Adaptive mesh refinement and multi-level iteration techniques

The governing equations for multi-phase fluid flow in porous media are usually written as a system of nonlinear conservation laws for the masses of the fluid components. It is often reasonable to separate the equations into a system of hyperbolic conservation laws coupled to a nonlinear elliptic/parabolic equation relating the pressure of the fluid to its composition through phase behavior equations and rock heterogeneity. This allows the use of numerical methods well-suited to each type of flow equation representing a distinct physical behavior. To achieve efficient, accurate field-scale simulations, it is important to concentrate the numerical effort locally to resolve certain details of the fluid flow. In the numerical approach developed in this thesis, local grid refinement is placed automatically and dynamically in regions requiring higher resolution. One of the primary issues in extending an adaptive mesh refinement methodology to flow in porous media is the treatment of the pressure equation. In this thesis, we extend an adaptive mesh refinement algorithm, originally developed for hyperbolic conservation laws, so that we may treat the coupled system of equations describing flow in porous media. We apply our techniques to a three-component, two-phase polymer flooding model. We use a second-order Godunov method to integrate the conservation laws. Multi-level iterative methods are used to solve the pressure equation in the adaptive mesh refinement setting. The adaptive grid is implemented in a highly structured, hierarchical fashion in which the mesh on each refinement level is the union of rectangular patches. The multi-level iteration method is well-suited to the subtle communication that must take place between patches at the same mesh level and between patches on different levels of refinement. Most importantly, our implementation provides an environment in which a variety of numerical methods can be tested and constructed for dynamically adaptive calculations involving flow in porous media.