SOCIETYOFPETROLEUMENGINEERSOFAlME 6200NorthCentralExpressway Dallas,Texas75206 PAPER NUMBERSPE4529 THISISA PREPRINT--- SUBJECTTOCORRECTION TheIsochronalTesti.ngofOiIWells By M.J.Fetkovich,MemberAIME,PhillipsPetroleumCo. Copyright1973 American In8titute of Mining, Metallurgical, and Petroleum Engineer8, Inc. Thispaperwaspreparedforthe48thAnnualFallMeetingoftheSocietyofPetroleum EngineersofAIME,tobeheldinLasVegas,Nev.,Sept.30-0ct.3,1973.Permissiontocopyis restrictedtoanabstractofnotmorethan300words.Illustrationsmaynotbecopied.The abstractshouldcontainconspicuousacknowledgmentofwhereandbywhomthepaperispresented. PublicationelsewhereafterpublicationintheJOURNALOFPETROLEUMTECHNOLQGYortheSOCIETYOF PETROLEUMENGINEERSJOURNALisusuallygranteduponrequesttotheEditoroftheappropriate journalprovidedagreementtogivepropercreditismade. Discussionofthispaperisinvited.Threecopiesofanydiscussionshouldbesenttothe SocietyofPetroleumEngineersoffice.Suchdiscussionmaybepresentedattheabovemeetingand, withthepaper,maybeconsideredforpublicationinoneofthetwoSPEmagazines. ABSTRACT Thispaperpresentstheresultsand me.thodeofanalyzingisochronalandflowafter flowmultipointback-preseuretestsconducted onoil wells.Testswereconductedin reser-voirswithpermeabilitiesrangingfrom6MDto > 1000MD.Reservoirsin whichoil well multipointback-pressuretestswereobtained rangedfromhighlyundersaturated,tosaturated atinitialreservoirpressure,toapartially depletedfieldwithagassaturationexisting abovethecritical.Eachofthesethree reservoirfluidstatescanresultindifferent interpretationmethods.Sack-pressuretests wereruntopseudo-steadystateinthefield wherethesaturation wasabovethecritical gassaturation. Inall cases,oil wellback-pressure curveswerefoundtofollowthesamegeneral formasthatusedtoexpresstherate-pressure relationshipofagaswell: JI(- 22)n qo=0PR- Pwf Fromsome40oil wellback-pressuretests examined,theexponentnwasfoundtolie between0.568and1.000,verynearthelimits commonlyacceptedfor"aswellReferencesandillustrationsatendofpaper. curves.Flowpointalignmenttoestablishan oil well curveonthecustomary logqvs.log6(p)plotisconsideredtobe asg08dasthatobtainedongaswellback-preseuretests. Thispaperdemonstratesthatgaswellsand oil wellsbehaveverysimilarlyandshouldbe testedandanalyzedusingthesamebasicflow equations. INTRODUCTION Multipointback-pressuretestingofgaswells isanacceptedprocedureforestablishing gas well'sperformancecurve.Flowafterflowand isochrona12 testingarethetwobasicmethods commonlyused.Inhighpermeabilityreservoirs, eithermethodcanbeemployed.Inlowper-meabilityreservoirs,theIsochronal methodoftestingeliminatesthetransient effectsthatcanseverelydistorttheresults obtainedfromaflowafterflowtest.Methods foranalyzingandcalculatinggaswell performancecurveshavebeenthesubjectof numerousinvestigations.Thebulkofthese investigationshaveexaminednon-Daroyflow behavior,theprimaryreasonthatmultipoint testsareconducted. Multipointtestingofoil wellsisnotnow acurrentpractice.Asearlyas1930,however, 2THEISOCHRONALTESTINGOFOILWELLS SPE4529 T.V.Moore3reportedtheresultsof anoil wellmultipointtestconductedontheHumble SmithA-2intheYatesField.Thepurposeof theback-pressuretest wastodemonstratea methodofestablishingawell ISopenflow potential withoutproducingthewell wideopen. Theneedforestablishinganaccurate performancecurveforanoil wellis as importantasdeterminingoneforagaswell. Inthesearchfornewoil,theindustryisturn-ingtoremoteareassuchastheArcticand offshore.Critical questionsofwhetherto develop,andif so,howtodevelopafield hingeontheabilitytoaccuratelypredicta wellisdeliverability.Often,becauseof equipmentlimitations,theratesofproduction obtainedduringdrillstemtestingaremuchless thanthoseplannedforfulldevelopment. Thetraditional methodforpredicting productionratesanddrawdownsforoil wells hasbeenbasedontheconceptoftheproductiv-ity index(PI),whichhasbeenusedintheoil industryformanyyears.Theusualformof theequation isvalidonlyforsystemsproducinganideal homogeneousliquidobeyingDarcy ISlaw.This conditionnormallyholdsforoil wellswhenthe oil isundersaturatedthroughouttheproducing formation.It haslongbeenrecognizedthat inreservoirsexistingator belowthebubble-pointpressure,producingwellsdonotfollow thissimpleequation.Actualfieldtests indicatethatoilflowratesobtainedat increasingdrawdownsdeclinemuchfasterthan wouldbepredictedbyEq.1. EvingerandMuskat4 firstderiveda theoreticalproductivityindexforsteadystate radialflowinanattempttoaccountforthe observednon-linearflowbehaviorof oil wells andarrivedatthefollowingequation q=7.08kh oIn(::) wheref(p)= k ro jPe f(p)dp Pwf

