Fetkovich Decline Curve Analysis Using Type Curves

SOCIETY OF FETROLEUM ENGINEERS CF AIME

=ER

6200 North C,?ntralExpressway Dallas, Texac K206

SPE

4629

THIS IS A PREPRINT --- SUBJECT TO CORRECTION

Decl

ine

Curve

Analysis

Using

Type

Curves

By

M. J. Fetkovich, Member AIME, Phillips Petroleum Co. @ Copyright 1973 American Institute of Mining,Metallurgical, and PetroleumEngineers,Inc.

This paper was prepared for the 48th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Las Vegasj Nev,j Sept. 30-Ott. 3, 1973. Permission to copy is restricted to an abstract of not more tham 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publicat.icnelsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal prcvijed agreement to give proper credit is made. Discussion of this paper is izv5ted. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. ABSTRACT

curve matching is demonstrated.

This Faper shows that the decline-curve analysis approa~h does have a solid fundamental basis. The exFonent.ialdecline is shown to be a longtime soluticm of the constant-pressure caie. The constant-pressureinfinite and finite reservcir solutions are placed on a common dimensionless curve with all the standard “empirical” expmential, hyperbolic, arid harmoni> decline-curve equations. Simple combinations of material balance equations and new forms of oil well rate equaticms fo: solution-gas drive reservoirs illustrate under what circumstances speoifis values of the hyperbolic de:line exponent (l/b) or “b” should result.

This paper demonstrates that decline-curve analysis not only has a solid fundamental base, but provides a tool with more diagnostic power than has previously been suspected. The type curve approach provides unique eolutions upon whish engineers can agree, or shows when a unique solution is not pssible with a type curve only. INTRODUCTION

Fiate-timedecline-curve extrapolation is one of the oldest and most often used tools of the petroleum engineer. The various methods used have always been regarded as strictly empirical and not very scientific. Results obtained for a well or lease are subject to a kg-log type curve analysis can be perwide range of alternate interpretations,mostly formed Gn declining rate data (constant-terminal as a function of the experience and objectives pressure case) completely analogous to the logof the evaluator. Recent efforts in the area log type curve matching procedure presently of decline-curve analysis have been directed teing employed with constant-rate case pressure towards a ~urely computerized statistical transient data. Production forecasting is done approach. Its basic objective being to arrive by extending a line drawn through the rate-time at a unique “unbiased” interpretation. AS data cverlain along the uniquely matched or best pointed out in a com rehensive review of the theoretical type curve. Future rates are then literature by R2msa , “in the period from simply read from the real time scale on which 1964 to date, (1968!?, several additional papers the rate-time data is plotted. The ability were published which contribute to the underto calzulate kh from decline-curve dzta by tyFe standing of decline-curves but add’litt.lenew .-.eferen:es and illustrations at end o? yapr,

JIECL.INE CURVE ANALYSIS USI

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TYPE CURVES —

nd for b = 1, referred tc as harmonic decline, e have

technology”. A new direction for decline-curve analysis was given by Slider2 with his development of an overlay method to analyze rate-time data. Because his method was rapid and easily apFlied, it was used extensively by Ikunsayin his evaluation of some 200 wells to determine the distribution o the decline-curve exponent term “b”. GentryIsJ Fig. 1 displaying the Arps’4 exponential, hyperbolic, and harmonic solutions all on one curve could also be used as an overlay to match all of a wells’ decline data. He did not, however, illustrate this in his example application of the zurve. The overlay method of SMder is similar in principal to the log-log tyFe zurve matching procedure presently being employed to analyze constant-rata Fressure build-up and drawdown data 5-$’. The exponential decline, often used in decliae-curve analysis, can be readily shown to be a long-time solution of the It followed then constant-pressure zase10-13 that a log-lo~pe curve matching procedure could be develcFed to analyze decline-zurve data. ●

This pap?r demonstrates that both the analytical 3onstant-pressure infinite (early transient pericd for finite systems) and finite reservoir solutions ~an be placed on a common dimensionless log-log type curve with all the standard “empirical“ exponential, hyperbolic, and ha~monic decline curve equations developed by Arps. Simple combinations of material balance equations and new forms of oil well ratt equations from the recent work of Fatkovich~ illustrate under what circumstances specific values of the hyperbolic decline expnent “b” should result in dissolved-gas drive reservoirs Log-log type curve analysis is then Ferformed using these curves with declining rate data completely analogous to the log-log type curve natcliingprocedure presently being employed with constant-rate case pressure transient data

(3)

y=.—

[l+;it]

i

“

“

“

A unit solution (Di = 1) of Eq. 1 was eveloped for values of “b” between O and 1 n 0.1 increments. The results are lotted as set of log-log type curves (Fig. 17 in t~~s f a decline-curve dimensionless rate ES&l ‘Dd

●

●

Nearly all conventional decline-curve analysis is based on the empirical rate-time 3quations qi.venby Arps4 as

For b = 0, we aan obtain the exponential declin aa.uationfrom E@. 1

●

●

“(k)

i

nd a decline-curve dimensionless time ‘Od

=Dit

.

.

.

.

.

.

.

.

(5)

From Fig. 1 we see that when all the basic lecline-curvesand normal ranges of “b” are Iisplayedon a single graph, all cur~es coincide =0.3. Any md become indistinguishable at t Iata existing prior to a t,~dof O. 9 till ,&P~T ;O be an exponential decline regardless of the ;rue value of b and thus plot as a straight lin< )n semi-log paper. A statistical or leastSquaresapproach could calculate any valus of J between O and 1. \NALYTICALSOLUTIONS (CONSTANT-Pi’iSSURE AT [NNER BOUNDARY) Constant well pressure solutions to predizieclining production rates with time were first pklished in 1933 b l;iore,Schilthui.sand {u::stlO,and HurstlI . fiesultswere pssented fcr infinite and finite, slightly compressible, ~in.gle-phase plane radial flow systems. The rf2SUltS were presented in graphical fcrm in Lerms of a dimensionless flow rate and a ji,mensionless time. The dimensionless flow rat ~D ,?anbe ex~ressed as

t = D

0.00634 kt

.

