SOCIETY OF FETROLEUM ENGINEERS CF AIME6200 North C,?ntralExpresswayDallas, Texac K206

THIS IS A PREPRINT --- SUBJECT TO CORRECTION

=ER SPE 4629

Decl ine Curve Analysis Using Type Curves

By

M. J. Fetkovich, Member AIME, Phillips Petroleum Co.

@ Copyright1973American Instituteof Mining,Metallurgical,and PetroleumEngineers,Inc.

This paper was prepared for the 48th Annual Fall Meeting of the Society of PetroleumEngineers of AIME, to be held in Las Vegasj Nev,j Sept. 30-Ott. 3, 1973. Permission to copy isrestricted to an abstract of not more tham 300 words. Illustrations may not be copied. Theabstract should contain conspicuous acknowledgmentof where and by whom the paper is presented.Publicat.icnelsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OFPETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriatejournal prcvijed agreement to give proper credit is made.

Discussion of this paper is izv5ted. Three copies of any discussion should be sent to theSociety 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

This Faper shows that the decline-curveanalysis approa~h does have a solid fundamentalbasis. The exFonent.ialdecline is shown tobe a longtime soluticm of the constant-pressurecaie. The constant-pressureinfinite andfinite reservcir solutions are placed on acommon dimensionless curve with all the standard“empirical” expmential, hyperbolic, aridharmoni> decline-curve equations. Simple com-binations of material balance equations andnew forms of oil well rate equaticms fo:solution-gas drive reservoirs illustrate underwhat circumstances speoifis values of thehyperbolic de:line exponent (l/b) or “b” shouldresult.

kg-log type curve analysis can be per-formed Gn declining rate data (constant-terminalpressure case) completely analogous to the log-log type curve matching procedure presentlyteing employed with constant-rate case pressuretransient data. Production forecasting is doneby extending a line drawn through the rate-timedata cverlain along the uniquely matched or besttheoretical type curve. Future rates are thensimply read from the real time scale on whichthe rate-time data is plotted. The abilityto calzulate kh from decline-curve dzta by tyFe

.-.eferen:esand illustrationsat end o? yapr,

curve matching is demonstrated.

This paper demonstrates that decline-curveanalysis not only has a solid fundamentalbase,but provides a tool with more diagnostic powerthan has previouslybeen suspected. The typecurve approach provides unique eolutions uponwhish engineers can agree, or shows when aunique solution is not pssible with a typecurve only.

INTRODUCTION

Fiate-timedecline-curve extrapolationisone of the oldest and most often used toolsof the petroleum engineer. The various methodsused have always been regarded as strictlyempirical and not very scientific. Resultsobtained for a well or lease are subject to awide range of alternate interpretations,mostlyas a function of the experience and objectivesof the evaluator. Recent efforts in the areaof decline-curve analysis have been directedtowards a ~urely computerized statisticalapproach. Its basic objective being to arriveat a unique “unbiased”interpretation. ASpointed out in a com rehensive review of the

!?literature by R2msa , “in the period from1964 to date, (1968 , several additional paperswere publishedwhich contribute to the under-standing of decline-curvesbut add’litt.lenew

JIECL.INECURVE ANALYSIS USI

technology”.

A new direction for decline-curve analysiswas given by Slider2 with his development of anoverlay method to analyze rate-time data.Because his method was rapid and easily apFlied,it was used extensivelyby Ikunsayin his evalu-ation of some 200 wells to determine the distri-bution o the decline-curve exponent term “b”.GentryIsJ Fig. 1 displaying the Arps’4 expo-nential, hyperbolic, and harmonic solutions allon one curve could also be used as an overlayto match all of a wells’ decline data. He didnot, however, illustrate this in his exampleapplication of the zurve.

The overlay method of SMder is similar inprincipal to the log-log tyFe zurve matchingprocedure presently being employed to analyzeconstant-rata Fressure build-up and drawdowndata 5-$’. The exponential decline, often usedin decliae-curveanalysis, can be readilyshown to be a long-time solution of theconstant-pressurezase10-13● It followed thenthat a log-lo~pe curve matching procedurecould be develcFed to analyze decline-zurvedata.

This pap?r demonstrates that both theanalytical 3onstant-pressureinfinite (earlytransient pericd for finite systems) and finitereservoir solutions ~an be placed on a commondimensionless log-log type curve with all thestandard “empirical“ exponential, hyperbolic,and ha~monic decline curve equations developedby Arps. Simple combinations of materialbalance equations and new forms of oil well rattequations from the recent work of Fatkovich~illustrate under what circumstances specificvalues of the hyperbolic decline expnent “b”should result in dissolved-gas drive reservoirsLog-log type curve analysis is then Ferformedusing these curves with declining rate datacompletely analogous to the log-log type curvenatcliingprocedure presently being employedwith constant-rate case pressure transient data

BASIC EQUATIONS

AIISl RATE-TIWW4UATIONS

Nearly all conventional decline-curveanalysis is based on the empirical rate-time3quations qi.venby Arps4 as

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

4JLJ_ ‘ “ “ “ “ ● ‘(2). D:t

TYPE CURVES SPE 4%—

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

y=.— (3)

i [l+;it] “ “ “ ● , ●

A unit solution (Di = 1) of Eq. 1 waseveloped for values of “b” between O and 1n 0.1 increments. The results are lotted as

7set of log-log type curves (Fig. 1 in t~~sf a decline-curve dimensionless rate

‘DdES&l ● ● ● * ● ● “(k)

i

nd a decline-curve dimensionless time

‘Od=Dit . . . . . . . . (5)

From Fig. 1 we see that when all the basiclecline-curvesand normal ranges of “b” areIisplayedon a single graph, all cur~es coincidemd become indistinguishableat t

9=0.3. Any

Iataexisting prior to a t,~dof O. till ,&P~T;Obe an exponential decline regardless of the;ruevalue of b and thus plot as a straight lin<)n semi-log paper. A statistical or least-Squaresapproach could calculate any valus ofJ between O and 1.

