In this work, an approach for estimating non-Gaussian permeability field is developed, through a probabilistic collocation based Kalman filter (PCKF). In this approach, the polynomial chaos expansion is used to parameterize the non-Gaussian permeability field. The state variables are expressed by the polynomial chaos expansion whose coefficients are sequentially updated when observations are available. The probabilistic collocation method is employed to solve for the coefficients of the polynomial chaos expansion. The probabilistic collocation method is non-intrusive to the reservoir model, thus it allows the forward simulations to be performed independently with existing reservoir simulators, as in the Monte Carlo simulation.
The applicability of the approach is demonstrated with black oil problems in spatially heterogeneous reservoirs. Continuous production data are used as observations for sequentially updating the permeability field. The PCKF approach is also compared with the ensemble Kalman filter (EnKF) for investigating the accuracy, efficiency and applicability. It reveals that the PCKF other than the EnKF can be used to efficiently estimate the non-Gaussian permeability field. While keeping the advantages of EnKF, such as the sequential updating and parallelism, the PCKF is more suitable for permeability fields characterized by non-Gaussian random fields.