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OntheuseofWirelineFormationTesting(WFT)data:2.ConsequencesofpermeabilityanisotropyandheterogeneityontheWFTresponsesinferredflowmodelingMathieuNoirota,b,n,GerardMassonnata,HerveJourdebaTotalS.A.,AvenueLarribau,64018Pau,FrancebUniversitéMontpellier2,PlaceEugèneBataillon,34090Montpellier,FrancearticleinfoArticlehistory:Received2July2011Receivedinrevisedform1July2013Accepted1August2013Availableonline8April2014Keywords:WFTpermeabilityheterogeneityanisotropycharacterizationsimulationabstractWFTtestsarecommonlyperformedtosamplefluidsandtoassesstheformationpressuresinreservoirsatdifferentdepths.Theinterpretationofthesepressurevaluesprovidesthefluidgradientsandthelocationoftheinterfacesbetweenthesefluids.TheWFTdataanalysiscanalsobeusedtoassessmobilityvaluesfromthedrawdown,thebuilduporboth,basedonanalyticalsolutionsofthediffusivityequationinitiallydevelopedforthewelltestanalysis.Variouspublications,comparingthepermeabilityvaluesderivedfromWFTdatawithplugmeasurements,NuclearMagneticResonance(NMR)interpretation,andlogmodeling,haveshownthatagoodmatchisinfrequentlyobtained.Formostauthors,thediscrepancybetweenpermeabilityvaluescouldberelatedtoeithertheuncertaintyonfluidviscosityorthepermeabilityanisotropy.Thisworkinvestigatesanewhypothesis–theheterogeneityofthepermeabilityfieldrelatedtodifferentinvestigatedvolumesduringtheWFT.Weuseflowmodelinginsyntheticreservoirstoexplainwhymobilityvaluesinferredfromdrawdownanalysisoftendifferfrommobilityvaluesinferredfrombuildupanalysis.ItisshownthatdrawdownmobilityvaluesareclearlycorrelatedtothemobilityofthecellsneartheWFTtool.Thesemobilityvaluesarecomparabletothemobilityinferredfromplugmeasurements.Onthecontrary,buildupmobilityvaluesinferredfromWFThydrodynamicresponseappeartobecontrolledbythehorizontalandverticalcorrelationrangesofthepermeabilityfield.Thispaperisthesecondpartofapaperentitled“OntheuseofWirelineFormationTesting(WFT)data.1.Fieldreviewof420WFTtests”thataddressestheinterpretationof420WFTanddiscussestheresults.&2014ElsevierB.V.Allrightsreserved.1.IntroductionTheWFTtoolisdeployedinthewellboretosamplefluid.Thetestgeneratesapressuredropintheformation.Thepressureisrecordedduringthedrawdownandbuildupperiods,untilthepressurestabilizesattheformationpressure.Theavailabilityofpressureandflowraterecordshasmotivatedreservoirengineerstodevelopequationstoassessthemobilityvaluesofformationsatdifferentdepths.Analyticalsolutionsofthediffusivityequationprovidemobilityvaluesassessedfromthedrawdownperiod,thebuildupperiodorboth.Variousauthors(Pelissier-Combescureetal.,1979;JensenandMayson,1985;Tangyanetal.,2005)comparedpermeabilityvaluesissuedfromWFTdatawithpermeabilityvaluesinferredfromclassicalmethodologies(plugmeasurements,NuclearMagneticResonanceinterpretations)andconcludethatthesevariousper-meabilityvaluesdonotmatcheachother.Similarly,themobilityvaluesestimatedfromdrawdownoftendifferfrommobilityvaluesestimatedfrombuildupanalysis.WFTtestshavebeensimulatedinasyntheticreservoirmodeltostudytheimpactsofthepermeabilityfieldanisotropyandheterogeneityontheWFTresponsesandtheirinterpretedmobi-lityvalues(drawdownandbuildupmobilities).First,Section2isdevotedtothepresentationofanalyticalsolutionsusedtoassessthemobilityvaluesandthemodelingapproachofWFTtests.Themodelingphaseiscomposedofseveralstepscorrespondingtoanincreasingcomplexityofthepermeabi-lityfield(homogeneousisotropicmedia,homogeneousanisotropicContentslistsavailableatScienceDirectjournalhomepage:www.elsevier.com/locate/petrolJournalofPetroleumScienceandEngineeringhttp://dx.doi.org/10.1016/j.petrol.2013.08.0550920-4105/&2014ElsevierB.V.Allrightsreserved.nCorrespondingauthorat:TotalS.A.,AvenueLarribau,64018Pau,France.Tel.:þ33559835837;fax:þ33559836007.E-mailaddresses:mathieu.noirot@total.com(M.Noirot),gerard.massonnat@total.com(G.Massonnat),herve.jourde@um2.fr(H.Jourde).JournalofPetroleumScienceandEngineering133(2015)776–784mediaandheterogeneousmedia).ThenSection3usesthesamestructuretointerprettheresultsoftheWFTsimulations.2.Methods2.1.MobilityassessmentThehydrodynamicresponsesoftheWFTtestsimulationsareinterpretedwithappropriateanalyticalsolutionsofthediffusivityequationtoassessthemobilityvaluesoftheformation.Inthiswork,itiseasytoestimatedirectlyapermeabilityvaluewhenavalueoftheviscosityisimposed.WithactualWFTtests,thejobislessthaneasyasitisnecessarytoestimatetheviscosityofthefluidbeforebeingabletoassessthepermeabilityvalue.Thustheremainderofthepaperfocusesonthedeterminationofmobility.2.1.1.DrawdownmobilityVariousauthorshavedevelopedanalyticalsolutionsofthediffusivityequation,insphericalcoordinates,(Eq.(1))todeter-mineamobilityvaluefromthehydrodynamicresponseofaWFTtestduringthedrawdownperiod.Inthiswork,thisdrawdownmobilityisestimatedwithEq.(2)proposedbyStewartandWittmann(1979),fromMoranandFinklea(1962).1r2∂∂rr2∂p∂rC18C19¼φμCk∂p∂tð1Þkddμ¼Cfq2ΠrpΔpddð2Þwhereqistheflowrate,mistheviscosity,rpistheproberadiusandCfistheflowshapefactor.ThisequationisvalidforallWFTtoolswhatevertheirradius.Thesolutionofthediffusivityequation(Eq.(2))isbasedonahypothesisofsphericalflow.DuringthedrawdownperiodofWFT's,theinvestigatedvolumeisnotperfectlysphericalbecausethewellboreandthepermeabilityanisotropydisturbtheflow.StewartandWittmann(1979)proposedinsertingaflowshapefactor(Cf)tocorrectthewellboreeffects;itcanvaryfrom0.5to1,respectively,forasphericalandahemisphericalflow.2.1.2.Buildupmobility(sphericalflow)ThehydrodynamicresponsesoftheformationtoaWFTcanalsobeinterpretedduringthebuildupperiodtoprovideabuilduppermeabilityvalue.Inthiswork,theWFTresponseshavebeeninterpretedusingthePressureTransientAnalysis,whichhasbeendevelopedsincethe1980stointerpretwelltests.Thismethodusesthederivativeofthepressurechange(Bourdetetal.,1983),whichisplottedonlog–logcoordinatesversustheelapsedtime(Δt)sincethebeginningoftheflowperiod:Δp0¼dpdlnðtpΔt=ðtpþΔtÞÞ¼tpþΔttpΔtdpdtð3ÞDifferentflowgeometriesoccurduringaWFTtest.Whenthederivativeofthepressurechangeisplottedagainstelapsedtime,thedatawillformstraightlines,theslopeofwhichisdeterminedbytheflowgeometry.Differentgradientscorrespondtoadifferentgeometryoftheflow.Acapacityeffectisobservableforearlytimebyastraight-lineslopeof1,whichcorrespondstothedecom-pressionoffluidsinthetool.Afteratransitionperiodasphericalflowgeometrycanbeidentifiedbyastraight-lineslopeofC01/2.Sometimesaradialflowoccurswhentwono-flowboundarieslimittheinvestigationvolume.Inthiscasethepressurechangederivativefollowsastraight-lineequalto0.Whenasphericalflowgeometryisidentified,theanalyticalsolutionofthediffusivityequation(Eq.