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.