SPE-464-G

8/9/2019 SPE-464-G

http://slidepdf.com/reader/full/spe-464-g 1/6

MODIFIC TIONS to DECLINE CURVE N LYSIS

HOMER

N.

MEAD

MEMBER AIME

VENEZUELAN ATLANTIC

REFINING

CO.

CARACAS, VENEZUELA

T.

P. 4200

ABSTRACT

This report develops equations for decline curve

analysis based upon the premise that the rate of change

of

the reciprocal

of

decline for succeeding time inter-

vals is constant when the reservoir s vroduced under a

fixed set of conditions. A method is shown for predict-

ing m ximum production rate against cumulative pro-

duction for any reservoir. A method is also presented

for predicting future production rate by an analysis of

past production performance after decline has heen es-

tablished.

I NTRODUCT ION

No

significant contribution to the analysis of decline

curves by the loss-ratio

method

has been

made

since

J. J. Arps

paper'.

The

concept

of

loss-ratio

that

was

developed by Arps and his contemporaries has been re

defined

in

another way and the concept of instantaneous

loss-ratio

at

time zero has been added.

In

the mathe

matical development by Arps, production rate was re

lated as a continuous function with time and his equa

tions were developed on that basis. In this paper, pro

duction is considered

to

be a series

of

segments

for

equal time intervals

and

equations have been developed

based

upon

these finite differences.

t is realized that decline curve analysis is not the

answer

to

all predictions

of

reservoir behavior. How

ever, as production

must

decline from

an

initial maxi

mum

rate

to

zero

in

any reservoir, if such decline can

be expressed as

an

infinite series, this series should ac

curately predict production. Decline curve analysis

should be considered a valuable tool

that

may be used

in

conjunction with predictions of future recoveries by

other

methods. Various uses of the decline curve meth

od

will be discussed

in

this paper. Equations predicting

Original

manuscr ipt

received in Petroleum

Branch

office on

Jan.

15, 1955.

Revised manuscr ipt received

on Dec 13 1955

: Homer N Mead is now enrolled in

the

School

of

Bu siness Admin-

istration a t the

University of California.

References given at end of paper. SPE 464-G

VOL.

2 0 7

1 9 5 6

production rate

at

any time

and

cumulative production

by exponential

and

hyperbolic decline will be devel

oped. Characteristic values of

b

related to various types

of

drive will be discussed.

After that

a hypothetical

reservoir will be studied

and

ultimate recovery by

nat-

ural

depletion

and

pressure maintenance compared,

then actual field examples will be shown and discussed.

The

rate-cumulative curve introduced by

H.

N.

Marsh in 1928 is an

important

one

for

use

in

esti

mating future production rate and ultimate recovery.

This graphical method

is the

one

that

is used

in

con

junction with

the

equations

to

be developed

for

predict

ing production behavior.

EXPONENT IAL

DECL INE

In

exponential decline

each

succeeding

production

rate per

unit

of time is a constant percentage of the

production

rate

just before it. Since

1

-

r is

the com

mon

ratio, where r is

the

constant decline rate, this con

dition may be expressed as a geometrical progression;

thus

P

n

=

P

1 -

r -1 •

Where

r

is less

than

unity,

P -

P

1 - r U

C =

Since

P,

1 -

r)n

=

Pn

1

C

= P

,

- P

n

1

r

r

LOSS -RAT IO

1)

(2)

In

this paper the calculation

for

loss-ratio has been

changed

from the method

presented

by

Johnson and

Bollens'.

The

first loss-ratio

to be

calculated is

a

·

it is

the initial production per unit of time after declin'e has

set in divided by the difference between the initial and

the second production rate.

This

loss-ratio

is for

the

first time interval.

After b

(which is

the

constant change

11

8/9/2019 SPE-464-G


in loss-ratio per unit of time) has been established, an

will

be

determined by subtracting b

from a

; a

o

is the

reciprocal of

the

instantaneous rate of decline at time

zero. Table 1 should

make

the above explanation clear.

I t is

the opinion of the author that decline curves

for

most

reservoirs

producing

under a given set

of

con

ditions will have a

constant

b.

This

factor will vary

between one and zero and will be constant for thai

reservoir when it

is

operating under a fixed set of condi

tions at

maximum

rates of production.

D I S C U S S I O N

O F

h

F A C T O R

At

this point it is well to discuss the factor b. Since

in exponential decline

the

reciprocal of r is constant,

it follows that b equals zero.

On

a rate-cumulative

plot the exponential decline curve is a straight line. For

hyperbolic decline, which assumes

b

to be constant,

b

has limits, 0 ::;;

b

<

1,

and

on

a rate-cumulative plot

the

curve

is concave upward see Fig. 1).

On

this

graph

it was assumed that

the

maximum initial rate was 1,000

units per unit time

for

all values of

b.

The

actual

initial

rate was

taken

to be 650 units per

unit

time. This initial

rate

of

650 units will be

maintained

for

each

value

of

h

until the dotted line at 650 units intersects

the

rate

cumulative plot for that particular value of b. For ex

ample, when

b is

equal to 0.75, approximately 1,825

units

can

be produced

before the

production

begins to

decline. As b approaches zero, for

the

same initial

maximum production rate and same ultimate recovery,

a greater and greater amount of oil

can

be produced

before a decline sets in.

When b is

equal to zero, ap

proximately 4,150 units

can

be

produced

at 650 units

per

unit time. For

the

same ultimate recovery and same

initial rate, the least time

for

recovery will be when

b

equals zero. The closer b is

to

1,

the

greater

the

time

needed to produce the same ultimate recovery. This is

shown clearly on Fig. 2. The data used for these curves

are the same as those used for Fig.

1. I t

was assumed

in Fig. 2

that

the ultimate recovery

and

initial rate per

unit time are

the

same for b equals 0, 0.25, 0.50, and

0.75. For example, at the

end

of

40

units of time, about

99

per

cent

of

the

ultimate recovery has been pro

duced

for h

equals zero while only about 57 per cent

of the

ultimate recovery has

been produced

for the

same time when b equals 0.75. It is true that if the

reservoir is produced to 1

unit

per

unit

time, eventually

the

same ultimate recovery will

be

obtained for all

values of

b;

however,

the

closer

b

is to 1

the

longer the

time needed to produce all the units.

D I S C U S S I O N

O F R A T E -

C U M U L A T I V E P L O T S

A rate-cumulative plot is a series of segments for

equal

time intervals including on

both

axes

the

produc-

0

\

K'

o

ASSUMED

lN lTI l

I'-.

i

'--

RATE OF

o

6 5 / . . . N ~ ~ E r \

1---

.:::,.

-f -..,

1 '-

f .. I' -

. '

'

5

,, -

I

0

r-.....

l '- I '-

0

6075

r--..

l '-

I' -

r

t-

r .

t--..

o

-

t

;:::::

-1 -

.-

4000 6000

7000

CUMULATIVE PRODUCTION

FIG. 1 - EFFECT OF B ON PRODUCTION RATE FOR SAME

INITIAL

RATE

OF

PRODUCTION AND SAME ULTIMATE

RECOVERY.

12

0

I--

-

---

I--

--

:..-f--

-

/

.- -1-1--

.-IJP

I-r-

--

1/

i. -

f- -

7

V

P

>

,

V

v

rl

V

I-- i-

/,

0

01

- f - -

f--

5

Y

1---: -

M v

VI

V I

3

}

I

2

If/

,

I

I

I

I

,

50

UNITS OF

TIME

FIG. 2 - EFFECT OF B ON TIME-RATE OF RECOVERY

FOR

SAME

INITIAL

RATE OF PRODUCTION

AND

SAME

ULTIMATE RECOVERY.

tion for that period of time. Fig. 3 shows this graphic

ally. There are three methods for making a rate-cumu

lative plot. In

the

first

method

the production per unit

of time

is

related to the cumulative production at the

end

of the period, as is shown

in

Fig. 3 in the upper

dotted

curve

marked

(1).

