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matchingCubillos,1.IntroductionHer-yuanandLawrance(2000)developedamethodtodiagnosetheexistenceofpermeabilityanisotropyfromdoesnotrequireneithertype-curvenorsimulationmatch-ingnorinformationofreservoirfluidproperties.Inthiswork,permeabilityanisotropyisobtainedbyAvailableonlineatwww.sciencedirect.comJournalofPetroleumScienceandEngineeri⁎Correspondingauthor.Multi-wellinterferencetestingisthebestandmostpopularmethodfordeterminationofpermeabilityanisot-ropy(magnitudeandorientation)atwell-reservoirscale.Thereexistafewmethodsforanisotropydeterminationfrominterferencetesting.PerhapsthemostrecognizedisthetechniqueproposedbyRamey(1975)whichusedtheline-sourcesolutionpresentedbyPapadopulos(1965).Thismethodisbasedupontype-curvematchingofpressureandallowstoobtainthemaximumandminimumpermeabilitysemi-axisandtheanisotropyangle.pressuretransienttestingbasedonalog–logplotofpressuredropvs.timeoverthesquaredistance,ΔPvs.t/r2.Additionally,themethodallowstoinferqualitativeaniso-tropyinrectangularcoordinates.Her-yuanandLawrance(2003)alsopresentedaquickmethodtoonlyestimatetheorientationofpermeability/flowanisotropyfrompressureinterferenceresponses.Itcanidentifyeithera90°rangeofanisotropyorientationfromtwoobservation-wellresponsesor,approximately,a45°(orlessunderfavorableconditions)rangefromthreeobservation-wellresponses.Iterrorprocedure.Thispaperhasbeenwrittenwiththepurposeofovercomingthisissue.Therefore,type-curvematchingprocedureisreplacedbyapressureandpressurederivativeinterpretationmethodwhichfollowstheTDStechniquephilosophy,sothatdirectionalpermeabilitycanbeeasilyestimated.Theprovidedmethodologyallowstopracticallyandaccuratelyestimateanisotropyandcarryoutaquantitativeestimationofboththemaximumandminimumdirectionalpermeabilityvalues,andtheangleformedbetweenthexaxisandthemaximumpermeabilitysemi-axis.Additionally,theaveragepermeabilityvalueisre-estimated.Afieldexampleproposedintheliteratureandasyntheticexamplehavebeensuccessfullysolvedforverificationpurposes.©2007ElsevierB.V.Allrightsreserved.Keywords:Angleofanisotropy;Degreeofanisotropy;Homogeneousreservoir;Averagehorizontalpermeability;Radialpermeability;Interferencetests;TDStechniquereservoirs.CurrentinterpretationtechniquesaremainlybaseduponAbstractNormally,interferencetestinginvolvingatleastthreeobservationwellsisusedforestimationofarealanisotropyinhydrocarbonbypressuretype-curvematchingwhichisbasicallyatrial-and-Estimationofhorizontalreservoirtype-curveFreddyHumbertoEscobar⁎,JorgeUniversidadSurcolombiana,Av.PastranaReceived26November2006;receivedinrevisedE-mailaddresses:fescobar@usco.edu.co(F.H.Escobar),jorgecubillos@gmail.com(J.Cubillos),matildemm@usco.edu.co(M.Montealegre-M).0920-4105/$-seefrontmatter©2007ElsevierB.V.Allrightsreserved.doi:10.1016/j.petrol.2007.05.003anisotropywithoutMatildeMontealegre-M-Cra.1,NeivaHuila,Colombiaform4May2007;accepted10May2007ng60(2008)31–38www.elsevier.com/locate/petrolinterferencetestingofthreeobservationwellsbyextendingtheTDStechniquetoobtaintheradialFig.1.Line-sourcesolution,afterOuandlous(1999).32F.H.Escobaretal./JournalofPetroleumScienceandEngineering60(2008)31–38permeabilityvaluesforeachpairofwellsbyusingtheintersectionpointofthepressureandpressurederivativecurve.Then,asystemofthreeequationswiththreeFig.2.