Tarek Ahmed. Reservoir engineering handbook
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Step 3. Generate the IPR data by applying the constant J approach for all
pressures above p
b
and Equation 7-11 for all pressures below p
b
.
P
wf
Equation # Q
o
3000 (7-6) 0
2800 (7-6) 100
2600 (7-6) 200
2130 (7-6) 435
1500 (7-11) 709
1000 (7-11) 867
500 (7-11) 973
0 (7-11) 1027
Case 2. The Value of the Recorded Stabilized p
wf
< p
b
When the recorded p
wf
from the stabilized flow test is below the bub-
ble-point pressure, as shown in Figure 7-8, the following procedure for
generating the IPR data is proposed:
Step 1. Using the stabilized well flow test data and combining Equation
7-10 with 7-11, solve for the productivity index J to give:
Step 2. Calculate Q
ob
by using Equation 7-10, or:
Q
ob
= J (p
–
r
- p
b
)
Step 3. Generate the IPR for p
wf
≥ p
b
by assuming several values for p
wf
above the bubble point pressure and calculating the correspond-
ing Q
o
from:
Q
o
= J (p
–
r
- p
wf
)
Step 4. Use Equation 7-11 to calculate Q
o
at various values of p
wf
below
p
b
, or:
J
Q
pp
pp
p
p
p
o
r
b
bwf
b
wf
b
=
-
(
)
+-
Ê
Ë
Á
ˆ
¯
˜
-
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
Í
˘
˚
˙
˙
18
102 08
2
.
..
(7 -12)
488 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 488
Example 7-4
The well described in Example 7-3 was retested and the following
results obtained:
P
wf
= 1700 psig, Q
o
= 630.7 STB/day
Generate the IPR data using the new test data.
Solution
Notice that the stabilized p
wf
is less than p
b
.
Step 1. Solve for J by applying Equation 7-12.
Step 2. Q
ob
= 0.5 (3000 - 21300) = 435 STB/day
Step 3. Generate the IPR data.
p
wf
Equation # Q
o
3000 (7-6) 0
2800 (7-6) 100
2600 (7-6) 200
2130 (7-6) 435
1500 (7-11) 709
1000 (7-11) 867
500 (7-11) 973
0 (7-11) 1027
J
STB day psi
=
-+ -
Ê
Ë
ˆ
¯
-
Ê
Ë
ˆ
¯
È
Î
Í
˘
˚
˙
=
630 7
3000 2130
2130
18
1
1700
2130
1700
2130
05
2
.
()
.
.//
QQ
Jp p
p
p
p
oob
bwf
b
wf
b
=+ -
Ê
Ë
Á
ˆ
¯
˜
-
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
Í
˘
˚
˙
˙
18
102 08
2
.
..
Oil Well Performance 489
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 489
Quite often it is necessary to predict the well’s inflow performance for
future times as the reservoir pressure declines. Future well performance
calculations require the development of a relationship that can be used to
predict future maximum oil flow rates.
There are several methods that are designed to address the problem of
how the IPR might shift as the reservoir pressure declines. Some of these
prediction methods require the application of the material balance equa-
tion to generate future oil saturation data as a function of reservoir pres-
sure. In the absence of such data, there are two simple approximation
methods that can be used in conjunction with Vogel’s method to predict
future IPRs.
First Approximation Method
This method provides a rough approximation of the future maximum
oil flow rate (Q
omax
)
f
at the specified future average reservoir pressure
(p
r
)
f
. This future maximum flow rate (Q
omax
)
f
can be used in Vogel’s
equation to predict the future inflow performance relationships at (p
–
r
)
f
.
The following steps summarize the method:
Step 1. Calculate (Q
omax
)
f
at (p
–
r
)
f
from:
where the subscript f and p represent future and present condi-
tions, respectively.
Step 2. Using the new calculated value of (Q
omax
)
f
and (p
–
r
)
f
, generate the
IPR by using Equation 7-9.
Second Approximation Method
A simple approximation for estimating future (Q
omax
)
f
at (p
–
r
)
f
is pro-
posed by Fetkovich (1973). The relationship has the following mathe-
matical form:
()()[()/()]
max max
.
QQpp
of oprfrp
=
30
()()
()
()
..
()
()
max max
QQ
p
p
p
p
of op
rf
rp
rf
rp
=
Ê
Ë
Á
ˆ
¯
˜
+
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
Í
˘
˚
˙
˙
0 2 0 8 (7 -13)
490 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 490
Where the subscript f and p represent future and present conditions,
respectively. The above equation is intended only to provide a rough esti-
mation of future (Q
o
)
max
.
Example 7-5
Using the data given in Example 7-2, predict the IPR where the aver-
age reservoir pressure declines from 2500 psig to 2200 psig.
Solution
Example 7-2 shows the following information:
• Present average reservoir pressure (p
–
r
)
p
= 2500 psig
• Present maximum oil rate (Q
omax
)
p
= 1067.1 STB/day
Step 1. Solve for (Q
omax
)
f
by applying Equation 7-13.
