Summary
The ensemble Kalman filter (EnKF) and related ensemble-based smoothers are well-suited to history match reservoir models that are multivariate Gaussian. Estimating categorical variables such as facies types is much more difficult with the EnKF, especially when the variables have complex transitional dependencies. In a previous study, the EnKF was used for updating third-order Markov-chain models in one dimension with an efficient post-processing step to ensure that the posterior samples are constrained by the prior. The efficiency of the post-processing step depended on the use of an optimization algorithm (Viterbi algorithm) that is not directly applicable in higher dimensions.
In this paper, the post-processing step is carried out with a sequential noniterative optimization algorithm that readily extends to higher dimensions. An iterative ensemble-based data-assimilation method by use of Levenberg-Marquardt (LM) regularization—namely, LM ensemble randomized maximum likelihood (LM-EnRML)—is used to update reservoir properties to honor production data disregarding the categorical feature of the facies model. The ensemble of realizations updated with LM-EnRML is used to approximate the likelihood of the model variables given data, and bivariate transition probability functions are used to represent the joint probability of the prior facies model. At the post-processing step, the facies type at each gridblock is determined sequentially to maximize the posterior probability, given the approximate prior and likelihood. We demonstrate the approach by conditioning three binary-facies models with channel structure and a three-facies model of sand dunes to nonlinear observations. Our results show the updated facies models honor production data very well, and the transitions among facies are consistent with the prior model.
An alternating optimization algorithm for two-channel factor analysis with common and uncommon factors