WC
b
61%J1U
M@wtJ@xan
Decline Curve Analysis by T.A. Blaslngame,
for Variable Pressu, J Drop/Variable
T,L. McCray, and
This
cop~hhl
t99\,
S&ety
This fmper WA8 prepeti TM paprx pmwnted, poslrim_Iof Permlwm ~~t~
W,J. Lee, Texas
is a preprint
EnLmm’s
Flowrate Systems
A&M U.
-- subject
to correction.
of Pe!rdeum Engineers, Inc. Psvpmsenlatim
a? dm SPE Qat Tedmbgy
Symposium held in Hwston,
Texas, January 23-24,1691.
was $elemed for presenlalicm by m SPE Program Commitrtoe following mtiew of inlormetkm conleined In an ●Mrnc4 wbmirted by the author(s), C%nmn,s of Um paper, ac ham not bean revkwed gmesoc*volPlrk e o urn I%qineem tnd are subject 10 C&motion by tie eulhor(t). The materiel, ● : presented, does not necosaanly rellecr any rho Sodaly of Petdeum nglneem, 11solfioerc, m members, Papera presented at SPE meetings are wbjact to publidon ravbw by Editorkil committees of the Sodary of &@neers, Permlmbn to copy is restdcled 10 an ab$tract o! not more thnn 303 words. Il[utl!arions may no! be copied. The abstract should conlain .%nsplcwous *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 arose from a need to analyze production dcclinc data where the flowing bottomhole pressure varies significantly, The varirtncc of the bottomhole rcssur@with time CXCIUS.ICS the use of the exponential decline mJ c1for conventional dcclinc curve analysis (scmilog plots arid type curves). Using pressure nonmalizcd flow rate rather than flow rate usually does not remedy this problem. The rncthmi wc present uscs a rigorous superposition function to account for the variance of rstc and pressure during production. This furwdon is the constant rate analog for vanablc.rate flow during post-tmnsicnt conditions and can bc used to develop a constant pressure analog for the dcclinc curve analysis of field data. The constant pressure antdog time function is computed from the constant rate function using the identity that cumulative production for both cases must be qud. Using the cumulative production identity, wc SOIVC recursively for the time function usin~ trapczoirlal rule integration and, as an altcrna!ivc, frnitc diffcrcncc formuhtc, Wc have also dcvclopcda const;mtpressure analog time rchuion which is rigorous for bounrky domimttcd flow and stxvcs as an accuraIcapproximationfor tmnsicntflow, We apply theserelationsto analyticalsolutionsfor verification and then usc the boundary dominntcdflow relation on simulated and ticld cascst These wmuhuioncases include htrgc rmdsmall step chun ‘CSin bottomhole tlowirtgpressures,and pcricxiicshutins. Finn]! y, wc apply these relationsto a gus w.’IIfield case,
The widespread usc of ty~rcCUI-VCSIJ to analyze mlc dcclinc data has motivated us to consldcr the implicationsof varying rate and pressure drop production. ‘1’hcorcticallys~aking, for the flow of a slightly compressibleliquid, the analytical stems cmthe Fctkovichl ty curve arc valid only for the ccm!ant wellbore pressure pr z uction CMC,+6 In previous worksv,a wc have shown that variablcwatdwtriablcpressuredrop data may bc transRcfcrcnccsand=tmtions
at cnd of paper
formed into an cquivslcmconstantrate ca.wfor both gas and liquid flow data. Camacho9indcpndcntly vcnficd that this equivalent constant rate formulation is exact for the constant pressure production of a sli htly comprtssiblc liquid during boundary dominri:ti flow con#‘mons, McCrayto sought to develop a method to transform variablcrate/vanablc pressure drop data into an equivalent constant pressure case. In doing this, McC!raydcvclopcd a recursion formula to compute an cquivdcnt time for constant wellbore pressure production, tcp, that could bc used with pressure drop normalized flowrate to perform dcclinc CUTVC analysis using type curves. Although the approach st,rggcstcdby McCray was verified using simulation, wc sought a rigorous foundation for the applicationof this result, As it turns out, a relativelysimple proof can bc shown for the application of the rcp function during boundary dominated flow conditions, This prcmf is given in AppendixA. In addition to the proof of McCray’sresult, wc also provide mcthmls to compute the constant pressure equivalent time, 2CP, using recursion formuhm in Appendix B. In Appendix C wc provide relations which can bc used to compute the constant pressure dimcsionlcss mtc solution given the ccnsutnt rate dimensionlesspressure solution, In the text wc will pmvc thtitthe computatiortalmethods wc provide ylcld csscntittllyexact results during boundarydominated flow and give very good pcrfornmncc during transientflow. ~ Our first goaJ is to establish that our new mcttisxi actually transforms a variable.ratclvariablc pressure drop systcm into an equivalent constant rcssure systcm, Wc begin with a proof of the validity of our soPution by transforminga constantrate system into a constant prcssutv s stcm. Because these solutions arc frqucntly used in dimonsrSMIMSformat, wc will perform this verificationusingdimcnsionhw variabkx.
