Geoscience Reference
In-Depth Information
Example 9.2. Power-form diffusivity
In the same way, in the case of the diffusivity (8.40), Equation (9.42) can be integrated
readily to yield a sorptivity
A
0
=
(2
H
b
(
θ
0
−
θ
i
)
k
0
/
b
)
1
/
2
(
n
−
b
−
1
+
0
.
5)
−
1
/
2
(9.49)
where
H
b
and
b
are the parameters of (8.14) and
n
is the power in (8.36).
With the diffusivity given by the somewhat more complex (8.41), the integral in (9.42)
is a complete beta function, which can also be written as follows
a
)
1
/
2
(
b
−
1
(
b
−
1
5))
1
/
2
A
0
=
(2
k
0
/
θ
0
−
θ
i
)(
(
n
−
+
0
.
5)
+
1)
/
(
n
+
0
.
(9.50)
In Equation (9.50)
a
and
b
are the parameters of (8.15),
n
is the power in (8.36) and
()
is the gamma function (see Abramowitz and Stegun, 1964). For most soils when
n
varies
between 2 and 10 and
b
between 1 and 10, the square root term in (9.50) containing the
gamma functions is likely to be of the order of unity. For instance, for a typical case of
n
=
b
=
3 it is equal to 0.7938. The position of the wetting front can be obtained from
Equation (9.43) by a similar integration, to produce
φ
f
=
(2
k
0
/
[(
θ
0
−
θ
i
)
a
])
1
/
2
(
n
−
b
−
1
1
/
2
+
0
.
5)
(
b
−
1
+
1)
(
n
−
0
.
5)
(
n
−
b
−
1
(9.51)
(
n
+
0
.
5)
−
0
.
5)
Comparison of (9.50) with (9.51) indicates that with this diffusivity function, the infiltrated
volume is proportional with the position of the wetting front as
F
=
C
(
n
,
b
)(
θ
0
−
θ
i
)
x
f
(9.52)
in which the proportionality constant is given by
C
(
n
,
b
)
=
(
n
−
b
−
1
−
0
.
5)
/
(
n
−
0
.
5); for
example,
C
(3
,
3)
=
0
.
867, which is similar to the result given by Equation (9.48).
9.2.4
A nearly exact solution for mildly nonlinear soils: linearization
For some soils the diffusivity can be assumed to be nearly independent of the water content
θ
; an example of this is shown in Figure 9.8. In such a case Equation (9.6) can be linearized
and it reduces to the linear diffusion equation; accordingly, (9.13) can be written as
D
0
d
2
θ
d
φ
+
2
d
θ
d
φ
=
0
(9.53)
2
where
D
0
is the constant soil water diffusivity. By means of the ad-hoc substitution
p
=
d
θ/
d
φ
, (9.53) can be integrated to yield
2
p
=
C
1
exp(
−
φ
/
4
D
0
)
(9.54)
A second integration yields
exp(
C
1
2
D
1
/
2
0
y
2
)
dy
θ
=
−
+
C
2
(9.55)
where
y
is the dummy variable of integration representing
φ/
2
D
1
/
0
and
C
1
and
C
2
are
constants to be determined from the boundary conditions (9.14). The integral term with
limits between zero and infinity equals (
π
1
/
2
/
2); therefore, imposing these conditions, one
