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 life-cycle production over an ensemble of given reservoirs models. Unfortunately, the “optimal” well controls obtained may be such that the variance in the values of 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 reservoir would not be commercially viable. Results to be presented elsewhere suggest that one way to manage risk is to consider the problem where the dual objectives are to maximize the expected value of NPV and to minimize the risk, i.e., maximize the minimum NPV. The algorithms presented in the other manuscript consider 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. The generation of an indviidual point on the Pareto front requires solving an appropriate constrained optimization problem using an augmented-Lagrange algorithm derived specifically for the problem considered here. To the best of our knowledge, this augmented-Lagrange implementation has not appeared previously in the scientific literature.

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