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Abstract An ensemble-based technique has been developed and successfully applied to simultaneously estimate the relative permeability and capillary pressure in a tight formation by history matching the conventional measurement data from displacement experiments. Relative permeability and capillary pressure curves are represented by the power-law model. Then, the to-be-estimated coefficients of the power-law model are tuned automatically and finally determined once the measurement data have been assimilated completely and history matched. This new technique has been validated by a synthetic coreflooding experiment and then extended to a real coreflooding experiment. Simultaneous estimation of relative permeability and capillary pressure has been found to improve, while standard deviation of the estimated coefficients is reduced gradually as more measurement data is assimilated. There exists an excellent agreement between both the updated relative permeability and capillary pressure and their corresponding reference values, once all the measurement data are assimilated. The relative permeability can be determined more accurately than the capillary pressure owing to the fact that it is more sensitive to the conventional measurement data. This newly developed technique has good computation efficiency and is suitable for performing uncertainty analysis under the same framework.
Abstract The relative permeability is a crucial parameter for accurately evaluating reservoir performance. Two-phase relative permeability curves are normally obtained by either directly or indirectly interpreting the displacement experiment data. As for the direct interpretation, the core samples are assumed to be homogeneous, while the capillary forces are normally neglected. Although the indirect interpreting approach is able to take heterogeneity of the core sample into account, calculating the derivatives of the objective functions through the graphical or numerical methods is prone to considerable errors. In this paper, a new method is developed to calculate the absolute and relative permeability from unsteady-state, two-phase immiscible displacement experiments. The permeability data is determined by history matching the experimentally observed pressure drop, production data and water saturation profiles via the ensemble Kalman filter (EnKF) algorithm. The power-law model is utilized to represent the relative permeability. Both the absolute and relative permeability are calculated simultaneously by assimilating the observed data. The newly developed method is validated using a numerical coreflooding experiment. It has been found that estimations of absolute and relative permeability are improved progressively as more observation data are assimilated. In addition, this method is convenient to be implemented as the derivative of the objective function is not required.
Abstract Because of its ease of implementation, computational efficiency and the fact that it generates multiple history-matched models, which conceptually allows one to characterize the uncertainty in reservoir description and future performance predictions, the ensemble Kalman filter (EnKF) provides a highly attractive technique for history matching production data. In this work, we apply EnKF with a recently proposed method of covariance localization to history-match production data from a real field to generate multiple realizations of the permeability field. A single manually history-matched model is available for comparisons. Only 7.6 years of the 10 years of history were matched with the remaining 2.4 years of history used to assess the predictive capability of the history-matched models. For this field case, covariance localization was necessary to avoid the propagation of spurious correlations and loss of variance and also resulted in better data matches and predictions than were obtained with EnKF without localization. EnKF with covariance localization also gave better data matches, more accurate "future" predictions and far more geologically realistic models than were obtained by manually matching production data. We also present results obtained using half-iteration EnKF (HI-EnKF) with covariance localization. For this field case, HI-EnKF gave a significant further improvement in the data match and predictions. However, as HI-EnKF requires rerunning the ensemble from time zero at every data assimilation step, it leads to a considerable increase in the computational time. The results for this field case indicate that we can reduce the computational cost of HI-EnKF, without compromising the quality of the results, by rerunning the ensemble from time zero only when "large" changes in the state vector occur.
Abstract In this paper, a modified ensemble randomized maximum likelihood (EnRML) algorithm has been developed to estimate three-phase relative permeabilities with consideration of hysteresis effect by reproducing the actualproduction data. Ensemble-based history matching uses an ensemble of realizations to construct Monte Carlo approximations of the mean and covariance of the model variables, which can acquire the gradient information from the correlation provided by the ensemble. A power-law model is firstly utilized to represent the three-phase relative permeabilities whose coefficients can be automatically adjusteduntil production history is matched. A damping factor is introduced as an adjustment to the step length since a reduced step length is commonly required if an inverse problem is sufficiently nonlinear. Arecursive approach for determining the damping factor has been developed to reduce the number of iterations and the computational loadof the EnRML algorithm. The restart of reservoir simulations for reducing the cost of reservoir simulations is of significant importance for the EnRML algorithm where iterations are inevitable. By comparing a direct-restart methodand an indirect-restart method for numerical simulations, we optimize the restart method used for a specific problem. Subsequently, we validate the proposed methodologyby using a synthetic water-alternating-gas (WAG) displacement experiment and then extend it to match laboratory experiments. The proposed technique has proved toefficientlydetermine the three-phase relative permeabilities for the WAG processes with consideration of the hysteresis effect, while history matching results are gradually improved as more production data are taken into account. The synthetic scenarios demonstrate that the recursive approach saves 33.7% of the computational expense compared to the trial-and-error method when the maximum iteration is 14. Also, the consistency between the production data and model variables has been well maintained during the updating processes by using the direct-restart method, whereas the indirect-restart method fails to minimize the uncertainties associated with the model variables representing three-phase relative permeabilities.
Zhang, Yin (University of Alaska Fairbanks) | Yang, Daoyong (Chenglin Hi-Tech Industry Co., Ltd. and University of Regina) | Li, Heng (Computer Modelling Group Ltd.) | Patil, Shirish (University of Alaska Fairbanks)
Abstract A damped iterative ensemble Kalman filter (IEnKF) algorithm has been proposed to estimate relative permeability and capillary pressure curves simultaneously for the PUNQ-S3 model, while its performance has been compared with that of the CMOST module. The power-law model is employed to represent the relative permeability and capillary pressure curves, while three-phase relative permeability for oil phase is determined by using the modified Stone II model. By assimilating the observed production data, the relative permeability and capillary pressure curves are inversely, automatically, and successively updated, achieving an excellent agreement with the reference cases. Not only are the associated uncertainties reduced significantly during the updating process, but also each of the updated reservoir models predicts the production profile that is in a good agreement with the reference cases. Although both the damped IEnKF and CMOST generate similar history matching results and prediction performance, the estimation accuracy of the damped IEnKF method developed in this study is generally much better than that of the CMOST. Besides, the variations in the ensemble of the updated reservoir models and production profiles of the damped IEnKF provide a robust and consistent framework for uncertainty analysis.