Updating Markov chain models using the ensemble Kalman filter

The problem we address is how to condition discrete variables with a prior described by a Markov chain model to a set of continuous, nonlocal observations. This is motivated by the need for improved approaches to condition geological facies descriptions of a reservoir to production data. In this study, we assume that the spatial arrangement of facies can be modeled by a Markov chain model. The conditioning to nonlocal observations consists of two primary steps. First, the ensemble Kalman filter is used to assimilate data without regard for the discrete nature of the variables, or the probability of transition from one state to another. Second, the updated realizations are projected onto the discrete state space using an algorithm for finding the sequence of states with maximum probability. The method, which is based on the Viterbi algorithm, is not iterative and is capable of assimilating nonlocal data such as production data with some limitations. We demonstrate the application of the method first with an example in which the data are nonlocal but linear. The second example is a nonlinear fluid flow example in which data are assimilated sequentially. For the linear problem, the distribution of conditional realizations from the approximate algorithm was found to be indistinguishable from the distribution of realizations from an exact sampling algorithm (McMC). Finally, we discuss how to generalize the methodology from the 1D example presented here to more general cases.