Using finite differences and the Crank-Nicholson implicit scheme for solving parabolic type partial differential equations, a computer program has been developed for solving the one-dimensional, vertical movement of water in soils. The formulation of the initial boundary value problem is obtained by introducing a new dependent variable through the Kirchoff transformation to replace the hydraulic head. Data relating saturation (or moisture content) to the capillary pressure in the soil are used to define the hydraulilc properties of the soil which are needed in order to obtain a solution. The Burdine Theory has been implemented in the program to obtain the needed relationship of hydraulic conductivity to capillary pressure. This formulation and solution method is consistent with the solution method developed earlier for three dimensional axisymmetric movement of water applied at the surface by a circular infiltrometer, so that comparisions of the solution results from the two different cases would indicate quantitative effects on the flow pattern of the component of radial moisture movement. A number of solutions have been obtained for several initial moisture contents and for several application rates (including variable rates equal to the intake capacity of the soil), for a soil at the Reynold's Creek experimental watershed. Saturation-capillary pressure data for this soil were obtained by the Agricultural Research Service in the laboratory. Laboratory measurements of the hydraulic conductivity corresponding to the number of capillary pressure were also obtained. Using the saturation-Capillary pressure data in the Burdine equations for evaluating the hydraulic conductivity gives good agreement with these latter laboratory measurements. The results from these solutions have been used to display the variations of hydraulic head, saturation and hydraulic gradient with time under varying conditions. By contrasting the results from these solutions with those from similar solutions for the axisymmetric case, the effects on the flow patters due to the radial component of moisture movement have been determined.
Jeppson, Roland W., "Solution to Transient Vertical Moisture Movement Based Upon Saturation--Capillary Pressure Data and Modified Burdine Theory" (1970). Reports. Paper 310.