Tarek Ahmed. Reservoir engineering handbook
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Step 2. Solve for t
DBT
by applying Equation 9-58.
Step 3. Estimate the time to the gas breakthrough t
BT
by using the gas
and oil densities in the following expression:
where t
BT
= time to gas breakthrough as expressed in days
r
o
= oil density, lb/ft
3
r
g
= gas density, lb/ft
3
Water and gas coning
For the two-cone case, the authors developed two graphical correla-
tions for determining the time to breakthrough and optimum placement
of the horizontal well. The proposed method is summarized below:
Step 1. Calculate the gas coning dimensionless flow rate by applying
Equation 9-60.
Step 2. Calculate the density difference ratio.
Step 3. Solve for the dimensionless breakthrough time by using Figure
9-25 or applying the following polynomial:
where U = ln (q
D
)
The coefficients c
0
- c
3
are tabulated in Table 9-4.
Step 4. Solve for the time to breakthrough by applying the gas-coning
Equation 9-61.
Step 5. Solve for the optimum placement of the horizontal above the
WOC by applying the following expression:
b
opt
opt
D
=hb (9 - 64)
l(
t
)=
cc
U
c
U
c
U
DBT 0 1 2
2
3
3
n ++ + (9 - 63)
y
rr
rr
=
wo
og
-
-
(9 - 62)
t
ht
k
BT
o DBT
vo g
=
-
22 758 528,.
()
fm
rr
(9 - 61)
628 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 628
where D
b
opt
= optimum distance above the WOC, ft
h = oil thickness, ft
b
opt
= optimum fractional well placement
The fractional well placement b
opt
is determined from Figure 9-26 or
the following relationship:
The coefficients of the above polynomial are given in Table 9-5.
Example 9-20
Resolve Example 9-18 by using Papatzacos’ method.
Solution
Step 1. Solve for the dimensionless flow rate by using Equation 9-57.
D
q
=
20,333.66 0.73 1.1 1000
1640 50 (63.76 47.5) 60 15
= 0.408
¥¥¥
¥- ¥
[]
opt
01 2
2
3
3
=
cc
U
c
U
c
U
b+++ (9 - 65)
Gas and Water Coning 629
Figure 9-25. Dimensionless time for two-cone case. (After Paptzacos, P. et. al.,
courtesy SPE, 1989.)
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 629
Table 9-4
Coefficients for Breakthrough Time, t
DBT
(Equation 4-64)
(After Papatzacos, P. et al., SPE Paper 19822, 1989)
y C
0
C
1
C
2
C
3
0.2 -2.9494 -0.94654 -0.0028369 -0.029879
0.4 -2.9473 -0.93007 0.016244 -0.049687
0.6 -2.9484 -0.9805 0.050875 -0.046258
0.8 -2.9447 -1.0332 0.075238 -0.038897
1.0 -2.9351 -1.0678 0.088277 -0.034931
1.2 -2.9218 -1.0718 0.091371 -0.040743
1.4 -2.9162 -1.0716 0.093986 -0.042933
1.6 -2.9017 -1.0731 0.094943 -0.048212
1.8 -2.8917 -1.0856 0.096654 -0.046621
2.0 -2.8826 -1.1103 0.10094 -0.040963
630 Reservoir Engineering Handbook
Figure 9-26. Optimum well placement for two-cone case. (After Paptzacos, P. et.
al., courtesy SPE, 1989.)
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 630
Table 9-5
Coefficients for Optimum Well Placement (Equation 4-66)
(After Papatzacos, P. et al., SPE Paper 19822, 1989)
y C
0
C
1
C
2
C
3
0.2 0.507 -0.0126 0.01055 -0.002483
0.4 0.504 -0.0159 0.01015 -0.000096
0.6 0.503 -0.0095 0.00624 -0.000424
0.8 0.502 -0.0048 0.00292 -0.000148
1.0 0.500 -0.0001 0.00004 0.000009
1.2 0.497 0.0042 -0.00260 0.000384
1.4 0.495 0.0116 -0.00557 -0.000405
1.6 0.493 0.0178 -0.00811 -0.000921
1.8 0.490 0.0231 -0.01020 -0.001242
2.0 0.488 0.0277 -0.01189 -0.001467
Step 2. Calculate the dimensionless breakthrough time from Equation 9-58.
