Tarek Ahmed. Reservoir engineering handbook
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Figure 14-28. Average water saturation after breakthrough.
Welge (1952) illustrated that when the water saturation at the produc-
ing well reaches any assumed value S
w2
after breakthrough, the fractional
flow curve can be used to determine:
• Producing water cut f
w2
• Average water saturation in the reservoir
w2
• Cumulative water injected in pore volumes, i.e., Q
i
As shown in Figure 14-29, the author pointed out that drawing a tan-
gent to the fractional flow curve at any assumed value of S
w2
greater
than S
wf
has the following properties:
1. The value of the fractional flow at the point of tangency corresponds
to the well producing water cut f
w2
, as expressed in bbl/bbl.
2. The saturation at which the tangent intersects f
w
= 1 is the average
water saturation
w2
in the swept area. Mathematically, the average
water saturation is determined from:
SS
f
df
dS
ww
w
w
w
S
w
22
2
1
2
=+
−
(
)
14 - 45
S
S
1
injector producer
breakthrough
0
S
wf
1-S
or
S
wBT
(S
wf
, f
wf)
= (S
wBT,
f
wBT
)
(S
w2
, f
w2
)
S
w2
918 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 918
Figure 14-29. Determination of average water saturation after breakthrough.
3. The reciprocal of the slope of the tangent is defined as the cumulative
pore volumes of water injected Q
i
at the time when the water satura-
tion reaches S
w2
at the producing well, or:
4. The cumulative water injected when the water saturation at the pro-
ducing well reaches S
w2
is given by:
or equivalently as:
where: W
inj
= cumulative water injected, bbl
(PV) = pattern pore volume, bbl
E
A
= areal sweep efficiency
E
V
= vertical sweep efficiency
WPVSSEE
inj w wi A V
=
()
−
()
()
2
14 - 48
WPVQEE
inj i A V
=
() ( )
14 - 47
Q
df
dS
i
w
w
S
w
=
(
)
1
2
14 - 46
1
1
0
0
S
wf
f
wf
f
w2
S
w2
S
wBT
S
w2
(df
w
/dS
w
)sw2
Principles of Waterflooding 919
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 919
5. For a constant injection rate i
w
, the total time t to inject W
inj
barrels of
water is given by:
Example 14-9
Using the data given in Example 14-7 for the linear reservoir system,
calculate the following when the water saturation at the producing well
reaches 0.70 (i.e., S
w2
= 0.7):
a. reservoir water cut in bbl/bbl
b. surface water cut in STB/STB
c. reservoir water–oil ratio in bbl/bbl
d. surface water–oil ratio in STB/STB
e. average water saturation in the swept area
f. pore volumes of water injected
g. cumulative water injected in bbl
Assume that the areal and vertical sweep efficiency are 100%, i.e., E
A
=
1.0 and E
V
= 1.0.
Solution
a. Results of Example 14-7 indicate that at a water saturation value of
70%, the corresponding water cut f
w
is 0.922, therefore:
f
w2
= 0.922 bbl/bbl
b. Calculate the surface water cut by applying Equation 14-31:
f
B
f
B
f STB STB
ws
o
w
w
o
ws
=
−
+
=
−
+
=
B
1
1
125
102
1
0 922
1125
0 935
.
.
.
.
. /
t
W
i
inj
w
=
()
14 - 49
920 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 920
c. Determine the producing water–oil ratio by using Equation 14-26:
d. Apply Equation 14-29 to determine the surface water–oil ratio:
e. Draw a tangent to the fractional flow curve at the coordinate of the
point (S
w
, f
w
) = (0.7, 0.922) and extrapolate to f
w
= 1.0 to give a corre-
sponding
w2
= 0.794. Equivalently, the average water saturation can
be calculated by determining the slope of the tangent and applying
Equation 14-45, to give:
f. From Equation 14-46, the cumulative pore volume of water injected is
the reciprocal of the slope of the tangent line:
Q
i
= 1/0.831 = 1.203
g. Calculate cumulative water injected by applying Equation 14-47:
Oil Recovery Calculations
The main objective of performing oil recovery calculations is to gener-
ate a set of performance curves under a specific water-injection scenario.
WPVQEE
W bbl
inj i A V
inj
=
()()
=
( )( )()()
=775 779 1 203 1 1 933 262,. ,
df
dS
S
w
w
S
W
w
=
=+
−
=
2
0 831
070
1 0 922
0 831
0 794
2
.
