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1、WATERRESOURCESRESEARCH,VOL.39,NO.6,1166,doi:10.1029/2001WR001018,2003FlowthroughporousmediawithmultifractalhydraulicconductivityDanieleVenezianoandAlbertK.EssiamDepartmentofCivilandEnvironmentalEngineering,MassachusettsInstituteofTechnology,Cambridge,Massachusetts,USAReceived17October20
2、01;revised16July2002;accepted28March2003;published26June2003.[1]WemakeanonlinearanalysisofflowthroughsaturatedporousmediawhenthehydraulicconductivityKisanisotropiclognormalfieldwithmultifractalscaleinvariance.DInthiscase,logKisisotropicGaussianwithspectraldensitySlogKðÞ/jkkj,whereDisth
3、espacedimension.OurmainresultisthatthehydraulicgradientrHandspecificflowq’arealsomultifractalfields,whoserenormalizationunderspacecontractioninvolvesrandomrotationofthefieldandrandomscalingofitsamplitude.ThescalingpropertiesandmarginaldistributionsofrHandqareobtainedanalyticallyasfuncti
4、onsofthespacedimensionDandamultifractalparameterofK(thecodimensionCK).ThefieldsrHand’q’areanisotropicatlargescalesbutapproachisotropyatverysmallscales.Usingscalingarguments,weobtaintheeffectiveconductivityofthemediumKeffasanexplicitfunctionofD,CK,andthescalingrangeofK.INDEXTERMS:1829Hyd
5、rology:Groundwaterhydrology;1894Hydrology:Instrumentsandtechniques;KEYWORDS:subsurfaceflow,multifractality,scaleinvariance,randomfields,stochastichydrology,effectiveconductivityCitation:Veneziano,D.,andA.K.Essiam,Flowthroughporousmediawithmultifractalhydraulicconductivity,WaterResour.Re
6、s.,39(6),1166,doi:10.1029/2001WR001018,2003.1.Introduction1997;LentandKitanidis,1996].Ingeneral,thesestudieshavefoundthatsecond-ordertermshavesignificanteffects[2]TheflowofaliquidthroughaporousD-dimensionalonrHandqwhenthevarianceoflnKexceedsunity.mediumisgovernedbyDarcy’slaw,However,for
7、largelogconductivityvariances,higher-ordertheoriesarenotnecessarilymoreaccuratethanfirst-orderq¼KrHð1Þsolutions[Ababouetal.,1988].Hence,forhighlyhetero-geneousmediaitisdesirabletodevisealternativestothewhereq(x)isthespecificdischarge,K(x)isthehydraulicperturbationapproachthat
