The regularizing Levenberg-Marquardt scheme for history matching of petroleum reservoirs,

Abstract: In this paper we study 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 $y^{\eta}$. In other words, we are interested in computing $u^{\eta}\equiv \arg\min_{u\in X}\frac{1}{2}\vert\vert \Gamma^{-1/2}(y-G(u))\vert\vert_{Y}^{2} $ where $\Gamma$ 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 in \cite{Hanke,Hanke2}. Under certain conditions on $G$, the theory of \cite{Hanke,Hanke2} ensures 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 of 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. The performance for recovering the true permeability with the regularizing Levenberg-Marquardt scheme is compared against the more standard techniques for history matching proposed in \cite{Li,Tavakoli,svdRML,Oliver}. 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