A trust region method for output-constrained reservoir optimization under geological uncertainty

Abstract We present an algorithm addressing well-control reservoir optimization problems in realistic situations. Primarily, we consider cases where the reservoir simulator is treated as a black box and the derivative information is unavailable. Then, in addition to the input constraints that can be imposed in the simulator, we also consider constraints that can only be evaluated after the simulation has been run, e.g., total water production and water cut. While reservoir simulators may be able to impose such output constraints for a single reservoir model, it is costly to configure for an ensemble of reservoir models. The proposed algorithm provides an efficient solution that deals with the situation above. The algorithm is based on a model-based trust region method that relies on building local polynomial models for both the objective functions and the constraints. The model building and the model improvement algorithms are based on the Lagrange polynomials, which provides information regarding the model quality solely from the geometry of the interpolation points. The next iterate is found by solving a subproblem based on the local polynomial models around the current iterate. Then, the models and the trust region are updated accordingly based on the quality of the actual evaluation. The notion of a filter complements the algorithm by handling the duality between the objective function and the constraints. The filter maintains a Pareto front of the best pairs of objective function and constraint violations, thus advancing the iterates in both directions simultaneously. We tested our algorithm on ensembles of reservoir models from two different benchmarks, investigating constrained and unconstrained optimization scenarios. The decision variables are the well controls (bottom-hole pressures and injection/production rates) at different time steps. In the constrained optimization scenario, we introduce a chance constraint to limit the total water production rate and the water cut. The results show that the proposed algorithm can find an optimal well-control schedule given the input and output constraints in all the cases studied. A comparison to other methods also shows that the proposed algorithm finds a locally near-optimal feasible solution with fewer simulation runs.