The regularizing Levenberg Marquardt scheme for history matching of petroleum reservoirs
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 ye. In other words, we are interested in computing an approximation to the minimizer of 12||G 1/2(ye G(u)|Y2$\frac {1}{2}\vert \vert \Gamma 1/2}(y^\eta G(u)\vert \vert _{Y}{2} where G 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 u \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.
Computational Geosciences