\NALYTICALSOLUTIONS (CONSTANT-Pi’iSSUREAT[NNERBOUNDARY)

Constant well pressure solutions to prediz-iecliningproduction rates with time were firstpklished in 1933 b l;iore,Schilthui.sand

I fiesultswere pssented{u::stlO,and Hurstl .fcr infinite and finite, slightly compressible,~in.gle-phaseplane radial flow systems. Therf2SUltS were presented in graphical fcrm inLerms of a dimensionless flow rate and aji,mensionlesstime. The dimensionless flow rat

~D ,?anbe ex~ressed as

%++&+&v””””(’)and the dimensionless time tu as

t = 0.00634 kt . . . . . (7)D

!J~~trw2

The original publication did not inuludetabular values of q and t . For use in this

??paper infinite SOIU ion va ues were obtainedfrom Ref. 15, while the finite values wereobtained from Ref. 16. The infinite solution,and finite solutions for r /r from 10 to100,000, are Tlot.tedon Fif$~.w2-a ant!~-b.

,PEf+629 M. J. FE’I

Most engineers utilize the constant-pressure solutionnot in a single constaRt-Fressure problem but as a series of constant-Iressure step functions to solve water influxprcblems using the dimensionless cumulativeproduction Q~ (13). The relationshipbetween{Q~ and qD is

d(QD) (8)—“=QD “ “ “ “ ● “ “dtfi

u

Fetkcvich’7 Fresented a simplified approaclto water influx calculations for finite systemsthat gave results whi,>hcompared favorablywith the more rigorous analytical sonstant-pressure solutions. Equation 3 of his Fa~er,~or a constant-pressurepwf, can be written as

Jo (pi - Pwf) . , . (9)q(t) =

[“[T 1t.

.e

butqi =Jo (pi - pwf) ● . . . ● (10)

Substituting I@. 11 into 10 we zan writs

Now substitutingEq. 10 and 12 into 9 we obtain

qit

Pwf

01l-— NPi

Pi● * (13)

Equation 13 can be considered as a deri-vation of the expmential decline equation interms of reservoir variables and the constant-~ressure imposed on the well. For the samewell, different values of a single constantoack-pressurep f will always result in anexponentialdecfine i.e., the level of back-pressure does not change the type of decline.For p f= O, a more realistic assumption fora wel!fon true wide-open decline, we have

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

q,. . .

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

k@w. ● ● ● ● ● (U)D: == NA pi

In terms cf a dimensionless time foriecline-curveanalysis we have from Eqs. 5 and15

‘~=[‘:-lt● ● ● ● ‘“)Defining Nvariables,pi an: ‘qi)- ‘n ‘em ‘f ‘eservoir

Tl(r- rw2) @ ct h pi . . (17)N

epi = 5.615 B

andkhpi

(A)= .* (18)‘=i’max

[()

IJJ*3AJ3 ~nbrw

1

1

--2

The decline-curve dimensionlessof reservoir variables, becomes

the, in terms

0.00634 kt‘m =

@ B ct rw2

J. . . (19)

..** (20)

To obtain a decline-curvedimensionlessrate qw in terms of q~

L DECIJNE CURVE ANALYSIS USING TYPE CURVES SPE L625

The published values of qD and tD for theinfinite and finite constant-pressureso~utionswere thus transformed into a decline-curvedimensionless rate and time, qusing Eqs. 20 and 21. Fig. 3 @ :;l:”?%fthe newly defined dimensionless rate and time,qm and tm, for various values of re/rw.

AL the onset of depletion, (a type ofpseudo-steady state) all solutions for variousvalues of re/rw develop exponential declineand converge to a single curve. Figure 4 isa combination of the constant-pressureanalytical solutions and the standard “empiri-cal” exponential, hyperbolic, and harmonicdecline-curve solutions on a single dimension-less curve. The exponential decline iscommon to both the analytical and empiricalsolutions. Note from the som~site curvethat rats data existing only in the transientperiod of the constant terminal pressuresolution, if analyzed by the em@rical ArPsapproach, would require values of “b” muchgreater than 1 to fit the data.

SOLUTIONS FROM RATE AND MATERIALBALANCEEQUAIIONS

‘he method of combining a rate equationand material balance equation for finitesystems to obtain a rate-time equation wasoutlined in Ref. 17. The rate-time equationobtained using this simple approach, whichneglects early transient effects, yieldedsurprisinglygood results when compared tothose obtained using more rigorous analyticalsolutions for finite aquifer systems. Thisrate-equationmaterial-balanceapproach wasused to derive some useful and instructivedecline-curve equations for solution-gas driveresewoirs and gas reservoirs.

RATE IJJUATIONS

Until recently, no simple form of a rateequation existed for solution-gasdrive reser-voir shut–in pressure. Fetkovich~ hasproposed a simple empirical rate equation forsolution-gasdrive reservoirs that yieldsresults which compare favorablywith computerresults obtained using two-phase flow theory.The proposed rate equation was given as

ijq. = J? () (~R2 -pwf2)n . . . (23)

‘i Ki

where n will be assumed to lie between 0.5 and1.0.

Although the above equation has not beenverified by field results, it offers theopportunity t.odeflna t,hedeeljw expnqn{.

(l/b) in terms of the back pressure curveslope (n) and to study its range of expectedvalues. Also, the initial decline rate Dican be expressed in terms of reservoirvariables. One further simplification‘lsedin the derivations is that pwf = O. For awell on decline, pwf will USUallY be ~intainedat or near zero to maintain maximum flow Pates.Equation 23 then becomes

The form of Eqs. 23 and 23A could alsobe used to represent gas well behavior witha pressure dependent interwell permeabilityeffect defined by the~R/FW). lhestandard form of the gas well rate equationis usually given as

Cg (j5R2-pwf2)n . . . . (24)‘i% =

I MATERIAL EAL!NCE OQUATIO’.;

Two basic forms of a material balan$eequation are investigated in this study~p~ is linear with N or G , and ~..2is linearw~th N or Gp (See ‘Figs.p5a and ~). Thelinearp~R relationship for oi2 is

I and for gas

Equation 25 is a good apFroXhnation fortotally undersaturated oil reservoirs,oris simply assuming that durinR the decline

12!@@-%

vs. N can be approximatedbya straigh line.p For gas reservoirs,Eq. 26is oorrect for the assumption of gas com-pressibility (Z) =1.

In terms of j5R2being linear with cumu-lative production, we would have

kLi2

()~R2=- N NF+~Ri2 . ● “ (27)

Pi

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

3PE 4629 M. J. FE!

drive reservoir as depicted in Fig. 5-b.Applications would be more appropriate in non-pmrated fields, i.e., wells are produced wide-open and go on decline from initial production.This would more likely be the case for much 18of the decline-curve data analyzed by Cutlerobtained in the early years of the oil induetrybefore proration.