o0 (2) CalculationsusingEq.2basedontypical reservoirandfluidpropertiesindicatedthat PIatafixedreservoirpressurePe(asdefined fromEq.1)decreaseswithincreasingdrawdown. InacomputerstudybyVoge15,results ba.sedontwo-phaseflowtheorywerepresented toindicatethatasingleempiricalinflow performancerelationship(IPR)equationmight bevalidformostsolution-gasdrivereservoirs. Hefoundthatasingledimensionless1PRequation approximatelyheldforseveralhypothetical solution-gasdrivereservoirsevenwhenusing awiderangeofoilPVTpropertiesandreservoir relativepermeabilitycurves.Thefactthathis studycoveredawiderangeoffluidproperties andrelativepermeabilitycurvestoobtaina singlereferencecurve,cannotbeover emphasized.Vogelthenproposedthathis equationbeusedtotaketheplaceofthelinear productivityindexrelationshipforsolution-gasdrivereservoirswhenthereservoirpressure isator belowthebubble-pointpressure. Theproposedempiricalreferenceequation (1PR)in dimensionlessformwasgivenas qo=1_0.200.00 . (3) (q)max )o A comparisonwasmadeof1PR ISforliquid flow,gasflow(n=l)andtwo-phaseflow(his referencecurve)onadimensionlessbasis, (Fig.1).AsisevidentfromFig.1the positionofthetwo-phasereferencecurve relativetoliquidandgasflowindicatesthat oil wellsproducingasif in asolution-gas drivereservoirshouldactuallymore likeagaswell,i.e.,(P.a2- Pwf)VB.qo shouldplotasastraightlineonlog-logpaper withaslope(n)nearunity. Thispaperpresentstheresultsofmulti-pointback-pressureteststakenatasingle reservoirpressurelevelTheseresults showthattheperformancecurveforanoil well canbeexpressedbyamoregeneralandfamiliar equationsimilartothatusedforgaswells, q= JI _Pf2)n.(4) ooltw Reservoirsin whichoil wellmultipoint back-pressuretestswereobtainedrangedfrom highlyundersaturated,tosaturatedatinitial reservoirpressure,toapartiallydepleted fieldwithagassaturationexistingabovethe critical(equilibrium)gassaturation.Equation 4wasfoundtobevalidfortestsconductedin allthreereservoirfluidstates,evenforthe conditionswhereflowingpressureswerewell abovethebubble-pointpressure.Permeabilities ofthereservoirsrangedfrom6to>1000 millidarcys.Flowpointalignmenttoestablish anoil wellback-pressurecurveonthecustomary log va.log6(p2)wasfoundtobeasgoodas thatobtainedongaswellback-pressuretests. BASICEQUATIONSANDPRESSUREFUNCTIONS ThebasicflowequationgivenbyEvinger andMuskat4 forsteady-stateflow,applicable toeitheroil orgasflow,is SPEL.'529M.J.FETKOVICH 3 (2) wheref(p)canbeanyfunctionofpressure. Usingthetypicalpressurefunctiondepicted inFig.2it isobviousthatwecanevaluate thetotalintegralintwopartsandwrite + l" Ib Forflowintheregionwherethepressures areabovethebubblepointpressureif we assumek= 1(neglectingthepressure dependenOpermeabilitytermforsimplicityof presentationonly)andtreat(uB)evaluatedat theaveragepressure write qo=[In:hS] dp (Pe- Ib)] +(6) Exceptfortheadditionofthenecessaryskin (discussed.laterinthepaper)Eq.6 isidenticaltothatderivedbyHandy21. Figure2illustratesaplotofllu Basa functionofpressureforanoil reservoir.Also,drawnonthisfigureisa dashedlinerepresentingtheeffectofrelative permeability(k)ondrawdownsbelowthe bubble-pointIt is assumedfor purposesofdemonstrationthatkI(uB)is linearandits interoeptis 0atrepr3sgure. Thesimplifyingassumptionofthe0intercept for kro/(uoB)approximatelydefinesVogel'sIPR curveanaexactlydefinesEq.4 whenn=l.)Also, drawnonFig.2isahypotheticalpressure functionkro/(uoBo)representedasaconstant forall pressures.It isolearthataconstant valueofkI(uB)overtheentirepressure