An Enriched Galerkin (EG) method for multicomponent porous media flow with nonlinear coupling

Abstract

In this paper, we discuss fully discrete numerical methods for the multicomponent compressible single-phase Darcy flow in porous media. The new compressible pressure equation is based on the volume balance approach while considering the decomposition of the diffusion-dispersion tensor into two parts to obtain a self-consistent coupled nonlinear model. Similar to conventional approaches, the numerical decoupling of the system is through a pressure equation, which can be used to calculate the pressure and Darcy velocity by using the concentration from the previous time step. The novelty of this work is a new formulation of the consistent pressure equation based on the multicomponent setting. The conventional approach, however, typically uses a pressure equation derived from a single-pseudo-component setting, which is not consistent physically; the physical inconsistency at the PDE level also yields certain inconsistencies at the numerical discretization. The new pressure equation formulated in this work overcomes the difficulties on both the PDE level and the discretization level. We use the enriched Galerkin (EG) methods to approximate the mass conservation law of each component, followed by substitution of the discrete Darcy law to obtain a numerical discretization of the species transport equations. We then carried out a weighted summation of the species transport equations to obtain the discrete pressure equation, which gave us a self-consistent decoupling numerical scheme. At the end of this article, error analyses for semi-discrete and fully discrete cases are presented, and the theoretical conclusions are verified through numerical experiments.