**SOCIETY** **OF FETROLEUM** **ENGINEERS** **CF AIME**

6200 North C,?ntralExpressway

Dallas, 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.

**@** **Copyright 1973**

**American Instituteof Mining,Metallurgical,and Petroleum Engineers,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

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

com-binations 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.

kg-log type curve analysis can be

per-formed Gn declining rate data (constant-terminal pressure case) completely analogous to the log-log type curve matching procedure presently teing employed with constant-rate case pressure

transient data. Production forecasting is done

by extending a line drawn through the rate-time data cverlain along the uniquely matched or best

theoretical type curve. Future rates are then

simply read from the real time scale on which

the rate-time data is plotted. The ability

to calzulate kh from decline-curve dzta by tyFe

.-.eferen:esand illustrations at end o? yapr,

curve matching is demonstrated.

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 wide range of alternate interpretations, mostly

as a function of the experience and objectives of the evaluator. Recent efforts in the area of decline-curve analysis have been directed towards a ~urely computerized statistical

approach. Its basic objective being to arrive

at a unique “unbiased” interpretation. AS pointed out in a com rehensive review of the

!?

literature by R2msa , “in the period from

1964 to date, (1968 , several additional papers were published which contribute to the

JIECL.INECURVE ANALYSIS USI 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 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 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

constant-pressure zase10-13● _{It followed then}

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

BASIC EQUATIONS

AIISl RATE-TIWW4UATIONS

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

### 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 was eveloped for values of “b” between O and 1

n 0.1 increments. The results are lotted as

7

set of log-log type curves (Fig. 1 in t~~s

f a decline-curve dimensionless rate

‘Dd

ES&l ● ● ● * ● ● “(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 md become indistinguishable at t

9

=0.3. Any

Iata existing prior to a t,~dof O. 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

least-Squaresapproach could calculate any valus of J between O and 1.

\NALYTICALSOLUTIONS (CONSTANT-Pi’iSSUREAT

[NNER BOUNDARY)

Constant well pressure solutions to prediz-ieclining production rates with time were first

pklished 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. The

rf2SUltS were presented in graphical fcrm in

Lerms of a dimensionless flow rate and a

ji,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 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

,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 influx prcblems using the dimensionless cumulative

production Q~ (13). The relationship between

{Q_{~ and qD is}

d(QD) _{(8)}

—“=QDdtfi “ “ “ “ ● “ “

*u*

Fetkcvich’7 Fresented a simplified approacl to water influx calculations for finite systems that gave results whi,>hcompared favorably with the more rigorous analytical sonstant-pressure solutions. Equation 3 of his Fa~er, ~or a constant-pressure pwf, can be written as

Jo (pi - Pwf) . , . (9) q(t) = [“[T . 1 t. e but qi =Jo (pi - pwf) ● . . . ● (10)

Substituting I@. 11 into 10 we zan writs

Now substituting Eq. 10 and 12 into 9 we obtain

qit

Pwf

01**l-—** Pi NPi

● * (13)

Equation 13 can be considered as a

deri-vation 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 back-pressure 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

,U)q, . . .

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

### k@w.

● ● ● ● ●### (U)

D: == N

A

pi

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

**‘~=[**

### ‘:-l

### t

● ● ● ●**‘“)**

Defining N

variables,pi an: ‘qi)- ‘n ‘em ‘f ‘eservoir

Tl(r- _{rw2) @ ct h pi} _{. . (17)}
N e
pi = 5.615 B
and
khpi
(A)= .* (18)
‘=i’max
[()
IJJ*3AJ3 ~nb
r
w
1
1
--2

The decline-curve dimensionless of reservoir variables, becomes

the, in terms
**0.00634** kt
‘m =
@ B ct rw2
J. . . (19)
..** (20)

To obtain a decline-curve dimensionless

* L* DECIJNE CURVE ANALYSIS USING TYPE CURVES SPE L625
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.

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 “empiri-cal” exponential, hyperbolic, and harmonic decline-curve solutions on a single

dimension-less 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.

SOLUTIONS FROM RATE AND MATERIALBALANCE

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.

RATE IJJUATIONS

Until recently, no simple form of a rate equation existed for solution-gas drive

reser-voir 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

ij

q. = J?

**()**

(~R2 -pwf2)n . . . (23)
‘i 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{.

