A prime example of multi-phase flow in karstic geometry is flow in natural karst aquifers. Karst aquifers supply about 40 percent of the drinking water in the United States, and are susceptible to contamination. During flooding seasons, the water pressure in the conduits is larger than that in the adjacent porous media so that conduit-borne contaminants are driven into the porous media. Likewise during dry seasons, contaminants sequestered in the porous media are released into flow in the conduits due to the pressure reversal, and exit through springs and wells into surface water systems. This exchange of flow between conduits and porous media poses an environmental issue in that sequestered contaminants may influence the quality of underground water sources and thus significantly decrease water availability. Besides applications in environmental science, multi-phase flows in karstic geometry are also important in oil recovery in petroleum engineering, in Polymer Electrolyte Membrane fuel cell technology, as well as in cardiovascular modeling and simulation in biomedical sciences. In these applications multi-phase flows in conduits and in porous media interact with each other, and therefore have to be considered together. Geometric configurations that consist of both conduits and porous media are termed as karstic geometry. Despite the importance of the subject, little work has been done in this direction, due to the nature of deforming boundary of the problem, the complex geometry, the coupling of different dynamics via domain interface, the vast disparity of spatial and temporal scales and so on.
The investigator will examine modeling and the design of hybridizable discontinuous Galerkin (HDG) methods for two-phase flows in karstic geometry. Based on the phase field formalism and Onsager's variational principle, the PI will derive a degenerate Cahn-Hilliard-Stokes-Darcy model for two-phase flow of arbitrary density and viscosity contrast in karstic geometry. The derivation seeks to overcome a number of obstacles in modeling of multiphase flows in karstic geometry, including maintaining a divergence-free velocity, deriving an explicit degenerate mobility function, and incorporating multiphysics such as wetting and solute. The PI then will introduce and analyze superconvergent HDG methods for solving the diffuse interface model by exploiting approximation via polynomials of mixed orders and by carefully stabilizing the nonlinear advection in the presence of high-order diffusion. Finally, the PI and his collaborators will develop scalable HDG multigrid solvers for diffuse interface fluid models. The practical solvers will further address the lack of efficient iterative solvers in the HDG community. Graduate students will participate in the work of this project.
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.