RATE-TIME EQUATIONS, OIL WELLS

Rate-time equations using various combi-nations of material balance and rate equationswere derived as outlined in Appendix B of Ref.17. Using Eq. 23A and @. 25 the res~tingrate-time equation is

Clo(t)—.

‘= [+J+,1% ● ● (y

A unit solution, (qoi/N .) = 1, ofEq. 28is plotted as a log-log ty~ curve forvarious values of n, Fig. 6, in terms of thedeel.ine-curvedimensionless lttie t (For

=0, q .%”(q .) .these de~ivations with pFor the limiting range o~fback-pr%ure %%?sloFes (n) of 0.5 and 1.0, the Arps empiricaldecline-curve exponent (l/b) is 2.0 and 1.5respectively or “b” =0.500 and 0.667respectively a surprisingly narl;owrange.To achieve an~pnential decline, n mustbe equal to zero, and a harmonic decltnerequires n+-. In Fractical applications,if we assume an n of 1.0 dominates in solution-gas (dissolved-gas)drive reservoirs and ~vs. N is l%near for non-uniquely definedrate-?ime data, we would simply fit the rate-time data to the n = 1.0 curve. On the Arpsjsolution type curves, Fig. 1, we would use(l/b) =20rb =0.667.

The rate-time equatior obtained usingAq. 23A and Eq. 27 is

qo(t) 1—= - ● ● (29)

‘oip(~)t+rThe unit solution of Eq. 29 is plotted

as a log-log type curve for various valuesof n, Fig. 7. This solution results in acomplete reversal from that of the previousone, n =0 yields the harmonic decline andn + .W gives the exponential decline. Forthe limiting range of back-Fressure curveslopes (n) of 0.$ and 1.0, the decline-surve

)VICH 5

exponent (l/b) is 2.0 and 3.0 m b = o.5~ ad0.333 respectively. This range of “bE valuesfits Arpsl ftidings using Cutlorls decline-curve data. He found that over 70 percentof the values of “b~ lie in the rangeO~b ~0.5. Ramsay found a differentdistribution of the value of “b” analyzingmodern rate decline data from some 202 leases.His distribution may be more a function ofanalyzing wells that have been subject toproration and are better representedby theassumptions underlying the rate-time solution

‘ver the decline period% ‘s” ‘pWas linear

given by Eq. 28, i.e.,

DECLINE-CURVE ANALYSIS OF GAS WELLS

Decline-curveanalysis of rate-time dataobtained from gas walls has been reported inonly afewinstances19S 20. Using Eq. 24with ~wf =0, and Eq. 26, the rate-timeequation for a gas well is

C@) 1 (3C)—— ● “’

for all back-pressure zurve slcFes wh~ren > 0.5.

For n = 0.5, the exponen~ial decline isobtained

q~

()*tC&) -.e ..0. ●

(31)

‘gi

The unit solutions of Eqs. 30 and 31 areplctted as a log-log type curve cn Fig. 8. Forthe limiting range of back-Fressure curveslopes (n) of 0.5 and l.oj the AW decline-curve expment (l/b) is 00 and 2, or b = O(exponential)and 0.500 respectively.

The effect of back-pressure on a gas wellis demonstrated for a back-prassure curveslope n =1.0 on Fig. 9. The back-pressureis expressed as a ratio of Pwf/Fi* Note that

-+ p. (Ap+O) the type curva approachesas pwf ,ex’ymentlalldecline,the liquid case solution.Whereas back-prsssure does not change the typeof decline for tha liquid-case solution itdoes change tha type of decline in this case.

Using the more familiar rate and materialbalance equations for gas wells, we can obtainthe cumulative-timerelationship5;’integratingthe rate-ttie equations 30 and 31 with

Gp= /t qg(t)dt . . . . . (32)o

r DECLINE CURVE ANALYSI

For n > 0.5 we oktain

&,..” :-’& 1 - [1 + (2n-1)()+t]e

● (33;

and n = 0.5

Lag-1og type curves of Eqs. 33 and 34 coukbe prepared for convenience in obtainingcumulative production.

Recent papers by Agarwal, et.al.5, E#mey6,Raghavan, et.al.7 and GringartenS et.al. ~have demonstrated or discussed the applicationand usefulness of a type-curve matchingpror.edureto interpret constant-rate pressurebu~ld-up and drawdown data. van Poolen21demonstrated the application of the type-c.urveprocedure in analyzing flow-ratedatzobtained from an oil well producingwith aconstant pressure at the well bore. All of hisdata, hcwever, were in the early transientperiod. No depletion was evident in hisexamples. This same type-curve matchingprocedure can be used for decline-curve-malysis.

The basic steps used in type-curvematchin~declining rate-time data is as follows:

1. Plot the actual rate versus time datain any convenient units on log-log tracingpaper of the same size cycle as the type curveto bs used. (For convenience all type curvesshould be plotted on the same log-log scale sothat various solutions can be tried.)

2. The tracing paper data curve isplaced over a tyFe curve, the coordinate axesof the two curves being kept parallel andshifted to a position which represents thebest fit of the datathan one of the typepaper may have to befit of all the data.

3. Draw a linebeyond the rate-time

to a type curve. Morecurves presented in thistried to obtain a best

through and extendingdata overlain along the

uniquely matched type curve. Future rates arethen simply read from the real-time scale onwhich the rate data is plotted.

4. To evaluate decline-curve constantsor reservoir variables, a matoh point isse~ezted anywhere on the over?.app?ng~rt$onof the curves and the coordir,atesof this

USING TYPE CURVES SPE 4629

common point on both sheets are recorded.

5* lf none of the type curves will.reasonably fit all the data, the departurecurve method 15$ 22 should be attempted.This method assumes that the data is a com-posite of two or more different decline-curves. After a match of the late time datahas been made, the matched curve is extra-@lated backwards in time and the departure,or difference, between the actual rates andrates detemi.ned from the extrapolated curveat coi-respondingtimes is replotted on thesame log-log scale. An attempt is then madeto match the departure curve with one of thetype curves. (At all times some considerationof the type of reservoir producing meck~tismshould be considered.) Future predictionsshould then be made as the sum of the ratesdetermined from the two (or more if needed)e:krapolated curves.

TYPE CURVE MATCHING EXAMPLES

Several,examples will be presented toillustrate the method of using type curvematching to analyze typical declining rate-time data. The type curve approach jmwidesunique solutions upon which engineers xmagree, or shows when a unique solution is notpossible with a type curve only. In the eventof a non-unique solution, a most probablesolution can be obtained if the pI’OdUCi.ngmechanism is known or indica~ed.