Step 2. Generate the IPR data by applying Equation 7-9.
p
wf
Q
o
= 849 [1 - 0.2 (p
wf
/2200) - 0.8 (p
wf
/2200)
2
]
2200 0
1800 255
1500 418
500 776
0 849
It should be pointed out that the main disadvantage of Vogel’s method-
ology lies with its sensitivity to the match point, i.e., the stabilized flow
test data point, used to generate the IPR curve for the well.
Wiggins’ Method
Wiggins (1993) used four sets of relative permeability and fluid prop-
erty data as the basic input for a computer model to develop equations to
predict inflow performance. The generated relationships are limited by
() . .. /
max
Q STB day
of
=
Ê
Ë
ˆ
¯
+
Ê
Ë
ˆ
¯
È
Î
Í
˘
˚
˙
=1067 1
2200
2500
02 08
2200
2500
849
Oil Well Performance 491
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 491
the assumption that the reservoir initially exists at its bubble-point pres-
sure. Wiggins proposed generalized correlations that are suitable for pre-
dicting the IPR during three-phase flow. His proposed expressions are
similar to that of Vogel’s and are expressed as:
where Q
w
= water flow rate, STB/day
(Q
w
)
max
= maximum water production rate at p
wf
= 0, STB/day
As in Vogel’s method, data from a stabilized flow test on the well must
be available in order to determine (Q
o
)
max
and (Q
w
)
max
.
Wiggins extended the application of the above relationships to predict
future performance by providing with expressions for estimating future
maximum flow rates. Wiggins expressed future maximum rates as a
function of:
• Current (present) average pressure (p
–
r
)
p
• Future average pressure (p
–
r
)
f
• Current maximum oil flow rate (Q
omax
)
p
• Current maximum water flow rate (Q
wmax
)
p
Wiggins proposed the following relationships:
()( ).
()
()
.
()
()
max max
QQ
p
p
p
p
of wp
rf
rp
rf
rp
=
È
Î
Í
Í
˘
˚
˙
˙
+
È
Î
Í
Í
˘
˚
˙
˙
Ï
Ì
Ô
Ó
Ô
¸
˝
Ô
˛
Ô
059 036
2
(7 -17)
()().
()
()
.
()
()
max max
QQ
p
p
p
p
of op
rf
rp
rf
rp
=
È
Î
Í
Í
˘
˚
˙
˙
+
È
Î
Í
Í
˘
˚
˙
˙
Ï
Ì
Ô
Ó
Ô
¸
˝
Ô
˛
Ô
015 084
2
(7 -16)
QQ
p
p
p
p
ww
wf
r
wf
r
=
(
)
-
Ê
Ë
Á
ˆ
¯
˜
-
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
Í
˘
˚
˙
˙
max
..1 0 72 0 28
2
(7 -15)
QQ
p
p
p
p
oo
wf
r
wf
r
=
(
)
-
Ê
Ë
Á
ˆ
¯
˜
-
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
Í
˘
˚
˙
˙
max
..1 0 52 0 48
2
(7 -14)
492 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 492
Example 7-6
The information given in Examples 7-2 and 7-5 is repeated here for
convenience:
• Current average pressure = 2500 psig
• Stabilized oil flow rate = 350 STB/day
• Stabilized wellbore pressure = 2000 psig
Generate the current IPR data and predict future IPR when the reser-
voir pressure declines from 2500 to 2000 psig by using Wiggins’ method.
Solution
Step 1. Using the stabilized flow test data, calculate the current maxi-
mum oil flow rate by applying Equation 7-14.
Step 2. Generate the current IPR data by using Wiggins’ method and
compare the results with those of Vogel’s.
Results of the two methods are shown graphically in Figure 7-9.
p
wf
Wiggins’ Vogel’s
2500 0 0
2200 216 218
1500 651 632
1000 904 845
500 1108 990
0 1264 1067
Step 3. Calculate future maximum oil flow rate by using Equation 7-16.
() . . /
max
Q STB day
of
=
Ê
Ë
ˆ
¯
+
Ê
Ë
ˆ
¯
È
Î
Í
˘
˚
˙
=1264 0 15
2200
2500
084
2200
2500
989
2
() . .
/
max
Q
STB day
op
=-
Ê
Ë
ˆ
¯
-
Ê
Ë
ˆ
¯
È
Î
Í
˘
˚
˙
=
350 1 0 52
2000
2500
048
2000
2500
1264
2
Oil Well Performance 493
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 493
Step 4. Generate future IPR data by using Equation 7-16
p
wf
Q
o
= 989 [1 - 0.52 (p
wf
/2200) - 0.48 (p
wf
/2200)
2
]
2200 0
1800 250
1500 418
500 848
0 989
Standing’s Method
Standing (1970) essentially extended the application of Vogel’s to pre-
dict future inflow performance relationship of a well as a function of
reservoir pressure. He noted that Vogel’s equation: Equation 7-9 can be
rearranged as:
Q
Q
p
p
p
p
o
o
wf
r
wf
r
()
.
max
=-
Ê
Ë
Á
ˆ
¯
˜
+
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
˘
˚
˙
1 1 0 8 (7 -18)
494 Reservoir Engineering Handbook
Figure 7-9. IPR curves.