ΔPvs.t/r2forahomogeneounknownsresultsandissolvedtoallowforthedeterminationofthemaximumandminimumperme-abilitiesandtheanisotropydegree.Consequently,usanisotropicreservoir.Fig.4.Wellconfiguration,afterHer-yuanandLawrance(2003).Fig.5.Wellconfigurationforfieldexample.Fig.3.Permeabilityconfiguration,afterHer-yuanandLawrance(2003).33F.H.Escobaretal./JournalofPetroleumScienceandEngineering60(2008)31–38Scienceaveragehorizontalpermeabilityanddegreeofanisotro-pyarealsocalculated.2.FormulationThepressuredistributionwithinareservoirisdescribedbytheexponential–integralsolutionofEq.(1)formulatedbyMathewsandRussel(1967).Itisalsocalledtheline-sourceortheTheissolution,PD¼C012EiC0r2D4tDC18C19ð1ÞThedimensionlessquantitiesareexpressedby:PD¼¯kh141:2qBlDPð2ÞtD¼0:0002637krt/lctr2wð3ÞrD¼rrwð4ÞNoticethatpermeabilityanisotropyisimplicitinPDbyk¯andintDbykr.Thepressurederivative(TiabandKumar,1976)isexpressedby:tDr2D⁎PDV¼12expC0r2D4tDC18C19ð5ÞOuandlous(1999)extendedtheTDStechnique(Tiab,1995),forinterferencetesting.Theradialpermeability,kr,canbefoundfromtheintersectionpointoftheΔPDand(tD/rD2⁎ΔPD′)vs.tD/rD2curvesplottedonalog-logscale,seeFig.1,¯k¼45:705qBlhðt⁎DPVÞint¼45:705qBlhðDPÞintð6Þandkr¼ϕctlr20:00045864tintð7ÞForisotropicsystems,porositycanbeestimatedfromEq.(7)oncepermeabilityisestimatedusingEq.(6).Ifradialflowdevelops,theaveragepermeabilitycanalsobeestimatedfrom:¯k¼70:6qlBhðt⁎DPVÞrð8ÞAccordingtoHer-yuanandLawrance(2000,2003),forahomogeneousisotropicreservoirkr=k¯=constant.Withoutconsideringthewelllocation,alog–logplotΔPvs.t/r2willprovideauniquecurveforallthewells.Incontrast,forananisotropicsystem,thesameplotwill34F.H.Escobaretal./JournalofPetroleumrevealavarietyofresponsesforeachobservationwellasdepictedinFig.2.Thesevariationsareafunctionoftheangleformedbetweeneachwellandthemaximumpermeabilitysemi-axis,kmax.Therefore,awellwiththelowestanglewillhavethehighestradialpermeabilityvalue,kr,andtheearliestpressureresponse.AspointedoutbyHer-yuanandLawrance(2003),oncethevaluesoftheradialpermeability,kr,arefoundforeachwell,kri,acoordinatesystemoughttobesetwiththeactivewelllocatedintheorigin.Then,theobservationwellsaresetandtheangleformedbetweenthewellandthex-axisismeasured.Thisβiangle(beingi=observa-tionwell1,observationwell2,andobservationwell3)ispositiveandmeasuredanticlockwise.SeeFigs.3and4.Aftersettingthesemi-axisofmaximumandminimumpermeability,theangle,θi,betweenthewellandthesemi-axisofmaximumpermeabilityisdetermined.Likeβi,θiisalsopositiveandismeasuredanticlockwise.Her-yuanandLawrance(2003)alsopresentedtherelationshipsamong(xi,yi),ri,βi,θiandα,asdescribedhere:xiyiC26C27¼ricosbirisinbiC26C27¼ricosðaC0hiÞrisinðaC0hiÞC26C27ð9ÞBaseduponFig.4,eachangleisrelatedtoothersas:hi¼biC0að10ÞBasedontheconceptofapermeabilityellipse,Her-yuanandLawrance(2003)presentedthefollowingexpressiontofindtheθiangleandmaximumandminimumpermeability:1kri¼cos2hikmaxþsen2hikminð11ÞAccordingtoEq.(10),Eq.(11)isreformulatedas:1kri¼cos2ðbiC0aÞkmaxþsen2ðbiC0aÞkminð12ÞSincekriandβiareknown,Eq.(12)isappliedtoeachobservationwell.Then,asystemofthreeequationswiththreeunknownswillresult.Itssolutionallowstoreadilyobtainα,kmax,andkmin.Therefore,thedegreeofanisotropyandtheaveragepermeabilitycanberespec-tivelyestimatedfrom:A¼kmax=kminð13Þ¯k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikmaxkminpð14Þ3.Examples3.1.FieldexampleAlog–logplotofpressureandpressurederivativeandEngineering60(2008)31–38vs.timeforanexamplereportedbyRamey(1975)ispresentedinFig.5.Reservoir,fluidandwelldataaregiveninTable1.Ramey(1975)solvedthisproblembytype-curvematching.Awaterfloodingprogramwasconductedinthisreservoirina9-normalpattern.3.1.1.SolutionTheinformationoftheinterceptpointbetweenpressureandderivativeforeachpairofwellsisreadfromFig.6asfollows:ΔPint5-E=(t⁎ΔP′)int5-E=12.8psi,tint5-E=53.0h,ΔPint1-E=(t⁎ΔP′)int1-E=11.0psi,tint1-E=73.0h,ΔPint1-D=13.0psi,tint1-D=43.0h.RadialpermeabilityisfoundforeachpairofwellsusingEq.