Analytical, numerical and geometric methods with applications to fractured reservoir modeling for Forchheimer flows

This dissertation is based on modeling and analysis of strongly coupled nonlinear flows with applications to fractured porous media. We model the flows in coupled fractured porous media system as a BVP, where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We present a low dimensional model to formulate the flow inside the fracture, which is coupled with the equation for flow in the reservoir. We estimate the difference between the solution of the original model and the reduced model, and prove that the solution of the reduced model can accurately approximate the solution of the original high-dimensional flow in the reservoir fracture system, because the thickness of the fracture is small. In the analysis, we consider two types of Forchheimer flows in the fracture: isotropic and anisotropic, which are different in nature. We develop a methodology to find an optimal design of the fracture, which maximizes the capacity of the fracture in the reservoir with fixed geometry. Our method, which is based on a set point control algorithm, explores the coupled impact of the fracture geometry and $\beta$ Forchheimer coefficient.
Moreover, we model the fracture as a Riemannian manifold immersed in porous media, and formulate the flow inside the fractures with complex geometries and variable thickness. The fracture is represented as the normal variation of a surface immersed in $R3$. Using operators of Laplace Beltrami type and geometric identities, we model an equation that describes the flow in the fracture. A reduced model is obtained as a low dimensional equation, and it is coupled with the flow in porous media. Theoretical and numerical analysis have been performed to compare the solutions between the original geometric model and the reduced model in reservoirs containing fractures with complex geometries. We prove that the two solutions are close, and therefore, the reduced model can be effectively used in large scale simulators for long and thin fractures with complicated geometry.