Civil Engineering Reference
In-Depth Information
For one-dimensional flow (assumption v) there is no component of hydraulic gradient in the x and y direc-
tions, and putting k
z
=
k the expression becomes:
∂
∂
2
h
Rate of change of volume
=
k
dx dy dz
.
.
2
z
The volume changes during consolidation are assumed to be caused by changes in void ratio.
Porosity
V
V
e
v
n
=
=
1
+
e
hence
e
V
v
=
dx dy dz
.
.
1
+
e
Another expression for the rate of change of volume is therefore:
∂
∂
e
t
dx dy dz
.
.
1
+
e
Equating these two expressions:
∂
∂
2
h
z
1
∂
∂
e
t
k
=
2
1
+
e
The head, h, causing flow is the excess hydrostatic head caused by the excess pore water pressure, u.
u
h
=
γ
w
k
∂
∂
u
z
2
1
∂
∂
e
t
⇒
=
γ
2
1
+
e
w
With one-dimensional consolidation there are no lateral strain effects and the increment of applied
pressure is therefore numerically equal (but of opposite sign) to the increment of induced pore pressure.
Hence an increment of applied pressure, dp, will cause an excess pore water pressure of du (
=−
dp). Now:
de
dp
a
=−
hence
de
du
a
=
(
see Section
11 3 3
.
.
)
or
de
=
a du
Substituting for de:
k
∂
∂
u
z
2
∂
∂
u
t
(
1
+
e
)
=
a
γ
2
w
