Geoscience Reference
In-Depth Information
depend on direction. Most soils and other water-bearing formations tend to be anisotropic
to some degree. In some cases larger permeabilities in the horizontal may be the result
of layering of sediments during their deposition in the soil formation process; in other
situations cracking during drying of clayey soils, and subsequent filling of the vertical
cracks by coarser wind-blown loess, may have resulted in relatively larger permeabilities
in the vertical.
In an anisotropic material the two vectors
q
i
and
−
∂
h
/∂
x
i
do not necessarily point in
the same direction. It can be shown that the only way to formulate a linear relationship
between such vectors is by means of a second-order tensor or a dyad. Thus Darcy's law for
anisotropic material must be of the form
q
x
=−
k
xx
∂
h
∂
x
−
k
xy
∂
h
∂
y
−
k
xz
∂
h
∂
z
q
y
=−
k
yx
∂
h
∂
x
−
k
yy
∂
h
∂
y
−
k
yz
∂
h
(8.25)
∂
z
q
z
=−
k
zx
∂
h
∂
x
−
k
zy
∂
h
∂
y
−
k
zz
∂
h
∂
z
Alternatively, this can be written more concisely in the subscript notation as
3
k
ij
∂
h
∂
x
j
q
i
=−
(8.26)
j
=
1
In general, the second-order tensor
k
ij
has nine components. Symmetry of such a tensor
quantity, i.e.
k
ij
=
k
ji
, is known to be a sufficient condition to allow it to be diagonalized,
such that it has only three components, along three principal axes. The hydraulic conductivity
tensor is usually assumed to be symmetrical.
Example 8.4. Directions of the gradient and of the flux vector
To bring out some of the implications of anisotropy, consider the flux vector
q
and the
negative hydraulic gradient vector
−∇
h
, which drives it, in a principal axes system, as
shown in Figure 8.22. The gradient consists of two components, and can therefore be
broken down as follows
∂
h
∂
x
i
+
∂
h
∇
h
=
∂
z
k
= |∇
h
|
(cos
α
i
+
sin
α
k
)
(8.27)
Similarly, the specific flux can be written as
q
=
q
x
i
+
q
z
k
= |
q
|
(cos
β
i
+
sin
β
k
)
(8.28)
These two vectors are related by Darcy's law (8.25), so that by virtue of (8.27) the flux can
also be written as
q
= −|∇
h
|
(
k
xx
cos
α
i
+
k
zz
sin
α
k
)
(8.29)
