{"id":{"repo_id":"houston","oai_identifier":"oai:uh-ir.tdl.org:10657/21434"},"canonical_url":"https://search.dev.ndltd.org/etd/houston/oai:uh-ir.tdl.org:10657/21434","repository":{"repo_id":"houston","name":"University of Houston","base_url":"https://uh-ir.tdl.org/server/oai/request"},"display":{"title":"Numerical Approximation of Multiphase Flows in Heterogeneous Porous Media","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Simmons, Mark"],"institution":"University of Houston","degree_name":"Doctor of Philosophy","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Cappanera, Loic"],"committee_chairs":[],"committee_members":["Timofeyev, Iliya","Rivière, Béatrice","Saylor, Giselle","Wang, Min"],"year":2026,"date_issued":"2026-05","date_published":"2026-05","updated_at":"2026-07-24T02:33:01Z","subjects":["Discontinuous Galerkin","Porous Media","Scientific Computing","Numerical Analysis"],"languages":["English"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10657/21434","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Cappanera, Loic"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Timofeyev, Iliya","Rivière, Béatrice","Saylor, Giselle","Wang, Min"]},{"key":"dc:creator","label":"Author","values":["Simmons, Mark"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2026-06-23T20:00:06Z"]},{"key":"dc:date.issued","label":"Date","values":["2026-05"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Houston"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Discontinuous Galerkin","Porous Media","Scientific Computing","Numerical Analysis"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["English"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10657/21434"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Numerical Approximation of Multiphase Flows in Heterogeneous Porous Media"]}]}],"canonical_facts":{"dc:contributor.advisor":["Cappanera, Loic"],"dc:contributor.committeemember":["Timofeyev, Iliya","Rivière, Béatrice","Saylor, Giselle","Wang, Min"],"dc:creator":["Simmons, Mark"],"dc:date.accessioned":["2026-06-23T20:00:06Z"],"dc:date.issued":["2026-05"],"dc:description.abstract":["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."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/10657/21434"],"dc:language.iso":["English"],"dc:subject":["Discontinuous Galerkin","Porous Media","Scientific Computing","Numerical Analysis"],"dc:title":["Numerical Approximation of Multiphase Flows in Heterogeneous Porous Media"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["University of Houston"]},"updated_at":"2026-07-24T02:33:01Z"}