2
SpE 21513
DeclineCurve Analysisfor VariablePressureDrop/VariableFlowrateSystcn?s
The computationalformulaearc givenin AppcndiccsB and C and wc will verify each. Thcsc include the fcilowing rccuraion formulae; the integral method proposed by McC!ray10and the 2and 3-point brickwarddtifcmncomethodsdcvclopcdin this work The recursion relations for this part of the verification arc dcvclopcdin AppendixB and summarizedin AppendixC, We will also use the boundarydominatedflow relationswhich result from tsating the constant rate and constant pressure 7 or..’,This dcvclopmcntand the pertinentrelations anal tics! SOIU for ti is psxtof the verificationam given in AppendixB. Fi . I shows the log-log behavior of the qo functions vcraus tD ant tcp~ function for tic cssc of a well ccntcrcd in abounded cimulaxreservoir (r~=NP), Duc to the number of mcthcds being cansidcm we will discuss the transientand boundarydominated flow behavior separately, First wc note that, during early times (transient flow), all of the tcpp methods yield a good approximation to tic q~(t~) sohmon, cxccpt at very early times (t@20). This implies that all of these methods yield a reasonable approximation to the analytical solution during transient flow, Obviously,the analyticalsolutionfor tmundarydominatedflow is not valid during transient flow as shown by the deviation of this solution and 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 D mcthds agrc.cwry closdy with the qD(lD) Solution.Athough &ls inspection,it dots appear that t D2 for scale prccludcs very CIOSC the dcnvativc mcth(xl1 dots show signi!lcantdeviation, ‘f%iswill bc invcstigsttcdmore closely when these dara arc rcplottcd on a scmilog q~ gmph in Fig. 2. Fig. 2 is a rcplot of Fig. 1 using a scmilog scale for the qD fLSnctiOnS and a cancsian scsdcfor the tD functions. Wc nOtCthat the scsulta for dcnvativc method 1 do begin to diverge from those r f the Othermcthtsds,which cIwly ovcr]ay the correct solution, Fig, 2 sug~sta that dtivativc rrscthod1 should not be used in racticc, but that the integraltncthw dcnvativc mcthcx!2, and the L undarydominatd flow methodshould give accuratewsults. Of these, the boundary dominatti flow method is the easiest to apply since it dots not rcqtlirc recursion calculations, but instead provides a dhtct Uansf{“mationbctwccn tD (or tc,D) and tcp (Eq.B-2), tc~o is the is the dimensionless matcnaI balance time function, introduced by Blasingamcand UC7 and later (and indcpcndcntly) b Camacho9as an quivalcnt constant rate time function for variai lc-ratchriablc pressuredrop flowconditions. Onc problcm that tic boundarydominatedflow method might b pcrccIvcdto have would bc that in field applications,formation proptics arc rquircd to computethe m and b constantsin the tcr tcftransform relation, Eq, IV1. However, these constants arc easily dctcm~incdfrom a cartesian plot of A@Qversus tc+=Q/q) during boundarydominatul flow. This proccdurc is verified for a variety of cases in Ref. 7, Also, bccausc this is a rigorous formulation for boundary dominated flow, the m and b constants arc unique and will provide theoreticallyconsistent results when used in Eq. B-1, Additionally,Mcthylo reportedinconsistentresults using the integral method to compute the rc function when Iargc pressure changes ancl/or shut-ins occurc# McCray was forced to usc cmpisicalextrapolationpointsto cause the imcgralmethodto yield correct results, Again, the boundarydominated flow method will not have these roblcms Iwausc tic m and b parameters arc unique and the formulation of the tq+ transfo~ 0%. B-1) iS rigorous, For these rcmons, wc rccommcnd that the bounda dominated flow rncthod which uscs Eqs. B-l (field) or B-T (dimcnsionlcss) be used for computation of the tcp or ?CPD
functions. Wc will demonstrate the ap~licationof the boundary dominatedflow mcthcxion a simulatedhquid productioIIaqucncc and a fic!dcaac for a gas WCIIthat has been analyzedpreviouslyin the literature.ZM
In this section wc will apply the tcP-tcrtransform describediri the previous section to a simulatedproduction squcrtcc in an Oil WCI1.The mscrvoirdata and flow historyarc given in Table 1. TABLE 1 WellimdReservdr Parameters (Well Centered in alloundedCircular Resemoir)
RB/STB cl, psia-l h, ft 4 \i, Cp rW, ft re, ft k, md s pi, psia CA
1,00
B,
15.0 X 1o-I5 0.:: 745 (40
0°2: acre) 1,0
48~ 31.62
Using the WC]!and rcscswoirparametersgiven above and Eqs. A-3 to A-5, wc computethem and b paramctwsand wc obtain m= b =
