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。甲烷水合物是一种类似冰的物质，在地球物理、气候研究和能源工程中引起了极大的兴趣，因为它可以释放出甲烷，这是一种强大的温室气体和钻井危险；水合物也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，从含有水合物的海底沉积物和北极永久冻土区的千米尺度，到人们观察沉积物孔隙的微观尺度。到目前为止，PI对水合物模型的数学分析，在一些简化的假设下，揭示了非常不寻常的特征，例如，在冰-水相变中。在这些分析的基础上，PI将为简化的和更复杂的现实模型制定更准确的算法，这反过来又可以加深我们对自然界中水合物形成和分解的理解。PI将开发的软件将与地球物理界共享。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，其解可以产生多个奇点，而不是已知的奇点，例如冰水相变的斯特凡问题。在这个项目中，PI将为简化模型开发新的技术，大大扩展已知的非参数情况下的经典结果，并研究关于非线性退化扩散和守恒律的基础研究问题，例如，具有特定类型的参数依赖单调算子的非线性退化扩散和守恒律。它将在抽象的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稳健算法。特别是，她将考虑她为宏观尺度所需的许多本构关系建立简化和动态模型的初级阶段。除了计算数学之外，该项目还将影响参与水合物建模的地球物理界，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.