Decline Curves Decline Curves that plot flow rate vs. time are the most common tools for forecasting production and monitoring well performance in the field. These curves quickly show by graphic means which wells or fields are producing as expected or under producing. Mainly used because they are easy to set up and to use in the field. They are not based on any of the physics of the flow of oil and gas through the rock formations, empirical in nature. The most common forms are daily flow rates vs. the month. Water and gas rates are commonly plotted along with the oil rate, or GOR and WOR. Cumulative production vs. the months is also very common, both oil and water can be plotted. These plots are plotted both on linear plots and semi-log plots with the q on the log scale.

Exponential decline (constant fractional decline)

Harmonic decline, and

Hyperbolic decline.

dbqdt

dq

q

1

b & d empirical constants based on production data

d= 0 exponential decline d=1 harmonic decline

0<d<1 hyperbolic decline

Exponential Decline

The relative decline rate and production rate decline equations for the exponential

decline model can be derived from volumetric reservoir model. Cumulative

production expression is obtained by integrating the production rate decline

equation.

Relative Decline Rate

Consider an oil well drilled in a volumetric oil reservoir. Suppose the well’s

production rate starts to decline when a critical (lowest permissible) bottom hole

pressure is reached. Under the pseudo-steady state flow condition, the production

rate at a given decline time t can be expressed as:

sr

rB

ppkhq

w

e

c

t wf

472.0ln2.141

)(

0

(2)

where tp = average reservoir pressure at decline time t, c

wfp = the critical bottom hole pressure maintained during the production

decline.

The cumulative oil production of the well after the production decline time t can be

expressed as:

dt

sr

rB

ppkhN

t

w

eo

c

t

p

wf

0 472.0ln2.141

)(

(3)

The cumulative oil production after the production decline upon decline time t can

also be evaluated based on the total reservoir compressibility:

t

o

itp pp

B

NcN 0 (4)

where tc = total reservoir compressibility,

iN = initial oil in place in the well drainage area,

0p = average reservoir pressure at decline time zero.

Substituting Eq (3) into Eq (4) yields:

t

o

it

t

w

eo

c

tpp

B

Ncdt

sr

rB

ppkhwf

0

0 472.0ln2.141

)(

(5)

Taking derivative on both sides of this equation with respect to time t gives the

differential equation for reservoir pressure:

dt

pdNc

sr

r

ppkht

it

w

e

c

t wf

472.0ln2.141

)(

(6)

Since the left-hand-side of this equation is q and Eq (2) gives

dt

pd

sr

rB

kh

dt

dq t

w

e

472.0ln2.141 0

(7)

Eq (6) becomes

dt

dq

kh

sr

rNc

qw

eit

472.0ln2.141

(8)

or the relative decline rate equation of

bdt

dq

q

1 (9)

where

sr

rNc

khb

w

eit

472.0ln2.141

. (10)

Production Rate Decline

Equation (6) can be expressed as:

dt

pdppb tc

t wf )( (11)

By separation of variables, Eq (8.11) can be integrated

t

wf

p

p

c

t

t

t

pp

pddtb

0)(

0

(12)

to yield an equation for reservoir pressure decline:

btcc

t eppppwfwf

0 (13)

Substituting Eq (13) into Eq (2) gives well production rate decline equation:

bt

w

eo

c

e

sr

rB

ppkhq

wf

472.0ln2.141

)( 0

(14)

or

btc

o

it eppB

Nbcq

wf

)( 0 (15)

which is the exponential decline model commonly used for production decline

analysis of solution-gas-drive reservoirs. In practice, the following form of Eq (15)

is used:

bt

ieqq (16)

where qi is the production rate at t = 0.

It can be shown that b

n

n eq

q

q

q

q

q

12

3

1

2 ...... . That is, the fractional decline is

constant for exponential decline.