2,3&/09x10-z@/S~/D/D 32.8948 pSi&~@ Flom History
flow scqucncc
r, days 180
; .. : .. 5 6, .. 7 8
180
1:0 180 3
180 S40
q,STBfD ——
p~,
psia
50
constantrate 2000 shut in constantp~ 4000 Constyt p)@ 3000 shut in 3s00 1500 shut in constantp~ 2500 const~x p~ 1500
const~t p~
The flow mtc profiles that WCIEobtained from the simulation cases arc shown in Fig, 3, Wc have inchsdcdan arbitrary base case (p~30Mt psia) to orient the analysisof all of tic data, That is, when wc have obtained the correct transformationof data, all cases including the base case should overlay the same trend, Although it is conceivable that these rate profiles could bc analyzed separatelyusing dcdinc curve analysis, the cawistcncy of the rcsuhs would b dcpcndcnton the ability of the analyst, In particular, the transient spikes caused by pressure changes and shut-ins would be difficult to interpret, and the non-uniform behavior after the effect of the spike has subsided would inevitablyyield ambiguousrcstdts. Onc method used to align variable rate pressure data with the correct cccnstantrate solution is rate norrnahsstionof the pressure drop. In dwlinc curve analysis, many analystsusc prcsstsrt dmp normalisation of the flow mtc profile in an attempt to obtain the
SPE 21513
T.A. B!usingamc,T.L. McCrayand W,J, La
“correct”constant pressure solution. It can be seen in Fig. 4 that II prcs: xc drop normalization does not yickl a constant nrcssure analog solution,
TABLE 2 Well~Reservoir par~ters (Assumed Geometry: Well Centered in a Bounded Circular Reservoir)
Clearly, wc must usc other techniques which are more rigorous than psessumdrop normalizationfor field applica{{,ons of decline curve analysis, The method of choice will be the onc proposed by Blasmgamc and Lec7 whit: converts variablcratchriablc rcssure tip data to the equiwdcm constant rare case. From t is anal sis wc will obtain the m and b parameters rquired by Eq. ‘! B-1 or transformation to an equivalent constant pressure system, Fig, 5 shows the cartesian plot of alp/q vs. tcr(=Q/q) rquircd to determine them and b parameters. m is the slope of this plot and b is the intcrcc t. Although thcte is some data scatter, it is ckar that them an1 b psramctcra do rcprmcnt a best fit trend of the dam. Therefore, the step of determining the m and b parameters is illustrtttcdas a simple and straightforward process. Fig. 6 shows the log-log plot of Ap/q vetmssrc, that could be used for type curve matching on constant rate type curves. Fig. 6 also shows that the concept of using Aplq and fcr appears to also be valid for transientflow, given the agreementbetweenthe constant rate and constant pressure base case (pW~3000 psia) during transientflow (rcrc50days). The next step is to usc them and b parameters in Eq. B-1 to convert from tcr (constant rate analog time) to tcp(constant pressure analog time), This is also a simple and straightforward procedure. Once tc is computed, a log-log plot of q/@ vs. Gp is made. Fig. 7 is suci a plot and wc immcxiiatclynote that all cases overlay the same trend during both transient and boundary dominated flow. obviously, the analyticalsolution for boundary dominatedflow (exponentialdecline) will not agree with transient flow solution. Fig, 7 represents the endpoint of our effort to determine art quivalent anstant pressuretransformationfor variablc-rate/vari~blcpressuredrop flow data, Wc are satisfiedthat this is a logical and consistent procedure that should yield accurate rcsuhs when applied to field data, The verification of this method is that all cases overlay the base case @~-30fXt ]’sia), where ~IAPand t were used as the plotting functions for I$C base case. At this point, Fig, 7 can & used for decline curt c analysis using type curves such as the one presentedby Fctkovich,l
In this section wc will apply the ICrtc, transform @q. B-1) to the analysis of vttriablc rate gas well data as described in rcfcrcnccs 2,3, and 8. Unfortunately, the complexity of this analysis is compounded because gas WCIIanalysis requires the used of paeudoprcssurcand pscudotime, Wc will refer to ref. 8 for the pseudopressure and pscudotimc functions as WCIIas for the msulta of an iterative procedure to dctcrminc the m and b parameters and the gas-in-place, G. This iterative procedure, as described in ref. 8, simulau+ntmuslydctcrmincs these parameters tsauac the pseudotime function requires Icnowlcdgcof the gasin-phw for matcdal balancecomputations.