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 494
Standing introduced the productivity index J as defined by Equation
7-1 into Equation 7-18 to yield:
Standing then defined the present (current) zero drawdown productivity
index as:
where J
*
p
is Standing’s zero-drawdown productivity index. The J
*
p
is related
to the productivity index J by:
Equation 7-1 permits the calculation of J
*
p
from a measured value of J.
To arrive to the final expression for predicting the desired IPR expres-
sion, Standing combines Equation 7-20 with Equation 7-18 to eliminate
(Q
o
)
max
to give:
where the subscript f refers to future condition.
Standing suggested that J
*
p
can be estimated from the present value of
J
*
p
by the following expression:
JJ
k
B
k
B
fp
ro
oo
f
ro
oo
p
**
=
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
mm
(7 - 23)
Q
o
=
J
f
*
( p
r
)
f
1.8
È
Î
Í
˘
˚
˙
1 - 0.2
p
wf
(p
r
)
f
È
Î
Í
˘
˚
˙
- 0.8
p
wf
(p
r
)
f
È
Î
Í
˘
˚
˙
2
Ï
Ì
Ô
Ó
Ô
¸
˝
Ô
˛
Ô
(7 - 22)
J
J
1
1.8
1 0.8
p
p
(7 - 21)
p
*
wf
r
=
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
˘
˚
˙
+
J 1.8
(Q )
p
r
(7 - 20)
p
*
o max
=
È
Î
Í
˘
˚
˙
J
Q
p
p
p
o
r
wf
r
=+
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
˘
˚
˙
()
.
max
1 0 8 (7 -19)
Oil Well Performance 495
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 495
where the subscript p refers to the present condition.
If the relative permeability data is not available, J
*
f
can be roughly esti-
mated from:
Standing’s methodology for predicting a future IPR is summarized in
the following steps:
Step 1. Using the current time condition and the available flow test data,
calculate (Q
o
)
max
from Equation 7-9 or Equation 7-18.
Step 2. Calculate J
*
at the present condition, i.e., J
*
p
, by using Equation
7-20. Notice that other combinations of Equations 7-18 through
7-21 can be used to estimate J
*
p
.
Step 3. Using fluid property, saturation and relative permeability data,
calculate both (k
ro
/m
o
B
o
)
p
and (k
ro
/m
o
B
o
)
f
.
Step 4. Calculate J
*
f
by using Equation 7-23. Use Equation 7-24 if the oil
relative permeability data is not available.
Step 5. Generate the future IPR by applying Equation 7-22.
Example 7-7
A well is producing from a saturated oil reservoir that exists at its satu-
ration pressure of 4000 psig. The well is flowing at a stabilized rate of
600 STB/day and a p
wf
of 3200 psig. Material balance calculations pro-
vide the following current and future predictions for oil saturation and
PVT properties.
Present Future
p
–
r
4000 3000
m
o
, cp 2.40 2.20
B
o
, bbl/STB 1.20 1.15
k
ro
1.00 0.66
JJp p
fprfrp
**
[( )
/
()]=
2
(7 - 24)
496 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 496
Generate the future IPR for the well at 3000 psig by using Standing’s
method.
Solution
Step 1. Calculate the current (Q
o
)
max
from Equation 7-18.
Step 2. Calculate J
*
p
by using Equation 7-20.
Step 3. Calculate the following pressure-function:
Step 4. Calculate J
*
f
by applying Equation 7-23.
Step 5. Generate the IPR by using Equation 7-22.
p
wf
Q
o
, STB/day
3000 0
2000 527
1500 721
1000 870
500 973
0 1030
J
f
*
=
Ê
Ë
ˆ
¯
=0 823
0 2609
0 3472
0 618.
.
.
.
k
B
ro
oo
f
m
Ê
Ë
Á
ˆ
¯
˜
==
066
22 115
0 2609
.
(.)(. )
.
k
B
ro
oo
p
m
Ê
Ë
Á
ˆ
¯
˜
==
1
24 120
0 3472
(.)(. )
.
J
p
*
=
È
Î
Í
˘
˚
˙
=18
1829
4000
0 823..
() ( .) /
max
Q STB day
o
=-
Ê
Ë
ˆ
¯
+
Ê
Ë
ˆ
¯
È
Î
Í
˘
˚
˙
=600 1
3200
4000
108
3200
4000
1829
Oil Well Performance 497
Reservoir Eng Hndbk Ch 07 2001-10-24 16:49 Page 497