凝胶资料123

LIEDGEOPHYSICS，Vo1．1，No．1(July2004))delingofRockswithSpatialDistributedFracturesbyRandomMediumTheory★ZengXinwu’，ZhangGuangying’，andWangXiumingAbstract：ByusingthespectrumexpandingtheoryofrandomprocessesandHudson’scrackmodel，WEdevelopedarandommediummodelforrockswithspatialrandomdistributednumberdensityofcracks．Thismodelcouldconnectthemicro—parametersofthecrackswiththemacro—mechanicalproperties．andcanbeeffectivelyappliedtomodeltherealinhomogeneousmedium．Numericalex—ampleindicatesthattherandomdistributioncharacterscouldbediflferentfordifferentelasticcon—stantsunderthesamerandomdistributionofnumberdensityofcracks．Bychangingthevalueoftheautocorrelationlengthpair,itispossibletomodelthedifierenceofthedistributioninthetwocoordi—natedirections．Numericalmodelingresultsforseismicwavepropagatinginrockswithrandomdis—tributedfracturesusingastaggeredhigh—orderfinite—differencefSHOFD1arealsopresented．Keywords：randommedium，autocorrelationfunction，rock，andfractureCracks，fracturesandfaultsarecom—moninthesubsurfaceoftheEarth’scrust．andtheycontrolmuchofthemechanicalstrengthandtransportpropertiesofthesolidstructure．Fracturesandfracturesystemsarea1socrucia1forhVdrocarbon，production．controlandmanipula—tionofwatersupplies，andthedis—ofpollutants．Thestudyofdynamicpropertiesof"edmasshaslongbeenanimportantprobleminthelfields．Asweknow．thesubsurfacerocksarenor—heterogeneouswithcracksandfractures．Theseplayanimportantroleonthepropagationofseis—·avesespeciallyforhighfrequencyorwidebands．Duetotheenormousamountandwidedistribu—Fcracks．itisdifficulttodescribethepropertiesof,ithsuchcracksbydeterminatemodels．Ontheotherhestatisticmethodmightbeaconvenientwayto2eatheoreticalmodelfordescribingtheheteroge—propertiesoftherocks．Itlcadstotherandomme—mode1．Bonneteta1．(200l、hadsummarizedthe}rrelationfunctionswhichcouldbeusedtodescribeachasticcharacteristicsofcracksinrocks．Theexponential，Gaussian，vonKarmanautocorrelationfunc—tionsareusuallyusedtomodelrandommediafDainty，l984；FrankelandClayton，l986；Ikelleet．a1．．1993：Hestholm．et．a1．，l994)．Muchresearchhasbeendonerelatedtotherandommediummode1．Ikelleet．a1．(1993)，ErgintavandCanitez(1997)usedexponentialandGaussiantypeautocorrelation，respectively，tomodel2一Drandommedium．Cowieet．a1．fl993)useastatisticalphysicalrupturemodeltosimulateantiplanesheardeformationandruptureofatectonicplatewithheterogeneousmaterialproperties．XiandYaof2002)discussedthecharacteristicsof2一Drandommediamod—elswithexponentialandGaussianellipticautocorrelationfunctions．Itisnotedthatmostofthecurrentresearchesonrandommediummodelingertiesofmedium，suchasfocusedonusingmacro—·prop—·wavevelocityanddensity，astherandomparameters．BasedonHudson’sfracturemodel(Hudson，1981，1986，1996)，LiuandZengf2001)devel—opedaformulatocalculatetheelasticconstantsoffrac—turedrocks．Intheirmodel，theelasticconstantsoftherockaredirectlyrelatedtothemicro—parametersofthefractures．Inthisresearch，basedonLiuandZeng’smodel，wedevelopedaspecialrandomdistributedfracturemodel，inwhichthenumberdensityofcracksischaracterizedby：riptreceivedbytheEditorMay18，2004；revisedmanuscriptreceivedJune9．2004．presentedattheCPS／SEGBeijing2004InternationalGeophysicalConference，March31-April3，2004．mmentofAppliedPhysics，NationalUniversityofDefenseTechnology，Changsha，Hunan410073，Chinat0Petroleum，ARRC，POBOX1130，TechnologyPark，Bentley,WA6102，Australia．