An integrated finite difference method for analyzing fluid flow in porous media

TLDR

IFDM and FEM are conceptually similar, differing mainly in how spatial gradients are measured. The authors present IFDM as a powerful numerical technique for groundwater flow and propose integrating its scheme into FEM to create an improved hybrid method. IFDM merges an integral formulation with finite‑difference gradients, includes a local‑stability criterion, and uses an efficient explicit‑implicit time‑marching scheme, making it suitable for multidimensional heterogeneous isotropic systems. Illustrative tests show that IFDM handles two‑ and three‑dimensional problems with equal ease.

Abstract

The theoretical basis for the integrated finite difference method (IFDM) is presented to describe a powerful numerical technique for solving problems of groundwater flow in porous media. The method combines the advantages of an integral formulation with the simplicity of finite difference gradients and is very convenient for handling multidimensional heterogeneous systems composed of isotropic materials. Three illustrative problems are solved to demonstrate that two‐ and three‐dimensional problems are handled with equal ease. Comparison of IFDM with the well‐known finite element method (FEM) indicates that both are conceptually similar and differ mainly in the procedure adopted for measuring spatial gradients. The IFDM includes a simple criterion for local stability and an efficient explicit‐implicit iterative scheme for marching in the time domain. If such a scheme can be incorporated in a new version of FEM, it should be possible to develop an improved numerical technique that combines the inherent advantages of both methods.