No abstract

Richards' equation of the unsaturated flow has been scaled in an invariant form with regard to the variation of boundary conditions within a defined class of flow problems. Two classes of problems were examined: (1) vertical infiltration into a homogeneous soil with a constant surface boundary flux, and (2) vertical infiltration into a soil topped by a seal layer of negligible thickness and with a given positive pressure head above the seal layer. The solution of the problem when plotted as function of scaled variables is unchanged for any variation at the boundary, i.e., variation of the flux in the first class and variation of the seal in the second class.

A recent solution determined by Babu for an arbitrary soil-water diffusivity is compared with two earlier solutions. It is shown that the perturbation solution is identical to the iterative result when Cisler's (1974) correction is used. It is also shown that a result obtained by optimization, which is numerically indistinguishable from either the perturbation or iterative result, has a much simpler analytical form. In a recent paper [Babu, 1976b] a perturbation method wasused to obtain the solution of the nonlinear one-dimensional diffusion equation. Earlier, Babu [1976a] had analyzed the one-dimensional movement of water for exponential diffusivity by the same perturbation method. For this special case it was shown [Parlange and Babu, 1977] that the perturbation result was identical to the earlier iterative result [Parlange, 1971], as corrected by Cider [1974]. It was also pointed out [Parlange and Babu, 1977] that the optimization result [Parlange, 1975a] leads to analytical forms that should be simpler to use, in general, than either the iterative or perturbation results (see also Brutsaert [1976]). In the following the perturbation and iterative methods are shown to be identical, and the greater simplicity of the optimal result is demonstrated for an arbitrary diffusivity. The corrected form [Cider, 1974; Parlange, 1975b] of the iterative result [Parlange, 1971] is, up to the second order, 1 )1/2 1 1 where D(O) is the diffusivity given as a function of the water content 0 and ß = xt -•/:, in which x is the distance from the

The full Saint Venant equations of overland flow on a plane are often replaced by simpler models. The errors in the kinematic and the diffusion models are estimated by comparing their predictions with the exact numerical solution of the Saint Venant equations under steady state conditions. It is shown that the two approximate models can have significant errors even for critical flow and fairly large kinematic wave numbers. When the kinematic approximation is inaccurate, the improvement of the diffusion approximation seems modest. For the same level of mathematical complexity as the diffusion approximation, a far more accurate approximation is proposed when the kinematic wave number is large. It is then possible to split the solution of the Saint Venant equation in two regions, one near the downstream end of the plane and the other covering most of the plane.

For specific soil properties the fundamental solution of Fujita (1952b) is extended to include the effect of air movement on horizontal absorption. This exact solution is then used to determine the accuracy and range of applicability of two approximations to the sorptivity with air effects previously presented by Parlange et al. (1982) and Sander and Parlange (1984). The optimal sorptivity of Parlange et al. has very good accuracy while the surface water content θs is near saturation but decreases in accuracy as θs decreases. The approximate sorptivity of Sander and Parlange, however, proves to be extremely accurate for all θs with a maximum error of only 0.5%. Sander and Parlange additionally give an approximation to the water profile which is also shown by comparison with the exact solution to have a maximum error of only 0.5%. The aim of this paper is primarily of a theoretical nature in that we use the exact solution to check the accuracy of the approximate solutions as stated above. Alternatively, the solution may be used for checking the accuracy of numerical models.