Mathematical and Computational Studies in Poroelasticity

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The PI will study mathematical models which describe the swelling and shrinking of fluid-saturated, elastic, porous media (elastic solids) and the interactions between the fluid and the elastic porous structures. The significance of the research is due to the fact that many natural substances, e.g., rocks, soils, and even biological tissues, as well as man-made materials such as foams, gels, concrete, and ceramics are such elastic porous media. Thus, research findings will be applicable in a multitude of areas (biomechanics, pharmacology, energy technology, geomechanics, geophysics, and materials science). The models developed can be used to describe various man-made materials and manufacturing processes. They can also be used to model the biomechanics of soft tissues and biological porous media, enabling the modeling and optimization of new therapies, development of new diagnostics tools, and advancing our understanding of human physiology. This research will also improve our capability to model, analyze, and assess the environmental impact of processes associated with hydraulic fracturing, wastewater injection, and carbon capture and storage, and activities associated with production of geothermal energy. Graduate and undergraduate students will be trained and will participate in the work. Research findings, and experience from it, will be incorporated into the graduate and undergraduate curricula.This research focuses on mathematical and computational issues arising in continuum models of poroelasticity and electroporoelasticity. These provide a unified and systematic treatment of various porous materials and processes which arise in diverse areas of science and engineering and a variety of applications. Additional physical phenomena, such as the electromechanical response of the medium, as well as chemical and/or thermal effects, may also be accounted for. Thus, poroelasticity, electroporoelasticity, or even electro-chemo-thermo-poroelasticity, are all complex coupled, multiphysics, multiscale, phenomena, where the swelling and shrinking of an elastic or viscoelastic deforming porous medium is coupled to the electromechanical (and/or the thermal, and chemical) response of the medium and saturating fluid. Poroelasticity also involves multiple scales; the micro-scale corresponding to the molecular scale, and the scale of continuum mechanics, the macro-scale. The PI will develop mathematical and computational tools and study, analytically and computationally, various mathematical problems arising in poroelasticity. Among the issues considered will be the well posedness of mathematical pde models and the derivation and analysis of efficient and accurate numerical algorithms for approximating solutions of these partial differential equations.