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.

Summary: The use of ensemble Kalman filter techniques for continuous updating of a reservoir model is demonstrated. The ensemble Kalman filter technique is introduced, and thereafter applied to a simplified 2-D field model, which are generated by using a single horizontal layer from a North Sea field model. By as similating measured production data, the reservoir model is continuously updated. The updated models give improved forecasts and the forecasts improve as more data is included. Both dynamic variables, such as pressure and saturations, and static variables, such as the permeability, are updated in the reservoir model.

The nonlinear filtering problem occurs in many scientific areas. Sequential Monte Carlo solutions with the correct asymptotic behavior such as particle filters exist, but they are computationally too expensive when working with high-dimensional systems. The ensemble Kalman filter (EnKF) is a more robust method that has shown promising results with a small sample size, but the samples are not guaranteed to come from the true posterior distribution. By approximating the model error with a Gaussian distribution, one may represent the posterior distribution as a sum of Gaussian kernels. The resulting Gaussian mixture filter has the advantage of both a local Kalman type correction and the weighting/resampling step of a particle filter. The Gaussian mixture approximation relies on a bandwidth parameter which often has to be kept quite large in order to avoid a weight collapse in high dimensions. As a result, the Kalman correction is too large to capture highly non-Gaussian posterior distributions. In this paper, we have extended the Gaussian mixture filter (Hoteit et al., Mon Weather Rev 136:317–334, 2008) and also made the connection to particle filters more transparent. In particular, we introduce a tuning parameter for the importance weights. In the last part of the paper, we have performed a simulation experiment with the Lorenz40 model where our method has been compared to the EnKF and a full implementation of a particle filter. The results clearly indicate that the new method has advantages compared to the standard EnKF.

Summary: The focus of this work is on an alternative implementation of the iterative-ensemble smoother (iES). We show that iteration formulae similar to those used by Chen and Oliver (2013) and Emerick and Reynolds (2012) can be derived by adopting a regularized Levenberg-Marquardt (RLM) algorithm (Jin 2010) to approximately solve a minimum-average-cost (MAC) problem. This not only leads to an alternative theoretical tool in understanding and analyzing the behavior of the aforementioned iES, but also provides insights and guidelines for further developments of the smoothing algorithms. For illustration, we compare the performance of an implementation of the RLM-MAC algorithm with that of the approximate iES used by Chen and Oliver (2013) in three numerical examples: an initial condition estimation problem in a strongly nonlinear system, a facies estimation problem in a 2D reservoir, and the history-matching problem in the Brugge field case. In these three specific cases, the RLM-MAC algorithm exhibits comparable or better performance, especially in the strongly nonlinear system.

The ensemble Kalman filter is a Monte Carlo method for state estimation of nonlinear models, developed as an alternative or improvement of the extended Kalman filter. In this technical note we introduce an iterative extension to the ensemble Kalman filter. Iterations are introduced to improve the estimates in the cases where the relationship between the model and observations is not linear. The iterations converge, but to a solution where the data are overfitted. An essential stopping criteria is therefore introduced for the proposed method.