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InterferenceAnalysisforAnisotropicFormations-ACaseHistoryH.J.Ramey,Jr.,SPE-AIME.StanfordU.IntroductionIthaslongbeenknownthatmanyformationsappeartoexhibitsimpleky-kcranisotropy.Thismodelalsomaybeapplicableforformationscontainingtrendingfracturepatterns.Knowledgeofdirectionalpermeabilityobviouslywouldhaveanimportanteffectonplanningreservoirdevelopment,particularlyforfluidinjectionoperations.ArecentstudybyPapadopulos1outlinesmethodsforinterferenceanalysisinanisotropicformations.Themethodisusedheretoanalyzeafieldwater-injectiontesttodeterminemajorandminorpermeabilitiesandtheorientationoftheunknownpermeabilitymatrix.Themethodalsomaybeextendedtopressurefalloff(orbuildup)interferenceanalysis.Thetheoryofflowofeitherfluidorheatthroughananisotropicmediumiswellestablished.2•3Abriefreviewofthisinformationissufficienthere.Weconsiderthatawellisproducedataconstantvolumetricrateinaninfinite,anisotropicmedium.Theformationhasaconstantthicknessandporosity,andthetotalsystemeffectivecompressibilityisconstant.Collins4haspresentedperhapsthebest-knownsolutiontothisproblemintheareaofoilproductiontechnology.Inthemid-1960's,aseriesofpapersappearedinthegroundwaterhydrologyliteraturedealingwiththeproblemofwelltestanalysisforanisotropicaquifers.StudiesbyHantush,5,6Papadopulos,1andWalton7areparticularlynotable.ThePapadopulosstudyappearstobewell-suitedforapplicationtoanalysisofoilandgaswelltestdataandservesasthebasisforthisstudy.ThepurposeofthispaperistopresentanalysisofactualfielddatawiththePapadopulosmethodinaformreadilyuseablebypetroleumengineers.Themethodisextendedtofalloffdata.TheoryTheCollins4solutionforthepressurefieldcausedbyawellproducingfromananisotropicreservoiriscorrect,butitassumesthedirectionsofthemajorandminorpermeabilityaxesareknownandarealignedwiththewell-locationcoordinatesystem.Inthegeneralwelltestanalysissituation,thedirectionofthemajorpermeabilityaxiswouldbeunknown.Fig.Ishowstheknownwelllocationx-ycoordinatesystemwiththeunknownpermeabilityaxes,X-Y,orientedatsomeunknownangle8.Thepressureat(x,y.t)causedbyaline-sourcewellattheoriginwaspresentedbyPapadopulos:Ykk-k2h(Pi-Pcr,y,t)=:r.ryyIU141.2qBp.,-lEi[-p.,c(k:r.ry2+kyyX2_2kcryX)·)Il...(I)20.001051kukyy-kcr/~kxx=t{(ku+kyy)+[(k:r:r-kyy)2+4kcr/Th}.(2)kl'}'=t{(ku+kyy)-[(k:r.r-kmJ2+4kIyJ1/2}.(3)8=arctan(kxx-kcrcr)................(4)kcryIfthepermeabilityaxeswerecoincidentwiththewellaxes,kxxwouldequalk,rcrandtheangle8wouldbecomeManyformations,suchaschannelsands,appeartoexhibitsimpleky-k;ranisotropy.Directionalpermeabilityhasanimportanteffectonplanningfluid-injectionoilrecovery.Themethoddiscussedinthispaperusesdataobtainedduringwaterinjectiontodeterminemajorandminorpermeabilityaxesandtheorientationoftheunknownpermeabilitymatrix.1290JOURNALOFPETROLEUMTECH;-";OLOGYzero,asindicatedbyEq.4.Inthiscase,Eq.1wouldproducetheCollinsresult:Ykxxkyyh(Pi-Px.y.t)141.2qB/L=-~El'[-cPf.LC(kxxy2+kyyx2)J24(O.OO0264)kxxkyyt'....