Analysis of geometric selection of the data-error covariance inflation for ES-MDA

Abstract

The ensemble smoother with multiple data assimilation (ES-MDA) has become a popular assisted history-matching method. In its standard form, the method requires the specification of the number of iterations in advance. If the selected number of iterations is not enough, the entire data assimilation must be restarted. Moreover, ES-MDA also requires the selection of data-error covariance inflations. The typical choice is to select constant values. However, previous works indicate that starting with large inflation and gradually decreasing during the iterations may improve the quality of the final models. This paper presents an analysis of the use of geometrically decreasing sequences of the data-error covariance inflations. In particular, the paper investigates a recently introduced procedure based on the singular values of a sensitivity matrix computed from the prior ensemble. Also, it introduces a novel procedure to select the number of iterations and the inflation factors. The procedures are evaluated in three reservoir history-matching problems with increasing level of complexity. The first problem is a small synthetic case used to illustrate that the standard ES-MDA scheme with constant inflations may result in overcorrection of the permeability field and that a geometric sequence can alleviate this problem. The second problem is a recently published benchmark (UNISIM-I-H) and the third one is a field case with real production data. The data assimilation schemes are compared in terms of a data-mismatch and a model-change norm. While the former evaluates the ability of the models to reproduce the observed data, the latter evaluates the amount of changes in the prior model. The results indicate that geometric inflations can generate solutions with a desirable good balance between both norms.