(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

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

Cg (j5R2-pwf2)n . . . . (24)

‘i% =

### 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 gasEquation 25 is a good apFroXhnation for totally undersaturated oil reservoirs, or is simply assuming that durinR the decline

12!@@-%

vs. N can be approximated by

a straigh line.p For gas reservoirs, Eq. 26

is 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 typical shape of the pressure (~ ) VS. cumul-ative

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 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 combi-nations 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**

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

deel.ine-curvedimensionless lttie t (For

=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

respectively a surprisingly narl;owrange.

To achieve an~pnential decline, n must

be equal to zero, and a harmonic decltne

requires n+-. In Fractical applications,

if we assume an n of 1.0 dominates in solution-gas (dissolved-solution-gas)drive reservoirs and ~

vs. N is l%near for non-uniquely defined

?ime data, we would simply fit the

rate-time 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

)VICH _{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

decline-curve data. He found that over 70 percent

of the values of “b~ lie in the range

O~b ~0.5. Ramsay found a different

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

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

given by Eq. 28, i.e.,

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

C@) 1 _{(3C)}

—— ● “’

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

For n = 0.5, the exponen~ial decline is obtained q~

**()*t**

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

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 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

-+ p. (Ap+O) the type curva approaches

as pwf ,

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 qg(t)dt . . . (32)

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 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 type-c.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

than one of the type paper may have to be fit of all the data.

3. Draw a line

beyond the rate-time

to a type curve. More

curves presented in this tried to obtain a best

through and extending 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.

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

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 departure curve 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 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

rate-time 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 “
@= 1000BO~
qm =0.33 = ~
j %
qi = 1000 BOPIVI
0.G33 = 30,303 BOPM
‘Dd =12.0 = Dit = Di 100MO.
.Di= ~=
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

**5PE** 4629 M. J. FETKOVICH 7

5.sequivalent to b = 0.5. Match pints

determined 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 individual well

raises an imprttint question. Should there be

a difference in results between analyzing eazh well individually and summing the results, or simply adding all wells production and analyzing the total lease production rate?

Consider a lease **or** field with fairly unifomn

reservoir properties, lib!:or n is similar for

eazh well, and all wells have been cn deckine at a similar terminal wellbore pressurey Fwf~ for a sufficient period of time to reach

FSeUdC-steady state. According td Matthews et.al.23

Iiat(Fseudo) steady state the drainage volumes in a bounded reservoir are proportional to the rates of withdrawal from each drainage volume.”

It follows then that the ratio q./N . will

be identical for each well and th$isthe sum of the results from each well will give the same results as analyzing the total lease or field

production rate. Some rather dramatic

illus-trations of how rapidly a readjustment in drainage volumes can take place by changing the production rate of an offset well or drilling an offset well is illustrated in a

paper by Marsha. Similar drainage volume

readjustments in gas reservoirs have also been demonstrated by Stewart25.

For the case where some wells are in different portions of a field separated by a fault or a drastic permeability change,

readjustment of drainage volumes proportional

to rate cannot take place among all wells. The

ratio ./N . may thenbe different for different

group %#f hlls. A total lease or field

production analysis would then give different results than summing the results from individual

well analysis. A stilar situation can also

exist for production from stratified reservoirs 26, 27, (no-cw~~flow)o

ARPS’ EXK)NENTIAL DECLINE EXAMPLE

Fig. 11 shows the results of a type curve analysis of Arpsl example of a well with an

apparent exponential decline. In this case,

there is not sufficient data to uniquely

establish a value of “b”. The data essentially

fall in the region cf the type curves where all

curves coincide with the expnmtial solution.

As shown on Fig. 11 a value ofb =0,

(~pnential) orb= 1.0 (harmonic) appear to

fit the data equally well. (Of course all

values in between would also fit the data.) The difference in forecasted results from the two ewtreme interpretationswould be great in

later years. For an economic limit of 20 BOP?4,

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

interpretations zan be conveniently placed on

one curve and forecasts made from them. A

statistical analysis would of course yiald a single answer, but it would not necessarily be

the correct or most probable solution.

Con-sidering the various producing mechanisms we ~ou>d select.

a) b = O, (exponential), if the reservoir is

highly undersaturated.

b) b = O, (exponential),gravity drainage with

no free surface28.

c) b = 0.5 gravity drainage with a free

5 surface 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 approximately

linear.

