Research paper Data assimilation for nonlinear problems by ensemble Kalman filter with reparameterization Yan Chen a, ⁎, Dean S. Oliver b , Dongxiao Zhang c,d a Chevron ETC, Houston, TX 77002, USA b Mewbourne School of Petroleum and Geological Engineering, The University of Oklahoma, Norman, OK 73019, USA c Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, China d Department of Civil and Environmental Engineering, and Mork Family Department of Chemical Engineering and Material Sciences, University of Southern California, Los Angeles, California, USA abstract article info Article history: Received 13 July 2006 Accepted 1 December 2008 Keywords: data assimilation non-Gaussian effect ensemble Kalman filter reparameterization Owing to its simplicity and efficiency, the ensemble Kalman filter (EnKF) is being used to assimilate static and dynamic measurements to continuously update reservoir properties and responses. Many EnKF implementa- tions have shown promising results even when applied to multiphase flow history matching problems. A Gaussian density for model parameters and state variables is an implicit requirement for obtaining satisfactory estimates through the EnKF or its variants. The EnKF may not work properly when the relationship between model parameters, state variables, and observations are strongly nonlinear and the resulting joint probability distribution is non-Gaussian. For instance, near the displacement front of an immiscible flow, use of the EnKF to directly update saturation may lead to non-physical results. In this work, we address the non-Gaussian effect through a change in parameterization. Instead of directly updating the saturation, the time of saturation arrival (at a particular saturation) is included in the state vector. The time variable is correlated with the reservoir properties and other reservoir responses and its density is better approximated by a Gaussian distribution. After updating the time of saturation arrival through the EnKF, the updated arrival time distribution is transformed back to estimate the saturation of the reservoir. The new approach has better performance in the presence of strong non-Gaussianities but requires a larger computation time than does the traditional EnKF, which works well when the Gaussian assumption is not strongly violated. In order to achieve both accuracy and efficiency, the EnKF with reparameterization can be used in conjunction with the traditional EnKF as an option to account for possible highly non-Gaussian densities. The EnKF with reparameterization is illustrated with a problem under highly non-Gaussian conditions, and the effectiveness of the combination of the new approach and the traditional EnKF is demonstrated with history matching of multiphase flow in a heterogeneous reservoir. © 2008 Elsevier B.V. All rights reserved. 1. Background 1.1. Ensemble Kalman filter The ensemble Kalman filter (EnKF) is a Monte Carlo data assimilation method that is able to incorporate available observations sequentially in time. The probability distribution of a model state (including both model parameters and model responses) is repre- sented empirically by an ensemble of realizations. The model responses (state variables) are propagated forward in time based on the model dynamics. This is called the forecasting step. The ensemble of the model state is adjusted by incorporating available observations. This is called the updating step. EnKF addresses two important factors in traditional inverse problems: the uncertainty of the model state and the sensitivity of the model state to the model parameters. The propagation of the model state distribution and the gradient calcula- tion require large computational efforts. In the EnKF scheme, both the model statistics and the sensitivity can be obtained from the ensemble in a straightforward and computationally efficient manner. By approximating the probability density function by an ensemble of the model states, EnKF reduces the dimension of the inverse problem from the number of the unknown variables to the number of realizations. Some other dimension reduced parameter estimation methods share the similar idea (Pham et al., 1998; Zhang et al., 2007). Since the original work of Evensen (1994) and subsequent improve- ments (Burgers et al., 1998; Evensen, 2003, 2004), EnKF has been widely used in oceanography (Keppenne and Rienecker, 2002; Bertino et al., 2003), meteorology (Hamill et al., 2000; Houtekamer et al., 2005), hydrology (Reichle et al., 2002; Chen and Zhang, 2006), and petroleum engineering (Nævdal et al., 2005; Gu and Oliver, 2005; Journal of Petroleum Science and Engineering 66 (2009) 1–14 ⁎ Corresponding author. E-mail address: yan.chen@chevron.com (Y. Chen). 0920-4105/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2008.12.002 Contents lists available at ScienceDirect Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol