Naturally occurring porous and granular materials, such as soil, sand, and clay, play a pivotal role in regulating the Earth's water resources by filtering contaminants and, over long timescales, supplying fresh water. Understanding of this contamination and filtration cycle relies on the phenomenon of transport and dispersion in porous media, with the added complexity that continually flowing groundwater can, over time, alter the details of the porous medium itself such as the size, shape, and position of individual sand grains. These effects are most noticeable during rapid events such as the gravitational collapse of a sinkhole, but can also occur due to the accumulation of slower processes, such as mechanical or chemical erosion. This project will leverage advanced computational methods to study in detail the interplay between fast and slow processes by which groundwater flow alters porous medium properties. By gaining a deeper understanding of the underlying physical processes, the project offers the societal benefit of better management of water resources in the face external factors, such as contamination or sinkhole formation. For example, the understanding developed herein may enable identification of specific regions most susceptible to contamination or to collapse. Graduate students will be involved and receive mentoring and interdisciplinary training in this project.

This project is to analyze a set of complex, dynamical problems that arise in geophysical porous-media applications by using a host of newly developed computational tools and reduced mathematical models. The particular problems of interest include: (1) the erosion of microscopic constituents of porous media leading to anisotropic macroscopic properties; (2) the modified transport of tracers through the medium, including anomalous dispersion; and (3) the occurrence of catastrophic events, such as sink hole collapse, resulting from interaction between groundwater seepage, erosion, and buoyancy forces. The project will address several computational challenges and opportunities. First the range of scales is vast: spatial scales range from microscopic granular constituents to large geological aquifers; timescales range from that of a sudden sinkhole collapse to years required mechanical and chemical erosion. The systems are inherently multicomponent, with coupling between the fluid and solid phases. Although the governing PDEs are linear, the presence of moving boundaries introduces nonlinear feedback between geometry and flow. One challenge in computational fluid dynamics is to obtain high-fidelity simulations of dense suspensions of dynamic bodies. Using integral equation methods in conjunction with accurate quadrature, fast summation methods, contact algorithms, and high-order time stepping, this project will use fast numerical methods to accurately simulate dense suspensions of anisotropic eroding, dissolving, and sedimenting bodies. Mixed-scale, deep neural networks will be used to learn from the data generated by high-fidelity numerical simulations to parameterize coarse-grained models based on the multiphase framework.

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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Malgorzata Peszynska
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Florida State University
United States
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