Environmental Engineering Reference
In-Depth Information
Fluid mass balance equations are obtained by combining the equations of mass
conservation for each fluid phase with that of the solid matrix:
[
]
n
∂
S
∂
⎛
1
⎞
∂
⎛
R
S
⎞
(
)
−
div
T
grad
p
1
+
ρ
1
gh
+
1
+
nS
⎜
⎜
⎝
⎟
⎟
⎠
+
n
⎜
⎜
⎝
s
1
⎟
⎟
⎠
m
1
B
∂
t
∂
t
B
∂
t
B
1
1
1
[14.9]
⎡
⎛
T
⎞
T
m
D
∂
ε
m
D
c
T
+
λ
⎜
⎜
⎝
m
−
T
⎟
⎟
⎠
+
T
⎢
⎣
f
3
∂
t
3
s
s
(
)
⎤
⎛
⎞
1
−
n
1
∂
p
⎜
⎜
⎝
⎟
⎟
⎠
+
−
m
T
D
m
=
0
⎥
⎥
( )
T
K
2
∂
t
3
⎦
s
s
where l = o,w,g. µ
l
is the fluid dynamic viscosity, B
l
the formation volume factor,
T
m
the total mobility of each fluid phase, and R
sl
the solubility of gas in the l phase.
The solubility of gas in water is neglected in the examples. As an initial condition,
we assume that the degrees of saturation of water and gas are above their critical
value so that these phases are mobile.
14.3.3.
The numerical model and its solution
A weak form of equations [14.8] and [14.9] is obtained by means of the Galerkin
procedure, while for the spatial discretization a finite element procedure is used
[ZIE 91]. The unknowns are displacements u and fluid pressures p
o
, p
w
and
p
g
.
These unknowns are expressed as functions of their nodal values through the shape
functions
N
, which are generally different for displacements and pressures:
l
l
[14.10]
p
=
N
ε
=
B
u
u
=
N
u
Matrix
B
is a differential operator. The linear momentum balance equation hence
becomes:
w
o
g
d
u
d
p
d
p
d
p
d
f
[14.11]
K
+
L
+
L
+
L
−
C
−
=
0
w
o
g
dt
dt
dt
dt
dt
The mass balance equation for water is:
w
o
g
d
p
d
p
d
p
d
u
[14.12]
W
p
w
+
W
+
W
W
+
W
+
F
=
0
p
w
o
g
u
w
dt
dt
dt
dt
