Numerical Simulation of 2-D Subsurface Reservoir

PTE 508 Project Two Report
Je↵rey Daniels
May 8, 2015
1 Reservoir Description
1.1 Reservoir Properties
The oil reservoir under study is approximated by a rectangular prism. The geometry
of this prism is length Lx = 1500 ft, width Ly = 1100 ft and height Z = 50 ft. The
reservoir has a reference pressure-dependent porosity of = 0.22 measured at 1000
psia, an isotropic permeability of k = 11 mD, and a formation compressibility factor of
ct = 2x10 6
psi 1
which is assumed to be constant.
1.2 Injection and Production Wells
Five wells, Well-A and Well-B, Well-C, Well-D and Well-E are drilled in the reservoir
first. Well-C serves as a producer and later as an injector in this reservoir simulation.
All wells have the same wellbore radius rw = 0.25 ft and the same skin factor of S = 0.
1.3 Reservoir Fluid Properties
The fluid flow is two-phase characterized by an initial oil saturation of Soi = 0.75 and
initial water saturation of Swi = 0.25. The fluid properties of oil are viscosity µo =
5 cp, oil compressibility co = 10 5
psi 1
and a reference formation volume factor Bo =
1.25 RB/STB measured at 1000 psia. The oil bubble point pressure is 1000 psia. The
fluid properties of water are viscosity µw = 1 cp, water compressibility cw = 10 6
psi 1
and a reference formation volume factor Bw = 1.02 RB/STB measured at 1000 psia.
2 Creation of Simulation cells
In this study, the reservoir is assumed to be two dimensional and was discretized into
165 simulation cells (i.e characterized by 11x15 grid blocks). Each cell with a well present
had a length of X = 100 ft, a width of Y = 100 ft and a height of Z = 50 ft.
1

3 Partial Di↵erential Equations (PDEs) With Initial and Bound-
ary Conditions
The PDEs to be discretized are as follows:
@
@x
( ox
@p
@x
) +
@
@y
( oy
@p
@y
) 887.53 (x xw) (y yw)
qo,wµB
z
= 158
@
@t
(
So
Bo
)
@
@x
( wx
@p
@x
) +
@
@y
( wy
@p
@y
) 887.53 (x xw) (y yw)
qw,wµB
z
= 158
@
@t
(
Sw
Bw
)
ox =
kxKro(So, Sw)
µoBo
, oy =
kyKro(So, Sw)
µoBo
wx =
kxKrw(So, Sw)
µwBw
, wy =
kyKrw(So, Sw)
µwBw
So + Sw = 1
Boundary conditions: @P
@x
|x=0= 0, @P
@x
|x=1500= 0 @P
@y
|y=0= 0, @P
@y
|y=1100= 0
Initial condition: P(x)|t0= 3000 psia
The PVT relations for formation volume factors Bo, Bw and are deﬁned by the fol-
lowing:
Bo(p) = 1.25 exp( Co(p 1000))
Bw(p) = 1.02 exp( Cw(p 1000))
(p) = 0.22 exp(C (p 1000))
The following plot shows the PVT relations graphically as the reservoir pressure de-
creases from 3000 psia to 1000 psia
1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000
1
1.05
1.1
1.15
1.2
1.25
FormationVolumeFactor,RB/STB
Plot of Bo
, Bw
and φ vs P
1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000
0.22
0.2202
0.2204
0.2206
0.2208
0.221
Porosity,v/v
Bw
Bo φ
2

Relative permeability curves, Kro and Krw are deﬁned by two-phase Corey’s model.
Swc = 0.2 and Sor = 0.
Sn =
Sw Swc
1 Sw
=
1 So Swc
1 Swc
Krw(Sw) =
(
S4
n , Sw > Swc
0 , Sw  Swc
Kro(Sw) =
(
(1 Sn)2
(1 S2
n) , So > SororSw < 1 Sor
0 , So  SororSw 1 Sor
The following plot shows the graphical relation between the relative perameabilities
and water saturation
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Water Saturation
RelativePermeability
Plot of Relative Permeability Curves
krw
kro
4 Implicit Block-Centered Grid System Discretization
4.1 General Discretized Equation
The following equation is used to solve for the grid cell pressures of the discretized
reservoir. It contains the PVT parameters for both water and oil.
SmPn+1
m Nx + WmPn+1
m 1 + CmPn+1
m + EmPn+1
m+1 + NmPn+1
m+Nx = Rm
where m = 1, 2, 3, ..., 165
Sm = ¯OS(
x
y
)2
Bo + ¯WS(
x
y
)2
Bw
Wm = ¯OW Bo + ¯WW Bw
3

