cop~hhl t99\, S&ety of Pe!rdeum Engineers, Inc.

This fmper WA8 prepeti Psvpmsenlatim a? dm SPE Qat Tedmbgy Symposium held in Hwston, Texas, January 23-24,1691.

TM paprx was $elemed for presenlalicm by m SPE Program Commitrtoe following mtiew of inlormetkm conleined In an ●Mrnc4 wbmirted by the author(s), C%nmn,s of Um paper, acpmwnted, ham not bean revkwed gmesoc*volPlrk e o urn I%qineem tnd are subject 10 C&motion by tie eulhor(t). The materiel, ●: presented, does not necosaanly rellecr anyposlrim_Iof rho Sodaly of Petdeum nglneem, 11solfioerc, m members, Papera presented at SPE meetings are wbjact to publidon ravbw by Editorkil committees of the Sodary ofPermlwm &@neers, Permlmbn to copy is restdcled 10 an ab$tract o! not more thnn 303 words. Il[utl!arions may no! be copied. The abstract should conlain .%nsplcwous~~t~ *m ●@ @ fim me paper is Pmmmtad. write puMic.atas hkwer, SPE, P.O Sex S3S9SS. Ridurd$on, TX T5C8S-21S6, Telex, 730369 SPEOAL

—

The motivationfor the workdescribedin this paper arosefrom formed intoan cquivslcmconstantrate ca.wfor bothgas and liquida need to analyze production dcclinc data where the flowing flow data. Camacho9indcpndcntly vcnficd that this equivalentbottomhole pressure varies significantly, The varirtncc of the constant rate formulation is exact for the constant pressurebottomhole rcssur@with time CXCIUS.ICSthe use of the exponential

Jdecline m c1for conventional dcclinc curve analysis (scmilogproduction of a sli htly comprtssiblc liquid during boundary

#‘dominri:ti flowcon mons,plots arid type curves). Using pressure nonmalizcd flow raterather than flow rate usually does not remedy this problem. The McCrayto sought to develop a method to transform variablc-rncthmi wc present uscs a rigorous superposition function to rate/vanablc pressure drop data into an equivalent constantaccount for the variance of rstc and pressure during production. pressure case. In doing this, McC!raydcvclopcd a recursionThis furwdon is the constant rate analog for vanablc.rate flow formula to compute an cquivdcnt time for constant wellboreduring post-tmnsicnt conditions and can bc used to develop a pressure production, tcp, that could bc used with pressure dropconstant pressure analog for the dcclinc curve analysis of fielddata.

normalizedflowrate to perform dcclinc CUTVCanalysisusing typecurves.

The constant pressure antdogtime function is computed from Although the approach st,rggcstcdby McCray was verifiedthe constant rate function using the identity that cumulative using simulation, wc sought a rigorous foundation for theproduction for both cases must be qud. Using the cumulative applicationof thisresult, As it turns out, a relativelysimple proofproduction identity, wc SOIVCrecursively for the time function can bc shown for the application of the rcp function duringusin~ trapczoirlal rule integration and, as an altcrna!ivc, frnitc boundary dominated flow conditions, This prcmf is given indiffcrcncc formuhtc, Wc have also dcvclopcda const;mtpressure AppendixA.analog time rchuion which is rigorous for bounrky domimttcdflow and stxvcsas an accuraIcapproximationfor tmnsicntflow, In addition to the proof of McCray’sresult, wc also provide

mcthmls to compute the constant pressure equivalent time, 2CP,

We apply theserelationsto analyticalsolutionsfor verification using recursion formuhm in Appendix B. In Appendix C wcand then usc the boundary dominntcdflow relation on simulated provide relations which can bc used to compute the constantand ticld cascst These wmuhuioncases include htrgc rmdsmall pressure dimcsionlcss mtc solution given the ccnsutnt ratestep chun ‘CSin bottomholetlowirtgpressures,and pcricxiicshut-

!dimensionlesspressuresolution, In the text wc will pmvc thtitthe

ins. Finn]y, wc apply theserelationsto a gus w.’IIfieldcase, computatiortalmethods wc provide ylcld csscntittllyexact resultsduring boundarydominated flowand give very good pcrfornmnccduring transientflow.

The widespread usc of ty~rcCUI-VCSIJto analyze mlc dcclinc ~data has motivated us to consldcr the implicationsof varyingrateand pressure drop production. ‘1’hcorcticallys~aking, for the Our first goaJ is to establish that our new mcttisxiactuallyflow of a slightly compressibleliquid, the analyticalstems cmthe transforms a variable.ratclvariablc pressure drop systcm into anFctkovichl ty curve arc valid only for the ccm!ant wellbore

zequivalent constant rcssure systcm, Wc begin with a proof of

pressure pr uction CMC,+6 In previous worksv,a wc have Pthe validityof our so utionby transforminga constantrate systemshown that variablcwatdwtriablcpressuredrop data may bc trans- into a constant prcssutv s stcm. Because these solutions arc

rfrqucntly used in dimons SMIMSformat, wc will perform thisRcfcrcnccsand=tmtions at cnd of paper verificationusingdimcnsionhw variabkx.

