Multi-Scale and Interface Mechanics for Porous Media: Mathematical Models and Computational Frameworks

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There are many challenges in subsurface modeling. First, many important subsurface processes occur at the interfaces, either the interface of two different porous media (e.g., layered media) or the interface of free-porous media (e.g., hyporheic zones, arterial mass transport). Second, these processes (flow, transport, and mechanical deformation) are complex, coupled, and multi-physics by nature. Third, natural geomaterials such as fissured rocks often exhibit a pore-size distribution with two dominant pore scales. Fourth, the practical problems are invariably large-scale by nature. Thus, successful modeling of such processes in complex porous media requires: (i) an accurate prescription of flow dynamics within each region and at the interface, (ii) development of robust and accurate computational methods, and (iii) implementation and understanding of these models in a parallel and scalable high performance computing (HPC) environment.

This dissertation develops modeling strategies to advance the current state-of-the-art in subsurface modeling to address the challenges mentioned above. The specific aims are three-fold: First, we develop a comprehensive mathematical framework that provides a self-consistent set of conditions for flow dynamics at an interface. It will be shown that many of the popular interface conditions form special cases of the proposed framework. The approach hinges on extending the principle of virtual power to account for the power expended at the interface and then appealing to the calculus of variations.

Second, we present a discontinuous Galerkin formulation for the double porosity/permeability (DPP) model. We present a numerical procedure to discretize the interface conditions accurately. We develop numerical strategies to simulate and study the flow of fluids in porous media with complex pore-networks by using the DPP model. We also devise solver and parallel computing strategies to solve large-scale practical problems.

Third, we address the coupling of mechanical deformation of the porous solid with transport processes. We assume the porous solid to be an elastoplastic material, and transport of chemical species to be Fickian and develop a mathematical model and a robust computational framework. These modeling tools can be applied to a variety of problems such as moisture diffusion in cementitious materials and consolidation of soils under severe loading-unloading regimes.

Interface mechanics, Double porosity/permeability model, Stabilized mixed formulations, Heterogeneous porous media, Block solvers for porous media, Coupled elastoplasticiy-diffusion
Portions of this document appear in: Joshaghani, M. S., S. H. S. Joodat, and K. B. Nakshatrala. "A stabilized mixed discontinuous Galerkin formulation for double porosity/permeability model." Computer Methods in Applied Mechanics and Engineering 352 (2019): 508-560. And in: Joshaghani, M. S., J. Chang, K. B. Nakshatrala, and Matthew G. Knepley. "Composable block solvers for the four-field double porosity/permeability model." Journal of Computational Physics 386 (2019): 428-466. And in: Nakshatrala, Kalyana B., and Mohammad S. Joshaghani. "On interface conditions for flows in coupled free-porous media." arXiv preprint arXiv:1902.02510 (2019).