Theoretical and Numerical Study in Porous Media: Solution Verification and Viscous Fingering Instability
The central objective of this dissertation is to develop predictive mathematical and numerical tools for modeling flow and transport in porous media. Specifically, the dissertation presents mathematical models for coupling flow and transport in porous media at the continuum scale; develops associated predictive numerical formulations to solve the resulting governing equations; derives mechanics-based verification methods to assess numerical accuracy; obtains scaling laws pertaining to mixing, miscible displacement and viscous fingering; and highlights the pitfalls of several popular stabilized finite element formulations in simulating physical instabilities like viscous fingering. Success of several important technological endeavors (e.g., hydraulic fracturing, geological carbon-dioxide sequestration) requires a fundamental understanding of coupled processes at multiple scales. Although tremendous progress has been made in the areas of flow and transport (and, of course, in the areas of mechanics, geochemistry, digital imaging, experimental techniques), time has come for another wave of intense research to model coupled processes. Future advancements certainly depend on predictive numerical simulations and careful experiments. The current modeling tools are good at providing qualitative trends, however they are not necessarily accurate to provide predictive quantitative results, which are required for the success of the aforementioned technological endeavors. The main motivation of this dissertation is to improve the predictive capabilities of the continuum modeling tools for flow and transport in porous media. First, a novel mechanics-based accuracy assessment methodology is developed for porous media models and is used to investigate the performance of finite element stabilized formulations with respect to accuracy and convergence. Second, a mathematical model is presented to study the combined effect of temperature and concentration on miscible displacement and viscous fingering. It is also shown that the popular numerical stabilized formulations, which are primarily developed to avoid numerical instabilities, may also eliminate physical instabilities. Hence, care should be exercised in using these formulations to study physical instabilities like viscous fingering. Third, scaling laws and reduced-order models are derived for double-diffusive miscible viscous fingering. One of the main findings is that the evolution of the variances of the concentration and temperature fields scales with the norm of the gradient of the velocity. Fourth, a theoretical and numerical study is performed on viscous fingering in a porous medium which has two dominant pore-networks with possible mass transfer across them. A linear stability analysis is also performed to understand the effects of various parameters (e.g., log-mobility ratio, permeabilities of micro- and macro-pore networks, mass transfer between the pore-networks) on the physical instability.