Data Assimilation Using the EnKF for 2-D Markov Chain Models

The ensemble Kalman filter (EnKF) is well-suited to update gaussian variables and can be used for updating continuous nongaussian variables either directly or after transformation. Categorical variables such as facies type are much more difficult for history matching, especially when the variables have complex transitional dependencies. In a previous paper we described a method for updating third order Markov chain models in one dimension using the ENKF, where its efficiency partially depends on the Viterbi algorithm that is not directly applicable in higher dimensions. In this paper, we develop a data assimilation method for updating categorical models using an approximation to the joint probability of facies types (Allard et al 2011) that can be used in a sequential algorithm without iteration. The ensemble of realizations after updating can be used to efficiently approximate the likelihood of the variables, while the categorical model provides an approximation to the transition probabilities. We demonstrate the approach with conditioning two synthetic channel models with two facies types to both linear and nonlinear observations. Our results show the distribution of facies after data assimilation honors data much better than before assimilation, and the transitions among facies are consistent with the prior model.