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been found useful to represent k ( θ ) by relatively simple parametric equations. As illustrated
in Figure 8.23, it is not surprising that the following has been used widely,
k = k 0 S e
(8.36)
where k 0 is the hydraulic conductivity at saturation and n is a constant; the effective
saturation S e is defined in (8.6). Equation (8.36) requires the determination of four param-
eters, namely k 0 0 r and n . Inspection of past determinations of n shows that it may be
as low as 1 and as high as 20, but that typical values lie around 3-5; n appears to be small
for materials with a narrow pore size distribution and larger for wider pore size distribu-
tions. Interestingly, this power form equation has been derived on the basis of some widely
different theoretical models. It is tempting, therefore, to conclude that the power form is
independent of its method of derivation. Some of these models are reviewed in Section 8.3.5.
For instance, Averyanov (Polubarinova-Kochina, 1952) proposed Equation (8.36) with
n = 3.5, and Irmay (1954) proposed it with n = 3; more recently, it was found (Brut-
saert, 2000) that n = (2 + 2.5 / b ), where b is the same as in Equation (8.14), provides the
best description of the available experimental data.
Equation (8.36) is one of the oldest and still among the most widely used expressions
today. Recently it has also received renewed theoretical interest because it arises naturally
in the fractal characterization of soils. Other parameterizations have been proposed for k ( θ ),
but they are all fairly similar to (8.36).
Since the water content θ is a function of the negative water pressure H ( =− p w w ), it
is also possible to express k as a function of H . Gardner (1958) has proposed an empirical
function, which can be fitted to data for many different soils, viz.
a
b + H c
k
=
(8.37)
where a , b and c are constants; note that ( a / b ) is equal to k 0 , the hydraulic conductivity at
satiation and b is the value of H c for k = k 0 / 2. The range of c was found to lie between
about 2 for clayey soils and 4 or more for sandy soils. It can be seen that Equation (8.37)
is of the right general shape to fit to experimental data such as shown in Figure 8.29. As
already mentioned and illustrated in Figure 8.25, however, k ( H ) normally exhibits marked
hysteresis, so that the constants have to be adjusted to reflect this.
In some applications it is convenient to describe the hydraulic conductivity by an expo-
nential function as follows
k = k 0 for H H b
k = k 0 exp[ a ( H H b )]
(8.38)
for H > H b
where a and H b are constants for a given soil; Equation (8.38) was introduced by Gardner
(1958) without H b ; this constant was added later by P. E. Rijtema to allow incorporation
of the capillary fringe. The spatial variability and physical significance of the parameter a
in (8.38) have been investigated (White and Sully, 1992); at the field scale, a appears to be
lognormally distributed, like k 0 .
Soil water diffusivity
A diffusivity function, which has been useful in the solution of a number of problems, is of
the following exponential form
D w = D wi exp[ β ( θ θ i ) / ( θ 0 θ i )]
(8.39)
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