The

advantage

of

using this

plot is that it is easier to take data directly from actual

production history. Because of the ease of plotting, this

method

is

the one used on the accompanying rate-cumu

lative graphs, in which actual

production

data have been

plotted. Another

method

for plotting a rate-cumulative

curve is to relate the production per unit of time to the

cumulative production existing at the

beginning

of the

period.

This

is shown in

the

dotted line on Fig. 3

marked

(2). Third, if it is desired to plot a curve show

ing the best approximation of instantaneous rate versus

cumulative production

at

any time, it will be necessary

to connect the midpoints

of

the segments. This

is

shown

in

the

dotted line on Fig. 3 marked

(3).

Since the

equations developed in this paper are based upon finite

differences in

production

rather than

making

rate a con

tinuous function with time, the

third method

described

will not be considered. The lines obtained by all three

methods converge at ultimate recovery. I t is

empha

sized that production for equal units of time must be

used in decline curve analysis.

H Y P E R B O L I C D E C L I N E

E Q U A T I O N S

In hyperbolic decline the reciprocal of the rate of

decline for each succeeding quantity of production per

unit

time increases by a constant factor called h. As

is shown in Table 1, a

o

cannot

be calculated

from

ac

tual production history but is equal to a

1

- b. The fol

lowing definitions are used to develop an equation for

cumulative

production

using a summation

of

produc

tion quantities for equal time intervals.

C = P

+ P +

p

+ P + P + ... + P

n

(3 )

(4)

TABLE 1

Production Rote

of

Reciprocal

Time er

Unit ecline of

r

nterval Time

(1 r)

r) (0)

b

-

-2.0

1

10,000

0.6000

0.4000 2.5

0.5

2 6,000 0.6667 0.3333 3.0 0.5

3 4,000

0.7143

0.2857

3.5

0.5

4 2,857 0.7500 0.2500 4.0

5

2,142

PETROLEUM T R A X S A C T I O X S

AIME

8/9/2019 SPE-464-G


I I I I I I I

i

I I I I I

:

" 1

7

I I I I I I 1

,

~ ' , ' , , : :

y--+t--t1+ 1t

r f

1Io' .1.14;J::+I_+-IH-+-HIH-+-+-++-H-+--l

, ' l ;k l I

, ,

' Irf, ' ,

I

I ,,, '' , ',>

, I I

0

:

I i

40 :;0 60

CUMULATIVE PRODUCTION

FIG,

3 - PLOTTING PRODUCTION PER UNIT

TIME WITH

CUMULATIVE PRODUCTION.

a

+

1 = a +

b

(

5 )

Substituting Eq. 4

and

5 into Eq. 3, we obtain,

C =

(a

1

PI

-

a

1

P,)

+

(a, P,

-

a, P,,)

+

(a p,

-

a

3

Pol)

+ . . . +

(a

P - a P + 1 )

= (a

o

+ b) PI - a

,

P, + (a

,

+ b) P, - a, P, +

(a,

+ b) P -

a P,

+ . . . + ( - 1 + b)

(a

O

PI + bPI) + ( - a

1

P, + a

,

P,) + bP,

+

(- a,p,

+

a,P

3

)

+

bP

+

...

+ (-

a

-,

P +

a

- 1 P,,) + b

P

-

a

P

+ 1

=aOP

'

+

b (PI +P,

+

P

3

+ ... +

P,,)

-

a P,,+,

= a

o

P

,

+ b

C -

a

P +,

Therefore,

1 -

b

(6)

Eq. 6

is considered the basic

equation

for decline

curve anaylsis as developed by the author.

