Tarek Ahmed. Reservoir engineering handbook
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Figure 14-22. Relative permeability ratio.
Step 2. Assume several values of water saturation and calculate the frac-
tional flow curve at its derivatives by applying Equations 14-37
and 14-38:
908 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 908
S
w
k
ro
/k
rw
f
w
, Equation 14-37 (df
w
/dS
w
), Equation 14-38
0.25 30.23 0.062 0.670
0.30 17.00 0.105 1.084
0.35 9.56 0.173 1.647
0.40 5.38 0.271 2.275
0.45 3.02 0.398 2.759
0.50 1.70 0.541 2.859
0.55 0.96 0.677 2.519
0.60 0.54 0.788 1.922
0.65 0.30 0.869 1.313
0.70 0.17 0.922 0.831
0.75 0.10 0.956 0.501
Step 3. Plot f
w
and (df
w
/S
w
) vs. S
w
on a Cartesian scale as shown in
Figure 14-23. Draw a straight line from S
wc
and tangent to the
f
w
curve. Determine the coordinates of point of tangency and the
slope of the tangent (df
w
/dS
w
)
S
wf
, to give:
This means that the leading edge of the waterfront (stabilized zone) has a
constant saturation of 0.596 and water cut of 78%.
Step 4. When constructing the water saturation profile, it should be
noted that no water saturation with a value less than S
wf
, i.e.,
59.6%, exists behind the leading edge of the water bank.
Assume water saturation values in the range of S
wf
to (1 – S
or
),
i.e., 59.6 to 75%, and calculate the water saturation profile as a
function of time by using Equation 14-36:
x
it
A
df
dS
x
t
df
dS
xt
df
dS
S
ww
w
S
S
w
w
S
S
w
w
S
w
w
w
w
w
w
()
=
()
=
()()
()( )
()
=
()
5 615
5 615 900
0 25 26 400
077
.
.
.,
.
φ
Sf
df
dS
wf wf
w
w
S
wf
, , and
(
)
=
(
)
=0 596 0 48 1 973.. .
Principles of Waterflooding 909
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 909
Figure 14-23. Water cut curve and its derivative.
910 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 910
Assumed S
w
(df
w
/dS
w
) x = 0.77t(df/dS
w
) x = 0.77t(df/dS
w
) x = 0.77t(df/dS
w
)
t = 60 days t = 120 days t = 240days
0.596 1.973 91 182 365
0.60 1.922 88 177 353
0.65 1.313 60 121 241
0.70 0.831 38 76 153
0.75 0.501 23 46 92
Step 5. Plot the water saturation profile as a function of distance and
time, as shown in Figure 14-24.
Figure 14-24. Water saturation profile for Example 14-7.
The above example shows that after 240 days of water injection, the
leading edge of the water front has moved 365 feet from the injection
well (235 feet from the producer). The water front (leading edge) will
eventually reach the production well and water breakthrough occurs.
The example also indicates that at water breakthrough, the leading
edge of the water front would have traveled exactly the entire distance
between the two wells, i.e., 600 feet. Therefore, to determine the time to
breakthrough, t
BT
, simply set (x)S
wf
equal to the distance between the
injector and producer L in Equation 14-34 and solve for the time:
L
it
A
df
dS
wBT w
w
S
wf
=
5 615.
φ
Principles of Waterflooding 911
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 911
Note that the pore volume (PV) is given by:
Combining the above two expressions and solving for the time to break-
through t
BT
gives:
where t
BT
= time to breakthrough, day
PV = total flood pattern pore volume, bbl
L = distance between the injector and producer, ft
Assuming a constant water-injection rate, the cumulative water injected
at breakthrough is calculated from Equation 14-39 as:
where W
iBT
= cumulative water injected at breakthrough, bbl
= total flood pattern pore volume, bbl
It is convenient to express the cumulative water injected in terms of
pore volumes injected, i.e., dividing W
inj
by the reservoir total pore vol-
ume. Conventionally, Q
i
refers to the total pore volumes of water inject-
ed. From Equation 14-40, Q
i
at breakthrough is:
where Q
iBT
= cumulative pore volumes of water injected at
breakthrough
PV = total flood pattern pore volume, bbl
Q
W
PV
df
dS
iBT
iBT
w
w
S
wf
=
(
)
=
(
)
1
14 - 41
φAL
5 615.
