Geoscience Reference
In-Depth Information
z
Fig. 8.22 Definition sketch of the directions of the specific flux and
gradient vectors in an anisotropic material, in which
k
xx
>
k
zz
, and
x
and
z
are principal axes.
−∇
h
q
α
β
x
Thus, since the
x
and
z
components must be the same in Equations (8.28) and (8.29),
respectively, one obtains
k
zz
k
xx
tan
β
=
tan
α
(8.30)
This shows that the directions of the specific flux and of the (negative) hydraulic gradient
can be the same only when the porous medium is isotropic. In an anisotropic material these
two vectors point in different directions.
Scale dependence of the hydraulic conductivity
It has been noted in numerous applications of Darcy's law to field conditions, that the
values of
k
, required to reproduce the observed flow rates, tend to depend on the size
of the domain over which Equation (8.19) is integrated or averaged. In other words, the
magnitude of
k
is scale dependent. Thus values of the hydraulic conductivity for a given
soil are often larger when obtained from auger hole measurements in the field, than from
small column measurements in the manner of Darcy (see Figure 8.21) or in other types
of permeameters. Values of
k
obtained by inversion methods with data from small river
catchments (see Brutsaert and Lopez, 1998) tend to be still larger. Permeameters involve
a flow domain with typical length scales of the order of 1.0 m at most, while auger hole
measurements and pumping tests tend to have zones of influence with length scales of
the order of 10-10
2
m, respectively; small catchments usually involve scales of the order
of 10
3
m or more.
There are several possible reasons for this scale dependence. One is that most per-
meameter measurements are carried out on disturbed samples; thus the soil is scooped
up and placed in the permeameter in a way that usually does not replicate the origi-
nal soil structure in the field. Moreover, under natural conditions in the field, most soils
have
macropores
and other additional conduits resulting from decaying plant roots, from
worms and from burrowing animals; even if undisturbed samples are used, it is nearly
impossible to include such macropores and other larger channels within the relatively
small confines of a permeameter. Finally, even within a supposedly homogeneous soil
type, all soil properties display pronounced spatial variability. At field scales of the order
of 10
2
m, the hydraulic conductivity is usually close to lognormally distributed (see
Rogowski, 1972; Nielsen
et al
., 1973; Hoeksema and Kitanidis, 1985); this means that
the larger
k
values in the domain have a relatively larger effect on the overall flow than
the smaller values. But regardless of the distribution, in a two- and three-dimensional
flow situation, regions of smaller conductivity can be bypassed by the flow, resulting
