Geoscience Reference
In-Depth Information
The sorptivity can be calculated by integration of (9.38) in accordance with (9.17); this
readily produces
⎛
⎞
1
/
2
1
⎝
2
D
w
S
1
+
b
n
⎠
A
0
=
(
θ
0
−
θ
i
)
dS
n
(9.39)
0
The optimal value of
b
can now be estimated by solving (9.39) with the special diffusivity
(9.26) and by comparing the result with the exact sorptivity (9.28); this yields
b
=
[(4
m
2
+
8
m
+
5)
1
/
2
−
(2
m
+
3)]
/
2
(9.40)
Because the numerical value 5 in Equation (9.40) may be approximated by 4, when
m
is
not small, it is clear that
b
is close to
−
0.5. As mentioned, with this value of
b
it can be
shown that the error in the sorptivity tends to be smaller than 1%.
Accordingly, in its most accurate form, i.e. with
b
=−
1
/
2, the solution (9.38) becomes
⎛
⎞
⎠
⎝
2
1
1
/
2
1
D
w
S
1
/
2
n
D
w
y
−
1
/
2
dy
φ
=
dS
n
(9.41)
0
S
n
Similarly, the sorptivity (9.39) becomes simply
⎛
⎞
1
/
2
1
⎝
2
D
w
S
1
/
2
n
⎠
A
0
=
(
θ
0
−
θ
i
)
dS
n
(9.42)
0
The position of the wetting front (9.22) follows directly from (9.41) with the lower limit of
the integral taken as
S
n
=
0, or
⎛
⎞
⎠
⎝
2
1
1
/
2
1
D
w
S
1
/
2
n
D
w
S
−
1
/
2
n
φ
f
=
dS
n
dS
n
(9.43)
0
0
Implementation with parametric diffusivity functions
Whenever it can be assumed that
θ
i
=
θ
r
(or
S
e
=
S
n
), which is usually a good approximation
when initially the soil is quite dry, the diffusivity functions (8.39) and (8.41) can be used to
perform the above integrations of (9.32) and (9.39).
Example 9.1. Exponential diffusivity
In the case of the diffusivity (8.39), the sorptivity (9.42) can be shown to be (Brutsaert,
1976)
D
1
/
2
A
0
=
w0
(
θ
0
−
θ
i
)
C
1
(
β
)
(9.44)
in which
D
w0
is the diffusivity at satiation, when
S
n
=
1.0. The term
C
1
(
β
) depends on the
value of the parameter
β
in (8.39); this can be calculated from the following
C
1
(
β
)
=
β
−
1
((2
β
−
1)
+
exp(
−
β
)M(
−
0
.
5
,
0
.
5
,β
))
1
/
2
(9.45)
in which M(
a
,
b
,
z
) is the confluent hypergeometric function, conveniently tabulated by
Abramowitz and Stegun (1964, pp. 516
−
535). The dependency of
C
1
(
β
)on
β
is illustrated
