A generalized fifteen-point finite-difference scheme for three-dimensional modeling of reservoir heterogeneities

Numerical studies with irregular shaped grid blocks may provide a better description of a geologically complex reservoir. A generalized nine-point scheme offers a method to calculated correct transmissibilities for such grid blocks and gridding flexibility without sacrificing solution accuracy and stability in two-dimensional (2-D) simulation studies. The current work investigates a numerical technique for three-dimensional (3-D) simulation studies.;The fifteen-point connectivity, or a so called generalized fifteen-point scheme, is considered as a 3-D extension of the generalized nine-point scheme. It can be achieved by applying the generalized nine-point scheme on two planes of the 3-D system. With a point distributed grid, this scheme offers similar gridding flexibility in 3-D that the generalized nine-point scheme offers in 2-D. The above-mentioned flexibility leads to fewer computational nodes and therefore less computational effort using more complicated grid block configurations such as irregular hexahedrons. Moreover, the fifteen-point scheme provides an appropriate means to confirm geologic findings such as the morphology of a highly complex reservoir with lenticular flow units.;A 3-D, two-phase numerical simulator with the generalized fifteen-point scheme was constructed. The newly developed simulator was implemented in the study of directional waterflood performance of crossbedded reservoirs using repeated patterns. In addition, a simplified problem was studied to predict the waterflood performance of a lenticular shaped reservoir.;The equations for geometric calculations of the generalized fifteen-point scheme are presented in somewhat more complex form than those of a conventional 3-D seven-point scheme. However, it is easily programmed in numerical simulators and results in a nominal increase in CPU-time when a fully-implicit solution method is used. An 80486-based micro-computer was used for the simulation studies throughout this research.