(7)tobekr5-E=14.758md,kr1-E=23.270mdandkr1-D=18.196md.TheβianglesarereadfromFig.5asβ1-D=90°,β5-E=0°,β1-D=47°.FromtheapplicationofEq.(12)toeachpairofwellswillresultinthefollowingsystemofequations:114:758¼cos2ð0-C0aÞkmaxþsen2ð0-C0aÞkmin123:270¼cos2ð47-C0aÞkmaxþsen2ð47-C0aÞkmin1¼cos2ð90-C0aÞþsen2ð90-C0aÞTable1Reservoir,wellandfluiddatafortheworkedexamplesFieldexampleSimulatedexampleParameterValueϕ(%)2012μ(cp)10.8ct(1/psi)7.95×10-68.3×10-6rw(ft)0.5630.27B(rb/STB)11.12Pi(psi)240q(STB/D)−115425r1D(ft)475r5E(ft)475r1E(ft)700rOW-1(ft)300rOW-2(ft)424.26rOW-3(ft)30035F.H.Escobaretal./JournalofPetroleumScienceandEngineering60(2008)31–38Theinjectoriswell5Dandtheproducersare5-E,1-E,1-D,1-C,5-C,9-C,9-Dand9-E.Onlywells5E,1Dand1Ewereusedasobservationwells.ThewellconfigurationisgiveninFig.5.FindthepermeabilityanisotropyandcomparetothesolutiongivenbyRamey(1975).Fig.6.Pressureandpressurederivative18:196kmaxkminInwhichsolutiongiveskmax=23.655md,kmin=12.428md,andα=54.78°.UsingEqs.(13)and(14),theestimateddegreeofanisotropyandaverageperme-abilityare1.9and17.14166md,respectively.dataforthefieldexample.3.2.SyntheticexampleThisexamplewaspreviouslyworkedbyOuandlous(1999)usingthedatafromTable1.Forthisexample,Ouandlous(1999)determinedanaveragepermeabilityplottedinFig.8.Findthepermeabilityanisotropyandcomparetothesimulationresults.3.2.1.SolutionFromFig.8thefollowingintersectionpointsareread:ΔPintOW-1=(t⁎ΔP′)intOW-1=33.496psi,tintOW-1=9.584h,ΔPintOW-2=(t⁎ΔP′)intOW-2=33.496psi,tintOW-211.858h,ΔPintOW-3=(t⁎ΔP′)intOW-3=33.496psi,tintOW-3=7.027h.FromEq.(7)theradialpermeabilityvaluesforeachpairofwellsarekrOW-1=16.313md,krOW-2=26.461md,andkrOW-3=22.249md.FromFig.7,theanglesareβOW-1=0°,βOW-2=45°,andβOW-3=90°.WiththisinformationandbaseduponEq.(12),thefollowingsystemofequationsresults:116:3136¼cos2ð0C0aÞkmaxþsen2ð0C0aÞkmin126:4617¼cos2ð45-C0aÞkmaxþsen2ð45-C0aÞkmin112:2498¼cos2ð90-C0aÞkþsen2ð90-C0aÞkFig.7.Wellconfigurationforsimulatedexample.36F.H.Escobaretal./JournalofPetroleumScienceandEngineering60(2008)31–38of17md.AssketchedinFig.7,thetestinvolvedoneproducerwellandthreeobservationshut-inwells.TheobservationwellswerelabeledasOW-1,OW-2andOW-3.ThepressureandpressurederivativedataareFig.8.PressureandpressurederivativemaxminThesimultaneoussolutionoftheabovesystemofequationsprovideskmax=27.9751md,kmin=14.1850mdandα=59.92°.UsingEq.(13),theestimateddegreeofdataforthesyntheticexample.Scienceanisotropy,A,is1.97andanaveragepermeability,k¯,of19.92mdiscalculatedwithEq.(14).4.AnalysisofresultsTable2summarizestheresultsfromnotonlythisstudybutalsoothersreportedintheliterature.Wecannoticethatallofthemarewithinareasonablerangeand,therefore,allofthemareacceptable.Forthesimulatedproblem,wereportedanaveragepermeabilityof19.9mdagainstavalueof17mddeterminedbyOuandlous(1999)whichalsofallswithinanexpectedrange.Theseworkedexamplesplusothers(withalsogoodacceptance)notreportedhereallowustorecommendthemethodologyproposedhereforitsapplicationforhorizontalpermeabilityanisotropydetermination.Forcaseswhenonlytwoobservationwellsexist,wesuggestthatreservoiraveragepermeabilitycouldbedeterminedfromeitherEq.(6)or(8)usingthevalueofeitherpressureorpressurederivativeattheinterceptpoint,ΔPint=(t⁎ΔP′)int,orfromthepressurederivativeduringradialflowregime,(t⁎ΔP′)r,respectively.Fromourobservations,wenoticethattheanisotropyisreflectedmuchmoreintimethaneitherpressureorpressurederivative,then,anaverageinterceptionpointoughttobeestimatedfromthetwoobservationwellsand,therefore,Eq.