Environmental Engineering Reference
In-Depth Information
2.1 Undeformable Porous Media
Consider first a porous medium that is statistically homogeneous and undeformable,
and assume that the system has four phases: a solid phase (
R
) and three fluid phases,
namely water (
W
), gas (
G
), and oil (
O
). The water phase wets the porous medium
more than the oil phase, and so it is called the
wetting phase
. In general, water is
the wetting fluid relative to oil and gas, while oil is the wetting fluid relative to gas.
Each phase has its own intrinsic mass density
ˁ
ʱ
, velocity
v
, and volume fraction
ʱ
ˆ
ʱ
, with the latter obeying the constraint
ˆ
ʱ
=
1
.
(3)
ʱ
The mass conservation equation for phase
ʱ
can be written as
∂ (ˆ
ʱ
ˁ
ʱ
)
∂
+∇·
(ˁ
ʱ
v
ʱ
)
=
I
ʱ
,
(4)
t
where
I
ʱ
.In
the absence of any external mass source or sink, the reaction rates must satisfy the
constraint
is an interfacial mass transfer rate from all other phases to phase
ʱ
I
ʱ
=
0
(5)
ʱ
in order to ensure mass conservation in the overall mixture. In addition, the fact
that all fluid phases jointly fill the void space in the solid matrix implies the further
relation
S
ʱ
=
,
S
W
+
S
O
+
S
G
=
.
1
or
1
(6)
ʱ
Using relations (
2
) and (
3
) and noting that
ˆ
+
ˆ
R
=
1, Eq. (
4
) can be expressed in
terms of the saturation and porosity as
∂ (ˆ
S
ʱ
ˁ
ʱ
)
∂
+∇·
(ˁ
ʱ
v
ʱ
)
=
I
ʱ
,
(7)
t
for the fluid phases, and
∂
[
(
1
−
ˆ) ˁ
R
]
∂
+∇·
(ˁ
R
v
R
)
=
I
R
,
(8)
t
for the solid (rock) phase. If the rock phase is chemically inert,
I
R
=
0. In addition,
if the solid medium is immobile then
v
R
=
0 and Eq. (
8
) reduces to
ˁ
R
=
const
.
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