维普资讯http://www.cqvip.comModelingofRockswithSpatialDistributedFracturesanellipsoidalautocorrelationfunction．Bythismodel，webuildthedirectrelationshipbetweenthecracknumberdensityandmacropropertiesofthefracturedrocks．Sev—eralrandommodelswithdifferentautocorrelationlengthsarepresented．1iraIiiii一-ElasticconstantsforrockswithparallelfracturesLiuandZeng(2001)developedaconstitutemodelforfracturedrockswithfracturemodeledasaplanardistribu—tionofsmallcracks．Inthismode1．assumingthatthereisonlyonesetofparalleldistributedcrackswiththecracknormalatn=(1，0，0)T，theelasticconstantscouldbewrit—tenas：c1．=[1+A／c22=c33=[+4(+)A～儿l+￡A～／】cl2=cl3=c2l=c3l=[1+A～／c23=c32=(1+2sA～)[1+￡Av／】～，c55=c66=P(1+sAr)～，c44=(c22一c23)／2=P．Theotherelasticconstantsarezero．Where九anduareLame’sconstantsfortheuncrackedrockand=+2，￡sisthenumberdensityofthecrack，and=[(X+2p)／p]“，=(U／P)“theP—waveandS—wavevelocitiesfortheuncrackedrocksrespectively，Pisthedensityfortheuncrackedrocks．where=Ul【11+(y02)U117r(3．／a)／41，AN=U3311+(ya2)U337r(1—／a)JUI1andU33areconstantsrelatedtothefractureproperties．Hudson，inaseriesofpapers，hasderivedexpressionsforUl_andU3forarangeofcrackmodels，suchasdrycracks，cracksfilledwithfluidorweakmaterials)．partialsaturation，andfluidflowbetweeninterconnectedcracksinporousmedia．Inthisstudy．assumingthecracksareisolatedanddry，thefollowingexpressionsareused(Hudson．1981：({(等)．(3)RandommodeIforrockswithdistributedfracturesInthisresearch，basedontheabovemode1．wearetry—ingtodeveloparandommodelforrockswiththecracknumberdensityrandomdistributedinthespace．andthenbuildadirectrelationshipbetweenthecracknumberden．sityandelasticconstants．Assumingthatthenumberden．sityofthecrackarerandomlydistributedinspace．andcouldbewrittenas：v(x)=V(X)+6V(X)，wherex=，X3)，v0(x)isthebackgroundnumberdensity，andAv(x)iStheheter0gene0usdistributesonv(x)．Inthispaper，v0(x)isaconstantoverthewholespace，andAv(x)iSasteadyspatialrandomprocesswithgivenmeanvalue．secondordermoment，andautocorrelationfunction．Forconvenience，werewriteequation(4intheformof：v(x)=V『1(1+(x))，where(x)=Av(x)(x)，(5)(6)Asweassumedthatthecracknumberdensity(xisastablespatialprocess．isalsostableinspace．Becausev(x)islinearwitho-(x)，basedontherandomprocesstheory．wecoulduse(x)todescribetherandommediumwithrandomdistributedcracks．TheautocorrelationfunctioniSonlythefunctionofx．Inthefollowingcontents．wewillexplainhowtoproducearandomprocess(x)basedonthespectrumexpandingtheory：ProductthepowerspectrumfunctionTakinga2一DFouriertransformationoftheautocorrelationfunction，weobtainedthepowerspectrumfunction：auto(kk)=￡aut。xI~x3似‘ldx。dx3ProducttherandompowerspectrumfunctionTherandompowerspectrumfunctionisnormallyob—tainedbyintroduceameanrandomvariable．Let0(k，k=)isarandomvariableevendistributedin(0，2)，thentheran—dompowerspectrumfunctioncouldbewrittenas：维普资讯http://www.cqvip.comZengXinwuetaRand(kk)=auto(kk)‘exp[一iO(kk)】．(8)ProductrandomseriesFromthespectrumexpandingtheoryofrandomprocess，takeainverse2．DFouriertransform0f(8)．wegottherandomdistribute：，、1一‘l，3(2丌)!J￡I’dx—dx(9)Smoothandfiltertotherandomseries．Althoughtheautocorrelationfunctionandrandomse—riesareallcontinuousfunctions．wecalculatethembydis—cretenumbers．whichcouldproducesomecomputationalerrorandmightresultintheobtainedrandomfunctionnotsatisfyingthepre—assumedautocorrelationfunction．ThiserrorcouldbereducedbytaperingfunctionandsmoothmethodasdiscussedbyIkellelf1993)．