Local Mesh Refinement for Finite Difference Methods

TL;DR: In this paper, a method of local mesh refinement was developed in which additional nodes are used only in regions where they are necessary, and the extra mesh lines are not extended across the entire simulator as in the usual current practice.

Abstract: A method of local mesh refinement has been developed in which additional nodes are used only in regions where they are necessary. The extra mesh lines are not extended across the entire simulator as in the usual current practice. Refinement of a two-dimensional element, herein defined as the rectangle formed by mesh lines with nodes at the vertices, consists of bisecting it in each coordinate direction, thus dividing the original element into four smaller, but similar ones, and creating five nodes. The resulting smaller elements may themselves be refined by the same procedure. An element must be refined if two adjacent elements of the same size have been refined or if one smaller adjacent element has been refined. Three configurations of five nodes each which arise in current refinement methods also appear in the new procedure. In addition, a configuration of six nodes also occurs. Finite difference analogs for all configurations were developed. An extensive study of local mesh refinement near a well was made for the repeated five spot in a homogeneous reservoir with unit mobility ratio. Results of the finite difference solutions were compared with the analytic solution. Generally, it was found there should be at least twomore » consecutive increments of the same size between changes in increment size. When this practice is followed, a local minimum in truncation error occurs at each node where there is a change in increment size. The more accurate solution obtained in the vicinity of the well by use of smaller increments there results in a more accurate solution also in the coarser unrefined mesh.« less