Developing Novel Numerical Methods for Flow and Transport in Porous Media

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Flow and transport in porous media arise from a wide variety of real world problems such as oil recovery, groundwater contaminant remediation, CO2 sequestration, wildfire, magma transport through the Earth crust, and viral protein trafficking inside host cells. All these problems have tremendous economic, environmental, medical, and social significance. Mathematical modeling and computer simulations will provide efficient and inexpensive tools to enhance our abilities in understanding, predicting, and controlling the aforementioned problems. Scientific challenges abound in the modeling and simulations of flow and transport, due to the heterogeneity and anisotropy of the media, multiple spatial and temporal scales, and uncertainty in these processes. This research project aims at developing a new class of efficient and robust numerical methods for coupled flow and transport problems. These methods will be implemented as computer software modules that can be used for a broad range of scientific computing tasks. This project will also provide hands-on training opportunities for graduate students, especially those from underrepresented groups.Specifically, this project focuses on development of novel finite element methods for solving coupled flow and transport problems in porous media. This will be accomplished by combining the weak Galerkin (WG) and Eulerian-Lagrangian approaches. The WG approach establishes a new type of approximations of differential operators by using degrees of freedom in element interiors as well as those on mesh skeleton. WG finite element schemes could offer preferred features such as local conservation and symmetric positive-definite discrete linear systems. The Eulerian-Lagrangian approach efficiently utilizes the flow information and produces small temporal truncation errors. This enables robust long-time simulations of transport problems. By incorporating these two approaches, the PI and collaborators will design, analyze, and implement a family of new finite element methods for solving the Darcy equation, convection-dominated transport equations, the miscible displacement problem, and two-phase flow problems. These new methods overcome disadvantages of existing methods but maintain those well-received advantages, for example, local conservation. Software modules (a Matlab toolbox and C++ libraries based on PETSc) will be developed to put these new methods into practical uses. PhD students will be trained through this project to gain integrated capabilities of mathematical modeling and algorithm development for challenging real world problems.