Adaptive finite difference methods for problems arising in flow of porous medium applications are being considered. Such methods have proved useful for overcoming limitations of computational resources and improving the resolution of the numerical solutions to a wide range of problems. By local refinement of the computational mesh where need to improve the accuracy of solutions, yield better solution resolution than is possible with traditional fixed-grid approaches, representing more efficient use of computational resources.
In this paper, we present a parallel adaptive cell-centered finite difference algorithm for black-oil reservoir simulation models. This is an extension of the adaptive mesh refinement methodology first developed by Berger and Oliger (1984) for the hyperbolic problem.
The material in this paper is subdivided into three parts. First we explain the methodology and intricacies of adaptive finite difference scheme. Then we extend a finite differential cell-centered approximation discretization to a multilevel hierarchy of refined grids, and finally we are employing the algorithm on parallel computers.
Our algorithm is fully adaptive in time and space through the use of sub-cycling, in which finer grids are advanced at smaller time steps than coarser one. When coarse and fine grids reach the same advanced time level, they are synchronized to ensure that the global solution is conservative and satisfy the divergence constraint across all levels of refinement.
The results in this work show that the approach presented is robust, and stable, thus demonstrating the increased solution accuracy due to local refinement and reduced computing resource consumption.