Developing Novel Numerical Methods for Flow and Transport in Porous Media

多孔介质中的流动和运输产生于各种各样的现实世界问题，如石油开采、地下水污染物修复、二氧化碳封存、野火、岩浆通过地壳的运输以及宿主细胞内的病毒蛋白质运输。所有这些问题都具有巨大的经济、环境、医疗和社会意义。数学建模和计算机模拟将提供高效和廉价的工具来提高我们理解、预测和控制上述问题的能力。由于介质的非均质性和各向异性、多时空尺度以及这些过程的不确定性，流动和输运的建模和模拟面临着大量的科学挑战。本研究项目旨在开发一种新的高效和鲁棒的流动和输运耦合问题的数值方法。这些方法将作为计算机软件模块实现，可用于广泛的科学计算任务。该项目还将为研究生，特别是那些来自代表性不足群体的研究生提供实践培训机会。具体来说，该项目侧重于开发新的有限元方法来解决多孔介质中耦合流动和输运问题。这将通过结合弱伽辽金（WG）和欧拉-拉格朗日方法来实现。该方法通过使用单元内部和网格骨架上的自由度，建立了一种新型的微分算子近似。WG有限元方案可以提供局部守恒和对称正定离散线性系统等首选特性。欧拉-拉格朗日方法有效地利用了流动信息，产生了较小的时间截断误差。这使得对运输问题的长时间模拟成为可能。通过结合这两种方法，PI和合作者将设计、分析和实现一系列新的有限元方法，用于解决达西方程、对流主导输运方程、混相位移问题和两相流问题。这些新方法克服了现有方法的缺点，但保留了那些广为接受的优点，例如局部保护。将开发软件模块（Matlab工具箱和基于PETSc的c++库）来将这些新方法应用于实际。通过该项目，博士生将获得数学建模和算法开发的综合能力，以解决具有挑战性的现实问题。

Flow and transport in porous media arise from a wide variety of real world problems such as oil recovery, groundwater contaminant remediation, CO2 sequestration, wildfire, magma transport through the Earth crust, and viral protein trafficking inside host cells. All these problems have tremendous economic, environmental, medical, and social significance. Mathematical modeling and computer simulations will provide efficient and inexpensive tools to enhance our abilities in understanding, predicting, and controlling the aforementioned problems. Scientific challenges abound in the modeling and simulations of flow and transport, due to the heterogeneity and anisotropy of the media, multiple spatial and temporal scales, and uncertainty in these processes. This research project aims at developing a new class of efficient and robust numerical methods for coupled flow and transport problems. These methods will be implemented as computer software modules that can be used for a broad range of scientific computing tasks. This project will also provide hands-on training opportunities for graduate students, especially those from underrepresented groups.Specifically, this project focuses on development of novel finite element methods for solving coupled flow and transport problems in porous media. This will be accomplished by combining the weak Galerkin (WG) and Eulerian-Lagrangian approaches. The WG approach establishes a new type of approximations of differential operators by using degrees of freedom in element interiors as well as those on mesh skeleton. WG finite element schemes could offer preferred features such as local conservation and symmetric positive-definite discrete linear systems. The Eulerian-Lagrangian approach efficiently utilizes the flow information and produces small temporal truncation errors. This enables robust long-time simulations of transport problems. By incorporating these two approaches, the PI and collaborators will design, analyze, and implement a family of new finite element methods for solving the Darcy equation, convection-dominated transport equations, the miscible displacement problem, and two-phase flow problems. These new methods overcome disadvantages of existing methods but maintain those well-received advantages, for example, local conservation. Software modules (a Matlab toolbox and C++ libraries based on PETSc) will be developed to put these new methods into practical uses. PhD students will be trained through this project to gain integrated capabilities of mathematical modeling and algorithm development for challenging real world problems.