‘

“

“

“

“

●

‘(2)

.

.

.

. (7)

!J~~trw2

The original publication did not inulude tabular values of q and t . For use in this paper infinite SOIU?? ion va ues were obtained from Ref. 15, while the finite values were obtained from Ref. 16. The infinite solution, and finite solutions for r /r from 10 to ant!~-b. 100,000, are Tlot.tedon Fif$~.w2-a

D:t

*

●

and the dimensionless time tu as

AIISl RATE-TIWW4U ATIONS

.

●

%++&+&v””””(’)

BASIC EQUATIONS

4JLJ_

,

●

M. J. FE’I

,PEf+629

Most engineers utilize the constantpressure solutionnot in a single constaRtFressure problem but as a series of constantIressure step functions to solve water influx prcblems using the dimensionless cumulative production Q~ (13). The relationship between {Q~ and qD is d(QD)

(8)

—“=QD dtfi

“

u

“

“

“

“

●

In terms of the empirical exponential ~ec~ne-curvej Eq. 2, Di is then defined as

k@w. D: == A

●

●

●

●

(U)

●

N pi

In terms cf a dimensionless time for iecline-curve analysis we have from Eqs. 5 and 15

“

‘~=[‘:-lt

‘“)

Fetkcvich’7 Fresented a simplified approacl to water influx calculations for finite systems Defining N that gave results whi,>hcompared favorably variables,pi an: ‘qi)- ‘n ‘em ‘f ‘eservoir with the more rigorous analytical sonstantpressure solutions. Equation 3 of his Fa~er, Tl(rrw2) @ ct h pi . . (17) e ~or a constant-pressure pwf, can be written as N 5.615 B pi = ●

Jo (pi - Pwf) q(t)

.

,

.

●

●

(9) and

= [“[T .

1

khpi

t.

(A)= ‘=i’max

e

IJJ*3AJ3

but qi =Jo

●

(pi - pwf)

●

.

.

.

(18)

.* ~nb [() rw

-- 1 2

1

(10)

●

The decline-curve dimensionless the, of reservoir variables, becomes

in terms

0.00634 kt

‘m

=

@ B ct rw2

Substituting I@. 11 into 10 we zan writs

J.

.

.

(19)

Now substituting Eq. 10 and 12 into 9 we obtain qit

Pwf l-— 01

Pi

NPi * (13)

●

Equation 13 can be considered as a derivation of the expmential decline equation in terms of reservoir variables and the constant~ressure imposed on the well. For the same well, different values of a single constant oack-pressure p f will always result in an exponential decfine i.e., the level of backpressure does not change the type of decline. For p f= O, a more realistic assumption for a wel!fon true wide-open decline, we have

~=;[(q(p]t q,

.

.

.

,U)

..**

(20)

To obtain a decline-curve dimensionless rate qw in terms of q~

SPE L625

DECIJNE CURVE ANALYSIS USING TYPE CURVES

L

The published values of qD and tD for the infinite and finite constant-pressure so~utions were thus transformed into a decline-curve dimensionless rate and time, q using Eqs. 20 and 21. Fig. 3 @ :;l:”?%f the newly defined dimensionless rate and time, qm and tm, for various values of re/rw.

(l/b) in terms of the back pressure curve slope (n) and to study its range of expected values. Also, the initial decline rate Di can be expressed in terms of reservoir variables. One further simplification‘lsed in the derivations is that pwf = O. For a well on decline, pwf will USUallY be ~intained at or near zero to maintain maximum flow Pates. Equation 23 then becomes

AL the onset of depletion, (a type of pseudo-steady state) all solutions for various values of re/rw develop exponential decline and converge to a single curve. Figure 4 is a combination of the constant-pressure analytical solutions and the standard “empirical” exponential, hyperbolic, and harmonic decline-curve solutions on a single dimensionless curve. The exponential decline is common to both the analytical and empirical solutions. Note from the som~site curve that rats data existing only in the transient period of the constant terminal pressure solution, if analyzed by the em@rical ArPs approach, would require values of “b” much greater than 1 to fit the data.

The form of Eqs. 23 and 23A could also be used to represent gas well behavior with a pressure dependent interwell permeability effect defined by the~R/FW). lhe standard form of the gas well rate equation is usually given as

‘i% =

Cg

(j5R2-pwf2)n

.

.

.

. (24)

SOLUTIONS FROM RATE AND MATERIALBA LANCE EQUAIIONS ‘he method of combining a rate equation and material balance equation for finite systems to obtain a rate-time equation was outlined in Ref. 17. The rate-time equation obtained using this simple approach, which neglects early transient effects, yielded surprisingly good results when compared to those obtained using more rigorous analytical solutions for finite aquifer systems. This rate-equation material-balance approach was used to derive some useful and instructive decline-curve equations for solution-gas drive resewoirs and gas reservoirs.

I

MATERIAL EAL!NCE OQUATIO’.; Two basic forms of a material balan$e equation are investigated in this study~ p~ is linear with N or G , and ~..2is linear w~th N or Gp (See ‘Figs.p5a and ~). The linearp~R relationship for oi2 is

I and for gas

RATE IJJUATIONS Until recently, no simple form of a rate equation existed for solution-gas drive reservoir shut–in pressure. Fetkovich~ has proposed a simple empirical rate equation for solution-gas drive reservoirs that yields results which compare favorably with computer results obtained using two-phase flow theory. The proposed rate equation was given as

q.

=

J?

‘i

ij

()

(~R2 -pwf2)n

.

.

.

(23)

Equation 25 is a good apFroXhnation for totally undersaturated oil reservoirs, or is simply assuming that durinR the decline vs. N can be approximated by 12!@@% line.p For gas reservoirs, Eq. 26 a straigh is oorrect for the assumption of gas compressibility (Z) =1. In terms of j5R2being linear with cumulative production, we would have

Ki

where n will be assumed to lie between 0.5 and 1.0. Although the above equation has not been verified by field results, it offers the opportunity t.odeflna t,hedeeljw expnqn{.

