AHMED, T.H - Reservoir Engineering Handbook 4ed-727-745
AHMED, T.H - Reservoir Engineering Handbook 4ed-727-745
AHMED, T.H - Reservoir Engineering Handbook 4ed-727-745
( We )n = ( We )n −1 + [(t D )n − (t D )n −1 ]
⎡ B Δp n − ( We )n −1 (p′D )n ⎤
⎢⎣ (p D )n − (t D )n −1 (p′D )n ⎥⎦ (10-34)
Example 10-9
Rework Example 10-7 by using the Carter-Tracy method.
Solution
Example 10-7 shows the following preliminary results:
Step 1. For each time step n, calculate the total pressure drop Δpn = pi − pn
and the corresponding tD
N t, days pn Δpn tD
0 0 2500 0 0
1 182.5 2490 10 180.5
2 365.0 2472 28 361.0
3 547.5 2444 56 541.5
4 730.0 2408 92 722.0
Step 2. Since values of tD are greater than 100, use Equation 6-92 to cal-
culate pD and its derivative p′D, i.e.,
pD = 0.5 [ln (tD) + 0.80907]
p′D = 1/(2 tD)
720 Reservoir Engineering Handbook
N t tD pD pD′
0 0 0 — —
1 182.5 180.5 3.002 2.770 × 10−3
2 365 361.0 3.349 1.385 × 10−3
3 547.5 541.5 3.552 0.923 × 10−3
4 730.0 722.0 3.696 0.693 × 10−3
−3
⎡ (20.4) (10) − (0) (2.77 × 10 ) ⎤
We = 0 + [180.5 − 0] ⎢
⎣ 3.002 − (0) (2.77 × 10 −3 ) ⎥⎦
We = 12,266 bbl
= 42,546 bbl
We = 104,406
We = 202,477 bbl
Water Influx 721
0 0 0
6 12,266 7080
12 42,546 32,435
18 104,400 85,277
24 202,477 175,522
Fetkovich’s Method
Equation 10-37 suggests that the maximum possible water influx occurs
if pa = 0, or:
Wei = ct Wi pi f (10-39)
Combining Equation 10-38 with 10-37 gives:
⎛ We ⎞
= pi ⎛1 − e ⎞
W
pa = p i ⎜ 1 − ⎟ (10-40)
⎝ c t Wi p i ⎠ ⎝ Wei ⎠
dWe W d pa
= − ei (10-41)
dt p i dt
Fetkovich combined Equation 10-40 with 10-36 and integrated to give
the following form:
−J pi t ⎞
(p i − p r ) exp ⎛
Wei
We = (10-42)
pi ⎝ Wei ⎠
Jp Δt
[( pa )n−1 − ( pr )n ] ⎡⎢1 − exp ⎛ − i n ⎞ ⎤⎥
Wei
(ΔWe )n = (10-43))
pi ⎣ ⎝ Wei ⎠ ⎦
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:
( pa )n −1 = p i ⎛ 1 −
( We )n −1 ⎞
(10-44)
⎝ Wei ⎠
– ) is estimated from:
The average reservoir boundary pressure (pr n
(Pr )n + (p r )n −1
( pr )n = (10-45)
2
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 be approximately constant throughout the productive life of the
reservoir. The productivity index J of the aquifer is given by the follow-
ing expressions.
a = 0.0142 kt / ( fm c t )
Wei = ct Wi pi f
Step 4. Calculate the incremental water influx (ΔWe)n from the aquifer
during the nth time interval by using Equation 10-42. For exam-
ple, during the first time interval Δt1:
Wei ⎡ ⎛ − Jp i Δt1 ⎞ ⎤
( ΔWe )1 = [ p i − ( p r )1 ] ⎢1 − exp ⎜
⎝ Wei ⎠ ⎥⎦
⎟⎥
pi ⎢⎣
with
p i + ( p r )1
( p r )1 =
2
For the second time interval Δt2
Wei ⎡ ⎛ − Jp i Δt 2 ⎞ ⎤
( ΔWe )2 = [( pa )1 − ( p r )2 ] ⎢1 − exp ⎜
⎝ Wei ⎠ ⎥⎦
⎟⎥
pi ⎢⎣
726 Reservoir Engineering Handbook
where (p– ) is the average aquifer pressure at the end of the first
a 1
period and removing (ΔWe)1 barrels of water from the aquifer to
the reservoir. From Equation 10-43:
⎛ ( ΔWe )1 ⎞
( pa )1 = p i ⎜1 − ⎟
⎝ Wei ⎠
Step 5. Calculate the cumulative (total) water influx at the end of any
time period from:
n
We = ∑ ( ΔWe )i
t =1
Example 10-102
0 2740
365 2500
730 2290
1095 2109
1460 1949
Solution
re = ⎛
140 ⎞ (2374) (43, 560)
= 9200 ft
⎝ 360 ⎠ π
ra = ⎛
140 ⎞ (1, 000, 000) (43, 560)
= 46, 000 ft
⎝ 360 ⎠ π
Step 3. Calculate the dimensionless radius rD.