ro00 drawdownrangerequiredtoaconstant productivityindex(PI). Figure3illustratesplotsofl/(u B)for twohighpressuregasreservoirs.gAwas obtainedfromcalculationsusingthereservoir gasanalysisandstandardcorrelationsofZand uasfunctionsofcriticalpressureandtem- CurveB wasobtaineddirectlyfroma PVTstudy.OnestrikingfeatureofcurveA is thefactthatit resemblesthatofanunder-saturatedoilreservoirwithanapparentor pseudobubble-pointpressurenear2500psia, thenormalinflectionpointofaZcurve.A furtherobservationthatcanbemadefrom curvesA andBisthataregionexistswhere agaswellcanbeconsideredtobehaveasa liquid,i.e.,l/(uB)isnearlyconstantor onlyslightly withpressureasis theoaseforthepressurefunctionofan undersaturatedoilreservoirabovethebubble-pointpressure. Fortheregionwherethepressure isaconstant,ornearlyso,wecanimmediately writeuponintegrationofEq.2thewellknown steady-statesinglephaseflowequation: _7.0Skh (Pe- Pwf) Ire)'J(uB) In\- +S rw Notethatthisequationwouldapproximately holdforgaswellsrepresentedbycurvesA and BinFig.3overaconsiderablerangeof pressuredrawdowns.qwillthenbepro-portionalto6pinsteaBof6(p2).This,in factwasfoundtobethecaseforisochronal testsconductedontwowellsinareservoir withfluidpropertiesrepresentedbycurveB. Nowconsideringtheentirepressurefunction fromPeto0,foreithertheoil orgascurves, (thedashedlineinFig.2)wenotethatf(p) canberepresentedapproximatelybytwosepa-ratestraightlinesegments.Theapproximate flowequationthen,overthetotalpressure interval,canbewrittenas:(SeeAppendix) q=7.0Skh rIn(:e )+s1Ut!) LwJPe'pt, "2 +(8) orq= J'(pt, 2 - Pwf2)+ J(Pe- pt,)..(SA) Fordrawdownsbothaboveandbelowthe bubble-pointpressure,aback-pressurecurve plotwillappearastwolinesegments,with theintersectionyieldinganapproximatevalue forthereservoirbubble-pointpressure.This thenoffersanapproachfordetermininga reservoir'sbubble-pointpressurefroman isochronaltest.Foranisoohronaltest,a constantreservoirradiusofinvestigationis obtainedforeachflow-aninsituconstant volumecell. Ifthedegreeofundersaturationisslight, thetwolinesegmentsmaynotbedefinable. Unstableflowconditionsinthetubingatthe THEISOCHRONALTESTINGOFOILWEllSSPE4529 lowflowratesnecessarytodefinethesingle-phaseflowconditionsmaypreoludedefiningtwo straightlines.Further,aswillbedemon-stratedlater,non-Darcyflowoanexisteven whenallflowingpressuresareabovethebubble-pointpressure.Conceivablythenthiscould leadtoeventhreelinesegments. Forthecaseofall drawdownsbelowthe bubble-pointpressureJ(Pe- Ph)isaconstant, whiletheremainingtermvariesnon-linearly withflowingpressure,Pwf. Thecompositeeffectresultsinanequation oftheform ( 22)n q= CPe- Pwf AsdecreasestothepressurePh,1.0and C->J'suchthatfortheoil wellcase,only thetwo-phaseflowtermremains.Wethusobtain thebasicequationsuggestedfromVogel's resultsforPe7.08kh[m(Pi)- m(pwf}] q wherem (p)canalso apressure dependentpermeability, m(p) I o p kro(S,p)dp uB (13) .(14) (Theeffectofapressuredependentpermeability couldreadilybedisplayedinFigs.2and3.) Equation12or13thenshouldbeappli-cableforanalyzingbothoil wellandgaswell back-pressuretests _J'(22)1.0 qo- 0Pe- Pwf (10)RATEANDTIMEDEPENDENTSKIN.s(g.t) A significantconclusiontobedrawnfromEq.9 isthatagaswelloranoil wellcanhave2a slopelessthan1.0onalogqvs.log6(p) plot withoutnon-Darcyflowexisting.The slope(n)inthiscaseisstrictlyaresultof theshapeofthewellspressurefunotion.This possibility,foragaswell,was6recognized andreportedbyRowanandClegg Eq.10mustbefurthergeneralizedwith anexponent(n)inlightofresultsobtained frommultipointback-pressuretestsconducted onoil wellsforbothsingle-phaseandtwo-phaseflowto q= J' o0 ( 22)n Pe- Pwf (11) Eq.11isidenticalinformtothegaswell back-pressureequation.Forconstantrate transientgasflow,thegaswellback-pressure equationisusuallyexpressedby7,8. 