Fast Robust Optimization Using Mean Field Bias Correction

Abstract

Summary Ensemble methods are remarkably powerful for quantifying geological uncertainty. However, robust optimization of a cost function for a problem in which uncertainty is characterized by a large ensemble size can be computationally demanding. In a straightforward approach, the computation of expected net present value (NPV) requires many expensive simulations. Several techniques (e.g., model selection, coarsening) have been proposed to reduce the cost but generally lead to a less accurate optimization. To reduce the amount of computation without sacrificing accuracy, we developed a fast and effective approach for computing the expected NPV by using only the reservoir mean model with a bias correction factor. At each iteration of the optimization procedure, we only require one additional simulation in the mean model with a different set of controls to obtain an initial approximate value through which the bias will be corrected with a multiplicative correction factor. Information from individual simulations with distinct controls and model realizations can be used to estimate the correction factor for different controls. The effectiveness of various bias-corrected methods is illustrated by the application of the drilling-order problem in the synthetic REEK Field model. Compared with the average NPV, the results show that the average error of estimated expected NPV from the mean model is reduced from -9% to 0.56% by estimating the bias correction factor. Distance-based localization with an appropriate taper length can further improve the accuracy of estimation. By adding a regularization term with a tuning parameter associated with the variance of the correction factor, the sensitivity of the estimates to the taper length is reduced such that the regularized estimate is potentially more accurate for a wider range of taper lengths. In previous work, we proposed a nonparametric online-learning methodology (learned heuristic search) to efficiently compute a sequence of drilling wells that is optimal or near-optimal. In this work, we apply the learned heuristic search (LHS) to the reservoir mean model with bias correction to optimize the drilling sequence and show that it leads to the same solution as the LHS with the average NPV. Moreover, we investigate the possibility of optimizing the first few wells without finding an entire drilling sequence. Our results show that LHS can optimize complete drilling sequences or only the first few wells at a reduced cost.