Modeling with Constraints and Phase Transitions in Porous Media

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The changes in sea temperatures and those in permafrost regions have influence on hydrological, climate, and human systems on the time scale of years and decades. The freezing and thawing of permafrost leads to subsidence and development of hills and depressions; these alter the direction of surface waters, and challenge the stability of construction. Warming of permafrost causes decay of biomass as well as release and transport of methane gas from hydrate and other stores underneath the frozen layers. Methane gas is a major contributor to greenhouse gas balances, and its presence, transport, and evolution, as well as that of methane hydrate, an ice-like substance, are of great interest in geophysics, climate studies, and energy engineering. Methane hydrate can dissociate to gas, and is a potential unconventional energy source, drilling hazard, and contributor to sub-sea slope instability. The realistic scenarios and case studies motivating this work include the melting of ice to water and dissociation of methane hydrate into gas in response to increased temperature or mechanical disturbance, with the liquid and gas phases traveling through the sediment. Models of similar nature also apply to bubble and steam transport, e.g., due to microbial activity, or in geysers. In this project the principal investigator will develop new mathematical results as well as those useful for geophysics, and continue interdisciplinary modeling efforts across the many fields. This project addresses mathematical and computational challenges arising in the models of phase change in porous media such as permafrost or sub-sea sediments, and the evolution and migration of the resident liquid and the dissociated gas phase. For these coupled processes of energy and mass transport, data and some nonlinear model systems of PDEs at the spatial and temporal (Darcy) scale of the reservoir are available. However, the new dynamics in the Earth's environment calls for further insights into the processes across the several interlinked spatial and temporal scales including the pore-scale.The models at the pore-scale bridge the physics between the interface and Darcy scales, account for confinement within the porous walls and complex geometries, and can be upscaled to Darcy scale. While x-ray micro-tomography can deliver unprecedented insight into pore-scale processes, at present, this data is rather sparse for conditions near phase transitions. These challenge the current knowledge and motivate the project. The research will advance the understanding of the micro-scale (pore-scale and interface scale) processes which inform the Darcy scale models. The project will develop and analyze algorithms for relevant models; many techniques and results are of independent interest. The evolution models the principal investigator will consider are complex and delicate, and involve pointwise constraints on the solutions in and out of the equilibrium.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.