This is a preprint -- subject to correction.

SPE 28688

SPESocIstu of Petmieun Engmers

Decline Cuwe Analysis Using Type Cuwes--Analysis of Oil Well Production Data Using Material Balance Time:Application to Field Cases

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● tJNOCAL-CoastalCalifornia,and T.A.Blasmgame,”Texas A&MU.

● SPE Merrtwa

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BRIEF SUMMARY Arps’ efforts provided a variety of results; including theThis paper presents rigorous methods to analyzeand interpret exponential,hyperbolic,and harmonicratedeclinerelationsthatproductionrateandpressuredatafromoilwellsusingtypecurves we use today for empiricaldecline curve analysis. Due to theto performdecline curveanalysis. Thesemethodsare shownto simplicityand consistencyof this empiricalapproach,the Arpsyield excellent results for both the variable rate and variablebottomholepressurecases,withoutregardto the structureof the

reistionsremaina benchmarkin the industryfor the analysisandinterpretationof productiondata.

reservoir (shape and size), or the reservoirdrive mechanisms.Remits of theseanalysesincludethefoilowing:

The utility of the Arps relations is the applicability of thehyperbolicfamilyof curvesto modela widevarietyof production

● Reaewoirpmpe!ti~ characteristics.In addition,thesimplifki ansiysisof exponential- Skinfactorfornearwelldamageor sdmtdation,s and hyperbolic data trends (such as the graphical techniques- Formationpermeability,k providedbyNind2)maintainthepopularityof theArpsrelations.

. In-placefluidvolumes: The applicationof the A@ relationstypicallyincludesa semilog

- Originaloil-in-place,N plot of rate versus time where the hyperboliccasesyieldgently- Movableoil at currentconditions,NP,mv decliningcurveswhichhavethestraight-line,exponentialdecline- Reservoirdrainagearea,A case as a lower limiL Nindz provides the development and

We have thoroughly verified these analyses and interpretationillustrationof plottingfunctionsfor the grsphicsisnsiysisof ratedata for the general hyperbolic decline case as well as the

methodsusing both syntheticdataand numerousfieldexamples.In addition,we provide illustrativeexamplesto demonstratethe

exponentialdeclinecase.

ease of analysisand interpretation,as well as to orientthe reader Anotherattractionof the Arpsrelationsis theirusein graphicalasas to whatam the benefitsof rigorousdeclinecum analysis. well as functionalextrapolation.Manyanalystsrelyuniquelyon

the Arps relations for performancepredictions,often withoutINTRODUCTION realizingtheempiricalnatmeof suchextrapolations.In thisworkThe importanceof performingaccurateanalysisandinterpretation we will use exponential decline case as a basis for estimatingof reservoir behavior using only rate and pressure data as a movableoil at currentconditions,NP,MV We willdemonstratefunction of time simply can not be overemphasized. In most that this approachcan be derivedtheoreticallyfor the case of acases, these will be the only data available in any significant weilproducedat a constantbottomholeflowingpressure.Wewillquantity,especiallyforolderwelisandmsrginsilyeconomicwells also show that this approach works for wells which are notwhere both the quantity and quality of ~ types of data are producedat suchrestrictiveconditions.iimited. The theoretics applicationof this techniqueis fornewer The Arps relations for flow rate and cumulativeproductionarewells, at pressuresabovethe bubblepoitt~aithoughweshowthat givenas foilowsthe methodsdescribedherecanbe accuratelyappliedat any timeduringthedepletionhistoryofa psrtictdarweIi. Arps Flow Rate Relations

The developmentof modemdeciinecurveansiysisbeganin 1944 -whenA@ pubiisheda comprehensivereviewof previousefforts *oncnriak (H) ~t) = ~ieXp(-D#) . . . . . . . . . . . . . . . . . . ...(1)

for the graphicsi anaiysisof productiondeclinebehavior. In that Hyperbalk (tkkl) ~t] = qiwork, Arps developeda familyof functionalrelationsbasedon .....................(2)thehyperbolicdeclinemodelfortheanalysisof flowrate&ts. [l+bD#]l/b

Harmonic: (b=l)g(f) = & .. . ... . . . . . . . . . .. . . . . . . ...(3)

Referencesandiiiustrsdonsat endof paper

.

2 DeclhteCurveAnalYsisUsingTypeCmwS-AIIdysiS ofOdWell~uction DataUsingMaterialBalanceTime: SPE 2t$bWApplicationto Fiild Cases

Arps Cumulative Production Relations

Exponential:(b=O) N~t) = ~[1 -exd-Dit~ . .. .. . .. . . ...(5)

or in terms of q@

Np(t) = ~[qi-q(t~ ..... ..............(6)

~yperbofic:(*1) N~t) = qi [] - (l+bDit)l-l/b].. .(7)(1-b)Di

or in termsof q(t)

Harmonic (b=l)N~t) = ~til+D$) ..................(9)

or in tmns ofq(r)

In additionto presentingthese fundamentalmlationa,A@ laterintroduced methods for the extrapolationof rate-time data toestimateprimaryoil reservesusingtheexponentialandhyperbolicdeclinecunfemodels.The use of “typecurves”(dimensionlessor normalimdflowratesolutionsplottedon a scaledgraph)foranalysisof productiondatawas introduced to the petroleumindustryin the late 1960’sandearly 1970’s.4$ In 1980 (preprint 1973)Fetkovichsintroducedthe most significant developmentin the type curvematchingofproduction data-tie creation of a unified analytical solution(exponentialdecline)for a wellproducedat a constantbottomhole

ee.m.a ABM%O Imnplwwdomkiated flow conditions.ph..” w..-.~ . . . .—, -.---–-Further, Fetkovichs plotted his unified exponential declinesolutionsimultaneouslywith the A@ hyperbolicdeclinestems,which are assumed to account for non-idealreservoirbehavior(changes in mobility, heterogeneous reservoir features, andreservoirlayering). The final result is the so-called“Fetkovich”type curve, which provides for the simultaneous analysis ofproduction data during transient and boundary-dominatedflowconditions. While the Fetkovichdeclinecurveis anextraordinarytool for reservoir engineering, this approach is not withoutlimitations.A particularlimitationarises in the analysisand interpretationofproductiondata which exhibit significantvariationsin wellborepresaum, as well as the effects of periodic shut-ins and otherconstraintsimposedby opimuionalconsiderations.To ita crectkithe Fetkovichdeclinecurveis themostpowerfultoolavailableforthe analysisof productiondata,as demonstratedin refs.6-10. Inthis ligh~ our presentefforts serveonly to extend the utilityandapplicabilityof this typecurveanalysisapproach.The initialeffort to incorporaterateandpressurechangesintotheanalysisand intapretation of productiondata was introducedin1986 by Blasingame and Lce.11 This work providesanalysismethods for determining drainage area size and shape fromvariable-rateproductiondatain closedreservoirsusinga Cartesianplotbaaedon thefollowingrelation

$= m-+bP,, .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . (11)

whereAp = pi - pwp ~d

.=& ... ...... ...... ..... ........................................ (12)

,.,=141.2#&(-$-&-]b . .. .. . .. . . . . . . . . . . . .. . . . . . (13)

andthedefiition of “materialbalancetime”is givenby

;= N~q ................................ ......... .......... .......(l4)The analysismethodderivedfromEq. 11was observedto workbeat when rate changes were small, that is, when the transientsinducedby rate changesdo not obscurethe boundary-dominatedflowbehaviorfor longperiodsof time. Eq. 11wasderivedusing.. . -,..- --...1.12 A. A- ~ tqm.t =Ie C&S, and verified byme UmLZKCMML - s~4 w,” -0..-comparison to the Muskatls solution for a bounded circularreservoirandby theanalysisof simulatedwellperformancedata.Continuing in a chronological fashion, we note that in 1987Fetkovich,et ap presenteda seriesof fieldcasestudiesevaluatedbydeclinecurveanalysisusingtypecum%. In additionto severalexcellent field examples, the authora also gave commentaryregardingthe analysisand intmpretationof productiondatausingdeclinetypecurves.One of the majorconclusionsof the Fetkovich,et af7 studywasLI=o~p~efi LMLMCantdvsis of transientproduction ddta usingthe Arps hyperbolic equatk ~ invalid. Transientflow theorystates that the flow rate profile should be concaveup, and as adeclining function, the Arps stems are concave down--whichclearly poses an inconsistency in both the analysis andinterpretationof transientflowdata. A curiousdevelopmentwasthe emergencein the industry of a “rule-of-thumb”during the1970’aand 1980’swhere it was suggestedthat an Arps stem ofb>l should be used for the analysia of transient flow data.However, from the previous arguments it is obvious that this“role”is withoutfoundationand willultimatelyleadto erroneousresultsas WMas incorrectinterpretations.Put in a practical sense, transient flow data (productiondatafunctionswhichareconcaveup) shouldneverbe usedtoestimateresmoir volume. Specifically, Fetkovich, et al suggest thatm.servoirvolumesandvolume-relatedflowcharacteristicsshouldnot be estimatedusing declinecurve analysisk.fore boundary-~- :-. -~ fIfiw fidlv exis~q (nrnductiondata exhibit a concaveuVmAiJaLu... . .-..= --.&—r...downwardsbehavior).In 1991 Blasingame, et ap expanded on the earlier work ofMcCray~ to develop a time function that would transformproductiondata for systems exhibitingvariablerate or pressuredropperformanceintoanquivalent systemproducedat a constantbottomholepressure. The motivationof this effortwas to createan equivalent constant pressure analysis formulation for theanrdyaisof variable-ratehsriablepressuredrop productiondata.Unfortunately,the solutionprovidedby Blasingame,et al, whiletheoreticallyconsistent is somewhatdifficultto applybecausetheapproachappearsto bevety sensitiveto ematicchangesin rateandpressure.However,the B1aaingame,et aP study providedboth insightandmotivation for the development of a more robust and lesscomplicated approach to analyze and interpret variable-ratehariable pressure drop production data, which ultimatelyresultedin ourpresentefforts.McCray$proposedthe followingrelationas a definitionforthe“quivalent constantpressuretime:tcp

NJ)=

/[1‘q& d. .... ...................................... (15)

b~t) o AddMcCrayprovideda recursive-typetrapezoidalmle formulationtosolve Eq. 5 for tcp In addition, Blaaingame,et aP providedaseriesof derivativefonmdationsforcomputingrcp As WSCtiVf2

as the concept of an equivalentconstant pressuremodel is, thecomputational aspects of its application are unsatisfactory,espccirdtyforapplicationto fielddatawitherraticvariationsin therateand bottomholepressureprofiles.The utility of the tc concept is aignificanlgiven the use of the

&Fetkovich6Wlquid OW)and Carter14Js(gas flow) type curvesfor analysis of production data, and given tlds potential, we

SPE 28688 L.E. DoybleLP.K. Pande,T.J. McCollum.@dT.~ Bl~ingme 3

recommend that the equivalent constant pressure concept beconsideredfor fmher study.In 1993,Palacio and Blasingame10developeda solutionfor thegeneralcaseof variableratdvariablepressuredropforthe flowofeither single-phaseliquid or gas. Theseauthorsshowedthat for~ny ~a,ti~cu!ar~rodu~ticn hismry using the pressure dropnormalized flow rate function and the material balance timefunction will yield a harmonic rate decline (b=l stem on aFetkovichdeclinecurve)for liquidflow.The authors derived this method rigorously from thepseudosteady-state (or boundary-dominated)flow equation asfollows. RecaUingthe paeudosteady-stateflowequation,Eq. 11,andthedefinitionof thematerialbalancetime,r,@q. 14)we have

~= m-+bP,, . . . ... . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. (11)

whereAp = pi - pwf ~d

i= Npl’q . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [M)

Takingthe reciprocalof Eq. 11gives

%.~@ [m-+bps]

. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . (16)

Rearranging.thisresultgives,

bpssAp‘=F&ior reducingto shorthandnotationwehave

_4k=-(q/Ap)inr [1+Dit]

... . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . (17)

wherethe(q/@)inrtermis definedss

andtheDi term is definedask

& = 7.9545 .10-2 41c+iDi = bp~~

%%%

....................(19)

Makingthe finalreductionofEq. 17wehave

q~=FkJ...................................................(20)

wherethe definitionsof;’ andq~ for thiscasearegivenby

;~=Di; . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(2l)and

-\ A-

‘W=(q;:x”nt.. .. . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)

RecallingtheArps“harmonic”declinerelation(b=l) asdefinedbyFetkovichc(andgivenas Eq.B-3 in AppendixB) wehave

‘m ‘& ..................................................(23)

Cqwing Eos. 20 and 23 we immediatelytecognizethat theserelauons are ‘~. And further, if we consider the baserelationfor variable-ratehriable ptessuredropperformance,Eq.16, we note ~at during boundary-dominatedflow q/Ap dataplotted versus r will exactly overlay the Arps b=l stem on theFetkovichdeclinecurve. This wasthe foundationof analysisforthe workby PalacioandBlasingarne1°as wellas the basisforoureffortsin this ptesentwork.In the presentworkwe focuson theanalysisandinterpretationofproduction data (flow rates and bottomhole pressures) for oil

wells in order to estimate reservoir voiuttiES art~ fiwcharacteristics.We focuson usingdata thatoperatorsacquireaspart of normalfield operations(e.g.,productionra~s froms~estickets and pressuresfrom permanentsurfaceand/orbouomholegauges). This approacheliminates the loss of productionthatoccurswhen wells are shut in for pressuretransienttesting.~d-*....:..-A :-*-m..a*.;nm-f .“*I *“Afi-ldprovidesSniuysls mm lIlbG1pHdA411 v, well UA.u . . . . . pe.tima!mat little or no cost to the operator. In addition,the methodsweintroducein this paperare not constrainedby the requirementofcerwm: rata w !mttornho!epresmes, as is the case for thepreviouslypubtishedmethods.Awe mentionedearlier,the analysismethodsthatwe presentinthisworkprovideestimatesof the following:

● Reservoir prOJ)Wtit?S

. Skinfactorfornearwelldamageor stimulation,s

. Formationpermeability,k● In-placefluidvolumes

- Origiiai r2i!4n=p!ace,N. Movableoil at cun’entconditions,Np,tMV- Reservoirdrainagearea,A

METHODS FOR THE ANALYSIS AND INTERPRET-ATION OF PRODUCTION DATAHarmonic Decline Case: General Approach forVariable-Rate/Vadable Pressure Drop Production DataAs we discussedin the Introduction,the rigoroussolutionforanyrate and pressureschedulefor thecaseof a wellproducingunderboundary-dominatedflowconditionsis givenby Eq. 16. Recall-ing Eq. 16we have

4L.~ .................................................(16)‘P [W-+ bpss]

We recognize that Eq. 16 is a “harmonic”type of equation inwhichthe “materialbalancetime”function,;, is givenby Eq. 14as

i= N~q . .. .. . ... . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . (14)As such, we simply plot the pressure drop normalized ratefunction,q/Ap, versusmaterialbalancetime,~,on a scaledlog-logplot and matchthesedataon theFetkovich/McCraytypecurve,lowith the boundary-dominatedflowdata beingforcematched(bydefinition) on the Arps b=l depletion stem. The type curvematchingproceduresand the associatedanalysismethodologiesarediscussedlaterin this tex~as wellas in AppendixC.Fetkovich-McCray Decline Type CurveTheso called“Fefkovich/McCraytypecurve”wasfirstpresentedas a single entity in ref. 10, althoughcomponentsof this curvewerepresentedby Fetkovich6(1980,preprint1973)andMcCray8(M.S. thesis 1990). The utility of the resulting “Fetkovich/McCray”solution is the ability to match flow rate functionsaswell as the flow rate integrai and integrai derivativefunctionssimultaneously. In addition, the integral functions providesmoother data trends for clarity and ultimately, improvedmatchingof dataandtypecutves.