3
R13/MSCF et, psia-l h, ft # & Cp Fw,ft B,
~, md p.~ psia pi, psia C* G, Bscf (!’ef,2) G, Bscf (rcf,3)
0.70942 1.870 X 10”4
0.;: 0,02167 0,354 -5,30 * 0,0786S * 710 4175 31,62 3.360 3.035
* Averageof valuesobtainedfrom ref. 2 and 3. From the results of ref. 8 wc have ma = b. =
G=
2,O5536X1O-3 ps@lSCFP/D 1.3094psi/MscF/D 2.6281 Bscf
Fig. 8, which is a log-log plot of @a/q versus tc~,a,is taken directly from rcf, 8 and shown here for complctcmess. Wc htwc included the computedresponseduting boundarydominated flow as prescribed by Eq. A-1, Thc Apa/4 and tcr,a variables arc the pseudotimc and pseudopressureas defined and computed in ref. 8, This nomenclature may seem awkward, but defining these variablesin this manner allowsus to usc liquid flow quations for analysis, Therefore, any equationswc preaem are valid for either liquid or gas flow as long as the correct time and pressure variablesarc used. Wc note that the boundary dominated flow sohstiondoss not model the transient flow behavior of the data in Fig. 8. This is cxpectcdand wc only state this observationfor completcncss, Wc could use Fig. 8 in a type curve matching analysis with constant rate type cwves. However, it is our objective to analyze these so wc must proeecd with data with a constant pressure typeCUIVC transfomningthe tcr~variableto yieldCcp,a. This analysisrquircs that wc modify the variables in Eqs, A1, A-3, A-4, and B.1 for analysis of gas WC]]test data using pscudoprcssurc and pscudotimc, For gas analysis Eq. A-1 becomes (1) Eq, A-3 becomes
~a=_&
(2)
Eq. A-4 becomes
The reservoirdata and flow hiworyarc given in Table 2. (3) For a closed cyhl.rkxd swemoir Eq, 3 becomes (4)
I
4
DcclincCur-wAnalysisfor VariablePressureDrop/VaririblcFlowrateSystetm
And finally,for gas weil test analysisEq. B-1 hxomes
Also, for a bounded cirat!ar reservoir, the fosmation permeability, k, can be computed using
(5), We have usedEq. 5 to eomjwtcthe tc~ functionsused in Fig. 9. Note that Fig, 9 is a log-log lot ot q fAP~ versus tcp,~and that the the boundary dominated h ow sOIWiOn(Computccfq14Pa function) agrees very wcli with the data during boundary dominated flow but not during transient flow. This is expected and 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. 10 reprc%ms this type curve match. Note that the data agree with the t curve during the transition from transient to boundary r ominated flow and throu hou; boundary dominated flow, The scamity of data for tc ~<1h days (4 points) iimits our interpretation of the transient Fiow portion of the data, but an estimateof r#W’=20 for the tmnsientstem seemsreasonable. Once the data are overlain and matched to the type curve, we will determine a match point from the coordinates of both plots, The match point for Fig, 10 is RateFunctionMatchPoint: qD= 1.0 q{dp = 0,78 MSCFiD/psi
-g_ k=14L2~
Polo” ~-z ~w
4
*
qdD
(lo)
Using Eq. 8, wc can estimate the gas-in-place, G, This calculation gives G=&[(w)(Hll G = 2.6278 B~f Using Eq, 10, wc can cs:imatc tthe formation permeability, k, This calculationyields ~ ~ ,41 ~~0.70942 0.02167
(;0) k=
+@o)-ii(w)
().()5432md
The computed values of gas-in-place and permeability compare very WC]] with those obtained in ref. 8, In fact, the results arc vimsally identical. The reason for this is quite simple, The constant rate analog method (using @a/q and tcr,a) and the constant pressure analog method presented in this work arc rigorously related, Therefore, if wc are consistent,both methods shouldyield the same results. Our estimate of gas-in-place, G, is 22 percent less than the estimate of Fctkovich, et al,q and is 14 percent Icss than the estimate given by Fraim and Wattcnbarger.j However, wc express confidence that our estimate is as acclqratt as the ones given by the other investigators. And wc feel that our approachis more rigorous, because of the pscudoprcssurc and pscudotimc soiutionfommiation.