~R2=-

kLi2 N () Pi

NF+~Ri2

.

●

“

(27)

‘I%isform of equation results in the typical shape of the pressure (~ ) VS. cumul-ative ~roductfon (Np) relaticn%ip of a s@lutfon-Fas

M. J. FE! )VICH

3PE 4629

drive reservoir as depicted in Fig. 5-b. Applications would be more appropriate in nonpmrated fields, i.e., wells are produced wideopen and go on decline from initial production. This would more likely be the case for much 18 of the decline-curve data analyzed by Cutler obtained in the early years of the oil induetry before proration. RATE-TIME EQUATIONS, OIL WELLS Rate-time equations using various combinations of material balance and rate equations were derived as outlined in Appendix B of Ref. 17. Using Eq. 23A and @. 25 the res~ting rate-time equation is Clo(t)

—.

‘= [+J+,1%

●

(y

●

5

exponent (l/b) is 2.0 and 3.0 m b = o.5~ ad 0.333 respectively. This range of “bE values fits Arpsl ftidings using Cutlorls declinecurve data. He found that over 70 percent of the values of “b~ lie in the range Ramsay found a different O~b ~0.5. distribution of the value of “b” analyzing modern rate decline data from some 202 leases. His distribution may be more a function of analyzing wells that have been subject to proration and are better represented by the assumptions underlying the rate-time solution given by Eq. 28, i.e., ‘ver the decline period% ‘s” ‘pWas linear DECLINE-CURVE ANALYSIS OF GAS WELLS Decline-curve analysis of rate-time data obtained from gas walls has been reported in only afewinstances19S 20. Using Eq. 24 with ~wf =0, and Eq. 26, the rate-time equation for a gas well is

1

C@) A unit solution, (qoi/N .) = 1, ofEq. 28 is plotted as a log-log ty~ curve for various values of n, Fig. 6, in terms of the (For deel.ine-curvedimensionless lttie t =0, q .%”(q .) . these de~ivations with p For the limiting range o~fback-pr%ure %%? sloFes (n) of 0.5 and 1.0, the Arps empirical decline-curve exponent (l/b) is 2.0 and 1.5 respectively or “b” =0.500 and 0.667 a surprisingly narl;owrange. respectively To achieve an~pnential decline, n must be equal to zero, and a harmonic decltne In Fractical applications, requires n+-. if we assume an n of 1.0 dominates in solutiongas (dissolved-gas)drive reservoirs and ~ vs. N is l%near for non-uniquely defined rate-?ime data, we would simply fit the ratetime data to the n = 1.0 curve. On the Arpsj solution type curves, Fig. 1, we would use (l/b) =20rb =0.667. The rate-time equatior obtained using Aq. 23A and Eq. 27 is qo(t)

—=

1

-

●

●

(29)

‘oi p(~)t+r The unit solution of Eq. 29 is plotted as a log-log type curve for various values of n, Fig. 7. This solution results in a complete reversal from that of the previous one, n =0 yields the harmonic decline and n + .W gives the exponential decline. For the limiting range of back-Fressure curve slopes (n) of 0.$ and 1.0, the decline-surve

——

(3C)

“’

●

for all back-pressure zurve slcFes wh~re n > 0.5. For n = 0.5, the exponen~ial decline is obtained

()*t q~

C&)

.e

-

..0.

●

(31)

‘gi The unit solutions of Eqs. 30 and 31 are plctted as a log-log type curve cn Fig. 8. For the limiting range of back-Fressure curve slopes (n) of 0.5 and l.oj the AW declinecurve expment (l/b) is 00 and 2, or b = O (exponential) and 0.500 respectively. The effect of back-pressure on a gas well is demonstrated for a back-prassure curve slope n =1.0 on Fig. 9. The back-pressure is expressed as a ratio of Pwf/Fi* Note that as pwf -+, p. (Ap+O) the type curva approaches ex’ymentlalldecline,the liquid case solution. Whereas back-prsssure does not change the type of decline for tha liquid-case solution it does change tha type of decline in this case. Using the more familiar rate and material balance equations for gas wells, we can obtain the cumulative-time relationship 5;’integrating the rate-ttie equations 30 and 31 with Gp=

/t o

qg(t)dt

.

.

.

.

.

(32)

r

DECLINE CURVE ANALYSI USING TYPE CURVES For n > 0.5 we oktain

&

1

-

[1 + (2n-1)

SPE 4629

common point on both sheets are recorded.

+t]e ()

&,..”

:-’ ●

(33;

and n = 0.5

Lag-1og type curves of Eqs. 33 and 34 couk be prepared for convenience in obtaining cumulative production.

Recent papers by Agarwal, et.al.5, E#mey6, Raghavan, et.al.7 and GringartenS et.al. ~ have demonstrated or discussed the application and usefulness of a type-curve matching pror.edureto interpret constant-rate pressure bu~ld-up and drawdown data. van Poolen21 demonstrated the application of the typec.urveprocedure in analyzing flow-rate datz obtained from an oil well producing with a constant pressure at the well bore. All of his data, hcwever, were in the early transient period. No depletion was evident in his examples. This same type-curve matching procedure can be used for decline-curve -malysis. The basic steps used in type-curve matchin~ declining rate-time data is as follows: 1. Plot the actual rate versus time data in any convenient units on log-log tracing paper of the same size cycle as the type curve to bs used. (For convenience all type curves should be plotted on the same log-log scale so that various solutions can be tried.) 2. The tracing paper data curve is placed over a tyFe curve, the coordinate axes of the two curves being kept parallel and shifted to a position which represents the best fit of the data to a type curve. More than one of the type curves presented in this paper may have to be tried to obtain a best fit of all the data.

