Assisted History Matching for Multi-facies Channelized Reservoir Using ES-MDA with Common Basis DCT

History matching a reservoir with multiple facies has always posed a great challenge to researchers. Most traditional history-matching techniques were designed to work with Gaussian distributed continuous variables instead of non-Gaussian distributed categorical variables like facies. Inspired by the previous researchers, we develop a workflow which combines Ensemble Smoother with Multiple Data Assimilation (ES-MDA) algorithm with common basis Discrete Cosine Transform (DCT) to conduct assisted history matching for multi-facies channelized reservoir, especially the 3D problems which has rarely been investigated before. In this work, an ensemble of geological realizations is first generated by using multi-point statistics (for 2D case) or object-based modeling (for 3D case). Then the DCT is implemented for each realization (facies field) to get the particular basis functions and their corresponding coefficients. For the purpose of extracting the general geological features among different realizations, we retain a series of common basis functions which are identical and fixed for all realizations. The corresponding coefficients of each realization are recomputed with respect to the common basis set in order to reconstruct the original facies field by minimizing the least square residual. Through history matching the observed data using ES-MDA, the DCT coefficients are updated and the facies field is renewed with the updated coefficients and the common basis set. The discrete facies field is obtained by applying an optimization algorithm to truncate the continuous values at the end of each ES-MDA iteration. We apply this procedure to both 2D and 3D synthetic problems considering complex three facies (shale, levee and sand) channelized reservoir. The results show that the proposed algorithm can provide good data matches and reduce the uncertainty in the prior ensemble significantly. Moreover, the posterior estimation of model parameters properly reflects the main geological features of the true model. Compared to previous studies, this work not only applies the ensemble-based method with common basis DCT for history matching and uncertainty quantification for 2D and 3D multi-facies reservoirs, but provides a robust and relatively easy approach to handle 3D cases which has very limited report in the literature.