Step 3. Estimate the time to breakthrough from Equation 9-59
Example 9-21
A 1,640-foot-long horizontal well is drilled in an oil reservoir with
developing gas and water cones. The following data are available:
h = 50 ft k
h
= 60 md k
v
= 15 md B
o
= 1.1 bbl/STB
m
o
= 0.73 cp r
w
= 0.3 ft r
o
= 47.5 lb/ft
3
r
w
= 63.76 lb/ft
3
r
g
= 9.1 lb/ft
3
f=15% Q
o
= 1000 STB/day
Determine the optimum well placement and calculate the correspond-
ing breakthrough time.
BT
t
=
22,758.528 50 .15 0.73 0.6191
15 (63.74 47.5)
= 316 days
¥¥ ¥ ¥
-
[]
DBT
t
= 1 (3 0.408 1) l
3 0.408
3 0.408 1
= 0.6191-¥ -
¥
¥-
È
Î
Í
˘
˚
˙
n
Gas and Water Coning 631
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 631
Solution
Step 1. Calculate the dimensionless flow rate from Equation 9-60.
Step 2. Calculate the density difference ratio from Equation 9-62.
Step 3. Read the fraction well placement b
opt
from Figure 9-26 by using
the calculated values of y and q
D
to give:
B
opt
@ 0.565
Step 4. Calculate the optimum well placement above the WOC from
Equation 9-64.
D
b
opt
= (0.565) (50) = 28.25 ft
Step 5. From Figure 9-25, for q
D
= 0.1728 and y=0.4234, find the
dimensionless breakthrough time t
DBT
:
Ln (t
DBT
) =-.8 (from Figure 9-25)
t
DBT
= 0.449
Step 6. Estimate the time to breakthrough by applying Equation 9-61.
t
BT
= 22,758.528 ¥ 50 ¥ 0.15 ¥ 0.73 ¥ 0.449/[15 (47.3 - 9.1)]
= 97.71 days
PROBLEMS
1. In an oil-water system, the following fluid and rock data are available:
h = 60¢ h
p
= 25¢r
o
= 47.5 lb/ft
3
r
w
= 63.76 lb/ft
3
m
o
= 0.85 cp B
o
= 1.2 bbl/STB r
e
= 660 r
w
= 0.25¢
k
o
= k = 90.0 md
y =
63.76 47.5
47.5 9.1
= 0.4234
-
-
D
q
=
20,333.66 0.73 1.1 1000
1640 50 (47.5 9.1) 60 15
= 0.1728
¥¥¥
¥- ¥
[]
632 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 632
Calculate the critical oil flow rate, by using the following methods:
• Meyer-Garder
• Chierici-Ciucci
• Hoyland-Papatzacos-Skjaeveland
• Chaney
• Chaperson
• Schols
2. Given:
Q
o
= 400 STB/day h = 60 ft h
p
= 25 ft r
w
= 63.76 lb/ft
3
r
o
= 47.5 lb/ft
3
m
o
= 0.85 cp B
o
= 1.2 bbl/STB
k
v
= 9 md k
h
= 90 md f=15% M = 3.5
Calculate the water breakthrough time by using the:
a. Sobocinski-Cornelius method
b. Bournazel-Jeanson correlation
3. The rock, fluid, and the related reservoir properties of a bottom-water
drive reservoir are given below:
well spacing = 80 acres
initial oil column thickness = 100 ft
h
p
= 40¢r
o
= 48 lb/ft
3
r
w
= 63 lb/ft
3
r
e
= 660¢
r
w
= 0.25¢ M = 3.0 f=14% S
or
= 0.25
S
wc
= 0.25 B
o
= 1.2 bbl/STB m
o
= 2.6 cp m
w
= 1.00 cp
k
h
= 80 md k
v
= 16 md
Calculate the water-cut behavior of a vertical well in the reservoir
assuming a total production rate of 500, 1000, and 1500 STB/day.
