Tarek Ahmed. Reservoir engineering handbook
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Equation 10-37 suggests that the maximum possible water influx occurs
if p
a
= 0, or:
W
ei
= c
t
W
i
p
i
f (10-38)
Combining Equation 10-38 with 10-37 gives:
Equation 10-37 provides a simple expression to determine the average
aquifer pressure p
–
a
after removing W
e
bbl of water from the aquifer to the
reservoir, i.e., cumulative water influx.
Differentiating Equation 10-39 with respect to time gives:
Fetkovich combined Equation 10-40 with 10-36 and integrated to give
the following form:
where W
e
= cumulative water influx, bbl
p
r
= reservoir pressure, i.e. pressure at the oil- or gas-water
contact
t = time, days
Equation 10-41 has no practical applications since it was derived for a
constant inner boundary pressure. To use this solution in the case in
which the boundary pressure is varying continuously as a function of
time, the superposition technique must be applied. Rather than using
superposition, Fetkovich suggested that, if the reservoir-aquifer boundary
pressure history is divided into a finite number of time intervals, the
incremental water influx during the n
th
interval is:
( ) ( ) ( ) exp∆
∆
W
W
p
pp
Jp t
W
en
ei
i
an rn
in
ei
=−
[]
−−
−1
1 (10 - 42)
W
W
p
pp
Jp t
W
e
ei
i
ir
i
ei
=−
−
( ) exp (10 - 41)
dW
dt
W
p
dp
dt
eei
i
a
=− (10 - 40)
pp
W
cWp
p
W
W
ai
e
tii
i
e
ei
=−
=−
1 1 (10 - 39)
708 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 708
where (p
–
a
)
n − 1
is the average aquifer pressure at the end of the previous
time step. This average pressure is calculated from Equation 10-39 as:
The average reservoir boundary pressure (p
–
r
)
n
is estimated from:
The productivity index J used in the calculation is a function of the
geometry of the aquifer. Fetkovich calculated the productivity index
from Darcy’s equation for bounded aquifers. Lee and Wattenbarger
(1996) pointed out that Fetkovich’s method can be extended to infinite-
acting aquifers by requiring that the ratio of water influx rate to pressure
drop to be approximately constant throughout the productive life of the
reservoir. The productivity index J of the aquifer is given by the follow-
ing expressions.
Type of Outer J for Radial J for Linear
Aquifer Boundary flow, bbl/day/psi Flow, bbl/day/psi Equation #
Finite, no flow (10-45)
Finite, constant pressure (10-46)
Infinite (10-47)
where w = width of the linear aquifer
L = length of the linear aquifer
r
D
= dimensionless radius, r
a
/r
e
k = permeability of the aquifer, md
t = time, days
θ=encroachment angle
h = thickness of the aquifer
f =θ/360
J
0.001 k wh
w
0.0633 kt/ (f
w
c
t
)
=
µµ
J
0.00708 kh f
w
ln (a / r
e
)
a
0.0142 kt / (f
w
c
t
)
=
=
µ
µ
J
0.001127 k wh
L
=
µ
w
J
0.00708 kh f
[ln (r )]
D
=
µ
w
J
0.003381 kwh
L
w
=
µ
J
0.00708 kh f
[ln r 0.75]
wD
=
−µ
()
() ()
p
Pp
rn
rn rn
=
+
−1
2
(10 - 44)
()
()
pp
W
W
an i
en
ei
−
−
=−
1
1
1 (10 - 43)
Water Influx 709
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 709
710 Reservoir Engineering Handbook
The following steps describe the methodology of using the Fetkovich’s
model in predicting the cumulative water influx.
Step 1. Calculate initial volume of water in the aquifer from:
Step 2. Calculate the maximum possible water influx W
ei
by applying
Equation 10-38, or:
W
ei
= c
t
W
i
p
i
f
Step 3. Calculate the productivity index J based on the boundary condi-
tions and aquifer geometry.
Step 4. Calculate the incremental water influx (∆W
e
)
n
from the aquifer
during the n
th
time interval by using Equation 10-42. For exam-
ple, during the first time interval ∆t
1
:
with
For the second time interval ∆t
2
where (p
–
a
)
1
is the average aquifer pressure at the end of the first
period and removing (∆W
e
)
1
barrels of water from the aquifer to
the reservoir. From Equation 10-43:
()
()
pp
W
W
ai
e
ei
1
1
1=−
∆
( ) ( ) ( ) exp∆
∆
W
W
p
pp
Jp t
W
e
ei
i
ar
i
ei
212
2
1=−
[]
−
−
()
()
p
pp
r
ir
1
1
2
=
+
( ) ( ) exp∆
∆
W
W
p
pp
Jp t
W
e
ei
i
ir
i
ei
11
1
1=−
[]
−
−
Wrrh
iae
=−
π
φ
5 615
22
.
()
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 710
Water Influx 711
Step 5. Calculate the cumulative (total) water influx at the end of any
time period from:
Example 10-10
2
Using Fetkovich’s method, calculate the water influx as a function of
time for the following reservoir-aquifer and boundary pressure data:
p
i
= 2740 psi h = 100′ c
t
= 7 × 10
−6
psi
µ
w
= 0.55 cp k = 200 md θ=140°
reservoir area = 40,363 acres aquifer area = 1,000,000 acres.