Hereweonlyquotetheformula．Theresultantrandompowerspectrumfunc—tioncanbewrittenas：J。。’。。。。。。。。。。。。‘‘。’’。。。。。。一Rand(k、．k；)x／auto(kk)k)exp[一iO(kk)I，(10)whereW(k一k)=(良蕊)，．c七，={。_5+。·46c。s(7r(1一七ifk≤kelsek))，wherekmaxisthelengthoftaperingfunctionNormalizationoftheobtainedrandomseriesTakingtheobtainedrandomseriesasatwodimensionalrandomprocess．itisnecessarytocheckthemeanvalueandthesecondordermomentoftheserieswiththepre—assignedvalues，Inthecomputingprogram，wenormallyusingthefollowingformula：52：ln×n∑∑，(13)l=，×，∑∑()，wherei1isthesamplingnumberinthetwocoordinatedirection．Sofar，wehaveobtainedastablerandomseries，whichsatisfiesthepresumedautocorrelationfunction，meanvalueandthesecondordermoment．Thisseriesisinhomoge—neousinthespace．Bysubstitutingtheobtainedrandomseriesintoequa—tion(2)，wecouldobtainthespatialdistributedrandomnumberdensityofcracks，whichsatisfiesthepre—assumeddistributioncondition．Therefore，byusingEquation(1)，theelasticconstantsforrockswithspatialdistributedran—domcrackscouldbecalculated．NumericalExamplesGaussianfunction，exponentialfunctionandVonKarmanfunctionarecommonlyusedautocorrelationfunc—tionsintheresearchofscattering．Inthispaper．weusedanexponentialellipsoidalautocorrelationfunctiontomodeltherandommediumwithdifferentinhomegeneousscaleinx1andxdirections：auto(x、=exp(一√xja+x3、／+(15)wherea=x．／．，b=x1／Ax1arecorrelationlengthsinxlandxdirection，respectively．AxlandAx3arespatialsteplengthduringthemodeling．Inthisresearch，alimitationofaAx．，bAx1mustbemuchlargerthanthewavelengthappliedtouseLiuandZeng’smode1．Here．assumedthemeanvalueofthenumberdensityofcrackisVn=<V>=0．1，themeanvalueofdistributeiszero，<A’，>=0，then<>=0，(<>)=l0％．Tak—ingdifferentautocorrelationlengthpair(a，b)，wecouldproducetherandommediumwithdifierentcharacters．Figure1isthespatialdistributionofcracknumberden—sityundertheellipsoidalautocorrelationfunctionwithautocorrelationlengthpair(a)=(20，1)，(20，10)，respectively．Theautocorrelationlengthpairs(a,b)are(20，1)(Figure1a)and(20，10)(Figurelb)respectively．Fromthisfigure．itisclearthatwecouldmodelthedifferentdistributionofnumberdensitybyselectingdifferentcor—relationlengthpairs．维普资讯http://www.cqvip.comModelingofRockswilhSpatialDistribuledFractures(a)c，，(b)c0F3RandomdistributedfracturemodelforAutocorrelationfunctionm¨l¨^⋯I√¨7Theautocorrelationlengthpairis(20，10)维普资讯http://www.cqvip.comZengXinwueialn’III1derMalKIIheeI"i~clsofraI1doradisitlbutedfrac—tule、t0seismicw【Vpropagation．weperformednUllleri—calimulatiollel~,eisnficwavespropagatinginitusingastaggeredhigh一[1rdelfin{te—dterencefSH0FD)．Figure4isanexainpleofthesnapshotsfirrtinle(】(132．()064s．()fI96sall(J¨l18srpectiveIv．IIIthiexalllpletherocki『I1ode1【wi【hanexpoocntialatltocorrelatioufulletL0n‘1lautocolreIati()i1IerigthpllirI2{I．IIasMlowninFiI．1le2Du¨rlgtheca】cu1atiOilthegridIenglhinboththex—【xiandv-axisistakentohc1nr．respecti,,elyThe6mestepbc)0gIlls．TheSOLlfCeanparticleveh：lcitvXOIIFCe1ocatedat{5Ⅲ)n1．7O0111i．wilhtheceriterI'rcqtiencoF1f)OHzNo1sccfctcausedhvdirbuledl"i-acill1ccallbeob—servedIearlvFromIhesjIllulatlI)llTCSU．dif1‘entSO；i[一tcringHtvIeswere0bservcd)^rdiricrntfr．ctIIredjstrIhkltlOriS，IVol‘Illesame蚍b^ution，thescatterinIhigherwithtileillcreasin(1rfluCltlalioi1ont'raClUlcdell—sityaslqOFiria]lyexpecledxA~lkmB0040Bx^sfkn’xAs(kmlConclusionIIIthispaper，basedOiltheelasticmodelfi>r1‘ockswilhhonlogencouscrack~；，wedeveltlped1ralldC,lllmedimnnlodel⋯sinllllateIheralldOlllspatialdistributionofthenllmberdensityofcracksina1ock．