Theoretical and Computational Study of Flow through Porous Media with Double Porosity/Permeability
Flow of incompressible fluids through porous media plays a crucial role in many technological applications such as enhanced oil recovery and geological carbon-dioxide sequestration. Numerous natural and synthetic porous materials contain multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores with different hydro-mechanical properties. Flow through such materials cannot be adequately described by the classical Darcy equations. Various mathematical models for fluid flow through media with multiple scales of pores are available in the literature. However, many of them are analytically intractable for realistic problems and lack a strong theoretical basis. This dissertation aims to fill this gap in knowledge by providing the much needed theoretical foundations. We present a mathematical model and a robust computational framework for studying flow through porous media that exhibit double porosity/permeability. On the theoretical front, we combine the theory of interacting continua and the maximization of rate of dissipation hypothesis to provide a firm thermodynamic underpinning. Several canonical problems are used to gain insights into the velocity and pressure profiles, and the mass transfer across the two pore-networks. Due to the importance of the coupling between structural mechanics and fluid dynamics in various industrial applications, the theoretical study of fluid flow is further extended to incorporate solid deformation. On the computational front, a mixed four-field finite element formulation is presented which is shown to be stable under convenient equal-order interpolation for all the field variables. The stabilization terms and the stabilization parameters are derived in a mathematically and thermodynamically consistent manner. A systematic error analysis is performed on the resulting stabilized weak formulation. Representative problems, patch tests and numerical convergence analyses are performed to illustrate the performance and convergence behavior of the formulation. The computational framework is further extended to the transient analysis and coupled flow-transport problems and the effect of velocity-dependent drag coefficient on the profiles of solutions is studied. The results of this study clearly highlight how the solutions under the double porosity/permeability differ from the corresponding ones under the Darcy equations. The importance of internal pore structure in characterizing flow through porous domains is emphasized. This pitches a case for using advanced characterization tools like micro-CT to measure the size distribution of pores within the solid, so that they could be modeled appropriately.