FRACTURED WiLL EXAMPLE

Fig. 12 is an example of type surve matching for a well with declining rate data available both before and after stimulation.

(The data was obtained from Ref. 1.) This

type problem usually presents some difficulties

in analysis. Both before and after frac.

log-log plots are shown on Fig. 12 with the after

frac. data reinitialized in time. These before

and after log-log plots will exactly overlay each other indicating that the value of “b” ——

did not than e for the well after the fracture

treatment. tThe before frac. plot can be

considered as a type curve itself and the after frac. data overlayed and matched on it.) Thus all the data were used in an attempt

to de~e “b”. When a match is attempted on the

Arps unit solution type curves, it was found that a “b” of between ~.6 and 1.0 could fit

the data. Assuming a solution-gas drive, a

match of the data was made on the Fig. 6 type curve with n = 1.0, b = 0.667.

**Using** **the** match Feints for the before
frac. data we have from the rate match mint,

qw “0.2.43 = w= 1000 BOPM

qoi q’oi

**qo~ =** 1000 BOPM_{●}

243 = 4115 BOPM

From the time match point,

= **0.60** **=** qoi **t** **=**

**%**

**()**

**0+**

115 BOPM)(1OO MO
r

N f4115 BOPM) (1OC MO.) = 685,833 BBL
pi = 0.60
then
q
~= 4115 BOPM
N_{...4} 685,833 = .006000 M0.-1

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

q~ =0.134 = ~ . 1000 BOPM

“Oi qoi

. 100C BOPM

q_{oi} _{0.134} = 7463 BOPM

From the time matsh pint

q.

‘Dd .1*13 . _{()}+

~ .m6W(loo MG.~

pi pi

N = (7463 BOPM) (loo MOJ = 660,4.42BBL

pi 1.13

then

q_{oi} _{7463 BOPM}

‘=N 660,w2 BBL = .011300 MC.-l

pi

We can now zheck the two limiting condi-tions 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?

**2.** Did the reserves hcrease h direzt

proportion to the increase in producing rate 3

as a resu t of a radius of drainage

read-justment ? BefGre treatment, N . was fOUnd

to be 685,833 BBL. Cumulative F~%duction

determined from the i-atedata prior to

stim-ulation was 223,500 BBL. N then at the

time of the fracture treatm~i%tis

N =685,833 EBL- 223,500 BBL =462,333 i3BI

pi

If only accelerated pradution was obtained and

the reserve” remained the same, q~ after

F pi the fracture treatment should have been

7&63 BOFM

f+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/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

N ‘J&J—-1**130PM** = 1,243,833 BBL

pi = .006000 MO.

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 treat-ment is 198,109 BBL, (660,442 BBL - 462,333 BBL; STRATIFIED r’SiRVOIR EXAMll&

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.

After Russell and Prats27 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

qi

**J%**

**t**

**-—**

**)**

### () t

~Npi 1**+-qi2e**

**N**

**qT(t)**

**= qi~**

**e**pi

**2**● ✎ ✎ .(35)

**‘r**

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

in another exponential.

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

**SPE L629** **M.** **J. FETKOVICH** 9

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

50sGOOBO~ e - (O*535)t

where t is in years.

30

Higgins and Le~htenberg 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 illu-strate a simple graphical forecasting

super-position 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-pressure case for a simple single pressure 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-t**

as a function of time is simply kde by

evaluating a single set of match points using the reservoir variables given in Table 2.

/r’ =100

At Pwfi and re w

t=lDAY; t

Dd =0.006967

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 super-postion)j this rate change retraces the qm

vs. t~ curve and is simply a constant fraction

of ql
[
Pwfl - pwf2
q2 =ql _{Pi}
- ~wfl

### 1

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

q2 = 697 BOPD _{4000 psi - 1000 psi}1000 Psi - 50 **Psi** = 221 BOPD

The total rate qT after the pressure shange is q

T

= ql + q2 as depicted in Fig* M*

All rates o 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** **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 match

curve 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 and

summing 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 wera

developed for de~line-curve analysis using

Pressure build-up and decliae-curve data some necessary simplifying assumptions. For

were available from a high-pressure, highly spe~ifi,creservoirs, when PVT data, reservoir

undersaturated, low-permeability sandstone variables, and back-pressure tests are

reservoir. Initial reservoir pressure was available, typs surves could be Eenerated fcr

estimated to be 5750 psia at -9300 ft. with various relative permeability wrves and

a bubble-point pressure of 2841 ~sia. Two ba>k-pressures. These curves developt~ for

field-wide pressure sarveys were condu?tsd while a given field ~iouldbe more accurate for

the reservoir was still undersaturated. lable analyzing fleclinedata in thak field.

