A computational framework for nonlinear multiphase flow in porous media using a meshfree method: numerical experiments and applications

Abstract

Meshfree methods offer several advantages, including their meshfree nature, capability to handle complex geometries, and simple programming implementation. This paper presents a numerical scheme extending the method of fundamental solutions (MFS) for a class of nonhomogeneous and nonlinear problems to investigate the flow through porous media. We first examine the existence and uniqueness of weak solutions by the theory of monotone operators for general nonlinear boundary value problems. In a particular case, the generalized equation turns out to be the governing equation for the flow through porous media, known as the Brinkman–Forchheimer equation. The equation effectively models diverse flow phenomena in porous media, including key aspects like nonlinearity and nonhomogeneity. The structure of the equation resembles that of the Helmholtz equation, suggesting an approach that extends by representing the nonhomogeneous term through the superposition of Green’s functions of the Helmholtz operator. To demonstrate the effectiveness of the approach, different numerical experiments and detailed error analyses are conducted. The nonlinearity in the equation is tackled by utilizing the fixed point iteration technique. Numerical simulations are performed to investigate flow through porous media within irregular channels, including the Stokes–Brinkman system, with different sets of boundary conditions to evaluate the performance of the approach. Numerical results are provided to validate the work with the finite-element method and existing literature by varying different parameters. © 2025 Elsevier B.V., All rights reserved.