The Local Discontinuous Galerkin Method for Coupled Unsaturated Flow and Chemical Transport in Porous Media

Abstract

The local discontinuous Galerkin (LDG) method offers many advantages over conventional finite element solutions of transport and mechanics problems in soils, namely more accurate simulation of sharp fronts, easier representation of domain discontinuities, algorithms that are more suited to h and p adaptivities and better mass balances between elements. The LDG has been applied to a large number of problems in soils, saturated and unsaturated, including the Richard’s equation, diffusion-advection equation and elastic and non-elastic mechanics. However, its application to coupled problems in porous media is much less widespread. When it has been used to solve problems such as poro-elasticity [1,2] and thermo-elasticity [3], it has been found to be computationally efficient. Coupled problems, thermal, mechanical, hydraulic and chemical, single or multi-phase have become highly significant in the last two decades and are indispensable to a number of current engineering challenges in waste management, biomedical engineering, mining, pollution control, and water resources management, among others. Multiphase thermo-chemo-hydro-mechanical problems frequently occur in problems of waste management and groundwater quality. A number of engineering material and subsurface environments experience unsaturated conditions where both air and water are present in the pores, with or without thermal gradients (e.g. exothermally decaying waste over clay liners, solar ponds, brine ponds). This paper presents an LDG solution to the problem of hydro-chemical transport in an undeformable, threephase medium containing solid, water and air. We build a weak formulation of the non-linear coupled equations of water flow (Richards’ equation) and chemical transport (diffusion-advection equation), in terms of the incremental variables. We present results for the one-dimensional, steady-state case. We implement both linear and higher order elements. We validate the formulation by comparing its predictions to known analytical solutions. We compare the performance of the LDG to that of a continuous Galerkin variant, and identify conditions under which the former is superior to the latter. The work is a prelude for the development of an hp adaptive Paper 16