Phase transitions in porous media across multiple scales

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In this project the PI will develop and analyze computational models describing the evolution of methane hydrate. Methane hydrate is an ice-like substance of great interest in geophysics, climate studies, and energy engineering, because it can release methane, a powerful greenhouse gas and drilling hazard; hydrates can also serve as a potential unconventional energy source. This work will involve several spatial scales from the kilometer scale of hydrate bearing subsea sediments and Arctic permafrost regions, to the micro-scale at which one looks at the pores of the sediments. The PI's mathematical analysis of hydrate models, under some simplifying assumptions, has so far revealed very unusual features absent, e.g., in ice-water phase transitions. Based on these analyses the PI will formulate more accurate algorithms for the simplified and for more complex realistic models, which in turn can further our understanding of hydrate formation and dissociation in nature. The software the PI will develop will be shared with the geophysics community.In PI's prior work, she has framed a simplified model of methane hydrate as a parameter-dependent free boundary problem, and her analyses show that its solutions can develop multiple singularities beyond those known, e.g., for Stefan problem of ice-water phase transitions. In this project the PI will develop new techniques for the simplified model extending substantially the classical results known for the non-parametrized case, and studying fundamental research questions concerning, e.g., nonlinear degenerate diffusion and conservation laws with the particular type of parameter-dependent monotone operators. The will extend the delicate time-stepping analyses in the abstract setting, develop a priori and a posteriori error analyses, and robust algorithms for the couplings across multiple time and spatial scales. In particular, the will consider the porescale at which she formulates reduced and dynamic models for many of the constitutive relationships needed for macroscale. In additional to computational mathematics, the project will impact the geophysics community involved in hydrate modeling, and more broadly other applications in porous media.