AlthoughbothFetkovich6andMcCraysprovidethedetailsof thedevelopmentof their respectivedeclinetype curves,we believethat a unifying discussion is in order, particularlyfor readersinterestedin fmher developmentsof this type.It is importantto mall that the “analytical”stems(transientstemsand the exponentialdeclinecase [b=o stem])on the FetkovictdMcCraytypecurvel”(orany“decline”typecurveforthatmatter)are solutions for a well producing at a constant bottomholeflowingpmssun?. However,the methodologyindicatedby Eqs.14and 16indicatethat the Fetkovich/McCraytypecurvecan beused to analyze any type of production data, including data

4 DeclineCurveAnalysisUsingTypeCurves-Analysisof OdWellproductionDataUsingMateriatBalanceTime: SPE 28688.. ...-. .An____Appltcauonto rlela uuses

exhibitingarbitrarychangeain rate and pressure,so long as theboundary-dominatedflow data are “forcematched”on the&=1

..:-~ .+.- -tn. mmmit diwu.ssions considertheapplication(&iiarmulIl&J awl,,. ..,” --.”... ----------- --

of the Fetkovictt/McCrsytypecurveonlyforcasesof radialflow,--**--S–-.----A. ...-11” ..jphin particular,verticaiweiii and vetticauy IrWUKU WGIIS w ~aellexhibit radialflow. The Fetkovich/McCraytypecutve approachwasrecentlyextendedto horizontalwellsas describedin ref. 16.In order to be consistent with cument literature we use “fieFetkovichsdefinitionsof the dimensionlessdeclinevariables(1LMand qm) which are given below. The tnjfunction is given intermsof dimensionlessvariablesas

t~=~ ~tf)& [lnr@-j

................ ..... ................... (24)

Slldht t&?ltS of ~ Vtititilf% %%%2ii~Vt?

tm= 0.00633~~@@ [lnr~ - *]

............. ............... .. (25)

In a similar fashion, the qm function is given in terms ofdimensionlessvariablesas

qDrf=[~rcD-+]~D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)

andin termsof realvariableswehaveq~=141.2#$[hreD-~] ............................... (27)

A minor discrepancy in these ...,,~~..~ .. . .. .a-fi~i*:~n. :~ f$~i Llhts1/7 term.W ., - .-- . .should actually be 3/4 as noted by Ehlig-Economides andRamey.17 We maintainthe conventionof using 1/2ratherthan3/4 for the purpose of type curve develo ments in order to be

fcompatiblewithexistingliteratuy. But in acLthis“discrepancy”rarely makes more than a few percent difference in theinterpretation,and is onlynotedhereforcompleteness.The rate integratand rateintegralderivativefunctionsintroducedby McCrays are given in dimensionless form below. Thedimensionlessrateintegralfunction,q~, is givenas

JW=L WWMi= ~w

W()q~?) dr .............................(28)

and the dimensionlessrate integralderivativefunction,q~, is~ptien~

..412(L=. tm*‘- d In(m)

.............................. (29)

whereEq. 29 can be reducedto the followingresultas showninAppendixB

qw = qlmi-qDd ........................... .................... (30)T. *tiriitiOn. we i~trodu~ &e dimensionlessratederivativefunc-*. ““”.- ... .7- .- -- —tion, q~, whichis definedas

qw=_J!mL=.tw$112!Ji ...............................(31)

Unfortunately,we do not expectEq. 31 to be of muchuse in theanalysis of productiondata due to the volume of randomemorfound in productiondat&wheretheserandomerrorswfiIordybemagnifiWbythedifferentiationprocess.In orderto developthe Fetkovich/McCraytypecurve,we requirevaluesof thesolutionfora wellproducedat a constantbottomholepressure, qD as a functionof dimensionlesstime,tD,which arec~n COnve@ to r~ and qm Using Eqs. 24 and 26 respectively.Ilmse q~@) valuescanbeobtainedfromtablesin vanEverdingenand Hurstls or using numerical inversionlg of the Laplacetransformsolution developedby Matthewsand Russell.~ ‘fheLrrptacetransformsolutionforconstantrateproductionfora wellcenteredin a boundedcimdar teservoiris givenby MatthewsandRussell~ as

However, we require the solution for a constant flowingbottomhole pressure rather tttan a constant flowrate. We cart..-..:1.. ki.: th rn~~tant hottomhcile pIIXSUrC Sohtion fromthe~ulay ouw~. -.e - -—.- __.-–..constantmtesolutionusingthefollowingrelationin Lsplacespacegivenby vanEvcrdingenandHurst.lg This resultis

~~u). LA (33)Uzpdu) .... ...... ........... ...........................

Once the q~t~) values are obtained from qdtD) values, theassociatedderivativeandintegralfunctionscanbecomputedusingstandard techniques, or these functions can be computedsimultaneously with the q~tm) values using the numericalLaplacetransfotminversionatgorithm.lg. “. A oinsl t%tknvich6 Iyy curve, slOngm rig. i we preen: tiie W..e.....- ----- ----with the derivativefunction,q~, as definedby Eq.31. We notein Fig. 1 that the q~ stemsshowa dramaticcharacterizationofthe transferfromtransientto boundary-dominati flow,however,as we suggestedbefore,we wouldnotexpect theq~ concepttobe particularlyapplicabledue to randomnoise present in field~~

Figure2 presentsthe Fetkovich/McCraytypecurvelowhereq~,q~, and q~ are all plottedversust~ on the type curvegrid.Althoughthii plot appearssomewhatbusy,we believethatFig.2provides all of the necessary functions for both rigorous andempiricalana@sisof productiondata. Figure2 is usedthroughout~tir pre=nt work for ~heanaiysis and interpretation of bothsimulatedandfielddata.

ANALYSIS OF OIL PRODUCTION DATA USINGTHE FETKOVICWMCCWY TYPE CURVESA step-by-stepprocedurefor the use of the Fetkovich/hfcCraytype curve is given in AppendixC, and is abbreviatedin thissection for referenceand use in applications. Our type curveanaIysistechniqueprovidesmethodsto estimatethe original-oil-in-placeand othervolume-relatedproperties,as wellas the flowcharacteristicsof theresetvoir.Our methodology is based on the use of the simple materialbalancetimefunction,t, thatyieldsa harmonicdeclineforthecaseof liquidproduction,regardlessof therateandpressuteschedule.We provide the following procedure for the analysis and,,,~,PR_U,, “f ~.wd-- -..:..—....*A.. . .rn nr~nn &@ us~.g &@ItC &p CUtWS.

1. ~ .

i=fV#q . . . .. .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(J4)

2. ~Our approach in this study is to work with the pressure dropnor@ized rate function, q/Ap, in order to be completely

.-* --- --- -. -:..-.. L.. c“ 1LconsMent wtui me.rneOrygivcit uy ~. ~u. -v ..nli. .o~~~~ V#iiifollowthis convenuonthroughoutthe text, includingcaseswherecontinuouslymeasumdbottomholepressuredataarenotavailable,and we use the initial reservoirpressure,pi, as the normalizingcondition.Thepressuredropnormalizedratefunctionis givenby

!q!Ap)= A = — (34)(pi - Pwfl G

. .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

when We use Ap = pi - Pw, as a shorthandnotation. The rateintegralfunctionis givenby

(q!l$)~=~~~d~.............................................(35)tjo AP

and therateintegralderivativefunctionis givenby

4h!M!..i4!i!M.(q/Ap)~= - d ~j . .. . . . . . . . . . . .. . . . . .. . . . . .di

(36)

SPE 28688 L.E. DoubIettP.K.Pande,T.J. McCollum,andT.A.Blasingame 5

The three plotting functions (Eqs. 34-36) are computed andplottedversus the materialbalancetime, t, then matchedon theFetkovich/McCray10typecurve,takingcareto “forcematch”theboundary-dominated portion of the data onto the Arps b= 1(harmonicdecline) stem. The “forcematching”of boundary-dominatedflow data is requiredby theoryand providesthe bestpossibleestimateof oil-in-place,N.3. ~-in-pbEstimating the reservoir volume or oil-in-place,N, from typeCtlrVSSnSiysis~ti thStWe.miiltt?ihediShiitttiSOff~”~id ij~(givenby Eqs.25 and27) to peld a “matchpomtwresultm termsof volume. Equatingand isolatingterms in Eqs. 25 and 27, weobtainthefollowingrelation

(9dMP(@hP=W(q@f@(h...................(37)

SolvingEq. 37 for the oil-in-place,N, weobtain

N.di.k!!@k%(nAJIP (9d’fP.. . . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . (38)

In orderto solvefor the pseudosteady-stateconstan~b ~, we willtuse the generalized definition of q~ given by q. C-5 in

AppendixC. RecallingEq.C-5wehave

We note that Eqs. 27 and 39 arcequivalent,but Eq. 27 is strictlyvalid only for the case of a well centeredin a boundedcirctdsrreservoir and Eq. W is vaiid for a generai reservoir/weiiCO~lgWStiOZI Using tie appropriak Shapf3 factor,CA.Recallingthedefinitionof bpss,Eq. 13,we have

““=14’”2W%%Ib . .. . .. . . . . . . . . . . . . . . . . . . . . (13)

Combining and solving Eqs. 13 and 39 for bPsSwe obtain thefollowingmatchpointrelation

bp,3.& ..................................................(40)

4. ~ . .

The relations given below are used to estimatevolumetricattdflow characteristicsof the reservoirbased on the resultsof thetypecurvematchandtheavaiiablewelldata.ReservoirDrainageArea:

A = 5.6148 ~ .. . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . (41)& (l-~wirr)

ReservoirDrainage Radiux

re = P . .. . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(42)

Effective ;ellbore Radiusrm= &

r& """""""""""""""""""""""'""""""""""""""""""""""""""'"""'""""(43)

FormationPermeability

~=1’+p[*]~:&] ....................(44)

or combiningI@. 40 and44 wehave

~=141.2~~[-]1~1 .......... ................. (44)

SkinFacto~

s = - ln(~) . . . .. . .. . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (45)

ANALYSIS AND INTERPRETATION OF LONG-TERM PRODUCTION DATAIn this sectionwe presentthe analysesand interpretationsof thesimulatedand field data cases that we consideredin this study.Ourgoalis to be ableto analyzecasesfor whichdatais plentiful,butalsoto be ableto accuratelyestimatemovableoilvolumesandfluid flow characteristicswhen high quality productiondata isscarce.We suggest that our proposedmethodsfor the analysisof long-term productiondata are easily transferableto any operator,inparticular, operators that iacicthe abiiity to perform periodicpressuretransienttestsor long-termproductiontests.We presentas completean analysisand interpretationas ssible

2for each data case. We are able to reducethe adverse fectsofproductionanomaliesthat occurduringthe life of a well, andweobtainedunique type curve matches using productionrate andpressure functions, material balance time, and the Fetkovich/McCrsylotypecurve. Theseproductionratefunctionsare

. pressuredropnormalizedme function,(q/Ap),

. m~ in@@ function,(q/AP)i,~d

● rateintegralderivativefunction,(q/Ap)~.

This processresultsin excellentestimatesof originalandmovableoil volumes,as well as goodestimatesof pmneability and akinfactor. The formationflowcharacteristicscan be calculatedwithmuchgreateraccuracyand confidenceif wehaveaccurateesrly-time(transient)data.Whenthe typecurvematchon eithera transientor depletionstemis indeterminate,anomaliesin theproductiondatacanbe removedby reinitirdizingthe data paata particularanomaly. Examplesofsuch “snomalka”are recompletion, mechanicalfailures,long-termshut-ins,and fluctuationsin flow rate and pressureat earlytimesin thelifeof thewell.Whendatareinitializationis requireddue to suchanomrdiesin theproductiondata, the cumulativeoil producedremainsconstan~regardlessof reinitislization.However,the reinitializationproccasrequireathatwe accou: forpriorproductionin thecalculationofmaterialbalancetime, t. This is accomplishedby computingibaaedon the~otslcumulativeproductionand currentrates, thenresealingthe I data to yield r =0 at the first data point. This is asimple procedureand can be easily implementedwith a smallcomputerprogramorspreadsheetapplicationmodule.

Data Preparation and Analysis ProcedureWe now provide the procedures that we use to interpret andanalyzeproductiondata. Theseproceduresare

1.

2.

3.