5* lf none of the type curves will. reasonably fit all the data, the departure curve method 15$ 22 should be attempted. This method assumes that the data is a composite of two or more different declinecurves. After a match of the late time data has been made, the matched curve is extra@lated backwards in time and the departure, or difference, between the actual rates and rates detemi.ned from the extrapolated curve at coi-respondingtimes is replotted on the same log-log scale. An attempt is then made to match the departure curve with one of the type curves. (At all times some consideration of the type of reservoir producing meck~tism should be considered.) Future predictions should then be made as the sum of the rates determined from the two (or more if needed) e:krapolated curves. TYPE CURVE MATCHING EXAMPLES Several,examples will be presented to illustrate the method of using type curve matching to analyze typical declining ratetime data. The type curve approach jmwides unique solutions upon which engineers xm agree, or shows when a unique solution is not possible with a type curve only. In the event of a non-unique solution, a most probable solution can be obtained if the pI’OdUCi.ng mechanism is known or indica~ed. ,ARPS’HYPKRBOLJC DECLINi EXAMPLE Fig. 10 illustrates a type curve match of Arpsl example of hyperbolic decline4. Every single data point falls on the b =0.5 type curve. This match was found to be unique in that the data would not fit any other value of “b”. Future producing rates can be read directly from the real-time scale on which the data is plotted. Ifwe wish to determine q. and D.. use the match mints indicated on “ Ftg. 10 *: follows

qm

=0.33

qi = 3. Draw a line through and extending beyond the rate-time data overlain along the uniquely matched type curve. Future rates are then simply read from the real-time scale on which the rate data is plotted.

‘Dd

4. To evaluate decline-curve constants or reservoir variables, a matoh point is se~ezted anywhere on the over?.app?ng~rt$on of the curves and the coordir,atesof this

@= ~ j

1000 BOPIVI = 0.G33

=12.0

.Di=

=

1000BO~ % 30,303 BOPM

= Dit = Di 100MO.

~= 100 Mo

0.12 MO.-l

The data could have also been matched ustng the ty~s ~urves on Pigs. 6 and 7. h both cases the mat~h would have oeen obtained with a back-pressure curve slcFe n = G.5 which

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M. J. FETKOVICH

7

the ex~nential interpret.ationgives a total life of 285 MONTHS, the harmonic ll@ MONTHS. ‘his points out yet a further advantage of the type curve approach, all pssible alternate pi“ interpretations zan be conveniently placed on one curve and forecasts made from them. A The fact that this example was for a lease, statistical analysis would of course yiald a a group of wells, and not an individual well single answer, but it would not necessarily be raises an imprttint question. Should there be the correct or most probable solution. Cona difference in results between analyzing eazh sidering the various producing mechanisms we well individually and summing the results, ~ou>d select. or simply adding all wells production and analyzing the total lease production rate? a) b = O, (exponential), if the reservoir is Consider a lease or field with fairly unifomn lib!: or n is similar for highly undersaturated. reservoir properties, eazh well, and all wells have been cn deckine b) b = O, (exponential),gravity drainage with at a similar terminal wellbore pressurey Fwf~ no free surface28. for a sufficient period of time to reach FSeUdCc) b = 0.5 gravity drainage with a free steady state. According td Matthews et.al.23 Iiat(Fseudo) steady state the drainage volumes surface58. in a bounded reservoir are proportional to the d) b = 0.667, solution-gas drive reservoir, rates of withdrawal from each drainage volume.” (n=l.O) if~Rvs. Npis linear. It follows then that the ratio q./N . will e) b = 0.333, solution-gas drive reservoir, be identical for each well and th$isthe sum of (n=l.0) if~k2 vs. Npis approximately the results from each well will give the same linear. results as analyzing the total lease or field production rate. Some rather dramatic illusFRACTURED WiLL EXAMPLE trations of how rapidly a readjustment in drainage volumes can take place by changing Fig. 12 is an example of type surve the production rate of an offset well or matching for a well with declining rate data drilling an offset well is illustrated in a available both before and after stimulation. Similar drainage volume paper by Marsha. (The data was obtained from Ref. 1.) This readjustments in gas reservoirs have also been type problem usually presents some difficulties demonstrated by Stewart25. in analysis. Both before and after frac. loglog plots are shown on Fig. 12 with the after For the case where some wells are in frac. data reinitialized in time. These before different portions of a field separated by a and after log-log plots will exactly overlay fault or a drastic permeability change, each other indicating that the value of “b” readjustment of drainage volumes proportional —— did not than e for the well after the fracture to rate cannot take place among all wells. The ratio ./N . may thenbe different for different treatment. tThe before frac. plot can be considered as a type curve itself and the group %#f hlls. A total lease or field after frac. data overlayed and matched on it.) production analysis would then give different results than summing the results from individual Thus all the data were used in an attempt to de~e “b”. When a match is attempted on the well analysis. A stilar situation can also Arps unit solution type curves, it was found exist for production from stratified reservoirs that a “b” of between ~.6 and 1.0 could fit 26, 27, (no-cw~~flow)o the data. Assuming a solution-gas drive, a match of the data was made on the Fig. 6 ARPS’ EXK)NENTIAL DECLINE EXAMPLE type curve with n = 1.0, b = 0.667. Fig. 11 shows the results of a type curve Using the match Feints for the before analysis of Arpsl example of a well with an frac. data we have from the rate match mint, apparent exponential decline. In this case, there is not sufficient data to uniquely 1000 BOPM establish a value of “b”. The data essentially qw “0.2.43 = w= qoi fall in the region cf the type curves where all q’oi curves coincide with the expnmtial solution. As shown on Fig. 11 a value ofb =0, 1000 BOPM = 4115 BOPM qo~ = 1.0 (harmonic) appear to (~pnential) orb= ●243 fit the data equally well. (Of course all values in between would also fit the data.) From the time match point, The difference in forecasted results from the two ewtreme interpretationswould be great in qoi t = = 0.60 = 0+115 BOPM)(1OO MO % later years. For an economic limit of 20 BOP?4, r N pi pi 5.sequivalent to b = 0.5. Match pints determined from these curves could have been ;sed to zalculate~ and ~/Npi and finally

()

-

N

f4115 BOPM) (1OC MO.) 0.60

pi =

=

685,833 BBL

then q ~=

4115 BOPM 685,833

N ...4

=0.134

=

~

. “Oi

.

qoi

100C BOPM 0.134

=

1000 BOPM qoi

‘Dd

N

pi

.

q.