WC b 61%J1UM@wtJ@xan EnLmm’s

Decline Curve Analysis for Variable Pressu, J Drop/Variable Flowrate Systems

by T.A. Blaslngame, T,L. McCray, and W,J. Lee, Texas A&M U.

This is a preprint -- subject to correction.

2 DeclineCurve Analysisfor VariablePressureDrop/VariableFlowrateSystcn?s SpE 21513

The computationalformulaearc givenin AppcndiccsB and Cand wc will verify each. Thcsc include the fcilowing rccuraionformulae; the integral method proposed by McC!ray10and the 2-and 3-point brickwarddtifcmncomethodsdcvclopcdin this workThe recursion relations for this part of the verification arcdcvclopcdin AppendixB and summarizedin AppendixC,

We will alsouse the boundarydominatedflowrelationswhichresult from

7tsating the constant rate and constant pressure

anal tics! SOIUor..’,This dcvclopmcntand the pertinentrelationsifor t is psxtof the verificationam given in AppendixB.

Fi . I shows the log-log behaviorof the qo functions vcrausttDan tcp~function for tic cssc of a well ccntcrcd in abounded

cimulaxreservoir (r~=NP), Duc to the numberof mcthcds beingcansidcm we will discuss the transientand boundarydominatedflow behavior separately, First wc note that, during early times(transient flow), all of the tcpp methods yield a goodapproximation to tic q~(t~) sohmon, cxccpt at very early times(t@20). This implies that all of thesemethodsyield a reasonableapproximation to the analytical solution during transient flow,Obviously,the analyticalsolutionfor tmundarydominatedflow isnot valid during transient flow as shown by the deviation of thissolutionand the transientflow sohstion.

Now if wc consider the late time (boundary dominated flow)portion of Fi&.1 (rD>3x105),wc find that virtually all of the to Dmcthds agrc.cwry closdy with the qD(lD) Solution.Athough&lsscale prccludcsvery CIOSCinspection,it dots appear that t D2 forthe dcnvativc mcth(xl1dots showsigni!lcantdeviation, ‘f%iswillbc invcstigsttcdmore closely when these dara arc rcplottcd on ascmilog q~ gmph in Fig. 2.

Fig. 2 is a rcplot of Fig. 1 using a scmilog scale for the qDfLSnctiOnSand a cancsian scsdcfor the tDfunctions. Wc nOtCthatthe scsulta for dcnvativc method 1do begin to diverge from thoser f the Othermcthtsds,which cIwly ovcr]ay the correct solution,Fig, 2 sug~sta that dtivativc rrscthod1 should not be used in

Lracticc, but that the integraltncthw dcnvativc mcthcx!2, and the

undarydominatd flow methodshould give accuratewsults.

Of these, the boundary dominatti flow method is the easiestto apply since it dots not rcqtlirc recursion calculations, butinstead provides a dhtct Uansf{“mationbctwccn tD (or tc,D)andtcp (Eq.B-2), tc~o is the is the dimensionlessmatcnaI balancetime function, introduced by Blasingamcand UC7 and later (andindcpcndcntly) b Camacho9as an quivalcnt constant rate time

ifunction for varia lc-ratchriablc pressuredrop flowconditions.

Onc problcm that tic boundarydominatedflow method mightb pcrccIvcdto have wouldbc that in fieldapplications,formationproptics arc rquircd tocomputethe m and b constantsin thetcrtcftransform relation, Eq, IV1. However, these constants arceasily dctcm~incdfrom a cartesian plot of A@Qversus tc+=Q/q)during boundarydominatul flow. This proccdurcis verified for avariety of cases in Ref. 7, Also, bccausc this is a rigorousformulation for boundary dominated flow, the m and b constantsarc unique and will provide theoreticallyconsistent results whenused in Eq. B-1,

Additionally,Mcthylo reportedinconsistentresultsusing theintegral method to compute the rc function when Iargc pressurechanges ancl/or shut-ins occurc# McCray was forced to usccmpisicalextrapolationpointsto causethe imcgralmethodto yieldcorrect results, Again, the boundarydominated flow method willnot have these roblcms Iwausc tic m and b parameters arc

funique and the ormulation of the tq+ transfo~ 0%. B-1) iSrigorous,

For these rcmons, wc rccommcnd that the boundaTdominated flow rncthod which uscs Eqs. B-l (field) or B-

(dimcnsionlcss) be used for computation of the tcp or ?CPD

functions. Wc will demonstrate the ap~licationof the boundarydominatedflowmcthcxion a simulatedhquidproductioIIaqucnccand a fic!dcaacfor a gas WCIIthat has been analyzedpreviouslyinthe literature.ZM

In this sectionwc will apply the tcP-tcrtransform describedirithe previous section to a simulatedproduction squcrtcc in an OilWCI1.The mscrvoirdata and flow historyarc given in Table 1.

