451.5691.6311.4521.8431.824601501.9051.6721.6501.4841.8381.735602001.9491.7281.6881.6131.8231.830602501.9661.7651.7111.6061.8661.80080501.7731.3181.7171.3941.7601.811801001.9221.4891.6281.2911.8661.783801501.9721.5701.6511.3841.8661.835802001.9861.6051.7411.4161.9351.906802501.9921.6271.6841.4461.8971.8452020pointpointpointpoint500.9991.3981.2971.5721.4802.1491001.4701.6271.7231.7611.8721.7011501.6661.7461.6451.7431.7142.015(#/domain)lineinXdirectionlinedirectionIntersectingMidIntersectingMidLength(m)fracturesetsinthedomainwithdirectionsN–S&E–WandvariablelengthwithDensityFD(box-counting)FD(sand-counting)ScanningScanningTableTwothatwasconsideredduringthisresearch.Thus,inordertopermeabilityineachmodel,thecombinationoffractureanddensitywasselectedinsuchawaytohaveatleastonecluster.1fractureaperture(e)withparallelwallsisdefinedascon-ductivity.TheintrinsicfracturepermeabilityandconductivityaccordingtoPoiseuille’lawareexpressedase2/12ande3/12respectively(Bourbiauxetal.,1998).Basedonthedifferentcombinationsoffracturelength,densityandorientation,totallyfourdifferentdatasetsweregenerated:(1)N–SandE–Wfractureswithfixedlength;(2)N–SandE–Wfractureswithvariablelength;(3)NW–SEandNE–SWfrac
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