Time MatchPoint: t&=l
tcP~= 630 days Cwve~
SPE21513
,.
Once the match point is determined for a data set on the Fctkovichl type curve, we can computethe volume of fluids in the reservoir and the formation permeability, k. We need computational relations for liquid (oil) and for gas where the correct pseudopressureand pseudotimefunctionsare used.
Our estimate of permeability is within 31 pcrccnt of the average of the estimates given in refs. 2 and 3. We feel that our permeability estimate is consistent with these vahsesof the other investigatorsand ccstainlyreasonable,even given the sparsedata trend in the transient flow region. These data arc used to identify ths rJrWc value finm the tmnsicntstem type curves.
The oil-in-place,N, can be computedusing
SJMMARYANDCOKLJJSIONS
~ N=.~ & h ct [()(1] 9dD mp rdD
rnp
(6)
The formationpermeability,k, can& computedusing
(7)
The gas-in-place,G, can bc computed using
‘=*[(~)mp(~~)mj
(8)
The formationpermeability,k, can be computedusing
(9)
This paper introduces a method that can be used to analyze variable-rate/variable pressure drop production data using a constant pressure analog time function, The most significant result is that of the boundary dominated time transformationfor constant mte or constant pressure flow, This transform, given by Eq. II-1, allows an analyst to compute an equivalent time for constant pressure production, quickly and easily, based on the parameters m and b, obtained using Eq. A-1, For gas wclis this prcdurc is lCSSstraightforwardand rquircs an iterativesolution developed in ref. 8. Wc considered four diffcmnt methods to tratyfomrtvariable. rate data into the constant pressure solution profile, Wc considered three recursion formulae which compute the constant pressure~uivalcnt time function, tc,by panel summationsbased on trapezoidalrule intcgrationloan{’fmitc diffcrcnccexpansions, Each of these relations was ap lied successfully to convert the constant rate dirncnsionlcss soPution into the constant pressure Iiowcvcr, McCraylo found that dimcnsionlcs solution. application of the trapezoidal rule integration method gives poor results when a plied to erratic data or data with cxtcncded shutins, This bc[ avior makes the general application of these rccrnsionfwrrwlacdifficultat best,
SPE21513
T,A. Blnsirsgime,T,L. McCray and W.J. Lee
The fourth method developed was a rigorous identity which equates the kxtndary dominated solutions for constant rate and constant pressure production. The resulting two-parameter relation @q. B-1) may bc used for dimensionless solutions or field data applications. When the m and b parameters are determined using the methods developed in ref, 7, data scatter should have Iittlc effect on the VSIUCS of the parametersbecause a best fit trend is established. These characteristics make the boundary dominated flow method the most useful product of this work.
Cp = D=
mp =
5
constantpressureor constantpressureanalog dirncnsionlcssvariable matchpoint on a typecurve
Wc gratefully acknowledge the assistance of Elizatwth BarbozattndJemniferJohnston for their hcIp in the preparationof dds manuscript.
Applications: Wc recommend using the methods prcscntcd in this work thr the type curve analysis of variable-rate/variable pressure drop production data, The methcd is relatively simple and should be applicable to a wide range of WCI1test problems, including the analysisof gas well test data demonstratedin this work.
1.
Fetkovich, M.J.: “Dcclinc Curve Analysis Using Type Curves,”JPT (June 1980) 1065-77,
2.
Fetkovich,M.J., et UL“Decline-CurveAnalysisUsing Type Curves.-Case Histories,”SPEFE (Dec. 1987) 637-56.
3.
Frairn, M.L. and Wattcnbmgcr, R,A.: “Gas Reservoir Decline-CusvcAnalysis Using Type Curves With Real Gas Pseudopressureand NormalizedTime,” WEFE (&c, 1987)
Conclusions: 1, The recursion fonnulae discussed in this work should not be applied in practice due to problems associatedwith the erratic nature of field data, which could cause poor results. 2. The boundarydominatedflow method is the methodof ckaicc to transform the constant rate analog time function ir,to a constant pressure analog time function, This mcthid is consistent, easy to apply, and should give accuratercsuits for a wide range of problcmtypes. 3. The boundary dominated transform method can be used co model constit wellborepressure productionbehaviorcxaciy during boundary dominated flow and should give accurate results during transient flow.
