Environmental Engineering Reference
In-Depth Information
=
/
correlation is positive (negative), while for
H
1
2 the correlation is random and
follows a Brownian motion.
For the heterogeneous porous medium described above, the equation of motion
for the elastic displacement
w
R
(
x
,
t
)
can be written in vector notation as follows
2
)
∂
ʻ(
ˁ
R
(
x
t
2
w
R
(
x
,
t
)
=
x
)
+
2
μ(
x
)
∇
[
∇·
w
R
(
x
,
t
)
]
∂
)
+∇
ʻ(
−
μ(
x
)
∇×∇×
w
R
(
x
,
t
x
)
∇·
w
R
(
x
,
t
)
+
∇
μ(
)
×
2
∇
μ(
)
·∇
w
R
(
x
[
∇×
w
R
(
x
,
t
)
]
+
x
x
,
t
)
+
F
(
x
,
t
),
(42)
where
F
is an external force per unit volume. If we assume mechanical equilibrium,
the governing equation for the total stress in the bulk material can be obtained from
Eq. (
42
) by setting
2
)
∂
ˁ
R
(
x
t
2
w
R
(
x
,
t
)
=
0
.
(43)
∂
The effects of the hydrostatic compression exerted by the fluid on the solid matrix
enter in Eq. (
42
) through the body force term as
F
=
ˆ
p
I
,
(44)
where
p
denotes the mean pore pressure exerted by the fluid phases on the solid
grains, defined by
p
=
S
p
ʱ
.
(45)
ʱ
ʱ
If the system is in thermodynamical equilibrium, the fluid is at rest and
p
is constant
throughout the body. We note that Eq. (
42
) with the body force as given by relation
(
44
) is equivalent to the concept of the modified effective stress employed in the
mechanics of partially saturated porous media to take into account the compressibil-
ity of the solid matrix. According to experimental observations, it is this modified
effective stress that governs the major deformation of the solid skeleton (Khoei and
Mohammadnejad
2011
).
If we define the displacement vector of the fluid phase
ʱ
as
w
, the displacement
ʱ
vector of the fluid phase relative to the solid is
w
ʱ
=
w
ʱ
−
w
R
and the variation in
the fluid phase content
ʾ
ʱ
can be defined as
=−∇·
ˆ
,
w
ʱ
ʾ
ʱ
=−∇·
[
ˆ (
w
ʱ
−
w
R
)
]
(46)
where
per unit pore volume. It is the
difference between the strain of the pore space and the strain of the volume of fluid
ʱ
ʾ
ʱ
is the variation of the volume of fluid phase
ʱ
in the pore space and it is therefore a dimensionless quantity. In order to complete
the description of multiphase flow in a deformable, heterogeneous porous medium,
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