Bayesian Optimization Algorithm Applied to Uncertainty Quantification

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 by 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. However, this requires 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 because of 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 that use models to estimate the probability distribution of promising solutions and then 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, the Bayesian optimization algorithm (BOA), which employs Bayesian networks for modeling. By considering possible interactions between variables and exploiting explicitly stored knowledge of such interactions, EDAs can accelerate the search process while preserving search diversity. Unlike most existing approaches applied to uncertainty quantification, the Bayesian network allows the BOA to build solutions using flexible rules learned from the models obtained, rather than fixed rules, leading to better solutions and improved convergence. The 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 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 with some other traditional population-based algorithms.

Assisted History Matching Using Pattern Recognition Technology

“…Generally these optimization algorithms have been coupled with a commercial simulator. The reported numbers of simulation runs for history matching the PUNQ-S3 reservoir model are in the order of thousands realizations Abdollahzadeh, et al, 2011). In this study, the required simulation runs to create and validate the SRM (eleven runs) are extremely less.…”

“…PUNQ, which stands for Production forecasting with Uncertainty Quantification, was a mutual study supported by European Union and conducted by 10 European companies, universities, and research centers (Floris, et al, 2001). The reservoir model is based on a real field operated by Elf Exploration and Production (Floris, et al, 2001;Barker, et al, 2001) and is used widely as a standard synthetic test case to investigate the cability of different methods of history matching and uncertainty quantification) (Floris, et al, 2001;Barker, et al, 2001;Gu, et al, 2005;Gao, et al, 2005;Abdollahzadeh, et al, 2011). …”

On Population Diversity Measures of the Evolutionary Algorithms used in History Matching

“…These algorithms are all stochastic, thus we repeated each tuned algorithm 10 times and averaged for the misfit and diversity measures to minimize the effects of randomness in the experiments. For more details on BHEDA and BOA refer to Abdollahzadeh et al (2011b), on PSO and hybrid BOA/PSO to Reynolds et al (2011). We used 3,000 function evaluations (simulation runs) in 60 generations where 50 new solutions were created in each generation.…”

Impact of Model Parameterisation and Objective Choices on Assisted History Matching and Reservoir Forecasting

“…AHM, middle block in Fig. 1, powered by stochastic populationbased sampling algorithms have been growing in popularity to handle history matching task (Romero et al 2000;Schulze-Riegert et al 2001;Christie et al 2002;Mohamed et al 2010a;Hajizadeh et al 2010;Hajizadeh et al 2011a;Abdollahzadeh et al 2012a;. These algorithms minimize a misfit function and thus obtain the reservoir model that best approximates the dynamic variable data recorded during reservoir life.…”

“…These algorithms minimize a misfit function and thus obtain the reservoir model that best approximates the dynamic variable data recorded during reservoir life. Powered by stochastic optimiser, AHM can be approached by Single Objective (SO) (Schulze-Riegert et al 2001;Ferraro and Verga 2009;Hajizadeh et al 2010;Mohamed et al 2010a;Mohamed et al 2010b;Hajizadeh et al 2011a;Abdollahzadeh et al 2012a; or Multi-Objective (MO) (Schulze-Riegert et al 2007;Ferraro and Verga 2009;Hajizadeh et al 2011b;Mohamed et al 2012;Sayyafzadeh et al 2012;Christie et al 2013;Park et al 2013;Min, B., et al 2014). The last block in the workflow is quantifying uncertainty of reservoir performance prediction from ensemble of matched models.…”