Environmental Engineering Reference
In-Depth Information
∂
∂
S
t
q
(
)
(
)
(
)
(
)
⎣
⎦
=
(5.110)
j
+∇
.
fsv
−
K
l
fs
∇+
P
r
−
r
g z
∇
w
o
c
o
w
r
w
This is the saturation equation. Similarly, the saturation equation for the
incompressible and immiscible two-phase flow under pressure and electri-
cal gradients can be derived as (Ghazanfari, 2013):
∂
∂
S
t
(
(
)
−∇
(
)
(
)
(
)
(
)
⎣
j
+∇
.
fsv
−
K
l
fs
∇+
P
r
−
r
g z
∇
k
e
E fsk
o
c
o
w
w
er ow
,
q
)
⎦
=
(
)
( )
w
(5.111)
+
fs
−
1
k
er ww
,
r
w
5.9.2
Pressure Equation for Two-Phase Incompressible
Immiscible Flow
Summing the Darcy's law over the two phases we get the following equation:
(
)
−∇−∇
(
)
(5.112)
vv v
=+=−
K
l
∇− ∇
P
r
gz
K
l
P
r
gz
wo
w w w
o
o
o
Taking divergence on both sides and considering the fact that
∇.
[
v
] =
r
,
we have:
{
}
=
∇
[]
=−∇
(
)
+∇−∇
(
)
(5.113)
.
v
K
l
∇
P
−
r
g
∇
z
K
l
P
r
g
z
q
w
w
w
o
o
o
This is the pressure equation. Similarly, the pressure equation for the
incompressible and immiscible two-phase flow under pressure and electri-
cal gradients can be derived as (Ghazanfari, 2013):
{
(
)
+∇−∇
(
)
+
∇
[]
=−∇
.
v
P
g
z
P
g
z
(5.114)
K
l
∇
−
r
∇
K
l
r
w
w
w
o
o
o
}
=
q
(
)
k
e
∇
Ek
+
k
w
erww
,
erow
,
The pressure and saturation equations (Eq. 5.111 and 5.114) are nonlin-
early coupled primarily through the saturation-dependent mobility (
l
w
, and
l
o
)
in the pressure equation, and through the pressure-dependent velocity (
v
w
and
v
o
)
in the saturation equation, and also through other terms that depend
on pressure or saturation (e.g., viscosity, capillary pressure). To reduce the
coupling, a global pressure term (
P
) can be introduced to replace the phase
pressures (
P
w
,
and
P
o
) (Chen et al., 2006; Aarnes et al., 2007). The global pres-
sure is defined as
PP P
o
, where the saturation-dependent complemen-
tary pressure (
P
C
)
is defined as (Chen et al., 2006; Aarnes et al., 2007):
=−
c
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