Environmental Engineering Reference
In-Depth Information
The mass balance equation for the oil phase reads:
w
o
g
d
p
d
p
d
p
d
u
o
[14.13]
H
p
+
H
+
H
+
H
+
H
+
F
=
0
p
w
o
g
u
o
dt
dt
dt
dt
and for the gas phase:
w
o
g
d
p
d
p
d
p
d
u
[14.14]
G
p
g
+
G
+
G
+
G
+
G
+
F
=
0
p
w
o
g
u
g
dt
dt
dt
dt
Equations [14.11] to [14.14] represent a set of ordinary differential equations in
time, and may be written for convenience in the following form:
0
0
0
0
K
L
L
L
⎡
−
d
f
/
dt
C
⎡
⎤
⎡
u
⎤
⎡
⎤
⎡
u
⎤
⎤
w
o
g
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
0
W
0
0
w
W
W
W
W
w
F
p
p
d
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎢
⎥
⎥
⎢
⎢
⎥
⎥
p
u
w
o
g
w
+
=
⎢
⎢
⎢
0
0
H
0
⎥
⎥
⎥
o
⎢
⎢
⎢
H
H
H
H
⎥
⎥
⎥
o
⎢
⎢
⎢
F
⎥
⎥
⎥
p
dt
p
p
u
w
o
g
o
⎢
⎥
⎢
⎥
0
0
0
G
p
g
G
G
G
G
p
g
F
⎣
⎦
⎣
⎦
⎣
⎦
⎣
⎦
⎣
⎦
g
u
w
o
g
g
[14.15]
For the time discretization, an implicit finite difference method is used. This
gives the final system to be solved:
K
L
L
L
⎡
⎤
⎡∆
u
⎤
w
o
g
⎢
⎥
⎢
⎥
I
p
W
W
W
W
w
p
⎢
⎥
⎢
⎢
⎥
⎥
u
o
g
⎢
⎢
⎢
H
H
H
I
p
H
⎥
⎥
⎥
o
p
u
w
g
⎢
⎥
I
p
[14.16]
G
G
G
G
p
g
⎣
⎦
⎣
⎦
u
w
o
t
+
∆
t
n
⎡
L
L
L
⎤
⎡
d
f
/
dt
−
C
⎤
w
o
g
⎡
w
⎤
p
⎢
⎥
⎢
⎥
II
p
W
W
W
F
⎢
⎥
⎢
⎢
⎥
⎥
⎢
⎢
⎥
⎥
o
g
w
o
=
p
+
∆
t
⎢
⎢
⎥
⎥
II
p
H
H
H
F
w
g
o
p
g
⎢
⎢
⎥
⎥
⎢
⎢
⎥
⎥
⎣
⎦
G
G
G
II
p
F
t
⎣
⎦
⎣
⎦
n
w
o
g
Equation [14.16] represents a fully coupled non-linear and non-symmetric
system of equations for a three-phase flow in a deforming porous medium. Since all
the coefficients depend on the unknowns, iterative procedures such as Newton-
Raphson-type procedures must be adopted within each time step to obtain the final
solution. For the analysis of the non-linear aspects and for the numerical properties
of the techniques used, see [LEW 98b], where the component matrices of system
[14.16] can be found.
