Tarek Ahmed. Reservoir engineering handbook
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3. Calculate the cumulative oil production N
P
at each of the four
water–oil ratios, i.e., 1, 5, 25, and 100 bbl/bbl, from:
4. If the water–oil ratio is plotted against the oil recovery on semi-log
paper and a Cartesian scale, the oil recovery at breakthrough can be
found by extrapolating the line to a very low value of WOR, as shown
schematically in Figure 14-55.
Figure 14-55. WOR versus E
V
relationship.
5. For a constant injection rate, adding the fill-up volume W
if
to the
cumulative oil produced at breakthrough and dividing by the injection
rate can estimate the time to breakthrough.
6. The cumulative water produced at any given value of WOR is
obtained by finding the area under the curve of WOR versus N
P
, as
shown schematically in Figure 14-56.
7. The cumulative water injected at any given value of WOR is calculat-
ed by adding cumulative oil produced to the produced water and fill-
up volume, or:
WNBWBW
inj P o P w if
=+ +
0
5
10
15
WOR
20
25
0.4 0.5 0.6 0.7
Ev
0.8 0.9 1
(E
v
)
BT
NNR
PS
=
1008 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1008
Figure 14-56. Cumulative water production from WOR vs. N
P
curve.
Example 14-23
A reservoir is characterized by the following parameters:
Initial oil-in-place, N
S
= 12 MMSTB
Permeability variation, V = 0.8
Mobility ratio, M = 2.0
Initial water saturation, S
wi
= 0.25
Predict the cumulative oil production as a function of the producing
water–oil ratio.
Solution
Using Johnson’s graphical approach, perform the required calculations
in the following worksheet:
WOR Figure 14-54 R N
p
= N
S
R
1 R (1 – S
wi
) = 0.049 0.065 0.78 MMSTB
5 R (1 – 0.72 S
wi
) = 0.100 0.122 1.464 MMSTB
25 R (1 – 0.32 S
wi
) = 0.200 0.217 2.604 MMSTB
100 R (1 – 0.40 S
wi
) = 0.377 0.419 5.028 MMSTB
Principles of Waterflooding 1009
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1009
Modified Dykstra–Parsons Method
Felsenthal, Cobb, and Heuer (1962) extend the work of Dykstra and
Parsons to account for the presence of initial gas saturation at the start of
flood. Assuming a constant water injection rate i
w
, the method is summa-
rized in the following steps:
Step 1. Perform the following preliminary calculations to determine:
• Pore volume PV and oil in place at start of flood N
S
• Water cut f
w
as a function of S
w
• Slope (df
w
/dS
w
) as a function of S
w
• Average water saturation at breakthrough
• Mobility ratio M from Equation 14-61
• Vertical sweep efficiency at breakthrough E
DAB
from Equation
14-9
• Areal sweep efficiency at breakthrough E
ABT
from Equation
14-64
• Permeability variation V from Equation 14-70
• Fill-up volume W
if
from Equation 14-82
Step 2. Using Equations 14-94 and 14-96, calculate the vertical sweep
efficiency at assumed water–oil ratios of 1, 2, 5, 10, 15, 20, 25,
50, and 100 bbl/bbl.
Step 3. Plot WOR versus E
V
on a Cartesian scale, as shown schematical-
ly in Figure 14-55, and determine the vertical sweep efficiency at
breakthrough E
VBT
by extrapolating the WOR versus E
V
curve to
WOR = 0.
Step 4. Calculate cumulative water injected at breakthrough by using
Equation 14-43:
where W
iBT
= cumulative water injected at breakthrough, bbl
PV = pattern pore volume
E
VBT
= vertical sweep efficiency at breakthrough
E
ABT
= areal sweep efficiency at breakthrough
WPVSSEE
iBT wBT wi ABT VBT
=
(
)
−
(
)
S
wB
T
1010 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1010
Step 5. Calculate cumulative oil produced at breakthrough from the fol-
lowing expression:
Step 6. Calculate the time to breakthrough t
BT
from:
Step 7. Assume several values for water–oil ratios WOR
r
, e.g., 1, 2, 5,
10, 15, 20, 25, 50, and 100 bbl/bbl.
Step 8. Determine E
V
for each assumed value of WOR (see step 3).
Step 9. Convert the assumed values of WOR
r
to water cut f
w2
and sur-
face WOR from Equations 14-25 and 14-29, respectively:
where f
w2
= water cut at the sand face of producer, bbl/bbl
WOR
s
= surface water–oil ratio, STB/STB
WOR
r
= reservoir water–oil ratio, bbl/bbl
Step 10. Determine the water saturation S
w2
for each value of f
w2
from
the water cut curve.