(1))(MoranandFinklea,1962;Culham,1974;RaghavanandClark,1975;Kohlhaasetal.,1982)providesasphericalbuildupmobilityvalue(ks/m):Δp'¼qmffiffiffiffiffiffiffiffiffiffiffiφμCtp2Π3=2k3=2sffiffiffiffiffiffiΔtpð4Þwhereqistheflowrate,mistheviscosity,φistheporosity,CtisthetotalcompressibilityandΔtistheelapsedtimesincetheendofthedrawdownperiod.Thisanalyticalsolutionshowsthatwhenthepressurederiva-tivevs.theelapsedtimecorrespondstoastraight-linewithaslopeofC01/2,itindicatesthattheflowgeometryisspherical.Whensuchbehaviorisidentified,theassumptionsfortheuseoftheanalyticalsolutionareverifiedandtheycanbeusedtofittherawdatabyvaryingthepermeabilityvalue.Thismethodisbetterthanthedrawdownmobilityestimationbecausethebuildupmobilityvalueiscalculatedusingananalyticalsolutiondefinedbythepressurederivativeevolution.Thebuildupmobilitycanbecom-putedonlyiftheflowdimensionisclearlyidentified(halfslopeforasphericalflow).2.2.Modelingapproach2.2.1.SyntheticreservoirmodelsSyntheticreservoirmodelshavebeencreatedwiththegeomo-delingtoolgOcad(v2.1.6).Thistoolallowsthecreationofastructuralgrid(mesh,fault,etc.)filledwithcategoricalproperties(faciesforexample)orcontinuousproperties(porosityforexam-ple).Thegridiscomposedofmorethanthreemillioncellswitharefinedmesh(Fig.1a).ThreedifferentvolumesarerepresentativeNomenclature(SIunits)Ccompressibility,1/PaCfflowshapefactorCttotalcompressibility,1/PalHhorizontalcorrelationrange,mlVverticalcorrelationrange,mkpermeability,m²kdddrawdownpermeability,m²(¼kdinFig.7)kHhorizontalpermeability,m²kVverticalpermeability,m²kssphericalpermeability,m²kxpermeabilityinthexdirection,m²kypermeabilityintheydirection,m²kzpermeabilityinthezdirection,m²ppressure,psiqvolumetricflowrate,cm3/srradius,mrpproberadius,cmttime,stptimeproduction,sΔpddpressuredropattheendofdrawdown,psiΔp'pressurederivative,psiΔtelapsedtime,sαgeostatisticalanisotropicfactor,fractionφporosity,fractionmviscosity,cpM.Noirotetal./JournalofPetroleumScienceandEngineering133(2015)776–784777successively–thereservoir,thewellandtheWFTtool.Severalcellsrepresenttheflowlineofthetool(Fig.1c).Onlyoneofthemallowsthecommunicationbetweenthetoolandthereservoir.ItisrepresentedinwhiteinFig.1c,whiletheredvolumecorrespondstothewellborevolume.ThevolumeoftheWFTtoolis113cm3,andthecontactareabetweenthistoolandtheremainingofthereservoircorrespondstoonecellof1.56cm².Thewellborehasaradiusof15.6cm.Table1alsoshowsthedifferentpropertiesofeachzoneofthemodel.Inordertotakeintoaccounttheimpermeablemudcakethatgeneratesano-flowboundary,thewellhaszeroporosity.TheWFTtoolhasporosityandpermeabilityequalto1and10D,respectively.Afluidflowsimulationofaconstantrate(0.5cm3/s)inthetoolissimulatedtoobtainthepressureevolutionasafunctionoftimewiththeECLIPSEtool(v2010.1).ECLIPSEisusuallyusedtosimulateflowsinreservoirgrid.Thesimulatedtestsaccountforapumpingassociatedwithadrawdownperiodof20s.Thepressureevolutionduringthebuildupperiodisrecordedduring100s.Atthebeginningofthesimulation,theformationpressureatthedepthoftheWFTtoolisequalto376bars.TheparametersoftheflowsimulationaresummarizedinTable1.ThehydrodynamicresponseoftheformationtoWFT'scalcu-latedfromthreesyntheticreservoirmodelsispresentedbelow.2.2.2.HomogeneousisotropicmediaWFTtestswerefirstsimulatedinahomogeneousandisotropicmediatoassesstheflowshapefactorvalueofthemodel.Inisotropicmediaonlythewellboreimpactstheflowgeometryandcontrolsthepressuredropinadditiontothediffusivityofthemedia.Thenthevalueofflowshapefactorcanbeusedtoassessthedrawdownmobilityforothermediausingthesamesyntheticreservoirmodel.Table2showsthepermeabilityvaluesofeachofthefourWFTtestssimulations.2.2.3.HomogeneousanisotropicmediaAsecondseriesofsimulationsinahomogeneousmediawithverticalpermeabilityanisotropyhasalsobeenperformed.Thehorizontalpermeabilityvalues(kxandky)arethesameforallmodels,whiletheverticalpermeabilitydecreasesfromSim1toSim9(Table3).2.2.4.HeterogeneousisotropicmediaInordertoanalyzetheimpactsofthereservoirheterogeneityontheWFTdataresponses,syntheticreservoirswithheteroge-neouspermeabilityfieldshavebeengeneratedfollowingseveralstages(Fig.2).Theinitialsyntheticmodeliscomposedofmore17millioncellsof2.5cmside.Thefirststagecorrespondstothesimulationofthepermeabilityfromthestatistical(meanandstandarddeviation)andgeostatistical(correlationranges).Thesecondstageistheupscalingofthemodeltodecreasethenumberofcellstoallowflowsimulations.Thelaststageisadownscalingofcellstoobtainthesamegeometryasthesyntheticmodelinhomogeneousmediaandtodefinethepropertiesofthecellscorrespondingtothewellbore(inred)andtheWFTtool.PermeabilityfieldsaregeneratedwithanunconditionalSequentialGaussianSimulationusingalog-normaldistributionwitha10mDmeananda10mDstandarddeviation.Thegeostatisticalparametersgoverningthespatialcorrelationofthepetrophysicsarethevariogramrangesthatdescribetheorganizationofthepermeabilityfield.Inthisstudy,weconsideredFig.1.3Dviewsof(a)thesyntheticreservoirmodel,(b)thecellssymbolizingthewellvolumeand(c)thecellssymbolizingthetool.Table1CharacteristicsoftheWFTmodelusedfortheflowsimulations.WellpropertiesFluidpropertiesRadius15.6cmViscosity1cPPorosity0Watercompressibility2.83EC0061/barWFTtoolproperties:FlowsimulationparametersWFTtoolvolume113cm3Initialpressure376barWellbore/WFTtoolcontactarea1.56cm²Flowrate0.5cm3/sPorosity1Drawdownduration20sPermeability10DarcyBuildupdurationrecord100sFormationpropertiesRockcompressibility3.65EC0061/barPorosity0.2Table2Permeabilityvaluesusedforsimulationsinahomogeneousandisotropicmedia.kxkykz(mD)(mD)(mD)Sim1202020215151530101Sim4555Table3Permeabilityvaluesusedforsimulationsinahomogeneousandanisotropicmedia.kxkykzkxkykz(mD)(mD)(mD)(mD)(mD)(mD)Sim1101010Sim610100.05Sim210105Sim710100.01