Ensemble-based optimization has recently received great attention as a potentially powerful technique for life-cycle production optimization, which is a crucial element of reservoir management. Recent publications have increased both the number of applications and the theoretical understanding of the algorithm. However, there is still ample room for further development since most of the theory is based on strong assumptions. Here, the mathematics (or statistics) of Ensemble Optimization is studied, and it is shown that the algorithm is a special case of an already well-defined natural evolution strategy known as Gaussian Mutation. A natural description of uncertainty in reservoir management arises from the use of an ensemble of history-matched geological realizations. A logical step is therefore to incorporate this uncertainty description in robust life-cycle production optimization through the expected objective function value. The expected value is approximated with the mean over all geological realizations. It is shown that the frequently advocated strategy of applying a different control sample to each reservoir realization delivers an unbiased estimate of the gradient of the expected objective function. However, this procedure is more variance prone than the deterministic strategy of applying the entire ensemble of perturbed control samples to each reservoir model realization. In order to reduce the variance of the gradient estimate, an importance sampling algorithm is proposed and tested on a toy problem with increasing dimensionality.@en

The purpose of this research is to determine an actual optimal solution through cyclic optimization of CO2-Gas Assisted Gravity Drainage (GAGD) process in a heterogeneous sandstone reservoir under geological uncertainties. We propose an integrated approach to optimize durations of gas injection, soaking, and oil production under geological uncertainties. Therefore, 100 stochastic reservoir realizations of the 3D permeability and porosity distributions were created honouring geological constraints. Ranking was applied through quantifying of reservoir oil response to select P10, P50, and, P90 that represent the overall reservoir uncertainty. More than 400 training simulation runs were created including the durations and geological uncertainty parameters through Latin Hypercube Design to build the second-order proxy model along with approximately 200 extra verification runs. The verification runs led to keep the solutions in global optima and obtain satisfactory proxy model through an iterative validation procedure. The cyclic optimization has shown its feasibility to increase oil recovery through the GAGD process from 71.5% to 75.5% with incremental cumulative oil production of 225 million barrels. The presented robust optimization workflow under geological uncertainties led to higher recovery factor than nominal realization optimization with providing degrees of freedom for the decision-maker to significantly reduce the project risk. @en