.
.
.
.
S
WOR
B WOR
B
WOR STB STB
S
or
w
S
=
=
(
)
(
)
=
125 1182
102
14 49
..
.
. /
WOR
f
f
WOR bbl bbl
r
w
w
r
=
−
=
−
=
1
0 922
1 0 922
11 82
.
.
. /
Principles of Waterflooding 921
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 921
A set of performance curves is defined as the graphical presentation of
the time-related oil recovery calculations in terms of:
• Oil production rate, Q
o
• Water production rate, Q
w
• Surface water–oil ratio, WOR
s
• Cumulative oil production, N
p
• Recovery factor, RF
• Cumulative water production, W
p
• Cumulative water injected, W
inj
• Water-injection pressure, p
inj
(discussed later in the chapter)
• Water-injection rate, i
w
(discussed later in the chapter)
In general, oil recovery calculations are divided into two parts: (1) before
breakthrough calculations and (2) after breakthrough calculations.
Regardless of the stage of the waterflood, i.e., before or after break-
through, the cumulative oil production is given previously by Equation
14-6 as:
where N
p
= cumulative oil production, STB
N
S
= initial oil in place at start of the flood, STB
E
D
= displacement efficiency
E
A
= areal sweep efficiency
E
V
= vertical sweep efficiency
As defined by Equation 14-10 when S
gi
= 0, the displacement efficiency
is given by:
At breakthrough, the E
D
can be calculated by determining the average
water saturation at breakthrough:
E
SS
S
DBT
wBT wi
wi
=
−
−
()
1
14 - 50
E
SS
S
D
wwi
wi
=
−
−1
NNEEE
pSDAV
=
922 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 922
where E
DBT
= displacement efficiency at breakthrough
= average water saturation at breakthrough
The cumulative oil production at breakthrough is then given by:
where (N
p
)
BT
= cumulative oil production at breakthrough, STB
E
ABT
, E
VBT
= areal and vertical sweep efficiencies at breakthrough
Assuming E
A
and E
V
are 100%, Equation 14-51 is reduced to:
Before breakthrough occurs, the oil recovery calculations are simple
when assuming that no free gas exists at the start of the flood, i.e., S
gi
= 0. The cumulative oil production is simply equal to the volume of
water injected with no water production during this phase (W
p
= 0 and
Q
w
= 0).
Oil recovery calculations after breakthrough are based on determining
E
D
at various assumed values of water saturations at the producing well.
The specific steps of performing complete oil recovery calculations are
composed of three stages:
1. Data preparation
2. Recovery performance to breakthrough
3. Recovery performance after breakthrough
Stage 1: Data Preparation
Step 1. Express the relative permeability data as relative permeability
ratio k
ro
/k
rw
and plot their values versus their corresponding
water saturations on a semi-log scale.
Step 2. Assuming that the resulting plot of relative permeability ratio,
k
ro
/k
rw
vs. S
w
, forms a straight-line relationship, determine values
of the coefficients a and b of the straight line (see Example 14-7).
NNE
p
BT
S DBT
()
=
()
14 - 52
N NEEE
p
BT
S DBT ABT VBT
()
=
()
14 - 51
S
wBT
Principles of Waterflooding 923
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 923
Express the straight-line relationship in the form given by Equa-
tion 14-36, or:
Step 3. Calculate and plot the fractional flow curve f
w
, allowing for
gravity effects if necessary, but neglecting the capillary pressure
gradient.
Step 4. Select several values of water saturations between S
wf
and
(1 – S
or
) and determine the slope (df
w
/dS
w
) at each saturation.
The numerical calculation of each slope as expressed by Equation
14-38 provides consistent values as a function of saturation, or:
Step 5. Prepare a plot of the calculated values of the slope (df
w
/dS
w
)
versus S
w
on a Cartesian scale and draw a smooth curve through
the points.
