Applying Support Vector Regression to Reduce the Effect of Numerical Noise and Enhance the Performance of History Matching
Authors: Guo Zhenyu , Chen Chaohui , Gao Guohua , Vink Jeroen 
Published: October 2017

Abstract
Numerical optimization is an integral part of many history matching (HM) workflows. To be efficient, these model-based optimization methods often use numerically computed gradients, which are difficult to calculate accurately due to numerical noise in simulation results. In this paper, Support Vector Regression (SVR) is integrated with a model-based optimization algorithm, Distributed Gauss-Newton (DGN), to provide accurate gradients that are insensitive to the negative influence of this type of numerical noise.
Previously we have developed a parallelized DGN optimization method, which uses an ensemble of reservoir simulation models to compute the required gradients with a simple linear interpolation or regression method. Numerical noise is unavoidable for reservoir simulations. More precisely, the allowed solver tolerances imply that simulation results no longer smoothly change with changing model parameters. By setting tight convergence criteria, these discontinuities can be reduced but then the overall simulation run time will increase and obviously jeopardize optimization efficiency. Furthermore, the inaccurate gradients degrade the convergence performance of the original linear DGN (L-DGN) significantly, or even worse; it may result in failure of convergence. In this paper, we use the simulation results to build SVR models, which are then used to compute the required gradients. The accuracy of the SVR models is further improved by reusing simulation results of preceding iterations. Starting from an initial ensemble of models, new search points for each realization are generated with a modified Gauss-Newton trust region method using the sensitivity matrix estimated with SVR. The SVR proxies are updated when simulation results of new search points are available. The procedure is repeated until the distributed optimization process has converged.
Both our L-DGN approach and the newly proposed SVR-DGN approach are first tested with a two-dimensional toy problem to show the effect of numerical noise on their convergence performance. We find that their performance is comparable when the toy problem is free of numerical noise. When the numerical noise level increases, the performance of DGN degrades sharply. In contrast, SVR-DGN performance is quite stable. Both methods are similarly tested using a real field history matching example. Also here, the convergence performance of SVR-DGN is not affected by different solver settings (i.e., noise levels), whereas the performance of L-DGN degrades significantly when loose numerical settings are applied. Moreover, the overall convergence rate is faster when the SVR-computed gradients are used.
Our results show that SVR can be utilizedefficiently and robustly to obtain accurate gradients from numerically computed, noisy simulation results. The SVR approach can also be integrated with other derivative-free optimization methods which require building accurate and robust proxy models that are used to guide iterative parameter updates.