Con-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

hydraulis fracture treatments. Also, (See their FiF. 11).

a~~earing on this table are results obtained

from an attenpt to calculate kh using decline- CONCLUSIGNS

curve 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

desline when they were first placed cn with more diagnostic power than has previously

production. 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 engineers

and a ty~e curve analysis IlsingYip. 3 was can agree, or shows when a unique solution is

matched to one of the r@/’rwst:ms. Gther wells not possible with a tyre zurve ocly. *in* ths

listed on the table, kh5ra an r ~rw mat:h is event of a non-unique sclution, a most ~robable

not indiczted, were Frorated wefls and began solution zan be obtained if the produ~ing

their 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 for

each well was taken from the pressure history lb =-l{~giro~al of detline >urve exponent

match 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-~urve

Ct = Total compressibility, psi-l

data 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, t

well is illustrated on Fip. 16. The constant- ~=

rate type curve of’Gringarten et.al.8 for =-Natural logarithm base 2.71828

fractured wells was used for mat~hing the pres- G = Initial pas-in-~dase, surface measure

sure build-up data. The build-up kh of47.5

G

rnd.-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 detarnrine_{d} Jo = Productivity index, STK BHL/DAy/pS~

from deuline-turve data and prsssure build-ur, J; = Productivity index ’(back-pressuresurve

data tabulated on Table 3 is sur~risingly good. coefficient) STK BBL/DAY/(psi)2n

(The pressure build-up analysis was performed k,

independently by another engineer.) One = Effective permeability, md.

fundamental observation to be made from the n = Expnent of bask-pressure curve

results obtained on wells where a match of

N = Cumulative oil ~roduction, S’fKBBL

**SPE L629** **M.** **J. FE!**
N
Fi
Fi
h
Fwf
%
(qi
q(t,
q~
‘Dd
QD
r
e
r
w
~f
tw
‘D
%
~
P

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

Initial pressure, psia

Reservoir average pressure (shut-in pressure), psia

Bottom-hole f’lowin~pressure, psia Initial surfase rate of flGw at t = O Initial wide-oFen surface flow rate at pwf = O

Surface rate of flow at time t Dimensionless rate, (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 Future

PrcxiuctionRates”, M.S. T?_lesis3U. of

Tulsa, Tulsa, Okla. (1968).

2. Slider, H. C.: ‘IASimplified Method of

Hyperboli~ 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 Test

Data Interpretation in the Precence of Skin EffeZt and Wellbore Storage”, J. Pet. Tech. (Jan., 1970) 97.

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. )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 presentad at the 47th b.nnualFall lfeethin, San

Y Antonio, Texas, (Gzt 8-11, 1972 .

MeKiriley,R. M.: “Wellbore Transmissibility from Afterflcw-DominatedPressure Buildup Data”, J. Pet. Tesh. (July, 1971) 863. Moore, T. V., Schilthuis, R, J. and Hurtitj w. : tl~e Deter~~natiOn of Permeability

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

~ 4.

Hurst, W.: “Unsteady Flow of Fluids in

Oil lieservoirs”, &sics. (Jan., 1934)

~, 20.

Hurst, W.: “l!aterInflux into a ileservoir and Its Application to the Zquaticm of

Volumetric Balance”, Trans., ?l~’~(1943)_{——}

~, 57.

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

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).

Ferrt~,,J.$,Knowles, D. B., i3rown,~:.J~.~ and Stallman$ R. W.: “’Ihecryof Aquifer

Tests”, 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’mir

Under Jlastie Conditlcns”, Problems @f

Rgservoiy Hvdrods’nami~sPart.I, Leningrad,

m 9-34●

~etkovi~h, M. Jo: “A Simplified Approach

to ~!at:?rInflux Calculations-Finite

Aquifer Systems”, J. Pet. Tech. (tJuIY~ 1971) Xl-4.

flr.lttlg]~,**W. W.,** Jr.: llEstimationof

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

Stewart, P. R.: “Low-PermeabilityGas

**Well** Porfomnanee at Constant.Pre9sure”,

**2****DECLINE** **CURVE** **ANALYSIS USING** **TYPE CURVES** **SPE 4629**
**20.**
**21.**
**22.**
**23.**
24.
25.
26.
——

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).