,ARPS’HYPKRBOLJC DECLINi EXAMPLE

Fig. 10 illustrates a type curve matchof Arpsl example of hyperbolic decline4.Every single data point falls on the b =0.5type curve. This match was found to be uniquein that the data would not fit any other valueof “b”. Future producing rates can be readdirectly from the real-time scale on whichthe data is plotted. Ifwe wish to determineq. and D.. use the match mints indicated onFtg. 10 *: follows “

@= 1000BO~qm =0.33 = ~j %

qi = 1000 BOPIVI0.G33

= 30,303 BOPM

‘Dd=12.0 = Dit = Di 100MO.

.Di=~=100 Mo

0.12 MO.-l

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

5PE 4629 M. J. FETKOVICH 7

5.sequivalent to b = 0.5. Match pintsdetermined from these curves could have been;sed to zalculate~ and ~/Npi and finally

pi“

The fact that this example was for a lease,a group of wells, and not an individualwellraises an imprttint question. Should there bea difference in results between analyzing eazhwell individually and summing the results,or simply adding all wells production andanalyzing the total lease production rate?Consider a lease or field with fairly unifomnreservoir properties, lib!:or n is similar for

eazh well, and all wells have been cn deckineat a similar terminal wellbore pressurey Fwf~for a sufficient period of time to reach FSeUdC-steady state. According td Matthews et.al.23Iiat(Fseudo) steady state the drainage volumes

in a bounded reservoir are proportional to therates of withdrawal from each drainage volume.”It follows then that the ratio q./N . willbe identical for each well and th$isthe sum ofthe results from each well will give the sameresults as analyzing the total lease or fieldproduction rate. Some rather dramatic illus-trations of how rapidly a readjustment indrainage volumes can take place by changingthe production rate of an offset well ordrilling an offset well is illustrated in apaper by Marsha. Similar drainage volumereadjustments in gas reservoirs have also beendemonstrated by Stewart25.

For the case where some wells are indifferent portions of a field separated by afault or a drastic permeability change,readjustment of drainage volumes proportionalto rate cannot take place among all wells. Theratio ./N . may thenbe different for differentgroup %#f hlls. A total lease or fieldproduction analysis would then give differentresults than summing the results from individualwell analysis. A stilar situation can alsoexist for production from stratified reservoirs26, 27, (no-cw~~flow)o

ARPS’ EXK)NENTIAL DECLINE EXAMPLE

Fig. 11 shows the results of a type curveanalysis of Arpsl example of a well with anapparent exponential decline. In this case,there is not sufficient data to uniquelyestablish a value of “b”. The data essentiallyfall in the region cf the type curves where allcurves coincide with the expnmtial solution.As shown on Fig. 11 a value ofb =0,

(~pnential) orb= 1.0 (harmonic)appear tofit the data equally well. (Of course all

values in between would also fit the data.)The difference in forecasted results from thetwo ewtreme interpretationswould be great inlater years. For an economic limit of 20 BOP?4,

the ex~nential interpret.ationgives a totallife of 285 MONTHS, the harmonic ll@ MONTHS.‘his points out yet a further advantage ofthe type curve approach, all pssible alternateinterpretationszan be conveniently placed onone curve and forecasts made from them. Astatistical analysis would of course yiald asingle answer, but it would not necessarily bethe correct or most probable solution. Con-sidering the various producing mechanisms we~ou>d select.

a) b = O, (exponential),if the reservoir ishighly undersaturated.

b) b = O, (exponential),gravity drainage withno free surface28.

c) b = 0.5 gravity drainage with a free5surface 8.

d) b = 0.667, solution-gas drive reservoir,(n=l.O) if~Rvs. Npis linear.

e) b = 0.333, solution-gas drive reservoir,(n=l.0) if~k2 vs. Npis approximatelylinear.

FRACTURED WiLL EXAMPLE

Fig. 12 is an example of type survematching for a well with declining rate dataavailable both before and after stimulation.(The data was obtained from Ref. 1.) Thistype problem usually presents some difficultiesin analysis. Both before and after frac. log-log plots are shown on Fig. 12 with the afterfrac. data reinitializedin time. These beforeand after log-log plots will exactly overlayeach other indicating that the value of “b”——did not than e for the well after the fracturetreatment. tThe before frac. plot can beconsidered as a type curve itself and theafter frac. data overlayed and matched on it.)Thus all the data were used in an attemptto de~e “b”. When a match is attempted on theArps unit solution type curves, it was foundthat a “b” of between ~.6 and 1.0 could fitthe data. Assuming a solution-gas drive, amatch of the data was made on the Fig. 6type curve with n = 1.0, b = 0.667.

Using the match Feints for the beforefrac. data we have from the rate match mint,

qw “0.2.43 = w= 1000 BOPMqoi q’oi

qo~ =1000 BOPM

●243= 4115 BOPM

From the time match point,

= 0.60 = qoi t =% () 0+115 BOPM)(1OO MO

rpi N

pi

N f4115 BOPM) (1OC MO.)= 685,833 BBL

pi = 0.60

then

q~= 4115 BOPMN 685,833

= .006000M0.-1...4

Now usin~ the match points for the afterfrac. data we have from the rate match Pint,

q~ =0.134 = ~ . 1000 BOPM“Oi qoi

. 100C BOPMqoi 0.134

= 7463 BOPM

From the time matsh pint

q.

‘Dd ().1*13 . + ~ .m6W(loo MG.~

pi pi

N (7463 BOPM) (loo MOJ= = 660,4.42BBLpi 1.13

then

qoi 7463 BOPM‘= 660,w2 BBLN

= .011300MC.-lpi

We can now zheck the two limiting condi-tions to be Considered following an inzreasein rate after a well stimulation. They are:

1. Did we simply obtain an accelerationof production, the well~ reserves remainingthe same?

2. Did the reserves hcrease h direztproportion to the increase in producing rate

3as a resu t of a radius of drainage read-justment ? BefGre treatment, N . was fOUndto be 685,833 BBL. Cumulative F~%ductiondetermined from the i-atedata prior to stim-ulation was 223,500 BBL. N then at thetime of the fracture treatm~i%tis

N =685,833 EBL- 223,500 BBL =462,333 i3BIpi

If only accelerated pradutionwas obtained andthe reserve” remained the same, q~ after

Fpi

the fracture treatment should have been

7&63 BOFMf+62,333BBL

= o.016J_42MO.-l

Actual (qoi/Npi)after treatment was 0.011300

MO.-l. If the resarves increased in direct

- “------ . ..-. ““.”,--, --- +Ws/

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

pi

N‘J&J 130PM—-1 = 1,243,833 BBL

pi = .006000MO.

