Geoscience Reference
In-Depth Information
60
x
f
40
(cm)
20
0
0
10
20
t
1/2
(min
1/2
)
Fig. 9.5 Distance to the wetting front
x
f
against the square root of time
t
for the three experiments of
horizontal infiltration into Columbia silt loam, already shown in Figure 9.3; the measurements
of the front position were made by visual inspection of the change in color of the soil in the
apparatus shown in Figure 9.2. The infiltration was allowed to proceed to distances of 25
(diamonds), 50 (triangles) and 75 cm (circles), respectively; the bulk densities were around
1.3 g cm
−
3
. (After Nielsen
et al
., 1962.)
in which
y
is a dummy variable of integration representing
θ
. As probably first shown by
Matano (1933), Equation (9.23) yields the following expression for the diffusivity,
θ
1
2
d
φ
d
θ
D
w
=−
φ
(
y
)
dy
(9.24)
θ
i
This integral has been useful in several ways, most notably in the experimental determination
of the soil water diffusivity, and also in the derivation of certain exact solutions for sorption
and horizontal infiltration capacity.
Direct measurement of the soil water diffusivity
Equation (9.24) was the basis for the method of Bruce and Klute (1956) to determine the
diffusivity
D
w
=
) directly from a sorption experiment. Substitution of the Boltzmann
transformation (9.11) into (9.24) produces the diffusivity in terms of the original variables
x
and
t
, as follows
D
w
(
θ
dx
d
θ
θ
1
2
t
D
w
=−
xd
θ
(9.25)
θ
i
This expression can be applied with a measured soil water content profile curve
θ
=
θ
(
x
)
obtained in a horizontal infiltration experiment of duration
t
, in a set-up like that shown
in Figure 9.2. The diffusivity
D
w
=
D
w
(
θ
) can be readily calculated from any curve like
those shown in Figure 9.3. As illustrated in Figure 9.6, this is done by estimating both the
area under the curve of
x
vs
θ
and the slope of that same curve, at a series of values
θ
,
for the given value of the elapsed time
t
. These numerical values of the integral and of the
