New Analytical Solutions to 2-D Water Infiltration and Imbibition into Unsaturated Soils for Various Boundary and Initial Conditions

Abstract

Fluid infiltration and imbibition into unsaturated soil are of vital significance from many perspectives. Mathematically, such transient flows are described by Richards’ equation, a nonlinear parabolic partial differential equation with limited analytical solutions in the literature. However, the choice of exponential model for water content and hydraulic conductivity linearizes the nonlinear Richards’ equation, making it possible to obtain an analytical solution via classical approaches. In this study, separation of variables and Fourier series expansion techniques are used to derive new analytical solutions to 2-D vertical and horizontal infiltration and imbibition into unsaturated soils for nonsymmetrical boundary and nonuniform initial conditions. A total of 11 cases are considered, where high water content is imposed on the top, side, or bottom edges of the sample and water is infiltrated (from the top and/or side boundaries) and imbibed (from the bottom boundary) into the sample. Residual water content and/or no-flow boundary condition are assumed on other edges of the sample. Initial conditions include 7 cases of constant residual water content, 2 cases of sinusoidal, and 2 cases of exponential water content functions over the sample. Presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed-form solution. Two-dimensional and 3-D plots of water content are presented for the transient as well as steady-state conditions. To illustrate the use of the derived equations, water content values from numerical solutions are compared to those from analytical solutions for four cases, showing a maximum error of <2 %. The presented analytical solutions may be used as a benchmark for verification and accuracy assessment of numerical approaches where nonsymmetrical boundary and/or nonuniform initial conditions exist.