Verificationof pertinentrock, fluid, and completiondatausingavailablefieldrecordsand fluidpropertycorrelations.Thecriticaldatatequiredforouranalysisinclude

●

●

●

●

Totalcompre.%sibility ● Porosity

Fluidviscosity . NetPayInterval

011formationvolumefactor . Wellboteradius

IrnxhtcibleWaterSaturation

Initialscreeningof fieldproductiondatausingsemilogandlog-logplots

● Identifyerrorsor anomaliesin theproductiondata

● ham andannotatechangesin thecompletionpractices

● Timeminitializationof theproductiondata

● performintegralandintegralderivativedstasmoothing

Performtype curve analysisusing the FetkovicMMcCraydecline type curve to determine the time and rate match

‘6 DeclineCurveAnalysisUsingType--Analysis of OilWellproductionDataUsingMaterialBshnceTime: SPE28688Applicationto Fieldcases

points. This typecurvematchingpmceaswasaccomplishedusing a commercialsoftware graphics package.zl Thesematchpointsare thenusedto estimatethefollowing● Oil-in-place,N

. Paeudosteady-stateflOWconstsnttbpss

● T~ent stemmatch,rm

These results are then used to estimatereservoirdrainagearea formationpermeability,andthenear-wellskinfactor.

4. To estimatethe movableoil, Np,mov, at cu~nt producingconditionsweusc thefollowing

●

●

●

Sm”ctlyrigorousapproach (requiresPWIdI@

Plot calculated average pressure, ~cal=Pw~+ @Ps,s.versuscumulativeoil production,Np, md exmpola~ toFca@

Sem--analyticalapproach:

Plot (q/Ap) versus cumulativeoil production,Np. andextmpolateto (q/Ap)=O

Analytical approach-constant bottomholepressure ca.wPlot the flow rate, q, versus cumulativeoil production,NP, and extrapolateto q=o. This methodis used whenbottomholepressuredataarenotavailable.

For a complete treatment of the proceduresused for theestimationof movableoilpleasereferto AppendixA.

Simulated Data CasesWe used a 2-D, radial, single-phaseblackoil simulatorwith 30geometricallyspacedgridsblocksto modelwellperformancein asingle-layerreservoirwithhomogeneousandisotropicpropmies.These cases are used for vetilcation of our type curveanalysisand interpretationmethods. A constantbottomholeprt%surecasewasusedas a benchmarkanda secondcasewithmultiplerateandpressurechanges(includingshut-ina)wasgeneratedto ver@ thevariable-ratdpreasutedropperformanceof ourapproach.The analysismethodwasverifiedusingsimulateddatacaseswitha wide range of permeabtity, and numerouschangesin rateandbottomholepmasure. Agreementbetweensimulatedperformanceand the results of decline curve analysis were checked forpermeabilitieaof 1, 10,and 100md. We presentthe analysisofsimulatedperformancefor thefollowingproductionhistories

Verkble pW.with O.olmlmultipleshut-ins llm.o

2C0.O210.0310.0410.06rn:o

520.0620.0630.0720.0

Iwo.o2m3.o

4000.O

vukbk1000

varkbk23CQ1300

vukble2000700

Varkbti1000300200100100

15.0vuiabk0.0

vukbk

.0.0vukbkvukbk

6.0vukbk

vuiebkvukbkverieble

The pertinent reservoir, rock, and fluid properties for thesevcsifkationrunsamsummarizedin thetablebelow.

Reserwir Properh”e.cWellboreradha, rw = 0.25 ft

Drainageradius,re = 744.7f[Net pay thickness,h = 10ftPorosity,#(fraction) = 0.20~ucible watersaturation,Swirr = 0.00OngmaInommalwellspacing = 40 acresFormationpermeability,k =lmdOriginal-oil-in-place,N = 564,210STB

FluidProperties:Oit formationvolumefactor,B = 1.1RB/STBOil viscosity,p = l.ocpTotalcompressibility,q = 2O.OX1O-6psi-l

ProductionParameterInitialmSCrVOirpESSUrC, pi = 4000 psia

Curve~The semilogand log-logproductionplots, togetherwith the ratefunctionplotsare shownfor the twosimulatedcasesin Figs.3-8.The rate function,(qhp). rate integml function.(@p)i, ad fsE

integralderivativefunction,(qhp)ti are plotted versusmaterialbalancetime,~,on the FetkovichtMcCraytypecurveas shownonFig. 9 (constantpressurecase)andFig. 10(variable-rate/pressurecase). The boundary-dominatedportionof the ratefunctionsareforcematchedon the 6=1 (harmonic)declinestemas dictatedbytheory for the use of materialbalancetime, and the appropriatematch points are taken. The dimensionless drainage radiusmatchingparameter,r~, is estimated fromthepositionof thedataon the transientflowtypecurvestems. me r~ parameteris thenusedto estimateformationpermeabilityandakmfactor.We obtainedexcellent type curvematcheson both the transientstems(forearly-timedata)as wellas thedepletionstems(forlatetimeor boundarydominatedflowdata),as shownon Figs.9 and10. The drainage area, total and movable oil volumes,permeability, and skin factor estimatedby type curve analysisexactly matched the input data to the simulator,verifyingourapproachforbothcases.

Type Curve Match FetkovichlMcCray Type Curve (RadialFlowin a BoundedReservoir).

~ ConstantBottomholePressure@lg.9)MatchingParamettxr~ = 3000(est.)

[tQ.&p = 1.0 [iMP = 1270.6days

[4*P = 1.0 MAPIMP = 0.00888 s~~/Psi

Cmi”gbia:-i2Wn-Haee:

N.dhd@d!&CtMMP (QM)MP

.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (38)

N = (1270.6days)(O.00888STB/D/psi)= Sa Zlo Sm*

20x 104 psi”lResemoirDrainageArea:

A .5.6148 ~ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . (41)@l(l%!iwi~)

~ = (s.6148 ft3/RB)[564,210STB)(l.1 RBNTB)(0.20)(10ft)(l -o)

A = (1,742359 ft~(l acn#43560ft? = 40.0 acresReservoirDrainage Radius

Fre= ~ . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (42)

re= ~(1,742,359 ft2)/~= 744.7ft

SPE 28688 L.E. Doublet,P.K.Pande,T.J. McCollum,andT.A.Blasingame “1

Effective Wellbore Radiur:

r~* . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . .. . ..(43)r~

~e ~ = 0.2482 ft

FormationPermeability

~=ldl.+~~[~]~:~] ....................(44.

k= 706 (1.0 cp)(l.1 RB/#?B)(loft) ‘“

In[

(4)(1,742,359ft2)][ 1

(ow888) = ~~d

(1.781)(31.62)(0.2482ft)2 (1)

SkinFacttx

s = - lj~) .. ... .. .. . . . . . .. . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . . . (45)

--- 1.u2wL\ s OO.”“ - -“t ().25 )

Since most weiis are not iisiialiy prorhdcedat 2 constantbottomhole pressure indefinitely, we developed our secondverification case with multiple rate and pressure changes(includingshut-ins). This casemorecloselymodelsactualfieldperformanceandshouldbeconsideredrepresentativeof the typesof production histories for which our methodologies weredeveloped.

_ v~~h!e B~ttomho!ehum withMultipleShut-ins(Fig. 10)

MatchingParatnetecr~ = 3000(esL)

[t&p = 1.0 [~p = 1270.6days

[4LnlMP= 1.0 [q/@~p = 0.00888S-fB/D/pal

Curve~Tlten%wdtaforthesecondcasearecalculatedsimilarly

N = 564,210STBA = 40.0 acresre = 744.7 ftrwa = 0.2482ftk = l.Omds = 0.0

~ (Figs.11-16)P1OLSof calculatedaveragepressure,~d, normalizeddaiiy rate,(q/Ap),and daily rate,q, versuscumulativeproduction,NP, wereconstructedto estimatethe movableoil volume,NP,mov Extra-polationof theplotteddatato theN axisinterceptyieldsmovable

(volumesof between46 and47 M TB forbothverificationcases.The simulated estimate for movable oil was slightly less(approximately45 MSTB).Theseextrapolatedvaluesrepresentthemovableoil volumeat thetimewhenall reservoirenergyhasbeendepleted.Thesevolumesare usuallyalightlyhigherthan the actualfieldvalueof movableoil due to the practicalandeconomicinabilityto producea welltosucha low pressurelevel.Whenbottomholepressuresareavailable,the~d or (q/Ap) plots

should be used to estimateNP,mow Even without bottomholepressuredata, the plot of q versusNP has been shown to yieldaccurateeatitnatesofNP,moW

Np,mov =45.0 MSTB(simulation)Np,mov =46.0 -47.0 MSTB(movableoil plots)

(Recovery Factor = ~~~o\~B1

100) = 8.33%.,

. . .Iscw.Thesimulatedcasesprovidean excellenttest for theutilityof thetypecurveanalysismethod.Theresultsof thetypecurveanalysisand materialbalanceanalysisareessentiallythesameas thedatainputto thesimulator. Ourmethodwasahownto workwellforavarietyof producingscenariosinvolvingboth variableratesandvariable bottomhole pressures, which gives us confidence inapplyingthesemethodsto fielddatacases.

Field Data CasesThisworkincludesfieldcasesfromthefollowingareas:kW2ti9n &elY&Li.tbJ@WestTexas Carbonate(Dolomite)SouthCentralTexas AustinChalk Carbonate(Chalk)WestTexas Sprdxrry clastic (TurWtite)OffshoreQdifornia LowerRepetto Clastic(TttrMdite)-. -....t:fi, .-d ~,tsi;wnf nrnductinn data vties for mch of the1116~Uf WULJ -*U ~---., “. y-”- ----- --— .— -

field cases, and the analysis of each case presents uniquechallenges. The types of fieiti production data rivaiiabk foranalysisirklude

● Singlewelldailyrateandbottomholepressuredata

. SinglewelldailyratedatawithsurfacetubhtgandcasingpltXs~ data

● Averagemonthly productiondata allocated on a tractbasis~th no boaomholepressuredata

For many of the wells we analyzed, the rock, fluid, and otherpertinent formation properties were unknown and had to beestimated.lle fluidpropertieswereestimatedusingtheavailablefield data and from cordations providedin the fiuitipropertiesmoduleof a commercialsoftwarepackage.nWe suggest that fluid properties be evaluated at an averagepreysurewhenthereservoiris betweenthe initialandbubblepointpre.saurea,and at a pressurejust abovethe bubblepointwhenthereservoirpressureis belowthe bubblepoint. Ourexperiencehasshownthat thesepracticesyield the best resultswhenusing thisapproach.Due to thedtificuhyin obtainingrepresentativevaluesof certain fluid properties,we suggest reportinga value for theNcf product. This approachallows each individualanalyst tosupplytheirownestimatesof fluidpropenies,andto providetheirowninterpretationof thecalculatedresults.In addition to difficulties in obtaining representative fluid

f-r ‘O report ~ Witt!2 fO~ the ~~e ~biiit~-poprrte s, ‘we a im fx e k. .

thicknessproduc~kh, in placeof permeabilitybecausewe lackaccurateestimatesof net pay tldcknessforeachof the ~aervoirsanalyzedin this work. However,to be consistent,we do presentpenneabilitiesanddrainageatwtabaaedonestimatedvaluesofnetpaythicknessforallcases.The inability to complete all results with a high degree ofconfidence is not related to the analysis or interpretationmethodologieswe present,but rather,to a lack of reservoirandfluiddatawithwhichto completethesecalculations.We use thisopportunityto pointout theimportanceofearlyandcompletedatacollection.

North Robertson Unit (Ch2arfc@, Mm Cc., TXThe NorthRobertson(Ckarfork) Field (Fig. 17)was developedon a nominal 40 acre well spacing beginning in 1956. Thedominant reservoirproducing mechanismfor the original 141wellswassolutiongasdrive. The initialreservoirpressurein theLowerClearfork(LCF)wasestimatedto be2800psia. As pan ofan infill drillingand waterfloodprojectbegunin 1987,116newwells were drilled, reducing well spacing to 20 acres, andresultingin uniform40 acre5-spotpatterns. Original-oil-in-placewas estimated to be approximately230 MMSTE,with primary

8 DeclineCumeAnatysisUsingTypeCurvca-Anrdysis ofOdWellProduction DataUsingMaterialBalanceTime: SPE28688ApplicationtoFieldCases

production before unitization in 1987 of 20.5 MMSTB.Individualwellprimaryrecoveryfactorsarelow,rangingbetween5 and 10pereent.The Lower Clearfork is a shallow-shelf carbonate composedprimarilyof a massivedolomitesectionwith varyingdegreesofanhydritecement. The geologicsettingat the timeof depositionandsubsequentdiagenesiscontributedto theheterogeneousnatureof the Clearforkformation,which is definedby extremelylargereservoir“pay”intends, poorvertiealand lateralcontinuity,andlowporosity(8%on average)andpermeability(often< 1red).The wells were initially completed in the Lower, Middle, andUpperClearfork,at measured depthsof between6200and7200feet. The majorityof the originalcompletionintenfalswerein theLowerCk?arfork,which is consideredthe main pay. Additionalcompletions were added in the Upper Clearfork and Glorietaduringworkoverprogramsin the 1970’s.At the inceptionof thewaterflood project in 1987, many of the original wells wereconverted to injeetors, and the remaining producerzwere re-completedup sauetum.Although the reservoir may be difficult to characterizegeologically, the Clearforkdoes behavelike a materialbalance~oir, andthe deelinecurvetechniquesoutlinedptevioualyareapplicable.Likemanyolderfields,thereamlimiteddataavailablefor analysis. Much of the fluid properdes data, as well as thecompletionintervalshave beenestimated. me oil flowratedatawas allocatedto individualwells on a tract basis, andmay be inerror, although the errorsare not likely to be significantbeeattsethe wells were tested for allocationon a semi-annualbasis. Inaddition, there are no bottomholepressuredata availablefor theNorth Robertson Unit and for analysis purposes we assumedpti= O, which means that the rate function term, (q/Ap), wasXtUSlly (q/’i).

Umt ~.. . .