Cm = Bo(¯OW +¯OE) (
x
y
)2
Bo(¯OS+¯ON ) Bw(¯WW +¯WE) (
x
y
)2
Bw(¯WS+¯WN )
158
x
t
Ct
Em = ¯OEBo + ¯WEBw
Nm = ¯ON (
x
y
)2
Bo + ¯WN (
x
y
)2
Bw
Rm = 158
x
t
CtPn
m + 887.53
qo,STBBo
x y
|ifwellincellm+887.53
qw,STBBw
x y
|ifwellincellm
To calculate the average mobility ratios
¯o,S =
kKro(So,up)
µoBo(Pm Nx+Pm
2
)
So,up =
(
So,m , Pm Pm Nx
So,m Nx , Pm < Pm Nx
¯o,N =
kKro(So,up)
µoBo(
Pm+Nx +Pm
2
)
So,up =
(
So,m , Pm Pm+Nx
So,m+Nx , Pm < Pm+Nx
¯o,W =
kKro(So,up)
µoBo(Pm 1+Pm
2
)
So,up =
(
So,m , Pm Pm 1
So,m 1 , Pm < Pm 1
¯o,E =
kKro(So,up)
µoBo(Pm+1+Pm
2
)
So,up =
(
So,m , Pm Pm+1
So,m+1 , Pm < Pm+1
Sw = 1 So
¯w,S =
kKrw(Sw,up)
µwBw(
Pm Nx +Pm
2
)
4

¯w,N =
kKrw(Sw,up)
µwBw(
Pm+Nx +Pm
2
)
¯w,W =
kKrw(Sw,up)
µwBw(Pm 1+Pm
2
)
¯w,E =
kKrw(Sw,up)
µwBw(Pm+1+Pm
2
)
4.2 Incorporating no flow boundary conditions
For no flow boundary conditions, the mobility ratios of water and oil are set to 0. At
the Southern Boundary:
¯os = ¯ws = 0
At the Western Boundary:
¯ow = ¯ww = 0
At the Eastern Boundary:
¯oe = ¯we = 0
At the Northern Boundary:
¯on = ¯wn = 0
4.3 Production term discretization
The production rate if a well exists in the grid cell is treated implicitly therefore
qSTB w = Jww(Pn+1
m Pn
m)
qo = Jwo(Pn+1
m Pn
m)
4.4 Finite-Di↵erence Equation for grid cells containing wells
The finite di↵erence equation for the grid cells with wells are as follows:
Well-A at m=34
S34Pn+1
19 + W34Pn+1
18 + C34Pn+1
34 + E34Pn+1
35 + N34Pn+1
49 = R34
S34 = ¯OS(
x
y
)2
Bo + ¯WS(
x
y
)2
Bw
W34 = ¯OW Bo + ¯WSBw
E34 = ¯OEBo + ¯WEBw
5

N34 = ¯ON (
x
y
)2
Bo + ¯WN (
x
y
)2
Bw
C34 = Bo(¯OW +¯OE) (
x
y
)2
x
y
)2
Bw(¯WS+¯WN )
158
x
t
Ct 887.53
Jwo well ABo
x y
887.53
Jww well ABw
x y
R34 = 158
x
t
CtPn
34 887.53
Jwo well APwf well ABo
x y
887.53
Jww well APwf well ABw
x y
Well-B at m=42
S42Pn+1
27 + W42Pn+1
41 + C42Pn+1
42 + E42Pn+1
43 + N42Pn+1
57 = R42
S42 = ¯OS(
x
y
)2
Bo + ¯WS(
x
y
)2
Bw
W42 = ¯OW Bo + ¯WW Bw
N42 = ¯ON (
x
y
)2
Bo + ¯WN (
x
y
)2
Bw
C42 = Bo(¯OW +¯OE) ( x
y
)2
Bo(¯OS +¯ON ) Bw(¯WW +¯WE) ( x
y
)2
Bw(¯WS +¯WN )
158
x
t
Ct 887.53
Jwo well BBo
x y
887.53
Jww well BBw
x y
R42 = 158
x
t
CtPn
42 887.53
JwoPwf well BBo
x y
887.53
Jww well BPwfABw
x y
Well-C at m=83
S83Pn+1
68 + W83Pn+1
82 + C83Pn+1
83 + E83Pn+1
84 + N83Pn+1
98 = R83
S83 = ¯OS(
x
y
)2
Bo + ¯WS(
x
y
)2
Bw
N83 = ¯ON (
x
y
)2
Bo + ¯WN (
x
y
)2
Bw
6