TABLE 1

WellimdReservdr Parameters(Well Centered in alloundedCircular Resemoir)

B, RB/STB 1,00cl, psia-l 15.0 X 1o-I5h, ft4 0.::\i, CprW, ft 0°2:re, ft 745 (40 acre)k, md 1,0spi, psia 48~CA 31.62

Using the WC]!and rcscswoirparametersgiven above and Eqs. A-3to A-5, wc computethem and b paramctwsand wc obtain

m= 2,3&/09x10-z@/S~/D/Db = 32.8948pSi&~@

Flom History

flow scqucncc

;. .

:. .56,. .78

r, days q,STBfD

180——

50180 const~t p~

1:0 constantp~180 Constyt p)@3

180 constantp~S40 const~x p~

p~, psia

constantrate2000

shut in40003000

shut in3s001500

shut in25001500

The flow mtc profiles that WCIEobtained from the simulationcases arc shown in Fig, 3, Wc have inchsdcdan arbitrary basecase (p~30Mt psia) to orient the analysisof all of tic data, Thatis, when wc have obtained the correct transformationof data, allcases including the base case should overlay the same trend,Although it is conceivable that these rate profiles could bcanalyzed separatelyusing dcdinc curve analysis, the cawistcncyof the rcsuhs would b dcpcndcnton the ability of the analyst, Inparticular, the transient spikes caused by pressure changes andshut-ins would be difficult to interpret, and the non-uniformbehavior after the effect of the spike has subsided wouldinevitablyyield ambiguousrcstdts.

Onc method used to align variablerate pressuredata with thecorrect cccnstantrate solutionis rate norrnahsstionof the pressuredrop. In dwlinc curve analysis,many analystsusc prcsstsrt dmpnormalisationof the flow mtc profile in an attempt to obtain the

SPE 21513 T.A. B!usingamc,T.L. McCrayand W,J, La 3

I “correct”constant pressure solution. It can be seen in Fig. 4 thatprcs: xc drop normalization does not yickl a constant nrcssure

I analog solution,

Clearly, wc must usc other techniques which are morerigorous thanpsessumdrop normalizationfor field applica{{,onsofdecline curve analysis, The method of choice will be the oncproposed by Blasmgamc and Lec7 whit: converts variablc-ratchriablc rcssure tip data to the equiwdcm constant rarecase. From t is anal sis wc will obtain the m and b parameters

‘!rquired by Eq. B-1 or transformationto an equivalent constantpressure system,

Fig, 5 showsthe cartesian plot of alp/q vs. tcr(=Q/q) rquircdto determine them and b parameters. m is the slope of this plotand b is the intcrcc t. Although thcte is some data scatter, it is

1ckar that them an b psramctcra do rcprmcnt a best fit trend ofthe dam. Therefore, the step of determining the m and bparameters is illustrtttcdas a simple and straightforwardprocess.Fig. 6 showsthe log-log plot of Ap/q vetmssrc, that could be usedfor type curve matchingon constant rate type curves. Fig. 6 alsoshows that the concept of using Aplq and fcr appears to also bevalid for transientflow, giventhe agreementbetweenthe constantrate and constant pressure base case (pW~3000 psia) duringtransientflow (rcrc50days).

The next step is to usc them and b parameters in Eq. B-1 toconvert from tcr (constant rate analog time) to tcp(constantpressure analog time), This is also a simple and straightforwardprocedure. Once tc is computed, a log-log plot of q/@ vs. Gp is

imade. Fig. 7 is suc a plot and wc immcxiiatclynote that all casesoverlay the same trend during both transient and boundarydominated flow. obviously, the analyticalsolution for boundarydominatedflow (exponentialdecline)will not agree with transientflow solution.

Fig, 7 represents the endpoint of our effort to determine artquivalent anstant pressuretransformationfor variablc-rate/vari-~blcpressuredrop flow data, Wc are satisfiedthat this is a logicaland consistent procedure that should yield accurate rcsuhs whenapplied to field data, The verification of this method is that allcases overlay the base case @~-30fXt ]’sia), where ~IAPand twere used as the plotting functions for I$Cbase case. At thispoint, Fig, 7 can & used for decline curt c analysis using typecurves such as the one presentedby Fctkovich,l

In this sectionwc will apply the ICrtc, transform @q. B-1) tothe analysis of vttriablc rate gas well data as described inrcfcrcnccs 2,3, and 8. Unfortunately, the complexity of thisanalysis is compounded because gas WCIIanalysis requires theused of paeudoprcssurcand pscudotime, Wc will refer to ref. 8for the pseudopressure and pscudotimc functions as WCIIas forthe msulta of an iterative procedure to dctcrminc the m and bparameters and the gas-in-place, G. This iterative procedure, asdescribed in ref. 8, simulau+ntmuslydctcrmincs these parameterstsauac the pseudotime function requires Icnowlcdgcof the gas-in-phw for matcdalbalancecomputations.

