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SPE-286-PA

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PETROLEUMTRANSACTIONS~flfi1"AStatisticalReservoir-ZonationTechniqueJ.D.TESTERMANABSTRACTAstatisticaltechniquetoidentifyanddescribenaturallyoccurringzonesinareservoirandtocorrelatethesezonesfromwelltowellisdescribed.Thetechniqueisparticularlyusefulindescribingareservoirwherecrossfiowbetweenadjacentstrataisimportantindeterminingreservoirbehavior.Althoughithasbeenusedprimarilyforpermeabilityzonation,thetechniqueisgeneralandcanbeusedtocorrelateanyreservoirpropertyorrelateddata,suchastheinformationcontainedinwelllogs.INTRODUCTIONOneofthefirstproblemsencounteredbythereservoirengineerinpredictingorinterpretingfluiddisplacementbehaviorduringsecondaryrecoveryprocessesisthatoforganizingandusingthelargeamountofdataavailablefromcoreanalysis.Permeabilitiesposeparticularproblemsinorganizationbecausetheyusuallyvarybymorethananorderofmagnitudebetweendifferentstrata.Duetothesheervolume,itisalmostalwaysnecessarytogroupdataandtouseanaveragevaluetorepresentanumberofmeasurements.Perhapsthemostcommonmethodnowusedtogrouppermeabilitydataisthecapacity-fractiontechnique,whichrankspermeabilitiesinorderofmagnitude,regardlessofthephysicallocationofthepermeabilitieswithinthereservoir.Thecumulativepercentcapacityisplottedagainstcumulativepercentthickness.Thisplotisdividedintoanarbitrarynumberofzones,generallyofequalthickness.Fivezones(oraveragedgroupsofdata)usuallyareobtained,eachofwhichistreatedashomogeneousinsubsequentcalculations.Thedivisionsoobtainedhasnophysicalmeaning;stratainthesamezone,calculation-wise,areusuallynotadjacentinthereservoir.Reservoirengineeringtechniquesbeingdevelopedwillhandlecrossflowthatoccursbetweenadjacentcommunicatingreservoirstratabecauseofimbibitionandgravitysegregation.Sincecrossflowoccursbetweenphysicallyadjacentlayerswithinthereservoir,anewzonationtechniquerecognizingtheactuallocationofstratawithinthereservoirisnecessary.Similarly,therecognitionofnaturalzonesisimportantforpredictionsofoilrecoverybyprocessesinvolvingdiffusion.Onesuchprocessismiscibledisplacement,wherepredictionsoflateraldiffusionwithinthereservoirmustrecognizetheactuallocationoftheOriginalmanuscriptreceivedinSocietyofPetroleumEngineersofficeFeb.28,1962.RevisedmanuscriptreceivedJune29,1962.PaperpresentedatSPEProductionResearchSymposium,April12-13,1962,inTulsa,Okla.Discussionofthisandallfollowingtechnicalpapersisinvited.Discussioninwriting(threecopies)maybesenttotheofficeoftheJournalofPetroleumTechnology.AnydiscussionofferedafterDec.31,1962,shouldbeintheformofanewpaper.Nodiscussionshouldexceed10percentofthemanuscriptbeingdiscussed.AUGUST,1962SPE286JERSEYPRODUCTIONRESEARCHCO.TULSA,OKLA.invadedzonesinrelationtotherestoftheformation.Naturalzonesmustalsobeadequatelyrecognizedtoaccountforheattransferwithinthereservoirduringthethermalexploitation.Becauseofthecomplexityoftheproblem,statisticsappeartooffertheonlypracticalhopeofdividingareservoirintophysically-meaningfulnaturalzones.Thispaperpresentsastatisticaltechniqueforidentifyingthesenaturalzonesandforascertainingwhichonesarelikelytobecontinuousbetweenadjacentwells.Thezonesdefinedhaveminimumvariationofpermeabilityinternallyandamaximumvariationbetweenzones.Thetechniqueisgeneralandcanthusbeappliedtoreservoirpropertiesotherthanpermeability.Themethodwillguidethereservoirengineerinestimatingwhichzonesarelikelytobecontinuousbetweenwells.However,astatisticalcorrelationbasedonpermeabilitiesintwodifferentwellsisnoguaranteethatthezonessodefinedare,infact,continuous.Rather,theassumptionofcontinuitymustbeconsistentwithgeologicaldataconcerningthedepositionalenvironment,aswellasjustifiedonthebasisofengineeringjudgmentincombinationwithstatistics,justasjudgmentisrequiredwithconventionalzonationmethods.CALCULATIONPROCEDUREThereservoirzonationtechniqueisatwo-stepoperation.Thestepsareindividuallydescribed,andasamplecalculationispresentedintheAppendix.ZONATIONOFINDIVIDUALWELLSFirst,thesetofpermeabilitydataatasinglewellisdividedintozones.Thesezonesareselectedsothatvariationisminimizedwithinthezonesandmaximizedbetweenthezones.Theequations'"usedtozonethedataare1[L__]B=L_1i::1nJ,(k,.-k.Y,.(1)1[L"--+-sz(v.p)'--~1(11)2nhn,(4)wherer:.=thearithmeticaverageofthepermeabilitydataofthehthzoneinonewellandk..=thearithmeticaverageofthepermeabilitydataoftheithzoneinanadjoiningwell,nhandn,=thenumberofdatainthehthandithzones,s=thestandarddeviationSofallthepermeabilitydatafromthereservoirandz=aconstanttabulatedasafunctionofthenumberofdata,thenumberofzonesandaprobabilitylevel.vandpareusedtoidentifyz-valuesasfunctionsoftheprobabilitylevel.Harter'providesatableofz-values.IftheleftsideofEq.4islargerthantherightside,thezonesrepresentedbythetwomeansareconsidered,onthebasisofstatistics,tobedifferent.However,iftheleftsideofEq.4issmallerthantherightside,thezonescorrelateandcanbeconsideredtobecontinuous.Forexample,inTable2thetopzoneofWell1hasanarithmeticaveragepermeabilityof90md.ThetopzoneofWell2alsohasapermeabilityaverageof90md.IfthetwoaveragesarecomparedbyEq.4,wenaturallyfindnosignificantdifference.Ontheotherhand,ifthe40-mdaverageforthesecondzoneofWell1andthe61-mdaverageforthesecondzoneinWell2arecompared,wefindadifferenceof21md.Whileweknowthatthesewell-zoneaveragesarenotalike,wewouldliketofindJOURNALOFPETROLEUMTECHNOLOGYoutifthedifferenceof21mdisduetodifferentstrata_ortonormalsamplingfluctuation.Thedifferenceof21mdcannotbeattributedtosamplingfluctuationbecauseitisgreaterthantherightsideofEq.4,*whichhasavalueof2.2md.Therefore,thesamplesprobablycamefromdifferentreservoirzones.WhenthemeanofthesecondzoneinWell1iscomparedwiththemeanofthethirdzoneinWell2,thedifferenceof11.5mdisgreaterthantherightsideofEq.4,whichhasavalueof2.2md.Thus,neitherZone2norZone3inWell2islikelytobecontinuouswithZone2inWelll.InTable2,asingle90-mdreservoirzoneatthetopofthesandisapparent.Statistically,thisisallwecansay.Practically,however,wecouldgroupthebottomtwozonesinWell2intoonezonewithanaveragepermeabilityof44.7mdandusethistocorrelatewithZone2ofWell1.Weemphasizethis"practical"pointinordertoshowhowengineeringjudgmentisusedwithresultsfromthestatisticalzonationtechnique.ACKNOWLEDGMENTTheauthorwishestoexpresshisappreciationtothemanagementofJerseyProductionResearchCo.forpermissiontopublishthispaper.AcknowledgmentisalsodueB.T.Willmanforhisassistanceinthepreparationofthispaper,andtoF.A.Graybillforhisideasandworkintheearlydevelopmentofstatisticalzonation.REFERENCES1.Beghtol,LeRoyA.:"AStatisticalApproachtotheZonationofaPetroleumReservoir",Master'sThesis,MissouriSchoolofMines(1958).2.Duncan,D.B.:"MultipleRangeandMultipleFTests",Biometrics(1955)II,1.3.Fisher,W.D.:"OnGroupingforMaximumHomogeneity",lour.ofAm.Stat.Assn.