C83 = Bo(¯OW +¯OE) (
x
y
)2
x
y
)2
Bw(¯WS+¯WN )
158
x
t
Ct 887.53
Jwo well CBo
x y
887.53
Jww well CBw
x y
R83 = 158
x
t
CtPn
83 887.53
Jwo well CPwf well CBo
x y
887.53
Jww well CPwf well CBw
x y
Well-D at m=124
S124Pn+1
109 + W124Pn+1
123 + C124Pn+1
124 + E124Pn+1
125 + N124Pn+1
139 = R124
S124 = ¯OS(
x
y
)2
Bo + ¯WS(
x
y
)2
Bw
N124 = ¯ON (
x
y
)2
Bo + ¯WN (
x
y
)2
Bw
C124 = Bo(¯OW +¯OE) (
x
y
)2
x
y
)2
Bw(¯WS+¯WN )
158
x
t
Ct 887.53
Jwo well DBo
x y
887.53
Jww well DBw
x y
R124 = 158
x
t
CtPn
124 887.53
JwoPwf well DBo
x y
887.53
Jww well DPwf well DBw
x y
Well-E at m=132
S132Pn+1
117 + W132Pn+1
131 + C132Pn+1
132 + E132Pn+1
133 + N132Pn+1
147 = R132
S132 = ¯OS(
x
y
)2
Bo + ¯WS(
x
y
)2
Bw
N132 = ¯ON (
x
y
)2
Bo + ¯WN (
x
y
)2
Bw
7

C132 = Bo(¯OW +¯OE) (
x
y
)2
Bo(¯OS+¯ON ) Bw(¯WW
¯WE) (
x
y
)2
Bw(¯WS+¯WN )
158
x
t
Ct 887.53
Jwo well EBo
x y
887.53
Jww well EBw
x y
R132 = 158
x
t
CtPn
132 887.53
Jwo well EPwf well EBo
x y
887.53
Jww well EPwf well EBw
x y
4.5 Well Productivity Index Calculation using Peaceman’s method
Radial reservoir
re = 0.14 ⇤
p
X2 + Y 2
re = 0.14 ⇤
p
1002 + 1002 = 19.799ft
Well productivity index
Jwo = 7.08 ⇤ 10 3 kkroh
µoBo
⇤
1
ln re
rw
+ S
Jww = 7.08 ⇤ 10 3 kkrwh
µwBw
⇤
1
ln re
rw
+ S
For all wells
S = 0
5 Leakproof Test
In this test performed, the well ﬂow rates are speciﬁed to be zero for all the wells and
a simulation is run for 100 days. This is done by setting both Jwoil
and Jwwater to zero.
The time step used in this simulation is 10 days. The objective of this test is to ascer-
tain that the PDEs for the simulation cells have been discretized correctly. The pres-
sure, oil saturation and water saturation distributions are to remain at initial reser-
voir values of 3000 psia, 0.75 and 0.25 everywhere in the reservoir at all the simulation
times as shown in the plots below.
8

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure distribution at t=0 days
Pressure,psia
2990
2992
2994
2996
2998
3000
3002
3004
3006
3008
3010
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
2990
2992
2994
2996
2998
3000
3002
3004
3006
3008
3010
9

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
2990
2992
2994
2996
2998
3000
3002
3004
3006
3008
3010
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Oil saturation distribution at t=0 days
OilSaturation,v/v
0.7
0.71
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
0.8
10

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
OilSaturation,v/v
0.7
0.71
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
0.8
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
OilSaturation,v/v
0.7
0.71
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
0.8
11

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Water saturation distribution at t=0 days
WaterSaturation,v/v
0.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
WaterSaturation,v/v
0.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
12

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
WaterSaturation,v/v
0.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
6 Symmetry Test
In this test performed, the bottomhole ﬂowing pressure for the wells are set to 1000
psia thereby conﬁguring all the wells to be producers. A simulation is then run for 100
days. The time step used in this simulation is 10 days. The reservoir pressure distribu-
tion for this test is expected to be symmetric about the center (x = 750ft, y = 550ft)
of the reservoir as shown in the plots below. The pressure results are also published in
the tables in the additional results section.
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
2999
2999.2
2999.4
2999.6
2999.8
3000
3000.2
3000.4
3000.6
3000.8
3001
13