HYPERBOLIC DECINE EQUATIONS

DERIVED

FROM

EQ. 6

It will be noted that when h equals 0, a

o

= a = 1/1',

Eq. 6 becomes:

C

=

[ > 1 _ - . - " ~

r

when P +l = 0 , C =

Q

so,

Q =

t - p ~ ·

Letting PI

-C P +l

= x and

1

b= ~ l x .

y x

(2)

(7)

C

= y

(8)

I f

it is assumed that when ultimate recovery is

obtained,

P

n

+

1

equals zero and if Q equals

the

re

covery left from and

including

P

and

P

1

and P, are the

production

for

two

equal, successive time intervals when

the reservoir is

producing

at

maximum

rates, then:

P

1

-

Q

p p,

b

(9)

P

1

-

P,

a

(

Q ) ( P, )

Q P

1

~ p

(10)

b

PI

(P,

+ r-=

b P,)

Q

-

---- --------

P

1

-

P,

(11 )

When a series of equal time intervals are taken with

P, the initial rate

and

P the last rate with C as the

cumulative production from

P

,

to

P

inclusive, then:

P

- -Q-----)

P

+

_

O C

,

b

-

; ~ ~ 1 ~ - = ~ r p " + 1

(12)

VOL. 2 0 7 , 1 9 5 6

P

,

Q

=

~

\ - Pn l

(13 )

I f k and n

are

two different numbers of

equal time

intervals,

then:

Q

.

(14)

SUMMARY

OF

CHARACTERISTIC

b

VALUES

An approach which uses a decline curve

method

is

based upon

the assumption that the reservoir

must be

produced

at

maximum

rates.

However,

a little

considera

tion given

to

this

problem

will indicate

to

the reader

that in the case of 100 per cent effective water drive, if

the wells are flowed at

MER

and the size of

choke

in

each well

remains

unchanged for the

period

of

produc-

tion

life

in

which

decline

has

been established,

the

rate

cumulative plot

will still follow

the curve

h

equals

zero.

I t

is

the

opinion of the author that even in solution gas

drive reservoirs, if the

choke

size of the wells remain

unchanged, the

decline

that

is established will have the

same b factor

as that

calculated

for

solution

gas

drive

using

maximum

rates.

Careful

consideration,

however,

must

be

given

to

the fact that

choke

size in wells

might

have been changed

during

the time the decline has been

established.

If

they have, the b factor that is calculatd

will not be the

same

as

that calculated for

maximum

rates. This is also assuming that no

methods

have been

used during the time decline has been established to

improve existing wellbore permeabilities. Such improve

ments

that

have been

made

must

be taken

into

consid

eration.

With the above in mind, the author presents a list of

ranges for b under various types of drive.

These

values

should not be considered as absolute; however, it is be

lieved that b will be

constant

when the

reservoir

is

pro

duced under

fixed conditions.

Even in

solution-gas drive

reservoirs

there

will be

some

gravity

drainage

and

per

haps a limited

water

drive, so actual conditions make

it impossible to determine one specific b factor for all

reservoirs of a given type. The

table

below shows the

general ranges

for b

that exist under various types of

drive.

\

I

3

\ \

\

PRESSURE

MAlrn:::NANCE

I

\

/1

I

I

ATURAL

Df?LETION

--

I

\

2 I

\

J.

,

5000

8/0 R ~ E

I

,

\

\

-

,

'

1 \

I

5

CUMULAT:VE PRODUCTION IN

MILLIONS

OF

BARRELS

FIG. 4 -

HYPOTHETICAL

EXAMPLE COMPARISON OF

PRESSURE

MAINTENANCE TO

NATURAL

DEPLETION.

13

8/9/2019 SPE-464-G


Solution Gas Drive

Gas Cap Drive

Gravity Drainage

. .

Pressure Maintenance by Gas

Pressure Maintenance by Water

RANGES FOR b

HYPOTHETICAL EXAMPLE

0.50 to 0.85

0.20 to 0.85

o to 0.40

0.20 to 0.50

o to 0.20

In

order to demonstrate

the

use

of

decline curve anal

ysis

for

predicting reservoir behavior on a reservoir that

has not been

produced

long enough to establish a de

cline rate,

the

following example is presented.

In

this

hypothetical case

it

is assumed

that

a reservoir

had

been

produced which by volumetric and materials balance

calculations

had

50 mililon bbl

of

stock-tank oil origi

nally in place.

t

was established that there was no water

drive and

no

original gas cap. All

of the

wells produc

ing from this reservoir were examined and it was esti

mated

a

maximum of

400,000 bbl

per month

could be

produced initially. It was determined that 20

per

cent

of

the total oil in place could be produced ultimately by

natural

depletion and 40

per cent of the

original oil in

place could ultimately be

produced

if the reservoir pres

sure were maintained

by

gas injection.

