Chemistry Reference
In-Depth Information
e
σ
=
H
A
⋅ ε
(3)
where
H
A
is the aggregate equilibrium modulus,
is strain, that is expressed through
ε
the solid phase displacement
u
(
x
,
z
,
t
)
as follows:
ε =
∂
u
x
∂
+
∂
u
z
∂
(4)
x
z
where
u
x
and
u
z
are components of
u
(
x
,
z
,
t
) .
The continuity equations for the À uid and solid phases, assuming (1), yield [10]:
(
)
= 0
v
f
v
s
∇ φ ⋅
+
(1
− φ
)
⋅
=
∂
u
∂
where
v
f
and
v
s
t
are the velocities of fluid and solid relative to a fixed representa-
tive element volume (REV) in the frame of reference associated with the fixed lower
plate [10]. Note, that
ε
z
=
∂
u
z
∂
is equal to the strain applied to the upper plate,
ε
⊥
(
t
) .
z
Strains in the matrix do not depend on
z
. Hence
u
z
=
z
⋅ ε
⊥
(
t
)
. Also the
z
-components
=
∂
u
z
∂
of solid and fluid velocities are equal:
v
z
f
. Combining the expressions in
v
s
=
t
this paragraph yields:
⎛
⎝
⎜
⎞
⎠
⎟
=
v
x
f
x
+
∂
v
z
f
z
2
+
∂
2
u
z
∂
(1
− φ
)
∂
u
x
+
0
φ
∂
x
∂
t
∂
z
∂
t
that may be written as:
⎛
⎝
⎜
⎞
⎠
⎟
=
∂
v
x
f
x
+
∂ε
⊥
∂
(1
− φ
)
∂
∂
∂
u
x
∂
0
.
+
+ ε
⊥
t
φ
t
x
=
∂
u
x
∂
By integrating this equation with respect to
x
with the condition that
v
x
f
=
0
at
t
0
we get:
x
=
(1
− φ
)
∂
u
x
∂
1
φ
x
∂ε
⊥
∂
v
x
f
=−
+
(5)
φ
t
t
The apparent fluid velocity relative to the solid matrix,
v
, is proportional to the gradi-
ent of pore pressure p as given by the Darcy's law[10]:
⋅
∂
p
∂
(
v
x
f
v
s
)
v
= φ ⋅
−
=−
k
x
(6)
Making use of Equation (2-6), we can rewrite Equation (1) as differential equation for
solid displacement:
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