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1650
1700
1750
1800
1850
1900
1950
2000
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1300
1320
1340
1360
1380
1400
1420
1440
1460
7 Oil Initially In Place Determination and Material Balance
Check
The wells are conﬁgured to be producers by setting their bottomhole ﬂowing pressures
Pwf = 1000psia. A simulation is then run for 100 days. The time step used in this
simulation is 10 days. The Oil-Initially-In-Place (OIIP) was calculated to be approx-
imately 1986558 STB. The cumulative production of the reservoir along with the oil
in place is calculated for each time step as well as the error between Oil-Initially-In-
Place and the sum of cumulative oil production and Oil-In-Place at the end of each
14

time step. It is expected that there should be oil mass conservation hence the error is
expected to be within 0.1%.
The oil initially in place was determined by the following equation with initial reservoir
and PVT parameters at Pi = 3000psia:
OIIP =
1
5.615
⌃
NxNy
m=1 [
x y z (Pi)So,i
Bo(Pi)
]m
Nx = 15
Ny = 11
The Oil In Place at the end of each time step was calculated using the following equa-
tion:
OIPn+1
=
1
5.615
⌃
NxNy
m=1 [
x y z n+1
Sn+1
o
Bn+1
o
]m
The cumulative oil production is calculated as follows
V n+1
o,prod = V n
o,prod + qn+1
o t = ⌃T
n=1qn+1
o t
The error from oil mass conservation is determined by the following equation:
Error =
|OIIP (V n+1
o,prod + OIPn+1
)|
OIIP
⇤ 100
Material Balance Check
Time (Days) Cumulative Pro-
duction (STB)
Oil In Place
(STB)
Error (%)
10 7557 1978998 0.000154
20 13785 1972767 0.000282
30 19097 1967453 0.000390
40 23644 1962904 0.000483
50 27541 1959005 0.000563
60 30886 1955664 0.000631
70 33746 1952798 0.000689
80 36202 1950341 0.000739
90 38308 1948234 0.000783
100 40115 1946426 0.000819
15

8 Primary Recovery Study
In this study recovery of hydrocarbons from the reservoir is simulated without water
injection. The wells are conﬁgured to be producers with their ﬂowing pressures set to
1000 psia. The simulation is run until the oil production rate from Well-A falls below 1
STB/D. This occurs after 313 days of producing the reservoir. A time step of 1 day is
used in the simulation.
From the simulation results obtained, all the wells have an initial oil production rate of
197 STB/D and an initial water production rate of 0.021 STB/D however, after about
11 days the the oil and water production rate of all the well C slightly decreases while
the other wells continue to have equivalent oil and water production rates. This trend
continues until t = 171days, after which all the wells have equivalent oil and water
production rates again.
0 50 100 150 200 250 300 350
0
20
40
60
80
100
120
140
160
180
200
Time, days
Oilproductionrate,STB/D
Oil production rate vs time
Oil producion rate for Well A
Oil producion rate for Well B
Oil producion rate for Well C
Oil producion rate for Well D
Oil producion rate for Well E
16

0 50 100 150 200 250 300 350
0
0.005
0.01
0.015
0.02
0.025
Time, days
Waterproductionrate,STB/D
Water production rate vs time
Water producion rate for Well A
Water producion rate for Well B
Water producion rate for Well C
Water producion rate for Well D
Water producion rate for Well E
The cumulative oil production rate of the reservoir after 313 days of production is 50684
STB as show in the plot below. The recovery factor is 2.5% which is signiﬁcantly low.
Such a scenario recommends the need for pressure maintenance.
0 50 100 150 200 250 300 350
0
1
2
3
4
5
6
x 10
4
Cumulativeproduction,STB/D
Time, days
Cumulative Production vs time
The reservoir pressure at the wells decreases from 3000 psia to 1260 psia while the pres-
sure at the boundaries drops to 1420 psia after 100 days of production. At the end of
production, the pressure drops to 1008 psia at the wells while it drops to 1013 psia at
the boundaries at the end of production. The pressure decline in the ﬁrst t=100 days
is the fastest decline as compared to other 100 day intervals. However, the decreasing
pressure drop in the reservoir as time progresses is expected and explains the decreas-
ing oil and water production rates as time progresses.
17

0 50 100 150 200 250 300 350
0
0.5
1
1.5
2
2.5
3
Recoveryfactor,%
Time, days
Recovery factor vs time
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
2999
2999.2
2999.4
2999.6
2999.8
3000
3000.2
3000.4
3000.6
3000.8
3001
18

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1260
1280
1300
1320
1340
1360
1380
1400
1420
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1050
1055
1060
1065
1070
1075
1080
19

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1010
1011
1012
1013
1014
1015
1016
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Pressure distribution at end of production
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1008
1008.5
1009
1009.5
1010
1010.5
1011
1011.5
1012
1012.5
1013
The simulation results suggest that the reservoir cannot produce for long under its pri-
mary recovery mechanism alone.
20

