Geometric model of the fracture as a manifold immersed in porous media

Abstract

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We develop a model to examine the flow inside fractures with complex geometries and variable thickness on a Riemannian manifold. 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 boundary value problem. We then couple the model with the porous media. Theoretical and numerical analyses 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.