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.

The topic of automatically history-matched reservoir models has seen much research activity in recent years. History matching is an example of an inverse problem, and there is significant, active research on inverse problems in many other scientific and engineering areas. While many techniques from other fields such as Genetic Algorithms, Evolutionary Strategies, Differential Evolution, Particle Swarm Optimisation, and the Ensemble Kalman Filter have been tried in the oil industry, some more recent and effective ideas have yet to be tested. One of these relatively untested ideas is a class of algorithms known as Estimation of Distribution Algorithms (EDAs). EDAs are population-based algorithms, which use probability models to estimate the probability distribution of promising solutions, then to generate new candidate solutions. EDAs have been shown to be very efficient in very complex high-dimensional problems. An example of a state of the art EDA is the Bayesian Optimisation Algorithm (BOA), which is a multivariate EDA employing Bayesian Networks for modelling the relationships between good solutions. The use of a Bayesian Network leads to relatively fast convergence as well as high diversity in the matched models. Given relatively limited number of reservoir simulations used in history matching, EDA-BOA offers the promise of high quality history matches with a fast convergence rate. In this paper, we introduce EDAs and describe BOA in detail. We shows results of EDA-BOA algorithm on two history matching problems. First, we tune the algorithm and demonstrate convergence speed and search diversity on the PUNQ-S3 synthetic case. Secondly, we apply the algorithm to a real, North Sea, turbidite field with multiple wells. In both examples, we show improvements in performance over traditional population-based algorithms.

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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.

The topic of automatically history-matched reservoir models has seen much research activity in recent years. History matching is an example of an inverse problem, and there is significant active research on inverse problems in many other scientific and engineering areas. While many techniques from other fields, such as genetic algorithms, evolutionary strategies, differential evolution, particle swarm optimization, and the ensemble Kalman filter have been tried in the oil industry, more recent and effective ideas have yet to be tested. One of these relatively untested ideas is a class of algorithms known as estimation of distribution algorithms (EDAs). EDAs are population-based algorithms that use probability models to estimate the probability distribution of promising solutions, and then to generate new candidate solutions. EDAs have been shown to be very efficient in very complex high-dimensional problems.An example of a state-of-the-art EDA is the Bayesian optimization algorithm (BOA), which is a multivariate EDA employing Bayesian networks for modeling the relationships between good solutions. The use of a Bayesian network leads to relatively fast convergence as well as high diversity in the matched models. Given the relatively limited number of reservoir simulations used in history matching, EDA-BOA offers the promise of high-quality history matches with a fast convergence rate.In this paper, we introduce EDAs and describe BOA in detail. We show results of the EDA-BOA algorithm on two historymatching problems. First, we tune the algorithm, demonstrate convergence speed, and search diversity on the PUNQ-S3 synthetic case. Second, we apply the algorithm to a real North Sea turbidite field with multiple wells. In both examples, we show improvements in performance over traditional population-based algorithms. MethodologyEstimation of Distribution Algorithms. Stochastic search algorithms, such as genetic and evolutionary algorithms, exploit knowledge of the distribution of good solutions, amongst those already visited, in order to wisely select new points in the search space to evaluate. In the case of genetic algorithms, this knowledge is stored implicitly in the current population. It is exploited through the application of crossover and mutation operators to the solutions in this population, a process which, it is hoped, will lead to the generation of improved solutions.The success of genetic algorithms is sometimes attributed to the so-called building block hypothesis (Goldberg 1989), whereby it is conjectured that the genetic algorithm efficiently identifies and recombines building blocks (i.e., solution components, or schemata) with above-average fitness. However, in practice, genetic operators often break such partial solutions, especially when these schemata are large, spread widely across the solution, or when operators such as uniform crossover are used.This raises the possibility of an alternative approach: one can attempt to explicitly identify high-quality building blocks and