Improvement of Ensemble Kalman Filter for Improper Initial Ensembles

Abstract

ABSTRACT Ensemble Kalman filter(EnKF) has been widely researched in petroleum industry due to uncertainty analyses and convenience to couple with commercial simulators. However, the EnKF has shown overshooting and filter divergence problems when ensemble size becomes smaller or initial ensembles are quite different from the true. These problems of the EnKF result from unified covariance between model parameters and dynamic variables. Since all ensemble members have different model parameters, especially initial ensembles are improper, it is difficult to calculate representative covariance. We introduce the concept of clustered covariance into the standard EnKF. On the assimilation step, each ensemble members can be updated using more appropriate Kalman gain rather than the unified one. From the comparison of the performances using a 2-dimensional synthetic reservoir model, the EnKF shows that root mean square (RMS) error of the logarithm of permeability increases as ensemble size becomes smaller. Furthermore, it cannot estimate the uncertainty due to incorrect dynamic prediction. It is also fluctuating with an increasing trend when assimilation time interval becomes longer. The proposed method overcomes the typical problems mentioned. It is insensitive to ensemble size and assimilation time interval, even for improper initial ensemble design cases. The RMS error is below 1 over all the cases examined including 50 ensemble size case and 100 days of assimilation time interval case. The dynamic prediction in small ensemble size is reliable with covering true performance trend despite the prediction band of initial ensemble member is failed to do so.