## 1. Introduction

[2] Infiltration is important in a broad range of applications, including irrigation, aquifer recharge, hydrologic budget, and climate change studies. It is a key component in the hydrological cycle, and it has a significant impact in flood forecasting models and water budget calculation. The incorporation of physically based infiltration expressions leads to a better quantification of the infiltration component in hydrologic models and a more reliable prediction of the runoff hydrograph. This work is an attempt to make useful contributions in that direction.

[3] The infiltration process is represented by a highly nonlinear parabolic partial differential equation known as Richards' equation. A series of papers dealing with exact nonlinear solutions to the Richards' equation has been published over the past two decades [e.g., *Sander et al.*, 1988; *Broadbridge and White*, 1988; *Warrick et al.*, 1990; *Kühnel et al.*, 1990; *Warrick et al.*, 1991; *Triadis and Broadbridge*, 2010]. The solutions are derived for constant boundary conditions and they are relatively complex to use since they are in parametric form and implicit in depth. In the present study, approximate solutions are presented that are simpler in form and adapt easily to changing boundary conditions. They consist of the kinematic wave approximation where the second-order diffusive term is ignored and the traveling wave approximation where the local time derivative in a moving coordinate system is neglected. They are validated against a numerical solution and an exact solution derived for the case of infiltration under a variable flux. The approximate solutions turn out to be highly accurate for specific flow conditions.

[4] The objective of the present study is to develop practical and theoretically sound expressions for hydrological applications that preserve the inherent nonlinearity of the infiltration phenomena while being simple and flexible. For hydrologic applications, proper expressions for the variation of the surface moisture content, the time to ponding, the infiltration rate, and the runoff rate must be available. Water balance models must be able to predict the infiltration and the redistribution component with sufficient accuracy while being simple to use at coarse time scales. Hydrologic models employ empirical equations to quantify some of these variables and the primary limitation of these equations is that the parameters are not physically well-defined.

[5] In the present work, simple expressions of theoretical and practical importance in hydrology and irrigation are presented. The results are stated in terms of well-defined soil water parameters and include the effect of the antecedent moisture content in an explicit fashion. These include simple expressions for the moisture content variation at the land surface for variable rainfall rates, time-to-ponding expressions and infiltration equations that account explicitly for the initial condition, formulas for the depth of the wetting zone, and algebraic equations for parameter estimation of the soil hydraulic parameters by inverse analysis. The above expressions describe the main components of hydrologic models and can lead to better prediction than models that assume instantaneous redistribution and neglect the effect of antecedent moisture content or capillary impacts.

[6] The paper is organized as follows. Section 2 introduces the governing nonlinear partial differential equation of one-dimensional nonsteady infiltration in transformed dimensionless form. Section 3 extends the exact solution of infiltration and redistribution to the case of a piecewise constant flux rate at the soil surface with a piecewise uniform initial condition. Sections 4 and 5 present the traveling and kinematic wave solutions, respectively, which constitute the basis from which infiltration models of hydrological interest are developed. Section 6 evaluates the solutions under various conditions and illustrates the effect of the hydraulic conductivity parameter and the antecedent moisture condition. Section 7 introduces algebraic models for the variation of the surface water potential with rainfall intensity, the time to ponding for predicting the onset of runoff, and the subsequent infiltration rate after ponding. Section 8 presents algebraic equations for parameter estimation of the soil hydraulic parameters by inverse analysis and section 9 concludes the study.