7.08kh(Pi- Pwf) . q(UB) +s+Dq (12. Otherthanfortheuniquefluidpropertycases discussedabove,orapressuredependentper-meabilityeffect,thenon-Daroyflowtermin Eq.12isrequiredtoobtainanexponent(n) lessthan1.0. Intermsofam(p) Slopesmuchlessthan1wereconsistently obtainedfromisochronaltestsconductedon oil wellsinsaturatedreservoirs.Forunder-saturatedreservoirs,theshapeofthepressure functionwasshowntobecapableofaccounting forslopeslessthan1.SinceVogel'swork basedontwo-phaseflowtheoryindicatedback-pressurecurveslopesshouldbeunityoreven greater,anearwellboreeffect wassuspected. (Allof Vogel'sresultsshowthefirstcalculatee IPRcurveafter 0.1%of originaloil-in-place isrecovered.Theeffectofinitialgassat-urationbuild-uparoundthewellboremayhavebeenpresentinhisresults.)Handy studiedtheadverseeffectonPIoftwo-phase flowinthevicinityofthewellboreforunder-saturatedoils.Muskat12,13presentedasimple approachtostudytheeffectoftwo-phaseflow aboutthewellboreforagascondensatewell thatcouldbeappliedtoasaturatedorunder-saturatedgascondensateoroil well. s(g,t)FOROONDENSATEWELLS Muskat'sequationtocalculatetherateof changeofliquidsaturationtakingplaceabout thewellboreforaproducingcondensatewell is: dSqQ dt=2f1rh!dr dc dp (15) Saturationisassumedtobuilduponlytothe limitingequilibriumliquidsaturation;its radiusthenexpandingwithtime.Forasteady statepressuredistribution,andsaturation Sequalto0att=O,wecanobtainanequation SPE4529M.J.FETKOVICH 5 intermsoftheapproximateradiusofthe equilibriumtwo-phaseflowregion.In engineeringunitsit is r a 2 "'".1135q/ uZyt.(16) h2 kpSclh whereY isexpressedasreservoircubicfeetof condensateaccumulationinthereservoirper Msafoffull wellstreamgasproducedperpsi, Y canbecalculatedusingtheretrograde liquidvolumedatadeterminedfromPITTstudies. ThetermSclhistheariticalhydrocarbonliquid saturationtoreachequilibrium,or mobilliquid saturation.TheotherpertinentunitsareMsafd cps.,days,ft.andDarcy. Thedefinitionofskineffect(s)interms oftheradiusofanalteredzoner(equilib-riumtwo-phaseflowregion),andtRereduced. permeabilityofthealteredzoneka,canbe expressedas2 s:::(k- ka)In (ra)(17) 2kr aw SubstitutingEq.16into17weobtain s(q, t) (18) Equation18definesarateandtimedepen-dentskintermthatcangivetheappearanceof non-Darcyflow.Theequation,although approximate,givesasimpleanalyticalexpres-sion withwhichtoestimatetheeffectsoftwo_ phaseflowinthevicinityofthewellbore. Thesignificanceofthiseffectinwellshasbeendemonstratedbyothers.14,,16, 17.Eq.18hasbeenusedtosuccessfully analyzetheresultsobtainedfromisochronal testsoncondensatewells.A significantpor-tionoftheskin wasattributedtos(q,t). s(g.t)FOROILWELLS 1819 Inthestudiesof Westetal,Perr1ne andWeller20,ananalogousbehavioraroundthe wellborehasbeenshowntoexistinanoil well.Underconstantrateproductionfor initiallysaturatedsolution-gasdrive reservoirs,theirresultsshowthatthegas saturationquicklybuildsuptotheequilibrium gassaturation(criticalgas)andremains constantatitsequilibriumvalue.Its'radius increaseswithtimeuntilthewellsdrainage volumeisabovethecriticalgassaturation. (See 4)This saturationbuild-upin thevicinityofthewellboreiscommonly referredtoas"gasblockllThecorresronding oilpermeabilityredUctiohinthisregionis thereforeconstant,withitsradiusincreasing withtime.Thisdamagedzonewithinwhichthe relativepermeabilityhasbeenreducedhasbeen referredtoasapseudo-skinbyWeller. Utilizing Eq.18 withtheappropreiatevariable substitution,therateandtimedependentskin s(q,t)foranoil wellis s(q,t)=2lre,aIn qoXt (19) (k- k)[2j hkScgrw whereX isexpressedasreservoircubicfeetof gasevolvedinthereservoirperstocktank barrelofoil producedperpsi, Xis readilyobtainedfromastandardPVTstudyusing theliberatedgasdataRLasafunctionof pressure.ScgistheeqUilibriumorcritical gassaturation,fractionofpore Other pertinentunitsareSTKBOPD,cps,DAY,FT, DARCYandRESBBL/STKBBL. Theresultsof Westetal werefirstused todeterminewhetherEq.19wouldreasonably predicttheradiusoftheIIpseudo-skinllfor timesbeforeboundaryeffectsbecameSignificant Usingthebasicdatagivenintheirpaperand Eq.19acalculatedr= 1.6FTversustheir 1.5FTwasobtainedat2.21days,andra=4.6 versustheir6.0FTat16.8days Eqi.180"- 19areapplicabletoinitially saturatedandpartiallyundersaturatedreser-voirs.Onceanoil well'sdrainagevolume exceedstheequilibriumgassaturationEq.19 