## 1. Introduction

[2] Analytical solutions of the nonlinear Richards' equation are relatively few. Most closed-form solutions are derived from the linearized version of Richards' equation, which is based on the assumption of constant soil water diffusivity and a linear dependency of the hydraulic conductivity on the moisture content. The latter assumptions also imply that the dependence of the hydraulic conductivity on the pressure head is exponential. The resulting linear form of the governing equation allowed the derivation of a number of solutions for one-, two- and three dimensional soil water flows subject to various initial and boundary conditions, and root uptake forcing functions [e.g., *Warrick*, 1975; *Philip*, 1986; *Basha*, 2000].

[3] However, nonlinear exact solutions of Richards' equation are quite few and they are all restricted to one-dimensional flows. There are mainly two classes of closed-form nonlinear solutions. The first class is based on the assumption of constant soil water diffusivity and a quadratic dependence of the hydraulic conductivity on the moisture content. The resulting equation is known as the Burgers equation for soil water flow [*Philip*, 1974; *Clothier et al.*, 1981]. The second class of exact nonlinear solutions is based on the *Fujita* [1952] functions for the soil hydraulic properties. The diffusivity is expressed by an inverse quadratic function of the water content and the hydraulic conductivity function is given in a rational form. The governing nonlinear partial differential equation is then reduced after a series of transformation into the weakly nonlinear Burgers' equation [e.g., *Sander et al.*, 1988]. All these transformations apply to one-dimensional flows only. However, the transformations used in the first class can handle flux and concentration boundary conditions and yield solutions that are explicit in depth. The Fujita-based solutions of the second class are based on three successive transformations that apply only to flux boundary conditions and yield solutions in parametric form and implicit in depth. The limiting cases of the Fujita-based solutions yield the Burgers' solutions or the linear solution depending on the limiting value of the parameter of the hydraulic conductivity function.

[4] In the first class, *Philip* [1974] presented solutions of J. H. Knight for infiltration into a semi-infinite medium subject to a constant flux or to constant water content at the soil surface. The constant flux solution was presented again with minor corrections by *Clothier et al.* [1981] and was compared with experimental data. *Philip* [1987] used the constant water content solution of Burgers equation to derive expressions for the infiltration rate and the cumulative infiltration at the soil surface. *Broadbridge and White* [1987] presented an expression for the time to ponding for a linearly varying rainfall using Burgers' equation. *Broadbridge and Rogers* [1990] presented a one-dimensional drainage solution of Burgers' equation for an initially saturated vertical column in a semi-infinite medium. *Hills and Warrick* [1993] published a solution of Burgers' equation for a finite region whereby the upper boundary condition is a time-dependent flux and the bottom condition is a constant water content. Finally, *Warrick and Parkin* [1995] derived the same one-dimensional drainage solution of *Broadbridge and Rogers* [1990] from the limiting cases of the Fujita-based solution and provided graphical comparisons and simplifying relationships for small and large times.

[5] In the present work, the class of analytical solutions of Burgers' equation is further explored. This class can be considered as a natural extension of the linear theory into the nonlinear one, since it is also based on a constant diffusivity but with a quadratic hydraulic conductivity function rather than a linear one. It therefore enables us to study the effect of the nonlinearity in the hydraulic conductivity on the moisture movement. The general solution of Burgers' equation in a bounded profile and in a semi-infinite medium is presented for the two types of boundary conditions at the soil surface: a time-dependent flux and a constant water content condition. The bottom condition is a constant water content to simulate the presence of a water table. The derived solutions extend the previous exact solutions of Burgers' equation to include the effect of an arbitrary initial profile and time-dependent surface flux in the presence of a shallow water table. It further allows the modeling of nonlinear infiltration from the preponding to the postponding stage.

[6] Section 2 presents Burgers' equation with the various initial and boundary conditions and section 3 derives the general solution for the cases of a prescribed flux and water content condition at the soil surface. Section 4 presents various particular analytical solutions while section 5 discusses the results pertaining to the time to ponding and postponding infiltration. Section 6 concludes the study.