Modeling and Hybridizable Discontinuous Galerkin Methods for Two-phase Flows in Karstic Geometry

岩溶几何中多相流的一个典型例子是天然岩溶含水层中的流动。喀斯特含水层提供了美国约40%的饮用水，很容易受到污染。在汛期，管道内的水压大于邻近多孔介质的水压，从而使管道携带的污染物被驱入多孔介质中。同样，在干旱季节，由于压力逆转，被隔离在多孔介质中的污染物被释放到管道中的流动中，并通过泉水和井进入地表水系统。管道和多孔介质之间的这种水流交换造成了一个环境问题，因为被隔离的污染物可能影响地下水源的质量，从而大大减少水的可用性。除了在环境科学中的应用外，岩溶几何中的多相流在石油工程中的采油、聚合物电解质膜燃料电池技术以及生物医学中的心血管建模和模拟中也很重要。在这些应用中，管道中的多相流和多孔介质中的多相流相互作用，因此必须一起考虑。由管道和多孔介质组成的几何结构称为岩溶几何。尽管这一课题具有重要意义，但由于问题边界变形、几何结构复杂、不同动力学通过域界面耦合、时空尺度差异大等特点，在这方面的研究还很少。研究者将研究岩溶几何中两相流的杂交不连续伽辽金（HDG）方法的建模和设计。基于相场形式论和Onsager变分原理，本文将推导出具有任意密度和粘度对比的岩溶几何两相流的简并Cahn-Hilliard-Stokes-Darcy模型。该推导旨在克服在岩溶几何中多相流建模中的许多障碍，包括保持无散度速度，推导显式简并迁移率函数，以及纳入多物理场（如润湿和溶质）。然后，PI将介绍和分析超收敛HDG方法，通过利用混合阶多项式近似和在高阶扩散存在下仔细稳定非线性平流来求解扩散界面模型。最后，PI和他的合作者将为扩散界面流体模型开发可扩展的HDG多网格求解器。实用的求解器将进一步解决HDG社区缺乏有效的迭代求解器的问题。研究生将参与该项目的工作。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.