Techniques for Making Finite Elements Competitve in Modeling Flow in Variably Saturated Porous Media

Abstract

A Galerkin finite element formulation is developed for the numerical simulation of water flow in variably saturated soil systems. Included in this formulation is a solution strategy based on Picard and Newton‐Raphson algorithms. Both algorithms are designed especially to cope with severely nonlinear field problems. The two algorithms are formulated for both rectangular and triangular elements. The element matrices are evaluated in a simple and efficient manner using a technique referred to as the “influence coefficient” technique. This technique avoids numerical integration and leads to a substantial saving of computational cost. Four examples are presented to demonstrate the effectiveness of the present finite element approach. These examples show that the nonlinear solution schemes are capable of accomodating cases involving large variations in the saturated hydraulic conductivity, as well as highly nonlinear soil moisture characteristics. A comparative study of the Picard and the Newton‐Raphson algorithms is also provided. The study indicates that despite the higher cost per iteration of the Newton‐Raphson scheme, it usually requires a substantially smaller number of iterations than the Picard scheme. In some instances where convergence difficulties are experienced with the latter scheme, it is desirable to use the Newton‐Raphson scheme in order to obtain a cost‐effective solution to the problem.