Preventing Ensemble Collapse and Preserving Geostatistical Variability Across the Ensemble with the Subspace EnKF

One of the key issues of the EnKF is the well known problem of ensemble collapse, which is particularly evident for small ensembles. This results in an artificial reduction of variability across the ensemble. The second, more important problem is that the EnKF is theoretically appropriate only if all ensemble members belong to the same multi-Gaussian random field (geological/geostatistical model). This is an important problem because for most real fields, we have more than one geological scenario, and ideally, we would like to obtain one or more history-matched models for each geological scenario. In this work, we propose the subspace EnKF to alleviate both problems mentioned above. The basic idea of the subspace EnKF is to constrain the different ensemble members to different subspaces of the same or different random field. This is accomplished by parameterizing the random fields and modifying the EnKF formulation with the gradients of the parameterizations. The subspace EnKF prevents ensemble collapse, providing a better quantification of uncertainty, and more importantly, retains key geological characteristics of the initial ensemble, even when each ensemble member belongs to a different geological model. The approach is demonstrated on a synthetic example with a multi-Gaussian permeability field.