Modeling with Constraints and Phase Transitions in Porous Media

海洋温度和多年冻土区温度的变化对水文、气候和人类系统的影响具有多年和几十年的时间尺度。多年冻土的冻融作用导致地表沉陷和丘陵、洼地发育，改变地表沃茨流向，对工程稳定性提出了挑战。永久冻土的变暖导致生物量的衰减以及从水合物和冻结层下的其他储存物中释放和运输甲烷气体。甲烷气体是温室气体平衡的主要贡献者，其存在、传输和演化，以及甲烷水合物（一种冰状物质）的存在、传输和演化，在物理学、气候研究和能源工程中具有极大的兴趣。甲烷水合物是一种潜在的非常规能源、钻井灾害和海底边坡失稳的重要因素。 激发这项工作的现实情景和案例研究包括冰融化成水和甲烷水合物分解成气体，以响应温度升高或机械扰动，液相和气相穿过沉积物。类似性质的模型也适用于气泡和蒸汽传输，例如，或是间歇泉中的微生物在这个项目中，首席研究员将开发新的数学成果以及对物理学有用的成果，并继续在许多领域进行跨学科的建模工作。 该项目解决了在多孔介质（如永久冻土或海底沉积物）中相变模型中出现的数学和计算挑战，以及驻留液体和解离气相的演变和迁移。对于这些能量和质量传输的耦合过程，数据和一些非线性模型系统的偏微分方程在空间和时间（达西）尺度的水库。然而，在地球环境中的新动力学要求进一步深入了解跨几个相互关联的空间和时间尺度的过程，包括孔隙scale.The模型在孔隙尺度之间的桥梁接口和达西尺度的物理，占限制内的多孔壁和复杂的几何形状，并可以升级到达西尺度。 虽然X射线显微断层扫描可以提供前所未有的洞察孔隙尺度的过程，目前，这一数据是相当稀疏的条件附近的相变。这些挑战了现有的知识，并激励了项目。该研究将促进对达西尺度模型的微观尺度（孔隙尺度和界面尺度）过程的理解。 该项目将开发和分析相关模型的算法;许多技术和结果是独立的兴趣。 首席研究员将考虑的演化模型是复杂和微妙的，并涉及对平衡内外的解决方案的逐点约束。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The changes in sea temperatures and those in permafrost regions have influence on hydrological, climate, and human systems on the time scale of years and decades. The freezing and thawing of permafrost leads to subsidence and development of hills and depressions; these alter the direction of surface waters, and challenge the stability of construction. Warming of permafrost causes decay of biomass as well as release and transport of methane gas from hydrate and other stores underneath the frozen layers. Methane gas is a major contributor to greenhouse gas balances, and its presence, transport, and evolution, as well as that of methane hydrate, an ice-like substance, are of great interest in geophysics, climate studies, and energy engineering. Methane hydrate can dissociate to gas, and is a potential unconventional energy source, drilling hazard, and contributor to sub-sea slope instability. The realistic scenarios and case studies motivating this work include the melting of ice to water and dissociation of methane hydrate into gas in response to increased temperature or mechanical disturbance, with the liquid and gas phases traveling through the sediment. Models of similar nature also apply to bubble and steam transport, e.g., due to microbial activity, or in geysers. In this project the principal investigator will develop new mathematical results as well as those useful for geophysics, and continue interdisciplinary modeling efforts across the many fields. This project addresses mathematical and computational challenges arising in the models of phase change in porous media such as permafrost or sub-sea sediments, and the evolution and migration of the resident liquid and the dissociated gas phase. For these coupled processes of energy and mass transport, data and some nonlinear model systems of PDEs at the spatial and temporal (Darcy) scale of the reservoir are available. However, the new dynamics in the Earth's environment calls for further insights into the processes across the several interlinked spatial and temporal scales including the pore-scale.The models at the pore-scale bridge the physics between the interface and Darcy scales, account for confinement within the porous walls and complex geometries, and can be upscaled to Darcy scale. While x-ray micro-tomography can deliver unprecedented insight into pore-scale processes, at present, this data is rather sparse for conditions near phase transitions. These challenge the current knowledge and motivate the project. The research will advance the understanding of the micro-scale (pore-scale and interface scale) processes which inform the Darcy scale models. The project will develop and analyze algorithms for relevant models; many techniques and results are of independent interest. The evolution models the principal investigator will consider are complex and delicate, and involve pointwise constraints on the solutions in and out of the equilibrium.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.