The

problems

are: 1) what are the time-rates

of

recovery by natural

depletion and pressure maintenance and how do they

compare,

2)

how long will it be possible to produce

at

a sustained rate

of

5,000

BID

before decline sets in.

t

was assumed

that b

equals 0.5

for natural

depletion

since this reservoir would

be produced in an

efficient

manner. A similar value

of

b equals 0.5 was used for

pressure maintenance so that the comparison with nat

ural depletion would be on a conservative basis.

Ultimate recovery in this

paper

is defined as

the

total

amount of

oil recovered when

the

reservoir rate has

decreased to zero. Obviously cost factors

of

production

would

make

this impossible to achieve. The economic

ultimate recovery

of

any reservoir after capital equip

ment has

been installed

is

when

the

direct

or

variable

cost

of

producing

one

barrel

of

oil

is

equal to the price

received by selling that barrel. This minimum produc

tion rate will consequently vary for each reservoir de

pending

upon the

factors

of

cost.

t

is possible for two

identical reservoirs

produced at

different times, places

and

depths

to

have different economic ultimate recov

eries yet both have the same total ultimate recovery if

costs were not

taken

into account.

t

is

for

this reason

that

ultimate recovery has been defined as total produc

ible oil.

It is

the obligation

of

the engineer to decide

from cost data to what minimum rate the reservoir can

be produced.

Under natural

depletion (see Tables 2 and 3) in 15

years

the

yearly production rate will have dropped to

74,000 bbl and 8,832,000 bbl

of

oil will have been pro

duced.

Under

pressure maintenance (see Tables 4 and

5) in 15 years

the production

rate will have dropped to

230,000 bbl

per

year and 15,508,000 bbl will have been

produced.

The calculations for Tables 2, 3, 4 and 5 were made

as follows:

t

was assumed that

the

production rate

would be zero

when

10 million

and

20 million bbl were

produced by natural depletion and pressure maintenance

respectively. Since P

n

, will be zero,

P,

o

C

=

0

1 -

b

Because it was assumed that b equals 0.5 both

cases,

14

0.5 C

P,

t

might be pointed

out that b is

a constant factor

regardless

of

the length

of

time given each time interval

as long as each time interval is equal. Therefore, if P,

is barrels produced per year,

o

is on a yearly basis. By

the

same token if

P,

is barrels

per

month,

o

is

on

a

monthly basis. Table 2

and

4 are the calculations

of

production

for the

first

year

by

natural

depletion and

pressure maintenance respectively. The first year s pro

duction was

then

used to start

out the

calculations in

Tables 3 and 5.

The

comparison

of

natural depletion to pressure main

tenance is shown

on the

rate-cumulative plot in Fig. 4.

f under proration the reservoir were produced at an

average rate

of

5,000 BID 4.8 million bbl

of

oil would

be produced from

the

reservoir

under natural

depletion

before a decline would set in. The reservoir would be

capable

of

producing

at

a sustained rate

of

5,000

BID

for 2.6 years.

f

the pressure maintenance project were

installed, and

the

reservoir were still

produced on

an

initial rate

of

5,000 BID it would be possible to pro

duce at this rate for 4.7 years, taking out 8.6 million

bbl before

the

production rate would decline.

FIELD EXAMPLE - RESERVOIR A

Reservoir A ultimately

had

10 wells drilled

in

it. The

initial productivity

of

these wells was such

that each

was

capable initially

of

producing about 1,000

BID.

The

wells have been flowed

at

maximum rates.

On

Fig. 5

(which is a rate-cumulative plot

of

oil production prior

to gas injection) a curve was

drawn

from

the

estimated

initial

maximum

production

of

300,000 bbl

per month

to the last point

on the

curve. P

,

is taken to be

the

pro

duction point

of

195.1M bbl per month.

