Geoscience Reference
In-Depth Information
1
2
0.9
1
0.8
0.7
0.6
3
0.5
0.4
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β
Fig. 9.12 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 solution (9.41), as functions of the parameter
β
in the exponential
diffusivity function (8.39).
D
w0
is the diffusivity at satiation.
in Figure 9.12; typical values are
C
1
(8)
=
0.4828 and
C
1
(3)
=
0
.
7256. Equation (9.44)
can provide a first estimate of
A
0
, when no other information is available. Because
D
w0
=
D
wi
exp(
is a constant, it follows that (9.44) also allows the formulation of the
diffusivity function (8.39) in terms of the sorptivity, as shown in Brutsaert (1979). Similarly,
substitution of the same diffusivity in Equation (9.43) produces immediately the position
of the wetting front, namely
β
), in which
β
φ
f
=
D
1
/
2
w0
C
2
(
β
)
(9.46)
in which
C
2
(
β
)
=
2 (1
−
exp(
−
β
)M(
−
0
.
5
,
0
.
5
,β
)
·
((2
β
−
1)
+
exp(
−
β
)M(
−
0
.
5
,
0
.
5
,β
))
−
1
/
2
(9.47)
This result is illustrated in Figure 9.12. Again, (9.46) with (9.47) shows how the diffusivity
function (8.39) for a given soil can be expressed in terms of the position of the wetting front,
once
β
is known (Miller and Bresler, 1977; Brutsaert, 1979). Recalling the definitions of
A
0
in (9.18) and of
φ
f
in (9.22), and also comparing (9.44) with (9.46), one can see that
A
0
/φ
f
=
(
); this means that when the soil water diffusivity is exponential, the
cumulative horizontal infiltration
F
is related with the position of the wetting front
x
f
,as
follows
θ
0
−
θ
i
)
C
1
(
β
)
/
C
2
(
β
F
=
C
3
(
β
)(
θ
0
−
θ
i
)
x
f
(9.48)
where
C
3
(
β
)
=
C
1
(
β
)
/
C
2
(
β
) is a number whose value depends on
β
. This result is also
illustrated in Figure 9.12; typical values are
C
3
(8)
=
0
.
862 and
C
3
(3)
=
0
.
626.