671-82,
4.
Ehlig-Economides, C.A. and Ramcy, H,J., Jr.: “Transient Rate Dcclinc Analysis for Wells Produced at Constant prCSSUR,” :PEJ (Fcbo1981)98-104,
5.
Ehlig-Economides, C.A. and Ramey, H,J,, Jr,: “pressure Buildup for Wells produced at a Constant pressure,” $PEJ (FcIJ. 1981) 105-114.
6.
Blasingamc, T,A. and Lee, W.J.: “Properties of Homogeneous Rcscrvoi.m,Naturally Fractured Reservoirs, and HydraulicallyFracturedReservoirsfrom Decline Curve Analysis,” paper SPE 15018 presented at the 1986 SPE PcrrnianBasin Oil and Gas RecoveryConfcrencc,Midland, TX, March 13-14, 1986.
7.
INasingamc,T.A. and Lee, W.J.: “Variable-RateReservoir Limits Testing,” paper SPE 1.5028presented at the 1986 SW? Permian Basin Oil and Gas Recovery Conference, Midland,TX, March 13-14, 1986.
8.
Blasingamc: T.A. and Lee, W.J.: “The Variable-Rate Reservoir Limits Testing of Gas Wcllst paper SPE 17708 prcscntcd at the 1988 SPE Gas Technology Symposium, Dallas, TX, June 13-15, 1988,
9.
Camacho-V,, R,G,: Well Pe@ormance Under Solution Gas Drive, Ph,D, Dissertation, University of Tulsa, Tulsa, OK (1987),
NO a Dimensionless Variables = dimensionlessconstant defined by Eq, B-4 bD
CA %’ f@ Y Field
= = = = =
Variables(Formationand Fluid ProDerliexJ fo~ation volume factor, RB/$IT3 ‘ constant as defined by Eq, A-4 (liquid) and Eq, 3 (gas) to:sI compressibility,psiai gas-in-place,MSCF (or 13scfas in the fieldexample) total formationthickness,ft formationpcrmeabllity,m? constant as defined by Eq. A-3(liquid) and F.q,2 (gas) oil-in-place,ST13 initial reservoirpsessurc.psi3 flowing bottomholepressure,psia flowrate,STB/Dor MSCWDfor the fieldexample cumulativeproduction,STB wcllbom radius, ft reservoir chainagcradius, ft re/rw, dimensionlessrescmoir drainageradius porosity, fraction viscosity, cp
Subscripts = a
&.
dirrmsionless shape factor dimensionlessconstant defined by FA.B-3 dimensionlesspressure (conslantmtc case) dimensionlessslltc(constantpressurecase) Euler’sconstant (0.577216 .,, )
“adjusted”variable for gas well test analysis. Usc of these variables in gas well test analysis yields an cquivalcrstliquidsys~cm. constantrate or conwmt rate analog
10, Iviccrav.T,L,: Reservoir AnalvsisUsing Productwn Decline Data &d Adjusted Time, ‘M,S, Tkis, Texas A&M University,College Station, TX (1990), Appendix A: ~
of v~
,&!&QWtm
.
We will start with a relationwhich allows for the constantrate analysis of vsiriablc-ratekwiablc-pressure drop test data during boundary dominated f!ow, This relation has been verified for liquid flow by Bhsingamc and Lec’;~ro?d”~~machog, This relation has also been verified for gas flow by IMsingarnc and IA@, The constantrate analog,wlationis given as
DeclineCurveAnalysisfor VariablepressureI)rop/VaricblcF!mvratesystems
6
4(0
—=mtcr+b N)
(A-1)
t J
where
= --1’q(t)dr = — ‘(t) (time in Days) ‘Cr f?(r) * q(t)
We need to prove the right-h~.,ld-side(RHS) of ~., A-8. This is done by integrating Eq. A-7 so it is the same form as the RHS of W,. A-8. This gives
(A-2) or
m=5.6l5--@--@hcfi
(A-3)
()
b=70.6#n~
SPE21513
eTCA r~2
(A-11)
(A-4)
and
CombiningEqns. A-7 and A-11 gives ,
rw = rw e-s
(A-5) We will also need the general solution for a well producing at a constant pressureduring boundarydominatedflow. Thk solution is given by Ehlig-Economides and Ramey~,Sand later by Blasingameand Lec.c This so!utionis
&)cp ‘row)
