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 eachcoordinate 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 two 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 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. Some improvement elements with the well node as a vertex. An effective refinement was one using increments which were one half and one fourth the size of those in the coarser mesh. Four small increments were used nearest the well, and four was obtained by one refinement of the four mid-sized ones were used between these and the coarsest mesh. Results are presented to aid the reader in selecting the extent of refinement he desires.
The storage requirement, computer time, and run cost of numerical simulators are controlled largely by the number of nodes. All three vary directly with atleast the first power of the number of nodes. Thus, it is desirable to use the smallest number of nodes which will yield the accuracy required for the solution. Often, it is necessary to use fine mesh spacing in parts of the simulator while much coarser spacing can be used in other areas. This practice is often employed in reservoir simulators where the fine spacing is used in the vicinity of wells, though other applications of variable spacing also occur. Currently, when variable mesh spacing is employed, the finely spaced mesh lines are extended throughout the simulator, as shown in Figure 1. Thus, nodes are used in some areas of the simulator where they are not necessary for an accurate solution. Recently, a method of localized mesh refinement for use with finite difference techniques was developed and tested. This method is described in some detail in this paper, and results of test computations are presented. presented.
The localized mesh refinement which was developed for finite difference methods was patterned after a similar procedure used in finite element methods. One example of local refinement is shown in Figure 2. The nodes are located at the intersections of the mesh lines, and a well is located at the node in the center of the mesh. In this mesh, sixteen nodes have been added to the original coarse mesh by halving the increment size for each of the four increments which includes the well node. If these increments are still too large, the process can be continued several more times. Local refinement can also be extended further from the well.