Abstract
We consider two procedures for multiobjective optimization, the classical weighted sum (WS) method and the normal boundary intersection (NBI) method. To enhance computational efficiency, the methods utilize gradients calculated with the adjoint method. Our objective is to develop implementations that can be applied for water flooding optimization under geological uncertainty where we wish to develop well controls that satisfy two objectives: the first is to maximize the expectation of life-cycle net-present-value (NPV) (commonly referred to as robust optimization) and the second is either to minimize the standard deviation of NPV over that set of plausible reservoir descriptions or to minimize the risk. Here, minimizing risk simply refers to maximizing the minimum value of the life-cycle NPV, i.e., is equivalent to a max-min problem. To avoid non- differentiability issues, we recast the max-min problem as a constrained optimization problem and apply a gradient-based version of either WS or NBI to construct a point on the Pareto front. To deal with the constraints introduced, we derive an augmented-Lagrange algorithm to find points on the Pareto front. To the best of our knowledge, the resulting versions of “constrained” WS and “constrained” NBI have not been presented previously in the scientific literature. The methodology is demonstrated for two synthetic reservoirs.
