Environmental Engineering Reference
In-Depth Information
ll l
=+
wo
(5.102)
The fractional flow for water and oil phases (function of water satura-
tion) are defined as (Aziz and Settari, 1979; Ertekin et al., 2001):
l
l
l
l
(5.103)
f
=
w
;
f
=
o
w
o
The total velocity and total flow are defined as (Aziz and Settari, 1979):
(5.104)
vv vqq q
wo
=+ =+
wo
The saturation equation for the case of two-phase incompressible
and immiscible flow under pressure gradient and elevation gradients
are extended to the case of two-phase incompressible and immiscible
flow under pressure, elevation, and electrical gradients. First, multiply-
ing Darcy's law for each phase with the mobility of the other phase and
ignoring the contribution of viscous coupling under pressure and elevation
gradients we have the followings:
(
)
(5.105)
l
v
=−
K
l l
∇
P
−
r
g
∇
z
ow
wo
w
w
(
)
(5.106)
l
v
=−
K
l l
∇
P
−
r
g
∇
z
wo
wo
o
o
Subtracting equation 5.105 from equation 5.106 gives:
(
)
(
)
(5.107)
l
v
−
l
v
−
K
l l
∇+−
P
r
r
g
∇
z
wo
ow
wo
c
o
w
To find an expression that only contains the total velocity and the satu-
ration of the wetting phase, we use
v
0
=
v
-
v
w
(Eq. 5.104) to get:
(
)
(
)
(
)
(5.108)
l
v
−+
l
l
v
=−
K
l l
∇
P
+
r
−
r
g
∇
z
w
w
o
w
w
o
c
o
w
l
ll
( )
=
Considering
s
=
s
w
and using the fractional flow of water as
fs
w
+
we have:
wO
(
)
( )
( )
(
)
(5.109)
vf
=
s v
−
K
l
f
s
∇+
P
r
−
r
g
∇
z
w
o
c
o
w
Inserting this equation into the continuity equation (Eq. 5.96) for the
wetting phase we get:
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