The reservoirdata and flowhiworyarc givenin Table 2.

TABLE 2

Well~Reservoir par~ters(Assumed Geometry: Well Centered in

a Bounded Circular Reservoir)

B, R13/MSCF 0.70942et, psia-l 1.870X 10”4h, ft# 0.;:& Cp 0,02167Fw,ft 0,354

-5,30 *~, md 0,0786S *p.~ psia 710pi, psia 4175C* 31,62G, Bscf (!’ef,2) 3.360G, Bscf (rcf,3) 3.035

* Averageof valuesobtainedfrom ref. 2 and 3.

From the resultsof ref. 8 wc have

ma = 2,O5536X1O-3ps@lSCFP/Db. = 1.3094psi/MscF/DG= 2.6281 Bscf

Fig. 8, which is a log-log plot of @a/q versus tc~,a,is takendirectly from rcf, 8 and shown here for complctcmess. Wc htwcincludedthe computedresponseduting boundarydominatedflowas prescribed by Eq. A-1, Thc Apa/4 and tcr,a variables arc thepseudotimc and pseudopressureas defined and computed in ref.8, This nomenclature may seem awkward, but defining thesevariablesin this manner allowsus to usc liquid flow quations foranalysis, Therefore, any equationswc preaem are valid for eitherliquid or gas flow as long as the correct time and pressurevariablesarc used.

Wc note that the boundarydominated flow sohstiondoss notmodel the transient flow behavior of the data in Fig. 8. This iscxpectcdand wc only state this observationfor completcncss, Wccould use Fig. 8 in a type curve matching analysis with constantrate type cwves. However, it is our objective to analyze thesedata with a constantpressuretypeCUIVCso wc must proeecd withtransfomningthetcr~variableto yieldCcp,a.

This analysisrquircs that wc modify the variables in Eqs, A-1, A-3, A-4, and B.1 for analysis of gas WC]]test data usingpscudoprcssurc and pscudotimc, For gas analysis Eq. A-1becomes

(1)

Eq, A-3 becomes

~a=_&(2)

Eq. A-4 becomes

For a closedcyhl.rkxd swemoir Eq, 3 becomes

(3)

(4)

I

4 DcclincCur-wAnalysisfor VariablePressureDrop/VaririblcFlowrateSystetm SPE21513

And finally, for gas weil test analysisEq. B-1hxomes

(5),

We have usedEq. 5 to eomjwtcthe tc~ functionsused in Fig.f9. Note that Fig, 9 is a log-log lot ot q AP~versus tcp,~and that

hthe the boundary dominated ow sOIWiOn(Computccfq14Pafunction) agrees very wcli with the data during boundarydominated flow but not during transient flow. This is expectedand we shouldnot be conccrncdabout this difference.

Once we have created Fig, 9 using the q/Apa versus rCPadata,wc will want ‘omate} this data upon the Fetkovichl type curve,Fig. 10reprc%ms this type curve match. Note that the data agreewith the t

rcurve during the transition from transient to

boundary ominated flow and throu hou; boundary dominatedhflow, The scamity of data for tc ~<1 days (4 points) iimits our

Fiinterpretation of the transient ow portion of the data, but anestimateof r#W’=20 for the tmnsientstem seemsreasonable.

Once the data are overlain and matched to the type curve, wewill determine a match point from the coordinates of both plots,The match point for Fig, 10is

RateFunctionMatchPoint:

qD= 1.0q{dp = 0,78 MSCFiD/psi

Time MatchPoint:

t&=ltcP~= 630 days

Cwve~ ,.

Once the match point is determined for a data set on theFctkovichl typecurve, we can computethe volumeof fluidsin thereservoir and the formation permeability, k. We needcomputational relations for liquid (oil) and for gas where thecorrect pseudopressureand pseudotimefunctionsare used.