(14)canbeusedalongwiththeapplicationofEq.(12)toobservationwells1and2,thus:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipTable2ComparisonofresultsforfieldexampleMethodofanalysiskmax(md)kmin(md)k¯(md)α(degree)Type-curvematching,Ramey(1975)21.113.516.962.1Non-regressionanalysis,EarlougherandKersch(1972)17.212.714.859.3Thisstudy23.612.417.154.7F.H.Escobaretal./JournalofPetroleum¯k¼kmaxkminð15Þ1kr1¼cos2ðb1C0aÞkmaxþsen2ðb1C0aÞkmin1kr2¼cos2ðb2C0aÞkmaxþsen2ðb2C0aÞkminThesimultaneoussolutionoftheabovesystemofequationswillleadtofindmaximumpermeability,minimumpermeability,angleofanisotropyandthedegreeofanisotropy,andofcourse,loweraccuracy.5.ConclusionsAnewquick,simpleandaccuratemethodologyispresentedhereforestimationofpermeabilityanisotropyusinginterferencetestingwiththreeobservationwells.Thiswassuccessfullyverifiedbysolvingfieldandsimulatedexamples.Themethodologycanalsobeappliedwithlessaccuracytocaseswhenonlytwoobservationwellsareconsidered.NomenclatureAAnisotropydegreeBOilformationfactor,bbl/STBctCompressibility,1/psihFormationthickness,ftkPermeability,mdk¯Averagehorizontalpermeability,mdkmaxMaximumpermeability,mdkminMinimumpermeability,mdkrRadialpermeability,mdqInjectionorproductionrate,STB/DrRadialdistancebetweenobservationandactivewells,ftrwWellboreradius,fttTime,ht⁎ΔP′Pressurederivativex,yRectangularcoordinatesPPressurePwfWell-flowingpressure,psiGreekΔChange,dropαAngleofthesemi-axisofmaximumperme-abilityandthexaxis,degreesβAngleformedbythewellandthexaxis,degreesθAngleofthesemi-axisofmaximumperme-abilityandeachobservationwell,degreesϕPorosity,fractionμViscosity,cpSubscriptsDDimensionlessiIndexintIntersectionmaxMaximumminMinimumrRadialflow37andEngineering60(2008)31–38riRadialflowforeachwelliAcknowledgmentsTheauthorsgratefullyacknowledgethefinancialsupportoftheColombianPetroleumInstitute,ICP,underthemutualagreementNumber008signedbetweenthisinstitutionandUniversidadSurcolombiana.ReferencesEarlougherJr.,R.C.,Kersch,K.M.,1972.Fieldexamplesofautomatictransienttestanalysis.J.Pet.Technol.1271–1277(Oct.).Her-yuan,C.,Lawrance,W.T.,2000.Aquickmethodtodiagnoseflowanisotropyusingpressureinterferencedata.PaperSPE60290presentedatthe2000RegionalSymposiumofLowPermeabilityreservoirsheldinDenver,CO.Her-yuan,C.,Lawrance,W.T.,2003.AQuickMethodtoDeterminePermeability-AnisotropyOrientationfromInterferenceTesting.PaperSPE84090presentedattheSPEAnnualTechnicalConferenceandExhibition,5–8October,Denver,Colorado.Mathews,C.S.,Russel,D.G.,1967.PressureBuildupandFlowTestsinWells.MonographSeries,SPEofAIME,Dallas,vol.1.Chap.2.Ouandlous,A.,1999.“InterpretationofInterferenceTestsbyTiab'sDirectSynthesisTechnique”,M.Sc.Thesis.TheUniversityofOklahoma,Norman,OK.Papadopulos,I.S.,1965.Non-steadyflowtoawellinaninfiniteanisotropicaquifer.InternationalSymposiumoftheAssn.Sci.Hydrologic,Dobrovinik,Yugoslavia.RameyJr.,H.J.,1975.Interferenceanalysisforanisotropicformations—acasehistory.J.Pet.Technol.1290–1298(Oct.).Tiab,D.,1995.Analysisofpressureandpressurederivativewithouttype-curvematching:1.SkinandWellboreStorage.J.Pet.Sci.Eng.12,171–181.TIAB,D.,KUMAR,A.,1976.“ApplicationofPD’FunctiontoInterferenceAnalysis”.JPTAug.1980,P.1465-1470;PaperSPEpresentedatthe51stAnnualFallTechnicalConferenceandExhibitionoftheSocietyofPetroleumEngineersofAIME,heldinNewOrleans,Oct.3-6.38F.H.Escobaretal./JournalofPetroleumScienceandEngineering60(2008)31–38
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