rD = ra / re
Wei = ct Wi pi f
Step 6. Calculate the productivity index J of the radial aquifer from Equa-
tion 10-45.
J=
0.00708 (200) (100) ( )
140
360 = 116.5 bbl/day/psi
0.55 ln (5)
211.9 × 106
(ΔWe )n = [( pa )n −1 − ( pr )n ] ( 0.4229 )
2740
t (ΔWe)n (We)
n days pr –)
(p ( –pa )n − 1 ( –pa )n − 1 − (p–r )n MM bbl MM bbl
r n
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
0 4000
120 3950
220 3910
320 3880
420 3840
Boundary pressure,
Time, months psi
0 2610
6 2600
12 2580
18 2552
24 2515
Reservoir Aquifer
radius, ft 2000 ∞
h, ft 25 30
k, md 60 80
φ, % 17 18
μw, cp 0.55 0.85
cw, psi−1 0.7 × 10−6 0.8 × 10−6
cf, psi−1 0.2 × 10−6 0.3 × 10−6
730 Reservoir Engineering Handbook
5. The following table summarizes the original data available on the West
Texas water-drive reservoir:
The aquifer geological data estimate the water influx constant at 551
bbl/psi. After 1,120 days of production, the reservoir average pressure
has dropped to 3,800 psi and the field has produced 860,000 STB of
oil. The field condition after 1,120 days of production is given below:
p = 3800 psi
Np = 860,000 STB
Bo = 1.34 bbl/STB
Bw = 1.05 bbl/STB
We = 991,000 bbl
tD = 32.99 (dimensionless time after 1120 days)
Wp = 0 bbl
It is expected that the average reservoir pressure will drop to 3,400 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:
Water Influx 731
0 2850
365 2610
730 2400
1095 2220
1460 2060
Given:
h = 120′ cf = 5 × 10−6 psi−1 cw = 4 ×10−6 psi−1
μw = 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.
732 Reservoir Engineering Handbook
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.
C H A P T E R 1 1
OIL RECOVERY
MECHANISMS AND THE
MATERIAL BALANCE
EQUATION
733
734 Reservoir Engineering Handbook
2. Provide the basic principles of the material balance equation and other
governing relationships that can be used to predict the volumetric per-
formance of oil reservoirs.
Both of the above two factors are the results of a decrease of fluid
pressure within the pore spaces, and both tend to reduce the pore volume
through the reduction of the porosity.
As the expansion of the fluids and reduction in the pore volume occur
with decreasing reservoir pressure, the crude oil and water will be forced
Oil Recovery Mechanisms and the Material Balance Equation 735
out of the pore space to the wellbore. Because liquids and rocks are only
slightly compressible, the reservoir will experience a rapid pressure
decline. The oil reservoir under this driving mechanism is characterized
by a constant gas-oil ratio that is equal to the gas solubility at the bubble
point pressure.
This driving mechanism is considered the least efficient driving force
and usually results in the recovery of only a small percentage of the total
oil-in-place.
gas will also begin a vertical movement due to the gravitational forces,
which may result in the formation of a secondary gas cap. Vertical per-
meability is an important factor in the formation of a secondary gas cap.
• Ultimate oil-recovery: Oil production by depletion drive is usually
the least efficient recovery method. This is a direct result of the forma-
tion of gas saturation throughout the reservoir. Ultimate oil recovery
from depletion-drive reservoirs may vary from less than 5% to about
30%. The low recovery from this type of reservoirs suggests that large
quantities of oil remain in the reservoir and, therefore, depletion-drive
reservoirs are considered the best candidates for secondary recovery
applications.