Algorithmic Advancements in the Numerics of Flow and Transport in Fissured Porous Media

Abstract

Research into modeling fluid flow and transport in fissured rocks has been ongoing for the better part of a century. Tremendous gains in our conceptual understanding, astute observations, and computing power have transformed the way we simulate fractured reservoirs, enabling us to model increasingly complex geological settings and predict reservoir behavior with greater fidelity. These developments have provided a robust foundation for understanding and predicting a range of subsurface flow phenomena, with burgeoning applications in subsurface energy storage, CO2 sequestration, and enhanced geothermal systems. Resolving the associated complex and highly nonlinear fluid dynamics at scale fundamentally requires the continuous development of advanced computational tools. This thesis presents new numerical treatments for simulating multiphase flow and transport in porous media, with particular attention to fractured reservoirs. A class of algorithmic monotone multiscale finite volume operators (AM-MsFV/MsRSB), specifically designed for the robust and accurate simulation of flow in heterogeneous porous media, is developed and presented. A key contribution lies in constructing algebraically modified multiscale coarse operators with favorable M-matrix properties. This algebraic construction is rigorously shown to preserve the fundamental mass conservation and near-null space properties characteristic of the original multiscale coarse system. Compared to existing ones, our framework provides both an accurate one-step approximate solver and a robust iterative linear solver while guaranteeing strictly monotone and bounded pressure solutions at all times. Two- and three-dimensional SPE10 benchmarks demonstrate that AM-MsFV/MsRSB operators consistently outperform classical MsFV and MsRSB methods. An extension to MPFA discretized systems is also given. For transport problems in fractured reservoirs, discrete fracture models specifically, we introduce an adaptive nonlinear elimination (NE) preconditioned Newton solver. The solver performs an inner elimination step which selectively neutralizes strong nonlinearities caused by sharp saturation fronts—the source of stiffness for the cases considered. Through a series of waterflooding examples, and an increasing EDFM configuration complexity, the global nonlinear iterations decreased by up to 25% with corresponding savings in CPU time by as much as 17%. On the subject of advanced discretization schemes, a comprehensive study of controlled inexact linear solves within the nonlinear Two Point Flux Approximation framework is presented. The findings suggest that relaxed early stage tolerances can accelerate global convergence without sacrificing accuracy. While a select few physical setups (two phase flow, discrete fracture representations, etc.) were used throughout this thesis work, the applications and use cases for the presented algorithms are far-reaching.

Date of Award	2025
Original language	American English
Supervisor	Mohammed Al Kobaisi (Supervisor)