The oil-in-place,N, can be computedusing

[()(1]

~

N=.~ & hct 9dD mp rdD rnp

The formationpermeability,k, can& computedusing

(6)

(7)

The gas-in-place,G, can bc computedusing

‘=*[(~)mp(~~)mj (8)

The formationpermeability,k, can be computedusing

(9)

Also, for a bounded cirat!ar reservoir, the fosmationpermeabi-lity, k, can be computedusing

-g_

Polo”k=14L2~ ~-z *~w 4 qdD (lo)

Using Eq. 8, wc can estimate the gas-in-place, G, This calcula-tion gives

G=&[(w)(Hll

G = 2.6278 B~f

Using Eq, 10, wc can cs:imatc tthe formation permeability, k,Thiscalculationyields

~ ~ ,41 ~~0.70942 0.02167(;0) +@o)-ii(w)

k = ().()5432md

The computed values of gas-in-place and permeabilitycompare very WC]] with those obtained in ref. 8, In fact, theresults arc vimsallyidentical. The reason for this is quite simple,The constant rate analog method (using @a/q and tcr,a) and theconstant pressure analog method presented in this work arcrigorouslyrelated, Therefore, if wc are consistent,both methodsshouldyield the sameresults.

Our estimate of gas-in-place, G, is 22 percent less than theestimate of Fctkovich, et al,q and is 14 percent Icss than theestimate given by Fraim and Wattcnbarger.j However, wcexpress confidence that our estimate is as acclqratt as the onesgiven by the other investigators. And wc feel that our approachismore rigorous, because of the pscudoprcssurc and pscudotimcsoiutionfommiation.

Our estimate of permeability is within 31 pcrccnt of theaverage of the estimates given in refs. 2 and 3. We feel that ourpermeabilityestimate is consistent with these vahsesof the otherinvestigatorsand ccstainlyreasonable,even given the sparsedatatrend in the transientflow region. These data arc used to identifyths rJrWcvalue finm the tmnsicntstem type curves.

SJMMARYANDCOKLJJSIONS

This paper introduces a method that can be used to analyzevariable-rate/variable pressure drop production data using aconstant pressure analog time function, The most significantresult is that of the boundary dominated time transformationforconstant mte or constant pressure flow, This transform,given byEq. II-1, allows an analyst to compute an equivalent time forconstant pressure production, quickly and easily, based on theparameters m and b, obtained using Eq. A-1, For gas wclis thisprcdurc is lCSSstraightforwardand rquircs an iterativesolutiondevelopedin ref. 8.

Wc considered four diffcmnt methods to tratyfomrtvariable.rate data into the constant pressure solution profile, Wcconsidered three recursion formulae which compute the constantpressure~uivalcnt time function,tc,by panel summationsbased

{’on trapezoidalrule intcgrationloan fmitc diffcrcnccexpansions,Each of these relations was ap lied successfully to convert the

Pconstant rate dirncnsionlcss so ution into the constant pressuredimcnsionlcs solution. Iiowcvcr, McCraylo found thatapplication of the trapezoidalrule integration method gives poorresults when a plied to erratic data or data with cxtcncded shut-

[ins, This bc avior makes the general application of theserccrnsionfwrrwlacdifficultat best,

SPE21513 T,A. Blnsirsgime,T,L. McCray and W.J. Lee 5

The fourth method developed was a rigorous identity whichequates the kxtndary dominated solutions for constant rate andconstant pressure production. The resulting two-parameterrelation @q. B-1) may bc used for dimensionless solutions orfield data applications. When the m and b parameters aredetermined using the methods developed in ref, 7, data scattershould have Iittlc effect on the VSIUCSof the parametersbecauseabest fit trend is established. These characteristics make theboundary dominated flow method the most usefulproduct of thiswork.

Applications:

Wc recommend using the methods prcscntcd in this work thrthe type curve analysis of variable-rate/variable pressure dropproduction data, The methcd is relatively simple and should beapplicable to a wide range of WCI1test problems, including theanalysisof gas well testdata demonstratedin this work.

Conclusions:

1,

2.

3.

The recursion fonnulae discussed in this work should not beapplied in practicedue to problems associatedwith the erraticnature of fielddata, whichcould causepoor results.

The boundarydominatedflow methodis the methodof ckaiccto transform the constant rate analog time function ir,to aconstant pressure analog time function, This mcthid isconsistent, easy to apply, and should give accuratercsuits fora widerange of problcmtypes.

The boundary dominated transform method can be used comodel constit wellborepressureproductionbehaviorcxaciyduring boundary dominated flow and should give accurateresults during transientflow.

