Environmental Engineering Reference
In-Depth Information
The phase velocity is given by the extended Darcy's law (
10
) for multiphase flow
and the temperature equation has the same form of Eq. (
96
) with
I
k
ʱ
replaced by
I
k
ʱ
. In the calculation of chemical flooding, a pressure equation for the acqueous
phase, say
A
, will be needed to account for the effects of adsorption. This equation
can be derived from Eq. (
57
) by replacing
I
k
by
I
k
and summing up over all volume-
occupying components. Noting from Eq. (
105
) that
m
c
k
=
,
1
(107)
k
=
1
and using the constraints (
61
) and (
106
), and making
ˁ
ₒ
ˁ
A
and
v
ₒ
v
A
,we
obtain the mass balance equation
m
∂ (ˆˁ
A
)
∂
+∇·
(ˁ
A
v
A
)
=
L
k
.
(108)
t
k
=
1
The compressibility of the flooding phase and rock requires introducing the following
definitions:
ˁ
A
∂ˁ
kA
1
C
A
=
p
A
,
(109)
∂
at a fixed temperature, for the acqueous phase, and
1
ˆ
ˆ
dp
A
,
d
C
R
=
(110)
for the rock. Assuming that the fluid and rock compressibilities are constant over a
certain range of pressures, the above equations can be readily integrated to give
ˁ
A
≈
ˁ
0
+
ˁ
0
C
A
(
p
A
−
p
0
) ,
(111)
ˆ
≈
ˆ
0
+
ˆ
0
C
R
(
p
A
−
p
0
) ,
(112)
where we have used Taylor series expansions to approximate the exact solutions of
Eqs. (
109
) and (
110
). Here
ˆ
0
are the density and porosity at the reference
pressure
p
0
of the acqueous phase. Expanding the time derivative in Eq. (
108
), using
relations (
106
) and (
107
), and replacing
ˁ
0
and
ˁ
A
v
A
by Darcy's law written in the form
ˁ
A
v
A
=−
ʻ
k
·∇
p
A
−
ʻ
ʱ
k
·
(
∇
p
c
ʱ
A
−
ˁ
ʱ
g
∇
z
) ,
(113)
ʱ
where
ʻ
ʱ
is the phase mobility defined as
n
k
r
ʱ
μ
ʱ
ʻ
ʱ
=
1
ˁ
k
c
k
ʱ
,
(114)
k
=
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