Numerical Approximation of Multiphase Flows in Heterogeneous Porous Media

This work provides an in-depth discussion and key contributions to numerical methods approximating multiphase flow problems in highly heterogeneous porous media. We first highlight important background information, motivations, governing equations, and past results to the numerical analysis of such problems. Second, we introduce a discontinuous finite element discretization, focusing on the specific case of three-phase immiscible flow with constant density and porosity. The resulting decouple system is combined to a stabilization scheme that results in time-independent stiffness matrices for two of the three governing equations. We derive time-error estimates for said scheme, and validate the method on a wide variety of numerical tests ranging from method of manufactured solutions, the quarter five-spot problem, and viscous fingering generation. The result is a robust convergent scheme achieving optimal levels of convergence to both small and large-scale problems. Furthermore, the scheme provides moderate to acceptable levels of improved computational efficiency when compared to the non-stabilized version of the scheme, laying the groundwork for higher dimensional applications. Finally, we study extensively viscous instabilities that arise in oil injection problems and provide valuable insight into key decision-making processes. Through detailed parameter studies and an analysis of element-order selection, we highlight the mechanisms that drive instability generation.

Name thesis:degree_name

Doctor of Philosophy

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2026

Author dc:creator

• Simmons, Mark

Advisor dc:contributor.advisor

• Cappanera, Loic

Committee members dc:contributor.committeemember

• Timofeyev, Iliya
• Rivière, Béatrice
• Saylor, Giselle
• Wang, Min

• Discontinuous Galerkin
• Porous Media
• Scientific Computing
• Numerical Analysis

Language dc:language.iso

English

Handle dc:identifier.uri

https://hdl.handle.net/10657/21434

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/21434