Evaluation of Gaussian approximations for data assimilation in reservoir models

Handle URI:
http://hdl.handle.net/10754/594222
Title:
Evaluation of Gaussian approximations for data assimilation in reservoir models
Authors:
Iglesias, Marco A.; Law, Kody J H; Stuart, Andrew M.
Abstract:
The Bayesian framework is the standard approach for data assimilation in reservoir modeling. This framework involves characterizing the posterior distribution of geological parameters in terms of a given prior distribution and data from the reservoir dynamics, together with a forward model connecting the space of geological parameters to the data space. Since the posterior distribution quantifies the uncertainty in the geologic parameters of the reservoir, the characterization of the posterior is fundamental for the optimal management of reservoirs. Unfortunately, due to the large-scale highly nonlinear properties of standard reservoir models, characterizing the posterior is computationally prohibitive. Instead, more affordable ad hoc techniques, based on Gaussian approximations, are often used for characterizing the posterior distribution. Evaluating the performance of those Gaussian approximations is typically conducted by assessing their ability at reproducing the truth within the confidence interval provided by the ad hoc technique under consideration. This has the disadvantage of mixing up the approximation properties of the history matching algorithm employed with the information content of the particular observations used, making it hard to evaluate the effect of the ad hoc approximations alone. In this paper, we avoid this disadvantage by comparing the ad hoc techniques with a fully resolved state-of-the-art probing of the Bayesian posterior distribution. The ad hoc techniques whose performance we assess are based on (1) linearization around the maximum a posteriori estimate, (2) randomized maximum likelihood, and (3) ensemble Kalman filter-type methods. In order to fully resolve the posterior distribution, we implement a state-of-the art Markov chain Monte Carlo (MCMC) method that scales well with respect to the dimension of the parameter space, enabling us to study realistic forward models, in two space dimensions, at a high level of grid refinement. Our implementation of the MCMC method provides the gold standard against which the aforementioned Gaussian approximations are assessed. We present numerical synthetic experiments where we quantify the capability of each of the ad hoc Gaussian approximation in reproducing the mean and the variance of the posterior distribution (characterized via MCMC) associated to a data assimilation problem. Both single-phase and two-phase (oil-water) reservoir models are considered so that fundamental differences in the resulting forward operators are highlighted. The main objective of our controlled experiments was to exhibit the substantial discrepancies of the approximation properties of standard ad hoc Gaussian approximations. Numerical investigations of the type we present here will lead to the greater understanding of the cost-efficient, but ad hoc, Bayesian techniques used for data assimilation in petroleum reservoirs and hence ultimately to improved techniques with more accurate uncertainty quantification. © 2013 Springer Science+Business Media Dordrecht.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Iglesias MA, Law KJH, Stuart AM (2013) Evaluation of Gaussian approximations for data assimilation in reservoir models. Computational Geosciences 17: 851–885. Available: http://dx.doi.org/10.1007/s10596-013-9359-x.
Publisher:
Springer Nature
Journal:
Computational Geosciences
Issue Date:
14-Jul-2013
DOI:
10.1007/s10596-013-9359-x
ARXIV:
arXiv:1212.1779v1
Type:
Article
ISSN:
1420-0597; 1573-1499
Sponsors:
MI, KJHL, and AMS gratefully acknowledge the support of EPSRC, ERC, ESA, and ONR for various aspects of this work.
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorIglesias, Marco A.en
dc.contributor.authorLaw, Kody J Hen
dc.contributor.authorStuart, Andrew M.en
dc.date.accessioned2016-01-19T14:43:42Zen
dc.date.available2016-01-19T14:43:42Zen
dc.date.issued2013-07-14en
dc.identifier.citationIglesias MA, Law KJH, Stuart AM (2013) Evaluation of Gaussian approximations for data assimilation in reservoir models. Computational Geosciences 17: 851–885. Available: http://dx.doi.org/10.1007/s10596-013-9359-x.en
dc.identifier.issn1420-0597en
dc.identifier.issn1573-1499en
dc.identifier.doi10.1007/s10596-013-9359-xen
dc.identifier.urihttp://hdl.handle.net/10754/594222en
dc.description.abstractThe Bayesian framework is the standard approach for data assimilation in reservoir modeling. This framework involves characterizing the posterior distribution of geological parameters in terms of a given prior distribution and data from the reservoir dynamics, together with a forward model connecting the space of geological parameters to the data space. Since the posterior distribution quantifies the uncertainty in the geologic parameters of the reservoir, the characterization of the posterior is fundamental for the optimal management of reservoirs. Unfortunately, due to the large-scale highly nonlinear properties of standard reservoir models, characterizing the posterior is computationally prohibitive. Instead, more affordable ad hoc techniques, based on Gaussian approximations, are often used for characterizing the posterior distribution. Evaluating the performance of those Gaussian approximations is typically conducted by assessing their ability at reproducing the truth within the confidence interval provided by the ad hoc technique under consideration. This has the disadvantage of mixing up the approximation properties of the history matching algorithm employed with the information content of the particular observations used, making it hard to evaluate the effect of the ad hoc approximations alone. In this paper, we avoid this disadvantage by comparing the ad hoc techniques with a fully resolved state-of-the-art probing of the Bayesian posterior distribution. The ad hoc techniques whose performance we assess are based on (1) linearization around the maximum a posteriori estimate, (2) randomized maximum likelihood, and (3) ensemble Kalman filter-type methods. In order to fully resolve the posterior distribution, we implement a state-of-the art Markov chain Monte Carlo (MCMC) method that scales well with respect to the dimension of the parameter space, enabling us to study realistic forward models, in two space dimensions, at a high level of grid refinement. Our implementation of the MCMC method provides the gold standard against which the aforementioned Gaussian approximations are assessed. We present numerical synthetic experiments where we quantify the capability of each of the ad hoc Gaussian approximation in reproducing the mean and the variance of the posterior distribution (characterized via MCMC) associated to a data assimilation problem. Both single-phase and two-phase (oil-water) reservoir models are considered so that fundamental differences in the resulting forward operators are highlighted. The main objective of our controlled experiments was to exhibit the substantial discrepancies of the approximation properties of standard ad hoc Gaussian approximations. Numerical investigations of the type we present here will lead to the greater understanding of the cost-efficient, but ad hoc, Bayesian techniques used for data assimilation in petroleum reservoirs and hence ultimately to improved techniques with more accurate uncertainty quantification. © 2013 Springer Science+Business Media Dordrecht.en
dc.description.sponsorshipMI, KJHL, and AMS gratefully acknowledge the support of EPSRC, ERC, ESA, and ONR for various aspects of this work.en
dc.publisherSpringer Natureen
dc.subjectData assimilationen
dc.subjectInverse modellingen
dc.subjectReservoir characterizationen
dc.subjectUncertainty quantificationen
dc.titleEvaluation of Gaussian approximations for data assimilation in reservoir modelsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalComputational Geosciencesen
dc.contributor.institutionUniversity of Warwick, Coventry, United Kingdomen
dc.identifier.arxividarXiv:1212.1779v1en
kaust.authorLaw, Kodyen
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