ReservoirPr9pcrties:Wellboreradms,rW = 0.31ftEstimatedgrosspayinterval = 1300ftEstimatednet pay thickness,h = 250ftAverageporosity,#(fraction) = 0.08Averageimeduciblewatersaturation,Swim = 0.25Averageformationperrncabfity,k < l.Omdoriginal nominalwellspacing = 4oacr&Curnmtnominalwellspacing = 20acres

FluidPropem”exAvrxue oil fortnationvolumefactor,B = 1.30RB/sTBAvera~eoil viscosity,P = 1.3ocpJnitiaItotalcompressibility,cri = 12.oxlti psi-lAveragetotalcompressibility,cl = 20.oxlo-~psi-i

ProductionParameterT.{*;.1 *c*rvnir nrecwm?fLCm. D;.,.AUUS.—. .“- y...””---- \—- ,.r. = 2800psiaFlowingbottomholepressure,pwf unknown

NRU Well No. 4202Figure 18 shows the locationof NRUWell 4202 with respecttoits weUpattern and the unit. Tlds well was drilledin 1962,andcompletedin both the LowerandUppmClearfork.Thewellwasstimulatedwith 3,000gallonsof acid,andhydraulicallyfmcturedwith 60,000gallonsof fracturingoil and90,000poundsof 20/40aand. The well initiallytestedat 141STBO/D. It had producedapproximately207 MSTBas of July 1994. Semilogand log-logproductionplotsshownin Figs. 19and20 indicatethattherewereno significant rate fluctuationsduring primaryproduction. Itisinterestingto note the decreasein declinerate at approximately5* days of producingtime. This stabilizingof the productionrate may be a responseto an adjacentwaterfloodprojeetthatwasinitiated during the same time period. The responseto the unitwaterfloodcan be seenat approximately9,000days,whentheoilrate incmsed sharply.

~ (Fig.22)

We nowconsiderthe typecurvematchingof the rate,(c@p), rateintegral,(q/@)i, and mte integral derivative,(q/Ap)i~,functionsplottedversusmaterialbalancetime,i, on the Fetkovich/McCraytypecurve. The threemtefunctionsareforcematchedon theArps6=1(harmonic)dedne stemas before,and the appropriatematchpointsareobtained.To obtainthebest typecurvematch,thedatawasreinitializedat atimeof 549days. Afterreinitiali=tion,weobtaineda goodmatchon the depletionstemsanda uniquematchon the transientstemsat an r~ value of 160. From the log-log productionplot (Fig.20),we notethat the transientflowperiodhadnotendedat a timeof 549 days,and the transientmatchshouldbe valid. Usingthisdimensionless radius and the time and rate match points, wecalculatevaluesfor in-placeoil, drainagearea,permeability,andskin.

Type Curve Match Fetkovich/McCmy Type Curve (RadialFlowin a BoundedReservoir).

MatchingPararnetecr~ = 160

[f*p = 1.0 [~p = 3300days

[9*P = 1.0 [q/Apkp = 0.019STB/D/psi

Curve~ : (’Rg.22)Baaedon our estimatedvalues for total compressibilityand netpay tidcknesswefind

Net = 62.7 STBlpsiN = 3.13MMSTB

A = 35.02SCKeS

re = 696.9 ftkh = 19.61 md-ftk =0.08 md

s = -2.6~ (Fig.23)

Due to the lack of bottomholepressuredata, it is not possibletouse ~mlplottedvemusNP to estimatemovableoil. Instead,weplot the daily oil rate, q, versus NP to find the movable oilvolume. The extrapolationof this line to the NPaxis intercepty&&d~p~eo~blevolumeat the timewhenall mswoir energyhas

Estimatesforprimaryandsecondaq movableoil were190MSTBand 130 MSTB, respectively. Our results indicate thatapproximately10,000STB of primarymovableoil remainedinthe drainageareaof ihe wciiwhenthewaterfhd was initiatedin1987. The analysisof the secondarydeelinetrendis difficultatpresentdue to a lackof aeeondaryproductionhistory. However,using the preseni SLXOiid~rj ddh fiit~ vw ~sti~.~i~ ih~!approximately113MSTBof reeovembleoil remainedas of July1994. Obviously,theactualmovableoil volumewillbe lessthanthevolumecalculatedif thewellwereproducedto zeromtc.

Np.mov= 190.0MSTB(primary)NP,mov= 130.0MSTB(seconda~)Recovery Factor =6.07% (primary)

= 4.15%(secondary)

The resultsof the typecutvematchandmaterialbalanceanalysisyield realistic estimates for original-oil-in-place,movable oil,drainage area, permeability, and skin factor. The primaryrecoveryfactorcalculatedusingthe valueof original-oil-in-placefromthe typecutvematchis typicalforwellsin thistUdL

A pressure build-up teat was performedon well NRU 4202 in1988,and the permeabilityto oil wasestimatedto be0.2 md,and

SPE28688 L.E.Doublet,P.ICPande,T.J. McCollum,andT.A. Blssingsme 9

the calculated skin factor was -3.7. Both of these values areconsistentwith the valuesobtainedfromouranalysis,althoughitshould be noted that the calculations for drainage area,permeability,and skinfactorareadverselyaffectedby thelackofanaccuratevalueforthenetpayinterval.NRU Well No. 1004Figure24 shows the locationof NRUWell 1004withrespecttoits well pattern and the North Robertson Unit. The wetl wasdrilled in 1960,andcompletedin the Lower,Middle,and UpperClearfork. It has producedapproximately135.5MSTBas ofJuly1994. The semilogand log-logproductionplots shownin Figs.25 and 26 indicatethat there wereseveralrate variationsand anextendedperiodof an apparentlyconstantproductionrateduringprimary depletion. Due to the fact that the productiondata isallocatedmonthlyon a tractbasis,webelievethattheratebehaviorbetween5,500 and 10,000daysmaynot representthe well’struedepletion behavior. In order to achieve the best estimate oforiginal oil-in-place, and the correct type curve match, onlyproductiondata priorto 5300 dayswasusedin ouranalysis.

curv~ . : (Fig.28)The p~oductionrate functionsareplottedversusmaterialbalancetime,r,on theFetkovich/McCraytypecurveandforcematchedonthe b=I (harmonic) decline stem. Upon further review, wereinitializedthe dataat a timeof 336days atwhichpointthewellachieveda stabledeclinerate. Afterminitialization,weobtainedagoodmatchon the b=l depletionatcmsaswelIas a uniquematchon the r~800 transientstem. Fromthe log-logproductionplot@lg.26),VVenote that the transientflow-periodhadnotyet endedat 336days,and thereforethe transienttypecurvematch-isvalid.

TyP Curve Match Fetkovich/McCray Type Curve (RadialFtowin a BoundedReservoir).

MatchingParametecr~ = 800

[tap = 1.0 [JMp = 2000days

[91MlMP= 1.0 [q/@kP = 0.013STB/D/psiCmve~

Fromour estimatesof totalcompressibilityandnet pay thicknesswe fmd

Net = 26.0 STi3ipsiN = 1.30MMSTBA = 14.52acresre = 448.7 ftkh = 18.41md-ftk = 0.07 md

s = -0.6~ ~lg. 29)As with well NRU 4202, we againplot the daily oil productionrate, q, versus NP to estimate the movable oil volume. Theextrapolationof thestraightlineportionof thisdatato theNPaxisinterceptyieldsthemovableoil volumeat thetimewhenatlof thereaetvoirenergyhas beendepleted. Ourresultsindicatethattherewere approximately105MSTB of primarymovableoil, and 75h4STBof secondary movableoil (using the averagesecondarydeclinefor the unit). The analysisof thesecondarydeclinetrendmay be inconclusive due to a lack of secondary productionhistory, however, we estimate that approximately44 MSTBof_ fi.,amhla nil mmdnd ae nf ]llly !$)$)4.WC” .- Q“.” “.. .“... ---- - “. .-.

Np.mov = 105.0MSTB(primary)Np,mov = 75.0MSTB(secondary)Recovery Factor = 8.08%(primary)

=5.77% (secondary)

. . .

The analysis techniquesused for this well show that the analystmustbe carefulwhenmajorevents,such as longshut-inperiods,or questionableproductiondataaffecta well’sproducinghistory.If a goodwell historyis available,the analysisand interpretationcanbe accuratelyperformed.Theresultsof ourtypecuwematchas well as our materiatbalanceanalysisindicatethat the well isdraininga verysmall areaand mayrequirestimulation,althoughthe primaryrecoveryfactorestimatedfromthis anatysisis typicalforwellsin the unit.Sprayberry Trend, West TexasThis particularSpraberryreservoirwas initiallydevelopedon anominal80 acrewell spacingand additionalout-of-patterninfillwetlsweresubsequentlydrilledthroughoutthe field. Theoriginatreservoir producing mechanismwas solution gas drive, but ispresentlygravitydrainageand waterfloodin certainareasof thefil-lfl..”.”.The SpraberryTrend in this field consists of two distinctzones(UpperandLower)withgrosssandintervalsof 150to 600ft and330 fq respectively. The averagetotat net sand intervalfor thewellsin thisfieldis approximately190ft. Theaverageporosityisabout9 percentandpermeabiliticsareextremelylow(<e 1red).After approximately30 years of primary production,a limitedwaterfloodwas initiatedin certainareasof the field,but hashadlimitedsuccessdue to the suspectedpresenceof preferentialflowpaths within this reservoir. While it is probablethat the lackofwaterfloodcontinuityis due to reservoirheterogeneity,it is alsoprobablethat thereis a low sweepeftlciencydue to communica-*A. fif hydmtjli~ frsc~m~ ~t’w~n indtidud We~S.-“s. “. . . -s”.. ----

l%eoriginal-oil-in-placefor thii reservoiris estimatedto be 112.8MMSTB. The estimates for primary and secondaryultimaterecoveriesare 1.9percentand 1.7percent,respectively,althoughindividualwell primaryrecove~ factorsrangeas highas 7 to 10percentforSprabemymacrvoirsin general Theinitiatpressureinthismaervoirwasestimatedto be 2650psia.In this case, only monthly oil production data is availableforanalysis. In addition,we haveno accuraterock,fluid,or bottomhole pressure data available for analysis. Since bottomholepressuredataarenot availableweassumedpWf= O,whichmeansthattheratefunctionterm,(q/Ap), was actually (q/Pi).

Pro~ReservoirPropemk.r

Estimatedwellboteradius,rWAveragenet paythickness,hAverageporosity,@(fraction)EstimatedirreduciblewaterSSL,Sw.~~Averageformationpermeability,kOriginalnomimdwellspacing

FluidProperties:Averageoil formationvolumefactor,BAverageoil viscosity,PInitialtotalcompreaaibitity,criAveragetotatcompressibility,c1

ProductionParontetersInitiatteservoirpressute,piFlowingbottomholepressure,p~~

Spraberry Well A

= 0.3 ft= 190ft= 0.09= 0.30

<< 1.0md= 80acres

= 1.33RB/sTB= 0.9Cp= 12.4x106psi-l= 18.3x10-6psi-l

= 2650psiaUnknown

Thiswellwasdrilledin 1957andcompletedin boththeupperandlower sections of the Spraberry. The well has producedapproximately123MSTB as of September 1993. The scmilogand log-log productionplots shown in Figs. 30 and 31 indicatethat the oil rate varied significantly during the later stages ofprimarydepletion. ‘llte rate integraland rate integralderivativefunctionsreducethe affectsof the datascatterevidenton the ratefunctionprofde@lg.32). This smoothingallowsfora bettertypecurvematchevenforratedatawitha highdegreeof scatter.

10 DeclineCurveAnalysisUsingTypeCurves-AnslysisofOitWellProduction DataUsingMaterialBalanceTime: SPE 28688—.. .—

Curve~ : @lg. 33)

The (q/Ap), (q/Ap)i, and-(ghp)id rate functionsare plottedversusmaterialbalance time, f, and then force matched on the b= 1(harmonic)declinestemas dictatedby theory. Uponobtainingamatchof the dataand thetypecmvetrends,theappropriatematchpdlrtt Va:iies are taken. WI=then nht~~n ~ va]~e for the. . ., .... .. . .dimensionlessdrainageradhIsmatchingparameter,r~, which isused to estimatepermeabilityand skin factor. The matchof thetatc functionson the r#2 transientstemis excellen~

Type Curve Match Fetkovich/’McCrayType Curve (RadialFlowin a BoundedReservoir).

MatchingParametecr~ = 12

[t&p = 1.0 [~p = 8500days

[qRflMP= 1.0 [q/ApkP = 0.0069 STB/D/psi

Curve~Usingourestimatesof totalcompre.ssiiiiiityandnetpay thicknesswe fmd

NC1= 58.65 STB/psiN = 3.20MMSTBA = 45.90 acresre = 797.8 ftkh = 2.024 md-ftk =0.01 mds = -5.4

~ (Fig.34)

Sincewe again lack bottomholepressuredata, we plot dailyoilproductionrate, q, versus IVpto estimatethe amountof movableoil. Theextmpolationof thislineto theNPaxisinterceptindicatesthat the total primarymovableoil volumeis 160MSTB,and titthere were approximately 35 MSTB of primary movable oilremaining in the well’s drainage area as of September 1993.presently,thereis insuffkientdataavailableforcommentas to thevolumeof secondaryoil thatmaybe produced.

Np,mov = 160.0MSTBRecovery Factor = 4.99%

The results of the typecurvematchandmaterialbalanceanrdysisyield realistic estimates for original-oil-in-place,movable oil,permeability, and skin. The recovery factor (4.99 percent)calculatedusingtheestimateof original-oil-in-placefromthe typecurvematchand materialbalanceanalysisis slightlyhigherthanaveragefor the field,and the reservoirqualityin the am of thiswellappearsto be high.Althoughwe had to estimatevirtuallyallof therockandfluiddatarequiredfor thecalculationof permeability,the resultingvalueofO.O!md is ~epm.wntativefor this extremely low permeability,turbiditcreservoir. The calculatedskin factorof -5.3 is whatwecould reasonablyexpect for a low permeability,hydraulicallyfracturedwell completion. As of September1993,the well hadproduced123MSTB,or 77 percentof the recoverableoil volumecalculatedfordepletionto zerorate.Giddings (Austin Chalk) Field, Burleson Co., TXThe Austin Chalk is an Upper Cretaceous,naturally fracturedreservoirconsistingof a homogeneousmicritic limestonechalkwith interbeddedblack shales. The reservoirhas a low matrixpermeability,with a dominantnatuml tlacmre system trendingfrom Northeast to SouthwesLbut the presenceand influenceofthis fracturesystem is not well correlated. The main producingoend parallelstheTexasGulfCoastbetweenthe PearsaUFieldtothe SouthweaGand the GiddingsField to the Northeas4although

significant exploration and production activities are presentlyoccuningin EastTexasandLouisiana.The AustinChalkconsistsof an immaturezoneabove6000ft. agenerationandaccumulationzonebetween6000and7000f~anda morematureoil generationand accumulationzonebelow7000fg in whichLhef.mcmesystemis mostdominant.~

TheGiddings(AustinChalk)Fieldwas firstdevelopedin the late1970’s. Initial field development used vertical wellbores,however, with the rapid development of horizontal welltechnologyin the early1980’s,almostall subsequentwellsdrilledin the field have beenhorizontalto take advantageof the AustinChalkfracturesystem.In theGiddingsField,theAustinChalkhasanaverageporosityofapproximately5 percentandan avemgepermeabilitybetween0.01and 1.3md,dependingon the relativecontributionsof thematrixand fmcturcsystems. The reservoirhas an avemgethicknessofbetween200 and 800f~ Totalcumulativeproductionas of 1993wasestimatedto be 150MMSTB. Theoriginalreservoirpressurefor theGiddingsFieldwasestimatedto be3326psiaThequantityandqualityof productiondatawasfairlygoodforthewells we analyzed. In particular,both daily rates and surfacepressuresareavailable. Theproblemwefacein theseanalysesisour inability to accumtelyconvert surface flowingpressure tobottomholeflowingpressure,as wellas the lack of accumtcrockandfluid data. To be consisten~surfacetubingP=WRP,P willbe used instead of pWffor both of the Austin Chw csscs wepresent.

ReservoirPropem.es:Estimatedwellboremdius,rW = 0.25 ftEstimatednetpaythickness,h = 3ooftAveragepotosity,# (fmction) = 0.05lMhtcxI irreduciblewaterSSt.,Swirl = 0.30Avemgeformationpermeabfity,k = 0.01-1.3 md

FluidPropem.eEAvemgeoil formationvolumefactor,B = 1.35RB/sTBAvemgeoilviscosity,P = 0.45CpInitialtotalcompmssibility,cfi = 16.4xI0-Spsi-lAvemgetotalcompressibility,cl = 21.OXIO-Spsi-l

n-- J..- --- 23....”-...-”.rrvuudiun rurumcux a.

Initi# reservoirp~ure, Pi = 3326psiaFlowtngsurfacembmgpressure,PWI = 80 psia(7/94)

Barton Lightsey Well No. 64This well was drilledand completedin 1991,and has producedapproximately330.5 MSTB of oil as of July 1994. The wellpresentlyhas a dailyoil mtc of 102STB/D,a producingGORof5275 scf/STB,and a watercut of 9%. The semilogand log-logproductionplots shown in Figs. 35 and 36 indicatethat the oilproductionmtestartedout veryhighand thendeclinedrapidly,aswould be expectedfroma dual porositysystem(fmcturdmatrixdrainage).Afterapproximately500 daysof production,the wellwasplaced. “mel:f% .-A *ha 4+1 nrmhmtinn mte hw.mav~ ~~~q)!~ f~~.rn. ~700,16$UAL&bWIUu.” “.. y.””**9.”.... . ...-.—

STB/D to about 400 STWDbeforeresumingthe initial declinerote. The mteintegralandrateintegralderivativefunctionsshownin Fig. 37 wereslightlyaffectedbyperiodicmtevariationsatearlyproducingtimes. Theavailabdityof dailyproductionandsurfacepressure data improvesour chancesof obtaininga unique typecurvematch.~ (Fig.38)

ASbfom, (q/~p), (q/Ap)i, ad (q/@)idSMplottedversusmaterialbalancetime,Z,andmatchedon theFetkovich/McCraytypecurve.Fmm Fig. 35, wesee thatdueto numerousratechangesandshut-in periodsearly in the well’slife, it is difficultto obtaina uniquematch on the transientflow stems. To improveour chancesforobtaininga matchof the transientdata,weminitialti thedatato

SPE28688 L.E. Doublet,P.K.Pande,T.J. McCollum,andT.A.Blasingame 11

a timeof 132days to removethe most significantpartof the ratedatascatter.After reinitialization,we obtaineda good match on the r =28

tfl’transientflowstem. It is interestingto note the effectthat e gaslift processhas on the flowrateprofile. The rateprofileshowsaspike-liketrendin Fig. 36but is smoothedto a pairof overlappingtrendsfor the (q/Ap) functionin Fig. 37. This behaviordoesnotaffecttheoverallqualityof the typecurvematch.

Type Curve Match FetkovichM4cCmyType Curve (RadialFlowin a BoundedReservoir).

MatchingPammettxra = 28

[t&p = 1.0 [t~p = 330days

[9AJMP = 1.0 [9/APhffp = 0“31s~~@i

:kxdour estimates of totalcompressibilityandnetpayttdcknesswe

Net = 102.3STB/psiN = 4.87MMSTBA = 80.73acresre = 1058.0ftkh = 68.67 md-ftk = 0.23 mds = -5.0

~ (Rgs. 39-41)

Plots of ~d (qlAp), and q versus Np are used to estimatethemovableoil volume. We assumethatbecauseflowingbottomholepressureis held constantaftergas tift is initiated,thestraightlineextrapolationof q to zeroyieldsaboutthesamevalueformovableoii ss rioesextrapo”mtionof~d or (gl~p)to ~he.NP~ ifi*=~ept.All of thematerialbalancemethodsyielda movableoil volumeof360MSTBwithgas lif~whichmeansthatthereareapproximately30 MSTB of movable oil remainingin the reservoirat presentCt?rtditkms:We also note that duringthe periodbeforeinstallationof gasliftjthat theextrapolatedmovableoil volumesfor all materialbalancemethodsare also quite similar (=310MSTB). The results of thevolumetricanalysisaregivenbelow.

Np,mov = 360.0MSTB(withgas lift)Recovery Facwr = 7.39%

The typecurve and material balanceanalysesyield acceptableresults for original-oil-in-place,movable oil, and the reservoirflowcharacteristics.Thecalculatedrecove~ factoris in therangeof whatwe wouldexpectfor AustinChalkwells,andwenotetheshortoperatinglife that is also characteristicof thesewells. Thecalculatedpermeabilityof0.23md andskin factorof -5.0arealsorepresentativevatues. ThecalculatedpermeabilitymayIMin errorsincewemayhavetmdereatimatedtheeffectivenetpayinterval.As tids is a horizontal well, it appears that we may be able toaccuratelymodel the behaviorof horizontalwells in the AustinChalk using the Fetkovich/McCray type curve which wasdeveloped for vertical wells (radial flow). In addition, thisanalysis technique may provide a method to estimate the welldrainagearea,whichis oftenunknownforAustinChalkwells.Searmardo Carrabba Well No. 225This well was drilled and completedin 1993,and has producedapproximately 92 MSTB of oil as of July 1994. The wellpresentlyhas a dailyoil productionrateof 58 STB/D,a producingGORof 5535SCUSTB,and a watercut of 11%. Thesemilogand

log-log productionplots shown in Figs. 42 and 43 exhibit thecharacteristicbehaviorof a dualporositysystem. Oil productionratedeclinesxapidlyas the fracturesystemis dmined,andthentherate of declineis reducedduring the period in whichthe matrixdominates.AswithwellBartonLightsey64,weagainhavedailyproduction rate and surface pressure data for more rigorousanalysis. The rate, rate integral, and rate integral derivativefunctionsare shownin Fig. 44. These pressurenormalizedratefunctionsate notgrearlyaffectedbyearly-timerateanomalies,andtherefore,datareinitializationis not required.

Curve~ : (Fig.45)The rate functionsare onceagainplottedversusmaterialbalancetime,i, andmatchpointsareobtainedusingtheFetkovichA4cCraytypecurve. Wehavea goodtransientmatchon ther~800 stem,andwewilluse thisdimensionlessradiusalongwiththe timeandratematchpointsto estimatevaluesforoil-in-place,dminage-permeability,and skin factor. This well is P=n~Y Producingunder boundary-dominatedflow conditions, attd is probablynearingtheendof its operatinglife.From our calculations,this weIl appearsto be draininga muchsmaller volume than the Barton Lightsey well, which is notsurprising considering the Scarmardo Carrabba well’s per-formanceto date. Aawe mentionedfor theBartonLightseywell,theanalysisandinteqxetationfromtypecurvematchingmaybe inerrorbecausewe areanalyzinga horizontalwellwithtypecurvesderivedfora verticalwell.

Type Gove Matclc Fetkovich/McCray Type Curve (RadialFlowin a BoundedReservoir).

MatchingParametecr~ = 800

[t&p = 1.0 [t~p =84 days

[91MlMP= 1.0 [9/APIMF’= 0.32 s~~/Psi

Curve~ 1“Forourestimatesof totalcompressibilityandnetpaythic-knesswefind

Net = 26.88STB/psiN = 1.28MMSTBA = 21.21acresre = 542.3ftkh = 162.90md-ftk = 0.54 mds = -1.0

~ (Figs.46-48)

Plots of id (qhp), and q versus Np are used to estimateN~mowand again the computedmovablevolume for all threemethodsis exactlythesame. Primarymovableoil for thiswellisestimatedto be 100MSTB,indicatingthatthe remainingmovableoil volume is less than 10,000 STB. The recovery factor isstightlyhigherthanfortheBartonLightsey64eventhoughno gaslift processwas initiated. The comptison of recovexyfactorsissomewhatmisleadingwhenwe considerthat the BartonLightseywell will recover approximately3.5 times as much oil as theScarmardoCarrabbawell. We assumethatthehigheroil recoveryfactor is due to better reservoir quality, if not bettercommunicationbetween the fracture and matrix systems. Allthings being equal, one possible recommendationwould be toperforma significantstimulationtreatmenton thiswelt.

Np,nIOV = 100.0MSTBRecovery Factor =7.8 1%

12 DeclineCurveAnatysisUsingTypeCuwea-Analysisof011WellProductionDataUsingMaterialBalanceTime: SPE2g688—Aj@catimi‘wFiild ‘-

The type curve matching and material balance analysesyieldconsistent results even though we have used a type curvedevelopedfor vertical wells to analyzehorizontalwells. Whilethereservoirqualitysumoundingthiswell appearsto bemuchhigherthan that of the BartonLightseywell, the movableoil volumeismuch lower,whichsuggestsless thanoptimalcommunicationofthe welland the l’CW’VOti.

Santa Clara (Lower Repetto) Field, Offshore, CAThe Santa Clara (LowerRepetto)Field (Fig.49) was developedon an approximate40 acre nominal well spacing beginningin1984. There are presently9 producingwells in the field at anaverage true vertical depth of 7500 fee~ The originalreservoirpressurein theLowerRcpettowasestimatedto be 5900paiaThe Lower Repetto reservoir is characterizedby four distinctzones consisting of thinly bedded turbidite sandstones, withinterbeddedsiltsandshaleswhichlimitboththeVC1’ti@ andlateralcontinuity of reamwoirproperties. Theseclasticturbiditeawerefotmedas a resultof densitycurrentswhichweredepositedonthemid andouterfanportionsof a tmlidite lobe.This depositional process resulted in the fortnation of poorlysorted, medium to very fine-grainedarkoaesand Iithicarkoaea.Thesesandahaveporoaiticsrangingfrom5 to 35 percen~withanaverage in-situ oil permeabilityof less than 3 md, and possiblymuchleaseven thoughcorepermeabilitiesfor theLowerRepettooftenaverage20 md or higher. The in-aimreservoirpermeabilityis much lower than the calculatedcore permeabilitydue to theunconsolidatednatureof the rock,andrelativelyhighoilviscosityat resenfoirconditions.-.. . .- AL:-1--1Due to tms lacKd corisci~ldrition,WUIU ~r-uuuull ~aa rnajer. . . . - ,...I..,.*:n”:. .problem and gravel-packedcompletionsusing slotted linersamrequired. l%e high viscosity of the oil at reservoirconditionsresults in the rapid depletion of reservoir energy, therefore,pmsaurecommunicationis limitedto withina fewhundredfeetofany particularwell. The LowerRepettoformationdipsat 10”to20” to the WIXLand due to the placementof the drillingplatformon the Upper Repetto structure,LowerRepettowells are inter-sectedat anglesbetween50”and6@relativeto horizontal.lle original-oil-in-placefor thereservoiris estimatedtobe greaterthan 300 MMSTB. Total productionfromthe LowerRepettoasof January 1994was 3.6 MMSTBoil and 3.9BCFgas. Ultimaterecoveryis expectedto be less than3%dueto theheterogeneous,low permeabilitynature of the reservoir,in additionto the highcoatof developmentdrilting.However,given the producibilityproblemsas wellas theexpenseof operation, the operator has elected to obtain continuousmeasurements of flow rate and bottomhole pressure. Sub-sequently, the quantity and quality of oil productiondata andbottomhole pressure data for the wells is very good, and weexpectto performa rigorousanalysisof thesedata.

