A new procedure for generating data covariance inflation factors for ensemble smoother with multiple data assimilation

Abstract

The ensemble smoother with multiple data assimilation (ES-MDA) has gained much attention as a powerful tool for history matching problems. Previous studies showed that it could provide both a good match of data and estimates of model parameters. In the original ES-MDA formulation, the number of data assimilation and covariance inflation factors are determined in advance. Selecting them in a decreasing order may improve the final results. Moreover, recent studies propose some theoretical and practical methods to select inflation factors based on the discrepancy principle. This work aims to introduce a new method for generating the data covariance inflation factors for ES-MDA. In the new method, the first inflation factor is generated using a Levenberg–Marquardt regularizing scheme. The last inflation factor is set by a parameter that limits its magnitude, computed using the singular values of the dimensionless sensitivity matrix estimated from the prior ensemble. As a result, the method computes the correct number of data assimilations that produces inflation factors such that the sum of their inverse is equal to one, as required by ES-MDA. It is shown through a synthetic two-dimensional water flooding history matching problem that the proposed methodology achieves both better model parameter match and data match with a smaller number of assimilations than the methods available in the literature.