P,

was deter

mined by following

the

rate-cumulative curve to that

point where

P,

added to the cumulative

production

at

point P, is

equal to the cumulative production shown on

TABLE 2 - NATURAL DEPLETION

CALCULATING FIRST YEARS MAXIMUM PRODUCTION

all :...::

Ultimate

Recovery X 1 - b) 10,000,000 X 1 - .5)

=

12.5

nitial

Monthly Rate

- - - - 4 0 0 ~ _ .

Monthly

Rate

Month

1M

bbl,)

a 1

- r

.

0 12.5

1

400

13.0

.0769

.9231

2

369

13.5

.0741

.9259

3 342 14.0

.0714

.9286

4

318

14.5 .0690 .9310

5 296

15.0

.0667

.9333

6 276

15.5

.0645 .9355

7

258 16.0 .0625 .9375

8 242 16.5 .0606 .9394

9

227

17.0 .0588 .9412

10

214

17.5

.0571

.9429

11

202

18.0

.0556 .9444

12 191

18.5

.0541

.9459

3,335

TABLE 3 -

NATURAL DEPLETION

CALCULATING YEARLY PRODUCTION MAXIMUM)

Ultimate

Recovery X 1 -

b)

-

10,000,000

X

1

-

.5)

= 1.5

0 =

- - - - - - - - - ~ - -

--3-,335,000-

nitial Yearly Rote

Yearly Prod

Cum Prod

Years

~ 1 i . ~ L

~ b b l }

a

r

1

- r

1.5

1 3,335 3,335

2.0

.5000

.5000

2

1,668 5,003 2.5

.4000

.6000

3

1,001 6,004

3.0

.3333

.6667

4 667 6,671 3.5

.2857

.7143

5

476 7,147 4.0

.2500

.7500

6

357 7,504

4.5

.2222

.7778

7

278

7,782

5.0

.2000

.8000

8

222

8,004 5.5

.1818

.8182

9

182

8,186 6.0

.1667

.8333

10

152

8,338 6.5

.1538

.8462

11

129

8,467 7.0

.1429

.8571

12

111 8,578 7.5

.1333

.8667

13 96 8,674 8.0

.1250

.8750

14

84 8,758 8.5

.1176

.8824

15

74

8,832 9.0

.1111

.8889

PETROLEUM TRANSACTIOl \ S ,

AIME

8/9/2019 SPE-464-G


TAF.LE 4 -

BY

PRESSURE MAl

NTENANCE

CALCULATING FIRST YEAR S PRODUCTION (MAXIMU M)

0 0 =

~ . O O O O O O

X_,5 =

250

400,000

.

Monthly

Role

Month

_ (M

bblsL.

a

r

1 - r

0 25.0

1

400

25.5

.0392 .9608

2

384 26.0 .0385

.9615

3

369

26.5

.0377

.9623

4

355

27.0 .0370

.9630

5

342

27.5

.0364 .9636

6

330

28.0 .0357

.9643

7

318

28.5 .0351

.9649

8

307

29.0

.0344

.9656

9 296 29.5 .0339

.9661

10

286 30.0 .0333

.9667

11

276

30.5

.0328 .9672

12

267

31.0

.0323

.9677

3,930

~ ~ ~ ~

TABLE 5 - BY

PRESSURE

MAINTENANCE

CALCULATING YEARLY PRODUCTION (MAXIMUM)

20,000,000

X .5

00 = - - 3 ; 9 3 0 , 0 0 0 - = 2.54 assume 2.50

Yearly Prod. Cum. Prod.

Yeors (M bbls) (M bbls)

1

- r

-- ------

0

2.50

1 3,930

3,930 3.0

.3333

.6667

2 2,620 6,550

3.5

.2857 .7143

3 1,871

8,421

4.0 .2500 .7500

4 1,403

9,824 4.5 .2222

.7778

5

1,091

10,915 5.0

.2000

.8000

6

873

11,788

5.5

.1818

.8182

7 714

12,502

6.0

.1667

.8333

8

595

13,097 6.5

.1538 .8462

9

503

13,600 7.0

.1429

.8571

10

431 14,031 7.5

.1333

.8667

11

374

14,405 8.0 .1250

.