~ .m6W(loo

+ () pi

MG.~ pi

= (7463 BOPM) (loo MOJ 1.13

= 660,4.42BBL

then q oi

‘= N pi

..-.

““.”,--,

---

+Ws

/

proprtion to the flow-rate, the ratio qoi/Kpi should have remained the sane as that obtained prior to treatment or 0.006000 MO.-l. This then would have indicated an N of pi

pi =

‘J&J 130PM —-1 .006000 MO.

= 1,243,833 BBL

Actual in~rease in reserves as a.rssult of’the fracture treatment a~pea,rsto lie between ths two extremes. Based on the msthod of al,alysisused, the astual increase in reserves attributable to the fracture treatment is 198,109 BBL, (660,442 BBL - 462,333 BBL; STRATIFIED r’SiRVOIR EXAMll&

7463 BOPM

From the time matsh pint .1*13

.

N

= .006000 M0.-1

Now usin~ the match points for the after frac. data we have from the rate match Pint, q~

“------

7463 BOPM = .011300 MC.-l 660,w2 BBL

We can now zheck the two limiting conditions to be Considered following an inzrease in rate after a well stimulation. They are: 1. Did we simply obtain an acceleration of production, the well~ reserves remaining the same?

This example illustrates a method of analyz5ng decline-curve data foi-a layered (no-crossflow) or stratified reservoir using type ~urvese The Qata is taken from i@f. 18 and is for the East Side Colinga Field. Ambrose2? presented a cross seztion of the fiel~ showing an upper and lower oil sand se~arated by a continuous blazk shale. This layered description for the field along with the predictive equation for stratified reservoir presented in Zef. 29 led to the idea of using the departure curve method (differencing)to analyzed dezlint?-curvedata. 27 After Russell and Prats the production rate of a well (or field) at p;eudo-steady state producing a single phase liquid at the same constant wellbore pressure, (Fwf = O for simplicity), from two stratified layers is

J% qT(t) = qi~

e

)

~Npi 1

() t

-—qi

t +-qi2e

N

●

Did the reserves hcrease h direzt proportion to the increase in producing rate as a resu t of a radius of drainage readjustment 3 ? BefGre treatment, N . was fOUnd to be 685,833 BBL. Cumulative F~%duction determined from the i-atedata prior to stimulation was 223,500 BBL. N then at the time of the fracture treatm~i%tis

pi 2 ✎

.(35)

✎

2.

N

pi

=685,833 EBL-

=

o.016J_42MO.-l

Actual (qoi/Npi) after treatment was 0.011300 MO.-l.

qT(t) =ql

(t) +q2

(t)

.

.

●

.

(36)

The total production from both layers then is simply the sum of two separate forecasts. Except for the special case of the ratio q /N ~ being equal for Loth layers, the sum o}tio expnentials will not in general result 223,500 BBL =462,333 i3BI in another exponential.

If only accelerated pradution was obtained and after the reserve” remained the same, q~ F pi the fracture treatment should have been 7&63 BOFM f+62,333BBL

‘r

If the resarves increased in direct

In attempting to match the rate-ttie data to.a type curve, it was found that the late time data can be matched to the exponential (b =0) type ourve. Fig. 13 shows this matcki of the late the data designated as layer 1. With this match, the curve was extrapolated backwards in time and the departure, or difference, between the actual rates determined from the ext~apolated curve was replotted on the same log-log scale. See TAB~ 1 for a

summary of the departure curve results. The difference or first departure curve, layer 2, itself resulted in a unique fit of the exponential type curve, thus satisfying Eq. 35 which can now be used to forecast the future production. Using the match points indioated on Fig. 13 to evaluate qi and Di for each layer the predictive equation becomes

qT(t)

9

M. J. FETKOVICH

SPE L629

= 58,824 BOPY e 50sGOOBO~

- (o.2oo)t + .*.*.

e - (O*535)t

where t is in years. Higgins and Le~htenberg30 named the sum of two expnentials the double semilog. They reasoned that the degree of fit of empirieal data to an equation increases with the number of constants. This interpretation is not claimed to be the only interpretation pX3sib16 for this S3t of data. A match with b =0.2 can be obtained fitting nearly all of the data points but can not be explained by any of the drive mechanisms so far discussed. The layered concept fits the geologis description and also offered the opportunity to”demonstrate the departure curve method. The deFarture curve method essentially places an infinite amount of combinations of type curves at the disposal of the engineer with which to evaluate rate-time data. EFFECT OF A CHANGE IN BACK-PRESSURE The effe~t of a change in back-pressure is best illustrated by a hypothetical single well problem. The reservoir variables and conditions used for this example are given illTable 2. The analytical single-phase liquid solution of Fig. 3 is used to illustrate a simple graphical forecasting superposition procedure. The inverse procidure, the departure or differencing method can be used to analyze decline-curve data affected by bask-pressura changes. After Hurst12, suparpsition for the constant-pressurecase for a simple single pressure change can be expressed by kh

or

(37)

Up tc the time of the pressure zhange pwf2 at t~l the well production is S~PIY he q fore-t ql - depicted on Fig. U. as a function of time is simply kde by evaluating a single set of match points using the reservoir variables given in Table 2. At Pwfi and re/r’ w =100 t=lDAY;

t =0.006967 Dd

ql(t) = 697 BOpD; q~

= 2.02

Plot the rate 697 BOPD and time of 1 day on log-log tracing paper on the same size cycle as Fig. 3. Locate the real-time points over the dimensionless time pints on Fig. 3 and draw in the re/r~ curve of 100 on the tracing paper. Read flow rates as a function of time directly from the real time scale. When a change in pressure is l~de to pwf2 at tl, t equal zero for the accompanying zhange in rate q29 (really a Aq for superpostion)j this rate change retraces the qm vs. t~ curve and is simply a constant fraction of ql

q2

=ql [

Pwfl - pwf2 Pi - ~wfl

1

- 1 day after the rate change is ::U:? :; t q2 = 697 BOPD

(pi - Pwfl)

