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THE FLOW ANALYSIS OF FLUIDS IN FRACTAL RESERVOIR WITH THE FRACTIONAL DERIVATIVE

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THE FLOW ANALYSIS OF FLUIDS IN FRACTAL RESERVOIR WITH THE FRACTIONAL DERIVATIVEIN,OF,of,WITH,THE,FLOW,with,the,flow,Flow

    THE FLOW ANALYSIS OF FLUIDS IN

    FRACTAL RESERVOIR WITH THE

    FRACTIONAL DERIVATIVE AvailableonlineatWWW.sciencedirect.com

    .ccE

    @……?

    JournalofHydrodynamics

    Ser.B,2006,18(3):287293

    287

    sdlj.chinajourna1.net.ca

    THEFLoWANALYSISOFFLUIDSINFRACTALRESERVoIRWITHTHE FRACTIONALDERIVATIVE

    TIANJi

    InstituteofMechanics,ChineseAcademyofScience.Beijing10008O,China GudongPetroleumFactoryofShengliPetroleumAdministration,Dongying256504,China

    TONGDengke

    DepartmentofAppliedMathematics,ChinaUniversityofPetroleum,Dongying257061,Ch

    ina,

    Email:tongdk@mail.hdpu.edu.cn

    (ReceivedApr.2,2004)

    ABSTRACT:Inthispaper.fractionalorderderivative.fractal dimensionandspectraldimensionareintroducedintotIle seepageflowmechanicstoestablishtheflowmodelsoffluids infractalreservoirswiththefractionaldeftvative.Theflow characteristiesoffluidsthroughafractalreservoirwiththe fractionalorderdeftvativearestudiedbyusingthefinite

    integraltransform.thediscreteLaplacetransformofsequential fractionaldeftvativesandthegeneralizedMittagLemer

    function.Exactsolutionsareobtainedforarbitraryfractional orderderivative.Thelongtimeandshorttimeasymptotic

    solutionsforaninfiniteformationarealsoobtained.The pmss~etransientbehavioroffluidsflowthroughaninfinite fractalreservoirisstudiedbyusingtheStehfest'sinversion methodofthenumericalLaplacetransform.Itshowsthatthe orderofthefractionaldeftvativeaffectthewholepressure behavior,particularly,theeffectofpressurebehaviorofthe early-timesmgeislargerThenewtypeflowmodeloffluidin fractalreservoirwithfractionalderivativeisprovidedanew mathematicalmodelforstudyingtheseepagemechanicsof fluidinfractalporousmedia.

    KEYWORDS:fractionalcalculus,porousmedia,fractal, integraltransformexactsolution.

    1.INTRoDUCTIoN

    ChangandYortsosILlpresentedthetheoretical

    modelforinfinitefractalreservoir.Acunaet

    a1.[zjexplainedthefractalcharacteristicofanaturally fracturedgeothermalfield.AcunaLJJreviewedthe theoreticalbackgroundoffractalanalysis.Theyalso demonstratedtheapplicationofvariousdiagnostic techniquesforfractalpressuretransientanalysisas developedbyChangandYortsos".

    Tong[4-71givethe

    analyticalsolutionofvariouscasesfortransientflow offluidthroughacylindersourcewelloffractal

    reservoir.

    Intheflowoffluidinfractalreservoir.the

dimensionlessflowequationiswrittenas

    杀鲁aaa(1)

    Forthemostporousmedia,theaboveequation

    canbasicallyreflecttheflowcharacteristicsoffluids infracta1reserveoir.Acuna[2,31studiedtheflow characteristicoffluidsinfractalreservoirfromtheory andproductionpracticebyusingtheaboveEq.f11.In recentyears.fractionalcalculusisasdynamicbasisof thefractalgeometryandfractionaldimension. Fractionalcalculushasencounteredmuchsuccessin describingtheconstitutiverelationshipofviscoelastic fluid.Thestartingpointoffractionalorderderivative modelofnonNewtonianfluidisusuallyaclassical

    differentialequationthatismodifiedbyreplacingthe timederivativeofanintegerOrderbythesocalled

    RiemannLiouvillefractionalcalculusoperators. ProjectsupportedbytheChinaNational973Program(GrantNo:2002CB211708)andtheNa

    turalScienceFoundationof

    ShandongProvince(GrantNo:Y2003F01).

    Boigraphy:TIANJi(1964),Male,SeniorEngineering

288

    Thusthedescriptionofthefractionalorder

    constitutiverelationshipofviscoelasticfluidismore extensive.Friedrich81,

    Huang91,

    Tan.o1,

    Xull'2andTong13introducedfractionalcalculusin therheologyandanalyzedvariousproblems.Thev thinkthatthefractionalcalculusapproachismore

    appropriatefortheviscoelasticfluid.Parkll'~l introducedfractionalcalculusintheflowequationof fluidinfractalreservoir.Hedeemthefractional calculusapproachtallywithrealfractalreservoir. particularly,thedescribedpressurebehaviorofthe earlytimestageismoreaccurate.Jiangapplied fractionalcalculustotheexperimentaldataofthe viscoelasticgluefluidtoobtainaverygoodfit. Generally.fractionalderivativeisintroducedinthe flowmodeloffluidinfractalreserveoir.theflow equationiswrittenas

    whereD:isfracti.nalcalculusoperator.f a

    orderwithrespecttotrespectivelyandis

    definedas

    D),(f)=(1

    O1

    wherefisGammafunction.While=1it

    maybesimplifiedasEq.fl1.