Actual in~rease in reserves as a.rssultof’the fracture treatment a~pea,rsto liebetween ths two extremes. Based on themsthod of al,alysisused, the astual increasein reserves attributable to the fracture treat-ment is 198,109 BBL, (660,442 BBL - 462,333 BBL;

STRATIFIED r’SiRVOIR EXAMll&

This example illustrates a method ofanalyz5ng decline-curvedata foi-a layered(no-crossflow)or stratified reservoir usingtype ~urvese The Qata is taken from i@f. 18and is for the East Side Colinga Field.Ambrose2? presented a cross seztion of the fiel~showing an upper and lower oil sand se~aratedby a continuous blazk shale. This layereddescription for the field along with thepredictive equation for stratified reservoirpresented in Zef. 29 led to the idea of usingthe departure curve method (differencing)toanalyzed dezlint?-curvedata.

After Russell and Prats27

the productionrate of a well (or field) at p;eudo-steadystate producing a single phase liquid atthe same constantwellbore pressure, (Fwf = Ofor simplicity), from two stratified layers is

qiJ% t -—) () t~Npi 1 +-qi2e

NqT(t) = qi~ e pi 2

● ✎ ✎ .(35)

‘r qT(t) =ql (t) +q2 (t) . . ● . (36)

The total production from both layers thenis simply the sum of two separate forecasts.Except for the special case of the ratioq /N ~ being equal for Loth layers, the sumo}tio expnentials will not in general resultin another exponential.

In attempting to match the rate-ttie datato.a type curve, it was found that the latetime data can be matched to the exponential(b =0) type ourve. Fig. 13 shows this matckiof the late the data designated as layer 1.With this match, the curve was extrapolatedbackwards in time and the departure, ordifference, between the actual rates determinedfrom the ext~apolated curve was replotted onthe same log-log scale. See TAB~ 1 for a

SPE L629 M. J. FETKOVICH 9

summary of the departure curve results. Thedifference or first departure curve, layer2, itself resulted in a unique fit of theexponential type curve, thus satisfying Eq. 35which can now be used to forecast the futureproduction. Using the match points indioatedon Fig. 13 to evaluate qi and Di for eachlayer the predictive equation becomes

qT(t) = 58,824 BOPY e- (o.2oo)t +

50sGOOBO~ e - (O*535)t

where t is in years.

30Higgins and Le~htenberg named the sumof two expnentials the double semilog. Theyreasoned that the degree of fit of empiriealdata to an equation increases with the numberof constants.

This interpretation is not claimed to bethe only interpretation pX3sib16 for this S3tof data. A match with b =0.2 can be obtainedfitting nearly all of the data points but cannot be explained by any of the drive mechanismsso far discussed. The layered concept fitsthe geologis description and also offered theopportunity to”demonstrate the departure curvemethod. The deFarture curve method essentiallyplaces an infinite amount of combinations oftype curves at the disposal of the engineerwith which to evaluate rate-time data.

EFFECT OF A CHANGE IN BACK-PRESSURE

The effe~t of a change in back-pressureis best illustratedby a hypothetical singlewell problem. The reservoir variables andconditions used for this example are givenillTable 2. The analytical single-phaseliquid solution of Fig. 3 is used to illu-strate a simple graphical forecasting super-position procedure. The inverse procidure,the departure or differencing method can beused to analyze decline-curve data affectedby bask-pressura changes.

After Hurst12, suparpsition for theconstant-pressurecase for a simple singlepressure change can be expressed by

kh (pi - Pwfl)q (t) =

[ ($) -$] ‘M(t~l

141.3(@) In

kh (Pwfl - Pwf2)+

‘103(~)[’n(~)-~] ‘~(tod-’~~)

or

.*.*. ..* (37)

Up tc the time of the pressure zhange

pwf2 at t~l the well production is S~PIY

ql - depicted on Fig. U. he q fore-tas a function of time is simply kde byevaluating a single set of match points usingthe reservoir variables given in Table 2.

/r’ =100At Pwfi and re w

t=lDAY; tDd

=0.006967

ql(t) = 697 BOpD; q~ = 2.02

Plot the rate 697 BOPD and time of 1 dayon log-log tracing paper on the same sizecycle as Fig. 3. Locate the real-timepoints over the dimensionless time pints onFig. 3 and draw in the re/r~ curve of 100 onthe tracing paper. Read flow rates as afunction of time directly from the real timescale.

When a change in pressure is l~de to pwf2at tl, t equal zero for the accompanyingzhange in rate q29 (really a Aq for super-postion)j this rate change retraces the qmvs. t~ curve and is simply a constant fractionof ql

[

Pwfl - pwf2q2

=qlPi - ~wfl 1

::U:? :; t- 1 day after the rate change is

q2 = 697 BOPD 1000 Psi - 50 Psi4000 psi - 1000 psi= 221 BOPD

The total rate qT after the pressureshange is q

T= ql + q2 as depicted in Fig* M*

All rates o flow for this example were readdirectly from the curves on Fig. 11+and summedat times past the pressure change pwf2.

The practical application of this exam~lein decline-curve analysis is that thedeparture or difference method can be usedon rate-time data affected by a change in

) lWf?T. TN17 flIIRVR AN?ITY.STS IISTNC TVPT7 PIIT+VF< .sPr? J. L’)G, - “...4.-1.- “...v- “.,., ----- ““J. ..” . ..- ““. .”-- “.- 6+”t.,~

back-pressure. The departure curve represented wallbore radius r ‘ (obtained from the build-by q2 on Fig. 14 should exactly overlay the up analysis) is u%d to obtain a good matchcurve represented by q . If it does in an between build-up ant decline-zurve ~alculated.actual field example, ~he future forecast is kh. ‘.correctly made by extending both curves andsumming them at times beyond the Fressure TYPE CUFWdS l~iiKNOkJNRESERVOIR AND FIJJ~change. PROPER71EL—

CALCULATION OF kh FROM DLCLINE-CURVE DATA All the type surves so far discussed weradeveloped for de~line-curve analysis using

