Algorithms for solving Richards' equation for variably saturated soils

Abstract

Previous work has shown that a one‐dimensional water content‐based finite difference algorithm can offer improved CPU efficiency in modeling infiltration into very dry heterogeneous soil when compared to similar pressure‐based algorithms. The usefulness of the water content‐based method is limited, however, since it cannot be applied to saturated soils. In this paper we develop two new methods which retain the advantages of water content‐based methods while still permitting fully saturated conditions. In the first method we define a new variable for the transformed Richards equation which has the characteristics of water content when soil is unsaturated and of pressure when soil is at or near saturation. In addition to the transformed Richards' equation method, an improved pressure‐based method which uses flux updating is presented. Both methods are implemented in algorithms using a preconditioned conjugate gradient method equation solver. The performance of the algorithms is compared to that of an algorithm based on a mixed form of Richards' equation using modified Picard iteration. Two test cases are examined. The first test case examines two‐dimensional infiltration into very dry heterogeneous soil which remains unsaturated throughout the simulation period. The second test case examines a two‐dimensional developing perched water table. Results indicate that the new methods retain the advantages of the water content‐based method for fully unsaturated heterogeneous problems while allowing for fully saturated conditions. For the unsaturated problem the new algorithms were from 16 to 59 times faster than the algorithm based on the mixed form of Richards' equation using modified Picard iteration. For the saturated problem the new algorithms were from 6 to 25 times faster than the mixed base algorithm. Results also indicate that the performance of the mixed base algorithm could be significantly improved if a more effective adaptive time step were developed.