Two phase flows in karstic geometry

我们建议使用物理激励的扩散界面(相场)模型的层次族来研究岩溶几何中的两相流。这些模型体现了几个挑战：两种类型的流体之间的界面移动导致模型的高度非线性，以及物理域不同部分的不同物理导致不同类型的子系统之间的耦合。虽然这两个难题以前是在不同的背景下研究的，但岩溶几何中两相流体流动的物理需要要求我们以耦合的方式研究这两个问题。这是一个到目前为止还没有得到解决的挑战。PI和合作者计划从几个不同的角度研究这些模型。首先，我们将研究模型的数学适定性。其次，我们将研究尖锐界面的极限。第三，我们将为模型设计和实现精确和高效的数值方法，以便将结果与实验结果进行比较。由于耦合系统的高度非线性，以及多孔介质和管道中物理和数学机制的差异，这些都是非常重要的任务。尖锐界面极限是一个高度非线性的奇异摄动问题，是一个具有挑战性的问题。我们将结合偏微分方程组、泛函分析、渐近分析、数值分析和计算以及实验室实验的工具来研究这些问题。同时包含管道(或洞穴)和多孔介质的几何形状称为岩溶几何。岩溶几何中多相流的研究在地下水研究、燃料电池技术、石油工程和二氧化碳封存等领域具有重要的应用价值。本文提出的模型有效性研究的成功完成，将有助于我们更好地理解岩溶几何学中的几个重要的双流体现象。我们还认为，将开发的方法可能会扩展到研究涉及相变和大密度比的更复杂的模型。对这些重要问题的更好理解可能导致更好的工程过程和更好的基于科学的环境政策。

We propose to study two phase flow in karstic geometry utilizing a hierarchical family of physically motivated diffuse interface (phase field) models. These models embody several challenges: moving interface between two types of fluids which leads to strong nonlinearity of the resulting models, and different physics in different parts of the physical domain which leads to the coupling of subsystems of different types. Although these two difficult issues have been studied before in separate contexts, the physical need of two phase fluid flow in karstic geometry requires us to investigate the two issues in a coupled fashion. This is a challenge that has not been addressed so far. The PI and collaborators plan to investigate the models from several different angles. Firstly, we will investigate the mathematical well-posedeness of models. Secondly, we will study the sharp interface limit. Thirdly, we will design and implement accurate and efficient numerical methods for the models so that the results can be compared to experimental results. These are highly non-trivial tasks due to the highly nonlinear nature of the coupled systems, and the disparity of physical and mathematical mechanism in the porous media and in the conduit. The sharp interface limit is a highly nonlinear singular perturbation problem which is known to be a challenge. We will combine tools from partial differential equations, functional analysis, asymptotic analysis, numerical analysis and computation, and laboratory experiments to investigate these problems.Geometric configurations that contain both conduit (or vug) and porous media is termed karstic geometry. It is known that the study of multiphase flow in karstic geometry is of great importance in many applications such as groundwater study, fuel cell technology, petroleum engineering and carbon-dioxide sequestration. The successful completion of the investigation on the validity of the models proposed here will help us better understand several important two fluid phenomena in karstic geometry. We also believe that the methodologies to be developed may be expanded to investigate more complex models that involve phase transition, and large density ratio. The better understanding of these important problems could lead to better engineering processes and better science based environmental policies.