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

Ensemble Optimization (EnOpt) is a rapidly emerging method for reservoir model 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 with a constant standard deviation, i.e. a diagonal covariance matrix with entries that remain constant during the optimization process. The Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) 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 has shown to be an efficient method for several difficult small dimension optimization problems and has recently been applied in the petroleum industry for well location and production optimization. In this study we investigated the scope to improve the computational efficiency of EnOpt through the use of covariance adaptation (CMA-EnOpt). We optimized water flooding of a multi-layer sector model containing multiple sealing and non-sealing faults. The controls used were inflow control valve settings at predefined time intervals for injectors and producers with undiscounted net present value as the objective function. We compared EnOpt and CMA-EnOpt starting from identical covariance matrices. We achieved slightly higher (0.7%-1.8%) objective function values and modest speed-ups with CMA-EnOpt compared to EnOpt, depending on choice of user-defined parameters in both algorithms. However, the major benefit of CMA-EnOpt 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. Copyright 2013, Society of Petroleum Engineers.