Step 11. Using Equation 14-67 or Figure 14-36, determine the areal
sweep efficiency E
A
for each value of f
w2
.
Step 12. Using Equation 14-67 or Figure 14-36, determine the areal
sweep efficiency E
A
for each value of f
w2
.
Step 13. Determine the average water saturation for each value of
f
w2
from Equation 14-45.
S
w2
f
WOR
WOR
WOR WOR
B
B
w
r
r
sr
o
w
2
1
=
+
=
t
W
i
BT
iBT
w
=
N
WWE
B
P
BT
iBT if VBT
o
(
)
=
−
(
)
14 - 97
Principles of Waterflooding 1011
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1011
Step 14. Calculate the displacement efficiency E
D
for each in step
13 by applying Equation 14-9.
Step 15. Calculate cumulative oil production for each WOR from:
Step 16. Plot the cumulative oil production N
p
versus WOR
s
on Carte-
sian coordinate paper, as shown schematically in Figure 14-56,
and calculate the area under the curve at several values of
WOR
s
. The area under the curve represents the cumulative
water production W
p
at any specified WOR
s
, i.e., (W
p
)
WOR
.
Step 17. Calculate the cumulative water injected W
inj
at each selected
WOR from:
where W
inj
= cumulative water injected, bbl
S
gi
= initial gas saturation
(N
P
)
WOR
= cumulative oil production when the water–oil
ratio reaches WOR, STB
(E
V
)
WOR
= vertical sweep efficiency when the water–oil
ratio reaches WOR
Step 18. Calculate the time to inject W
inj
:
Step 19. Calculate the oil and water flow rates from Equations 14-55 and
14-56, respectively:
Q
i
B B WOR
Q Q WOR
o
w
ow S
wo S
=
+
=
t
W
i
inj
w
=
WN BW BPVSE
inj P
WOR
oP
WOR
wgiv
WOR
=
(
)
+
(
)
+
(
)
(
)
(
)
14 - 99
NNEEE
PV S E E
B
PSDAV
gi A V
o
=−
(
)
−
(
)
(
)
1
14 - 98
S
w2
1012 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1012
Craig–Geffen–Morse Method
With the obvious difficulty of incorporating the vertical sweep effi-
ciency in oil recovery calculations, Craig et al. (1955) proposed perform-
ing the calculations for only one selected layer in the multilayered sys-
tem. The selected layer, identified as the base layer, is considered to
have a 100% vertical sweep efficiency. The performance of each of the
remaining layers can be obtained by “sliding the timescale” as summa-
rized in the following steps:
Step 1. Divide the reservoir into the appropriate number of layers.
Step 2. Calculate the performance of a single layer, i.e., the base layer,
for example, layer n.
Step 3. Plot cumulative liquid volumes (N
P
, W
P
, W
inj
) and liquid rates
(Q
o
, Q
w
, i
w
) as a function of time t for the base layer, i.e., layer n.
Step 4. For each layer (including the base layer n) obtain:
• (k / φ)
• (φ h)
• (k h)
Step 5. To obtain the performance of layer i, select a succession of times
t and obtain plotted values N
p
*
, W
p
*
, W
*
inj
, Q
o
*
, Q
w
*
, and i
w
*
by read-
ing the graph of step 3 at time t*:
Then calculate the performance of layer i at any time t from:
WW
h
h
Pp
i
n
=
(
)
(
)
(
)
*
φ
φ
14 -102
NN
h
h
Pp
i
n
=
(
)
(
)
(
)
*
φ
φ
14 -101
tt
k
k
i
i
n
*
=
(
)
φ
φ
14 -100
Principles of Waterflooding 1013
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1013
where n = base layer
i = layer i
N
p
*
, W
p
*
, W
*
inj
= volumes at t
*
Q
o
*
, Q
w
*
, and i
w
*
=rates at t
*
Step 6. The composite performance of the flood pattern at time t is
obtained by summation of individual layer values.
Example 14-24
Results of Example 14-16 are shown graphically in Figure 14-47.
Assume that the reservoir has an additional four layers that are character-
ized by the following properties:
Layer k, md h, ft
“Original (base)” 1 31.5 5 0.20
2 20.5 5 0.18
3 16.0 4 0.15
4 13.0 3 0.14
5 10.9 2 0.10
Calculate N
P
, W
P
, W
inj
, Q
o
, Q
w
, and i
w
for the remaining four layers at
t = 730 days.
ii
k
k
ww
i
n
=
(
)
(
)
(
)
*
/
/
φ
φ
14 -106
QQ
k
k
ww
i
n
=
(
)
(
)
(
)
*
/
/
φ
φ
14 -105
QQ
k
k
oo
i
n
=
(
)
(
)
(
)
*
/
/
φ
φ
14 -104
WW
h
h
inj inj
i
n
=
(
)
(
)
(
)
*
φ
φ
14 -103
1014 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1014
Solution
Step 1. Calculate k/φ, φ h, and kh for each layer.
