History Matching With Parameterization Based on the Singular Value Decomposition of a Dimensionless Sensitivity Matrix

Abstract

Summary In gradient-based automatic history matching, calculation of the derivatives (sensitivities) of all production data with respect to gridblock rock properties and other model parameters is not feasible for large-scale problems. Thus, the Gauss-Newton (GN) method and Levenberg-Marquardt (LM) algorithm, which require calculation of all sensitivities to form the Hessian, are seldom viable. For such problems, the quasi-Newton and nonlinear conjugate gradient algorithms present reasonable alternatives because these two methods do not require explicit calculation of the complete sensitivity matrix or the Hessian. Another possibility, the one explored here, is to define a new parameterization to radically reduce the number of model parameters. We provide a theoretical argument that indicates that reparameterization based on the principal right singular vectors of the dimensionless sensitivity matrix provides an optimal basis for reparameterization of the vector of model parameters. We develop and illustrate algorithms based on this parameterization. Like limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS), these algorithms avoid explicit computation of individual sensitivity coefficients. Explicit computation of the sensitivities is avoided by using a partial singular value decomposition (SVD) based on a form of the Lanczos algorithm. At least for all synthetic problems that we have considered, the reliability, computational efficiency, and robustness of the methods presented here are as good as those obtained with quasi-Newton methods.