..*

1000 psi Psi - 1000 50 Psi psi = 221 BOPD 4000

q (t) = 141.3(@)

In [

($)

-$]

‘M(t~l

kh (Pwfl - Pwf2) + ‘103(~)[’n(~)-~]

‘~(tod-’~~)

The total rate qT after the pressure shange is q = ql + q2 as depicted in Fig* M* All rates o T flow for this example were read directly from the curves on Fig. 11+and summed at times past the pressure change pwf2. The practical application of this exam~le in decline-curve analysis is that the departure or difference method can be used on rate-time data affected by a change in

,)

-

lWf?T. TN17 “...4.-1.-

flIIRVR “...v-

AN?ITY.STS

“.,.,

-----

back-pressure. The departure curve represented by q2 on Fig. 14 should exactly overlay the curve represented by q . If it does in an actual field example, ~he future forecast is correctly made by extending both curves and summing them at times beyond the Fressure change.

IISTNC ““J.

..”

TVPT7 .

..-

PIIT+VF< ““.

.”--

.sPr?

“.-

J. L’)G

6+”t.,~

wallbore radius r ‘ (obtained from the buildup analysis) is u%d to obtain a good match between build-up ant decline-zurve ~alculated. kh. ‘. TYPE CUFWdS l~iiKNOkJNRESERVOIR AND FIJJ~ PROPER71EL —

CALCULATION OF kh FROM DLCLINE-CURVE DATA

All the type surves so far discussed wera developed for de~line-curve analysis using Pressure build-up and decliae-curve data some necessary simplifying assumptions. For spe~ifi,creservoirs, when PVT data, reservoir were available from a high-pressure, highly undersaturated, low-permeability sandstone variables, and back-pressure tests are reservoir. Initial reservoir pressure was available, typs surves could be Eenerated fcr various relative permeability wrves and estimated to be 5750 psia at -9300 ft. with ba>k-pressures. These curves developt~ for a bubble-point pressure of 2841 ~sia. Two field-wide pressure sarveys were condu?tsd while a given field ~iouldbe more accurate for analyzing fleclinedata in thak field. Conthe reservoir was still undersaturated. lable 3 summarizes vl,ereservoir properties and ventional material balance programs or more basic results obtained from the rressure kuild- sophisticated simulation models could be used UF analySiS on %azh Well. Note that nearly to develop dimensionless sonstant-pressure 31 all wells had n8ga.tiveshins as a result of type zurves as was done bv Levine and Pratts (See their FiF. 11). hydraulis fracture treatments. Also, a~~earing on this table are results obtained CONCLUSIGNS from an attenpt to calculate kh using declinecurve data available for eazh of tk,ewells. Decline-zurve analysis not only has a Ten of the twefity-twowells started on solid fundamental base, but providss a tool with more diagnostic power than has previously desline when they were first placed cn been suspe~ted. The ty~!e:urve ap~roach ~:roproduction. As a result, the early production vides unique solutions u~on whi~h engineers desl:ne data existed in the transient pmiod can agree, or shows when a unique solution is and a ty~e curve analysis IlsingYip. 3 was matched to one of the r@/’rwst:ms. Gther wells not possible with a tyre zurve ocly. in ths event of a non-unique sclution, a most ~robable listed on the table, kh5ra an r ~rw mat:h is solution zan be obtained if the produ~ing not indiczted, were Frorated wefls and began meshanism is known or indicated. their de:line several montk,safter they were first put on production. For the de2lineNO~NCLATU~, zurve determination cf kh, the reservoir pressure existing at the beginning of decline for =-l{~giro~al of detline >urve exponent each well was taken from the pressure history lb (1/b~ match of the two field-wide pressure surveys, The zonstant bottom hole flowing ~ressure for ~ = Formation volume factor, res. vol.,’ the wells ranged between 80G and 900 psia. surface vol. A type curve mat~h using deeline-~urve data to talculate kh for well No. 13 is illustrated on Fi~. 15. A type ?urve matsh using pressure build-up data obtained on this same well is illustrated on Fip. 16. The constantrate type curve of’Gringarten et.al.8 for fractured wells was used for mat~hing the pressure build-up data. The build-up kh of47.5 rnd.-ft.cmmpares very well with the kh ofl+O.5 MD-FT determined by using the rate-time dezlinezurve data. In general, the comparison of kh detarnrine d from deuline-turve data and prsssure build-ur, data tabulated on Table 3 is sur~risingly good. (The pressure build-up analysis was performed independently by another engineer.) One fundamental observation to be made from the results obtained on wells where a match of re/rw was not ~ssible is that the effective

Ct (-J= D! ~=

Gas well ba~k-pressure :urve coefficient -1 = Initial Decline rate, t =-Natural logarithm base 2.71828 = Initial pas-in-~dase, surface measure

G G

= Total compressibility, psi-l

p

= CUmu].ativegas production, SUrfaCe measure

h

= l%i~knsss, ft.

Jo

= Productivity index, STK BHL/DAy/pS~

J;

= Productivity index ’(back-pressuresurve coefficient) STK BBL/DAY/(psi)2n

k,

= Effective permeability, md.

n

= Expnent of bask-pressure curve = Cumulative oil ~roduction, S’fKBBL

N F

SPE L629

N

Fi

M. J. FE!

Cumulative oil production to a reservoir shut-in pressure of O, STK BBL

Fi

Initial pressure, psia

h

Reservoir average pressure (shut-in pressure), psia

Fwf

Bottom-hole f’lowin~pressure, psia

)VICH 7.

