Geoscience Reference
In-Depth Information
Although numerical methods of solution can be obtained (Rubin, 1966; Smith, 1972),
the problem remains a difficult one. A number of simpler approaches have provided ways
of obtaining more concise parameterizations for certain aspects of the phenomenon.
These have consisted of extensions of the approach of Green and Ampt (Mein and Larson,
1973; Swartzendruber and Hillel, 1975; Chu, 1978); empirical equations derived from
the numerical solution of Equation (9.1) (Smith, 1972; Smith and Chery, 1973); and
equations derived by the analytical solution of (9.1) on the basis of the quasi-steady state
or other approximations (Parlange, 1972; Smith and Parlange, 1978; Broadbridge and
White, 1987; White and Broadbridge, 1988; White et al. , 1989).
Probably the most important part of any solution, for practical purposes, is the deter-
mination of the time to ponding t p and the subsequent infiltration rate. In the following
two sections, parameterizations are developed for this purpose.
9.4.2
Time to ponding
Consider for this analysis the simplest possible case of precipitation on the surface of a
deep uniform soil profile. One of the oldest approximate methods for solving nonlinear
diffusion equations like (9.1) consists of considering the problem as a succession of
steady states. In groundwater theory it was used as early as 1886 by K. E. Lembke
(Polubarinova-Kochina, 1962, p. 573) to approximate the Boussinesq equation (10.30),
that is (9.6) with D w θ
, in the analysis of the drainage or desorption problem. Later,
essentially the same method was applied by Landahl (1953) in the solution of the linear
diffusion equation and was then generalized by Macey (1959) to the nonlinear diffusion
equation (9.6) for sorption; Parlange (1971) applied the concept to derive a first estimate
of the soil water profile
0.
Parlange (1972) and Parlange and Smith (1976) then explored the same approach to
study rainfall infiltration; this quasi-steady state approach is described next.
φ = φ
(
θ
) for sorption, that is (9.32) with a
=
1 and b
=
Sharp front approach
The approach is based on the assumption of a sharp wetting front, such that, once this front
has passed a point, the water content θ is already so close to satiation that it does not change
much from then on. Thus the term ( ∂θ/∂ t ) in the Richards equation (9.1) can be neglected,
so that the term on the right-hand side becomes zero; this means that the specific flux is the
same at all z , including at the surface, z = 0, where it is equal to the precipitation intensity P .
Therefore, after one integration, (9.1) in its diffusion form (9.61) yields
P =− D w ∂θ
z + k
(9.77)
This is in accordance with the second of (9.5), and can be integrated a second time to yield
θ
D w
P k d θ
z =−
(9.78)
θ s
where θ s is the water content at the soil surface z = 0, which changes with time as the
precipitation proceeds. In Equations (9.15) and (9.63) the infiltrated volume F is obtained
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