A robust adaptive iterative ensemble smoother scheme for practical history matching applications

Abstract

Much of the recent work on history matching reservoir models has focused on the iterative Ensemble Smoother (iES) method. This is well suited for practical applications because it avoids frequent simulation restarts which can be expensive for traditional Ensemble Kalman Filter (EnKF) approaches. In this paper, we derive a novel, adaptive iES method (ES-LM) based on the Levenberg-Marquardt algorithm for nonlinear least squares optimization. We demonstrate that the solution of the linearized least squares subproblems from each iteration of ES-LM have a similar structure to those of the standard ensemble smoother update equation. This allows us to use the ensemble smoother as an approximate linear least squares solver and thus avoid expensive adjoint calculations. The proposed algorithm may therefore be seen as a variant of iES whose regularization parameter can be updated using the usual trust region method. The resulting parameter update equation is similar to those in existing iES methods, but our derivation provides new insights into iES that enable us to extend the adaptive ES-LM to two new formulations: (1) Auto ES-LM, where the measurement error has unknown variance that will be estimated by the algorithm; and (2) Robust ES-LM, where the measurement error follows a Cauchy distribution. Robust ES-LM, in particular, reduces the effect of outliers in the data, which can be critical for real-world applications. Although estimating measurement error and adding robustness are common for data assimilation with EnKF, they have not been incorporated into iES within the reservoir engineering community. The three resulting algorithms are easy for reservoir engineers to use for practical history matching applications without the need for deep understanding of the computational details. Finally, we demonstrate the effectiveness and robustness of these algorithms on two synthetic reservoir models.