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1、M.Sc.inPetroleumEngineeringFlowinPorousMediaSection12002-2003Dr.R.W.ZimmermanPage11.DiffusionEquationforFluidFlowinPorousRocks1.1.Darcy’slawandthedefinitionofpermeabilityThebasiclawgoverningtheflowoffluidsthroughporousmediaisDarcy’slaw,whichwasformulated
2、bytheFrenchcivilengineerHenryDarcyin1856onthebasisofhisexperimentsonverticalwaterfiltrationthroughsandbeds.DarcyfoundthathisdatacouldbedescribedbyCAΔ(P−ρgz)Q=(1)Lwhere:P=pressure[Pa]3ρ=density[kg/m]2g=gravitationalacceleration[m/s]z=verticalcoordinate(me
3、asureddownwards)[m]L=lengthofsample[m]3Q=volumetricflowrate[m/s]2C=constantofproportionality[m/Pas]2A=cross-sectionalareaofsample[m]AnyconsistentsetofunitscanbeusedinDarcy’slaw,suchasSIunits,cgsunits,Britishengineeringunits,etc.Unfortunately,intheoilindu
4、stryitiscommontouse“oilfieldunits”,whicharenotconsistent.Darcy’slawismathematicallyanalogoustootherlinearphenomenologicaltransportlaws,suchasOhm’slawforelectricalconduction,Fick’slawforsolutediffusion,orFourier’slawforheatconduction.DepartmentofEarthScie
5、nceandEngineeringImperialCollegeM.Sc.inPetroleumEngineeringFlowinPorousMediaSection12002-2003Dr.R.W.ZimmermanPage2•Whydoestheterm“P−ρgz”governtheflowrate?RecallfromfluidmechanicsthatBernoulli’sequation(whichessentiallyembodiestheprincipleof“conservationo
6、fenergy”)containstheterms2⎛2⎞Pv1ρv−gz+=⎜P−ρgz+⎟,(2)⎜⎟ρ2ρ⎝2⎠whereP/ρisrelatedtotheenthalpyperunitmass,gzisthegravitationalenergyperunitmass,2v/2isthekineticenergyperunitmass.Asfluidvelocitiesinareservoirareusuallyverysmall,thethirdtermisnegligible,andwese
7、ethatthecombination“P−ρgz”representsanenergy-typeterm.Itseemsreasonablethatthefluidwouldflowfromregionsofhighertolowerenergy,and,therefore,thedrivingforceforflowshouldbethegradient(i.e.,rateofspatialchange)ofP−ρgz.SubsequenttoDarcy’sinitialdiscovery,itha
8、sbeenfoundthat,allotherfactorsbeingequal,Qisinverselyproportionaltothefluidviscosity,μ[Pa⋅s].Itisthereforeconvenienttofactoroutμ,andputC=k/μ,wherekisknownasthepermeability,with2dimensions[m].Itisalsomoreconvenienttoworkwit