9 Secondary Recovery Study
In this study recovery of hydrocarbons from the reservoir is simulated with water injec-
tion for pressure maintenance. Initially, the wells are configured to be producers with
their flowing pressures set to 1000 psia. However after the oil production rate from
Well-A falls below 1 STB/D, Well-C is converted to an injector with a flowing pres-
sure of 2950 psia. The simulation is run until 10000 days. A time step of 1 day is used
in the simulation.
Water injection is started at t = 313days. It is observed from the oil saturation distri-
bution below that there is still a significant amount of hydrocarbons in the reservoir.
However, the due to a low pressure drop as shown in the pressure distribution plot, the
reservoir cannot produce at a prolific rate.
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Oil saturation distribution at start of injection
Reservoir Length,ft
ReservoirWidth,ft
OilSaturation,v/v
0.7485
0.7485
0.7485
0.7485
21

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Pressure distribution at start of injection
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1008
1008.5
1009
1009.5
1010
1010.5
1011
1011.5
1012
1012.5
1013
At the end of the simulation (t = 10000days), the oil saturation distribution plot shows
that a signiﬁcant amount of the reservoir oil has been produced with most of the re-
maining oil in place located at the corners or the reservoir. The pressure distribution
shows a large pressure drop which is able to support the reservoir to produce at a pro-
liﬁc rate. The average pressures at the producing well locations are 1700 psia while the
average pressure at the injection well location is 2400 psia.
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
OilSaturation,v/v
0.1
0.2
0.3
0.4
0.5
0.6
0.7
22

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1700
1800
1900
2000
2100
2200
2300
2400
The cumulative oil production at the end of 10000days is 969687 STB. The recovery
factor is 48.81% which is signiﬁcantly higher compared to when the reservoir was only
under primary recovery.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0
1
2
3
4
5
6
7
8
9
10
x 10
5
Cumulativeproduction,STB/D
Time, days
Cumulative Production vs time
23

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0
5
10
15
20
25
30
35
40
45
50
Recoveryfactor,%
Time, days
Recovery factor vs time
Also the time for which watercut increased above 20% at Well-A was determined to
be at t=3094 days.The oil saturation distribution shows that a signiﬁcant amount of
oil remained in the reservoir so continued production with an increasing water cut is
justiﬁed. At the end of the simulation the water cut was approximately 90%.
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Oil saturation distribution at Water Breakthrough Time
Reservoir Length,ft
ReservoirWidth,ft
OilSaturation,v/v
0.1
0.2
0.3
0.4
0.5
0.6
0.7
24

200 400 600 800 1000 1200 1400
100
200
300
400
500
600
700
800
900
1000
Pressure Distribution at Water Breakthrough Time
Reservoir Length,ft
ReservoirWidth,ft
Pressure,psia
1800
1900
2000
2100
2200
2300
2400
2500
The simulation results suggest that the reservoir produces for much longer when there
is pressure maintenance to support production.
10 Conclusion
Water Injection aids in realizing signifcantly higher recovery factors and slower oil pro-
duction rates as time progresses. The simulation studies show that under primary re-
covery alone, the recovery factor was 2.5% as compared to the 60% when there was wa-
ter injection
However, with water injection as serving as pressure maintenance, there is a risk of
higher water cut as time progresses. In the primary recovery study, the water cut was
almost negligible while during the secondary recovery study, the water cut kept increas-
ing after water injection was initiated.
25

11 Additional Results
Pressure distribution at t = 0days
x(ft) y(ft) P(x,y,t)
50 50 3000
150 50 3000
250 50 3000
350 50 3000
450 50 3000
550 50 3000
650 50 3000
750 50 3000
850 50 3000
950 50 3000
1050 50 3000
1150 50 3000
1250 50 3000
1350 50 3000
1450 50 3000
50 150 3000
150 150 3000
250 150 3000
350 150 3000
450 150 3000
550 150 3000
650 150 3000
750 150 3000
850 150 3000
950 150 3000
1050 150 3000
1150 150 3000
26

1250 150 3000
1350 150 3000
1450 150 3000
50 250 3000
150 250 3000
250 250 3000
350 250 3000
450 250 3000
550 250 3000
650 250 3000
750 250 3000
850 250 3000
950 250 3000
1050 250 3000
1150 250 3000
1250 250 3000
1350 250 3000
1450 250 3000
50 350 3000
150 350 3000
250 350 3000
350 350 3000
450 350 3000
550 350 3000
650 350 3000
750 350 3000
850 350 3000
950 350 3000
1050 350 3000
1150 350 3000
1250 350 3000
27

