Geoscience Reference
In-Depth Information
obtains finally the solution
θ
0
−
θ
i
) erfc
φ/
2
D
1
/
0
+
θ
i
θ
=
(
(9.56)
in which the complementary error function is, by definition,
∞
2
z
2
)
dz
erfc(
y
)
=
exp(
−
(9.57)
π
1
/
2
y
The normalized water content
S
n
=
(
θ
−
θ
i
)
/
(
θ
0
−
θ
i
) given by this solution is shown graph-
ically in Figure 9.10. Application of (9.20) with Leibniz's rule (see Appendix) to (9.57),
and comparison with (9.19) produce the following expression for the sorptivity
A
0
=
2(
θ
0
−
θ
i
)(
D
0
/π
)
1
/
2
(9.58)
Most natural soils have a soil water diffusivity, which is markedly dependent on the
water content; therefore, the results obtained in this section with a linearized soil may
appear suspect at first sight. However, as illustrated in Figure 9.8, this
θ
-dependency is
not always strong, so that a linear model may still come close to describing the situation.
Indeed, linearization tends to simplify the analysis considerably, and should therefore be
of interest. The question remains what value should be assigned to the constant diffusivity
D
0
, to ensure that the linear model will reproduce the more important sorption features
of the prototype. One possibility is to assign the value, which would reproduce the same
infiltration rate and volume with the linearized soil as the nonlinear prototype soil. In this
case one obtains immediately from Equation (9.58)
π
A
0
4(
θ
0
−
θ
i
)
2
D
0
=
(9.59)
in which
A
0
, the sorptivity of the prototype, is to be determined independently. As another
possibility, if no independent estimates of
A
0
are available, one can use an empirical approx-
imation proposed by Crank (1956, p. 256); from his calculations he had found that the
weighted mean
θ
0
D
0
=
(5
/
3)(
θ
0
−
θ
i
)
−
5
/
3
(
θ
−
θ
i
)
2
/
3
D
(
θ
)
d
θ
(9.60)
θ
i
can yield initial rates with good accuracy for diffusivity functions
D
=
D
(
θ
) which increase
with
θ
over several orders of magnitude.
9.3
INFILTRATION CAPACITY
The infiltration capacity, or the potential infiltration rate, was defined above as the max-
imal rate at which the soil surface can absorb water. Such conditions prevail when the
soil at the surface is saturated, that is whenever its water pressure is at least atmospheric.
The problem is usually analyzed by assuming that the surface is ponded with a very thin
layer of water, so that the water pressure is essentially atmospheric.
