Ensemble Kalman Filtering for State and Parameter Estimation on a Reservoir Model

Abstract

In reservoir management it is important with reservoir models that have good predictive abilities. Since the models initially are based on measurements with high uncertainties it is important to utilize new available data. Ensemble Kalman Filter (EnKF) is a new method for history matching that has received a lot of attention the last couple of years. This method is sequential and continuously update the reservoir model states (saturations, pressures etc.) and parameters (permeabilities, porosities etc) as data become available. The EnKF algorithm is derived and presented with a different notation, similar to that of the Kalman Filter (KF) used in control engineering. This algorithm is also verified on a simple linear example to illustrate that the covariance of the EnKF approaches that of the linear KF in case of an infinite ensemble size. In control theory this method falls under the category of parameter and state estimation of nonlinear large scale systems. Interesting aspects as observability and constraint handling arises, and these are linked to the EnKF and the reservoir case. To determine if the total problem is observable is a nearly impossible task, but one can learn a lot from introducing this concept. The EnKF algorithm was implemented on a simple shoe box reservoir model and four different problem initializations were tested. Although decent results were achieved from some of the simulations other failed completely. Some strange development in the ensemble when little information is available in the measurements was experienced and discussed. An outline was presented for a reservoir management scheme where EnKF is combined with Model Predictive Control (MPC). Some challenges was pointed out and these involve computation time, predictive ability, closed-loop behavior etc.