Tarek Ahmed. Reservoir engineering handbook
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598 Reservoir Engineering Handbook
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b
c
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e
E
D
C
B
A
Distance from Top Perforation to Top
of Sand or Gas-Oil Contact–In Feet
Critical Production Rates–Reservoir bbl/day
Figure 9-19. Critical production rate curves. (After Chaney et al., courtesy OGJ,
May 1956.)
Critical-production-rate curves for sand thickness of 12.5 ft., well radius of 3 in.,
and drainage radius of 1,000 ft. Water coning curves: A, 1.25 ft. perforated inter-
val; B, 2.5 ft.; C, 3.75 ft.; D, 5.00 ft.; and E, 6.25 ft. Gas coning curves: a, 1.25 ft.
perforated interval; b, 2.5 ft.; c, 3.75 ft.; d, 5.00 ft., and e, 6.25 ft.
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 598
Gas and Water Coning 599
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a
b
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d
e
E
D
C
B
A
Distance from Top Perforation to Top
of Sand or Gas-Oil Contact–In Feet
Critical Production Rates–Reservoir bbl/day
Figure 9-20. Critical production rate curves. (After Chaney et al., courtesy OGJ,
May 1956.)
Critical-production-rate curves for sand thickness of 25 ft., well radius of 3 in., and
drainage radius of 1,000 ft. Water coning curves: A, 2.5 ft. perforated interval; B, 5
ft.; C, 7.5 ft.; D, 10 ft.; and E, 12.5 ft. Gas coning curves: a, 2.5 ft. perforated inter-
val; b, 5 ft.; c, 7.5 ft.; d, 10 ft., and e, 12.5 ft.
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 599
600 Reservoir Engineering Handbook
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a
b
c
d
e
E
D
C
B
A
Distance from Top Perforation to Top
of Sand or Gas-Oil Contact–In Feet
Critical Production Rates–Reservoir bbl/day
Figure 9-21. Critical production rate curves. (After Chaney et al., courtesy OGJ,
May 1956.)
Critical-production-rate curves for sand thickness of 50 ft., well radius of 3 in., and
drainage radius of 1,000 ft. Water coning curves: A, 5 ft. perforated interval; B, 10
ft.; C, 15 ft.; D, 20 ft.; and E, 25 ft. Gas coning curves: a, 5 ft. perforated interval; b,
10 ft.; c, 15 ft.; d, 20 ft., and e, 25 ft.
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 600
Gas and Water Coning 601
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a
b
c
d
e
E
D
C
B
A
Distance fom Top Perforation to Top
of Sand or Gas-Oil Contact–In Feet
Critical Production Rates–Reservoir bbl/day
Figure 9-22. Critical production rate curves. (After Chaney et al., courtesy OGJ,
May 1956.)
Critical-production-rate curves for sand thickness of 75 ft., well radius of 3 in., and
drainage radius of 1,000 ft. Water coning curves: A, 7.5 ft. perforated interval; B,
15 ft.; C, 22.5 ft.; D, 30 ft.; and E, 37.5 ft. Gas coning curves: a, 7.5 ft. perforated
interval; b, 15 ft.; c, 22.5 ft.; d, 30 ft., and e, 37.5 ft.
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 601
602 Reservoir Engineering Handbook
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d
e
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D
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B
A
Distance from Top Perforation to Top
of Sand or Gas-Oil Contact–In Feet
Critical Production Rates–Reservoir bbl/day
Figure 9-23. Critical production rate curves. (After Chaney et al., courtesy OGJ,
May 1956.)
Critical-production-rate curves for sand thickness of 100 ft., well radius of 3 in.,
and drainage radius of 1,000 ft. Water coning curves: A, 10 ft. perforated interval;
B, 20 ft.; C, 30 ft.; D, 40 ft.; and E, 50 ft. Gas coning curves: a, 10 ft. perforated
interval; b, 20 ft.; c, 30 ft.; d, 40 ft., and e, 50 ft.