Pressure build-up and decliae-curve data some necessary simplifying assumptions. Forwere available from a high-pressure, highly spe~ifi,creservoirs,when PVT data, reservoirundersaturated,low-permeabilitysandstone variables, and back-pressure tests arereservoir. Initial reservoir pressure was available, typs surves could be Eenerated fcrestimated to be 5750 psia at -9300 ft. with various relative permeability wrves anda bubble-pointpressure of 2841 ~sia. Two ba>k-pressures. These curves developt~ forfield-wide pressure sarveys were condu?tsd while a given field ~iouldbe more accurate forthe reservoir was still undersaturated. lable analyzing fleclinedata in thak field. Con-3 summarizes vl,ereservoir properties and ventional material balance programs or morebasic results obtained from the rressure kuild- sophisticated simulation models could be usedUF analySiS on %azh Well. Note that nearly to develop dimensionless sonstant-pressure 31all wells had n8ga.tiveshins as a result of type zurves as was done bv Levine and Prattshydraulis fracture treatments. Also, (See their FiF. 11).a~~earing on this table are results obtainedfrom an attenpt to calculate kh using decline- CONCLUSIGNScurve data available for eazh of tk,ewells.

Decline-zurve analysis not only has aTen of the twefity-twowells started on solid fundamentalbase, but providss a tool

desline when they were first placed cn with more diagnostic power than has previouslyproduction. As a result, the early production been suspe~ted. The ty~!e:urve ap~roach ~:ro-desl:ne data existed in the transient pmiod vides unique solutions u~on whi~h engineersand a ty~e curve analysis IlsingYip. 3 was can agree, or shows when a unique solution ismatched to one of the r@/’rwst:ms. Gther wells not possible with a tyre zurve ocly. in thslisted on the table, kh5ra an r ~rw mat:h is event of a non-unique sclution, a most ~robablenot indiczted, were Frorated wefls and began solution zan be obtained if the produ~ingtheir de:line several montk,safter they were meshanism is known or indicated.first put on production. For the de2line-zurve determination cf kh, the reservoir pres- NO~NCLATU~,

sure existing at the beginning of decline foreach well was taken from the pressure history lb =-l{~giro~al of detline >urve exponentmatch of the two field-wide pressure surveys, (1/b~The zonstant bottom hole flowing ~ressure for ~the wells ranged between 80G and 900 psia.

= Formation volume factor, res. vol.,’surface vol.

A type curve mat~h using deeline-~urveCt

= Total compressibility,psi-ldata to talculate kh for well No. 13 is illu- (-J=strated on Fi~. 15. A type ?urve matsh using

Gas well ba~k-pressure :urve coefficient

D!-1

pressure build-up data obtained on this same = Initial Decline rate, twell is illustrated on Fip. 16. The constant- ~=rate type curve of’Gringarten et.al.8 for

=-Natural logarithm base 2.71828

fracturedwells was used for mat~hing the pres- G = Initial pas-in-~dase, surface measuresure build-up data. The build-up kh of47.5

Grnd.-ft.cmmpares very well with the kh ofl+O.5 p

= CUmu].ativegas production, SUrfaCe

MD-FT determined by using the rate-time dezline-measure

zurve data. h = l%i~knsss, ft.

In general, the comparison of kh detarnrinedJo = Productivity index, STK BHL/DAy/pS~

from deuline-turve data and prsssure build-ur, J; = Productivity index’(back-pressuresurvedata tabulated on Table 3 is sur~risinglygood. coefficient)STK BBL/DAY/(psi)2n(The pressure build-up analysis was performed k,independentlyby another engineer.) One = Effective permeability,md.

fundamentalobservation to be made from the n = Expnent of bask-pressure curveresults obtained on wells where a match of

N = Cumulative oil ~roduction,S’fKBBLre/rw was not ~ssible is that the effective F

SPE L629 M. J. FE!

NFi

Fi

h

Fwf

%

(qi

q(t,

q~

‘Dd

QDrerw~f

tw

‘D

%

~

P

Cumulative oil production to areservoir shut-in pressure of O, STKBBL

Initial pressure, psia

Reservoir average pressure (shut-inpressure), psia

Bottom-hole f’lowin~pressure, psia

Initial surfase rate of flGw at t = O

Initial wide-oFen surface flow rateat pwf = O

Surface rate of flow at time t

Dimensionlessrate, (Eq. 6)

Decline wrve dimensionless rate,(Eq. &)

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

Vis20sity, cp.

REFERENCES

1. Ramsay, H. J., ~Jr.:“The Ability of Rat@-Time )ecl.ineCurves to Predizt FuturePrcxiuctionRates”, M.S. T?_lesis3U. ofTulsa, Tulsa, Okla. (1968).

2. Slider, H. C.: ‘IASimplified Method ofHyperboli~ Decline Curve Analysis”,J. Pet. Tech. (March, 1968), 235.

3. Gentry, R. W.: llDe~line-curVe Analysis”>

J. Pet. Tech. (Jan., 1972) 38.

4. Arps, J. J.: IiAnalysisof Decline Curves”$

Trans. AIME (1945) &@, 228.

5* Ramey, H. J., Jr.: “Short-TimeWell TestData Interpretationin the Precence ofSkin EffeZt and Wellbore Storage”, J. Pet.Tech. (Jan., 1970) 97.

6. Agavwal, R., A1-Hussainy,Il.and Ramey,H. J., Jr.: llAnInvestigationof ~~elibore

Storage and Skin Effect in Unsteady LiquidFlow: I. Analytical Treatment:j Sot. Pet.Eng. J. (Sept., 197G) 279.

)VICH 11

7.

8.

9.

10.

11.

12*

13.

:4.

15.

16.

17.

18.

19.

Raghavan, R., Cady, G. V. and Ramey,H. J., Jr.: i,~;ell-TestAnalysis for

Vertically Fractured Walls””,J. Pet. Ta?&~(hlgo 1972) 10L4.

Gringarten, A. C., Ra!!ey,H. tJ.~Jr. and.Raghavan, R.: tgpres~ureAnalysis for

Fractured ~lells”,paper SPE 4051 presentadat the 47th b.nnualFall lfeethin, San

YAntonio, Texas, (Gzt 8-11, 1972 .

MeKiriley,R. M.: “Wellbore Transmissibilityfrom Afterflcw-DominatedPressure BuildupData”, J. Pet. Tesh. (July, 1971) 863.

