Ensemble-Based Methods for Data Integration and Uncertainty Quantification

Recently, Emerick and Reynolds (2012, 2013) introduced the ensemble smoother with multiple data assimilations (ES-MDA) for assisted history matching. Via computational examples, they demonstrated its superiority over the ensemble Kalman filter (EnKF) and the ensemble smoother (ES) in that ES-MDA provides both a better data match and a better quantification of the uncertainty in the reservoir description and future performance predictions than are obtained with either the ES or EnKF. However, like the ES and EnKF, ES-MDA can experience near ensemble collapse for complex problems with large data sets. This negative characteristic of ES-MDA can be ameliorated by a judicious choice of the inflation factors used in ES-MDA. Here, we provide two automatic procedures for choosing the inflation factor for the next data assimilation step adaptively as the history-match proceeds. Both methods are motivated by knowledge of regularization procedures; the first method is intuitive and heuristical; the second procedure is based on existing, but not very well known, theory on the regularization of least-squares inverse problems involving compact operators. We illustrate that the adaptive ES-MDA algorithm is superior to the original ES-MDA algorithm by history matching three-phase flow production data for a complicated synthetic problem.