Assessing the performance of the ensemble Kalman filter for subsurface flow data integration under variogram uncertainty

[1] The ensemble Kalman filter (EnKF) has recently been proposed as a promising parameter estimation approach for constraining the description of rock flow properties, such as permeability and porosity, to reproduce flow measurements that are modeled as nonlinear functions of these properties. One of the key factors that strongly affect the performance of the EnKF is the quality or representativeness of the prior ensemble of property fields used to initialize the EnKF assimilation procedure. The initial ensemble is commonly constructed by assuming a known geological continuity model such as a variogram. However, geologic continuity models are derived from incomplete information and imperfect modeling assumptions, which can introduce a significant level of uncertainty into the produced models. Neglecting this important source of uncertainty can lead to systematic errors and questionable estimation results. In this paper, we investigate the performance of the EnKF under varying levels of uncertainty in the variogram model parameters. We first attempt to directly estimate variogram model parameters from flow data and show that the complex and nonunique relation they have with the flow data provides little sensitivity for an effective inversion with the EnKF. We then assess the performance of the EnKF for estimation of permeability values under uncertain and incorrect initial variogram parameters and show that any bias in specifying variogram parameters tends to persist throughout the EnKF analysis even though locally reasonable permeability updates may be obtained near observation points. More importantly, we show that when variogram parameters are specified probabilistically to account for the full range of structural variability in the initial permeability ensemble, the EnKF update results are quite promising. The results suggest that under uncertain geologic continuity, the EnKF tends to perform better if a very diverse set of property fields is used to form the initial ensemble than when a deterministic and potentially erroneous variogram model is used. Therefore, in applying the EnKF to model calibration problems, it is preferable to overestimate the uncertainty in geologic continuity and to initialize the EnKF procedure with a wide range of variability in property description than to overlook the variogram uncertainty at the risk of introducing systematic bias that cannot be corrected by the EnKF updates. The practical implications of the results in this paper are significant for designing the EnKF for realistic ensemble model calibration problems where the level of uncertainty in the initial ensemble is usually not known a priori.

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