Mixed Lagrangian-Eulerian and Eulerian Approach to Discretizing Richards' Equation

Abstract:

This paper presents a robust, efficient numerical solution involving the use of the mixed Lagrangian-Eulerian (LE) method and the Eulerian (L) approach for three dimensional simulations of variably saturated subsurface flow that is described by Richard’s equation. The LE approach with its particle tracking algorithm and/or finite element methods (FEM) were employed to discretize interior nodes while the finite element method is selected to set up algebraic equations for boundary nodes. The use of FEM for boundary nodes alleviate the difficulty in dealing with flux and gradient types of boundary conditions. Extrapolations are no longer needed to handle flux or gradient type boundary conditions. In this new mixed LE&E approach, subsurface flow in variably saturated media can be efficiently dealt with. Three examples are provided to demonstrate the efficiency of the proposed approach. First, a one-dimensional column problem is used to compare the accuracy of the mixed LE&E approach versus the traditional Eulerian approach. Second, a three-dimensional drainage problem was simulated to compare the CPU time between two approaches. Third, a three-dimensional pumping well problem was simulated. In all three examples, the mixed LE&E using relatively large time steps yielded superior results in terms of the accuracy and computational efficiency in comparison with the conventional Eulerian approach. The proposed mixed LE&E approach may contribute to the efficient numerical solutions of problems involving moving sharp fronts problems such as groundwater in real-world watersheds.