Civil Engineering Reference
In-Depth Information
Fig. 2.6
Element in an orthotropic soil.
The rate of change of the hydraulic gradient i
x
along the length of the element in the x direction
will be:
∂
∂
i
x
=−
∂
∂
2
h
x
x
2
Hence the gradient at the face of the element nearest the origin
h
x
i
x
dx
=−
∂
∂
+
∂
∂
−
x
2
=−
∂
∂
h
x
+
∂
∂
h
x
2
dx
2
2
From Darcy's law:
h
x
2
h
x
dx
dy.dz
−
∂
∂
+
∂
∂
Flow Aki
=
=
k
(1)
x
2
2
The gradient at the face furthest from the origin is
−
∂
∂
h
x
+
∂
∂
i
x
dx
x
2
h
x
2
h
x
dx
=−
∂
∂
−
∂
∂
2
2
Therefore
−
∂
∂
h
x
−
∂
∂
h
x
2
dx
dy.dz
Flow k
=
(2)
x
2
2
Expressions
(1) and (2)
represent respectively the flow into and out of the element in the x direction, so
that the net rate of increase of water within the element, i.e. the rate of change of the volume of the
element, is
(1) - (2).
Similar expressions may be obtained for flow in the y and z directions. The sum of the rates of change
of volume in the three directions gives the rate of change of the total volume:
