Mathematical modeling and numerical simulation of multiscale and multiphysics processes are essentially involved in a large number of scientific and engineering problems. Particularly in many applications in environmental sciences and geosciences, one is concerned with the modeling of flow and transport in porous media containing fractures and faults. There the spatial and temporal scales associated with various geological layers and fractures or different physical processes may vary with several orders of magnitude. The goal of this project is to enhance the efficiency of numerical techniques for fractured porous medium applications by designing and analyzing novel computational methods based on parallel global-in-time domain decomposition. These methods facilitate the coupling of different models and enable the use of different time step sizes and spatial mesh sizes in different regions of the computational domain. Thus the proposed methods can be used as an efficient and accurate computational tool for solving large-scale, strongly heterogeneous, coupled evolution partial differential equations arising from diverse application fields such as groundwater flow and contaminant transport, hydraulic fracture, geological disposal of nuclear waste and geological carbon sequestration. The numerical simulations carried out in this project would also provide new insights to the understanding of the long-term behavior and performance of geological nuclear waste repositories. Graduate students will be involved in this project and will be offered a great opportunity to participate in an interdisciplinary research environment.
Although domain decomposition methods have been well studied for many scientific and engineering problems, no enough attention and work have been devoted to fractured porous medium applications with local time stepping. This project focuses on the design and analysis of efficient global-in-time domain decomposition methods for reduced fracture models, in which the fractures are treated as manifolds of one dimension less than the medium. Three model problems will be considered: the linear transport problem, the multiphysics flow and the incompressible two-phase flow, respectively. The developed methods are based on either physical transmission conditions or optimized transmission conditions on the space-time interface fractures; the latter conditions involve more general transmission operators, motivated by the physics of the underlying problem, with some coefficients that can be optimized to improve the convergence rates of the iterations. Importantly, the proposed methods make possible the use of different time step sizes and spatial grids in the interface fractures and in the surrounding medium. The PI will also study the application of the proposed methods to numerical simulation and investigation of fluid flow and contaminant transport in fractured porous media arising from the framework of geological nuclear waste disposal.
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.
