This paper presents a review of ensemble-based data assimilation for strongly nonlinear problems on the characterization of heterogeneous reservoirs with different production histories. It concentrates on ensemble Kalman filter (EnKF) and ensemble smoother (ES) as representative frameworks, discusses their pros and cons, and investigates recent progress to overcome their drawbacks. The typical weaknesses of ensemble-based methods are non-Gaussian parameters, improper prior ensembles and finite population size. Three categorized approaches, to mitigate these limitations, are reviewed with recent accomplishments; improvement of Kalman gains, add-on of transformation functions, and independent evaluation of observed data. The data assimilation in heterogeneous reservoirs, applying the improved ensemble methods, is discussed on predicting unknown dynamic data in reservoir characterization.

Ensemble Kalman filter (EnKF) has the limitation of applications for multi-point geostatistics because it assumes Gaussian random field. It also uses all ensembles to get covariance matrix, even though they have different permeability field each other, resulting in filter divergence. The proposed method suggests the concept of clustered covariance by grouping initial ensembles using a distance-based method. Hausdorff distance is used for calculating similarity between permeability fields and they are separated by k-means clustering. When EnKF is applied to a 2D channel field, it shows overshooting problem and mismatches the true production data. The proposed method gives better history matching and future performance prediction without overshooting problems. Furthermore, it shows stable results for sensitivity analyses over the number of total ensembles. The more accurate covariance is calculated by clustering, the better results are obtained.

Ensemble Kalman filter (EnKF) has been researched for reservoir characterization in petroleum engineering. However, the repeated assimilation causes lots of simulation cost. Ensemble smoother (ES) assimilates all available data once. It has advantages over EnKF: efficiency and simplicity. The two ensemble methods are based on the same assumptions: Gaussian distribution and trust in the mean of all ensembles. Many researchers have pointed out that EnKF gives overshooting and filter divergence problems when the two key assumptions are not satisfied. This paper presents characterization of channel fields using ES with the concept of clustered covariance, especially improper ensemble design. The standard EnKF, ES, and the proposed method are applied to a 2D synthetic channel field with 200 ensembles. From distance-based clustering method, we separate initial ensembles into 10 groups based its on similarity. The proposed method uses 10 Kalman gains, since each cluster has own Kalman gain. They can represent ensembles properly by using similar ensembles instead of 200 different ensembles. For the channel fields, the standard EnKF and ES show overshooting and filter divergence problems. Updated permeability fields have extreme values and lose the continuity of channel stream. However, the proposed method manages those two problems and provides reasonable results. We can get future prediction with reliable uncertainty. The proposed method only requires about 5% of simulation time compared to EnKF, since it is based on ES. It can be applied to the characterization of channel fields, even though we have improper ensembles due to limited information.

Ensemble smoother (ES) assimilates all available dynamic data without iterations as global update. Therefore, ES is much faster than ensemble Kalman filter (EnKF), which uses recursive updates. Iterative concepts are introduced for ES to increase accuracy of history matching. However, they lose advantages of simulation time and cost over EnKF. We propose ES with selective use of observation data in assimilation to improve history matching results and to keep simulation time short. Three methods, EnKF with all data, ES with all data, and the proposed, are applied to 2D synthetic channelized reservoirs with a nine spot waterflooding. Ensemble-based methods interpret the reason of reduced oil production rate after water breakthrough as low permeability. In this research, we suggest selective measurement data to manage misinterpretable data for the ensemble-based methods. As a logical choice, oil production rate before water breakthrough and water cut after water breakthrough are used for assimilation. EnKF with all data cannot predict true performances of oil and water productions on each well. ES with all data shows severe overshooting and filter divergence problems, which are two typical problems in the ensemble-based methods. However, the proposed method overcomes the two problems and shows good history matching results. It provides reliable uncertainty quantification of reservoir performances for both each well production and field total productions. Simulation cost of the proposed method is about 2.2% of that of EnKF, which uses 45 times update. It has clear advantage over EnKF or iterative ES methods.