1350 350 3000
1450 350 3000
50 450 3000
150 450 3000
250 450 3000
350 450 3000
450 450 3000
550 450 3000
650 450 3000
750 450 3000
850 450 3000
950 450 3000
1050 450 3000
1150 450 3000
1250 450 3000
1350 450 3000
1450 450 3000
50 550 3000
150 550 3000
250 550 3000
350 550 3000
450 550 3000
550 550 3000
650 550 3000
750 550 3000
850 550 3000
950 550 3000
1050 550 3000
1150 550 3000
1250 550 3000
1350 550 3000
28

1450 550 3000
50 650 3000
150 650 3000
250 650 3000
350 650 3000
450 650 3000
550 650 3000
650 650 3000
750 650 3000
850 650 3000
950 650 3000
1050 650 3000
1150 650 3000
1250 650 3000
1350 650 3000
1450 650 3000
50 750 3000
150 750 3000
250 750 3000
350 750 3000
450 750 3000
550 750 3000
650 750 3000
750 750 3000
850 750 3000
950 750 3000
1050 750 3000
1150 750 3000
1250 750 3000
1350 750 3000
1450 750 3000
29

50 850 3000
150 850 3000
250 850 3000
350 850 3000
450 850 3000
550 850 3000
650 850 3000
750 850 3000
850 850 3000
950 850 3000
1050 850 3000
1150 850 3000
1250 850 3000
1350 850 3000
1450 850 3000
50 950 3000
150 950 3000
250 950 3000
350 950 3000
450 950 3000
550 950 3000
650 950 3000
750 950 3000
850 950 3000
950 950 3000
1050 950 3000
1150 950 3000
1250 950 3000
1350 950 3000
1450 950 3000
50 1050 3000
30

150 1050 3000
250 1050 3000
350 1050 3000
450 1050 3000
550 1050 3000
650 1050 3000
750 1050 3000
850 1050 3000
950 1050 3000
1050 1050 3000
1150 1050 3000
1250 1050 3000
1350 1050 3000
1450 1050 3000
50 50 2011
150 50 1987
250 50 1949
350 50 1919
450 50 1936
550 50 1964
650 50 1985
750 50 1992
850 50 1985
950 50 1964
1050 50 1936
1150 50 1919
1250 50 1949
31

1350 50 1987
1450 50 2011
50 150 2005
150 150 1974
250 150 1914
350 150 1845
450 150 1897
550 150 1943
650 150 1969
750 150 1977
850 150 1969
950 150 1943
1050 150 1897
1150 150 1845
1250 150 1914
1350 150 1974
1450 150 2005
50 250 2002
150 250 1961
250 250 1861
350 250 1626
450 250 1836
550 250 1915
650 250 1944
750 250 1950
850 250 1944
950 250 1915
1050 250 1836
1150 250 1626
1250 250 1861
1350 250 1961
32

1450 250 2002
50 350 2011
150 350 1979
250 350 1918
350 350 1843
450 350 1882
550 350 1910
650 350 1914
750 350 1906
850 350 1914
950 350 1910
1050 350 1882
1150 350 1843
1250 350 1918
1350 350 1979
1450 350 2011
50 450 2022
150 450 1999
250 450 1960
350 450 1921
450 450 1914
550 450 1903
650 450 1869
750 450 1818
850 450 1869
950 450 1903
1050 450 1914
1150 450 1921
1250 450 1960
1350 450 1999
1450 450 2022
33

50 550 2027
150 550 2007
250 550 1974
350 550 1942
450 550 1922
550 550 1892
650 550 1815
750 550 1604
850 550 1815
950 550 1892
1050 550 1922
1150 550 1942
1250 550 1974
1350 550 2007
1450 550 2027
50 650 2022
150 650 1999
250 650 1960
350 650 1921
450 650 1914
550 650 1903
650 650 1869
750 650 1818
850 650 1869
950 650 1903
1050 650 1914
1150 650 1921
1250 650 1960
1350 650 1999
1450 650 2022
50 750 2011
34

150 750 1979
250 750 1918
350 750 1843
450 750 1882
550 750 1910
650 750 1914
750 750 1906
850 750 1914
950 750 1910
1050 750 1882
1150 750 1843
1250 750 1918
1350 750 1979
1450 750 2011
50 850 2002
150 850 1961
250 850 1861
350 850 1626
450 850 1836
550 850 1915
650 850 1944
750 850 1950
850 850 1944
950 850 1915
1050 850 1836
1150 850 1626
1250 850 1861
1350 850 1961
1450 850 2002
50 950 2005
150 950 1974
35