Raghavan, R., Cady, G. V. and Ramey, Analysis for H. J., Jr.: i,~;ell-Test Vertically Fractured Walls””,J. Pet. Ta?&~ (hlgo 1972) 10L4.

8.

Gringarten, A. C., Ra!!ey,H. tJ.~Jr. and. Raghavan, R.: tgpres~ureAnalysis for Fractured ~lells”,paper SPE 4051 presentad at the 47th b.nnualFall lfeethin, San Antonio, Texas, (Gzt 8-11, 1972Y .

9.

MeKiriley,R. M.: “Wellbore Transmissibility from Afterflcw-DominatedPressure Buildup Data”, J. Pet. Tesh. (July, 1971) 863.

10.

Moore, T. V., Schilthuis, R, J. and Hurtitj w. : tl~e Deter~~natiOn of Permeability from Field Data”, Bull..ApI (~ky, 1933) ~ 4.

11.

Hurst, W.: “Unsteady Flow of Fluids in , &sics. (Jan., 1934) Oil lieservoirs” ~, 20.

12*

Hurst, W.: “l!aterInflux into a ileservoir and Its Application to the Zquaticm of Volumetric Balance”, —— Trans., ?l~’~(1943) ~, 57.

13.

van Everdingen, A. F. and Hurst, ~~.:“The AFpli2ation of’the Laplaee Transformation to Flow Proble.nsin :eszrvcirs”, Trans. AIME (1949) 186, 3050

:4.

Fetko\fi~h,M. J.: “The Isochronal Testing of Oil v~ells”,Papr SPE 4529 presented at the 48th Annual Fall Meetinz, las Vepas) !~~vada,(S~~t. 30-G~t. 3, 19?3).

15.

Ferrt~,,J.$,Knowles, D. B., i3rown,~:.J~.~ and Stallman$ R. W.: “’Ihecryof Aquifer Tests”, g. S. Gecl, Surv., ‘:later 29FF1% Pap3r 1.53~, 109.

16.

Tsarevi~h, K, A. and Kuranov, I. J’.: l’f~a~culation of the I’1ow l:atesfor the Ce:lt3r‘Jellin a Circular Seser’mir Under Jlastie Conditlcns”, Problems @f Rgservoiy Hvdrods’nami~sPart.I, Leningrad, m 9-34

Initial surfase rate of flGw at t = O

% (qi

Initial wide-oFen surface flow rate at pwf = O

q(t,

Surface rate of flow at time t

q~

Dimensionless rate, (Eq. 6)

‘Dd

Decline wrve dimensionless rate, (Eq. &)

QD r e r w ~f tw ‘D

lXLmensionlessmxnulative production External boundary radius, ft. Wellbore radius, ft. Effective wellbore radius, ft. Time, (Days for tD) Dimensionless time, (@. 7)

%

Decline nrve dimensionless time, (Eq. 5)

~

Porosity, fraztion of bulk volume

P

Vis20sity, cp.

REFERENCES 1.

2.

11

Ramsay, H. J., ~Jr.:“The Ability of Rat@Time )ecl.ineCurves to Predizt Future PrcxiuctionRates”, M.S. T?_lesis3 U. of Tulsa, Tulsa, Okla. (1968). Slider, H. C.: ‘IASimplified Method of Hyperboli~ Decline Curve Analysis”, J. Pet. Tech. (March, 1968), 235.

3.

Analysis”> Gentry, R. W.: llDe~line-curVe J. Pet. Tech. (Jan., 1972) 38.

4.

Arps, J. J.: IiAnalysisof Decline Curves”$ Trans. AIME (1945) &@, 228.

17.

5*

Ramey, H. J., Jr.: “Short-TimeWell Test Data Interpretation in the Precence of Skin EffeZt and Wellbore Storage”, J. Pet. Tech. (Jan., 1970) 97.

~etkovi~h, M. Jo: “A Simplified Approach to ~!at:?r Influx Calculations-Finite Aquifer Systems”, J. Pet. Tech. (tJuIY~ 1971) Xl-4.

18.

W. W., Jr.: llEstimationof Lhderflr.lttlg]~, ground Oil Reserves bv Oil-well P~duction Curves”, Bull., USBM”(1924)~

●

6.

Agavwal, R., A1-Hussainy, Il.and Ramey, H. J., Jr.: llAnInvestigation of ~~elibore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment:j Sot. Pet. Eng. J. (Sept., 197G) 279.

19.

Stewart, P. R.: “Low-PermeabilityGas Well Porfomnanee at Constant.Pre9sure”,

J. Pet. Tech.

(Gept. 1970) 1149.

2

DECLINE CURVE ANALYSIS USING TYPE CURVES

20.

Gurley, J.: 11Aproductivity and EsOnOnd.C Projection Method-Ohio Clinton Sand Gas Walls”, Paper SPE 686 presented at the 38th Annual Fall Meeting, New Orleans, La., (Oct. 6-9, 1963).

21.

van Poollen, H. K.: “HOW to Analyze Flowing Well-Test Data... with Constant Pressure at the Well Bore)’,Oil and Gas Journal (Jan. 16, 1967).

22.

Withers~on, F. A., Javandel, I., Neuman, S. P. and Freeze, P. A.: “Interpretation of Aquifier Gas Storage Conditions from Water Pumpinp Tests”, Monopraoh AGA~ New York (1967) 110.

23.

Matthews, C. S., Brons, I!.and Hazebroek, P“ “A Method for Determination of A&age Pressure in a Bounded Reservoir”, Trans., AIME (?954) 201, 182.

24.

hlarshjH. N.: llMethodof Appraising Result! of Production Control of Oil Wells”, ~. API (Sept., 1928) 2@, 86.

25.

tewart, P. R.: “Evaluation of Individual Gas Well Reserves”, Pet. Eng. (Mayj 1.566) 85.

26.

——

SPE 4629

27.

Russel, Q. G. and Prats, M.: “Performance of Layered Reservoirs with Crc)seflow---Single-Ccimpressible Fluid Case”, SOS. P@. Eng. J. (March, 1.962)53. —

28.