NO a

Dimensionless VariablesbD = dimensionlessconstantdefinedby Eq, B-4CA = dirrmsionless shape factor

= dimensionlessconstantdefined by FA.B-3%’ = dimensionlesspressure(conslantmtc case)f@ = dimensionlessslltc(constantpressurecase)Y = Euler’sconstant (0.577216 .,, )

Field Variables(Formationand Fluid ProDerliexJfo~ation volumefactor, RB/$IT3 ‘constant as defined by Eq, A-4 (liquid)and Eq, 3 (gas)to:sI compressibility,psiaigas-in-place,MSCF(or 13scfas in the fieldexample)total formationthickness,ftformationpcrmeabllity,m?constant as defined by Eq. A-3(liquid)and F.q,2 (gas)oil-in-place,ST13initial reservoirpsessurc.psi3flowing bottomholepressure,psiaflowrate,STB/Dor MSCWDfor the fieldexamplecumulativeproduction,STBwcllbomradius, ftreservoir chainagcradius, ftre/rw, dimensionlessrescmoirdrainageradiusporosity, fractionviscosity, cp

Subscriptsa = “adjusted”variable for gas well test analysis. Usc of

these variables in gas well test analysis yields ancquivalcrstliquidsys~cm.

&. constantrate or conwmt rate analog

Cp = constantpressureor constantpressureanalogD= dirncnsionlcssvariablemp = matchpointon a typecurve

Wc gratefully acknowledge the assistance of ElizatwthBarbozattndJemniferJohnstonfor their hcIp in the preparationofdds manuscript.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10,

Fetkovich, M.J.: “Dcclinc Curve Analysis Using TypeCurves,”JPT (June 1980) 1065-77,

Fetkovich,M.J., et UL“Decline-CurveAnalysisUsing TypeCurves.-CaseHistories,”SPEFE (Dec. 1987) 637-56.

Frairn, M.L. and Wattcnbmgcr, R,A.: “Gas ReservoirDecline-CusvcAnalysis Using Type Curves With Real GasPseudopressureand NormalizedTime,”WEFE (&c, 1987)671-82,

Ehlig-Economides, C.A. and Ramcy, H,J., Jr.: “TransientRate Dcclinc Analysis for Wells Produced at ConstantprCSSUR,” :PEJ (Fcbo1981)98-104,

Ehlig-Economides, C.A. and Ramey, H,J,, Jr,: “pressureBuildup for Wells produced at a Constant pressure,” $PEJ(FcIJ. 1981) 105-114.

Blasingamc, T,A. and Lee, W.J.: “Properties ofHomogeneous Rcscrvoi.m,Naturally Fractured Reservoirs,and HydraulicallyFracturedReservoirsfromDecline CurveAnalysis,” paper SPE 15018 presented at the 1986 SPEPcrrnianBasin Oil and Gas RecoveryConfcrencc,Midland,TX, March 13-14, 1986.

INasingamc,T.A. and Lee, W.J.: “Variable-RateReservoirLimits Testing,” paper SPE 1.5028presented at the 1986SW? Permian Basin Oil and Gas Recovery Conference,Midland,TX, March 13-14, 1986.

Blasingamc: T.A. and Lee, W.J.: “The Variable-RateReservoir Limits Testing of Gas Wcllst paper SPE 17708prcscntcd at the 1988 SPE Gas Technology Symposium,Dallas, TX, June 13-15, 1988,

Camacho-V,,R,G,: WellPe@ormance Under Solution GasDrive, Ph,D, Dissertation, University of Tulsa, Tulsa, OK(1987),

Iviccrav.T,L,: Reservoir AnalvsisUsing Productwn DeclineData &d Adjusted Time, ‘M,S, Tkis, Texas A&MUniversity,College Station,TX (1990),

Appendix A: ~ ,&!&QWtmof v~ .

We will start with a relationwhich allowsfor the constantrateanalysis of vsiriablc-ratekwiablc-pressure drop test data duringboundary dominated f!ow, This relation has been verified forliquid flow by Bhsingamc and Lec’;~ro?d”~~machog,This relationhas also been verified for gas flow by IMsingarnc and IA@, Theconstantrate analog,wlationis givenas

6 DeclineCurveAnalysisfor VariablepressureI)rop/VaricblcF!mvratesystems SPE21513

4(0—=mtcr+bN)

where

Jt

‘(t) (time in Days)= --1’- q(t)dr = —‘Cr f?(r) * q(t)

(A-1)

(A-2)

m=5.6l5--@---@hcfi (A-3)

()b=70.6#n~eTCA r~2 (A-4)

and,

rw = rw e-s (A-5)We will also need the general solution for a well producing at aconstantpressureduring boundarydominatedflow. Thk solutionis given by Ehlig-Economides and Ramey~,Sand later byBlasingameand Lec.c This so!utionis

&)cp‘row)(A-6)

Our objecciveis to develop a general time function that allows usto use Eq. A-6 to model a variable-rate/vanable-pressur~ dropprocess, This generalrelation is

(A-7)

where tcp is the time at which the constant pressure solution isvalid for a general variable-rate/variable-pressuredrop response.In this sense, ICPis an unknown whichmust lx dctcrmincri.

