NDLI: The regularizing Levenberg–Marquardt scheme for history matching of petroleum reservoirs

Content Provider	Springer Nature Link
Author	Iglesias, Marco A. Dawson, Clint
Copyright Year	2013
Abstract	In this paper, we study on a history matching approach that consists of finding stable approximations to the problem of minimizing the weighted least-squares functional that penalizes the misfit between the reservoir model predictions G(u) and noisy observations y η . In other words, we are interested in computing an approximation to the minimizer of $\frac {1}{2}\vert \vert \Gamma ^{-1/2}(y^{\eta }-G(u))\vert \vert _{Y}^{2} $ where Γ is the measurements error covariance, Y is the observation space, and X is a set of admissible parameters. This is an ill-posed nonlinear inverse problem that we address by means of the regularizing Levenberg–Marquardt scheme developed by Hanke (Inverse Probl. 13:79–95, 1997; J. Integr. Equ. Appl. 22(2):259–283, 2010). Under certain conditions on G, the theory of Hanke (Inverse Probl. 13:79–95, 1997; J. Integr. Equ. Appl. 22(2):259–283, 2010) ensures the convergence of the scheme to stable approximations to the inverse problem. We propose an implementation of the regularizing Levenberg–Marquardt scheme that enforces prior knowledge on the geologic properties. In particular, the prior mean $\overline {u}$ is incorporated in the initial guess of the algorithm, and the prior error covariance C is enforced through the definition of the parameter space X. Our main goal is to numerically show that the proposed implementation of the regularizing Levenberg–Marquardt scheme of Hanke is a robust method capable of providing accurate estimates of the geologic properties for small noise measurements. In addition, we provide numerical evidence of the convergence and regularizing results predicted by the theory of Hanke (Inverse Probl. 13:79–95, 1997; J. Integr. Equ. Appl. 22(2):259–283, 2010) for a prototypical oil–water reservoir model. The performance for recovering the true permeability with the regularizing Levenberg–Marquardt scheme is compared to the typical approach of computing the maximum a posteriori (MAP) estimator. In particular, we compare the proposed application of the regularizing Levenberg–Marquardt (LM) scheme against the standard LM approach of Li et al. (SPE J. 8(4):328–340, 2003) and Reynolds et al. (2008) for computing the MAP. Our numerical experiments suggest that the history matching approach based on iterative regularization is robust and could potentially be used to improve further on various methodologies already proposed as effective tools for history matching in petroleum reservoirs.
Starting Page	1033
Ending Page	1053
Page Count	21
File Format	PDF
ISSN	14200597
Journal	Computational Geosciences
Volume Number	17
Issue Number	6
e-ISSN	15731499
Language	English
Publisher	Springer Netherlands
Publisher Date	2013-09-29
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Data assimilation History matching Inverse problems Reservoir modeling Earth Sciences Geotechnical Engineering & Applied Earth Sciences Hydrogeology Mathematical Modeling and Industrial Mathematics Soil Science & Conservation
Content Type	Text
Resource Type	Article
Subject	Computers in Earth Sciences Computational Theory and Mathematics Computer Science Applications Computational Mathematics

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