Collaborative Research: Development of Reduced Order Models for Poroelasticity and Related Problems

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Poroelasticity is a framework of continuum mechanics models for problems involving a porous elastic medium and a fluid flow. Poroelasticity problems have real-world applications such as hydrocarbon extraction in petroleum engineering, physiological processes such as the blood flow in the human body, groundwater contamination in environmental engineering, and modeling magma and mantle migration in geophysics. There is a real need to obtain high-resolution numerical solutions for the poroelasticity but these require large computational resources, even with theoretically optimal algorithms. This project will develop methods that can provide high accuracy solutions for large poroelasticity problems with feasible computational costs. Students will be involved and trained in interdisciplinary applications. Poroelasticity problems are parametric partial differential equations (PDEs) which are described by a set of coupled PDEs with various physical parameters. Developing reliable and efficient numerical methods for these problems involving complex forms of physical parameters is crucial to obtain high-resolution solutions in a relatively short time. The project will use the reduced basis method (RBM) approach which can provide fast algorithms for parametric partial differential equations of elliptic and parabolic types and has a solid mathematical framework to derive reduced order models with rigorous, certified error bounds. However, developing an RBM for a system of coupled PDEs is nontrivial because it must be devised to preserve the stability structure of the coupled system. The techniques and analysis developed in this project will advance the understanding of parametric poroelasticity and its related problems. The results of this project can also provide a general guideline in developing RBMs for other coupled parametric PDEs. Once developed, these algorithms can be employed for complex problems that are currently intractable with traditional numerical algorithms. These solutions will lead to advances in fields such as petroleum engineering, environmental engineering, and computational biomechanics with potential beneficial impacts on human life and health.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.