One of the goals of science is to understand and explain natural phenomena in order to predict and forecast outcomes. The continual development of mathematical tools to express many of the fundamental laws of nature has resulted in models with unparalleled predictive power. Mathematical models involve complex systems of equations expressing the physical processes of interest, and form the conceptual foundation of modern engineering and science. These equations exhibit all of the depth, difficulties, and subtleties of the physical systems being modeled, and sophisticated computational methodologies are required for their solution. This project focuses on the development of computational tools and software that engineers and scientists need to simulate problems arising in geology. One of the physical problems motivating this work is the need to model deep sea beds and permafrost regions where vast quantities of frozen methane, carbon dioxide, and other gases are trapped. These gases diffuse from deep within the earth and get trapped in permafrost, under glaciers, and in the cold depths of the deep ocean where they may freeze. Predicting the evolving state of these regions through geologic cycles and changes in climate is essential for environmental modeling. This project will develop and analyze computational tools and software required to model these regions where multiple fluids and gases undergo freezing, thawing, and dissolution as they flow. In addition to the technological developments, this project will also support the education and training of the next generation of scientists needed to sustain discovery and scientific leadership in these disciplines.
The work proposed centers around the development and analysis of computational tools needed to simulate flows in porous media containing multiple fluids, which may combine to form multiple phases and states (solid, liquid, gas). The thermodynamics of mixtures and phase changes is used to model the properties and states of the fluids at the pore scale, which appear constitutively in the macroscopic balance law of mass, momentum, and energy. This proposal focuses on the challenging problem of integrating pore scale and continuum models of these systems into a consistent framework where computational tools can be utilized to simulate these complex physical problems. Convexity and homogeneity of the free energy (or concavity of entropy) enter these models in a subtle way, and are essential for proving stability of solutions. Numerical schemes will be developed that faithfully inherit these important attributes; in addition, convexity properties will be exploited to facilitate robust algorithms for the solution of the nonlinear systems that arise. Development of mathematical tools required to analyze the stability, robustness, convergence, and correctness of these algorithms will be undertaken.
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.