Wit
PV
df
dS
iBT w BT
w
w
S
wf
==
()
()
14 - 40
t
PV
i
df
dS
BT
w
w
w
S
wf
=
(
)
(
)
1
14 - 39
PV
AL
()
=
φ
5 615.
912 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 912
Example 14-8
Using the data given in Example 14-7, calculate:
• Time to breakthrough
• Cumulative water injected at breakthrough
• Total pore volumes of water injected at breakthrough
Solution
Step 1. Calculate the reservoir pore volume:
Step 2. Calculate the time to breakthrough from Equation 14-39:
Step 3. Determine cumulative water injected at breakthrough:
Step 4. Calculate total pore volumes of water injected at breakthrough:
Q
df
dS
Q
iBT
w
w
S
iBT
wf
=
==
1
1
1 973
0 507
.
. pore volumes
Wit
W bbl
iBT w BT
iBT
=
=
()( )
=900 436 88 393 198.,
t
PV
i
df
dS
t
BT
w
w
w
S
BT
wf
=
()
=
=
1
775 779
900
1
1 973
436 88
,
.
. days
PV bbl
(
)
=
(
)
(
)
(
)
=
0 25 26 400 660
5 615
775 779
.,
.
,
Principles of Waterflooding 913
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 913
A further discussion of Equation 14-40 is needed to better understand
the significance of the Buckley and Leverett (1942) frontal advance theo-
ry. Equation 14-40, which represents cumulative water injected at break-
through, is given by:
If the tangent to the fractional flow curve is extrapolated to f
w
= 1 with a
corresponding water saturation of S
*
w
(as shown in Figure 14-25), then
the slope of the tangent can be calculated numerically as:
Combining the above two expressions gives:
The above equation suggests that the water saturation value denoted as
S
*
w
must be the average water saturation at breakthrough, or:
where
= average water saturation in the reservoir at
breakthrough
PV = flood pattern pore volume, bbl
W
iBT
= cumulative water injected at breakthrough, bbl
S
wi
= initial water saturation
S
wBT
WPVS S PVQ
iBT wBT wi iBT
(
)
−
(
)
=
(
)
(
)
14 - 42
WPVSSPVQ
iBT W wi iBT
=
(
)
−
(
)
=
(
)
*
df
dS
SS
w
w
S
wwi
wf
=
−
−
10
*
WPV
df
dS
PV Q
iBT
w
w
S
iBT
wf
=
()
=
()
1
914 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 914
Figure 14-25. Average water saturation at breakthrough.
Principles of Waterflooding 915
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 915
Two important points must be considered when determining :
1. When drawing the tangent, the line must be originated from the initial
water saturation S
wi
if it is different from the connate water saturation
S
wc
, as shown in Figure 14-26.
Figure 14-26. Tangent from S
wi
.
2. When considering the areal sweep efficiency E
A
and vertical sweep
efficiency E
V
, Equation 14-42 should be expressed as:
S
wBT
916 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 916
or equivalently as:
where E
ABT
and E
VBT
are the areal and vertical sweep efficiencies at
breakthrough (as discussed later in the chapter). Note that the average
water saturation in the swept area would remain constant with a value of
until breakthrough occurs, as illustrated in Figure 14-27. At the
time of breakthrough, the flood front saturation S
wf
reaches the produc-
ing well and the water cut increases suddenly from zero to f
wf
. At break-
through, S
wf
and f
wf
are designated and f
wBT
.
Figure 14-27. Average water saturation before breakthrough.
After breakthrough, the water saturation and the water cut at the pro-
ducing well gradually increase with continuous injection of water, as
shown in Figure 14-28. Traditionally, the produced well is designated as
well 2 and, therefore, the water saturation and water cut at the producing
well are denoted as S
w2
and f
w2
, respectively.
1
injector producer
breakthrough
0
S
wf
1-S
or
S
wBT
S
wf
= S
wBT
S
wBT
S
wBT
WPVQEE
iBT iBT ABT VBT
=
() ( )
14 - 44
WPVSSEE
iBT wBT wi ABT VBT
=
()
−
()
()
14 - 43
Principles of Waterflooding 917
Reservoir Eng Hndbk Ch 14 2001-10-25 17:37 Page 917