Matthews, C. S. and Lefkovits, H. C.: llGra~~,y~rainage Performance of Depletion‘l&peReservoirs in the Stripper Stage”, Trans., AIME (1956) ~, 265.

29.

Ambrose, A. W.: !!underg~und ~enditions in Oil Fields”, Bull., USBM (1921) 195, 151.

30.

Higgins, R, V. and Lechtenberg, H. J.: !!Meritsof ~e~line Equations Based on Production History of 90 Reservoirs”, Paper SPE 2450 presented at the Rocky Mt. I@ional Meeting, Denver, Colo., (My 2527, 1969).

310

I&wine, J. S. and Prats, M.: “’TheCal:u~ated performance of Solution-Gas Drive Reservoirs”, Sot. Pet. Eng. J. (s3pt., 1961) 142.

ACKNOtilEIXXMENT I wish to thank phillipS Petroleum Co. for permission to publish this FaFer.

Lefkovi%, H. C. and Matthews, C. S.: NOTE : Tle author has a limited quantity of Curves to Gratity“Applisaticm of Desline full size type curvas with grid suitable Drainage Reservoirs in the Strippr Stagel’ for actual use which are avzilable on Trans., AIME (1958) ~, 275. written request. _——. .. ..

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Q,

l+

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[

4 3

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1

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r Fig. 1 - Type curves for Arps empi.

cal

rate-time

D1l’

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‘D

9 8 7 -

-

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6 –

–

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4

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infinite and finite Fig. 2A - Dimensionless flow rate functions olane radial system 10,11,15,16 ~uter boundary, constant pres-ure at inner bmndary.

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finite system with constant Fig. 6 - Dissolved gas drive reservoir rate - decline type curves effects not included. pressure at inner boundary (pI+T= O @ rlJ). Early transient

45678102

1$’ . 87 6 5 4 SLOPE n

3

2

2“+

0.5

2.000

0.6

2.200 2.404)

0.454

2.600 2.800

0.365

3.mo

0.333

21.000

0.046 0.005

0.8 0.9 1.0 10.0

-

7\

o

1 .mo

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0.1 9 8

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100.0

–

3

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—

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–

7

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7

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6

–

5

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824 ,1

—

S3 u

MATERIAL 2

‘R’= IATE

BALANCE

-(*)

T

‘“+””

EQUATIONS:

TWO-PHASE [OIL&GAS):

6R ~

GO = Jo,

(

0.01 :

RATE-TIME EQUATION

(“~

,!/

EQUATION:

q. = O): ~

~R’

\ %4 n 5 !-

‘W*

)[

1

(t)

-

2n+l

7 t+l

6 w)’

5 4 3

SINGLE-PHASE

LIQUID:

RATE.TIME EQUATION fexwnentiall

l~f

1

GO = JO (6R - Pw$ q. = 01: ~

(t) “ — .(i)’

2

\ 0.00

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Fig. 7 - Dissolved gas drive reservoir rate - decline type curves finite s~stem with constant pressure at inner bwndary (ph~ = O @ rl,). E:rN transient effects not included.

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back pressure finite system with constant Fig. 9 - Gas reservoir rate decline type cur-~eswith pressure at inner boundary (pVfl= constant Z r,,). Early transie,;teffects not included and z = 1 (based ~n gas uell back pressure curve slope, n = 1.

‘Dd

= 0,1

1000

EXPONENTIAL

—

I

MATCH POINTS EXPONENTIAL, t=loo

EXPONENTIAL,

b =O

b= 1

t= looh.hO. ;tDd=l.60

MO, ;tDd=l.lo

J(t) = 100 BOPM ; qDd ‘ o.2~2 ECONOL~lC LIMIT 20 BOWVIm 1480 MO.

q(t) = 100 BOPM ; qDd = o.212 ECONOMIC LIMIT 20 BOPM @ 285 L1O.

\

EXPONENTIAL

-–L 1480klo,

1 285 fI~O.

100

10

1000

FFIACTURE TREATED

\l AFTER FRAC

BEFORE FRAC

()

J

—

\, “

I

‘“h \

MATCH POINTS \ \

AFTER FRAC

BEFORE FRAC

1.0, (b = 0.667)

\

n = 1.0, (b = 0.667)

n =

t = 100 MO. ;tD~ = 0.60

t= loo MO. ;tD,j=

q(t) = 1000 BOPM ; qDd = 0.243

q(t) ‘ 1000BOPM ; qDd

1.13

\ = o.134

\ \ \

\

100

10

1

1

t- MONTHS Fig. 12 - Type-curve

analysis

of a st imulatw!

well before

and after fracture

treatment.

..-.

1000

Ioo,ooo

O-q’=q’+q’ / /

\

EXPONENTIAL

>

$ , 10,03( $’

_

MATCH POINTS

b=O,

EXPONENTIAL LAYER 1

t=loy

LAYER 2

Rs. ;tDd =2.0

qz=’+-ql

Cl(t) = 1000 BOPY ;qDd = 0,017 LAYER 2 t= loy

Rs. ;tDd =5.35

q(t) = 1000BOPY

; qDd = 0.020

●

1,00

100

10

1 t - YEARS ~ig, 13 - Tj’pe-curie

anal:.’si s of .s layered

reser,mir

(no crossflm.:) t,::di ffcrcncin~.

+ Q(2)

< * \

-=2

\* \ \ \ \

\

.-. .-

lUW

100

10

t - DAYS Fig. 14 - Effect

of a change

in back pressure

on decline

using

graphical

supe~osition.

I

11 1

I ing e Fig. 15 - Tj~e-curve xatck.

‘ Dd

I

I

F- :,., ..> :’:

.

..

L~,,;; . . .. . .

Pressure

at

0.00634 kt ‘3L

10 0.1

10

1

t-

‘

~ucl

~Z

Iw

HRS.

Fig. 16 - Type-curve ::atckingexample f~r calculating Kb from pressure buildup data, :Jell13, ~iel[iA (type curve from Refl.9).