Use of Phase Streamlines for Covariance Localization in Ensemble Kalman Filter for Three-Phase History Matching
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Ensemble Kalman Filter (EnKF) has gained increased popularity for history matching and continuous reservoir model updating. It is a sequential Monte-Carlo approach that works with an ensemble of reservoir models. Specifically, the method utilizes cross-covariance between measurements and model parameters estimated from the ensemble. For practical field applications, the ensemble size needs to be kept small for computational efficiency. However, this leads to poor approximations of the cross-covariance and can cause loss of geologic realism from unrealistic model updates outside the region of the data influence and/or loss of variance leading to ensemble collapse. A common approach to remedy the situation is through covariance localization to limit the influence of the data. In this paper, we show that for three phase flow conditions, the region of covariance localization strongly depends on the underlying flow dynamics and also, on the particular data type that is being assimilated viz. water-cut or gas-oil ratio. This makes the traditional distance-based localizations sub-optimal and often, ineffective. Instead, we propose the use of water and gas phase streamlines as a means for covariance localization for water-cut and GOR data assimilation. The phase streamlines can be computed based on individual phase velocities which are readily available after flow simulation. Unlike the total streamlines, phase streamlines can be discontinuous. We show that the discontinuities in water phase and gas phase streamlines naturally define the region of influence for water-cut and GOR data and provide a flow-relevant covariance localization during EnKF updating. We first demonstrate the validity of the proposed localization approach using a waterflood example in a quarter five-spot pattern. Specifically, we compare the phase streamline trajectories with cross-covariance maps computed using ensemble size of 2000 for both water-cut and GOR data. The results show a close correspondence between the time evolution of phase streamlines and the cross-covariance maps of water-cut and GOR data, a small-size industrial reservoir engineering PUNQ-S3 model application shows that our proposed localization outperforms distance-based localization method. The updated models show improved forecasts while preserving geological realism. Copyright 2011, Society of Petroleum Engineers.
author list (cited authors)
Watanabe, S., & Datta-Gupta, A.