Bayesian Optimization Algorithm Applied to Uncertainty Quantification

Abstract

Abstract Prudent decision making in subsurface assets requires reservoir uncertainty quantification. In a typical uncertainty quantification study, reservoir models must be updated using the observed response from the reservoir via a process known as history matching. This involves solving an inverse problem, finding reservoir models that produce, under simulation, a similar response to that of the real reservoir, requiring multiple expensive multiphase flow simulations. Thus uncertainty quantification studies employ optimization techniques to find acceptable models to be used in prediction. Different optimization algorithms and search strategies are presented in the literature, but they are generally unsatisfactory due to slow convergence to the optimal regions of the global search space, and, more importantly, failure in finding multiple acceptable reservoir models. In this context, a new approach is offered by Estimation of Distribution Algorithms (EDAs). EDAs are population-based algorithms, which use probability models to estimate the probability distribution of promising solutions, and then to generate new candidate solutions. This paper explores the application of EDAs including univariate and multivariate models. We discuss two histogram-based univariate models, and one multivariate model, Bayesian Optimization Algorithm (BOA), which employs Bayesian Networks for modelling. By considering possible interactions between variables and exploiting explicitly stored knowledge of such interactions, EDA can accelerate the search process, while preserving search diversity. Unlike most existing approaches applied to uncertainty quantification, the Bayesian Network allows BOA to build solutions using flexible rules learned from the models obtained, rather than fixed rules, leading to better solutions and improved convergence. BOA is naturally suited to finding good solutions in complex high-dimensional spaces, such as those typical in reservoir uncertainty quantification. We demonstrate the effectiveness of EDA by applying to the well-known synthetic PUNQ-S3 case with multiple wells. This allows us to verify the methodology in a well controlled case. Results show better estimation of uncertainty when compared to some other traditional population-based algorithms.