isnolongerapplicable.Forcondensatewells, Eq.18 willapplyforamuchlongerperiodof time,atleastuntilrevaporizationbeginsto takeplace.Thenra willbegintorecede. Onlyinthecaseofundersaturatedreser-voirs,wecouldassumethatthetwo-phaseregion isattheequilibriumgassaturationandexists outtowherethepressureisequaltothebubble pointpressure.Thissimplerapproach,developec byHandy2lforwellsproducingfromunder-saturatedreservoirs,leadstothemaximum reductionofPIwhichcouldbeexpectedfroma gassaturationbuild-uparoundawellproducing withaflowingpressurebelowthebubble-point pressure.Byanalogy,thesameapproachcould beusedfortreatingundersaturatedgascon-densatewells. Forcompletenessthen,Eqs.12and13shoulc bewrittentoincludearateandtimedependent skin,s(q,t).Wewouldthenhave 7.08yill(Pi-Pwr)=In q(UB) +s+s(q,t)+ Dq (20) 6THEISOCHRONALTESTINGOFOILWELLS SPE4529 and - m q (p,)1(I wf .)14.23kit In2+ i!(uct)irw s+s(q,t)+Dq (21) 22 AfterRamey,wecandefine s'=s+Dq(22) and sIr=S+s(q, t)+Dq (23) WELLTESTRESULTS Thebasicresultsobtainedfromisochronal back-pressuretestsandflowafterflowmulti-pointtestsconductedonoil wellsare summarizedinTables1and2. Reservoirfluidstatesin whichmultipoint welltestswereobtainedare,inchronological order, 1.Gassaturationexistedthroughoutthe reservoirabovethecriticalorequilibrium gassaturation. 2.Undersaturatedreservoirwithflowing pressuresobtainedbothaboveandbelow thebubble-pointpressure. 3.Saturatedreservoirswiththereservoir pressureatorverynearthebubble-point pressure. 4.Undersaturatedreservoirwithall flowingpressuresabovethebubble-point pressure. GASSAWRATIONABOVEEQUILIBRIUM Stabilizedflowafterflowmultipoint back-pressuretestswereavailableon16wells producingfromasolution-gasdrivecarbonate reservoir,FieldA.Reservoirconditionswere idealfortestingthehypothesisthatqovs. (- 22) PR- Pwfwouldplotasastraightlineon log-loggraphpaperwithaslope(n)of1.The reservoirvariablesinthisfieldclosely approximatedthoseusedbyVogelinhisstudy, (SeeTable3).Averagegassaturationinthe reservoiratthetimethetestswereconducted wasestimatedtobebetween10or12percent. Producinggas-oilratioswhencomparedtothe initialsolutiongas-oilratioof684SCF/BBL indicatesthatthereservoirwaswellabove abovetheequilibrium(criticalgas)saturation atthetimethetestswereconducted.Gas-oil ratiosincreasedonlymoderatelyatincreasing drawdownsformosttests. Althoughtheunitslopedidpredominate, fourwellsexhibitedback-pressurecurveslopes muchlessthan1.A slopelessthan1results inanevenmorerapiddeclineinrateqwith drawdownthanwouldbepredictedfromVogel's IPRequation. ThetestonWell6,FieldA(?ig.5) consistedofsevenindividualflows,eachto apparentstabilization.Thefirstfourflow rateswereruninanormalincreasingsequence. Followingthefourthflowat229BOPD,therate wasreducedto93BOPDthenagainfollowedby anincreasingsequenceofflows.Allpoints essentiallyfellonthesameline,indicating thattransienteffectswerenotthecauseof thedeviationfromthelinearrelationship predbtedbytheproductivityindexconcept. Notethattheflowpointsdefineaperformance curvewithaslopeof1almosttoitsabsolute openflowpotential(AOFP).Table1showsthat forall wellstestedinthisfield,themaximum flowrate wasveryneartheextrapolatedabsolute openflowpotential.Intheotherfieldsin whichmultipointtestswereconducted,equipment limitationprecludeddefiningtheentirecurve, requiringagreaterdegreeofextrapolationto AOFP. WellNo.3,FieldA,(Fig.6)illustrates themostsignificantresultofthisfirstgroup oftests.Withanexcellentalignmentoffive stabilizedflows,theslopeoftheback-pressure curveis0.648.Theresultsobtainedfromthis testfirstsuggestedthepossibleexistanc8of thesamelowerlimitoftheexponent(n)as existsforgaswells(n= 0.500),andafloweffect. WellNo.14,FieldA,(Fig.7)exhibited themaximumincreaseingas-oilratiowith increasingdrawdownofallthewellstested. Evenwiththegas-oilratioincreasingwithrate, theslopenoftheperformancecurvewas1.0. Inanattempttoutilizetheoil wellback-pressuretestingmethodtomoreaccuratelypre-dictfulldevelopmentwellperformancefrom wildcatwelltests,anisochronaltestprogram wasinitiated.Thefirstknownoil