We]lresellledtilede—tailprI)eess【t’huiIdsucharalldolllmediunlnlodelThemodel@',elopedinthispapeltonileftstherandimllysF,a—tia【dIsir】htiledIlliCR~一ptopelliest“therock(thenlll／lherdellsitvIll"crackI【otilenltl~romeuhariicalplODert1eNttheelastirt．'Ollstallls}．Asancxalllpleby．us{ngallexponelltialelIipsoidaltiLltocorreIatioi1funeti【)n．wctil1．>deIeda!一DspatiuIrail—domrYdistf。fbutedriledi“HIBthesefectH1JT¨fatllOCOlTe]ation1rUllCliOllandseveral『；al'alllelei、suchau【ocorreIa【ioi1IengIhIIICallvaliientadsecond—Il1de1iriOlllent．itjD；s：．ibletodescribethesnlalIscale1nhomt)一gell~(1URralldolllpropeFliesoftherealkBychoc,shlgdilt'erelllau【oct|℃1叭j0I1lengthpair,wecouldeas1lvnlodelthedferelltllleallcaleol’therandoliidistribuli011iI1theDA't)CO0rdinaledirectio__ItialsojI1ulldthattheYarldonldistiibuliollcharacterscouldbedifl~：rentfbrdit'l；eI-elltdiets—I】ccellstaff“tlI1dertilesallle1．aRdoIll(1LribuIjOll0Iltill[11berdells】tvofcracksAcknOwIedgementSThisWOIkwasstlpp(~rtedbvtheNationalSeientIticF()tlndaiionofChinaundercontractor40074025．Wethai1kEnruLluol’B(SofLfK1rhjHLIsefu【di0USSIOI1andforCOlllillelltsonthjwork．Thispaperisptl[~lishedwiththeapovalottheNationa1Universib．olDeferiseTechnol一0v‘1rChin．ReferencesBOllllelE．BOLiIOOdlln业N．EDvPMailllCowiePalldBelkowlI7B．2()0ISeaJin‘】rI-rac【ure、vstelllsiIIe0一logicalmedia：RevieworGeoph$,sIc～．347—358(’)wkPA，ViiilaesleC，till[1SolnetteE．1993．Stalistica】ph1．sIcaliriodeItbrthespatiotempora]evollll10[]州L_】【：JouIGltth1Igec．phvqcaJresearc_1-9B12}!I809—2I81IDainl~AM．1984High—frequent],．actltlSticbackscatteringan(IseismlcatlellalllOlHJGeophv、．Res8．3l7—】76．ErgiIltavSandCan1llezN．．I997Modelingt；fmllhi—scalereedaii1discFeteI‘_rn_：JSel．Explor，6、77-96．FrallkeIAalldClavtonRⅥ】986FinIle—dilferenceSillla—Iati‘)Orseis~n1cscattering：hnplicationsrthepropaga—li[’nofshort—periodcISir{iFk'VLI"e'CglI1thecltlstandmode(’1．Cl。tlSialheterogeneity：J．GeophysRes．9l646564㈣HcstholmSOHuseybeE．Sall(】rutldBO一1994．Seismic,lavepropagatiollincomplexcrusl—upperhi,lille]mediaLJNlngfinite—di[telencesynthetic'<Geoph)．sl『】I1[．IlH．643—67()HtldsOn，『A】98JWcsspeedsl¨it．Iattelluation‘1relasticwHveI11lllaleriaIcontaIi1IngcracksGe【，nh”jRastr．．I33l5【)Hlldson．JA—l986AhigherordeIapproxilllation10tilewavepmpagatl¨『1COIlSt~lIltSforcracked('liclOeopllysJRas【r．St．87265一174Hudson．JALiLIEnlli，：II1dCnIn]plIlS．『qq6，Tla『_n1in1]r’pcr【i‘1f'pjancfule：GeI'1)_1vJ+【，【：559—56^维普资讯http://www.cqvip.comModelingofRockswithSpatialDistributedFractureslkelleLT．YungSK．DaubeF．．1993．2一Drandommediawithellipsoidalautocorrelationfunction【J】．Geophysics，58(9·)．1359一l372．Liu，E，andZeng，X．，20HD1．E髓ctiveelasticconstantsforfrac—turedmedium：OilGeophysicalProspecting，36(1)，37—44．XiX．andYaoY．2002．Simulationofrandommediummodelandintermixedrandommedium：EarthScience_Journal0fChinaUniversityofGeosciences．27．67—71．THEAUTHORSZengXinwu，Professor，obtainedhisBachelordegreeinap—pliedmechanicsfromtheNationalUniversityofDefenseTechnology(NUDT,China)in1983，andPh．DinGeo—physicsfromEdinburghUniversity(UK)in1994．Froml995．heworkedintheFacultyofScience，NUDTaspro—fessorinMechanics．Hiscurrentresearchinterestsincludedynamicresponseofmaterials，multicomponentdatapro—cessingforseismicwavespropagatinginanisotropicmedia，andelasticwavespropagationinfracturedrocks．