Resenwir Propertie~Wellboreradius,rw = 0.146ftNet pay thickness,h = 120-150 ftAverageporosity,#(fraction) = 0.25Averageimeduciblewatersaturation,SWim= 0.35Averageformationpermeability,k < 3.0 mdOriginalnominalwellspacing = 40 acres

FluidProperll”eXAverageoil formationvolumefactor,B = 1.42RBISTBAverageoil viscosity,K = 2.ocpInitialtotalcompressibility,cfi = 10.OX1O6psi-lAveragetotalcompressibility,c1 = 1LOxl@ psi-1

Prd4ction PamnuterKIflitialXWMXVOh P-lWC, pi = 5900paiaWelldeviation = 50”-60”

S. Gilds Well S-42Figure50 showsthe locationof WellS-42withinthe SantaClaraField (Lower Repetto Reservoir). Well S-42 was drilled andcompletedin 1986andhasproducedapproximately620MSTBofoil as of January1994. At present,thepresentdailyoil rateis 113STB/D, with a producing GOR of 737 scf/STB, a flowingbottomholepressureof 2126psia, and a watercut of c 5 percent.This well intersectsthe Lower.Repettoresmoir at 56.6°andhasanestimatednet verticalpaythicknessof 150ft.The aemilogand log-logproductionplots are shownin Figs. 51and 52 and indicate that the oil rate is decliningsmoothly,butquite rapidly, which is probably a result of the producibilityproblemsmentionedabove. The rate integral and rate integralderivative functions, as seen in Fig. 53, show no instancesoferraticratevariationsin theproductionhistory.~ (Fig.54)

The - mk functions,(q/Ap~, (q/Ap)i, and (q/Ap)~~ plot~dversusmaterialbalancetime, L and matched on the FetkovicldMC@I ~ curve, as shown in Fig. 54. We have obtainedavexygoodmatchon the transientflowstemsat a valueof r+,While this is a goodmatch,we mustspeculateas to whyther~ isso low, which indicates an extremely high level of near-wellStidion. The obviousexplanationis thatWellS-42is highly

.The rate functions also indicate that the well is just beginningboundary-dominated flow, and this behavior may adverselyinfluencethe analysisand interpnxationof the wellperformance,However, we believe that the results of this analysis arerepresentative and consistent with boundary-dominatedflowtheory. ~a ~ei,lt= nf th;c analvc{e akn hdicat$ ~h~~1~~ wC!! isAs.” SUu”.w “. M..” . ....”..- ..-.” . ..-.draininga much largervolumethan wouldbe indicatedby a 40acre well spacing. This interpretationof a largerdrainageareamaybe dueto the uncertaintyof the netverticalpaythicknessandthe significant deviation of the well. Given the difficultiesassociated with interpreting the perfmttmiiceof tiii~ -wdi, ‘werecommend the development and application of decline typecurvesfor the analysisof horizontalwells(seeRef. 16).

Type Curve Match Fetkovich/McCray Type Curve (RadialFlowin a BoundedReservoir).

MatchingPatametecra = 4

[f&p = 1.0 [z~p = 5900days

[q&P = 1.0 [@PhtP = o.068 s~~lwi

Using the results of our type curve analysis along with ourestimatesof total compressibilityand net pay thicknesswe havedevelopedthe followingresults

Net =401.2 STWpsiN = 36.5MMSTBA = 273.82acresre = 1948.5ftkh = 17.36 md-ftk =0.12 mds = -8.1

~ (Figs.55-57)In thiscase,we estimateconsistentvaluesof movableoil fromtheplots of (q/Ap) and q versusNP,whichyield about 1.0MMSTBtotal recovery. However, the ~d versus NP plot predictsapproximately1.7MMSTBof movableoil volume, In an attemptto maolvethis discrepancy,we considerthat the~d functionisdifficult to interpret relative to the actual pressurelevel in thereservoir.

SPE 28688 L.E. llouble~ P.K.Pande,T.J. McCollum,snd T.A.B1asingame 13

Is there really 1.7 MMSTB of movable oil? Probably not atcuxrentoperatingconditions,especiallywhenweconsiderthatthebottomholepressure level has risen and stabilized for the pastthreeyeara. This rise andstabilizationin the bottomholepressuresuggeststheneedforstimulationandprobablyMlcisl lit.Duetotheincmase andstabfizationofpWlat a highpressureleveland since the well has only just entered the pseudoateady-stateflow regime, the rate and pressure drop normalized rate-cumulative oil plots probablyyield the most accuratevalue ofmovable oil for this case, and will be used for referencein ouranalysis.

.10 nll~s~.Vp,mov– ..” ... ----Recovery Factor = 2.74%

The type curve and materialbalanceanalysesyield acceptableresults for estimatesof original-oil-in-placeandmovableoil, andthe calculatedrecoveryfactoris reasonablefor wellsproducingfrom the LowerRepetto. The calculateddrainageareais muchlargerthanwhatweexpected,butwebelievethiscanbe attributedto a lack of knowledge of net vertical pay tldckness and thedeviationof thewell.Perhaps the most intriguing result of this entire analyaisis thematch of transient data on the r- stem, which yields anestimated permeabilityto oil of 0.12 md and a calculatedskinfactorof-8. 1. This estimateof skin factoris unrealisticfor anyvexticalwellcase,withthepossibleexceptionbeingthecaseofanextremely large, high conductivityverticalfracture. As this isclearlynot the case,we can onlyassumethat the akinfactorcanbeattributedto welldeviation.In contrast, the aemilog and log-log analysis performedon apressure build-up test taken in 1992 gave an estimatedpermeabilityto oil of 0.8 md and gavea skin factorof +2.0. Ifthe permeabilityand skin factorare correctedfor the effectsof

*:.. ..~ ~.11Amviatinn we twrwrr that @c txtmp~~par-i penetrhml us,- “.AA -..-”.., ..- *..r-. -.– . . . . . . .

valuea would be comparable to the values obtained fromproductiondataanalysis.

- -.. .-” ●. .,.-..-+.91.,-.*;-m+a+-m-_,;nn flnwiii order i~ USe tjTk Gui%a w SbbUL aLGtJ WLUUaW AU. . .. WUW.. ..v r.characteristicsfor the LowerRepettowells we shouldprobablyuse a matchmg parameterthat incorporatesdeviatedhorizontalwell length, instead of effective wellbore radius. Thedevelopmentand applicationof type curves for the analysisofproductiondata for horizontalwellswill aid in both the analysisand interpretationof problemslikethis.lcSUMMARY AND CONCLUSIONSIn this work,we havedeviseda rigorousandconsistentprocedurefor theanalysisand interpretationof long-termoil wellproductiondata *uaittg- “’wv*rnfit~hinotechniques. Specifically,weproposethe G“o~”;lti;Z;%;lcCray type curveto estimatetotal and movable reservoir volumes, as well as the flowcharacteristicsof the reservoir. Further,givena limitedquantityof productiondata, we ahowthatwe can accuratelyinterpretandpredictreservoirbehavior.We also note that the use of rate integral and rate integralderivative functionsallow for the analysisand interpretationof“noisy” field productiondata. In addition,the integralfunctionsprovidebetter typecurvematchesand increaseconfidencein ourinterpretations.The analyais techniquesthat we proposealwaysyield excellentestimates of original and movable oil volumes, and accurateestimatesof reservoirflow characteristics,providedgoodearly-time dataare available. Ouranalyaistechniqueswereveflled byevaluationof the simulateddatacases, and we againrecommendthatqualitydata be takenearlyand oftento ensuremoreaccurate~ldy~ and iltte~t’etZtiOItS.

Themainconclusionsof thisworkare:

1.

2.

3.

4.

5.

6.

7.

For the case of single-phaseliquid flow, the analysisof anyproductionrate and bottomholepressurescheduleis possibleprovidedthat we use the materialbalancetime function,andthe appropriate rate functions for data matching duringboundary-dominatedflowontotheb=] stemof theFetkovicldMcCraytypecurve.*,-,__ -. .--. L..A...1“.. *,-.““.1.”* .mrl ;sltD1.nr,atusing UW nlGUI WU@y UJ aJ14uJ&- Q!lU ..1*’ p. “. p.““””””..rdw.linn

data is relatively straightforwardand can providethe sameinformation as conventional pressure transient analysis,without the as”. . ..eneia~d ~o~!of data acquisition, Or]05SOfproduction.The flow rate integral and flow rate integral derivative---—-.:----11-...s.’.------ . -..-*n A-l:”- *- P,,luflcuu~ ~UUWiu[ Illul6 acbu~abu-~.,~- .J p ““N” . . . . . . . ..-

n matrhee

than would be possible using flow rate data alone. Theseintegralfunctionsalsoeliminateproblemsassociatedwiththe~n~y~ of field nmdUC~~n &@ wigh erratic pK)dUC1.iOII rste. . . . . . ~.-and bottomholepressurebehavior.The use of data reinitializationfor the removalof early-timerate variationscan yield improvedtype curvematches. Theanalyat must be aware of major events in the productionhistory that mighthavechangedthe producingconditionsofthe wellor reservoir.The calculationof movableoil volumeusing theq verausNPplot yields acceptableresultsunlesspw, variessignificantly.The simulatedcases verify that the q verausNp plot yieldsresultssimilarto thosepredictedby themorerigorousplotsof(q/Ap) verausNP, and ~d versus NP. This conclusionhasalsobeenconfirmedforfielddatacasesforwhichsurfaceandbottomholepressuredataareavailable.The techniques introduced in this work give excellentestimates of maervoir volumes (total and movable), andreasonable estimates of formation flow characteristics.However, all of these estimates could be significantlyimproved if high quality transient production data areavailable,as wellas accuraterock,fluid,andcompletiondata.Additionalworkshouldbe developedfor theanalysisof long-tem. ~roduction data frQm.horizonta! wells. In addition,present decline type curve analysis concepts should beextendedfortheanalysisof multiphaseflowdata.

NOMENCLATURE

Formationand FluidPammeters:A=B=cl =C(i =

=!=stir~ =k =re =rW =rmu =

P =

drainagearea,ft2formationvolumefactor,RB/STBtotalsystemcompreaaibility,psi-linitialtotalsystemcompmsibility,psi-lporosity,fractionformationthickness,ftinducible watersaturation,fractionformationpermeability,mdreservoirdrainageradius,ftwellboreradius,ftapparentwellboreradius(includesformationdamageoratimuhuioneffects),ftfluidviscosity,cp.

Pressur@ate/Tii ParameteWb = Fetkovich/Aspsdeclinecurveexponentbpss = constantin thepseudoateady-stateequationforliquid

flow,as definedby Eq. 13or Eq. A-4Di = constantdefinedby Eq. 19,D-1m = constantin thepseudoateady-stateequationforliquid

flow,as definedby Eq. 12,psilSTB(q/AP)iti = constantdefinedby @. 18,STB/D/psi

= oil flowrate,mmk = originaloil in ptace,STB

DeclineCum AnalysisUsingTypeCurves-AnalysisofOilWellProductionDataUsingMaterialBalanceTime: SPE 28688Applicationto Fieldcases

)+/P .Np,mov =

E==

~

Pabn =

Pi =Pwl =Ptt =Ap =r =t =i=tq =

r =

cumulativeoil production,STBmovableoil, STBpressure,psiaaveragereservoirpressure,psiaaveragereservoirpressureat abandonmentconditions,psiainitialreservoirpressure,psiaflowingbottomholepressure,psiaflowingsurfacetubingpreawm,psiaPrPwf*P~u~ dropspsiradialdistance,rtime,daysN#q, materialbalancetime,daysequivalentconstantpmasuretimeas definedbyMcCray8,daysdummyvariableof integration

. .~ RcaJDomain

lr

PD

md

hi

mid

reservoirshap factorEuler’sConstant= 0.577216 ...dimensiordeasdeclinecumulativeproductionfunctioncircumferenceto diameterratio= 3.1415926 ..* Ap, ~enaiortks pressure functionfor

the;onstant flowratecaseBP

141”2kh(p,.pw)q, dimensionlessflowrate function

fortheconitait wellborepressurecasedimensionlessdeclineratefunctionasdefinedbyFetkovichdimensionlessdeclinerateintegralasdefinedbyMcCraydimensiotdmsdeclinerateintegralderivativetimction as definedby McCray*= dimensionlessradius&mensionkssdrainageradiusof rcaervoirakinfactorfornearwelldamageor stimulationdimensionlesstimebasedon drainagemadimensionlesstimebasedon wellboreradiusdimensionlessdeclinetimeas definedbyFetkovich

. .~ bphlce TmnsformDomain

~D = LapIacetransformof dimensionlesspressun fortheconstantflowratecase

~D = Laplacetsansformof dimensionlessratefortheconstantwellborepressurecase

u = Laplacespacevariable,dimensionless

Zo(x) = modifiedBesselfunctionof the 1stkind,zeroorderIi(x) = mod~ledBesselfunctionof the 1stkind, 1storderKo(x) = modilledBesselfunctionof the2ndkind,zeroorderKI(x) = moditledBesselfunctionof the2ndkind, 1storder

= calculatedE = dimensionlessdeclinevariableMP = matchpointpss = pacudosteady-statei = ~&@id = integal derivative

ACKNOWLEDGMENTSWe acknowledgethepermissionto publishfielddataprovidedby

● FhtaOil andChemical,Co. (’westernDivision,USA),

● Mobil Explorationand Producing,U.S., Inc.,● UNGCALCorporation(CoastalCaliforniaDivision),and

● UnionPacificResourcesCo. (UPRC).

We alsoacknowledgethe technicalassistanceof Dr.AnilKumarof Mobil Explorationand Producing,U.S., Inc., and Mr. DavidElmer of ~PRC regardingthe acquisitionand interpretationoftheirrespectivefielddatacases.Andfinally,weacknowledgethe technicalandcomputingsupportservicesprovidedby theDepartmentof PetroleumEngineeringatTexas A&M University,as well as the financialsupport of theUnitedStatesDepartmentof Energy(DOE)for fundingprovidedthroughtheDOEClassII OilProgram.REFERENCES1.

2.

3.

4.

5.

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12.

13.

14.

15.

16.

97Al.

18.