(1958)53,No.284.4.Graybill,F.A.:AnIntroductiontoLinearStatisticalHypo·thesis(1961)257.5.Harter,H.L.:"CriticalValuesforDuncan'sNewMultipleRangeTest",Biometrics(1960)16,671.6.Kramer,C.Y.:"ExtensionofMultipleRangeTeststoGroupMeanswithUnequalNumbersofReplications",Biometrics(1956)12,307.APPENDIXANEXAMPLEOFSTATISTICALZONATIONFig.A-Ishowsthelocationoffourwellsselectedtoillustratetheuseofthestatisticalzonationtechnique.Thedatachosenfortheillustrationarepermeabilitiesfromaconsolidatedsandstonereservoir.TableA-Iliststhepermeabilitydataandtheircorrespondingdepthforeachofthefourwells.STEPI-ZONATIONOFINDIVIDUALWELLSToillustratethecalculations,weselectedthedatainWell11becauseithaslessdatathantheotherwells.TableA-2illustratesthedivisionofthedataintotwozones(theasteriskmarksthepointofdivisioninTablesA-2,A-3andA-4).WeuseEqs.1and2inthefollowingmore-convenientformforthecomputations.1[L__]B=L_1i:1m,(k..-k.Y·SeetheAppendixforthecalculationofEQ.4.AUGUST,1962WELLNO.11•NWELLNO.8WELLNO.18••I_[000fT_IWELLNO.37•FIG.A·I-LoCATw:-;OFWELLS.1[L(.~'k,;)'(.~.~ikij)'1=__~3-1_t-13-1L-1i=1miN(5)w=N~L[.~.1'(k'j-ki)']t=13=11[L11"L(.n~kijr1=___~~k';j-~--=-3_-_1_-,--_N-Lic~1j=1i=1mi(6)Eqs.1,2and3arecomputedforeachdivisionintotwozonesandtabulatedinthecolumnslabeledB,WandR.NotethatanynegativevaluesinRarereplacedbyzeroinordertoconformtothedefinitionofR.Forexample,thefirstlineinTableA-2iscomputedasfollows.TABLEA·I-RESERVOIRPERMEABILITYDATAWellNo.8'WellNo.11WellNo.18WellNo.37DepthPerm.DepthPerm.Depthferm.Depth'Perm......J!!L(md)-...i!!L..(md)~(md)~(md)1917.5'111906.5'101973.5*201922.5'341918.5271907.5521974.5401923.5671919.51571908.52761975.51901924.5201920.52341909.51401976.51461925.51971921.53901910.51391977.5531926.51861922.51101911.51561978.54.81927.5331923.51921912.53421979.50.01928.5301924.52181913.5871980.5451929.5211925.5421914.50.01981.5141930.51171926.51201915.50.01982.50.01931.5271927.51581983.5841932.5271928.53161984.5281933.5261929.5201985.50.0A934.5611930.5991986.50.01931.51211987.50.01932.5431988.50.01933.5681934.57.41935.51491936.50.0*Topofproductiveinterval.TABLEA·2-DIVISIONOFDATAINTOTWOZONESCum.GrandSumSampleSumofMinusNo.PerlPerm.IPerm.Cum.Su.mBW~(md)(md)~~(md2)R'0101,19213,49313,6000252621,14019,89212,8000.35327633886424315,25604140478724315,2860513961758510215,273061567734291,11815,146073421,1158735,64610,8300.698871,202036,12010,7710.70*901,202016,05313,2800.171001,20201,202(GrandSum)891Eq.1:B=~[(10)'+(1192)'_(1202)']1191013,493.Eq.2:W=M(lOr+(52)'+(276),+(l40),+(139)'(10)'(1192)']+(342)'+(87)'+(0)'+(0)'----~-1913,600.Eq.3:R=13,493-13,600=-000813,493.,whichisreplacedby0;andtheotherlinesarecomputedinthesameway.Thebestdivisionintotwozonesoccursafterthepermeabilityvalue87asshowninTableA-2.Sincewehavealreadyseparatedtheoriginaldataintotwogroups,ourproblemisnowthatofseparatingeitherGroup1orGroup2intotwoadditionalgroups.TableA-3givesdetailsofthecalculations.Thesecondpointofdivision(thefirstwasafterthevalue87)occursafterthevalue52,anddefinesourdataasasetofthreegroups,orzones.Theresultsofthecalculationsforthedivisionofthedataintofourzones(notgiven)indicatethatthelargestfour-zoneindex,0.79,issmallerthanthethree-zoneindexof0.81.Therefore,thewellisbetterdescribedasthreezones.Theotherwellsweredividedintozonesinthesamemanner,althoughthedetailsarenotgiveninthispaper.ThefinalresultsofourworkareshowninTableA-4,wheretheasterisksmarkthedivisionsintozones.TheidentificationforthezonesinTableA-4givesthezonenumberwithinthespecifiedwell.Thus,Zone(2,11)isZone2inWell11.STEPII-CORRELATIONOFWELLZONESOnthebasisofprimarypressureandproduction-ratehistoriesforthefourwells,thereiseveryreasontobelievetheformationiscontinuousovertheareaunderstudy.Thesecondstepofthestatisticalzonationtechniqueisthecorrelationofzonesinonewellwithzonesinadjacentwellstodeterminethesizeandlocationofthecontinuouszoneswheretheyintercepttheareas.Eachofthefourwellshasbeendividedintozones.TABLEA·3-DIVISIONOFDATAINTOTHREEZONESSampleGrandSumNo.PerPerm.Cum.SumIMinusCum.BWGro'lJp{md)(md)Sum(md)(md')(md')RGroup1110101,19229,3009,0980.68252621,14037,0216,8930.81.327633886421,45011,3410.47414047872421,84211,2290.48513961758522,86610,9370.52615677342923,56410,7370.5473421,1158720,34611,6570.428871,2020Group20.00018,06012,3100.3120.000H92TABLEA-4-FINALZONATIONOFRESERVOIRPERMEABILITYDATAWellNo.8We'!No.11We'lNo.18WellNo.37Perm.Perm.Perm.Perm.Zone(md)lone(md)ZOl"e(md)Zone(md)11102034(1,8)(1,11)(1,18){1,37)2752406720157276190234140(2,18)197390139146(2,37)90(2,11)186192156(2,8)3422188753424.8331200.015845213160.01411713,i1)(3,18)0.0(3,37)0.0208427202826990.0611210.0430.0(3.8)0.0887.41490.0Eq.4isappliedtothezonedatainTableA-4.Thefollowingissuggestedasaconvenientandefficientman-nerofapplyingEq.4.1.Rankwell-zonemeansinorderofdecreasingmag-nitude,asfollows.Zone(2,8)(2,37)(2,11)(2,18)(3,8)(3,37)(1,37)(1,11)(1,18(3.18)(1,8)(3,11)WELLNO.·I[NumberofZoneMean(md)DatainZone19210191219061682668428403312302191219202WELLNO.8WELLNO.37WELLNO.18"}------c")-----,-ZONEIZONEIIZONEIIIZONEI..~",...',-c·~.:.'.ZONEm>,.:.:.......,/.F~_--.:'="·:.:5:.':AVERAGETHICKNESS(FT)AVERAGEPERMEABILITY(MO)'2.15,07.53318936FIG.A·2-CROSS-SECTIONSHOWINGFINALZONATION.JOURNALOFPETROLEUMTECHNOLOGY2.CalculateEq.2usingallpermeabilitydataintheentirereservoir.W=59~12[(11)"+(27)"+(157)"+...+(61)"(3~)2_(l~~7)'_..._(3~5r]=3,964(md").3.CalculatethestandarddeviationfromStep2.s=y3,964=62.96(md).4.Selectz-values'fora95percentprobabilitylevel(zv,p)'p234567S910II12Z47,p2.S63.013.103.173.223.273.303.333.353.373.395.Multiplythez-valuesinStep4bythestandarddeviationinStep3,e.g.,F'p=SZ47,P'P234567S910II12F'pISO19019520020320620S2102112122136.Testforsignificantdifferencesamongwell-zonemeans.Firstthelargestmeaniscomparedwitheachofthesmallermeans.InorderforthemeansofZones(2,8)and(3,11)tobesignificantlydifferent,(192-0)J2(10)(2)=350"10+2mustexceedF'12=213.Itdoes;therefore,thewellZones(2,8)and(3,11)representedbythemeans192and0aresignificantlydifferent.InorderforthemeansofZones(2,8)and(1,8)tobesignificantlydifferent,(192-19)J2(10)(2)=316"10+2AUGUST,1962mustexceedF'n=212.Sinceitdoes,thewellzones(orthemeans)aresignificantlydifferent.Completingthetest,wecomparethemeanofZone(2,8)withthemeanoftheotherzonesuntilthedifferenceisnotsignificant.WefindthecomparisonsaresignificantuntilZone(2,18)isreached.Atthispoint,webegintocomparethenextlargestmeanwiththesmallermeanasfollows.(191-0)~2~2~(~)=191(1.414)=270>F'1landthusissignificant,andsoforth,untilanonsignificantdifference,(191-168)~2~2~(~)=23(1.414)=32
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