McCray10proposed [he following rcl.~tionas a defining refwionfor tcp

1

tpQ(t)—. = &l ~T

M(t) o A’P(o(A-8)

McCray developed Eq, A-8 via induction from the constantpressure solution where ~(?) would be constant and could befactoredout,

Although McCray proved Eq. A-8 cmpincal;y using simulation,wc sought a more rigorous justification. In the fcJlowingdcnvtitions we provide a rigorous proof of Eq, A-8 for boundarydominate ! flow.

Frst we rquire the coustantrate analogresult, Eq, A-1, andQ. A-2, CombiningEqs. A-1 and A-2 gives

9#1.mQ~+h(A-9)

Solving Eq. A-9 for Q(t)/dp(t) gives

[1

Q(O .1 ,- b-@-

AP(t) m Ap(t) (A-10)

Eq, A-10 providesus with the Icft-’:and.sidc(I,HS) of Eq, A-l?.

We need to prove the right-h~.,ld-side(RHS) of ~., A-8. This isdone by integratingEq. A-7 so it is the same form as the RHS ofW,.A-8. This gives

or

CombiningEqns. A-7 and A-11 gives

(A-11)

(A-12)

Notice that the right-hand-sides of Eqs. A-10 and A-12 arcidentical, This result proves that Eq. A-8 is exact for boundarydominatedflow,

AppendixB: ‘ “~J%!@Jl

arv 130ml~The objectiveof this swion is to developa methodto computethercpfunction. A relatively simple relation is obtained by quatingEqs. A-1 and A-7 and solvingfor tcp.This gives

~ Exp(~@), “ *

or

%/2=$ql +fpcr) (B-1)or in termsof dimcnsimless variables

(Jmq..td‘c/)D=mD ) (R-2)where

mp=L. tirW2r@2 A (B-3)

whew(B-4)

Although Eqs. B-1 and B-2 are strictly valid for boundaryd~ot~a$d flow, we find that, for short times, these relations

tcp Z tcr (K3-5a:and, in dimcnsiortlcssfomr

tq)J)= t@ (}1-5b

Althoughwed? not present a rigorous proof, J@ B-5a and B:51do suggest thut Khctcpfunction can M approximated by the tcfunctionduring transient flow: General applicationof I@ B-5and B-5brequiresfurlhcr invcstigaticm,

SPE21513 T,A.131asingamc,T,L,McCray and W,J, lee 7

I And combining Eqns. B-8, B-9, and B-11, and solving for AtCP,i

Integral hiethod gives

McCmylo proposed to compute thetc functionby approximationi [

. ~ @(ti) Q(ti.$ 4Q(ti-1)+~of the integral in Eq. A-8 using t c trapezoidal rule, This ‘rep’l %(ti) Ap(ri.2) @(ti-1) AP(ti)1 (B-13)essentiallyresults in a recursion formulawhere the tcpfunciion iscomputeda:

‘Cp= f At.p,iAppendix C: ~~

i= 1

The AtCP,iterms are computedusingindividualtrapumid panelsofIn the calculations, the constant pressure dimensionless rate

*C q(1) solutionis definedas— function. For an individualtrapezoidwc haveAPO)

qcpD = ~(c-1)

[

Theequivalentdimensionlesstime isIi .%!?L * + ~ti-l)

2 @(ti) &(ti. 1) 1 tcpD = i~l AtcpD,i(c-2)

also where the ZMCpDjare given for the integralmethodas

[

/i - Q(tJ Q(ti.]).—. . —

AP(ri) Wi-1) 12[%%+1AtcPD,i= ~ ~ ~-

for a givenpane!. [PD,i F’D,i-l1 (c-3)

Combiningand solvingfor AtcP,igivesand fol dxivative method 1 as

tD,i pfi,iAtcPD,i = tD,i - ——

[ J

‘ @2iL. aLIL PD,i-1 (c-4)

t~p,i =~(ti) A~ti- and for derivativemethod2 as

[&,. 4U.L

1 [

,- pll,i tD,i-2 . 4tD,i- 1 &AlcpD,I

@(ti) AP(~i.] )2 PD,i-2 PD,i-1 PE,;

(B-7)1 (c-5)

DerivativeMetlwdsOther methods,which are basedon the dcnvativc of Eq. A.7, canbe developedto compute the tcpfunctionusingEq.B-6.Differen-tiationof Eq. A-7 with respect to rCPyields

q(t)

()- d Q(O

dtcp Ap(i)I

Ret%)(B-8)

.,ICgeneral flnitc diffcrcncc cxp~nsion for the firstdcrivati’.

f’(x) s kdxi (B-9)

whwc

~(x) = fi~t derivative

Af = differenceexpansion

a = constant for a given differenceexpansion

‘xi = panel width for the differenceexpansion.in our case we will use the first and second backwardsdiffcrcnccExpansions,These arc

CaSCl:&=$i -fi.l, U=l m-lo)

case 2 Zw= fi.z - 4fi.1 + 3fi , CS~2 (B-11) ICombining I@. B-8, B-9 and I3-10and solving for Mclj,igives

[ 1.= MQ.L?U. M!ULAtcp,l ~(ti) A/l(li)*(ti.1) (I-.12)

We can also u~eihc Imund.uydominatedflow methodtocomputethe tcpDfunct]on, In this case, tcpD is obtained using E+ R-2,B-3 and B-4.