(5)wherekxxandkyyaretheprincipalpermeabilities.Ref.1providesaderivationofEqs.1through5.Eq.5showsthatawelltestconductedinananisotropicreservoirandanalyzedintheconventionalmannerforaconstantpermeabilitycasewouldyieldthegeometricmeanpermeability.Eq.5alsocanbeusedtoanalyzefielddataforanunknownpermeabilityorientationbyatrial-and-errormethod.Thex-ycoordinatesystemistakenasalignedwiththemajorandminorpermeabilitydirections,andthewellpatternisrotatedabouttheoriginuntilaminimumbetweencomputedandmeasuredpressuresisfound.PapadopulossuggestedthatEq.1couldbeusedintwoways.First,itmightbeuseddirectlywiththetype-curvematchingtechniqueonlog-logcoordinates.Papadopulosalsodescribedtheuseofaserniloggraphingmethodcorrectforconditionswhentheline-sourcesolutioncanberepresentedbythelogarithmicapproximation.Ineithercase,pressuredatafromthreedifferentobservationwellslocatedondifferentraysfromtheproducing(orinjecting)wellwouldberequiredifcfx:werenotknown.Thatis,pressuredatafromthreeobservationwellsalongdifferentlinesfromtheproducingwellwouldyieldkxx,kyy,8,andcfx:.Ifcfx:wereknown,datafromonlytwoobservationwellsaresufficient.Itmightappearthatthepressuredatafromtheproducingwellwouldprovideadditionalinformationtodetermineoneoftheunknowns.Thisisnotcorrectbecausetheskineffectattheproducingwellconstitutesanadditionalunknownintroducedbytheproducer.FieldExampleInplanningaprojectinawatered-outformation,aninjectiontestwasruntodeterminewhetherdirectionalpermeabilitywouldinfluencetheproject.Aninjectiontestwasselectedbecauseproducingrateswereverylow,causinglowpressuredrawdowns.Fig.2presentsanisopachmapforthewellpatternusedinthistest.Fig.3presentsthewellpatternonly.InFig.3,thenumbersinparenthesesaftereachwelldesignationrepresentthecoordinatesofthewellwiththeinjectorasorigin(unitsareinfeet).Ascanbeseen,thewellpatternisanisolatednine-spot.Thetestwasrunasasingle-wellinjectiontest,withthesurroundingwells(Wells5-E,I-E,I-D,I-C,5-C,9-C,9-D,and9-E)usedasobservationwellsonly.Thetestwasrunbyinjecting115BIDofwaterintothecentralwell,WeIl5-D,andmeasuringthepressureriseatthesurroundingeightwells.TableIpresentsthepertinentwellandreservoirdataandthepressure-timedataatthesurroundingobservationwells.Injectionwasstoppedat101hoursandthesucceedingfalloffdataarealsopresentedinTableI.Table1presentsthecompletefieldtestdata,whichismoreinformationthanisrequiredtoestablishtheutilityofthePapadopulosmethodforthiscase.Papadopulosillustratedhismethodwithsyntheticinformation.Aliter-n(,TClRFR107~y\\\xWELLPATTERNCOORDINATESMAJORANDMINORPERMEABILITYAXESFig.l-Coordinatesforanisotropicpermeabilitysolution.N1•25'\oo'"\10\"oI""'00'Fig.2-Netsandisopach.I-e(-4-70,490)•'\,.fJt1-0(0,475)eI-El475,514)If/69570~5-C(-455'O)~/5-E(47.3,0)----c.¢5-0(l/lJ.)_~l>X/1"-640'G09/••g-C(-470,-4GO).9-0(0,-455)9-E(470,-415)IFig.3-Fieldexamplewellpattern(distancesandcoordinatesinfeet).1291aturesearchindicatedthatcompleteinterferencedataofthetyperequiredforsuchastudyarerare.Forthisreason,Ihavetriedtopresentallinformationgatheredinthistestthatmightbeofinterestinfuturestudies,aswellasthatrequiredforpresentpurposes.Forexample,theinjectionwelldataaregivenalthoughtheyarenotnecessaryforthisstudy,andthedataforbothwellsinanoppositepair(suchasWells5-Cand5-E)aregivenalthoughbothshould(anddo)yieldthesameresultsiftheanisotropicmodeliscorrectforthefielddata.Thefalloffdataarealsogiven.Afewcommentsconcerningtheapplicabilityofsingle-phaseflowtheorytothiswaterinjectioncaseareinorder.Inthiscase,theformationinvolvedwasathoroughlywatered-outoilreservoirwithaveryhighproducingwatercut.Thus,theoilcontentoftheformationhadanappreciableeffectonlyonthetotalsystemeffectivecompressibility.Thisisanearlyperfectapplicationofsingle-phaseflowtheorytoareservoircontainingtwophases.Onlywaterhadsignificantmobilityandtherewasnoknowngassaturation.Inthiscase,thetotalsystemeffectivecompressibilitycanberelatedtotheporespacesaturationsofthetwophases:8c=Soco+Su·cu·+Cf......................(6)Or,c=Co+Su,(cu--co)+cf=So(co-cu')+cu'+Cf'..................(7)becausetherewasnogassaturation.Ifitcanbeassumedthatthecompressibilitiesareallknown,determinationofthetotalsystemcompressibilityfromaninterferencetestcanprovideanestimateoftheoilsaturationusingEq.7.Returningtothequestionoftheapplicabilityofsingle-phaseflowtheorytorealmultiphaseflowcases,itisnotnecessarythatotherpotentialapplicationsbeasidealastheoneinthispaper.Iftherewereagassaturationatorbelowthecriticalgassaturationforflowofagasphase.thegaswouldaffectonlythetotalsystemeffectivecompressibilityandwouldthusmodifyEqs.6and7.Intheextremecasewhereoil,water,andgasweremobile,themethodstilIwouldbeapplicableinviewofthePerrineI7speculationonmultiphaseflow,laterexplainedbyMartin.18(SeealsoMatthewsandRussell.ll)Perhapsoneofthegreatestconcernsinwaterinjectiontestingisthemobilitybankscausedbyinjection.Eveninthisextremecase,single-phaseflowtheoryisusuallyappropriatebecausethemobility-bank(singleormultiple)effectisincorporatedintotheapparentskineffectontheinjectionwell.ThetotalmobilityratiosuggestedbyPerrinewouldbe,inthiscase,thatofthemultiphaseflowinthereservoiraheadofthebanksneartheinjectionwell.Actually,waterinjectiontestingcanhaveratherbroadapplicationwithoutseriousconcernformobility-ratioeffects.Inastudyofwellborestorageandskineffect,WattenbargerandRamey9showedthat.undercertainconditions,annularregionsofdifferentmobilityofradiiTABLEl-FIEI'1EXAMPLEINTERFERENCETESTDATAi"=115STB/Dh=25ftB"=1resbbl/STB11-.,.=1cp=20percentc=8X1O-6psi-'c.,.=3.3x10-0Cf=3.7X10-6(Ref.13)C"=7.5x10-6(Ref.12)Oilgravity36°APIat60°/60°Formationtemperature=72°FInitialpressure=240psir"=0.563ftmean,openholeshotwithnitroInjectionWellData:Totaldepth,1,011ft;co.mpletedwith2-in.EUEtubing,bottom·holepackerPressureData(::'pispressureriseaboveinitialpressure)Well1-Ct(hours)113125146195215249295Well5-E22221916141411t(hours)21477294/).p(psi)115122140188210285129241116.321.2222522.319.21815____""--elJ~__t(hours)::'p(ps.0__23.56.728.57.25115772095251192412523.2141191631818814215122651029010Well9-Ct(hours)::'p(psi)24447872139417.711518126181451719411215132451129510Well1-EWell5·Ct(hours)~2J~~_t(hours)~£..