Improving the ensemble optimization method through covariance matrix adaptation (CMA-EnOpt)

Ensemble optimization (referred to throughout the remainder of the paper as EnOpt) is a rapidly emerging method for reservoirmodel-based production optimization. EnOpt uses an ensemble of controls to approximate the gradient of the objective function with respect to the controls. Current implementations of EnOpt use a Gaussian ensemble of control perturbations with a constant covariance matrix, and thus a constant perturbation size, during the entire optimization process. The covariance-matrix-adaptation evolutionary strategy is a gradient-free optimization method developed in the "machine learning" community, which also uses an ensemble of controls, but with a covariance matrix that is continually updated during the optimization process. It was shown to be an efficient method for several difficult but small-dimensional optimization problems and was recently applied in the petroleum industry for well location and production optimization. In this study, we investigate the scope to improve the computational efficiency of EnOpt through the use of covariance-matrix adaptation (referred to throughout the remainder of the paper as CMA-EnOpt). The resulting method is applied to the waterflooding optimization of a small multilayer test model and a modified version of the Brugge benchmark model. The controls used are inflow-control-valve settings at predefined time intervals for injectors and producers with undiscounted net present value as the objective function. We compare EnOpt and CMA-EnOpt starting from identical covariance matrices. For the small model, we achieve only slightly higher (0.7 to 1.8%) objective-function values and modest speedups with CMAEnOpt compared with EnOpt. Significantly higher objective-function values (10%) are obtained for the modified Brugge model. The possibility to adapt the covariance matrix, and thus the perturbation size, during the optimization allows for the use of relatively large perturbations initially, for fast exploration of the control space, and small perturbations later, for more-precise gradients near the optimum. Moreover, the results demonstrate that a major benefit of CMAEnOpt is its robustness with respect to the initial choice of the covariance matrix. A poor choice of the initial matrix can be detrimental to EnOpt, whereas the CMA-EnOpt performance is near-independent of the initial choice and produces higher objectivefunction values at no additional computational cost.