Sim310101Sim810100.005Sim410100.5Sim910100.001Sim510100.1M.Noirotetal./JournalofPetroleumScienceandEngineering133(2015)776–784778varioussimulationsforwhichhorizontalcorrelationrangesvaryfrom50mto2mandverticalcorrelationrangesvaryfrom10to0.01m(Table4).Foreachcoupleofhorizontalandverticalcorrelationranges(lH–lV)threestatisticallyequivalentrealizationsofthepermeabilityfieldhavebeensimulated.Eachcasecorre-spondstoapossibledistributionofthepermeabilityfieldinagreementwiththeinputparameters.Thesesyntheticpermeabilityfieldsareconsideredasinputfortheflowsimulations.Thesesyntheticreservoirmodelsdonotpresentanystatisticalanisotropy(kV/kH¼1)atthecellscale.Theheterogeneousmodelshavedifferenthorizontalandverticalcorre-lationrangesthatallowdefininga“geostatisticalanisotropy”thatcontrolsthepermeabilityfield;however,thelocalpermeabilityineachgridcellofthesyntheticreservoirmodelisanisotropicscalar(kX¼kY¼kZ).3.Results3.1.HomogeneousisotropicmediaTheaimoftheseWFTsimulationsinanisotropichomogeneousmediaistodeterminethevalueoftheflowshapefactor(C).Itisusedintheanalyticalsolutionofthediffusivityequation(Eq.(2))toassessthemobilityvaluefromthedrawdownperiod.Thisfactorcorrectstheeffectsofthewellboreonflowgeometry.Conse-quentlyinanisotropicmediawiththecorrectvalueoftheflowshapefactor,thedrawdownandbuilduppermeabilityvaluesareequal.Sotheflowshapefactor(Cf)isestimatedusinganiterativemethodtoobtainthebestlinearcorrelationbetweendrawdownandbuildupmobilityvalues(Fig.3).Fig.4showstheisopressurefrontrelatedtotheinvestigatedvolumeoftheformationfromthewellboreinahomogeneousandisotropicmedia.3.2.HomogeneousanisotropicmediaThehydrodynamicresponsesofWFTtestsimulationshavebeeninterpretedtoassessthedrawdownandbuildupmobilityvalues.Forthedrawdownmobility,weusedtheflowshapefactorestimatedinhomogeneousisotropicmediabecausethewellboreshapeandradiusarethesameinbothisotropicandanisotropicmedia.Fig.5showstherelationshipbetweenthedrawdownmobilityandthebuildupmobilityinferredfromthehydrodynamicresponsesanalysisoftheninesimulationsofWFT'sperformedinahomogeneousanisotropicmedium.Theimpactofthepermeabilityanisotropyontheinterpreteddrawdownmobilityvaluesisclearlyapparent(Fig.5).ThedirectconsequenceofthepermeabilityfieldanisotropyisanFig.2.Constructionofthesyntheticreservoirmodelwithheterogeneouspermeabilityfield.Table4Horizontalandverticalcorrelationrangesofeachsimulation.lxlylzlxlylz(m)(m)(m)(m)(m)(m)Sim1505010Sim18555Sim250505Sim19551Sim350501Sim20550.5Sim450500.5Sim21550.25Sim550500.25Sim22550.1Sim650500.1Sim23550.05Sim750500.05Sim24550.025Sim850500.025Sim25550.01Sim950500.01Sim26221Sim1010105Sim27220.5Sim1110101Sim28220.25Sim1210100.5Sim29220.1Sim1310100.25Sim30220.05Sim1410100.1Sim31220.025Sim1510100.05Sim32220.01Sim1610100.025Sim1710100.01M.Noirotetal./JournalofPetroleumScienceandEngineering133(2015)776–784779overestimationofdrawdownmobilityvalues.Thepermeabilityanisotropyaffectstheflowgeometryanddecreasesthenormalpressuredropusedtoassessthedrawdownmobility.Onthecontrary,theper
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