The Ensemble Kalman Filter in Reservoir Engineering--a Review

Introduction and Background There has been great progress in data assimilation within atmospheric and oceanographic sciences during the last couple of decades. In data assimilation, one aims at merging the information from observations into a numerical model, typically of a geophysical system. A typical example where data assimilation is needed is in weather forecasting. Here, the atmospheric models must take into account the most recent observations of variables such as temperature and atmospheric pressure for better forecasting of the weather in the next time period. A major challenge for these models is that they contain very large numbers of variables. The progress in data assimilation is because of both increased computational power and the introduction of techniques that are capable of handling large amounts of data and more severe nonlinearities. The aim of this paper is to focus on one of these techniques, the ensemble Kalman filter (EnKF). The EnKF has been introduced to petroleum science recently (Lorentzen et al. 2001a) and, in particular, has attracted attention as a promising method for solving the history matching problem. The literature available on the EnKF is now rather overwhelming. We hope that this review will help researchers (and students) working on adapting the EnKF to petroleum applications to find valuable references and ideas, although the number of papers discussing the EnKF is too large to give a complete review. For practitioners, we have cited critical EnKF papers from weather and oceanography. We have also tried to review most of the papers dealing with the EnKF and updating of reservoir models available to the authors by the beginning of 2008. The EnKF is based on the simpler Kalman filter (Kalman 1960). We will start by introducing the Kalman filter. The Kalman filter is an efficient recursive filter that estimates the state of a linear dynamical system from a series of noisy measurements. The Kalman filter is based on a model equation, where the current state of the system is associated with an uncertainty (expressed by a covariance matrix) and an observation equation that relates a linear combination of the states to measurements. The measurements are also associated with uncertainty. The model equations are used to compute a forward step (Eqs. 1 and 2) where the state variables are computed forward in time with the current estimate of the state as initial condition. The observation equations are used in the analysis step (Eqs. 3 through 5) where the estimated value of the state and its uncertainty are corrected to take into account the most recent measurements See, e.g., Cohn (1997), Maybeck (1979), or Stengel (1994) for an introduction to the Kalman filter.