Bayesian Reservoir History Matching Considering Model and Parameter Uncertainties
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Bayesian Reservoir History Matching Considering Model and Parameter Uncertainties A.H. Elsheikh · M.D. Jackson · T.C. Laforce
Received: 20 November 2011 / Accepted: 2 April 2012 / Published online: 24 April 2012 © International Association for Mathematical Geosciences 2012
Abstract This paper presents a consistent Bayesian solution for data integration and history matching for oil reservoirs while accounting for both model and parameter uncertainties. The developed method uses Gaussian Process Regression to build a permeability map conforming to collected data at well bores. Following that, an augmented Markov Chain Monte Carlo sampler is used to condition the permeability map to dynamic production data. The selected proposal distribution for the Markov Chain Monte Carlo conforms to the Gaussian process regression output. The augmented Markov Chain Monte Carlo sampler allows transition steps between different models of the covariance function, and hence both the parameter and model space are effectively explored. In contrast to single model Markov Chain Monte Carlo samplers, the proposed augmented Markov Chain Monte Carlo sampler eliminates the selection bias of certain covariance structures of the inferred permeability field. The proposed algorithm can be used to account for general model and parameter uncertainties. Keywords Bayesian Inference · Markov Chain Monte Carlo · Inverse problem · Gaussian process regression · Model comparison · Uncertainty quantification 1 Introduction Inference of subsurface geological properties is essential for prediction and optimization of production rates in oil reservoirs. Modeling oil reservoirs is a challenging problem since the models are generally large and the collected data is limited. This results in an ill-posed inverse problem. Standard methods for solving the history matching problem rely on sequential application of different techniques (Oliver and Chen A.H. Elsheikh () · M.D. Jackson · T.C. Laforce Department of Earth Science and Engineering, Imperial College London, Prince Consort Road, London, SW7 2BP, UK e-mail: [email protected]
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Math Geosci (2012) 44:515–543
2011). Geo-statistical analysis is commonly used to generate a set of subsurface models assuming a correlation length between the samples. These geo-statistical models are then used for forward simulation, and as an input for history matching algorithms. Randomized Maximum Likelihood (RML) (Kitanidis 1995; Oliver et al. 1996) is an effective method for both static and dynamic data integration. RML is a sequential algorithm where at each iteration three steps are performed. First, a simulation-based method (Davis 1987; Alabert 1987) is used to generate a realization of the permeability map conditioned to the hard data. The corresponding covariance matrix is also generated. Following that, a realization of the production data is generated from the production probability distribution function while accounting for measurement errors. Finally, the permeability map is found by minimizing an objecti
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