Numerical analysis and simulations of coupled free flow and dual porosity models

Abstract

Coupled free-flow and porous media flow can be found in many applications in science and engineering, including oil extraction, groundwater contamination, etc. However, these problems are hard to solve exactly and fast, accurate numerical methods are necessary to solve them computationally. The coupled problems we consider in this thesis are the dual-porosity Stokes problem and its nonlinear version dual-porosity Navier-Stokes problem that model flow in regions partially occupied by naturally fractured porous media. These problems combine flow in microfractures and the porous matrix, governed by the dual-porosity equations, with free flow in macrofractures/conduits, governed by the (Navier-)Stokes equations. The dual porosity and (Navier-)Stokes equations are connected through four interface conditions: mass conservation, balance of forces, the Beavers-Joseph-Saffman condition, and a no-exchange condition between the matrix and the conduits. This thesis introduces, analyzes, and tests through numerical experiments hybridizable discontinuous Galerkin methods for solving Stokes-dual-porosity and Navier-Stokes-dual-porosity problems. We prove existence, uniqueness, and stability of weak solutions to both problems under appropriate data restriction in the nonlinear case. Then, we introduce hybridizable discontinuous Galerkin (HDG) methods, prove well-posedness of the numerical schemes, and present a priori error analyses. Our error estimates show optimal convergence in the energy norm for the Stokes velocity and pressures in Stokes, microfacture, and the martix. In addition, our HDG method is strongly mass conservative and has the typical advantage of computational efficiency compared to the standard discontinuous Galerkin methods. Our numerical examples not only support our findings but also show optimal convergence rates in all variables in the L² norm and show that our method performs well with highly discontinous permeability fields in the microfractures and the porous matrix