well isochronaltest wasconductedonApril14,1970 onthePhillipsEkofisk2/4-2Xwell.Surprising resultswereobtainedfromthesefirsttests. Twostraightlineswereobtainedwhenalogq vslog plotwasprepared.Figure 8illustratestheresultsobtainedfroma6 hourisochronaltestconductedonzone2. Handy1s21,workledtotheconclusionthat thetwostraightlineswerearesultofthe reservoirbeingundersaturated,withtheinter-sectionpointindicatingtheapparentreservoir bubble-pointpressure.Usingthefirsttwo SPE4.529M.J.FETKOVICH 7 flowratesandtheconstantPIapproach,an apparentabsoluteopenflowpotentialof13,000 BOPDisindicated.Thetruepotentialestab-lishedbyextrapolationofdrawdowndatabelow thebubble-pointpressureis5200BOPD. Calculatedpermeabilityfrombuild-updata followingthefirstsinglephaseflowwas6.1 MDwithaskins=O.Forflowsatpressure drawdownsbelowthebubble-pointpressure,a ratedependentskin wasTherate dependentskinsextrapolatedtoaskinof 0at thepointsinglephaseflowended,q 2100 BOPD,asshouldbeexpected,(See9). Single-phaseskinsof-4arenormallyobtained fromtestsfollowingacidstimulations.This favorableresponseusuallyprecludedobtaining drawdownsbelowthebUbble-pointpressureafter acidbecauseofequipmentlimitations.Asa result,noafteracidisochronaltestshave beenobtainedwhichcoulddemonstratewhether thenatureoftheperformance'curveissubstan-tially differentthanthatobtainedbeforeacid. Isoohronaltestsconductedontwootherzones inthiswell,beforestimulation,yielded similarresults. StartingwithEq.8A,weoanoutlinethe procedureusedtocalculatethebubble-point pressurefromthepre-acidtest If wethendefine and q(2)= JI o q(l)=JI o 22 (rb- Pwf) (Pe-%) thenqo=q(2)+q(1) (8A) (24) (26) (Nophysicalsignificanceshouldbe attachedtoq(l)orq(2)sinceit isobvious thatforthesteadystateassumptionuponwhich it wasderived,thetotalqmustbeflowing throughbothregions.)0 Whencombinedtwo-phaseandsinglephase flowareoccuringinawell q(l)= CONSTANT= J(p-n)(25) o'Pe'Ibe'0 thereforeq(2)= q(measured)- q(1) o JI(22) o%- Pwf (27) Withthecorrectvalueofabubble-pointpre8-sure,Pb'aplotofq(2)vs.(%2_PWf2)should plotastraightlineoneithercartesianora log-logplot.Onalog-logplot,theslopeis 1.0andtheinterceptJI. o The1.0slopewasassumedforthetwo-phase termatthisstageofdevelopmentbecauseofthe computerresultsobtainedbyVogelandthe resultsobtainedfromtestsin FieldA.However, thefactthatslopeslessthan1.0areindicated fromothertestswheretwo-phaseflowexisted inthereservoir,suggeststhemoregeneralform of Eq.8A.tobe q=JI(n2_Pf2)n+ J(p-n)..(28) o0'0w0e'o A trialanderrorcalculationassuming variousvaluesofPbwasperformeduntila slopeof1was (SeeFig.10).This resultedinacalculatedbubble-pointpressureof 5874psia.A bubble-pointpressureof5885 psiawasdeterminedfromaPVTstudyofthe reservoirfluidobtainedfromthiswell. A simplegraphical ofthebubble-pointpressurefromtheapparentintersection pointisprobablyadequatebecauseofthe unoertaintiesintroducedbyn,theexponentof thetwo-phaseterm,beingavariable.Once thetruebubble-pointpressureisdetermined fromPV'I'data,ncanbedirectlycalculated. SATURATEDRESERVOIRS Mostofthereservoirsin FieldsCthru H aresaturatedatinitialreservoirpressure. Thereservoirsareverysimilarinnatureat correspondingdepthssincethefieldsarein closeproximitytoeachother.Allreservoirs arerelativelycleanTertiarysandstonesranging indepthoffrom7800to11200feet.Perme-abilitiesdeterminedfrombuild-uptestsranged from130to2500MDwithnetpaysrangingfrom 20to180feetinthickness.Typicalporosities are22percentwithwatersaturationsofaround 30percent.Relativepermeabilitymeasurements exhibitedcriticalgassaturationsrangingfrom 7to13percent. Humpingeffects,wellborestorage,flat pressurebuild-upcurvesandtheshortduration ofthebuild-upsmadethedeterminationof permeabilitiesdifficultonseveralwells. Forthosewellsnothavingpermeabilitieslisted inthetables,itsorderofmagnitudeis reflectedbythewellsA.OFP.A summaryofall theisochronaltestresultsobtainedappearin Table2. Thestandardisochronaltestinthesefields consistedofafourhourflowfollowedbyafour hourshut-in.aflowafterflow testwasalsoconducted.Increasingand