Abstract
The ensemble smoother with multiple data assimilation (ES-MDA) has been shown to outperform EnKF for both synthetic and field problems. Specifically, ES-MDA gives better data matches than EnKF, maintains the correct geology and appears to provide a better quantification of uncertainty than EnKF. However, if ES-MDA (or EnKF) is applied to update the rock-property fields where the underlying geological model is a facies model, then the boundaries between facies are not preserved. Here, we couple an ES-MDA update of the permeability field with an update of the distribution of facies for cases where both the distribution of geological facies and the distribution of permeability within each facies are unknown. To do this, the permeability field is represented as a Gaussian mixture model (GMM), where the permeability within each facies is represented by a different Gaussian probability distribution. ES-MDA is applied to update the permeability field in the normal way, but after each ES-MDA iteration, the facies value in each reservoir simulator gridblock is updated by calculating the probability of each possible facies with respect to the GMM. In addition, the updated permeability distributions for each facies are remapped to the original Gaussian distribution. To keep the facies distribution consistent with the underlying geological model, which in this work is based on multi-point statistics (MPS), every several ES-MDA iterations, the facies distribution is regenerated by using the facies probability map as soft data and by using certain permeability values as the hard data to avoid destroying the data match. For the example considered in this paper, the procedure is able to provide good data matches as well as posterior facies maps and permeability fields that reflect the main geological features of the true model.