250 950 1914
350 950 1845
450 950 1897
550 950 1943
650 950 1969
750 950 1977
850 950 1969
950 950 1943
1050 950 1897
1150 950 1845
1250 950 1914
1350 950 1974
1450 950 2005
50 1050 2011
150 1050 1987
250 1050 1949
350 1050 1919
450 1050 1936
550 1050 1964
650 1050 1985
750 1050 1992
850 1050 1985
950 1050 1964
1050 1050 1936
1150 1050 1919
1250 1050 1949
1350 1050 1987
1450 1050 2011
36

50 50 1472
150 50 1461
250 50 1443
350 50 1429
450 50 1436
550 50 1449
650 50 1458
750 50 1462
850 50 1458
950 50 1449
1050 50 1436
1150 50 1429
1250 50 1443
1350 50 1461
1450 50 1472
50 150 1470
150 150 1455
250 150 1427
350 150 1394
450 150 1418
550 150 1439
650 150 1451
750 150 1455
850 150 1451
950 150 1439
1050 150 1418
1150 150 1394
1250 150 1427
1350 150 1455
37

1450 150 1470
50 250 1468
150 250 1449
250 250 1402
350 250 1292
450 250 1389
550 250 1426
650 250 1439
750 250 1442
850 250 1439
950 250 1426
1050 250 1389
1150 250 1292
1250 250 1402
1350 250 1449
1450 250 1468
50 350 1472
150 350 1457
250 350 1428
350 350 1393
450 350 1411
550 350 1424
650 350 1425
750 350 1421
850 350 1425
950 350 1424
1050 350 1411
1150 350 1393
1250 350 1428
1350 350 1457
1450 350 1472
38

50 450 1478
150 450 1467
250 450 1448
350 450 1430
450 450 1426
550 450 1420
650 450 1404
750 450 1380
850 450 1404
950 450 1420
1050 450 1426
1150 450 1430
1250 450 1448
1350 450 1467
1450 450 1478
50 550 1480
150 550 1471
250 550 1455
350 550 1439
450 550 1430
550 550 1415
650 550 1379
750 550 1281
850 550 1379
950 550 1415
1050 550 1430
1150 550 1439
1250 550 1455
1350 550 1471
1450 550 1480
50 650 1478
39

150 650 1467
250 650 1448
350 650 1430
450 650 1426
550 650 1420
650 650 1404
750 650 1380
850 650 1404
950 650 1420
1050 650 1426
1150 650 1430
1250 650 1448
1350 650 1467
1450 650 1478
50 750 1472
150 750 1457
250 750 1428
350 750 1393
450 750 1411
550 750 1424
650 750 1425
750 750 1421
850 750 1425
950 750 1424
1050 750 1411
1150 750 1393
1250 750 1428
1350 750 1457
1450 750 1472
50 850 1468
150 850 1449
40

250 850 1402
350 850 1292
450 850 1389
550 850 1426
650 850 1439
750 850 1442
850 850 1439
950 850 1426
1050 850 1389
1150 850 1292
1250 850 1402
1350 850 1449
1450 850 1468
50 950 1470
150 950 1455
250 950 1427
350 950 1394
450 950 1418
550 950 1439
650 950 1451
750 950 1455
850 950 1451
950 950 1439
1050 950 1418
1150 950 1394
1250 950 1427
1350 950 1455
1450 950 1470
50 1050 1472
150 1050 1461
250 1050 1443
41

350 1050 1429
450 1050 1436
550 1050 1449
650 1050 1458
750 1050 1462
850 1050 1458
950 1050 1449
1050 1050 1436
1150 1050 1429
1250 1050 1443
1350 1050 1461
1450 1050 1472
Pressure and Saturation distribution at start of injection
x(ft) y(ft) P(x,y,t) So(x,y,t)
50 50 1013 0.74851
150 50 1013 0.74851
250 50 1012 0.74850
350 50 1012 0.74850
450 50 1012 0.74850
550 50 1012 0.74850
650 50 1013 0.74850
750 50 1013 0.74851
850 50 1013 0.74850
950 50 1012 0.74850
1050 50 1012 0.74850
1150 50 1012 0.74850
1250 50 1012 0.74850
1350 50 1013 0.74851
1450 50 1013 0.74851
42

