Ensemble optimization is a rapidly emerging method for reservoir model-based production optimization. In this paper, we compared a line search method and a trust-region method to design an algorithm that retains the excellent convergence rate, but is more economical to implement when the number of variables is large. Here, the mathematics (or statistics) of ensemble-based optimization with several mathematical treatments is studied. Conjugate gradient is carried out within the general optimization framework that employs trust-region methods, aiming at delivering a faster convergence approach for reservoir management. For general benchmarks, the Rosenbrock function and five-spot waterflooding are tested for both two methods. To the best of our knowledge, the embedment of the Steihaug conjugate gradient in solving the sub-problem of ensemble-based optimization using trust-region methods is studied for the first time for reservoir management. The conjugate gradient approach is known for its prescriptive convergence theory, in which the progress can be observed at each iteration for a quadratic programming. With numerical experiments, we illustrate that trust-region method is competitive against line search method in ensemble-based production optimization.

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