4. A 2,000-ft-long horizontal well is drilled in the top elevation of the
pay zone in a water-drive reservoir. The following data are available:
h = 50 ft k
h
= 80 md k
v
= 25 md B
o
= 1.2 bbl/STB
m
o
= 2.70 cp r
w
= 0.3 ft D
b
= 50 ft r
o
= 48.5 lb/ft
3
r
w
= 62.50 lb/ft
3
y
e
= 1320 ft
Gas and Water Coning 633
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 633
Calculate the critical flow rate by using:
a. Chaperson’s method
b. Efros’ correlation
c. Karcher’s equation
d. Joshi’s method
5. A 2,000-foot-long horizontal well is producing at 1500 STB/day. The
following data are available:
h = 60 ft k
h
= 80 md k
v
= 15 md B
o
= 1.2 bbl/STB
m
o
= 2.70 cp r
w
= 0.3 ft r
o
= 47.5 lb/ft
3
r
w
= 63.76 lb/ft
3
z
wD
= 1 f=15% S
wc
= 0.25 S
or
= 0.25
Calculate the time to breakthrough by using the:
a. Ozkan-Raghavan method
b. Papatzacos’ method
REFERENCES
1. Bournazel, C., and Jeanson, B., “Fast Water Coning Evaluation,” SPE Paper
3628 presented at the SPE 46th Annual Fall Meeting, New Orleans, Oct.
3–6, 1971.
2. Calhoun, John, Fundamentals of Reservoir Engineering. Norman, OK: The
University of Oklahoma Press, 1960.
3. Chaney, P. E. et al., “How to Perforate Your Well to Prevent Water and Gas
Coning,” OGJ, May 1956, p. 108.
4. Chaperson, I., “Theoretical Study of Coning Toward Horizontal and Vertical
Wells in Anisotrophic Formations: Subcritical and Critical Rates,” SPE Paper
15377, SPE 61st Annual Fall Meeting, New Orleans, LA, Oct. 5–8, 1986.
5. Chierici, G. L., Ciucci, G. M., and Pizzi, G., “A Systematic Study of Gas
and Water Coning by Potentiometric Models,” JPT, Aug. 1964, p. 923.
6. Efros, D. A., “Study of Multiphase Flows in Porous Media” (in Russian),
Gastoptexizdat, Leningrad, 1963.
7. Hoyland, L. A., Papatzacos, P., and Skjaeveland, S. M., “Critical Rate for Water
Coning: Correlation and Analytical Solution,” SPERE, Nov. 1989, p. 495.
634 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 634
8. Joshi, S. D., “Augmentation of Well Productivity Using Slant and Horizontal
Wells,” J. of Petroleum Technology, June 1988, pp. 729–739.
9. Joshi, S., Horizontal Well Technology. Tulsa, OK: Pennwell Publishing
Company, 1991.
10. Karcher, B., Giger, F., and Combe, J., “Some Practical Formulas to Predict
Horizontal Well Behavior,” SPE Paper 15430, presented at the SPE 61st
Annual Conference, New Orleans, Oct. 5–8, 1986.
11. Kuo, C. T., and Desbrisay, C. L., “A Simplified Method for Water Coning
Predictions,” SPE Paper 12067, presented at the Annual SPE Technical Con-
ference, San Francisco, Oct. 5–8, 1983.
12. Meyer, H. I., and Garder, A. O., “Mechanics of Two Immiscible Fluids in
Porous Media,” J. Applied Phys., Nov. 1954, No. 11, p. 25.
13. Muskat, M., and Wyckoff, R. D., “An Approximate Theory of Water Coning
in Oil Production,” Trans. AIME, 1953, pp. 114, 144.
14. Ozkan, E., and Raghavan, R., “Performance of Horizontal Wells Subject to
Bottom Water Drive,” SPE Paper 18545, presented at the SPE Eastern
Regional Meeting, Charleston, West Virginia, Nov. 2–4, 1988.
