History Matching With Parametrization Based on the SVD of a Dimensionless Sensitivity Matrix

Abstract

In gradient based automatic history matching, calculation of the derivatives of all production data with respect to gridblock rock properties (sensitivities) and other model parameters is not feasible for large-scale problems. Thus, the Gauss-Newton method and Levenberg-Marquardt 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 as 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 reduced the number of model parameters.We provide a theoretical argument which indicates that reparametrization based on the principal right singular vectors of the dimensionless sensitivity matrix provides an optimal basis for re-parametrization of the vector of model parameters. We present and apply to example problems two algorithms for using this parametrization. Like LBFGS, these algorithms avoid explicit computation of individual sensitivity coefficients. Explicit computation of the sensitivities is avoided by using a partial singular value decomposition 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.

Monte Carlo simulation of permeability fields and reservoir performance predictions with SVD parameterization in RML compared with EnKF

“…As shown in Tavakoli and Reynolds [27], the search direction in the modified LM algorithm for computing a MAP estimate in the transformed model domain is given by…”

“…The preceding LM can be applied successfully [17] if the number of data is sufficiently small so that it is computationally feasible to calculate all entries of the sensitivity matrix with an adjoint method. Here, following Rodriques [24] and Tavakoli and Reynolds [27], we seek to improve computational efficiency by expanding δm l+1 in terms of the right singular vectors of the dimensionless sensitivity matrix at each iteration for cases where the computation of all entries of the sensitivity matrix is not feasible.…”

“…As shown in Tavakoli and Reynolds [27], with this parameterization the LM step of Eq. 10 simplifies to…”

“…Recently, Tavakoli and Reynolds [27] provided a theoretical basis for parameterization of reservoir model parameters based on the truncated singular value decomposition (SVD) of the dimensionless sensitivity matrix [33]. Specifically, we showed that parameterization based on the principal singular triplets of the dimensionless sensitivity matrix is ideal if the objective is to minimize the posterior uncertainty in model parameters.…”

“…Specifically, we showed that parameterization based on the principal singular triplets of the dimensionless sensitivity matrix is ideal if the objective is to minimize the posterior uncertainty in model parameters. In Tavakoli and Reynolds [27], we implemented two algorithms for generation of only the MAP estimate. The generation of multiple realizations was not considered.…”