Phase transitions in porous media across multiple scales

在这个项目中，PI将开发和分析描述甲烷水合物演化的计算模型。甲烷水合物是一种冰状物质，在地球物理学、气候研究和能源工程中具有重要意义，因为它可以释放甲烷，一种强大的温室气体和钻井危险;水合物还可以作为一种潜在的非常规能源。这项工作将涉及几个空间尺度，从含水合物的海底沉积物和北极永久冻土区的千米尺度，到观察沉积物孔隙的微观尺度。PI对水合物模型的数学分析，在一些简化的假设下，迄今为止已经揭示了非常不寻常的特征，例如，在冰水相变中。基于这些分析，PI将制定更准确的算法，简化和更复杂的现实模型，这反过来又可以进一步了解水合物的形成和分解的性质。PI将开发的软件将与地球物理学社区共享。在PI之前的工作中，她将甲烷水合物的简化模型构建为参数依赖的自由边界问题，她的分析表明，其解决方案可以发展出超出已知的多个奇点，例如，冰水相变的Stefan问题。在这个项目中，PI将开发简化模型的新技术，大大扩展了非参数化情况下已知的经典结果，并研究有关的基本研究问题，例如，非线性退化扩散和守恒律与特定类型的参数依赖单调算子。这将在抽象环境中扩展精细的时间步进分析，开发先验和后验误差分析，以及跨多个时间和空间尺度的耦合的鲁棒算法。特别是，将考虑孔隙尺度，她制定了减少和动态模型的许多宏观所需的本构关系。除了计算数学之外，该项目还将影响水合物建模所涉及的物理学社区，以及更广泛的多孔介质中的其他应用。

In this project the PI will develop and analyze computational models describing the evolution of methane hydrate. Methane hydrate is an ice-like substance of great interest in geophysics, climate studies, and energy engineering, because it can release methane, a powerful greenhouse gas and drilling hazard; hydrates can also serve as a potential unconventional energy source. This work will involve several spatial scales from the kilometer scale of hydrate bearing subsea sediments and Arctic permafrost regions, to the micro-scale at which one looks at the pores of the sediments. The PI's mathematical analysis of hydrate models, under some simplifying assumptions, has so far revealed very unusual features absent, e.g., in ice-water phase transitions. Based on these analyses the PI will formulate more accurate algorithms for the simplified and for more complex realistic models, which in turn can further our understanding of hydrate formation and dissociation in nature. The software the PI will develop will be shared with the geophysics community.In PI's prior work, she has framed a simplified model of methane hydrate as a parameter-dependent free boundary problem, and her analyses show that its solutions can develop multiple singularities beyond those known, e.g., for Stefan problem of ice-water phase transitions. In this project the PI will develop new techniques for the simplified model extending substantially the classical results known for the non-parametrized case, and studying fundamental research questions concerning, e.g., nonlinear degenerate diffusion and conservation laws with the particular type of parameter-dependent monotone operators. The will extend the delicate time-stepping analyses in the abstract setting, develop a priori and a posteriori error analyses, and robust algorithms for the couplings across multiple time and spatial scales. In particular, the will consider the porescale at which she formulates reduced and dynamic models for many of the constitutive relationships needed for macroscale. In additional to computational mathematics, the project will impact the geophysics community involved in hydrate modeling, and more broadly other applications in porous media.