Solutions are obtained to the problem of steady-state partially saturated infiltration of moisture applied over a horizontal source circle which moved through homogeneous soils toward a water table. A commonly accepted relationship between relative permeability and capillary pressure has been utilized in conjunction wit Darcy’s law to formulate the mathematical model. The solutions have utilized an inverse formulation and have been obtained by finite difference. The inverse formulation considers the magnitudes of the cylindrical coordinates r and z as the dependent variables and the potential function and the Stokes’ stream function as the independent variables (i.e the problem is solved for r and z in the plane). The approach used for solving the problems is practical with modern digital computers. The computer output gives the r and z coordinates at each finite difference grid point. These values can readily be plotted in flow net form to show the characteristics of the flow pattern at a glance. From the solution results, the distribution of capillary pressure, relative permeability, or effective saturation over any surface or plane of interest can be obtained. The solutions indicated that significant radial movement of moisture (or spreading effect) occurs causing higher infiltration rate at the edge of the source circle than near the center. The infiltration rate is closely related to carious soil parameters which characterize the hydraulic properties of soils. Also presented are several distributions of the relative permeability r effective saturation on the surface, along the axis of symmetry, and on the place including the axis of symmetry and how these distributions are related to the sole parameters.
Wei, Chi-Yuan and Jeppson, Roland W., "Finite Difference Solutions of Axisymmetric Infiltration Through Partially Saturated Porous Media" (1971). Reports. Paper 16.