ZhangGuangying，Ph．D，obtainedhisbachelordegreeinphysicsfromNankaiUniversity(China)in1996．andPh．DinEngineeringMechanicsfromNUDTin2003．ShenowisaresearchscientistinBeijingInstituteofDesigna—tionforspecialengineering．Hercurrentresearchinterestiselasticwavespropagationinfracturedrocks．WangXiuming．Professor,obtainedhisbachelordegreeinPetroleumEngineeringf1984)．andPh．DinGeophysicsf19961fromthePetroleumUniversityofChina．HeisnowaSeniorScientificOfficerinCSIR0．Australia．Hiscur-rentresearchinterestsincludingnumericalmodelingofseismicwavepropagationandseismicmigration．Thetheory“Explorationofoilandgassubtletrapsinthecontinentalfaultedbasins”guidingtheexplorationforoilandgassubtletrapsFromtheScienceandTechnologyDaily(April5)inBeijing，throughlongperiodofhardstrugglethegeolo—gistsoftheShengliOilfieldshaveestablishedatheorysystemforoilandgasexplorationofthesubtletraDsinthecontinentalfaultedbasin．Ithasansweredthescientificquestionsabouthowoilandgaswasaccumulatedinthesubtletraps，andputforward4patternsof“faults—slopecontrolssands”．Ithasestablishedthefourbasictransmittingsystems，“carpettype”，“Ttype”，“terracetype’’and“fracturetype”：andthecompositetransmittingsystemformedbyallofthem．Ithasalsoestablishedthebasicpatternsofoilandgasaccumulatedinthecontinentalfaultedbasins，andthequantitativedescriptionmethodof“faciesandpotentialcontroltheoilandgasaccumulations”．Foursystematictechnologiesare，thenformed，whichcouldsolvetheproblemsinsearchingforoilandgassubtletrapsinthefourtypesofsandstonebodies：fluvialchannelsands．conglomeriticsands，turbiditesandbankandbarsands．ItwasestimatedbytheAppraisalCommitteethatthetheoryisoneoftheimprovementsoftheChinesepetroleumgeo—logictheoryafterthetheoryofoiloriginatedfromconti—nentalsedimentaryrocksandthetheoryofoilandgasac—cumulatedinthecompositezones．Thewholetheoryhasreachedthefrontoftheinternationalleve1．Dependingonthistheory，theShengliOilfieldshavemadecleartwotargetareas．inwhichoilfieldswithre—serveoverhundredmilliontonsareexpectedtofindout；discoverandfindout5targetareaswithreserveof50mil—liontonsineach．5targetareaswithreserveof30milliontonsineach．andseveraltargetareaswithreserveofl0milliontonsineach．SincelastWinterandthisSpringanoilfieldwithreserveupto50milliontons，twooilfieldswithreserveof30milliontonsineach，andthreetofouroilfieldswithreserveof10milliontonsineachhavebeendiscoveredintheShengliOilfieldssuccessively．andtheyareallbelongtothetypeofsubtletraps．Undertheguidanceofthenewtheorytheexplorationf0rsubtletrapshasrealizedagreatleapforwardinthequalitythatisfrom“blindlyfortuitous“to”regularopera—tionundertheguidanceofscientifictheory，andfrom“qualitativeexpectation”to“quantitativeestimation”．Theexplorationtheoryforsubtletrapswillnotonlyplayanimportantroleintheresourcesstrategyof“stabilizetheeasternpart”ofourcountry．butalsowillguidetheexplo—rationforthesamekindofoilandgastrapsabroad．ItisestimatedthattherecoverableoilandgasresourcesinsubtletrapsintheShengliOilfieldsaremorethan3．3billiontons．Thereareseveraloil—and—gas—bearingbasinswithsimi—largeologichistoryandbasictectonicframeworkinChinaasintheJiyangsag．TheyaretheSongliao，Bohaibay，Nanxiang，JianghanandSubeibasins．InthesebasinstherecoverableoilandgasresourcesinthesubtletrapsareestimateduDto9．95billiontonswithpotentialbenefituptotrillionyuan．55维普资讯http://www.cqvip.com