Collaborative Research: Adaptive Mixed-Dimensional Modeling and Simulation of Porous Media

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The overall goal of this research project is to develop, analyze, and implement stable numerical methods for the simulation of flow in fractured porous media, based on mixed-dimensional modeling. Fractured porous media flow is multi-physics and multi-scale, and the development of robust and effective numerical tools for such systems represents a class of important challenges in computational mathematics. For example, in many practical applications fractures or other features, such as capillaries in the brain, are lower-dimensional and interact in a complex way through the three-dimensional domains encapsulating them. Important applications of fractured porous media include hydraulic fracturing, waste deposition, and models from biomechanics. The project aims to provide new computational paradigms, promote the usage of mixed-dimensional modeling in these and related fields, and alleviate current limitations in computer simulations. The techniques will be implemented in an open-source software package and will be made available to the scientific community. In this project, three important aspects of mixed-dimensional modeling and simulation will be investigated. The first research objective is to design advanced stable discretizations, which couple stabilized schemes for linear elasticity with structure-preserving discretizations for Darcy flow in a mixed-dimensional setting. Stability and mass conservation will be achieved using a minimal number of degrees of freedom. The second objective is adaptive approximations in the mixed-dimensional framework. Space-time adaptivity will be developed and implemented to improve accuracy with solid theoretical foundations. The third objective is to study robust linear solvers for the resulting discrete linear systems. Based on physical and mathematical properties of the mixed-dimensional models and their numerical discretizations, new block preconditioners and monolithic multigrid methods will be developed. Robustness with respect to physical and discretization parameters will be justified both theoretically and numerically. The main application of this project is linear poromechanics in fractured porous media, but possible generalizations to nonlinear formulations will also be investigated, including several applications in physics and engineering.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.