Moore, T. V., Schilthuis, R, J. and Hurtitjw. : tl~e Deter~~natiOn of Permeability

from Field Data”, Bull..ApI (~ky, 1933)~ 4.

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

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

van Everdingen, A. F. and Hurst, ~~.:“TheAFpli2ationof’the Laplaee Transformationto Flow Proble.nsin :eszrvcirs”, Trans.AIME (1949) 186, 3050

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

Ferrt~,,J.$,Knowles, D. B., i3rown,~:.J~.~and Stallman$ R. W.: “’Ihecryof AquiferTests”, g. S. Gecl, Surv., ‘:later29FF1%Pap3r 1.53~, 109.

Tsarevi~h, K, A. and Kuranov, I. J’.:l’f~a~culationof the I’1ow l:atesfor the

Ce:lt3r‘Jellin a Circular Seser’mirUnder Jlastie Conditlcns”, Problems @fRgservoiy Hvdrods’nami~sPart.I, Leningrad,m 9-34●

~etkovi~h, M. Jo: “A Simplified Approachto ~!at:?rInflux Calculations-FiniteAquifer Systems”, J. Pet. Tech. (tJuIY~1971) Xl-4.

flr.lttlg]~,W. W., Jr.: llEstimationof Lhder-

ground Oil Reserves bv Oil-well P~ductionCurves”, Bull., USBM”(1924)~

Stewart, P. R.: “Low-PermeabilityGasWell Porfomnanee at Constant.Pre9sure”,J. Pet. Tech. (Gept. 1970) 1149.

2 DECLINE CURVE ANALYSIS USING TYPE CURVES SPE 4629

20.

21.

22.

23.

24.

25.

26.

——

Gurley, J.: 11Aproductivityand EsOnOnd.CProjectionMethod-Ohio Clinton Sand GasWalls”, Paper SPE 686 presented at the38th Annual Fall Meeting, New Orleans,La., (Oct. 6-9, 1963).

van Poollen, H. K.: “HOW to AnalyzeFlowing Well-Test Data... with ConstantPressure at the Well Bore)’,Oil and GasJournal (Jan. 16, 1967).

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

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

hlarshjH. N.: llMethodof Appraising Result!

of Production Control of Oil Wells”, ~.API (Sept., 1928) 2@, 86.

tewart, P. R.: “Evaluationof IndividualGas Well Reserves”, Pet. Eng. (Mayj 1.566)85.

Lefkovi%, H. C. and Matthews, C. S.:“Applisaticmof Desline Curves to Gratity-

Drainage Reservoirs in the Strippr Stagel’Trans., AIME (1958) ~, 275.

I

27.

28.

29.

30.

310

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

Matthews, C. S. and Lefkovits, H. C.:llGra~~,y~rainage Performance of Depletion-

‘l&peReservoirs in the Stripper Stage”,Trans., AIME (1956) ~, 265.

Ambrose, A. W.: !!underg~und~enditions

in Oil Fields”, Bull., USBM (1921) 195,151.

Higgins, R, V. and Lechtenberg, H. J.:!!Meritsof ~e~line Equations Based onProduction History of 90 Reservoirs”,Paper SPE 2450 presented at the Rocky Mt.I@ional Meeting, Denver, Colo., (My 25-27, 1969).

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

ACKNOtilEIXXMENT

I wish to thank phillipS Petroleum Co. forpermission to publish this FaFer.

NOTE: Tle author has a limited quantity offull size type curvas with grid suitablefor actual use which are avzilable onwritten request.

_——. .. ..

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ERATE - TIME EQUATIONS

1Qlt) = ;FORb>o

Q,l+ bD,,]~

[Qlt) 1‘= —; FORb=OQ1 ~ D,t

lDd. D1l’

Fig. 1 - Type curves for Arps empi.r cal rate-time decline equations, unit solution (Di .’1).

‘D

9 - - 8

8 - - 7

7 - – 6

6 – – 5

5 - – 4

4 -– 3

3 -

2 -

0.1967

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I I I I I I I0.001 I I I I [ [ [ I I r

: j:;~[, 011

1 2 3 4587810 2 34 567810] 2 34 5678103 234 5678104 2 3

Fig. 2A -

‘D

Dimensionless flow rate functionsolane radial system infinite and finite

~uter boundary, constant pres-ure at inner bmndary.10,11,15,16

1o“ 23 456781O* 2 3 455781O’ 2 34 56781O’ 2 34 567810’ 2 3

1.0 I I I I I I I I I I 1 I I I 1 I I I I i I I I I I I I I I I I I

9I [ I I I I [ I

I87

6

E 1

6

55

44

3

t

2

0.1 —9 -8 -7 -

6 -

5 –

4 -qD

3 :IzIzzl2

0.01 —9 -8 -7 –

6 -

5 -

4

3

2

1

13

2

fj- 6

‘%– 5

4

– 3

\ – 2

~ ~

0.01987

6

5

4

3

2

LI I I I I I I I I I I \llllll I ,\ll, llll \ I I i I I r i I I I I 1 1 I I

0.0010.001

I@ 2 3 4567810S 2 3 45678106 23 4567810’ 23 45678106 23 4567810’

‘D

Fig. 2B - Dimensionless flow rate functions plane radial system infinite and finiteouter boundary, constant pressure at inner boundary.

10,11,15,16

lo.t& I I ! [ I I I 1 I I I 1 I I I I I I I I I [ I I [ I I I I I [ I

2 -

1O,ooi)

41.0 i----

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98

E

L7

100,OQO

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i i I I I I I 1

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2 -

0.1 —9 -8 -7 -

6 -

5 -

4 -

3 -

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I 0.00634kt -1)

~;

1qDd . ~-’Dd

L~~:’lllll I I I I I I I I

2 34 567810”’ 567891 2 3 4567810

‘Dd

Fig. 3 - Dimensionlessflovjrate functions florplane radial system, infinite andfinite soter boundary> constant Pressure at inner bo~dary”

,Oln. 23 4567s10~ 23 4567810’3 23 45e78~v~ 23 ~56781 23 4567810

1 ! I 1 1 I I I \ I \ I I I \ 1 I 1 ! I I 1 I 1 1 [ 1 I 1 1 I

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1

ANALYTICAL TY~E_C#RVE SOI.UTION

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0001 I I I I 1 I I I I 1 ( I 1,8I I ! ! I [I11 1 ! 1 1 k

,“. 2 3 4567810) 234 5678 10: 23 4 5.? 870 .?3 .356781 23 4S 67810 23 4s6.