15. Papatzacos, P., Herring, T. U., Martinsen, R., and Skjaeveland, S. M., “Cone
Breakthrough Time for Horizontal Wells,” SPE Paper 19822, presented at
the 64th SPE Annual Conference and Exhibition, San Antonio, TX, Oct.
8–11, 1989.
16. Pirson, S. J., Oil Reservoir Engineering. Huntington, NY: Robert E. Krieger
Publishing Company, 1977.
17. Schols, R. S., “An Empirical Formula for the Critical Oil Production Rate,”
Erdoel Erdgas, A., January 1972, Vol. 88, No. 1, pp. 6–11.
18. Slider, H. C., Practical Petroleum Reservoir Engineering Methods. Tulsa,
OK: Petroleum Publishing Company, 1976.
19. Sobocinski, D. P., and Cornelius, A. J., “A Correlation for Predicting Water
Coning Time,” JPT, May 1965, pp. 594–600.
Gas and Water Coning 635
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 635
636
Nearly all hydrocarbon reservoirs are surrounded by water-bearing
rocks called aquifers. These aquifers may be substantially larger than the
oil or gas reservoirs they adjoin as to appear infinite in size, or they may
be so small in size as to be negligible in their effect on reservoir perfor-
mance.
As reservoir fluids are produced and reservoir pressure declines, a
pressure differential develops from the surrounding aquifer into the reser-
voir. Following the basic law of fluid flow in porous media, the aquifer
reacts by encroaching across the original hydrocarbon-water contact. In
some cases, water encroachment occurs due to hydrodynamic conditions
and recharge of the formation by surface waters at an outcrop.
In many cases, the pore volume of the aquifer is not significantly larg-
er than the pore volume of the reservoir itself. Thus, the expansion of the
water in the aquifer is negligible relative to the overall energy system,
and the reservoir behaves volumetrically. In this case, the effects of water
influx can be ignored. In other cases, the aquifer permeability may be
sufficiently low such that a very large pressure differential is required
before an appreciable amount of water can encroach into the reservoir. In
this instance, the effects of water influx can be ignored as well.
This chapter focuses on those those reservoir-aquifer systems in which
the size of the aquifer is large enough and the permeability of the rock is
high enough that water influx occurs as the reservoir is depleted. This
chapter also provides various water influx calculation models and a
detailed description of the computational steps involved in applying
these models.
CHAPTER 10
WATER INFLUX
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 636
CLASSIFICATION OF AQUIFERS
Many gas and oil reservoirs produced by a mechanism termed water
drive. Often this is called natural water drive to distinguish it from artifi-
cial water drive that involves the injection of water into the formation.
Hydrocarbon production from the reservoir and the subsequent pressure
drop prompt a response from the aquifer to offset the pressure decline.
This response comes in a form of water influx, commonly called water
encroachment, which is attributed to:
• Expansion of the water in the aquifer
• Compressibility of the aquifer rock
• Artesian flow where the water-bearing formation outcrop is located
structurally higher than the pay zone
Reservoir-aquifer systems are commonly classified on the basis of:
• Degree of pressure maintenance • Flow regimes
• Outer boundary conditions • Flow geometries
Degree of Pressure Maintenance
Based on the degree of the reservoir pressure maintenance provided by
the aquifer, the natural water drive is often qualitatively described as:
• Active water drive
• Partial water drive
• Limited water drive
The term active water drive refers to the water encroachment mecha-
nism in which the rate of water influx equals the reservoir total produc-
tion rate. Active water-drive reservoirs are typically characterized by a
gradual and slow reservoir pressure decline. If, during any long period,
the production rate and reservoir pressure remain reasonably constant,
the reservoir voidage rate must be equal to the water influx rate.
or
e
w
= Q
o
B
o
+ Q
g
B
g
+ Q
w
B
w
(10-1)
water lux
rate
oil flow
rate
free gas
flow rate
water production
rate
inf
=
+
+
Water Influx 637
Reservoir Eng Hndbk Ch 10 2001-10-25 14:22 Page 637