Development and Analysis of Algorithms to Simulate Multi-Component Multi-Phase Porous Flows

科学的目标之一是理解和解释自然现象，以便预测和预测结果。数学工具的不断发展，表达了许多基本的自然法则，使得模型具有无与伦比的预测能力。数学模型涉及表达感兴趣的物理过程的复杂的方程式系统，并构成现代工程和科学的概念基础。这些方程展示了被建模的物理系统的所有深度、困难和微妙之处，并且它们的解需要复杂的计算方法。这个项目专注于开发计算工具和软件，工程师和科学家需要这些工具和软件来模拟地质学中出现的问题。推动这项工作的物理问题之一是需要对深海海床和永久冻土区进行建模，在这些地区，大量冻结的甲烷、二氧化碳和其他气体被捕获。这些气体从地球深处扩散，被困在永久冻土、冰川下和深海寒冷的深处，在那里它们可能会冻结。通过地质旋回和气候变化预测这些地区的演化状态对于环境模拟是必不可少的。该项目将开发和分析所需的计算工具和软件，以对这些区域进行建模，在这些区域中，多种流体和气体在流动时经历冻结、融化和溶解。除了技术开发，该项目还将支持对下一代科学家的教育和培训，这些科学家需要维持在这些学科中的发现和科学领导地位。建议的工作围绕开发和分析计算工具，以模拟包含多种流体的多孔介质中的流动，这些流体可能结合形成多个相和状态(固体、液体、气体)。混合物和相变的热力学被用来模拟流体在孔隙尺度上的性质和状态，这些性质和状态在质量、动量和能量的宏观平衡定律中是结构性的。这一建议集中在一个具有挑战性的问题上，即将这些系统的孔隙尺度和连续介质模型集成到一个一致的框架中，在该框架中可以使用计算工具来模拟这些复杂的物理问题。自由能的凸性和齐性(或熵的凹性)以一种微妙的方式进入这些模型，并且对于证明解的稳定性是必不可少的。将开发出忠实地继承这些重要属性的数值格式；此外，将利用凸性属性来促进求解所产生的非线性系统的稳健算法。将致力于开发分析这些算法的稳定性、稳健性、收敛和正确性所需的数学工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

One of the goals of science is to understand and explain natural phenomena in order to predict and forecast outcomes. The continual development of mathematical tools to express many of the fundamental laws of nature has resulted in models with unparalleled predictive power. Mathematical models involve complex systems of equations expressing the physical processes of interest, and form the conceptual foundation of modern engineering and science. These equations exhibit all of the depth, difficulties, and subtleties of the physical systems being modeled, and sophisticated computational methodologies are required for their solution. This project focuses on the development of computational tools and software that engineers and scientists need to simulate problems arising in geology. One of the physical problems motivating this work is the need to model deep sea beds and permafrost regions where vast quantities of frozen methane, carbon dioxide, and other gases are trapped. These gases diffuse from deep within the earth and get trapped in permafrost, under glaciers, and in the cold depths of the deep ocean where they may freeze. Predicting the evolving state of these regions through geologic cycles and changes in climate is essential for environmental modeling. This project will develop and analyze computational tools and software required to model these regions where multiple fluids and gases undergo freezing, thawing, and dissolution as they flow. In addition to the technological developments, this project will also support the education and training of the next generation of scientists needed to sustain discovery and scientific leadership in these disciplines.The work proposed centers around the development and analysis of computational tools needed to simulate flows in porous media containing multiple fluids, which may combine to form multiple phases and states (solid, liquid, gas). The thermodynamics of mixtures and phase changes is used to model the properties and states of the fluids at the pore scale, which appear constitutively in the macroscopic balance law of mass, momentum, and energy. This proposal focuses on the challenging problem of integrating pore scale and continuum models of these systems into a consistent framework where computational tools can be utilized to simulate these complex physical problems. Convexity and homogeneity of the free energy (or concavity of entropy) enter these models in a subtle way, and are essential for proving stability of solutions. Numerical schemes will be developed that faithfully inherit these important attributes; in addition, convexity properties will be exploited to facilitate robust algorithms for the solution of the nonlinear systems that arise. Development of mathematical tools required to analyze the stability, robustness, convergence, and correctness of these algorithms will be undertaken.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.