8750

12

327

14,732

8.5

.1176

.8824

13

289 15,021 9.0 .1111

.8889

14

257 15,278

9.5

.1053

.8947

15

230 15,508

10.0

.1000

.9000

the curve for P . a was calculated using

P,

and P . On

Fig. 6 the points from C to D were used to calculate b.

These

data

are shown

in

Table 6. Using this information

the entire curve was plotted as shown in Fig. 6.

I t is of interest to note that

the

amount of oil to be

attributed to gas injection has been calculated by

the

decline curve to be 191,000 bbl.

This

figure is in line

with an estimate made by a previous reservoir engineer

ing study.

FIELD

EXAMPLE

-

RESERVOIR

B

In this reservoir

the maximum

rate attained was

264.2 M bbl per month

after

965.7 M bbl had been

pro-

duced. A reservoir study was made and it was estimated

that the ultimate recovery would be

11

million bbl of

oil. By

taking the

two productions per month with 264.2

M bbl as P,

and

239.4 M bbl

per

month as P, and sub-

r

o

0

0

0

I

I I

,

I

i

I--

~ S T

MAX POSSIBLE

INITIAL RATE

I

300,000 81M)

I

I

i

I

1

I

i ,

r,

:

1 ,

P,

:

I

I

1'\.0

I

i

I ~ P

,

I

I

I

1- 1 -.

1

I

,

I

1

f'.. I

,

P,

f- -

I

I

I I

05

CUMUL.GTIVE PRODUCTION IN MILLIONS OF BARRELS

i

I

I

I

i

I

i

I

I

,

,

I

I

I I

i

I

I

I

i

I

i

i

I

I I

I

i

FIG. 5 - RESERVOIR A

RATE

CUMULATIVE PLOT USED

FOR

DETERMINING A.

VOL.

207, 1956

TABLE

6 - NATURAL DEPLETION,

RESERVOIR

A (DETERMINATION

OF CONSTANTS)

Determining

01

from Graph

IV

Assume Dec.,

1948

is PI then P1 = 195,100 bbls and

the

cumulative pro

duction to

end

of PI is 698,700

bbls.

The next point pz on the curve is that

point

P2,

(698,700 + P::: .

170 000

BIM is 840,000 bbls cumulative

or

160 000

BIM

is

900,000 bbls cumulative

698,700 + (160,000 +

X)

= 900,000 - 6x

7x = 900,000 - 858,700 = 41,300

x

=

5,900

p,

= (160,000

+ 5,900) or

165.9 M

bbl

195.1

01

= ~ 1 6 5 9

=

6.68

The

Interval

C to 0 was used to

determine

b

P

=

195.1

M

bbl/month

Pn+l = 59.8 M bbl/monlh

n = 11

01 = 6.68

C = 1,225.2 M

bbl

Pl

-

Pn l

x =

C

_ n

Pn 1

y 1 - C

.1104

.4631

tracting 965.7 M bbl from 11 million bbl for

the

ulti

mate recovery that was left to be produced including P

Eq. 6 was used and

b

calculated to be 0,74,

a

o

was cal

culated

from

Eq. 7

and

equals 9.92. With these data

the

decline curve was plotted as shown in Fig. 7. I t will be

observed that this

curve

follows quite closely the actual

production

data

C O N C L U S I O N

A

m a t h ~ m a t i c a l

approach

by

infinite series has been

developed ~ i t h

the

basic assumption

that rate of change

of the

reciprocal

of

decline for succeeding

equal

time

periods is constant if

the

reservoir is produced under

a fixed set of conditions. I f this assumption is valid, the

derived equation relating the

four

factors (ultimate re

covery

to zero production

rate, initial production rate

per unit time,

band

a

o

is true.

I f

the

ultimate recovery and maximum initial

produc-

tion

rate

are known and b is estimated for the different

types

of

recovery mechanism,

au

can be calculated for

each.

It

will

then

be

possible

to

compare what

the

reser

voir would do under

each

recovery

mechanism

with re

spect to rate of production, time and cumulative produc-

tion.

