Bang-Bang Control in Reservoir Flooding

Various studies have shown that dynamic optimization of waterflooding using optimal control theory has a significant potential to increase Net Present Value (NPV). In these studies, gradient-based optimization methods are used, where the gradients are usually obtained with an adjoint formulation. However, the shape of the optimal injection and production settings is generally not known beforehand. The main contribution of this paper is to show that a whole variety of reservoir flooding problems can be formulated as optimal control problems that are linear in the control and that, if the only constraints are upper and lower bounds on the control, these problems will sometimes have bang-bang (on-off) optimal solutions. This is supported by a waterflooding example of a 3-dimensional reservoir in a fluvial depositional environment, modeled with 18.553 grid blocks. The valve settings of 8 injection and 4 production wells are optimized over the life of the reservoir, with the objective to maximize NPV. For various situations, the optimal solution is either bang-bang, or a bang-bang solution exists that is only slightly suboptimal. This has obvious practical implications, since bang-bang solutions can be implemented with simple on-off control valves.