Geoscience Reference
In-Depth Information
θ
0
tan
−
1
(
δθ
/
δ
x
)
θ
θ
i
x=
0
x=x
f
Fig. 9.6 Graphical illustration of the numerical calculation of the soil water diffusivity
D
w
=
D
w
(
θ
)by
means of Equation (9.25) with the soil water profile
θ
=
θ
(
x
) obtained from a horizontal
infiltration experiment of duration
t
. In (9.25) the integral is the shaded area and (
dx
/
d
θ
)isthe
inverse of the slope at
θ
.
derivative can then be used in Equation (9.25) to calculate the diffusivity at each of these
values of
θ
. Another illustration of the method is the study by Clothier and White (1982);
with the profile data shown in Figure 9.7 they applied it to compare the diffusivity function
D
w
=
D
w
(
θ
) in undisturbed and repacked soil columns, as shown in Figure 9.8.
An exact solution for soil water sorption
Equation (9.24) indicates that
D
w
=
D
w
(
θ
) can be determined when
φ
is known; in other
words, (9.24) can also be applied in an inverse mode, to derive the form, which the diffusivity
D
w
=
D
w
(
θ
) must have, to produce any assumed functional form of the solution
φ
=
φ
(
θ
).
This way Philip (1960) was able to list a large class of exact solutions of the nonlinear dif-
fusion equation (9.6) subject to (9.7) for corresponding functional forms of the diffusivity.
At the time, none of the obtained
φ
(
θ
) functions seemed to be applicable to infiltration into
soils, and they received relatively little attention in the hydrologic literature. However, it was
subsequently shown that in certain cases by proper scaling, one such solution is adaptable
to describe sorption in real soils, and that it can thus be made compatible with experi-
mental data (Brutsaert, 1968; 1976). The simplest form of that solution is
φ
=
(1
−
S
n
),
which corresponds according to (9.24) with a diffusivity
D
w
=
mS
n
[1
−
S
n
/
(1
+
m
)]
/
2.
To obtain
D
w
=
D
w0
, that is the diffusivity at satiation for
S
n
=
1, this result must be scaled
as follows
D
w
=
D
w0
(1
+
m
)
S
n
−
S
2
n
(1
+
m
)
/
m
(9.26)
and the corresponding exact solution becomes
φ
=
2
D
w0
(1
/
m
2
1
/
2
1
−
S
n
+
m
)
(9.27)
