Augmented Lagrangian Method for Maximizing Expectation and Minimizing Risk for Optimal Well-Control Problems With Nonlinear Constraints

Abstract

Summary Robust waterflooding optimization commonly refers to the problem of estimating well controls (wellbore pressures or rates at specified control steps) that maximize the expectation of net-present-value (NPV) of the life-cycle production over an ensemble of given reservoir models. Unfortunately, if the reservoir is operated under the “optimal” well controls obtained, the variance in NPV may be large; more importantly, if the smallest NPV obtained is close to the one that would be obtained for the true reservoir, the development of the reservoir might not be commercially viable. Liu and Reynolds (2015b) suggested that one way to manage risk was to consider the problem in which the dual objectives were to maximize the expected value of NPV and to minimize the risk in which the risk was defined as the minimum NPV from an ensemble of models spanning the uncertainty in reservoir description. However, the algorithms presented in Liu and Reynolds (2015b) considered only bound constraints. Here, we develop algorithms to generate points on the Pareto front when nonlinear state (output) constraints are present. The Pareto front is generated either by a constrained weighted-sum (WS) method or a constrained normal-boundary-intersection (NBI) method. In this paper, we extend the augmented Lagrangian approach given in Liu and Reynolds (2015b) for biobjective optimization with bound constraints to biobjective optimization problems in which nonlinear state constraints are present. We provide a detailed derivation of how to incorporate nonlinear constraints for multiobjective optimization problems (MOOPs) and illustrate, by means of example, that the methodology is viable for biobjective optimization with nonlinear constraints.