Comparison of Stochastic Sampling Algorithms for Uncertainty Quantification

Abstract

History matching and uncertainty quantification are two important research topics in reservoir simulation currently. In the Bayesian approach, we start with prior information about a reservoir (e.g., from analog outcrop data) and update our reservoir models with observations (e.g., from production data or time-lapse seismic). The goal of this activity is often to generate multiple models that match the history and use the models to quantify uncertainties in predictions of reservoir performance. A critical aspect of generating multiple history-matched models is the sampling algorithm used to generate the models. Algorithms that have been studied include gradient methods, genetic algorithms, and the ensemble Kalman filter (EnKF).This paper investigates the efficiency of three stochastic sampling algorithms: Hamiltonian Monte Carlo (HMC) algorithm, Particle Swarm Optimization (PSO) algorithm, and the Neighbourhood Algorithm (NA). HMC is a Markov chain Monte Carlo (MCMC) technique that uses Hamiltonian dynamics to achieve larger jumps than are possible with other MCMC techniques. PSO is a swarm intelligence algorithm that uses similar dynamics to HMC to guide the search but incorporates acceleration and damping parameters to provide rapid convergence to possible multiple minima. NA is a sampling technique that uses the properties of Voronoi cells in high dimensions to achieve multiple history-matched models.The algorithms are compared by generating multiple historymatched reservoir models and comparing the Bayesian credible intervals (p10-p50-p90) produced by each algorithm. We show that all the algorithms are able to find equivalent match qualities for this example but that some algorithms are able to find good fitting models quickly, whereas others are able to find a more diverse set of models in parameter space. The effects of the different sampling of model parameter space are compared in terms of the p10-p50-p90 uncertainty envelopes in forecast oil rate.These results show that algorithms based on Hamiltonian dynamics and swarm intelligence concepts have the potential to be effective tools in uncertainty quantification in the oil industry.

Uncertainty Quantification in Reservoir Prediction: Part 2—Handling Uncertainty in the Geological Scenario

“…SVM classification of the model space has already been used to improve the search for better-fitting models in a HM exercise, for instance, Demyanov et al (2010). In this work we used the adaptive stochastic algorithm particle swarm optimisation (PSO) (Mohamed et al 2010) to navigate the combination of the SVM-classified metric space of geological scenarios and the space of the geomorphic model parameters for each reservoir description.…”

Recent Developments in Closed-Loop Approaches for Real-Time Mining and Petroleum Extraction

“…In addition to these 'classic' data assimilation methods there are 'non-classic' techniques from the machine learning or artificial intelligent community such as genetic algorithms, particle swarms, or simulated annealing; see, e.g., Schultze-Riegert et al (2002), Mohamed et al (2010) and Jin et al (2012). Because these methods typically require a very large number of function evaluations (i.e.…”

“…All rigorous methods for uncertainty quantification use some variety of the Monte Carlo method, which is, however, in its full form computationally completely infeasible for realistically sized reservoir models. Short-cuts, either in the form of 'surrogate' reservoir models or through drastically simplified uncertainty handling (of which the EnKF method is an example) lead to results with varying trustworthiness; see, e.g., Mohamed et al (2010). Practical approaches for quantifying the uncertainty in future reservoir performance therefore still rely on a mixture of human judgement and the use of ensembles of geological realizations.…”

History matching of naturally fractured reservoirs based on the recovery curve method

“…Different techniques have been used for uncertainty forecast in history matching problems including the maximum likelihood model (Subbey et al, 2004), hamiltonian Monte Carlo algorithm, Particle Swarm Optimization algorithm, and the NAB algorithm (Mohamed et al, 2010). The NAB, introduced by Sambridge (1999aSambridge ( , 1999b, has been confirmed to be a powerful uncertainty quantification tool (Mohamed et al, 2010). The NAB is a subset of the Markov chain Monte Carlo (McMC) method which builds an approximation for the real posterior probability distribution by using a Gibbs sampler.…”