Optimization Based Scheme for Solving Richards' Equation

Abstract

The water movement through porous media is governed by Richards’ equation, an over-damped and energy-driven partial differential equation. The energy of the system is determined by the distribution of water inside the soil and the area and material properties of the water-soil contact. Several methods have been used to solve Richards’ equation numerically. Numerical solutions based on the standard soil moisture-based (or θ-based) form of Richards’ equation generally yield poor results for saturated flow problems, due to layers becoming oversaturated. Decreasing the time step or the grid spacing does not help with this deficiency. Here, a new θ-based method is proposed, in which the water distribution in each time step is updated as a solution to a certain optimization problem, with the oversaturation problem being treated through inequality constraints.