Stage 2: Recovery Performance to Breakthrough (S
gi
= 0, E
A
, E
V
= 100%)
Step 1. Draw a tangent to the fractional flow curve as originated from
S
wi
and determine:
• Point of tangency with the coordinate (S
wf
, f
wf
)
• Average water saturation at breakthrough by extending
the tangent line to f
w
= 1.0
• Slope of the tangent line
Step 2. Calculate pore volumes of water injected at breakthrough by
using Equation 14-41:
Q
df
dS
SS
iBT
w
w
S
wBT wi
wf
=
=−
()
1
df
dS
w
w
S
wf
S
wBT
df
dS
abe
ae
w
w
w
o
bS
w
o
bS
w
w
=
−
+
µ
µ
µ
µ
1
2
k
k
ae
ro
rw
bS
w
=
924 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 924
Step 3. Assuming E
A
and E
V
are 100%, calculate cumulative water
injected at breakthrough by applying Equation 14-42:
or equivalently:
Step 4. Calculate the displacement efficiency at breakthrough by apply-
ing Equation 14-50:
Step 5. Calculate cumulative oil production at breakthrough from Equa-
tion 14-52:
Step 6. Assuming a constant water-injection rate, calculate time to
breakthrough from Equation 14-40:
Step 7. Select several values of injection time less than the breakthrough
time, i.e., t < t
BT
, and set:
W
inj
= i
w
t
Q
o
= i
w
/B
o
WOR = 0
W
p
= 0
N
p
=
Step 8. Calculate the surface water–oil ratio WOR
s
exactly at break-
through by using Equation 14-28:
where f
wBT
is the water cut at breakthrough (notice that f
wBT
= f
wf
).
WOR
B
B
f
s
o
w
wBT
=
−
1
1
it
B
W
B
w
o
inj
o
=
t
W
i
BT
iBT
w
=
NNE
p
BT
S DBT
()
=
E
SS
S
DBT
wBT wi
wi
=
−
−1
WPVQ
iBT iBT
=
()
WPVSS
iBT wBT wi
=
()
−
()
Principles of Waterflooding 925
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 925
Note that WOR
s
as calculated from the above expression is only correct
when both the areal sweep efficiency E
A
and vertical sweep efficiency
E
V
are 100%.
Stage 3: Recovery Performance After Breakthrough (S
gi
= 0, E
A
, E
V
= 100%)
The recommended methodology of calculating recovery performance
after breakthrough is based on selecting several values of water satura-
tions around the producing well, i.e., S
w2
, and determining the corre-
sponding average reservoir water saturation for each S
w2
. The spe-
cific steps that are involved are summarized below:
Step 1. Select six to eight different values of S
w2
(i.e., S
w
at the produc-
ing well) between S
wBT
and (1 – S
or
) and determine (df
w
/dS
w
)
values corresponding to these S
w2
points.
Step 2. For each selected value of S
w2
, calculate the corresponding reser-
voir water cut and average water saturation from Equations 14-37
and 14-45:
Step 3. Calculate the displacement efficiency E
D
for each selected value
of S
w2
:
Step 4. Calculate cumulative oil production N
p
for each selected value of
S
w2
from Equation 14-6, or:
N
P
= N
s
E
D
E
A
E
V
E
SS
S
D
wwi
wi
=
−
−
2
1
f
ae
SS
f
df
dS
w
w
o
bS
ww
w
w
w
S
w
w
2
22
2
1
1
1
2
2
=
+
=+
−
µ
µ
S
w2
926 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 926
Assuming E
A
and E
V
are equal to 100%, then:
N
P
= N
s
E
D
Step 5. Determine pore volumes of water injected, Q
i
, for each selected
value of S
w2
from Equation 14-46:
Step 6. Calculate cumulative water injected for each selected value of
S
w2
by applying Equation 14-47 or 14-48:
Notice that E
A
and E
V
are set equal to 100%
Step 7. Assuming a constant water-injection rate i
w
, calculate the time t
to inject W
inj
barrels of water by applying Equation 14-49:
Step 8. Calculate cumulative water production W
P
at any time t from the
material balance equation, which states that the cumulative water
injected at any time will displace an equivalent volume of oil and
water, or:
Solving for W
p
gives:
or equivalently in a more generalized form:
W
WSSPVEE
B
p
inj w wi A V
w
=
−−
(
)
(
)
(
)
2
14 - 54
W
WNB
B
p
inj p o
w
−
(
)
14 - 53
WNBWB
inj p o p w
=+
t
w
i
inj
w
=
WPVQ WPVSS
inj i inj w wi
=
()
=
()
−
()
or
2
Q
df
dS
i
w
w
S
w
=
1
2
Principles of Waterflooding 927
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 927