- “Analysis of Decline Curves,” Trans.,fi#&(;;45) 160,228-247.Nind,T.W.:Principles of Oil Well Production, 2ndEdition,McGraw-Htil(1981).Arps, J.J.: “Estimationof Primary 011Reserves,”Trans.,AIME(1956) 207, 182-91.Slider,H.C.: “ASimplifiedMethodof HyperbolicDeclineCurveAnalysis,”JPT(March 1968) 235-236.Gentry,R.W.: “Decline-CurveAnalysis,”JPT (Jan. 1972)38-41.Fetlcovich,M.J.: “Decline Curve Analysis Using TypeCurves,”JPT (June 1980)1065-1077.Fetkovich,M-J.,●t UL“DeclineCurveAnalysisUsingTypeCUNCS - Case HKtories7SPEFE (Dec. 1987) 637-656.McCray, T.L.: Reservoir Analysis Using ProductionDecline Data and Adjusted lime, M.S. Thesis, TexasA&MUniversity,CollegeStation,TX (1990).Blasingame,T.A., McCray, T.C. and Lee, W.J.: “DeclineCurve Analysis for Variable Pressure Drop/VariableFlowrateSystcms,wpaperSPE21513presentedat the 1991SPE Gas TechnologySymposium,Houston,TX, January23-24.Palacio, J.C. and Blasingame, T.A.: “Decline CurvesAn-1..c.:. TTe:m- Tmw P,,rUUL4J DL~ W*..86 . J ~“ “u%~~ : .AMq~!y~~~ of ~~~ ~~!!

ProductionData ,“ paperSPE 25909 presentedat the 1993SPE Rocky Mountain Regional/Low Permeability~~sw~~~ &mnnsium. Denver. CO. Amil 12-14,, ...r --------—---.-., __, --r--- __Blasingame,T.A. and Lee,WJ.: “Variable-RateReservoirLimits Testing,” paper SPE 15028 presentedat the 1986SPE Permian Basin Oil & Gas Recovery Conference,Midland,TX, March13-14.Dietz,D.N.:“Determinationof AverageReservoirpressurefromBuildupSurveys: SPEFE (August1965)955-959.MuskaLM.: Flow of Homogeneous Fluids ThroughPorousMedia, McGraw-HillBookCo., Inc.,NewYork(1937).Catter,R.D.: “CharacteristicBehaviorof FiniteRadialandLinear Gas Flow Systems - Constant Terminal PressureCase,” paper SPE 9887 presented at the 1981SPE/DOELowpermeabilitySymposium,Denver,Colorado,May27-29.Carter,R.D.:“TypeCurvesforFiniteRadialandlinearGasFlow Systems:ConstantTerminal PressureCase,” SPEJ(Oct. 1985) 719-728.Shih, M.Y.:Decline Curve Analysis for Horizontal Wells,M.S. Thesis,Texas A&MUniversity,CollegeStation,TX[i994j.lx.1: C.n.nMsA.. P A c+.A Ramnu U T 1.. ‘~pmp~i~n:Unug-muuumn.a, L.n., Uaau ..-,, -,, * *..., d,..

Rate Decline Analysis for Wells Produced at ConstantPresaureySPEJ (Feb. 1981)98-i04.van Everdingen,A.F. and Hurst, W.: “TheApplicationofthe Laplace Transformation to Flow Problems inReservoirs,”Trans., AIME (1949), 186,305-324.

SPE28688 L.E. Doublet,P.K.Pande,T.J. McCollum,andT.A. Blasingarne 15

19.

20.

21.

22.

23.

24.

25.

StehfesGH.: “NumericalInversionof LaplaceTransforms,”Communications of the ACM (January 1970),13, No. 1,47-49.(Algorithm368withcorrection)Matthews,C.S. and Russell,D.G. : Pressure Buildup andFlow Tests in Wells, Monograph Series, Society ofPetroleumEngineersof AIME,Richardson(1967)1.Igor-Graphingand Data AnalysisProgram(Version2.7),WaveMetrics,LakeOswego,OR,USA, 1992.PanSystemTM-WellTest AnalysisProgram(Version1.8),EdinburghPetroleumServices,Ltd., Edinburgh,Scotland,UK, April 1991.Hinds,G.S. and Berg, R.R.: “EstimatingOrganicMaturityFrom Well Logs, Upper CretaceousAustin Chalk,TexasGulf CoaaL”Trans., GCAGS (190) a. 295-300.Dake, L.P.: Fundamentals of Reservoir Engineering,Elsevier Scientific Publishing Company, Amsterdam(1978).Johnston, J.L.: Variable-Rate Analysis of Transient WellTest Data Using Semi-Analytical Methods, M.S. Thesis,TexasA8cMUniversity,CollegeStation,TX (1992).

APPENDIX A - DER1VATION OF MATERIALBALANCE PLOTTING FUNCTIONS FORPRODUCTION DATAIn this appendix,westartwiththe materialbalanceequationforaslightlycompressibleliquidwhichis givenby Dake~ as

~Np .................................................(A.l)?=Pi-Net

Wenotethatif we plot~ versusNPthenwe wiii obtaina straight*.—.–m-s-—-s llr-une or slope JIfVCfad fii*WC2pij+. we am .l.A Dvtrm+te the Z, .“” “a- .y” .- . .. y

versus NPtrend to’@ in orderto estimatethe “movable”liquid(oil)volume,NP,mov Of course,~ is typicallynot availableinpractice,so we-mustuse an aitemateapproachto appiyingthisconceptWe now considerthe so called“oilflowequation”whichrelatesrates and pressuredropsduringboundary-dominated(orpseudo-Steady-5tattj fiow”. This eqmaakxi is gii;efias

F = Pwf+ dress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(A-2)CombiningE@. A-1 and A-2 and solvingfor the pressuredrop,Ap = pt-pwfi we obtain

AP = P,-pwf= & Np + qbpss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(A-3). . . ..

wherethe paeudosteady-stateconstamtbpss,is givenby

bp’’=1412w%k)l.......... ...............(A-4)

For the interestedreader,a completederivationof Eq. A-3fromfundamental principles is given in Appendix A of ref. 25.Nommlizingbothsidesof Eq.A-3by theflowrate,q, we have

*.~~+’pss ..............................................f? Nc,(A-5)

where

i.+ . .. . . . . .. .. . . .. . . .. . .. .. .. .. .. . . .. . .... .. .. .. .. ... . .. .. .. ..(A-6)

Eqs.A-5and A-6 weredevelopedandverifiedbyBlasingarneandbet 1for the analysis of oil well productiondata. Taking thereciprocalof bothsidesof Eq.A-5andreamanginggives

~=*~ ........ .. .............................(A-7)Ap bpss 1 +-i

IVC@pSS

Eq. A-7showsthat a of plot q/Ap versus ~willyielda “hatmonic”declineon a Fetkovich/McCraytypecurveas dkuased byPalacio

and Blasingamelofor the analysisof oil and gas wellproductiondata.

Movable~ . .

SolvingEq. A-3for the flowmte,q. givm

q=$ (P-Pwh-&NP .. .. ..- .. .. . . . . ..”. .. .. .. . . . . . . ..(A-8)

Weimmediitclynotethatifpwl = constan4thena plotof q versusNpwillyielda straightlineof thefollowingcharacter

slope . .1NC#pSS

. .. . . .. . . . .. . . ... . ... . .. .. .. .. . . ... ..(A-9)

y-int.mept = #PI-Pw) .. . . .. . . . . . . . . .. . . . . . . . . . . . . . .. (A-IO)

x-intercept = Npmy=Np atq=O........................(A-l U

This result has considerable implications from a practicalstandpoint. In partictdar,we can use a plot of q versusNp as ameansto estimatethemovableoil for thecaseof a wellproducedat an approximatelyconstantbottomholepressure. For casesofvariablebottomholepressures,Eq. A-8 becomesless applicable,but we can still use the q versus Np plot as a “~mi-an~Ytic~”methodto predictmovableoil.Aninterestinghistoricalfootnoteis thatNind2developedEq.A-8froma comp~etely_ perspective.Hisgoalwasto developthe~ usingtheobservationof a lineartrendof q versus Np. In this light, we recall that the at@.yl@development of the exponential decline solution for a wellproduced at constant bottomhole pressure is given by Ehlig-EconomidesandRamey.17

Moe Otl. V~. . .

Thedevelopmentof a variable-rat.dvariablepressuredropformofEq. A-8 can be derivedby simply dividing throughEq. A-8 byL6ep~SUE drnp: Aps pi.pw$ This gives

~=~ 1 Np ..................................... (A-12)Ap bpss N@pss AP

Eq. A-12 and other variations of this result are developedanddiSCUSsdill tietSiiin R%. 8 Sttd if).

Consideringthe form of Eq. A-12, we note that a plot of q/Ap

versus N~Ap will yield a straight line with the following-em

slope = -- .... ...............................(A-13)N@p$f

y-intercept = ~bpss

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-14)

x-intercept =[1% =%atq/Apa . . . . . . . . . . . . . . . ..(A-15)Ap *OV Ap

Unfortunately,this method does not yield a direct estimateofPmow However,we can employa “semi-empirical”approachN

that uses a plot of q/Ap versus Np from which the movableoil,Np,moWMestimated from the linear extrapolationof the q/Ap

ttend to thex-axisinterceptat q/Ap=O. l%is approach,whilenotcompletelyrigorous,shouldprovideaccurateestimatesofNP,mOvwhilealso “fdtenng”the influenceof variableratesandpressuresThis is simply an intermediate recommendationand furtherresearchon this topicis warranted.

To developa straightfonvardand rigorousapproachto estimatethe movable oil, Np,mov we can use the material balancequation, Eq. A-1, as a plotting function where the averagereservoirpressure,~, is computedfrom Eq. A-2. RecallingEq.A-1wehave

16 DeciineCurveAnaiysisUsingTypeCurves-hsdysis ofOdWeiiPrmluctionDataUsingMatexiaiBakmceTime: SPE28688Applicationto FieldCases

..~J$~ ‘k? Ncl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(A-1)

If we have an estimateof the pseudostidy-smte constant,bpsf.fromsay, typecurveanaiysisor usingtheqhp versusN~Ap plotas described above, we can calculate the average reservoirpressure,jL as

Fcal=Pwf+9bpss """"""""""""""""""""""""""""""'"""""""""""""(A-l6)Plotting~~ versusNpgi~ tie foliowing~uIts

1slope

= -g””” ”””””””””””””””””””””””””””””””””””””@:~y-intercept = pi ............................................ (A-18)x-intercept = NPmOV=Npatw ......................(A-19)

While Eqs. A-16 to A-19 provide the most rigorous andcomprehensiveanaiysisof movableoii, this analysisrequiresacertain degree of interpretation. For example, we will neverproducean oil reservoirto the@condition, so werealiywanttodetermineN*,moVat some~~, whichdependaon the producingconditions.Obviouaiythis methodassumesthat themeasuredflowratesandbottomholepressuresare msonably accurate,which is usuaiiym the case in practice. So again,we havea tail forvigiiantdataaquisition--if we want to performstate-of-the-artanaiysisandinterpretationof productiondata.

APPENDIX B - THE ARPS EMPIRICAL RATEDECLINE FUNCTIONSThis appendix summarizesthe A@ semi-empiricaisolutions(depletionstems) used in the Fetkovich/McCray1°type curves.These solutionsare derivedfromthe Arps13empiricairesultsforflowrate,presentedin the formof theplottingfunctionsgivenbyFetkovich6 and McCray.8 A complete developmentof thesesolutionscan be foundin AppendixB of mf. 16.in presenting tie AVS solutions we provide ae-vemiaitxiiiar-yfunctions based on the flow rate, or in this case dimensionlessflow rate function. The rate and auxiiiaryfunctionsaregivenasfoiiows

YatiWle =Onlw ~@ Function

$:M DimensionlessCumulativeProduction91mi DimensionlessRateIntegraiFunction9W DimensionlessRateIntegraiDerivative

Aa a prelude to these developments, we acknowledgethat aspeciai~~~~n~!a~~r~ has been adoilted for the Arpssolutions. Inparticular,the term “exponentialdecline”refersto thecasewherethe flow rate decays in an exponentialfashion with respect totime.The exponentialdeclinecaseis the~ “ for theratebehavior in a well producinga singlephase liquid at a constanthnftnmlplp nrecmwe AC shnwn ~y Eh!~g-ECO~Qrn.~&ZS MIC!. . . . . . .. . “.” y.””” ”.= -“ “.. - . . ..Rarney.17 The term “harmonic decline” refers to the case wherethe fiowratevariesin a reciprocalfashionwithtimeor sometimefunction for intermediate to large times. This case is also“anaiyticai”in the sense that flow rate normalizedby pressuredrop plotted versus the “materialbaiance”time functionyieldsexactiy a harmonic decline during boundary-dominatedflowconditions,as shownby Eq. A-7.The “hyperbolic”declineis thegeneral term givento anydeclinecurve case lying betweenthe exponentialand harmonicdeclinecases. Hyperbolic cases generailyhave iittie if any analyticalbasis,the most notableexceptionsbeingcertainideaiandreaigasflow cases as describedby Fetkovich.c The hyperbolicdeclinecaseaaretypicaiiyuaedto~ datairendsandmostattempts to correlate “hyperbolic” behavior with physicalphenomena(e.g.,changesin mobiiity,layerfeatures,andspecific

drivemechanisms)are aisoempirical,basedmoreon speculationthantheory.The purpose of this appendix is to collect the pertinent Arpsrelations and to provide an introduction to the auxiliary ratefunctions so that interested readers may create their own typecurves. Starting@I the Arpsdlmens:onlessratefunction,q~,we havethe followingcasesArps Dimensionless Fiow Rate ReiationsThe differentcasesfor thedimensionlessflowrate,qM, functionrm=oivt=n rIc-“ ~. .-.. -

Ihponentiak (b=O) q~ = exp(-t~) . ... . . . . . . . . . . . . . . . . ..(B-l)

Hyperbolic (04<1) 9W . 1 . .. . .. .. . . . . . . . ..(B-2)[1+ bt~llb

Hamwnic: (b=l)‘m= * ........”....3)”...””.(B-3)

Arps Dimensionless Cumulative Production ReiationsThe definitionof thedimensionlesscumulativeproduction,NPW,is givenby

J

t~NPZM= q~?) dt ............. ...................... .....(B-4)

oThe differentcasesfor the dimensionlesscumulativeproduction,NPLM,functionaregivenas

Ihponentiak (b=O)

Hyperbolic: (Ocbcl)

Harnwnic (b=l)

Arps Dimensionless

NPLM= [1-ex~-t~~ ..............(B-5)or in termsof q~

NPLM= [1-q~] . . . . .. . . . . . . . . . . . . ..(B-6)

Np~ = #l. [l+b#-@] . . . ..(B-7)1-”

or in termsof ?Mand q~

Np~ = *[1 ‘q~d (1 +brM)l...(ItI)I).

or in termsof q~

NPW = ~[1 -q&b] . .. .. . .. . . . . ..(B-9).

NPW = in(l+tw) ................(B-lO)or in termsof q~

~pDd = Mi;9w)= -wg~)... (iijij

Rate Integral RelationsThe definitionof the dimensionieasrate intcgraifunction,q~, isoivm hv~. . . . . “=

●✌❞

❞

!W=L9Ddi ’ ~m

hi oq~r) dr . ......................... (B-12)

Thedifferentcases for thedimensionlessrateintegralfunctionaregivenbeiow

&ponentiak (b=O) ~[1-exp(-t~)l . ... ...(B-13)9Lkii = ~w

or in termsof q~

-L[l. qM].............(B-14)Wklt = *m

SPE28688--- c.-. - .. ----- ms - 1--..11 . . . . .-,4 T A Rl!i@ino!tm PLb UOUDlek r.h. ranae, L .J. m.umm, cwu . ..=. A-M-..fF..-

17

Hyperbolic: (kb<l) ~w = -1_ d_[I. [ l+btj(l- I@]trn l-b. ........................... (B-15)

or in termsof q~

L(l-qM . . . . . . . /1* h\ll-b Itm ‘u Viii“ ‘1 ~ 16)............................ .

Mmttont”c(b=l) ~M = _L~l+t~) . . . . .. .. ..(B.17)tm

or in termsof q~

9Ddi = &@qN) . . .. . . .. ..(B-18)

Arps Dimensionless Rate Integral Derivative RelationsThe definition of the dimensionless rate integral derivativefunction,q~ti, whichwe wume to be @tive~ iSgivenby

.d!mL=.tw*‘- d ln(t~] ()

..tM-A..## ..(B.19)

Or if we use thedef~ition of thecumulativeproductionfunction,NPLM,we have .

Expandingthe derivativeandredueinggives

1t~

..1 q~t) dr- q~ . . . . . ... . . . . . . . . . . . . . . . . . (B-20)!?LMdmo

CombiningEqs. B-12 withB-20gives the mostusefuldefinitionof thedimensionlessrate integralderivativefunction,q~. Thisresultisgivenas

redid = 9LM-9LM... ......................................... (B-21)ApplyingEq. B-21 to our previousresultsfor the qmi functionsyields

13ponentiak (b=O) 9M = _&[l- ex~-hi~ - qDd

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-22)Hyperbolic (Ml)

‘== i%[ii-@4&f+b)l-q~’............................(B-23)Harmonic (b=l)

qDfM = -&(l+t Dd)-qDd...24)4)

APPENDIX C - PROCEDURE FOR THE ANALYSISOF PRODUCTION DATA USING THE FETKOVICWMCCRAY TYPE CURVESIn this appendix we develop analysis relations for theFetkovich/McCraylo type curves. In order to generalizethe---1..-:- ------- c-. ---! :-..:--- ●,. . . . ..:-..1... -.a-.a:. “1..,.la”arwysm wncwpt Iur appuwwwn w IIUII-UUU414U I GWI VUII CIIICLFwe have defined modified expressions for dimensionless“deeline”variables.RecallthatFetkovich,aas wellas laterefforts(refs.7-10),all considerthe caseof a boundedcircularreservoir.Whilethissolutionis usuallyacceptableforanalysisof productiondata from vertical wells, we must understandhow to interpretperformanceresponsesfromnon-circularreservoirshapes. Theuse of the reservoir shape factor, CA,permits interpretationofotherreservoirgeometries.Startingwith thedimensionless“decline”timefunction,wehave

‘“=@&J2mDA=l*2”DbycAdtz] ,---

. .. . .. .. . . . .. . .. . .. . . .. . . .. . . .. . .. . . .. . . .. . .. . .. .. . . . .. .. . .. . . .. . . . (c-l)

wherethe dimensionlesstimebasedon drainagearea,A, is givenby

tDA =0.m33* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(C.2)@c/i

and thedimensionlesstimebasedon wellbore.radius,rW, is given~;

tD=0.~633& .. .. . .. .. . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . ..(C.3)4tctr~

Combiningeitherdefinitionof dimensionlesstime(Eq.C-2or C-3) yields the followingexpressionfor the dimensionlessdeclinetime

t~ = 0.00633~ 2

[1

.......................(C-4)#/@@pl -L

eYCAr&Similarlythedefinitionof thednensionless “decline”flowrateisgivenby

‘w=141”2aw%k?)l*””””McCray8definedtie dimensionl~s “decline”integralflowratefunctionas

/

.=l ‘mqm&m.......................................(C-6)9Ddl ,m o

and McCraysalso definedthe dimensionless“decline”flowrateintegraldeuivstivefunctionas

.A12rL=. t&R&‘- d ln(tm)

.............................(c-7)

Thedimensionalformsof thesemhtions aregivenby

[1qWi=141.2&~h ~ (q/@)i . ... ................(C-8)

eyCAr$aand

[1qDdid=141.2#~ln~ (q/Ap)id. . . . .. .. . . . . . . . . ..(C-9)

where

Ji

(q/Ap)i= ~ ~d~ ..... ........ .............. .. ...... ....... (c-lo)toAp

and

4@Ml=.#@Ap)J.......................(C-11)(q/Ap)~= - d ~(dd;

Curve Ma@@@mWE .Thisprocedureassumesthatwe”haveaccuratemeasuredratesandpressures as a function of time. Unfortunately,pressuresareusually not available, so for the purposes of analysis and;-+--.**.*;*- ...- -.., h..,- *na.m,,mm* *n”@*s”tnmee,lm ArnnMILG&~lGUXUU1l, W- LIMAJ U-VU ●U -O USbSb.E -u1.a_alb ~SVU-UL- U. VP

@IIIL Ap=pl-pW(.when PWIis assumedto be constantwith time.The assumptionof a constantpressuredropposeslittledifficultyin the analysis--althoughthis assumptionmay cause errors ininterpretation.

1. Compute the material balance time function from theproductionratedata. This functionis givenby

i= N~q . . . ... . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . (C-12)2. Computethe flowrateandflowrateintegralfunctionsusing

the material balance time function. These functions aregivenby

(q/Ap)= h= $ ..................................(C-13)

18 DeclineCurveAnalysisUsingTypeCurves-AnalysisofOilWellProductionDataUsingMaterialBalanceTime SPE 28688Application to Field Ca.wxI

3.

[

ift@p)j= & ~ dt ...................................... (C-14)

CJO4P

(q;@)ti= - d@APll = ~#?/~P)il [P-1 wd ln(~ &

....... ..... ....... .. --,

A minorcomputationalissueis that the datamustbeSQU@._ . . c..-..,:-- tm- ..-~~-- ..eln,,ls~;tim fif thmm teriiis Of ‘he i UIIIWIUII Iul PIUIJCU -albulmL.ull w. ~.wintegral andintegralderivativefunctions.Plot q/Ap, (q/Ap)i, and (ghpkj versus ~on a scaledlog-loggrid. Force match the data trends onto the Arps b= 1(harmonic) stem on the Fetkovich/McCraytype curve.-oral the “time”and“rate”axismatchpointsaswellas thematched transient r~ stem.

Xfthematerialbalancetimefimction,~,is correctlycalculatedthena scaledlog-log plot of q/Ap versus z will exactly overlaytheq~versus r~ trend for a harmonic decline on the Fetkovich/Mccraylotypecurve. Oncea matchof thedataandthetypecurvehas beenObtied: the timeandrateaxismatchpointscank usedto developthe followingrelationsforbPWandN

~ -Ja2fk‘P$z- [q/Ap~

. .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (C-16)

N= l&(q/Apbc1(bJIVtP (9DJMP

.. . .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . (C-17)

whereM.P.refersto the “matchpoint”vrdue.We then may solve for the drainagearea using the estimateofOxiginaloil-in-place

A=5.6148~ . . . .. . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . ..(C.l8)#/l (l-sw~w)

wherethe effectivedrainageradius,re, canbe estimatedfromthefollowingidentity

Pre= = ........... ............ .......... ..................... (C-19)From the rate match point, we can solve for the formationpermeability,k

k=141.2~;h[~]~~~] . . . . . . . . . . . . . . . ..(C-20)

-Fromthematchof thedataon a particulartransientstem(auniquevalueof r~), we can solve for the effectivewellboreradius,rW,and theskin factor,s. Theseestimatesamobtainedusing

rfi *r~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (C-21)

and

s =- 4)~rw . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . (c-22)

SPE 28688 L.E. DoubleL P.K. Psnde, T.J. McCollum, snd T.A. Blssingsme19

,N* I* 10’ d Idlti

1 ‘1”’”:‘“”””’”:“:-~-%I!?SS-=2E2=’==

Figural.Fatkwkhqodarld%ddvJP@QJf’J-.

. -------- ---------- .- . . . . . . . ..- ---------- . . . . . . . . . .lti 1 I1 1 1 I

ZomI 0

0 SOO lWO t’c% - 3mo

F19re3-serIIuog Prod@on PlotforSirnutatedCase #l (ConstantfJ~.

d=ppk-1I I’k

Figure4- LqFLqI ProdxUm Pbf forSlrnulatedCaee #l (Constantpd.

W. 1 I I I 1 11 v ‘1 1 1 .-1-

W’ ld 10’ d lti dw%-rFbure5— Rate FurElwe for Sknufated_ 81 (~ pd.

i- k--------- --------------------------- o,.O

I I 1 t 1

0 1000 zooo moo 40C0

towFigure6- SernUogPmduotimplotforSirnutafedCeee #2 (Vadebfep~with SM-ine).

1$~10’ 10’

towFtgure7- Log-LogProductionPtotIor SlrnutatedCaee #i?(VariabfePdvhfh Shut-ha).

10+I I 1 ,# 1 1 7 -1 1 --i-

16’ ld 10’ d 10’ d~N$ ~

FlgJre 8- RateFmoflonsforSbnufatedCeeeW(variablePWWIUI Shut-ha).

---- .... .

$0’

J

~

W’

10” 10*

Figure9—Metchot~ “-Oateforsirnutatad cuomlComtaIM P- . FtadM FiOWType ~M.

I&

DeclineCurveAnalysisUsingTypeCurves-AnalysisofOdWellProductionDataUsingMaferialBalanceTime: SPE 28688ApplicationtoFieldCases

tm. 1 # I t

IE!!Ezl:.

0 wooo N,,STS

F-11 - Movsble Oii EsfimstiofIffan RetoHietofy.

1O.mot I 1 1 I

E!!Eiim

Figum 12- Movsble Oil EsfimstimfromNonnsliARsteHistoty.

q+_T!f.-0

0 mom N,.STS

Figure 14- Mwsble Oil EsUmstbn from Rste Hktofy.

,

i loioo S&m ___S&o 4&o ‘Soioo%.~

Figure 15- Movs14e Oit Eetimstfon fromNonmtized Rate Hiototy.

F@m16-Mcwsbfe Oil EsffmetionfromCaic@Uedpti.

.- .

---- w “,

c WA a 8.aJl.a-

mm~ L-m

,,. . . . . . . . . . . . . . .

w.: ----- -. -

..

~ &’ Ie4“’&i:_J2.:eLKtm4

i 1604

1° ‘4202 4

I @Of

A1901

1

I●

● l SEC. 8

IeL ----

i I I i tlfl-

10’ ldLx

Figum20-Log-tog Pmddion Ptotfor NRU Wd 4202 (Cbadodt).

SPE 28688 L.E. Double4 P.K. Pande, T.J. McCollum, and T.A. Blaaingame21

1o”’ I 1 1 1

3 I I

.I

Ill Ill

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10’ # lo~ d‘ObJ.s04°4F@UM 21- Rate FunctionEfor NRU Well 4202 (ClamfoIIC).

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m I I t 1 I

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F~ra 25- Sadlog Pmductica plot forNRU Wall 1004 (Clewfofk).

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Figure27- Rata Functionsfof NRU Wail 1004- (Cbarfwk).

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Rgum 29- Movable (2II EatimatJon fromRateI+Mxy.

22 Decline Cunfe Analysis Using Type Curves-Analysis of Ofl Well Production Dafa Using Material Balance Time: SPE 28688Application to Fmld Cases

d1

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. -d-lfl

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o zooo am am am 10MO nooO 94001W18

Fi~m 30- Sernibg Pmdmlion Pfof for Well A (Sprabarry).

Fq)ura31- Log-@ Produdbn Pfot for Wafl A (Sfxabarry).

F@m 33- Rata Fundona for Wall A (Sprabany)..-. _. -..

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.

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1500m—. b-*am

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tow

Rglna35—serrlilog PlOdUcbm“ plot for Sartm LigfItaay Well S4 (Austin Chalk)..—

12 “,10 & ld 10’tow

fi~m 3S- Log-LogProductionPtoffor SartonLightaayWall S4 (Austin Chatk).

Figure 37- Rate Funofiona for Mton Ugfrfaay WeMS4 (Auafin Chafk).

F~m 3S. fMcflofPmduoumoalatorsd onLigtucqwdla4(Auam-)”w*TYwh.

zooo I * 1

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i

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F@um39-Movabie 011EafimaUmffofnRate~.

SPE 28688 L.E. Doublet. P.K. Psnde, T.J. McCollum, snd T.A. BlssingsJne23

~ 41- Mwsbls Oil t&fnStiOfI frWII@kUhtSd ~.

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~ 43- ~ pmductianPM for S Csnsbbs Well 225 (AustinChslk).

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Figure 47- Moveble Oil Estimtbn fmm NOnndizsd ReteHistmy.

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Figure 52- Log-l-q Production PM for Gilds Well S-42 (Lower Repelto)

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11 I /1

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1000 I

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00

Fqum 55- Mmrsbls Oil EsUmstion trorn Rste History.I I 1

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F~m E4 - Movsble Oil Estirnstb from NorrnsliZed Rate History.

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Figure 57- Movsble~ &lrnsthmfm~tsdpbu.