Transient multi-phase flow problems in porous media are ubiquitous in engineering and environmental systems and processes;​ examples include heat exchangers, reservoir simulation, environmental remediation, magma flow in the earth crust and water management in porous electrodes of PEM fuel cells. This thesis focuses on the development of accurate and computationally efficient numerical models to simulate such flows. The research challenges addressed in this work fall in two areas. For a numerical standpoint, conventional numerical methods including Newton-Raphson linearization and a simple upwind scheme do not always provide the required computational efficiency or sufficiently accurate resolution of the flow field. From a modelling perspective, closure schemes required in volume-averaged formulations, such as the generalized Leverett J function for capillary pressure, are specific to certain media (e.g. lithologic media) and are not valid for fibrous porous media, which are of central interest in fuel cells.

This thesis presents a set of algorithms that are integrated efficiently to achieve computations that are more than two orders of magnitude faster compared to traditional techniques. The method uses an adaptive operator splitting method based on an a posteriori criterion to separate the flow from the transport equations which eliminates unnecessary and costly solution of the implicit pressure-velocity term at every time step;​ adaptive meshing to reduce the size of the discretized problem;​ efficient block preconditioned solver techniques for fast solution of the discrete equations;​ and a recently developed artificial diffusion strategy to stabilize the numerical solution of the transport equation. The significant improvements in accuracy and efficiency of the approach is demosntrated using numerical experiments in 2D and 3D. The method is also extended to advection-dominated problems to specifically investigate two-phase flow in heterogeneous porous media involving capillary transport. Both hydrophilic and hydrophobic media are considered, and insights relevant to fuel cell electrodes are discussed.