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).

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.

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.

hlarshjH. N.: llMethodof Appraising Result! of Production Control of Oil Wells”, ~.

API (Sept., 1928) 2@, 86.

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

Lefkovi%, H. C. and Matthews, C. S.:

“Applisaticm of **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.: “Performance of 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 on Production 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 Drive

Reservoirs”, Sot. Pet. Eng. J. (s3pt.,

1961) 142. ACKNOtilEIXXMENT

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

NOTE : Tle author has a limited quantity of

full size type curvas with grid suitable for actual use which are avzilable on written request.

**(1)**
TotalField
Rate-?T
90,GO)
6L,,LW
lb8,uc
36,(ALI
27,5L0
*2J ,250*
16,2S
I?,L.CA
10,:Kl
8,Sk
*6,%*
5,LM
*L,55G*
3,t3L&
3,**):J.**
*7,75L*
**(2)**
L3yer**1**
Late-ql
~
52,(KKM
L2,5W.
3k,5W*
28,%@
**:3,(LAN**
**18,(YW**
**15,cw***
1Z,5GW
lb,5(IJ
8,503
6,$’w
5,4X
L,55G
3,W.6
3,2cL
2,:%
(l)-(2)
layer2
Iate.qz
80FY
**38WL**
2;5L.G
135(L
75(XI
k5CG
2650
1250
yxl
G
‘.el:
**!.0 .**
**—**
**1**
**?**
3
b
5
1>
7
F
9
IL
11
12
13
I&
15
16
I-I
ia
19
26
21
22
**t!** *J*C*
**$** s
—.
**5.4** **J?, Y**
10.5 l&.3
$.$ ?b.1’
5.5 18.6
IL..? :5.1
:L.3 12.(.
IL.[ 1’7.5
s.] 2J..2
1(,,? IP.L
JL.1. >!.-l
11.5 19.2
.1,1 1-I.C
iC.2 18.8
5*.L* *2ti.~*
*10.Y* *2e.6*
*IG.b* *25.6*
8.8 22.;
G.& **]r.1**
3.\ ;9.2
5.6 2L.6
11.1 16.5
8.9 2.?.5
.
-(,,?3
-?.65
-3.71
-3.Ll
-[.29
-?.G7
*-;. L1*
*-3.7L*
-L,]>
-;.K
-1.W
**-3. ~,?**
**-3.85**
**-&.lLl**
**-3.59**
-L.5--3.:2
-1.5G
-2.11
-5.L8
-2.1$
-3.?9
lre.s.re}**,~:d-1:** *..esu:t:*
*Win* *M*
*rw*
*J1.*
L.3
3.>
lf;.3
‘7.6
18.3
2 .“
-.L
IL.(.
16.5
02.9
.2.(,
13.3
;I.P
I>.&
9.1
2L.2
5.?
?.7
2.1
&.1
2.2
11.2
m. **H.**
1.7L.5
\6.?
L3,(.
2?.5
*LL.L*
57.\
IL.e
:t.6
*:GL.?*
*31.3.2*
55.9
8.[,
L7.5
2?J+.8
101.5
IL.3
‘?7.2
65.1
*Lf.I.5*
*88.1*
**3$.1**
116.L
s
rw
%tzhed

**1.:**
**8**
**7**
**6**
**5**
**4**
**3**
**2**
**0.1**
**9**
**8**
**7**
**6**
**:IF** **:**
**,!** **3**
**u**
**2**
**2**
**O.OJ**
**6**
**7**
**6**
**5**
**4**
**3**
**L**
**0.00**

**E**

**RATE**

**- TIME**

**EQUATIONS**

**1**

**Qlt)**

**=**

**;FORb>o**

**Q,**

**l+**

**bD,,]~**

**[**

**Qlt)**

**1**

**‘=**

**—;**

**FORb=O**

**Q1**

**~**

**D,t**

*lDd.*D1l’