50 150 1013 0.74851
150 150 1013 0.74850
250 150 1012 0.74850
350 150 1011 0.74850
450 150 1012 0.74850
550 150 1012 0.74850
650 150 1013 0.74850
750 150 1013 0.74850
850 150 1013 0.74850
950 150 1012 0.74850
1050 150 1012 0.74850
1150 150 1011 0.74850
1250 150 1012 0.74850
1350 150 1013 0.74850
1450 150 1013 0.74851
50 250 1013 0.74851
150 250 1012 0.74850
250 250 1011 0.74850
350 250 1008 0.74849
450 250 1011 0.74850
550 250 1012 0.74850
650 250 1012 0.74850
750 250 1012 0.74850
850 250 1012 0.74850
950 250 1012 0.74850
1050 250 1011 0.74850
1150 250 1008 0.74849
1250 250 1011 0.74850
1350 250 1012 0.74850
1450 250 1013 0.74851
50 350 1013 0.74851
43

150 350 1013 0.74850
250 350 1012 0.74850
350 350 1011 0.74850
450 350 1011 0.74850
550 350 1012 0.74850
650 350 1012 0.74850
750 350 1012 0.74850
850 350 1012 0.74850
950 350 1012 0.74850
1050 350 1011 0.74850
1150 350 1011 0.74850
1250 350 1012 0.74850
1350 350 1013 0.74850
1450 350 1013 0.74851
50 450 1013 0.74851
150 450 1013 0.74851
250 450 1012 0.74850
350 450 1012 0.74850
450 450 1012 0.74850
550 450 1012 0.74850
650 450 1011 0.74850
750 450 1011 0.74850
850 450 1011 0.74850
950 450 1012 0.74850
1050 450 1012 0.74850
1150 450 1012 0.74850
1250 450 1012 0.74850
1350 450 1013 0.74851
1450 450 1013 0.74851
50 550 1013 0.74851
150 550 1013 0.74851
44

250 550 1013 0.74850
350 550 1012 0.74850
450 550 1012 0.74850
550 550 1012 0.74850
650 550 1011 0.74850
750 550 1008 0.74849
850 550 1011 0.74850
950 550 1012 0.74850
1050 550 1012 0.74850
1150 550 1012 0.74850
1250 550 1013 0.74850
1350 550 1013 0.74851
1450 550 1013 0.74851
50 650 1013 0.74851
150 650 1013 0.74851
250 650 1012 0.74850
350 650 1012 0.74850
450 650 1012 0.74850
550 650 1012 0.74850
650 650 1011 0.74850
750 650 1011 0.74850
850 650 1011 0.74850
950 650 1012 0.74850
1050 650 1012 0.74850
1150 650 1012 0.74850
1250 650 1012 0.74850
1350 650 1013 0.74851
1450 650 1013 0.74851
50 750 1013 0.74851
150 750 1013 0.74850
250 750 1012 0.74850
45

350 750 1011 0.74850
450 750 1011 0.74850
550 750 1012 0.74850
650 750 1012 0.74850
750 750 1012 0.74850
850 750 1012 0.74850
950 750 1012 0.74850
1050 750 1011 0.74850
1150 750 1011 0.74850
1250 750 1012 0.74850
1350 750 1013 0.74850
1450 750 1013 0.74851
50 850 1013 0.74851
150 850 1012 0.74850
250 850 1011 0.74850
350 850 1008 0.74849
450 850 1011 0.74850
550 850 1012 0.74850
650 850 1012 0.74850
750 850 1012 0.74850
850 850 1012 0.74850
950 850 1012 0.74850
1050 850 1011 0.74850
1150 850 1008 0.74849
1250 850 1011 0.74850
1350 850 1012 0.74850
1450 850 1013 0.74851
50 950 1013 0.74851
150 950 1013 0.74850
250 950 1012 0.74850
350 950 1011 0.74850
46

450 950 1012 0.74850
550 950 1012 0.74850
650 950 1013 0.74850
750 950 1013 0.74850
850 950 1013 0.74850
950 950 1012 0.74850
1050 950 1012 0.74850
1150 950 1011 0.74850
1250 950 1012 0.74850
1350 950 1013 0.74850
1450 950 1013 0.74851
50 1050 1013 0.74851
150 1050 1013 0.74851
250 1050 1012 0.74850
350 1050 1012 0.74850
450 1050 1012 0.74850
550 1050 1012 0.74850
650 1050 1013 0.74850
750 1050 1013 0.74851
850 1050 1013 0.74850
950 1050 1012 0.74850
1050 1050 1012 0.74850
1150 1050 1012 0.74850
1250 1050 1012 0.74850
1350 1050 1013 0.74851
1450 1050 1013 0.74851
47

Numerical Simulation of 2-D Subsurface Reservoir

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