Layer k/hk h
1 157.5 1.00 157.5
2 113.9 0.90 102.5
3 106.7 0.60 64.0
4 92.8 0.42 39.3
5 109.0 0.20 21.8
Step 2. At t = 730, calculate t*:
Layer
1 1.000 730
2 0.723 528
3 0.677 495
4 0.589 430
5 0.692 505
Step 3. Read the values of N
p
*
, W
p
*
, W
*
inj
, Q
o
*
, Q
w
*
, and i
w
*
at each t
*
from
Figure 14-47:
Layer t
*
N
P
*
W
P
*
W
*
inj
Q
o
*
Q
w
*
i
w
*
1 730 81,400 61,812 206,040 51.5 218.4 274.5
2 528 68,479 19,954 153,870 65 191 261
3 495 63,710 21,200 144,200 68 175 258
4 430 57,000 9,620 124,000 75 179 254
5 505 74,763 41,433 177,696 58 200 267
Step 4. Calculate N
P
, W
P
, W
inj
, Q
o
, Q
w
, and i
w
for each layer after 730
days by applying Equations 14-101 through 14-106:
Layer (h)
i
/(h)
1
N
P
W
P
W
inj
1 1.00 81,400 61,812 206,040
2 0.90 61,631 11,972 138,483
3 0.60 38,226 12,720 86,520
4 0.42 23,940 4,040 52,080
5 0.20 14,953 8,287 35,539
Total 220,150 98,831 466,582
t
k
k
i
n
*
/
/
==
(())
(())
730
φφ
φφ
k
k
i
n
/
/
φφ
φφ
(())
(())
Principles of Waterflooding 1015
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1015
Layer (h)
i
/(h)
1
Q
o
Q
w
i
w
1 1.000 51.5 218.4 274.5
2 0.651 42.3 124.3 169.9
3 0.406 27.6 71.1 104.7
4 0.250 18.8 44.8 63.5
5 0.138 8.0 27.6 36.8
Total 148.2 486.2 649.4
Producing .
Step 5. Steps 2 and 3 are repeated for a succession of times t and the
composite reservoir performance curves are generated to describe
the entire reservoir performance.
PROBLEMS
1. A saturated oil reservoir exists at its bubble-point pressure of 2840 psi.
The following pressure-production data are given:
PN
P
, STB G
P
, Mscf B
t
, bbl/STB B
g
, ft
3
/scf R
S
, scf/STB
2840 0 0 1.528 — 827
2660 36,933 37,851 1.563 0.00618 772
2364 65,465 74,137 1.636 0.00680 680
2338 75,629 91,910 1.648 0.00691 675
2375 85,544 115,256 1.634 0.00674 685
2305 96,100 148,200 1.655 0.00702 665
The following information is also available:
S
wi
= 25% and S
or
= 35%.
a. Calculate the reduction in the residual gas saturation if a waterflood-
ing project were to start at 2364 psi.
b. Calculate the injection that is required to dissolve the trapped gas.
WOR STB STB==
486 2
148 2
328
.
.
./
1016 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1016
2. The relative permeability data for a core sample taken from the Vu-
Villa Field are given below:
S
w
k
rw
k
ro
0.16 0 1.00
0.20 0.0008 0.862
0.26 0.0030 0.670
0.32 0.0090 0.510
0.40 0.024 0.330
0.50 0.064 0.150
0.60 0.140 0.040
0.66 0.211 0.010
0.72 0.30 0.00
1.00 1.00 0.00
S
wi
= 0.16 S
or
= 0.28
w
= 0.75 cp
o
= 2.00 cp
w
= 1.0 g/cm
3
o
= 0.83 g/cm
3
a. Neglecting gravity and capillary pressure terms, develop the fractional
flow curve.
b. Assuming a water-injection rate of 0.08 bbl/day/ft
2
and a dip angle of
15°, develop the fractional flow curve.
c. Determine from both curves:
• f
w
at the front
• S
w
at the front, i.e., S
wf
• Average water saturation behind the front
3. A linear reservoir system is characterized by the following data:
L = 500 ft A = 10,000 ft
2
S
wi
= 24% B
w
= 1.01 bbl/STB
S
or
= 20% i
w
= 100 ft
3
/hr B
o
= 1.25 bbl/STB φ = 15%
S
w
f
w
0.30 0.0181
0.40 0.082
0.50 0.247
0.60 0.612
0.70 0.885
0.75 0.952
0.80 1.000
Principles of Waterflooding 1017
Reservoir Eng Hndbk Ch 14 2001-10-25 17:38 Page 1017