Geoscience Reference
In-Depth Information
Fig. 8.23 Experimental values of relative permeability
k
/
k
0
(or relative conductivity) of water (wetting fluid -
circles) and of air (non-wetting fluid - triangles), as a function of the degree of saturation
S
=
θ/θ
0
,in
a sandy soil and in a silty clay loam. The curves for water represent Equations (8.36) with (8.45) and
those for air represent the analogous expression for a non-wetting fluid. (After Brooks and Corey,
1966.)
in a larger effective conductivity for the entire flow domain, than a simple average
value would suggest (see El-Kadi and Brutsaert, 1985). In flow simulation calcula-
tions, the variability of the soil characteristics is usually avoided by the assumption that
the entire flow domain is homogeneous and that effective parameters can be used (see
Section 1.4.3), to represent the flow domain. The scale dependency of the hydraulic
conductivity is a direct result of this assumption.
8.3.2
Partly saturated flow
Extension of Darcy's law
It was probably Buckingham (1907) who first postulated that Darcy's law (8.19) is also
valid for a soil that is only partly saturated with water, and that in this case the hydraulic
conductivity is a function of water content, or
k
). As the water content of the soil
is reduced,
k
decreases. Reasons for this are that fewer pores are available for flow, and
that the flow paths become more tortuous and thus longer, as the water can no longer
move through emptied pores and must move around them. Because the larger pores are
emptied first, the initial decrease in conductivity for a certain decrease in water content
is larger than that later on at lower water contents.
Actually, the water of the empty pores has been replaced by air. Under such conditions
Darcy's law is also valid to describe the flow of air; but it must be applied with an “air
head” gradient as given by Equation (8.21) with the density of air. This is illustrated in
Figure 8.23, which shows the relative permeabilities
=
k
(
θ
k
0
) for water and for air in
two soils, as examples. In what follows
k
0
and
k
0
will usually denote the hydraulic con-
ductivity and permeability, respectively, at saturation or satiation. Under partly saturated
conditions the hydraulic conductivity
k
is also often called the
capillary conductivity
.In
hydrology mainly the water is of interest, because it can usually be assumed that the air
movement takes place under negligibly small pressure gradients.
(
k
/
κ
r
=