[

I

I

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1..,.,,,,,,,,,,,,!,,,,,,,,,,,,,,,,,,,,,,.,,

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Oawq (-4 Iww)w (’4 -$+U2 ‘q% ‘CJ

q)‘~ ~

Gc)

r-l

%

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● I-.4

WJ

T1.-

.ij--

&’

8.-

%’.4c)

-4

!sd/u/tiJs ‘dV/[)

. 0 ~(jo 1000 1500 2000 2500120 I

30001 1 I I ~ 120

~p=+()!ltl psia ~

j)o-. — —.—. ..-..--—- .-.—-.-—_______...................... ............... ..... ....... ......... .................. ........................... . .. ....... .... ......_.-...-.__

~ 80- –—— .---.-+..._-._-..._.-.....-....-.-+....................................+.............................

~

~

2 40 ............. ....-.-_.—.&______

q=50 S?’B/l) ~?.~z~ p~a ! pWf=15U0 psia ~@W;] ; (ficw 2) ~ (fl~v~ @ ;

i \o

i500

11000

i1500 2000 2500 3000

L (=Q/q),days

\‘

F@ITe 5- ~afi~sja~ plot of Ap/q versus ta for the liquid simulation cases.

‘~

l“,.

o

i,,,,,,,,.,..,,,,,,,,

.,!,,,,,,,,,,,,

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

?m7

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w

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,0’’-”-{II.,,,.,,..........................,,!!,.

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,,,,,,,,,,.,.............,,,,,,,,,....,, “t’ ,,...,,,,,,,~

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l-f-l 1 1QWw

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... .. ....,,,,,,,,,,,,,,,.- “o

‘ane 1 i>Wv

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‘b

t00

r+

cI/LnswsLl ‘I)/”dv

sPE 21513.

10° 101 102 103 104

‘“’d---’”” ‘ ‘ “’’’” ‘ ‘ “’’’” * 1 n 1 1 1 1q7J01

6

4

1

2{

““-------”-”------------------”---”--”””-s,““”----"---""-"-*.-..-----------------------------------------------------------------------------------------100-

6 /\

~ f

44 Bmmdaiiy Dominated Flow Solution

@APa=(l/ba) El:p[-(rnJbJtcP,.l-). \:

or 2

@Pa=[% 1.,,, + baJ-l............ .... .............................+......... ................................ ................................................... ......................................

10-]6 Results of Gas ~aterial J3alance Iterative

4- solution ( 131asinmrne and Lee8~

ma =2.05536x10-3 Psi/MSCF~~2- ba =1.3094 @./lvlSCF/D 2

G=2.6281 Bscf [tW,.= (bJmJ ~[1 + (m@.J&J I~o-2

i 1 # I1#I#J 10-210° 101 102 103 “410

t.P,a,days

Figure 9- Plot of q/Apa versus tCP,afor d~.ta of Fetkovich, et aL2

t~P,afunction computed using IC,, and the equation shown..

102

101

10°

10-]

~o-2

rr=k”’= 1

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

!

.- 1.........................<6............

g~

z;

5’m

1a n 1 4 13if

I I I 3 I 1 11 I a I I ! Ill I I 1 I t f 111 a 1 1 t I I I

10°1r..8....................................6

4

e

6j

4-

2-

1

L

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

..8.. . . . . . . . .. . .. . . . . . . . . . .. .. . . . .

6

4

2

2

1 8 1 8 x JI.A.Iu

10’ : 102 ; 103JII 1 1 * 1 r~

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

W

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

.. . . ..... . .. .. . .. . . . . . . . . .. . . . . . . . . . ...?......

-l?~ I # 8 I 1. t 1(

-loiI &I Iv1II

‘ ,1io4 10-’ tm,a,days 10-L 10-’

1 1 1 1 1:1 111 1 1 1 I::

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

Mat~h #oint(@A)m; = 0.78 MSCF/Dfi(qdD)mp ~ = 1-0

(~q,~w : = 630days(td~):np ; = l.~

------rJr’w : = 20.0.................................+...........................................

a I 9 I I I u .

104

1 !

10]...-.................................

i

2

~~ 10°...... ................................

102

@

1no

10°‘u 1011“ 102

Figure 10- Type curve match of q/Apa versus tCP,,for data from Fetkovich, et al.2

Fetkovich type curve for radial flow.