A Modified Gradient Formulation for Ensemble Optimization under Geological Uncertainty

In this dissertation we have investigated theoretical and numerical aspects of the Ensemble Optimization (EnOpt) technique for model based production optimization. We have proposed a modified gradient formulation for robust optimization which we show to be theoretically more robust than the earlier existing formulation. Through a series of numerical experiments we illustrate the impact of ensemble size on the quality of an ensemble gradient and illustrate the superior performance of the modified gradient formulation. We also show that this modified gradient formulation hereafter referred to as Stochastic Simplex Approximate Gradient (StoSAG) shows comparable performance to an Adjoint based robust optimization. Additionally we have investigated the impact of a Covariance Matrix Adaption procedure to improve the EnOpt technique. This new CMA-EnOpt was shown to improve robustness of the method to an initial user defined choice of the covariance matrix. Most real world problems need multiple objectives to be optimized, in this dissertation we have investigated the applicability of EnOpt for multi-objective optimization and generation of Pareto trade-off curves. Finally many of the new proposed modifications were applied to a sector model of a real field case where we demonstrate the flexibility of EnOpt (StoSAG) as well as the significant practical value which can be achieved when using Ensemble Optimization for model based production optimization especially under geological uncertainty