(A-6) Our objecciveis to develop a general time function that allows us to use Eq. A-6 to model a variable-rate/vanable-pressur~ drop process, This generalrelation is (A-7) where tcp is the time at which the constant pressure solution is valid for a general variable-rate/variable-pressuredrop response. In this sense, ICPis an unknown which must lx dctcrmincri. McCray10proposed [he following rcl.~tionas a defining refwion for tcp tp Q(t)= —. M(t) o
&l
~T
A’P(o
1
(A-8) McCray developed Eq, A-8 via induction from the constant pressure solution where ~(?) would be constant and could be factored out, Although McCray proved Eq. A-8 cmpincal;y using simulation, wc sought a more rigorous justification. In the fcJlowing dcnvtitions we provide a rigorous proof of Eq, A-8 for boundary dominate ! flow. Frst we rquire the coustant rate analogresult, Eq, A-1, and Q. A-2, CombiningEqs. A-1 and A-2 gives
Solving Eq. A-9 for Q(t)/dp(t) gives ,- b-@m [1
Ap(t)
AppendixB: ~J%!@Jl ‘ “ arv 130ml~ The objectiveof this swion is to developa methodto computethe rcpfunction. A relatively simple relation is obtained by quating Eqs. A-1 and A-7 and solving for tcp.This gives ~ Exp(~@), “ *
or
%/2 =$ql +fpcr) or in termsof dimcnsimless variables
Jmq.. td ‘c/)D =mD
(
where mp=L. tirW2 r@2 A
)
(B-1) (R-2) (B-3)
(B-4) whew
Although Eqs. B-1 and B-2 are strictly valid for boundary d~ot~a$d flow, we find that, for short times, these relations (K3-5a:
and, in dimcnsiortlcssfomr (A-9)
Q(O .1
Notice that the right-hand-sides of Eqs. A-10 and A-12 arc identical, This result proves that Eq. A-8 is exact for boundary dominatedflow,
tcp Z tcr
9#1.mQ~+h
AP(t)
(A-12)
(A-10) Eq, A-10 provides us with the Icft-’:and.sidc(I,HS) of Eq, A-l?.
tq)J) = t@
(}1-5b
Although wed? not present a rigorous proof, J@ B-5a and B:51 do suggest thut Khctcpfunction can M approximated by the tc function during transient flow: General applicationof I@ B-5 and B-5b requiresfurlhcr invcstigaticm,
SPE21513
T,A. 131asingamc, T,L, McCray and W,J, lee
I Integral hiethod McCmylo proposed
to compute the tc functionby approximation of the integral in Eq. A-8 using ti c trapezoidal rule, This essentiallyresults in a recursion formula where the tcpfunciion is computeda:
7
And combining Eqns. B-8, B-9, and B-11, and solving for AtCP,i gives
‘rep’l
. ~ @(ti) Q(ti.$ 4Q(ti-1)+ ~ %(ti) [Ap(ri.2) @(ti-1) AP(ti)
1
(B-13)
Appendix C: ~~
‘Cp= f At.p,i i= 1
The
AtCP,i terms are computedusing individualtrapumid panelsof
*C q(1) — function. For an individualtrapezoidwc have APO) Ii .%!?L * 2 [@(ti) also
+ ~ti-l) &(ti.
1)
Wi-1)
AtcPD,i =2[%%+1 ~ ~ ~-
aLIL A~ti- J
&,.
4U.L AP(~i.] ) 1
[PD,i F’D,i-l1 and fol dxivative method 1 as tD,i pfi,i
f’(x) s k dxi whwc ~(x) = fi~t derivative Af = differenceexpansion = constant for a given differenceexpansion a
(B-7)
case 2 Zw= fi.z - 4fi.1 + 3fi , CS~2
m-lo) (B-11) I
Combining I@. B-8, B-9 and I3-10and solving for Mclj,igives Atcp,l
.= MQ.L?U. M!UL ~(ti)[A/l(li)*(ti.1)1
and for derivativemethod2 as ,- pll,i tD,i-2 . 4tD,i- 1 & AlcpD,I 2 [PD,i-2 PD,i-1 PE,;
(c-4)
1
(c-5)
We can also u~eihc Imund.uydominatedflow method to compute the tcpDfunct]on, In this case, tcpD is obtained using E+ R-2, B-3 and B-4.
I
(B-9)
‘xi = panel width for the differenceexpansion. in our case we will use the first and second backwardsdiffcrcncc Expansions,These arc U=l
(c-3)
AtcPD,i = tD,i - —— PD,i-1
DerivativeMetlwds Other methods, which are based on the dcnvativc of Eq. A.7, can be developedto compute the tcpfunctionusing Eq. B-6.Differentiationof Eq. A-7 with respect to rCPyields q(t) - d Q(O I dtcp ()Ap(i) (B-8) Ret%) .,ICgeneral flnitc diffcrcncc cxp~nsion for the first dcrivati’.
