Global-in-Time Domain Decomposition Methods for Flow and Transport Problems in Fractured Porous Media

Abstract

This thesis contributes to the development of numerical methods for the reduced fracture models of flow and transport problems in porous media containing fractures. In particular, our goal is to construct numerical algorithms that enable different time steps on the fracture and on the surrounding area by utilizing global-in-time domain decomposition (DD) methods. In this work, we focus on two types of methods: the first one is based on time-dependent Steklov-Poincare operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. Each method is formulated in a mixed formulation which is suitable for handling problems arising in the modeling of flow and transport in porous media.

We first consider the compressible fluid flow in a fractured porous medium in which the fracture represents a fast pathway (i.e., with high permeability) and is modeled as a hypersurface embedded in the porous medium. Three different global-in-time DD methods are derived using the pressure continuity equation and the tangential PDEs in the fracture-interface as transmission conditions. Each method leads to a space-time interface problem which is solved iteratively and globally in time. Efficient preconditioners are designed to accelerate the convergence of the iterative methods while preserving the accuracy in time with nonconforming grids. Numerical results for two-dimensional problems with different types of fractures and with different number of subdomains are presented to show the improved performance of the proposed methods.

We then focus on constructing efficient numerical methods for the reduced fracture model of the advection diffusion equation. We develop three global-in-time DD methods coupled with operator splitting to treat the advection and the diffusion with different numerical schemes and with different time steps. For each method, separate transmission conditions are formulated for the advection and the diffusion and are combined together to write the discrete space-time interface system. Numerical results for two-dimensional problems with various Peclet numbers and different types of fracture are presented to illustrate and compare the convergence and accuracy in time of the proposed methods with local time-stepping.

We finally reconsider the reduced fracture model of the advection-diffusion equation and aim to tackle the case when the advection is strongly dominated. Three upwind methods are constructed in the content of mixed hybrid finite elements. The first method is a monolithic scheme obtained by fully discretizing the reduced model directly. To incorporate local time-stepping technique, two global-in-time DD methods are derived by decoupling the monolithic solver and imposing appropriate transmission conditions. Several numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Peclet numbers.