Numerical algorithms for nonlinear subsurface flow and transport

对涉及地下水库和含水层的人类活动进行评估、设计和监测，需要对长时间内的水流和输送过程进行大规模的计算机模拟。这种模拟用于指导科学家和工程师，例如，石油生产，地下水管理，以及二氧化碳和核污染物等废物的安全储存。地下环境的计算机模拟尤其重要，因为它在很大程度上无法直接观察，因此也很难或不可能纠正人为失误。该项目涉及非混相两相地下流的基础研究和教育。通过适当地考虑控制方程中包含的物理过程的非线性耦合行为，该项目具有显著提高保真度模拟的潜力，并且数值算法应该在现代超级计算机上运行良好。此外，许多科学和工程感兴趣的模型由类似的非线性耦合流动和输送系统组成，因此这里的一般进展可能支持更广泛的努力。该项目将资助两名STEM博士研究生和至少两名本科生的研究。他们将在计算工程与科学研究所的地下建模中心工作，该中心提供了一个跨学科的环境，融合了数学、计算科学、石油工程和地质科学的专业知识。学生将为在学术界、政府实验室和工业界的就业机会做好准备。外部合作将增强项目的影响。该项目涉及对具有代数约束的非线性、耦合偏微分方程组的新算法的基础研究和开发。要解决的目标系统是非混相的两相地下流，用于地下流体运动的数值模拟。该项目强调流动和运输的高度非线性物理过程，以及它们如何相互影响。该项目的目标是：(1)改进的混合型数值算法，当耦合到输运模型时，用于近似多尺度非均质多孔介质内的非线性流动；(2)发展了欧拉-拉格朗日加权本质非振荡数值算法，用于耦合到非线性流动模型的多维非线性输运；(3)算法在石油生产、地下水管理、碳封存等具体应用中的有效性论证；(4)在跨学科环境中教育和培训两名研究生和两名本科生。该项目预计将显著改善地下流量和输送的数值近似，即使在很长一段时间内也是如此。项目的成功将取决于数值算法的发展，这些算法在局部保持质量，几乎不产生过冲或过冲，表现出低水平的数值扩散，并且在粗计算网格上具有高阶和精确。为了稳定，它们将需要大幅放宽CFL时间步长限制。最重要的是，这些方法将被证明可以很好地用于非线性耦合系统，因为这两个系统将在精度和效率方面相互补充。

Assessment, design, and monitoring of human activities involving reservoirs and aquifers in the Earth's subsurface require large-scale computer simulation of flow and transport processes over long time periods. Such simulations are used to guide scientists and engineers regarding, e.g., petroleum production, groundwater management, and secure storage of wastes such as carbon dioxide and nuclear contaminants. Computer simulation of the subsurface environment is especially important because it is largely inaccessible to direct observation and therefore also difficult or impossible to rectify human failures. The project involves fundamental research and education on immiscible, two-phase subsurface flow. This project has the potential for significantly greater fidelity simulations by properly accounting for the nonlinearly coupled behavior of the physical processes embodied in the governing equations, and the numerical algorithms should work well on modern supercomputers. Moreover, many models of scientific and engineering interest consist of similar nonlinearly coupled flow and transport systems, and so general progress here is likely to support efforts more broadly. The project will fund the research of two STEM Ph.D. graduate students and involve at least two undergraduate students. They will work in the Center for Subsurface Modeling of the Institute for Computational Engineering and Sciences, which provides an interdisciplinary environment mixing expertise in mathematics, computational science, petroleum engineering, and geological science. The students will be prepared for employment opportunities in academia, government laboratories, and industry. External collaborations will enhance the impact of the project.The project involves fundamental research on and development of new algorithms for nonlinear, coupled systems of partial differential equations with algebraic constraints. The target system to be addressed is immiscible, two-phase subsurface flow, which is used for numerical simulation of the movement of underground fluids. The project emphasizes the highly nonlinear physical processes of flow and transport, and how they influence each other. The objectives of the project are: (1) Improved numerical algorithms of mixed type for approximation of nonlinear flow within a multi-scale, heterogeneous porous medium when coupled to a transport model; (2) Development of Eulerian-Lagrangian Weighted Essentially Non-Oscillatory numerical algorithms for multi-dimensional, nonlinear transport when coupled to a nonlinear flow model; (3) Demonstration of the effectiveness of the algorithms in specific applications such as petroleum production, groundwater management, and carbon sequestration; and (4) Education and training of two graduate and two undergraduate students in an interdisciplinary setting. The project is expected to result in significantly better numerical approximations of subsurface flow and transport, even over very long time periods. Project success will be seen by the development of numerical algorithms that preserve mass locally, produce little to no overshoots or undershoots, exhibit low levels of numerical diffusion, and are of high order and accurate on coarse computational meshes. They will require a significantly relaxed CFL time-step restriction for stability. Most importantly, the methods will be shown to work well for the nonlinearly coupled system, in that the two systems will complement each other in terms of accuracy and efficiency.