The ensemble Kalman filter (EnKF) has recently received significant attention as a robust and efficient tool for data assimilation. Although the EnKF has many advantages such as ease of implementation and efficient uncertainty quantification, it is technically appropriate only for random fields (e.g., permeability) characterized by two-point geostatistics (multi-Gaussian random fields). Realistic systems however are much better described by multi-point geostatistics (non-Gaussian random fields), which is capable of representing key geological structures such as channels. Furthermore, the updating step in the EnKF can lead to non-physical updates of model states and other variables (such as saturations), and this problem is evident in highly nonlinear problems like compositional simulation.

In a recent paper (Sarma and Chen, 2009), we formulated a generalized EnKF using kernels (KEnKF), capable of representing non-Gaussian random fields characterized by multi-point geostatistics. In this work, we further extend the KEnKF to efficiently handle constraints on the state and other variables of the simulation model, arising from physical or other reasons. In the standard EnKF, the usual approach to handle constraints is through truncation or variable transformation, which has been shown to be problematic for highly nonlinear problems. In the KEnKF, because the Kalman update equation is solved as a minimization problem, constraints can be easily and rigorously handled via solution of a constrained nonlinear programming problem. We propose a combination of fixed point iteration and the augmented Lagrangian method to solve this problem efficiently. Note that with a kernel of order 1, the KEnKF is equivalent to the standard EnKF, and therefore, the proposed approach to handle constraints is applicable even if high order kernels are not used. The procedure is demonstrated on an example case, and is shown to better handle various state constraints compared to the standard EnKF with truncation.

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