Geoscience Reference
In-Depth Information
1
8
4
2
0.8
S
n
0.6
0.4
1 .5 .
5
0.2
m
=16
0
0
1
2
3
φ
(
θ
0
− θ
i
)/
A
0
Fig. 9.10 The soil water content distribution obtained with the exact solution (9.27) in the scaled form
x
(
S
n
) for different values of the parameter
m
. The
heavy line represents the solution of the linear case (9.56).
(
A
0
t
1
/
2
)
θ
0
− θ
i
)
/
=
[(1
+
m
)
/
m
](1
−
1
0.9
2
1
0.8
0.7
3
0.6
0.5
0.4
123456789 0
m
Fig. 9.11 The scaled sorptivity
A
0
/[(
θ
0
− θ
i
)
D
1
/
2
w0
] (curve 1), the position of the wetting front
φ
f
/
D
1
/
2
w0
(curve 2), and the ratio of the infiltrated volume over the wetting front position
F
/
[(
θ
0
− θ
i
)
x
f
]
(curve 3), obtained with the exact solution (9.27), as functions of the parameter
m
in the
diffusivity function (9.26).
D
w0
is the diffusivity at satiation.
These expressions for the sorptivity
A
0
, the position of the wetting front
φ
f
, and the ratio
x
f
, obtained with the exact solution (9.27), are illustrated in Figure 9.11 as functions of
the shape parameter
m
.
F
/
Integral constraint for approximate solutions
In some methods of solution the governing equation (9.13) is changed slightly to make it
more amenable to mathematical analysis. Because physically (9.13) is based on the validity