Cumulative Production

Integration of Eq (16) over time gives an expression for the cumulative oil

production since decline of

t

bt

i

t

p dteqqdtN00

(17)

i.e.,

btip e

b

qN 1 . (18)

Since bt

ieqq , Eq (18) becomes

qqb

N ip 1

. (19)

8 Determination of Decline Rate

The constant b is called the continuous decline rate. Its value can be determined

from production history data. If production rate and time data are available, the b-

value can be obtained based on the slope of the straight line on a semi-log plot. In

fact, taking logarithm of Eq (8.16) gives:

btqq i lnln (20)

which implies that the data should form a straight line with a slope of -b on the

log(q) versus t plot, if exponential decline is the right model. Picking up any two

points, (t1, q1) and (t2, q2), on the straight line will allow analytical determination of

b-value because

11 lnln btqq i (21)

and

22 lnln btqq i (22)

give

2

1

12

ln1

q

q

ttb . (23)

If production rate and cumulative production data are available, the b-value can be

obtained based on the slope of the straight line on an Np versus q plot. In fact,

rearranging Eq (19) yields:

pi bNqq (24)

Picking up any two points, (Np1, q1) and (Np2, q2), on the straight line will allow

analytical determination of b-value because

11 pi bNqq (25)

and

22 pi bNqq (26)

give

12

21

pp NN

qqb

. (27)

Depending on the unit of time t, the b can have different units such as month-1

and

year-1

. The following relation can be derived:

dma bbb 36512 . (28)

where ba, bm, and bd are annual, monthly, and daily decline rates.

Effective Decline Rate

Because the exponential function is not easy to use in hand calculations,

traditionally the effective decline rate has been used. Since xe x 1 for small x-

values based on Taylor’s expansion, be b 1 holds true for small values of b.

The b is substituted by 'b , the effective decline rate, in field applications. Thus Eq

(16) becomes

ti bqq '1 (29)

Again, it can be shown that '1......12

3

1

2 bq

q

q

q

q

q

n

n

.

Depending on the unit of time t, the 'b can have different units such as month-1

and

year-1

. The following relation can be derived:

36512'1'1'1 dma bbb . (30)

where 'b a, 'b m, and 'b d are annual, monthly, and daily effective decline rates.

Example Problem

Given that a well has declined from 100 stb/day to 96 stb/day during a one-month

period, use the exponential decline model to perform the following tasts:

a) Predict the production rate after 11 more months

b) Calculate the amount of oil produced during the first year

c) Project the yearly production for the well for the next 5 years.

Solution:

a) Production rate after 11 more months:

m

m

mm

mq

q

ttb

1

0

01

ln1

/month04082.0

96

100ln

1

1

Rate at end of one year

stb/day 27.61100 1204082.0

01 eeqq

tb

mmm

If the effective decline rate b’ is used,

/month04.0100

96100'

0

10

m

mmm

q

qqb .

From

/year3875.0'

gets one

04.01'1'11212

y

my

b

bb

Rate at end of one year

stb/day 27.613875.01100'101 ybqq

b) The amount of oil produced during the first year:

/year48986.01204082.0 yb

stb 858,2836548986.0

27.61100101,

y

pb

qqN

or

stb 858,281001342.0

100

day

1001342.0

42.30

1

96

100ln

365001342.0

1,

eN

b

p

d

c) Yearly production for the next 5 years:

stb681,171001342.0

27.61 365001342.0

2, eN p

stb/day 54.37100 )2(1204082.0

2 eeqq bt

i

stb834,101001342.0

54.37 365001342.0

3, eN p

stb/day 00.23100 )3(1204082.0

3 eeqq bt

i

stb639,61001342.0

00.23 365001342.0

4, eN p

stb/day 09.14100 )4(1204082.0

4 eeqq bt

i

stb061,41001342.0

09.14 365001342.0

5, eN p

In summary,

Year Rate at End of

Year (stb/day)

Yearly

Production (stb)

0

1

2

3

4

5

100.00

61.27

37.54

23.00

14.09

8.64

-

28,858

17,681

10,834

6,639

4,061

68,073

Go do,

Celtic #3

Acme Southeast San Andres, Ponderosa Petroleum