The basic developed

equation

is:

(6)

T ~ S T

Lx

l oss lLE

i

-

\ INITIAL RATE

I

:

I

(300,000 B/M)

I

,

i

0

1\

:

i

\

I

0

I

CIRCLES ARE

~ R O D U T I O N

FIGURE

I 1

~ ~ ~ ~ ~ I S C U ~ \

I

1\

0

1::-

Gl 3 INJECTION

~

2

STARTED

NOV

194

J

OCi

l94

j-1.-'O

I

00

0

GAS INJECTION

r-...::

c ~ ~ 4 N r 3

f - - f - -

0

0

0

~

2 3

4

CUMULATIVE PRODUCTION IN MILLIONS OF BARRELS

FIG. 6 - RESERVOIR A

RATE CUMULATIVE

PLOT,

8/9/2019 SPE-464-G


'

1-1-t-,\\ -i0ECLlNE CURVE ANIILYS S +--t-H--+-+-++-t-H-t--t---t-t-1

~ 2 0 0 ~ - - \ ~ - - t - ~ - - - - t - H - t - - t - - - t - t - H - t - t - ~

o

o

z

I

, o o l - - - - - - - - , K ~ o ; t - - - i - - - l - - - - - t - - - t - - i

CUMULATIVE PRODUCTION IN

MILLIONS

O BARRELS

FIG. 7 - RESERVOIR

B

RATE

CUMULATIVE CURVE

SHOWING

How ACTUAL PRODUCTION

COMPARES

TO

DECLINE

CURVE

ANALYSIS.

PROJECT.

When b equals zero,

Eq.

6 becomes the same as Eq.

2. When b equals

1,

it is impossible to solve

for

C by

Eq. 6; therefore, according to Eq. 6, b has limits, 0 s;

b

<

1.

In

the case

of

a reservoir

that

has been

produced

until

a definite decline in rate has

been

established and con

tinues to be produced

at

maximum rates after decline

has set in,

the production data

during decline

may

be

used to calculate ultimate recovery, band a

o

• Since b is

considered to be constant, a rate-cumulative plot for

future production can be drawn. I f the reservoir con

tinues to be

produced at maximum

rates,

production

rate

versus time

and

cumulative production versus time

curves can be drawn.

6

SYMBOLS

PI

=

Initial

production

in barrels du ring first time

period t.

P

n

=

Production

in barrels during

anyone

of

a

constant series of time intervals which are

equal in length to

the

original time pe

riod t.

n

=

Number

of

equal time intervals from

the

be

ginning

of

initial production to

the

end

of

that time for

Pn

C = Cumulative production in barrels from the

beginning

of

initial

production

to the

end

of

the P

n

interval.

r

=

Instantaneous decline rate at time o.

r, = Decline rate for time interval 1.

rn = Decline rate for time interval

n.

a,,, a

an

=

Reciprocal

of

rate

of

decline or loss-ratio.

b

=

Constant difference between successive loss

ratio.

PI P n l

X = ---c --

_ 1 n P

O

1

Q

=

Ultimate recovery in barrels left to be pro

duced at the beginning of time interval in

which P bbl were produced.

ACKNOWLEDGMENT

The writer wishes to thank the Venezuelan Atlantic

Refining Co.

for

permission to publish this paper. Spe

cial expression of appreciation is given to

A.

Chatas,

reservoir engineer for Socony Vacuum in Caracas, for

a careful examination of this paper together with several

valuable suggestions; and to

T.

Hutchinson, research

engineer

for the

Atlantic Refining Co. in Dallas,

for

the

development of the derivation of the basic formula by

series.

REFERENCES

1.

Arps, J. J.: Analysis

of

Decline Curves,

Trans.

AIME

(1945) 160, 228.

2. Johnson, R. H. and Bollens, A. L.: "The Loss-Ratio

Method of

Extrapolating Oil Well Decline Curves,

Trans. AIME

(1927) 77,771.

3. Marsh, H. N.: "Method of Appraising Results of

Production Control of Oil Wells,

PI

and Prod.

Eng. Bull. 202

(1928).

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Transcript of SPE-464-G