Environmental Engineering Reference
In-Depth Information
s
s
π
w
s
s
π
w
S
c
D
ρ
D
D
c
(
)
w
1
n
+
n
c
π
w
S
+
n
S
w
w
s
Dt
Dt
Dt
ρ
(
)
[14.19]
π
w
ws
π
w
s
+
div
n
S
c
v
+
S
c
div
v
w
w
(
)
π
π
w
π
w
=
I
+
div
ρ
D
grad
c
w
and the solute conservation in air may be written as:
s
S
c
π
g
D
s
c
π
g
s
s
D
ρ
D
(
)
g
1
n
+
n
c
π
g
S
+
n
S
g
w
s
Dt
Dt
Dt
ρ
(
)
[14.20]
π
g
ws
π
g
s
+
div
n
S
c
v
+
S
c
div
v
g
g
(
)
π
π
g
π
g
=
I
+
div
ρ
D
grag
c
g
where D w and D g are the water and air diffusion coefficients respectively, c w and c g
the concentrations of the contaminants, and I w and I g the exchange terms between
phases. The discretization in space of the complete set of equations is carried out as
in the previous section and yields the following coupled, non-symmetric and non
linear system of equations:
&
&
T
w
g
& σ
&
&
B
'
d
+
C
p
+
C
p
+
C
T
=
f
sw
sa
st
s
&
&
&
&
&
w
&
g
π
π
w
C
u
&
+
P
p
+
C
p
+
C
T
+
C
C
+
C
C
+
H
p
=
f
ws
ww
wa
wt
wc
w
wc
g
ww
w
w
a
&
&
&
&
w
g
π
π
g
C
u
&
+
C
p
&
+
P
p
&
+
C
T
+
C
C
+
C
C
+
H
p
=
f
as
aw
aa
at
ac
w
ac
g
aa
a
w
a
&
&
&
&
w
g
π
π
C
p
&
+
C
p
&
+
P
T
+
C
C
+
C
C
+
H
T
=
f
tw
ta
tt
tc
w
tc
g
tt
t
w
a
&
&
&
&
w
g
π
π
π
&
&
C
p
+
C
p
+
C
T
+
P
C
+
C
C
+
H
C
=
f
c
w
c
a
c
t
c
c
w
c
c
g
c
c
w
c
w
w
w
w
w
w
a
w
w
w
&
&
&
&
w
g
π
π
π
&
&
C
p
+
C
p
+
C
T
+
C
C
+
P
C
+
H
C
=
f
c
w
c
a
c
t
c
c
w
c
c
g
c
c
g
c
[14.21]
a
a
a
a
w
a
a
a
a
a
The time integration is carried out with an implicit scheme. The component
matrices of system [14.21] and further details can be found in [SCH 99].
14.3.5. Examples
14.3.5.1. Compaction of a petroleum reservoir
This example has been selected to show the need for including deformation of
the porous medium in the simulation of three-phase flow in a reservoir rock. The
reservoir shown in Figure 14.6 has four layers that are non-homogenous as far as
 
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