(ps!2._27.5347104757117.272119424951311325.111516124261251614624142131921719210210152151024015.22406260142955.828513Well9-DWell9-Et(hours)::'p(psi)t(hours)f).p(psi)23.58.221328.59.347351177137523.294109527.211512.512027125131432114312.81901619513215142151327013240102851229510JOl'R:--;ALOFPETROLFt'\lTfTH,\()!()(;Yupto1,000rU'mayonlychangetheapparentskineffectofaproducingorinjectionwell.Thisresultedbecausewellborestoragetendstodominatethetimeperiodnormallyaffectedbythemobilitybanknearawell.Finally,anotherpracticalcasewheresingle-phaseflowtheoryisapplicableisinjectionofgasintoaheavyoilortarreservoir.Becausedisplacementissopoor,gasflowsmainlythroughexistinggasspace.Interferencetestingcanprovideestimatesofthemeangassaturation.ResultsInthefollowing,onlyWells5-E,I-E,andI-Dwereused.Itcanbeverifiedthatoppositepairsofwellsyieldthesameresults,indicatingthattheanisotropicmodelisreasonablycorrect.Fig.4presentsalog-loggraphofthefield-datapressurerisevstheinjectiontimeforWells5-E,I-E,andI-D.Fig.5presentstheanalyticalsolutionfortheconventionalline-sourcewell(exponentialintegralsolution).PD=-~Ei(-~~:),..................(8)wherekhPD=(Pi-Pr,t).................(9)141.2qBILrD=r/ru'...............................(10)tD=0.000264kt........................(11)cPILcrU'2Fig.5alsorepresentstheanisotropicsolutiongivenbyEq.I.Thedimensionlessquantitiesfortheanisotropiccaseshouldbedefinedas100l'"0ll.00>Atismistakenfortheconditionthattheproducingtimeislongenoughtoreachpseudosteadystate.ItshouldbecomeevidentfromFig.8thatthisisnotthecase.Theline-sourcesolutionisforawellinaninfinitelylargesystem-onethatcanneverbecomepseudosteady.Italsoshouldbeclearthatitisnotnecessarythattheproducingtimetbelargeatall;itisonlynecessarythatthemaximumAtbelessthan10percentoft.Thetype-curveanalysisforbuildup(orfalloff)representedbyFig.8isanalternativetothetedioustrialand-errormethodcommonlyused.IIThetype-curvemethodisvalidforisotropiccases,ofcourse.Itrepresentsasortof"'desuperposition.,.Thatis.thepressure500•400wcr::>U>U>~300c.w-'o::t:200"~IoID1000~".;;:0_010SHUTINAT101HR.6060MIN--,.0o,«?~0-0'0""'10c",00-0-~M'60PSI/CY.,..o---t::p·330PSI~oIHR100TIME.MINUTES100010.000Fig.9-Well5·0injectiontest.OCTOBER.197~falloffmeasured(likethecommoncaseofpressurebuildup)canberepresentedbyasuperpositionofaproductionwellandaninjectionwell.ThegraphicalprocedureinFig.8involvesextractingaportionofthedatafromthesuperposedanswer.Thisprocessofseparatingasuperpositionanswerintocomponentsmaybetermed"desuperposition."Table2presentsinjectionandfalloffpressuresfortheinjectionwell,Well5-D.Fig.9presentsaconventionalsemiloggraphoftheinjectiondata.Thedatadoappeartoformastraightlinewithaslope,m,of60psi/cycle.Thiscorrespondstoapermeabilityof12.5md-lessthantheminorpermeabilityof13.7mdobtainedfromtheobservation-welldataanalysis.Althoughnotshown,thefalloffdataproduceastraightlineofslope42psi/cycleandacorrespondingpermeabilityof17.8mdveryclosetothegeometricmeanvalueof16.92mdobtainedfromtheinterferencedata.Thereasonforthedifferencebetweentheinjectionandfalloffdataisnotknown.Thefalloffdataclearlyshowtheeffectofwellborestoragecausedbyfallingliquidlevelinthetubing,anditappearsthatthereissomesortofsimilarstorageeffectintheinjectiondatainFig.9.Becausethedepthofthewellis1,011ft,abottom-holepressureof438psicorrespondstohydrostaticpressureforthatdepth.ThefinalbendupwardsintheinjectionpressuresinFig.9appearstocoincidewithbottom-holepressurereachinghydrostaticpressure.Thus,thewaterlevelrosecontinuouslythroughouttheinjectionuntilthistime.Itisbeyondthepurposeofthispapertodiscusstheinjectionwelldatafurther.Theinjectionwellispresentedmainlyforthesakeofcompletedisclosureofthisremarkablesetofdata.ConclusionsItisclearthatthefieldcasepresentedisnotanidealcaseofsimpleanisotropy.Attheleast,itisknownthatthethicknessvariesthroughoutthepattern.Itispossiblethattheporosity-compressibilityproductalsovaries.Isayonlythatahomogeneousformationmodelwithanisotropicpermeabilitymatchesthefielddatabetterthanamodelwithconstantpermeability.Ontheotherhand.itisnotlikelythattheporosity-compressibilityproductvariesasmuchaspermeabilitydoesintheusualcase.Isuspectthissortoftestmaybecomeimportantinplanningfluidinjectionoperations.Anotherimportantapplicationofthismodelisapttobethefracturedformationcontainingparallelverticalfractures.Thepermeabilitynormaltothefractureplaneswouldbeessentiallymatrixpermeability,whilethepermeabilitywouldappearhigherthanmatrixpermeabilityinthedirectionofthefractureplanes.ElkinsandSkovl4appeartohavebeenfirsttousethissortofmodelfortheverticallyfracturedSpraberryreservoir.Elkinsl5appearstohaveusedtheline-sourcetypecurve(likeFig.5)firstinapetroleumstudyin1946,althoughthemethodwasproposedbyTheisl6in1935inthegeophysicalliteratureandwaswidelyusedinpump-testanalysisinhydrology.Log-logtypecurveshavetremendousdiagnosticpowersbecausefielddatagraphsmustappearliketheproperanalyticsolutionforalltimes.Fig.5isausefultypecurveforhomogeneousaswellasanisotropicreservoirs.Whenwellsarelocatedwithinboundedreservoirs,itisasimplemattertoextendprocedures.2oThesizeofthereservoirmaybefoundinidealcases.1297Finally,somecommentsconcerningpulsetesting21areinorder.Theinjection-falloffcycledescribedinthispaperisthefirstpulseinapulsetest.Fundamentally,thereisnodifferencebetweenaninterferencetestandapulsetest,assumingpressuregaugesofthesameprecisionareusedinbothtests.Thissubjectstudywasperformedwithsonicliquid-levelpressuremeasurements.New,high-precisionpressureinstrumentsareamajorbreakthroughininterferencetesting.ThesametestcouldhavebeenperformedinmuchlesstimeusinganyofseveraldevicesthatmeasurepressuretolessthanIpsiwithprecision.Pulsetestingcanhaveanadvantagewhenunknownpressuretrendsexistbeforethestartoftheinterferencetest,however.NomenclatureB=formationvolumefactor,resbbI/STBC=totalsystemeffectivecompressibility,psi-Icf=isothermalporespacecompressibilityofrock,psi-ICo=isothermalcompressibilityofoil,psi-Icu'=isothermalcompressibilityofwater,psi-IEi=exponentialintegralh=netformationthickness,ftiu'=injectionrateofwater,STB/Dk=permeability,mdkxx=maximumprincipalpermeabili
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