‘Od

Fig. 4 - Composite of analytical and e~irical type curves of Fi&s. 1 and 3.

oNp OR Gp NPi OR G

~i~. 5A - Graplical representation of material balance equation,

I

TR VS Np APPROXIMATELY LINEARBEGINNING OF DECLINE

PRORATED WELLWHEN DECLINE BEGINS

oN~Gp Npi OR G

Fig. 5B - Graphical representation of mteri al balance equation.

flATERIAL BALANCE EQUATION:

()

PifiR=. — NP+P,

Npi

IATE EQUATIONS:

~R\ - ,

([

11$

TwO.PHASE IOIL & GAs): Qo = ‘oi T) ‘R - ‘@j

Cl. it)RATE.TIME EQUATION [PM = 01: ~ = 2n+l

P“(i) l–~+1 zn

23 4567810z1.0987

7

6 -

5 –

4 - EXPONENTIAL

3 -

2 -

0.1 —9 -8 –7 -

6 -

“

g3u

2 –

0.019-8-

7-

6-

~: ~

SINGLE.PHASE LKIUIO: q. = JO6R - Pwf)

2

I I I I I [ [ [ I I I I I I I I 0.001I

0.00 1I I I I I 1 I I I I

4567810’45678102

,0.3 2 34 567810”’ 2 3 2 34 56781 23 4567810 23

co o2.667 0.375

2.000 0.5001.714 0.583

1.500 0.6671,333 0.7501.250 0.8001.125 0.6891.050 0.9521,0Q5 i 0.995

‘od ‘(~)’

Fig. 6 - Dissolved gas drive reservoir rate - decline type curvespressure at inner boundary (pI+T= O @ rlJ). Early transient

finite system with constanteffects not included.

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T,!/

MATERIAL BALANCE EQUATION:

‘R’= -(*) ‘“+””\

IATE EQUATIONS: %4

TWO-PHASE 6Rn

( )[ 1

5[OIL&GAS): GO = Jo, ~ ~R’ ‘W* !-

– 3

– 2

— 0.1

– 7

– 6

– 8

– 7

– 6

– 5

– 4

1 I 1 I I 1 I I i I I I I \l,l I I I I I I23 45678103 4~67612 3 4 5678 10”’ 23 2 3 4567810 23 45676102

RATE-TIME q. (t)EQUATION (“~ = O): ~ -

2n+l

w)’ 1t+l

SINGLE-PHASE LIQUID: GO = JO (6R - Pw$

RATE.TIME q. (t)EQUATION l~f = 01: ~ “ —

fexwnentiall.(i)’

\

‘~~-(%3’

Fig. 7 - Dissolved gas drive reservoir rate - decline type curvespressure at inner bwndary (ph~ = O @ rl,). E:rN transient

finite s~stem with constanteffects not included.

6114,1, .9 I. . .

u6

I

-—

U-I.. 0000Ooo-1-+v~g

I

‘Ptd

r+CJo

g II

1

co

G.,-1L4

1U3 2 3 4567810+ 2 3 4 567810”’ 2 3 456781 2 3 4567810 23 4 567810~,o

1.0 I I I I I I 1 I [ I [ I I I I I- 9

8 -

– 8

7 -

- 7

6 -- 6

5 -– 5

4 -– 4

3 -– 3

– 2

‘t

0.1 L

\\ \ \2

\ \ \.F

\\\\

-%

o-@.

0,% i

0.1987

6

5

4

7

~

“

2

:01

; -

I ~ \\,\,,l,,Ji

7 -7

6 -6

5 -5

4 -4

3 -3

2 -2

0.001

-,0-3 23 45678101 2 3 4567810’ 2 34 56/81 234 567810 23 46678102illl \ IO.CO1 I I I I I 1 I II I I I I I 1 I I I I I I I.-.

Fig. 9 - Gaspressure at

reservoir rate decline type cur-~eswith back pressure finite system with constant

inner boundary (pVfl= constant Z r,,). Early transie,;teffects not included andz = 1 (based ~n gas uell back pressure curve slope, n = 1.

‘Dd = 0,1

1000

EXPONENTIAL

— I

MATCH POINTS

EXPONENTIAL, b = O EXPONENTIAL, b = 1

t=loo MO, ;tDd=l.lo t= looh.hO. ;tDd=l.60

q(t) = 100 BOPM ; qDd = o.212 J(t) = 100 BOPM ; qDd ‘ o.2~2

ECONOMIC LIMIT 20 BOPM @285 L1O. ECONOL~lC LIMIT 20 BOWVIm 1480 MO.

EXPONENTIAL

1

\

-–L10 100 285 fI~O. 1000 1480klo,

\lFFIACTURE

AFTER FRAC

BEFORE FRAC()

J

TREATED

—\, “

I ‘“h

MATCH POINTS

\

\

\

BEFORE FRAC AFTER FRAC \

n = 1.0, (b = 0.667) n = 1.0, (b = 0.667) \

t = 100 MO. ;tD~ = 0.60 t= loo MO. ;tD,j= 1.13 \

q(t) = 1000 BOPM ; qDd = 0.243 q(t)‘1000BOPM ; qDd = o.134 \

\

\

1 ..-.1 10 100 1000

t- MONTHS

Fig. 12 - Type-curve analysis of a st imulatw! well before and after fracture treatment.

Ioo,ooo

>$, 10,03(

$’

1,00

/

\ /EXPONENTIAL

_ MATCH POINTS

b=O, EXPONENTIAL

LAYER 1

t=loy Rs. ;tDd =2.0

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

LAYER 2

t= loy Rs. ;tDd =5.35

q(t)=1000BOPY ; qDd = 0.020

●

O-q’=q’+q’

LAYER 2qz=’+-ql

1 10t - YEARS

~ig, 13 - Tj’pe-curie anal:.’sis of .s layered reser,mir (no crossflm.:) t,::di ffcrcncin~.

<

-=2*

\\*

.-. .-

100

+ Q(2)

\

\\

\

\

10 100 lUW

t - DAYS

Fig. 14 - Effect of a change in back pressure on decline using graphical supe~osition.

I ‘ Dd I I

111

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

F- :,., ..>:’: L~,,;;. . . . . . . . . Pressure at

0.00634 kt‘3L ‘ ~ucl ~Z

100.1 1 10 Iw

t - HRS.

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