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 602
where r
o
= oil density, lb/ft
3
r
w
= water density, lb/ft
3
Q
oc
= critical oil flow rate, STB/day
k
o
= effective oil permeability, md
In gas-water systems
where r
g
= gas density, lb/ft
3
r
w
= water density, lb/ft
3
Q
gc
= critical gas flow rate, Mscf/day
b
g
= gas FVF, bbl/Mscf
k
g
= effective gas permeability, md
In gas-oil systems
Example 9-8
In an oil-water system, the following fluid and sand data are available:
h = 50¢ h
p
= 15¢
r
o
= 47.5 lb/ft
3
r
w
= 63.76 lb/ft
3
m
o
= 0.73 cp B
o
= 1.1 bbl/STB
r
w
= 3≤ r
e
= 1000¢
k
o
= 93.5 md
Calculate the oil critical rate.
Solution
Step 1. Distance from the top of the perforations to top of the sand = 0¢
oc
o
og
o
o
curve
Q
= 0.2676
k
()
B
Q
¥
-
È
Î
Í
˘
˚
˙
-
10
4
rr
m
(9 -16)
gc
g
wg
g
g
curve
Q
= 0.5288
k
()
B
Q
¥
-
È
Î
Í
Í
˘
˚
˙
˙
-
10
4
rr
m
(9 -15)
Gas and Water Coning 603
(text continued from page 597)
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 603
Step 2. Using Figure 9-20, for h = 50, enter the graph with 0¢ and move
vertically to curve C to give:
Q
curve
= 270 bbl/day
Step 3. Calculate critical oil rate from Equation 9-14.
The above method can be used through the trial-and-error procedure to
optimize the location of the perforated interval in two-cone systems. It
should be pointed out that Chaney’s method was developed for a homo-
geneous, isotropic reservoir with k
v
= k
h
.
Chaperson’s Method
Chaperson (1986) proposed a simple relationship to estimate the criti-
cal rate of a vertical well in an anisotropic formation (k
v
π k
h
). The rela-
tionship accounts for the distance between the production well and
boundary. The proposed correlation has the following form:
where Q
oc
= critical oil rate, STB/day
k
h
= horizontal permeability, md
Dr = r
w
-r
o
, density difference, lb/ft
3
h = oil column thickness, ft
h
p
= perforated interval, ft
Joshi (1991) correlated the coefficient q
*
c
with the parameter a≤ as
q
*
c
= 0.7311 + (1.943/a≤) (9 - 18)
¢¢
a =(
r
/
h)
k
/
k
evh
()919-
oc
h
2
p
o
o
c
Q
= 0.0783
k
(h
h
)
B
[ ]q¥
-
-
10 9 17
4
m
rD
*
()-
oc
Q
= 0.5288
93.5 (63.76 47.5)
(1.1) (0.73)
270 = 27 STB/day¥
-
È
Î
Í
˘
˚
˙
-
10
4
604 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 604
Example 9-9
The following data are available on an oil-water system:
h = 50¢ r
e
= 1000¢m
o
= 0.73 cp
B
o
= 1.1 bbl/STB r
w
= 63.76 lb/ft
3
k
h
= 100 md
r
o
= 47.5 lb/ft
3
k
v
= 10 md h
P
= 15¢
Calculate the critical rate.
Solution
Step 1. Calculate a≤ from Equation 9-19.
Step 2. Solve for q
*
c
by applying Equation 9-18.
q
*
c
= 0.7311 + (1.943/6.324) = 1.0383
Step 3. Solve for the critical oil rate Q
oc
by using Equation 9-17.
Schols’ Method
Schols (1972) developed an empirical equation based on results
obtained from numerical simulator and laboratory experiments. His criti-
cal rate equation has the following form:
oc
wo
o
2
p
2
o
o
ew
0.14
e
Q
= 0.0783
()
k
(
hh
)
u
B
0.432 +
3.142
l(
r
/
r
)
(h /r )
¥
--
È
Î
Í
˘
˚
˙
¥
È
Î
Í
˘
˚
˙
-
10
4
rr
n
(9 - 20)
oc
2
oc
Q
= 0.0783
(100)
(50 15)
(0.73) (1.1)
[63.76 47.5] (1.0383)
Q = 20.16 STB/day
¥
-
-
-
10
4
¢¢
a = (1000/50) 10 /100 = 6.324
Gas and Water Coning 605
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 605
where k
o
= effective oil permeability, md
r
w
= wellbore radius, ft
h
p
= perforated interval, ft
r=density, lb/ft
3
Schols’ equation is only valid for isotropic formation, i.e., k
h
= k
v
.
Example 9-10
In an oil-water system, the following fluid and rock data are available:
h = 50¢ h
p
= 15¢r
o
= 47.5 lb/ft
3
r
w
= 63.76 lb/ft
3
m
o
= 0.73 cp B
o
= 1.1 bbl/STB r
e
= 1000¢ r
w
= 0.25¢
k
o
= k = 93.5 md
Calculate the critical oil flow rate.
Solution
Applying Equation 9-20, gives
BREAKTHROUGH TIME IN VERTICAL WELLS
Critical flow rate calculations frequently show low rates that, for eco-
nomic reasons, cannot be imposed on production wells. Therefore, if a
well produces above its critical rate, the cone will break through after a
given time period. This time is called time to breakthrough t
BT
. Two of
the most widely used correlations are documented below.
The Sobocinski-Cornelius Method
Sobocinski and Cornelius (1965) developed a correlation for predict-
ing water breakthrough time based on laboratory data and modeling
results. The authors correlated the breakthrough time with two dimen-
oc
22
.14
oc
Q
= 0.0783
(63.76 47.5) (93.5) (
50 15
)
(0.73) (1.1)
0.432 +
3.142
l (1000 /.25)
(50/1000
)
Q = 18 STB/day
¥
--
È
Î
Í
˘
˚
˙
¥
È
Î
Í
˘
˚
˙
-
10
4
0
n
606 Reservoir Engineering Handbook
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 606
sionless parameters, the dimensionless cone height and the dimensionless
breakthrough time. Those two dimensionless parameters are defined by
the following expressions:
Dimensionless cone height Z
where r=density, lb/ft
3
k
h
= horizontal permeability, md
Q
o
= oil production rate, STB/day
h
p
= perforated interval, ft
h = oil column thickness, ft
Dimensionless breakthrough time (t
D
)
BT
The authors proposed the following expression for predicting time to
breakthrough from the calculated value of the dimensionless break-
through time (t
D
)
BT
:
where t
BT
= time to breakthrough, days
f=porosity, fraction
k
v
= vertical permeability, md
M = water-oil mobility and is defined by:
with (k
ro
)
swc
= oil relative permeability at connate water saturation
(k
rw
)
sor
= water relative permeability at residual oil saturation
a=0.5 for M ≤ 1
a=0.6 for 1 < M ≤ 10
Joshi (1991) observed by examining Equation 9-22 that if Z = 3.5 or
greater, there will be no water breakthrough. This observation can be
M=
k
k
rw sor
ro swc
o
w
()
()
È
Î
Í
˘
˚
˙
Ê
Ë
Á
ˆ
¯
˜
m
m
(9 - 24)
t=
20,325 h (
t
)
()
k
(1+ M )
BT
o
D
BT
wo
v
m
f
rr
a
-
(9 - 23)
()t=
4 Z 1.75
Z
0.75 Z
72 Z
DBT
2
+-
-
3
(9 - 22)
Z = 0.492
khh h
BQ
woh p
oo o
¥
--
-
10
4
()()rr
m
(9 - 21)
Gas and Water Coning 607
Reservoir Eng Hndbk Ch 09 2001-10-25 08:37 Page 607