Numerical reservoir models are used to predict, optimise and improve production performance of the oil and gas reservoirs. History matching is required to calibrate reservoir models to dynamic behaviour of the reservoir. On the one hand, history-matching does not have a unique solution and multiple models can fit observation data, on the other hand, history-matching is a tedious and time-consuming trial and error process as it involves numerous reservoir simulation runs. Modern history matching techniques use optimisation algorithms aim at providing a set of good fitting models in an efficient time.
Many optimisation algorithms are applied in history-matching. Of them, Evolutionary Algorithms (EAs), inspired by natural evolution, do not use gradient information from the optimisation problem and only require the fitness function, usually defined as the sum of squares root deviation of model response from the observation data. Estimation of distribution algorithms (EDAs) are a novel class of EAs developed as a natural alternative to genetic algorithms in the last decade. To date, many EDAs are introduced which differ in the probabilistic model that guides the search process. Most of the EDAs are designed for discrete problems and require discretisation of search space when used for continuous problems, e.g. in history matching. In some cases, discretization error can be significant and deteriorate the search process.
Gaussian-based EDAs use characteristics of Gaussian distribution for multivariate continuous problems. i.e. they make use of mean and covariance matrix of the variables in the promising solutions to generate new solutions which fit better the observation data. In this paper, we introduce and for the first time apply four Gaussian-based EDAs to assisted history-matching of a standard synthetic case. We show our proposed algorithms may produce results more accurately and more efficiently for the continuous problems.
Abdollahzadeh, Asaad (Heriot-Watt University) | Reynolds, Alan (Heriot-Watt University) | Christie, Michael (Herit-Watt University) | Corne, David W. (Heriot-Watt University) | Davies, Brian J. (BP) | Williams, Glyn J.J. (BP)
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.
Abdollahzadeh, Asaad (Heriot-Watt U.) | Reynolds, Alan (Heriot-Watt University) | Christie, Michael A. (Heriot-Watt U.) | Corne, David (Heriot-Watt University) | Williams, Glyn (BP) | Davies, Brian James (BP Exploration & Production)
In history matching, the aim is to generate multiple good-enough history-matched models with a limited number of simulations which will be used to efficiently predict reservoir performance. History matching is the process of the conditioning reservoir model to the observation data; is mathematically ill-posed, inverse problem and has no unique solution and several good solutions may occur.
Numerous evolutionary algorithms are applied to history matching which operate differently in terms of population diversity in the search space throughout the evolution. Even different flavours of an algorithm behave differently and different values of an algorithm's control parameters result in different levels of diversity. These behaviours vary from explorative to exploitative.
The need to measure population diversity arises from two bases. On the one hand maintaining population diversity in evolutionary algorithms is essential to detect and sample good history-matched ensemble models in parameter search space. On the other hand, since the objective function evaluations in history matching are computationally expensive, algorithms with fewer total number of reservoir simulations in result of a better convergence are much more favourable. Maintaining population's diversity is crucial for sampling algorithm to avoid premature convergence toward local optima and achieve a better match quality.
In this paper, we introduce and use two measures of the population diversity in both genotypic and phenotypic space to monitor and compare performance of the algorithms. These measures include an entropy-based diversity from the genotypic measures and a moment of inertia based diversity from the phenotypic measures.
The approach has been illustrated on a synthetic reservoir simulation model, PUNQ-S3, as well as on a real North Sea model with multiple wells. We demonstrate that introduced population diversity measures provide efficient criteria for tuning the control parameters of the population-based evolutionary algorithms as well as performance comparison of the different algorithms used in history matching.
To make prudent decisions regarding the exploitation and management of hydrocarbon reservoirs, we need to carry out history matching, a process for conditioning the reservoir simulation model to observation data collected over time. History matching is an inverse problem which requires an optimisation technique to match the simulation results to the measurements. Many techniques have been applied to address this optimisation problem effectively and in efficient time since reservoir simulation runs are computationally expensive.
Genetic algorithms (GAs) and Estimation of Distribution Algorithms (EDAs) are two popular types of evolutionary algorithms. In GAs, new candidate solutions are obtained by applying crossover and mutation operators to a population of feasible solutions according to the principle of ‘survival of the fittest' in natural evolution. The Estimation of Distribution Algorithm (EDA) is a modern class of EA in which new candidate solutions are generating by sampling from a probability distribution inferred from the better members of the population.
A suitable hybrid of the GA and EDA algorithms can combine beneficial characteristics from each of GA and EDA, while addressing each other's sources of inefficiency. The main difference between these two EAs is the way they generate new individuals, which results in different exploration/exploitation properties. GAs may sample bad representatives of good search regions and good representatives of bad regions, while the EDA may suffer from fitting a single probability distribution to diverse and distinct regions of good solutions. The hybrid algorithm performs a cooperative search that improves the exploitation and the exploration power of both algorithms.
In this paper, we applied GA, EDA, and a new hybrid GA-EDA algorithm to optimisation of three cases, a test function, the IC-Fault synthetic reservoir model, and one real reservoir, Teal South. The results show that each of these algorithms can be used for exploring the parameter search space in history matching problem. Depending on the problem type, GA, EDA, and Hybrid GA-EDA can achieve good quality matches while they perform a global seach in the space.