Summary
A method based on the ensemble Kalman filter (EnKF) for continuous model updating with respect to the combination of production data and 4D seismic data is presented. When the seismic data are given as a difference between two surveys, a combination of the ensemble Kalman filter and the ensemble Kalman smoother has to be applied. Also, special care has to be taken because of the large amount of data assimilated. Still, the method is completely recursive, with little additional cost compared to the traditional EnKF. The model system consists of a commercial reservoir simulator coupled with a rock physics and seismic modeling software. Both static variables (porosity, permeability, and rock physic parameters) and dynamic variables (saturations and pressures) may be updated continuously with time based on the information contained in the assimilated measurements. The method is applied to a synthetic model and a real field case from the North Sea. In both cases, the 4D seismic data are different variations of inverted seismic. For the synthetic case, it is shown that the introduction of seismic data gives a much better estimate of reservoir permeability. For the field case, the introduction of seismic data gives a very different permeability field than using only production data, while retaining the production match.
Introduction
The Kalman filter was originally developed to update the states of linear systems (Kalman 1960). For a presentation of this method in a probabilistic, linear least-squares setting, see Tarantola (2005). However, this method is not suitable for nonlinear models, and the ensemble Kalman filter (EnKF) method was introduced in 1994 by Geir Evensen for updating nonlinear ocean models (Evensen 1994). The method may also be applied to a combined state and parameter estimation problem (Evensen 2006; Lorentzen 2001; Anderson 1998). Several recent investigations have shown the potential of the EnKF for continuous updating of reservoir simulation models, as an alternative to traditional history matching (Nævdal et al. 2002a, b; Nævdal et al. 2005; Gu and Oliver 2004; Gao and Reynolds 2005; Wen and Chen 2005). The EnKF method is a Monte Carlo type sequential Bayesian inversion, and provides an approximate solution to the combined parameter and state-estimation problem. The result is an ensemble of solutions approximating the posterior probability density function for the model input parameters (e.g., permeability and porosity), state variables (pressures and saturations), and other output data (e.g., well production history) conditioned to measured, dynamic data.
Conditioning reservoir simulation models to seismic data is a difficult task (Gosselin et al. 2003). In this paper, we show how the ensemble Kalman filter method can be used to update a combined reservoir simulation/seismic model using the combination of production data and inverted 4D seismic data. There are special challenges involved in the assimilation of the large amount of data available with 4D seismic, and the present work is based on the work presented by Evensen (2006, 2004) and Evensen and van Leeuwen (2000). In the following, the combined state and parameter estimation problem is described in a Bayesian framework, and it is shown how this problem is solved using the EnKF method, with emphasis on the application to 4D seismic data. When the seismic data are given as a difference between two surveys, a combination of the ensemble Kalman filter and the ensemble Kalman smoother has to be applied. Special challenges involved when the amount of data is very large are discussed. The validity of the method is examined using a synthetic model, and finally, a real case from the North Sea is presented.
Technical Note: Comparison of ensemble Kalman filter and variational approaches for CO<sub>2</sub> data assimilation