Bang-bang control and singular arcs in reservoir flooding

Over the past few years, dynamic optimization of reservoir flooding using optimal control theory has received significant attention. Various studies have shown that dynamic (time-varying) injection and production settings can yield a higher Net Present Value (NPV) than conventional reactive settings. In these studies, the optimization procedure itself is always gradient-based, where the gradients are 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 investigate why and under what conditions reservoir flooding problems can be expected to have bang-bang (on-off) optimal solutions. A major practical advantage of bang-bang controls is that they can be implemented with simple on-off valves. Furthermore, there are sufficient optimality conditions that are tailor-made for bang-bang solutions, meaning we can actually check whether a solution is locally optimal or not. These results are illustrated by a water flooding 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. It turns out that optimal settings are sometimes bang-bang, and sometimes bang-bang in combination with so-called singular arcs. For the latter situations, however, bang-bang solutions are found that are only slightly suboptimal.