Reservoir simulation models often require grids sufficiently resolved to capture the complexities of the geological structures present so that pressure and saturation profiles are deemed reasonable enough for decisions to be made with confidence. However, high resolution often implies long simulation run times, which is compounded in optimization problems. Reservoir optimization problems usually involve localized decisions, e.g., production control for a specific time-dependent objective function. With this tighter focus on optimization objectives, the reservoir simulator can often be replaced by an approximate proxy function, e.g., decline curves, neural networks and polynomial response surfaces. This accelerates the run time, allowing the optimizer to make more calls to the simulator, but at the cost of simplified physics and some consequent loss of accuracy.
Recent work in numerical solutions of partial differential equations (PDE) by Druskin and Knizhnerman (2000) introduced a finite-volume optimal grid approach to improve solution accuracy of linear PDEs at targeted points of a computational domain. This approach was applied to the solution of electromagnetic and acoustic forward and inverse problems arising in oil exploration and medical imaging. We extend this work to non-linear PDEs by introducing a reservoir simulator that is used as its own proxy. We distinguish our optimal grid from an optimal upscaling in that it may not preserve local features of the model, but will preserve some aggregate objective value used in production optimization. This narrow focus allows greater coarseness than a properly upscaled grid by replacing the fine-grid model with a coarser grid whose coarsening is determined by an optimizer that preserves the accuracy of a predefined aggregate objective value.
In this work we optimize injection and production rates to maximize the field oil production total (FOPT) over a fixed period of time. The proxy optimizer will seek the best coarsening that preserves this objective value over a range of injection/production rates for each well. The resulting coarse-grid model has a similar physical behavior in these output values to that of the fine-grid model. The grid coarsening problem is a non-linear discrete optimization problem, where the optimizer adjusts the grid coarsening while preserving the fine-grid response of interest. Although we focus on FOPT here, other reservoir objectives can also be handled.