Data assimilation for nonlinear problems by ensemble Kalman filter with reparameterization

Abstract

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 implementations 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.