Time, days p
r
, psi
0 2740
365 2500
730 2290
1095 2109
1460 1949
Figure 10-17 shows the wedge reservoir-aquifer system with an encroach-
ment angle of 140°.
Solution
Step 1. Calculate the reservoir radius r
e
:
Step 2. Calculate the equivalent aquifer radius r
a
:
rft
a
=
=
140
360
1 000 000 43 560
46 000
(, , )( , )
,
π
rft
e
=
=
140
360
2374 43 560
9200
()(,)
π
WW
eei
t
n
=
=
∑
()∆
1
2
Data of this example is given by L. P. Dake, Fundamentals of Reservoir Engineering,
Elsevier Publishing Company, 1978.
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 711
712 Reservoir Engineering Handbook
Figure 10-17. Aquifer-reservoir geometry for Example 10-10.
Step 3. Calculate the dimensionless radius r
D
.
Step 4. Calculate initial water in place W
i
.
Step 5. Calculate W
ei
from Equation 10-38.
W
ei
= c
t
W
i
p
i
f
Step 6. Calculate the productivity index J of the radial aquifer from Equa-
tion 10-45.
Therefore, Jp
i
/W
ei
= (116.5 × 2740)/(211.9 × 10
6
) = 1.506 × 10
−3
J
Ln
bbl day psi=
=
0 00708 200 100
140
360
055 5
116 5
.()()
.()
.//
W MM bbl
ei
=× ×
=
−
7 10 28 41 10 2740
140
360
211 9
69
(. )( ) .
Wrrh
MMM bbl
iae
=−
=
−
=
πφ
π
().
(, , )( )(.)
.
.
22
22
5 615
46 000 9 200 100 0 25
5 615
28 41
rrr
Dae
=
==
/
,/,46 000 9 200 5
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 712
Water Influx 713
Since the time step ∆t is fixed at 365 days, then
Equation 10-42 can be reduced to:
Step 7. Calculate cumulative water influx as shown in the following table:
t(∆W
e
)
n
(W
e
)
n days p
r
(p
–
r
)
n
(p
–
a
)
n − 1
(p
–
a
)
n − 1
− (p
–
r
)
n
MM bbl MM bbl
0 0 2740 2740 2740 0 0 0
1 365 2500 2620 2740 120 3.925 3.925
2 730 2290 2395 2689 294 9.615 13.540
3 1095 2109 2199 2565 366 11.970 25.510
4 1460 1949 2029 2409 381 12.461 37.971
PROBLEMS
1. Calculate the cumulative water influx that results from a pressure drop of
200 psi at the oil-water contact with an encroachment angle of 50°. The
reservoir-aquifer system is characterized by the following properties:
Reservoir Aquifer
radius, ft 6000 20,000
porosity 0.18 0.15
c
f
, psi
−1
4 × 10
−6
3 ×10
−6
c
w
, psi
−1
5 × 10
−6
4 × 10
−6
h, ft 25 20
2. An active water drive oil reservoir is producing under the steady-state
flowing conditions. The following data is available:
p
i
= 4000 psi Q
w
= 0R
s
= 500 scf/STB
Q
o
= 40,000 STB/day p = 3000 psi T = 140°F
GOR = 700 scf/STB B
o
= 1.3 bbl/STB B
w
= 1.0 bbl/STB
z = 0.82
Calculate Schilthuis’ water influx constant.
()
.
[( ) ( ) ]
() [() ()]
(. )
∆
∆
Wpp
Wpp
en an rn
en an rn
=
×
−
=−
−
−
211 9 10
2740
32705
6
1
0 4229
1
1 1 0 4229
1 506 10
3
365
−=− =
−
−×
−
×
ee
Jp
i
tW
ei
∆ /
.
.
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 713
714 Reservoir Engineering Handbook
3. The pressure history of a water-drive oil reservoir is given below:
t, days p, psi
0 4000
120 3950
220 3910
320 3880
420 3840
The aquifer is under a steady-state flowing condition with an estimat-
ed water influx constant of 80 bbl/day/psi. Using the steady-state model,
calculate and plot the cumulative water influx as a function of time.
4. A water-drive reservoir has the following boundary-pressure history:
Boundary, pressure,
Time, months psi
0 2610
6 2600
12 2580
18 2552
24 2515
The aquifer-reservoir system is characterized by the following data:
Reservoir Aquifer
radius, ft 2000 ∞
h, ft 25 30
k, md 60 80
φ, % 17 18
µ
w
, cp 0.55 0.85
c
w
, psi
−1
0.7 × 10
−6
0.8 × 10
−6
c
f
, psi
−1
0.2 × 10
−6
0.3 × 10
−6
If the encroachment angle is 360°, calculate the water influx as a
function of time by using:
a. The van Everdingen-Hurst Method
b. The Carter-Tracy Method
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 714
Water Influx 715
5. The following table summarizes the original data available on the West
Texas water-drive reservoir:
Oil Zone Aquifer
Geometry Circle Semi-circle
Area, acres 640 Infinite
Initial reservoir
pressure, psia 4000 4000
Initial oil saturation 0.80 0
Porosity, % 22 —
B
oi
, bbl/STB 1.36 —
B
wi
, bbl/STB 1.00 1.05
c
o
, psi
−1
6 × 10
−6
—
c
w
, psi
−1
3 × 10
−6
7 × 10
−6
The aquifer geological data estimates the water influx constant at 551
bbl/psi. After 1120 days of production, the reservoir average pressure has
dropped to 3800 psi and the field has produced 860,000 STB of oil. The
field condition after 1120 days of production is given below:
p = 3800 psi
N
p
= 860,000 STB
B
o
= 1.34 bbl/STB
B
w
= 1.05 bbl/STB
W
e
= 991,000 bbl
t
D
= 32.99 (dimensionless time after 1120 days)
W
p
= 0 bbl
It is expected that the average reservoir pressure will drop to 3400 psi
after 1,520 days (i.e., from the start of production). Calculate the
cumulative water influx after 1,520 days.
6. A wedge reservoir-aquifer system with an encroachment angle of 60°
has the following boundary pressure history:
Time, days Boundary pressure, psi
0 2850
365 2610
730 2400
1095 2220
1460 2060
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 715
Given:
h = 120′ c
f
= 5 × 10
−6
psi
−1
c
w
= 4 ×10
−6
psi
−1
µ
w
= 0.7 cp k = 60 md φ=12%
reservoir area = 40,000 acres aquifer area = 980,000 acres T = 140°F
Calculate the cumulative influx as a function of time by using
Fetkovich’s Method.
REFERENCES
1. Allard, D. R., and Chen, S. M., “Calculation of Water Influx for Bottom Water
Drive Reservoirs,” SPE Reservoir Engineering, May 1988, pp. 369–379.
2. Carter, R. D., and Tracy, G. W., “An Improved Method for Calculations
Water Influx,” Trans. AIME, 1960.
3. Chatas, A., “A Practical Treatment of Nonsteady-State Flow Problems in
Reservoir Systems,” Petroleum Engineering, May 1953, 25, No. 5, B-42;
No. 6, June, p. B-38; No. 8, August, p. B-44.
4. Coats, K., “A Mathematical Model for Water Movement about Bottom-Water-
Drive Reservoirs,” SPE Jour., March 1962, pp. 44–52; Trans. AIME, p. 225.
5. Craft, B., and Hawkins, M., Applied Reservoir Engineering. Prentice Hall,
1959.
6. Craft, B., Hawkins, M., and Terry, R., Applied Petroleum Reservoir Engi-
neering, 2nd ed. Prentice Hall, 1991.
7. Dake, L. P., Fundamentals of Reservoir Engineering. Amsterdam: Elsevier,
1978.
8. Dake, L., The Practice of Reservoir Engineering. Amsterdam: Elsevier, 1994.
9. Edwardson, M. et al., “Calculation of Formation Temperature Disturbances
Caused by Mud Circulation,” JPT, April 1962, pp. 416–425; Trans. AIME,
p. 225.
10. Fetkovich, M. J., “A Simplified Approach to Water Influx Calculations-
Finite Aquifer Systems,” JPT, July 1971, pp. 814–828.
11. Hurst, W., “Water Influx into a Reservoir and its Application to the Equation
of Volumetric Balance,” Trans. AIME, Vol. 151, pp. 57, 1643.
12. Lee, J., and Wattenbarger, R., Gas Reservoir Engineering. SPE Textbook
Series, Vol. 5, SPE, Dallas, TX, 1996.
13. Schilthuis, R., “Active Oil and Reservoir Energy,” Trans. AIME, 1936, pp.
37, 118.
14. Van Everdingen, A., and Hurst, W., “The Application of the Laplace Transfor-
mation to Flow Problems in Reservoirs,” Trans. AIME, 1949, pp. 186, 305.
716 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 10 2001-10-25 14:23 Page 716
Each reservoir is composed of a unique combination of geometric
form, geological rock properties, fluid characteristics, and primary drive
mechanism. Although no two reservoirs are identical in all aspects, they
can be grouped according to the primary recovery mechanism by which
they produce. It has been observed that each drive mechanism has certain
typical performance characteristics in terms of:
• Ultimate recovery factor
• Pressure decline rate
• Gas-oil ratio
• Water production
The recovery of oil by any of the natural drive mechanisms is called
primary recovery. The term refers to the production of hydrocarbons
from a reservoir without the use of any process (such as fluid injection)
to supplement the natural energy of the reservoir.
The two main objectives of this chapter are to:
1. Introduce and give a detailed discussion of the various primary recov-
ery mechanisms and their effects on the overall performance of oil
reservoirs.
717
CHAPTER 11
OIL RECOVERY
MECHANISMS AND THE
MATERIAL BALANCE
EQUATION
Reservoir Eng Hndbk Ch 11 2001-10-25 15:59 Page 717