Two phase flows in karstic geometry

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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.