This paper compares two ensemble-based data-assimilation methods when solving the history-matching problem in reservoir-simulation models. The methods are the Ensemble Kalman Filter (EnKF) and the Ensemble Smoother (ES). Several publications have discussed the use of EnKF in petroleum applications while ES is now used for the first time for history matching. ES differs from EnKF by computing a global update in the space-time domain, rather than using recursive updates in time as in EnKF. Thus, the sequential updating of the realizations with associated restarts is avoided. EnKF and ES provide identical solutions for state estimation with linear dynamical models. However, for nonlinear dynamical models, and in particular models with chaotic dynamics, EnKF is superior to ES, due to the fact that the recursive updates keep the model on track and close to the true solution. Thus, ES is not much used and EnKF has been the method of choice in most data assimilation studies where ensemble methods are used. On the other hand, reservoir simulation models are rather diffusive systems when compared to the chaotic dynamical models that were previously used to test ES. If we can assume that the model solution is stable with respect to small perturbations in the initial conditions and the history-matching parameters, then ES should give similar results to EnKF, and ES will be a more efficient and much simpler method to implement and apply. The technical advantages of using ES compared to EnKF are severe, especially when the methods are applied with complex real reservoir models. ES provides a significant reduction in simulation time. Furthermore, a more flexible parameterization is possible, which makes it easier to handle structural and geological model parameters in the history-matching process. In this paper we compare EnKF and ES and show that ES indeed provide for an efficient ensemble-based method for history matching.

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.

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 to a rock physics and seismic modelling software. Both static variables (porosity, permeability, rock physic parameters, etc.) 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.[1] For a presentation of this method in a probabilistic, linear least-squares setting, see e.g., Tarantola.[2] However, this method is not suitable for non-linear models, and the ensemble Kalman filter (EnKF) method was introduced in 1994 by Geir Evensen for updating non-linear ocean models.[3] It may also be applied to a combined state and parameter estimation problem.[4] Several recent investigations have shown the potential of the EnKF for continuous updating of reservoir simulation models, as an alternative to traditional history-matching.[5–10] 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.[11] 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. Special challenges are 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,[4,12] and Evensen and van Leeuwen.[13] 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.

This paper demonstrates the potential and advantages of the Ensemble Kalman filter (EnKF) as a tool for assisted history matching, based on its sequential processing of measurements, its capability of handling large parameter sets, and on the fact that it solves the combined state and parameter estimation problem. A method and a thorough workflow for updating reservoir simulation models using the EnKF is developed. In addition, we present a method for updating relative permeability curves, as well as an improved approach for updating fault transmissibility multipliers. The proposed workflow has been applied on a complex North Sea oil field. The EnKF successfully provides an ensemble of history matching reservoir models. A significant improvement in the history match is obtained by updating the relative permeability properties in addition to porosity and permeability fields and initial fluid contacts. Fault multipliers are estimated, and it is shown how the use of transformations, which handles non-Gaussian model variables, makes it possible to determine if a fault is open, closed, or partially closed with respect to flow. The presented method is an innovative contribution to reservoir management workflows, which show growing interest in real time applications and fast model updating. Sequential data assimilation provides an updated reservoir model conditioned on the most recent production data. The updated ensemble is used to predict the uncertainty in future production and it is demonstrated that the EnKF leads to improved predictions with reduced uncertainty. Introduction Reservoir modelling and history matching aim to deliver integrated reservoir models for reservoir management purposes. These reservoir models must not only reproduce the historical field performances, they must also be consistent with all available static data (such as core data, well logs, seismic data) and dynamic data (such as well production data, tracer concentration, 4D seismic data). Furthermore they should integrate the most current information about the reservoir and the associated uncertainty to allow for real-time decisions. During the last years, high drilling activity, use of permanent sensors for monitoring pressure and flow rates and developments in 4D seismic monitoring, have considerably increased the data output frequency of producing fields. Thus growing need for fast and continuous model updating calls for alternative solutions to traditional history matching methods, which are often unacceptably time-consuming. Sequential data assimilation appears ideally-suited for addressing the new challenges within reservoir management.

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