Generating Facies Maps by Assimilating Production Data and Seismic Data with the Ensemble Kalman Filter
Authors: Zhao Yong , Reynolds Albert C. , Li Gaoming 
Published: April 2008

Abstract
Truncated pluri-Gaussian models have proven to be useful for generating realistic geological models of facies distributions. In this approach, truncation maps are applied to the underlying Gaussian random fields to generate the facies distribution on a reservoir simulator gridblock basis. Here, we investigate methods to condition such a facies map to production and seismic data by the sequential assimilation of data using the ensemble Kalman filter (EnKF). Although EnKF is a highly efficient data assimilation scheme, our experience is that it can be difficult to match watercut data with EnKF, and for highly nonlinear/non-Gaussian problems, EnKF may provide only a very approximate quantification of uncertainty. Moreover, EnKF is designed to generate an approximation to the conditional mean, but for a pluri-Gaussian model, the mean does not represent a plausible geological model, so it is important that the ensemble gives a reasonable sampling of the conditional probability density function. For two and three-dimensional pluri-Gaussian models, we present a new procedure to ensure that facies observations at wells are honored at each data assimilation step. The difficulty in matching water cut data is more clearly seen for the channel model considered where the EnKF update of water saturation is inconsistent with the updated location of the channel system. The result is that in the updated state vector, much of the injected water is moved into gridblocks occupied by the low permeability facies.
As the erroneous saturation distribution obtained with EnKF can only result from nonlinearity or the failure of the assumption that the ensemble of predictions is approximately Gaussian, we investigate the application of a global and local normal score transform to transform water saturation to a Gaussian variable before applying the EnKF analysis step. We also consider matching breakthrough time directly before matching watercut data as well and also apply an iterative EnKF scheme to obtain more plausible saturations distributions.
For the first time, we also show results on applying EnKF to generate realizations of both the distribution of facies and the permeability and porosity within each facies.
The integration of seismic data poses problems because of the large number of data that are assimilated. With a global assimilation procedure based on subspace projection, filter divergence becomes severe. On the other hand, our implementation of a local updating method to reduce filter divergence results in an unrealistic rough facies map. We introduce a projection method to obtain a more realistic map of the distribution of facies, which retain the inherent smoothness of the underlying geological model.
After assimilating data with EnKF, it is common to rerun the ensemble of models from time zero when predicting future reservoir performance and assessing its uncertainty. Our results indicate that more accurate predictions of future reservoir performance will be obtained by simply making predictions forward from the last data assimilation time.