Using the Ensemble Kalman Filter with Kernel Methods for History Matching of Petroleum Reservoirs
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The problem of conditioning a petroleum reservoir model to production data, so called history matching, is considered. The history matching problem is formulated within a Bayesian framework, where prior geological knowledge, well logs and seismic surveys are consistently honered. The resulting posterior distribution is in general analytically untractable, and Monte Carlo methods must be applied to infer about it's properties. To this end, the ensemble Kalman filter (EnKF) is used to generate a sample, or ensemble, of reservoir models conditioned to all available data. The EnKF is derived under the assumption of Gaussian forecast distributions, which makes estimation of non-Gaussian variables such as multi-modal porosity fields difficult. To overcome this problem, we propose to modify the EnKF with a kernel function. By introduction of a kernel, the reservoir state is implicitly mapped to a high-dimensional feature space where the Gaussian assumption might be more plausible, and as a result have a favorable effect on the EnKF performance. The proposed method have been applied to a synthetic case and a North sea field case, both with a channel description in the prior, which makes the porosity and permeability fields multi-modal. It is shown that the introduction of a kernel does not break the EnKF. However, it is also shown that introduction of a kernel is not sufficient to capture the true underlying uncertainty in a channelized reservoir, and alternative approaches should also be considered.