A novel localization scheme for scalar uncertainties in ensemble-based data assimilation methods

Abstract

History matching, also known as data assimilation, is an inverse problem with multiple solutions responsible for generating more reliable models for use in decision-making processes. An iterative ensemble-based method (Ensemble Smoother with Multiple Data Assimilation—ES-MDA) has been used to improve the solution of history-matching processes with a technique called distance-dependent localization. In conjunction, ES-MDA and localization can obtain consistent petrophysical images (permeability and porosity). However, the distance-dependent localization technique is not used to update scalar uncertainties, such as relative permeability; therefore, the variability for these properties is excessively reduced, potentially excluding plausible answers. This work presents three approaches to update scalar parameters while increasing the final variability of these uncertainties to better scan the search space. The three approaches that were developed and compared using a benchmark case are: binary correlation coefficient (BCC), based on correlation calculated by ES-MDA through cross-covariance matrix C_{text{MD}}^{text{f}} (BCC-CMD); BCC, based on a correlation coefficient between the objective functions and scalar uncertainties (R) (BCC–R); and full correlation coefficient (FCC). We used the work of Soares et al. (J Pet Sci Eng 169:110–125, 2018) as a base case to compare the approaches because although it showed good matches with geologically consistent petrophysical images, it generated an excessive reduction in the scalar parameters. BCC-CMD presented similar results to the base case, excessively reducing the variability of the scalar uncertainties. BCC–R increased the variability in the scalar parameters, especially for BCC with a higher threshold value. Finally, FCC found many more potential answers in the search space without impairing data matches and production forecast quality.