**‘D**
**9 -** **-** **8**
**8 -** **-** **7**
**7 -** **–** **6**
**6 –** **–** **5**
**5 -** _{–}_{4}**4 **
**-–** **3**
**3 **
**2 **
**-0.1**
**9**
**6**
**7**
\ \]
4**qD****3**
**2**
**6**
**5**
**qD** **4**
**F**

**L-Bsl**

6
**0.01**

**—**

**,\,ll,,\\,,,,,~l**

**\**

**\**

**‘-J**

**!i”**

**:**

**-7 -**

**1;**

**:**

**6 -**

**5**

**5 -**

_{4}4 –

**3**

**3**

**-2**

**2 –**

**I**

**I**

**I**

**I**

**I**

**I I**

**0.001**

**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 functions olane radial system infinite and finite

1**o“** **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**
**9** **I** **[** **I** **I** **I** **I** **[** **I** _{I}**8**
**7**
**6**
**E** **1**
**6**
**5**
**5**
**4**
**4**
**3**
**t**
**2**
**0.1** **—**
**9 **
**8 **
**7 **
**6 **
**-5 –**
**4 **
**-qD**
**3 :IzIzzl**
**2**
**0.01** **—**
**9 **
**8 **
**-7 –**
**6 **
**5 **
**-4**
**3**
**2**
**1**

**1**

**32**fj -

**6**

**‘%**

**–**

**5**

**4**

**–**

**3**

**\**

**–**

**2**

**~**

**~**

**0.01**

**9**

**8**

**7**

**6**

**5**4

**3**

**2**

### L

**I**

**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**

**0.001**

**0.001**

**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 finite

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)
4
**1.0 i----** **\** **L**
**9**
**8**
**E**
**L**
**7** **100,OQO**
**6**
**5**
i i I **I** **I** **I** **I** **1**
**4 **
**-‘Dd**
**3 **
**2 **
-0.1 —
**9 **
**8 **
**7 **
**6 **
**5 **
**4 **
**3 **
**2 **
-.01
10-’

### I

0.00634**kt**

**-1**

**)**~;

**1**

qDd . ~-’Dd
L~~:’lllll I I I I I I I I
2 34 567810”’ 567891 2 3 4567810
**‘Dd**

Fig. 3 - Dimensionless flovjrate functions florplane radial system, infinite and finite 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**
**:**
**I** **1** **I** **I** **1 !** **)** **I**
**7**
**6**
**5 **
*-4*
*2 *
*-Um*
*10.CC.J*
*1:* *t*
*0 *
*-7*
*6 *
*5 *
*-4*
*23* *456781O’* *23* *45678101*
*1 )* *1 ,.10*
*1* *1* *1* *1 I* *I I* *I* *I* *I* *I* *-* *9*
*-* *8*
*7*
*-* *6*
*-* *5*
*-* *4*
*-* *3*

**!**

*2*1.0

*9*

*8*

*1*... .. ...

**I**

**U(II**

**1**

**Wd”~=e~**

**.FORb.0**

### I

### I2

.t**Qo.j**01

_{\\\Y}

*7*

*6*

_{I}*5*

*1*

**ANALYTICAL** **TY~E_C#RVE SOI.UTION**
**4**
**3**
**2**
**001**
**9**
**8 1**
**7**
**e**
**5**
**4**
**3**
**2**
**I**

**\\YNsN\\\**

~
**j-3**

**2**

**‘e**

**00**

**6.,**

**0>**

**0**

**00**

**001**

**%**

**9**

**0**

**IJ**

**.**

**7**

**O@**

**6**

*0*

**5**

**.**

_{4}*0*

*.*

_{3}**.**

**2**

**0 COT**

**7 8**

**.O>**

**23**

**456761W**

**1**

**I**

**1 1**

**I**

**1**

**1**

**1**

**1**

**1 1 I )\l**

**1**

**l\**

**1 \l**

**1**

**0001**

**I**

**I**

**I**

**I**

**1 I**

**I**I I 1 ( I 1,8I I ! ! I [I1

*1 1*

*!*

*1*

**1**

**k**

**,“.**

**2**

**3**

**4567810)**

**234**

**5678**

**10:**

**23**

**4**

**5.?**

**870**

**.?3**

**.356781**

**23**

**4S**

**67810**

**23**

**4s6**.

**‘Od**

**o**

Np OR Gp NPi OR G

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

**I**

TR VS Np APPROXIMATELY LINEAR BEGINNING OF DECLINE

PRORATED WELL WHEN DECLINE BEGINS

**o**

N~Gp Npi OR G

**flATERIAL** **BALANCE** **EQUATION:**
**()**
Pi
fiR=. — 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** **4567810z**
**1.0**
**9**
**8**
**7**
**7**
**6 **
**-5** **–**
**4** **-** **EXPONENTIAL**
**3**
**2 **
**-0.1** **—**
**9**
**-8** **–**
**7 **
**6 **
**-“**
**g3**
**u**
**2** **–**
**0.01**
**9-**
**8-**
**7-**
**6-~:** **~**

