Multi-scale methods for the numerical simulation of flow and reactive transport in porous media

Abstract

This thesis concerns the design, analysis and application of numerical methods applied to mathematical models over several scales. First, we propose an efficient numerical strategy for solving non-linear parabolic problems defined in a heterogeneous porous medium. The scheme presented here is based on the classical homogenization theory and uses a locally mass-conservative formulation at different scales. Besides, we discuss some properties of the proposed non-linear solvers and use an error indicator to perform a local mesh refinement. The main idea is to compute the effective parameters so that the computational complexity is reduced but preserving the accuracy. We perform a benchmark study of two multi-scale methods. The parameters of the system are obtained by using multi-scale local basis functions and by homogenization over local domains. Both sets of local basis functions and effective parameters are used afterwards in an algebraic dynamic multilevel (ADM) solver. The results reveal an insightful understanding of the two approaches and qualitatively address their performance. It is emphasized that the test cases considered here include permeability fields with no clear scale separation. This development sheds new light on advanced multi-scale methods for simulation of coupled processes in porous media. In addition, we present the details of the implementation of the hybridizable discontinuous Galerkin method (HDG) to solve the porous medium equation (PME). This is a representative example of a degenerate parabolic equation appearing in the last century as a mathematical model for the flow of an ideal gas in a porous medium. We combine the HDG scheme with a robust non-linear solver to efficiently approximate the solution and give rigorous proofs for the existence and uniqueness of the fully discrete solutions and the convergence of the scheme. Finally, we consider mineral precipitation and dissolution processes in a porous medium. Such processes alter the structure of the medium at the scale of pores and make numerical simulations a challenging task as the pores’ geometry changes in time. To deal with such aspects, we adopt a two-scale phase-field model and propose a robust scheme for the numerical approximation of the solution. The scheme takes into account both the scale separation in the model, as well as the non-linear character of the model. After proving the convergence of the scheme, an adaptive two-scale strategy is incorporated, which improves the efficiency of the simulations.