History Matching With Parameterization Based on Singular- Value Decomposition of a Dimensionless Sensitivity Matrix
Authors: Denney Dennis 
Published: February 2015

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 118952, ’History Matching With Parameterization Based on the SVD of a Dimensionless Sensitivity Matrix,’ by Reza Tavakoli and Albert C. Reynolds, SPE, University of Tulsa, prepared for the 2009 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2-4 February.
In gradient-based automatic history matching, calculating the derivatives (sensitivities) of all production data with respect to gridblock rock properties and other model parameters is not feasible for large-scale problems. The quasi-Newton and nonlinear-conjugate-gradient algorithms do not require explicit calculation of the complete sensitivity matrix (the Hessian). Another possibility is to define a new parameterization to reduce the number of model parameters.
Introduction
In automatic history matching, an optimization algorithm is applied to minimize an objective function that includes the mismatch between observed and predicted data. In the Bayesian approach used here, a model-mismatch term is included in the objective function. In gradient-based optimization methods, the required gradient information depends on the optimization algorithm used. In the Gauss-Newton method, the sensitivity matrix, which involves the derivative of predicted data with respect to all model parameters, is required to form the Hessian matrix. For the nonlinear-conjugate-gradient method, explicit computation of the full sensitivity matrix is not necessary. See the full-length paper for history, references, and detail.
Reducing the number of estimated parameters reduces the computational costs. However, to preserve geological reality, reparameterization should yield a model that is qualitatively similar to the one that would be obtained with no reparameterization (e.g., reparameterization should not result in excessive smoothness and artifacts inconsistent with the geological model).
The simplest form of reduced parameterization is use of zonation. The reservoir is divided into a small number of zones, and the properties in each zone are assumed to be uniform, which however, introduces modeling error. The gradzone method and adaptive-multiscale methods represent more-modern variations of zonation; however, these methods can introduce nongeological discontinuities between zones.
The expansion of unknown model parameters in terms of the right-singular vectors of a linear-data kernel operator for linear inverse problems is optimum in the sense that it maximizes the amount of information passed from the solution space to the data space. For parameterization, right-singular vectors of the dimensionless sensitivity matrix were used. A theoretical argument is presented that the principal right-singular vectors of the dimensionless sensitivity matrix form an optimal basis because eliminating those vectors corresponding to smaller singular values has a negligible effect on the reduction in uncertainty obtained by conditioning a reservoir model to dynamic data.