    Theaimofthispaperistostudytheflowoffluid infractalreservoirwiththefractionalderivative. Wheisarbitraryfraction.theexactsolutionforthe flowequationisobtainedbyusingtheLaplace transfornloffractionalorderderivativeandthe generalizedMittagLeffierfunction.Manyprevious

    andclassicalresultscanbeconsideredasspecialcases ofourresults.Forexamplewhen=1,d,=2,

    J

    =

    0ourresultsbecomeTong'ssolutionforthe

    flowoffluidinfractalreservoir.Theflow characteristicandpressuresensitivetoparametersof thefluidsthroughfractalreservoirwithfractional derivativearediscussedbyusingthenumerical inversionofLaplacetransformandasymptotic solutions.Itshowsthatthepressurecharacteristicsof thefluidissensitivetotheorderofthefractional derivative.

    2.THEFLoWMoDELoFFLUIDIN

    FRACTALRESERVoIRWITH

    FRACTIoNALDERIVATIVE

    Weassumethefractalpermeablenetwork

    embeddedinimpermeableEuclideanmatrix.where thefractalnetworkdimensionisd,,andtheJ Euclideanmatrixdimensionisd(d=l'2,3). Fluidflowonlyoccursinafractalnetwork.The materialbalanceequationcanbederivedusing integralrelationllq'J

    

    DO(to,"~D)drD=

    .(f.,.)d..(,.)d(3)

    Thedimensionlesstotalradialflowis

    ,

    dr-0-1

    .(ro,to=-r~j)pD

    Equationf3)expressesthetotalradialflow passeduptotimetDatdistancerofromthewellas aconvolutionintegraloftotalpressurewiththe diffusionkerne1.d,isfractaldimensionand0is diffusionexponent.Assumingthestationaryprocess,

K(tD,)=K(tD)

    Weappliedthetheory

    incorporatethememory. and(4gives

    K(tD

    offractionalderivativeto CombinationofEqs.(3)

    )(rD,)=

    d

    1

    rD,一一l

    f-I~IDUI,

    D

    (ID

    Eq.(5)canbeinterpretedasthesecondtypeof

    fractionaldiffusionequationI4.

    Therefore,Eq.(5)canbeexpressedasthe

    followingintegrodifferentialequation =

    ro

    afarDarD

    where

rD=r

    ,

    =

    01

    withtheinitialcondition

.=o

    theinnerboundarycondition

    l,b=l:1arD' andtheouterboundarycondition

    limPD

    rD_?

    or

    pD

    arD

    =

    0

    IrD:=o I:=o

    mt

    .

    cre'

    _D=ro0ro (0ro

    )J

    D

    arDrD:1 lim

    ro——o.

    (7)Assume 1

    =

    0

    

    1

    -

    #

    =ro(),P=

    (8)Itcanbeshownthat

    (9)

    f=+望一,.

    PPpLpL

    where=1d/2(d=

    +

    3.THEEXACTSoLUTIoNFoRTHEFLoW

    MoDELoFFLUmINFRACTALRESERVoIR

    WITHFRACTIoNALDEVATIVE

    3.1Exactsolutionforthe/towmodeloffluidinan infinitefractalreservoir

    TheinitialandboundaryvalueproblemIthatiS madeupofEqs.(6),(7),(8)and(9)representsthe caseofconstantproductionratefromaninfinite reservoir.Weconsidertheinitialvalueconditionof integerOrderforfractionaldifferentialequation. TheoperatorD"iStakenastheMiller.ROSS sequentialfractionalderivative.Let (rD,)=e-St~p.(ro,to)dt.

    ApplicationoftheLaplacetransformationto Eqs.(6),(7),(8)and(9)yields

    (1)

    2

    ]

    

    SK(bs)

    2d,

+2

    1

    289

    (12)

    (13)

    (14)

    (15)

    (16)

    (17)

    (18)

    Eq.(18)isreducedtothefollowingformatthe

    wellbore(ro=1)

    K(bs)

    pwD_互——

    SK(bs)

    TheWebertransformationisgivenby =

    W[f]=IPf(P)~Pl(P,)

    where

    (19)

    .

    (P,)=Jr()

    .(b2)()

    .(b2)

    ApplyingthegeneralizedWebertransformationto

    Eqs.(16),(17)yields

f=2

    z2s(2+S,(20)

    TheinverseofWebertransformationcanbeexpressed as16

    )=f2

    .

    f~~l(P

    T

    '2)

    .

    d2

    Substituting(21)into(15)thefollowing expressioncanbeobtained

    (rD,S)=

    2ro(,2)d2

    [Jv2_

    .()+.(b2)]zs(2+Sct)

    1-8

    2G

    b2).()+(]

    whereA(,S)=

    s(s+1

    Analyticalsolutionfortheinitialandboundaryvalue problemcanbeobtainedaslongastheinverseof LaplacetransformofthefunctionA(A,s).Wewill applythediscreteinverseLaplacetransformmethod togivetheanalyticalsolution.Firstly,weusedthe followingpropertyofMLfunction【】

    S+

    a-1)

    inwhichE(z)denotesMittagLefflerfunction

    UsingLaplacetransformtheintegralofconvolution,

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