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.
Similar content being viewed by others
References
Abacioglu, Y., Oliver, D., Reynolds, A.: Efficient reservoir history matching using subspace vectors. Comput. Geosci. 5(2), 151–172 (2001)
Chen, Z., Huan, G., MA, Y.: Computational methods for multiphase flows in porous media. In: Society for Industrial and Applied Mathematics, Philadelphia (2006)
Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin (1992)
Deutsch, C.V.: Geostatistical Reservoir Modeling. Oxford University, Oxford (2002)
Engl, H.W., Flamm, C., Kgler, P., Lu, J., Mller, S., Schuster, P.: Inverse problems in systems biology. Inverse Probl. 25(12), 123014 (2009)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, vol. 375. Springer, Dordrecht (1996)
Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston (1984)
Hanke, M.: A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Probl. 13, 79–95 (1997)
Hanke, M.: Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. Numer. Funct. Anal. Optim. 18(9–10), 971–993 (1997)
Hanke, M.: The regularizing Levenberg-Marquardt scheme is of optimal order. J. Integr. Equ. Appl. 22(2), 259–283 (2010)
Iglesias, M.A., Dawson, C.: An iterative representer-based scheme for data inversion in reservoir modeling. Inverse Probl. 25, 035006 (2009)
Iglesias, M.A., Law, K.J.H., Stuart, A.M.: Evaluation of Gaussian approximations for data assimilation in reservoir models. Comput. Geosci. (2013). doi:10.1007/s10596-013-9359-x
Iglesias, M.A., McLaughlin, D.: Level-set techniques for facies identification in reservoir modeling. Inverse Probl. 27, 035008 (2011)
Iglesias, M.A., McLaughlin, D.: Data inversion in coupled subsurface flow and geomechanics models. Inverse Probl. 28, 115009 (2012)
Isakov, V.: On inverse problems in secondary oil recovery. Eur. J. Appl. Math. 19, 459–478 (2008)
Jensen, T.K., Hansen, P.C.: Iterative regularization with minimum-residual methods. BIT Numer. Math. 47(1), 103–120 (2007)
Katltenbacher, B., Neubauer, A., Scherzer, O.: Iterative regularization methods for nonlinear ill-posed problems.Radon Series on Computational and Applied Mathematics, 1st edn.Walter de Gruyter, Berlin (2008)
Kravaris, C., Seinfeld, J.H.: Identification of parameters in distributed parameter systems by regularization. SIAM J. Control. Optim. 23(2), 217–241 (1985)
Li, R., Reynolds, A.C., Oliver, D.S.: History matching of three-phase flow production data. SPE J. 8(4), 328–340 (2003)
Nagy, J.G., Palmer, K.M.: Steepest descent, CG, and iterative regularization of ill-posed problems. BIT Numer. Math. 43, 1003–1017 (2003)
Nocedal, J.A., Wright, S.J.: Numerical Optimization: With 85 Illustrations. Springer Series in Operations Research Series. Springer, New York (1999)
Reynolds, A.C., Oliver, D.S., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching, 1st edn. Cambridge University, Cambridge. ISBN: 9780521881517 (2008)
Oliver, D., Chen, Y.: Recent progress on reservoir history matching: A review. Comput. Geosci. 15, 185–221 (2011). doi:10.1007/s10596-010-9194-2
Tavakoli, R., Reynolds, A.C.: History matching with parametrization based on the SVD of a dimensionless sensitivity matrix. SPE J. 15(2), 495–508 (2010)
Tavakoli, R., Reynolds, A.: Monte Carlo simulation of permeability fields and reservoir performance predictions with SVD parameterization in RML compared with EnKF. Comput. Geosci. 15, 99–116 (2011). doi:10.1007/s10596-010-9200-8
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Iglesias, M.A., Dawson, C. The regularizing Levenberg–Marquardt scheme for history matching of petroleum reservoirs. Comput Geosci 17, 1033–1053 (2013). https://doi.org/10.1007/s10596-013-9373-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-013-9373-z