CaSCl:&=$i -fi.l,
(c-2)
where the ZMCpDj are given for the integralmethod as
1
@2iL. [ ~(ti)
[@(ti)
I
tcpD = i~l AtcpD,i
1
Combiningand solving for AtcP,igives
t~p,i =
(c-1)
Theequivalentdimensionlesstime is
for a given pane!.
‘
In the calculations, the constant pressure dimensionless rate solution is definedas qcpD = ~
/i .—. - Q(tJ . — Q(ti.]) [AP(ri)
[
(I-.12)
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-4
-$ ‘q
q) ~
.
~(jo
0
1000 1
I
120
1500
1
2000 I ~ p=+()!ltl
3000 ~ 120
psia ~
............. ........ ............... ..... ....... ......... .................. ........................... . .. ..... — —.—. ..-..--—- .-.—-.-—_______ .. .... ......_.-...-.__
j)o-.
~
2500 I
.---. -+..._-._..._. -..... -....-.-+ ....................................+.............................
80- –——
~
~ 2
40
............. .....-_. —.&______
q=50 S?’B/l) ~?.~z~ @W;]
o
;
p~a ! pWf=15U0 psia ~ (ficw 2) ~ (fl~v~ @ ;
i
\
500
1000
i 1500 L
F@ITe 5-
~afi~sja~
1 2000
i 2500
3000
(=Q/q),days
plot of Ap/q versus ta for the liquid simulation cases.
\
‘
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.,
!,,,,,,,,,,,,
u
‘~
,,
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l“,.
o
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w
to
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r-
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,,,,,,,,,,.,.............,,,,,,,,,....,,
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m
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l-f-l
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sPE 21513 .
10°
101
‘“’d---’””
102
103
‘ ‘ “’’’” ‘ ‘ “’’’”
6
*
104 1
n
1
1
11
q7J01
4 1
2 {
"---""-"-*...---------------------------------------------------------------------------------------100
““-------”-”------------------”---”--”””-s, ““”----
-
6
/
4
\
4
Bmmdaiiy Dominated Flow Solution
-).
@APa=(l/ba)
El:p[-(rnJbJtcP,.l
\:
or
2
@Pa=[% 1.,,, + baJ-l ............ .... .............................+.... ..... ................................ ................................................... .................... .......... ........
6 42-
~o-2
10°
Results of Gas ~aterial J3alance Iterative ~ solution ( 131asinmrne and Lee8~ ma =2.05536x10-3 Psi/MSCF~~ ba =1.3094 @./lvlSCF/D G=2.6281 Bscf
10-]
f 2
[tW,.= (bJmJ ~[1 + (
[email protected]&J
I
i 1 # I1#I#J
101
10-2 102 t.P,a, days
103
“4
10
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.
r
102
101
I
I
I
3
I
1
..8. .r.......
a
I
I
Ill
!
1
JII
I
1
I
I
t
f
1
1
.. .... ........ ........ ..........
*
1 r~ ... . ... .......
6
1
1
t
I
a
I I
; 103 111
1:1 1 1 1 I :: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1
1
(qdD)mp
1
I
9
I
I
I
u .
102
104 10] ...-................................. @
~ = 1-0
630days (td~):np ; = l.~ ------rJr’w : = 20.0 .................................+..... ...................................... (~q,~w
e W
6j
a
Mat~h #oint (@A)m; = 0.78 MSCF/Dfi i
4
10°
111
: 102
10’
10°
r=k”’= 1 ..........................
I
11
:
=
2
................................! 1......~~ 10°
1
4!
2-
.-
1
.........................<6............
10-]
1
.. . ..... . ..
..8..
... . ... ..... .
.... ........ . .... ...... ......
... ... ...... .. .. ... ... ....... ........ ....... . ..... ............ ... ...... . ....... . ..... ....?......
6
g ~
4
z ;
2
5’ m
~o-2
io4
a
1 n 1 4 13if
2
-l?~
L 1 .A
.Iu
10-’
8
1
8 x JI
I
#
8 I 1. t 1(
I &I Iv1II ‘ ,1 -loi
tm,a,days 10-L
10-’
10°‘u
1011“
Figure 10- Type curve match of q/Apa versus tCP,,for data from Fetkovich, et al.2 Fetkovich type curve for radial flow.
102
no