decreasingsequencesofflowswereperformedon mostteststocheckreproductability.Because oftheratherhighpermeabilitiesinthese reservoirs,flowafterflowtestsoften duplicatedtheisochronaltestperformance 8THEISOCHRONALTESTINGOFOILWELLSSPE4529 ourve.Performanoecurveslopesobtainedfrom thesetestsareseentorangefrom0.568to 0.875.Notonesinglewellexhibitedthe1 slopethatwassopredominantinFieldA. Severalofthewelltestperformanceourves obtainedininitiallysaturatedreservoirsare showninFigs.11- 21.Ingeneral,flow pointalignmenttoestablishanoil wells performanoeourveisasgoodasthatobtained fromgaswellback-pressuretests.Shut-in pressurerecoverybetweenisochronalflowson thesetestsissufficienttoestablishtrue isochronalconditions.Gas-oilratiovariations areconsideredtobemoreafunctionofsep-aratorpressurethanreservoirdrawdown pressureeffeots.Themostsignificantobser-vationtobemadefromthesetestsisthatflow afterflowdatafallonthesameperformance curveasthatestablishedbyisochronaldata points.Thelowestpermeabilityofthisgroup of wellsis130MD.TestresultsforWellNo. 3-C,Field0,(Fig.12andTable4)demonstrate theflowafterflowandisochronaltestper-formancecurvereproduoabilitybytwoseparate testsconductedoneweekapart. ThetestonWellNo.5-CinFieldD was selectedtoapplyEq.20toanalyzethewell performancedata.Thefourhourisochronalwell performancecurvewasestablishedbytwo separatetestssix monthsapart.Nearly100 psireservoirpressuredropoccuredbetween thesetwotests.Nodetectableshiftinthe positionofthewell'sperformancecurvewas noted.WellNo.5-Cwastheonlyoneofthe saturatedreservoirwellsthathadafully perforatedinterval,thuseliminatingyet anothervariable,partialpenetrationeffects. Further,thepermeabilitycalculatedfrombuild-updatawasconsistentwithmeasuredooreper-meabilitiesforthis well.Tables5and6 summarizethereservoirandtestdataused inthecalculationsandtheresultsobtained. Theratedependentskinterms(q,t),forthis well,wasfoundtobeinsignifioantateventhe highestflowrateof2308BOPD.Boths'and stlwereplottedasafunotionof%.Ineither case,alinecanbedrawnthrutheplotted pointstoq= 0yeildingaformationskins= O. Non-Darcyflowappearstobesignificantfor thiswell. Theisochronalperformancecurveobtained onWellNo.7-e,FieldD,(Fig.16),exhibits thesteepestslopeofallthetestsconducted inasaturatedreservoir.Anyoftheflow rateswouldbereasonableforanormalsingle flowdrillstemtest.A comparisonofcalculated absoluteopenflowpotential(AOFP)ismade usingthePImethodandVogel'sIPRmethodfor eachoftheflowrates.Themaximumerrorin AOFPisofcourseobtainedwiththelowestflow rate- AOFP= 57,200BOPDPImethod,31,990 BOPDIPRmethodandactualisoohronalAOFP= 7250BOPD.EventhoughtheerrorinAOFP,using thePIorIPRmethodsisreducedwhendetermined atthehighestflowrate,theerrorin evaluatingskinandflowefficiencywillbe increased. WellNo.8-e,FieldD,(Fig.17andTable7) demonstratesthechangeinthewellsperformance curveasaresultofincreasingtheperforated intervalfrom20Ft.to60Ft.;netpayis182 Ft.Thewellspotentialn9arlydoubledand theslopeoftheperformancecurveincreased onlyslightly. UNDERSAWRATEDRESERVOIR(pwrIb) Ofallthetestsconducted,the mostsurprisingresults 'werethoseobtained onWellsl-aand2-binFieldG(Figs.22and 24respectively).Withallflowingpressures wellabovethereservoirbubble-pointpressure, (single-phaseliquidflow),slopesof0.813 and0.712wereobtainedfromalogqvs.log 1I(p2)plot. Conolusiveevidenceoftheocourenceof non-Daroyflowinanoil wellisdemonstrated fromadetailedanalysisoftheisochronal testdataobtainedonWellNo.1-a.PVT studies ontwobottom-holesamples andarecombinationofsurfacesamplesbubble-pointpressuresof4495,4756and4785 psia Thelowestflowingpressure obtainedonthistest was5669psiaataflow rateof2973STKBOPD.Netpayforthiswell is25feetwithaperforatedintervalof10 feet. TheisochronalperformancecurveforWell No.l-a(Fig.22)indicatesaslopenof0.813 withanexcellentalignmentof8separateflow rates.Threedecreasingsequenoeflowswere followedbyfivemoreflows.Table 8summarizesthedataobtainedforeachflow rate.Thefactthatalignmentwasobtained followingrepeatedflowsandshut-ins,and