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

**2**
**I** **I** **I** **I** **I** **[** **[** **[** **I** **I** **I** **I** **I** **I** **I** **I** **0.001**
**I**
**0.00 1** **I** **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** o
**2.667** **0.375**
**2.000** **0.500**
**1.714** **0.583**
**1.500** **0.667**
**1,333** **0.750**
**1.250** **0.800**
**1.125** **0.689**
**1.050** **0.952**
**1,0Q5** **i 0.995**
**‘od**

### ‘(~)’

Fig. 6 - Dissolved gas drive reservoir rate - decline type curves

pressure at inner boundary (pI+T= O @ rlJ). Early transient

finite system with constant effects not included.

–**6**
**– 5**
**–** **4**
**– 3**
**-** **2**
**3.G.D.** **-$1**
**–B**
**-7**
**–6**
**-5**
**–4**
**–3**
{
**2**
**}01**
**‘8**
**7**
**6**
**5**
f
*4*
*3*
*i*
*2*

1$’ .
**8-7**
**6**
**5**
**4**
**3**
**2**
**0.1**
**-9**
**8**
**7**
**6**
**~..5**
**824**
**,1**
**S3**
**u**
**2**
**0.01**
**:**
**7**
**6**
**5**
**4**
**3**
**2**
*0.00*
**SLOPE**
**n**
*o*
**0.5**
**0.6**
**0.7**
**0.8**
**0.9**
**1.0**
**10.0**
**100.0**
—
**2“+** **1=;**
**1 .mo**
**2.000**
**2.200**
**2.404)**
**2.600**
**2.800**
**3.mo**
**21.000**
**201.000**
\

**7\**

**1.000b**

**0.500**

**0.454**

**0.417**

**0.365**

**0.357**

**0.333**

**0.046**

**0.005**

**T**

**,!/**

**MATERIAL** **BALANCE** **EQUATION:**
**‘R’=** **-(*)** **‘“+””**
**\**
**IATE** **EQUATIONS:** _{%4}**TWO-PHASE** **6R** **n**
**(** **)[** **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 I

**23**

**45678103**

**2**

**3**

**4**

**5678**

**10”’**

**23**

**4~6761**

**2**

**3**

**4567810**

**23**

**45676102**

**RATE-TIME**

**q.**

**(t)**

**EQUATION**

**(“~**

**= O): ~**

**-2n+l**

**w)’**

**1**

**t+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 curves

pressure at inner bwndary (ph~ = O @ rl,). E:rN transient

finite s~stem with constant effects not included.

6114,1, .9
I . . .
u
6
I
-—
U-I.. 0000O
oo-1-+v~g
**I**
**‘P**
**td**
r+
*CJo*
*g* **II**
*1*
co
G
.,-1
**L4**

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 0.1

**9**

**8**

**7**

**6**

**5**

**4**

**7**~ “

**2**

**:01**; -I ~

**\\,\,,l,,J**

**i**

**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**

**46678102**illl \ I O.CO1

**I**

**I**

**I**

**I**

**I**

**1**

**I**II I I I I I 1 I I

**I**

**I**

**I**

**I**

**I**

**.-.**Fig. 9 - Gas pressure at

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

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

\

**-–L**

10 100 285 fI~O. 1000 1480klo,

**\l**

AFTER FRAC FFIACTURE
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

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 2*qz=’+-ql*1 10 t - 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
### I

**‘ Dd**

### I

### I

**11**

### 1

F- :,.,..>:’: L~,,;;
. . . **Pressure** **at**
0.00634 kt
**‘3L** **‘** **~ucl** **~Z**
10
0.1 **1** **10** **Iw**
t - **HRS.**

Fig. 16 - Type-curve ::atckingexample f~r calculating Kb from pressure buildup