flowreversalstendstoindicatethatapressure dependentpermeabilitywouldnotaccountfor thenon-linearflowbehaviour23Thenormal hysteresiseffect24,25in a pressuredependent permeabilitycausedbyrepeatedpressure reversals,asoocuredduringthisisoohronal test,shouldnothaveallowedtheflowsto retracetheback-pressurecurve. Analysesperformedonbuild-upsobtained afterfouroftheflowsyieldedconsistent permeabilitiesof222MD.Theskineffect calculatedfromthesebuild-upanalyseswas foundtoberatedependent.Whenswas plottedasafunctionofq,askinatq= 0 of+2.2andanon-Darcyflowcoeffioient Do=.00233BOPD-lwasobtained(Fig.23). FromReferenoe26,apartialpenetrationskin sbwascalculatedtobe+2.5,inveryclose SPE4529M.J.FETKOVICH 9 agreementwiththatobtainedto qo= O.Thereservoirskindamage(s)therefore isconcludedtobeO. A furtherverificationofnon-Darcyflow inthiswellwasmadebycheckingtheReynolds number R=?-Vd e)J. A Reynoldsnumberof8wasobtainedforthewell underitsflowingconditionof2973BOPDwith aflashformationvolumefactorof2.70,a densityof0.48gm/cc,0.22cps.viscosity,and asswningagraindiameterof 0.5mm.According toMuskat12 turbulentflowcanbeexpectedfor Reynoldsnumbersgreaterthan1. Thenecessityfor conducting multi-rate tests dnoil wellsforthecorrectevaluation ofwellperformance,PI,reservoirdamage,flow efficiencyandpotentialisparticularly emphasizedbythisexample.Onecanalso concludethatnon-Darcyflowwouldalsoexist inthepresenceofagassaturationaroundthe wellboreandwouldbeevenmoreseverethanis indicatedforthesinglephaseliquidflows. For WellNo.1-a,abreakintheperformance curveshouldoccurforflowsbelowthebubble-pointpressure,withtheabsoluteopenflow potentialbeingevenlessthanthatindicated bytheextrapolationonFig.22. CHANGEINPERFORMANCECURVESWITHDEPIETION Perhapsthebiggestimpedimenttoan earlier developmentofmultipointtestingof oil wellswastherealizationthatawell's performancecurvechangeswithchangingoil saturationandpressureinacomplexmanner. Standing27 extendedtheutility ofIPR equation(performancecurve)byillustrating asimplemethodtocorrectaknownIPRcurve positiontosomefuturepositionasaresult ofachangeinkThefuturevalueofkin roro uB o0 hisexamplewasobtainedfromaTamermaterial balancecalculationusingaCorey-type correlationforkTheinabilitytodefine arealkcurveroforaspecificwellstill makesthfgapproachonlyapproximate. It hasbeenobservedthatinmanymaterial balancecalculationsforsolution-gasdrive reservoirs,kisapproximatelylinear with reservoirAsanapproximationtothe changeinoilpermeabilitywithpressure depletionwecouldthenwrite (30) or k ro -=% =--PRi (31) wherekrois withrespecttoki andis defined atavanishingt.p,zerodrawdown.%iisalso tobeequalto bubble-pointpressure. orlessthanthe Thenkro(PR)plotted : t; "' .. !1. , "'O' 'G. OR w' PSIGPStG 3403.73394.9 l082.8 3387.23376.5 3391,23062.6 3394.13199.6 3405,53306.5 3402.03385.9 3398.73393,3 2482 1000 STKSOPO ~,-1398 ." '" 4 4 _2 PR= 3420.2 pSia I AOfP - 7800SOPO n = 0.59610,000 GOOSEP. PRESS. SCF/STKSBLpS'G '" --,-.--." ,]5 "

'" :I: t: 'T N_ "' 2 1"-2 "-No: ,"-10001000 ..8 88 44 oISOCHRONAL FLOWAFTERFLOW 2 100 24 100 10010.000 Fig.22- Four-hourisochronalperformancecurveofWellI-a, FieldG,Jan.18.1972. 18 16 14 12 10 'en 8 RESULTS FROMBUILD-UP ANALYSIS FLOW STKqSOPO K NO.MO ....,. 2i6 2120 ,236 '" 2Z2 5001000 S' + 9.ff 6.65 5.08 '.U CALCULATEDPARTIALPENETRATIONSKIN. Sb-+2.5(REf.26) 15002000 qo -STKBOPD 250030003500 8 6 4 2 10,000 8 6 4 2 oISOCHRONAL pSI' AOFP-I3,300BOPD;I nO.1IZ,/ I' FLOWAFTER FLOW BUBBLEPOINT PRESSURE= 5035 pSI. Fig.23_Non-Darcyfloweffect,single-phaseliquidflow,Weill-a,FieldG. 1000 8 6 FLOWlJRPwf:!1.< SEP. PREss. PSIG

e PRLPwf2) .,J'ei""=TO 6SBL/DAY/PSI2 4 2 10,000 e e 4 2 1000 e II 4 2 DRAW DOWNATINITIAL CONDITIONS(PRi=2100PSI ... ) VOGE'Q;IG.7 PSIA(THOUSANDS)BBL/DAYDAY Ii"Oii"BOO --,,- --,-,-1400102 VJ1000J.HO140 o6004050165 Z04410184 OEPLETION MAXAT :;)(.!a.),-P')EQ.32VOGELFIG.7 oPR\:P"R iRqoqo :t:PSI ... (THOUSANDS)BeL/DAYBBL/DAY t- """'i'iSO--,,-,- '''' ..... N 'i Q. I N", IQ. '-' r--. "'I.-IQ. '---" '.50 '50. 12SO""21]9 1607 .,. 34' " . .. 57 " no .. n =1.000