Estimation of Three-Phase Relative Permeabilities for a Water-Alternating-Gas Process by Use of an Improved Ensemble Randomized Maximum-Likelihood Algorithm
- Zhaoqi Fan (University of Regina) | Yin Zhang (University of Regina) | Daoyong (Tony) Yang (University of Regina)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- October 2016
- Document Type
- Journal Paper
- 683 - 693
- 2016.Society of Petroleum Engineers
- Water-alternating-gas process, Three-phase relative permeability, Restart method, Ensemble-based algorithm
- 5 in the last 30 days
- 367 since 2007
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In this paper, a modified ensemble randomized maximum-likelihood (EnRML) algorithm has been developed to estimate three phase relative permeabilities with consideration of the hysteresis effect by reproducing the actual production 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 first used to represent the three-phase relative permeabilities, the coefficients of which can be automatically adjusted until production history is matched. A damping factor is introduced as an adjustment to the step length because a reduced step length is commonly required if an inverse problem is sufficiently nonlinear. A recursive approach for determining the damping factor has been developed to reduce the number of iterations and the computational load of 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 method and an indirect restart method for numerical simulations, we optimize the restart method used for a specific problem. Subsequently, we validate the proposed methodology by use of a synthetic water-alternating-gas (WAG) displacement experiment and then extend it to match laboratory experiments. The proposed technique has proved to efficiently determine the three-phase relative permeabilities for the WAG processes with consideration of the hysteresis effect, whereas 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 with 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 use of the direct-restart method, whereas the indirect-restart method fails to minimize the uncertainties associated with the model variables representing three phase relative permeabilities.
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Aanonsen, S. I., Nævdal, G., Oliver, D. S. et al. 2009. The Ensemble Kalman Filter in Reservoir Engineering-A Review. SPE J. 14 (3): 393–412. SPE-117274-PA. http://dx.doi.org/10.2118/117274-PA.
Anderson, J. L. and Anderson, S. L. 1999. A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts. Mon. Weather Rev. 127 (12): 2741–2758. http://dx.doi.org/10.1175/1520-0493(1999)127%3C2741:AMCIOT%3E2.0.CO;2.
Baker, L. E. 1988. Three-Phase Relative Permeability Correlations. Presented at the SPE Enhanced Oil Recovery Symposium, Tulsa, 16–21 April. SPE-17369-MS. http://dx.doi.org/10.2118/17369-MS.
Beygi, M. R., Delshad, M., Pudugramam, V. S. et al. 2014. Novel Three-Phase Compositional Relative Permeability and Three-Phase Hysteresis Models. SPE J. 20 (1): 21–34. SPE-165324-PA. http://dx.doi.org/10.2118/165324-PA.
Burgers, G., van Leeuwen, P. and Evensen, G. 1998. Analysis Scheme in the Ensemble Kalman Filter. Mon. Weather Rev. 126 (6): 1719–1724. http://dx.doi.org/10.1175/1520-0493(1998)126%3C1719:ASITEK%3E2.0.CO;2.
Chen, S., Li, H., Yang, D. et al. 2010. Optimal Parametric Design for Water-Alternating-Gas (WAG) Process in A CO2-Miscible Flooding Reservoir. J Can Pet Technol 49 (10): 75–82. SPE-141650-PA. http://dx.doi.org/10.2118/141650-PA.
Chen, Y. and Oliver, D. S. 2012. Ensemble Randomized Maximum Likelihood Method as An Iterative Ensemble Smoother. Math. Geosci. 44 (1): 1–26. http://dx.doi.org/10.1007/s11004-011-9376-z.
Chen, Y. and Zhang, D. 2006. Data Assimilation for Transient Flow in Geologic Formations via Ensemble Kalman Filter. Adv. Water Resour. 29 (8): 1107–1122. http://dx.doi.org/10.1016/j.advwatres.2005.09.007.
Chen, Y., Oliver, D. S. and Zhang, D. 2009. Efficient Ensemble-Based Closed-Loop Production Optimization. SPE J. 14 (4): 634–645. SPE-112873-PA. http://dx.doi.org/10.2118/112873-PA.
Christensen, J. R., Stenby, E. H. and Skauge, A. 2001. Review of WAG Field Experience. SPE Res Eval & Eng 4 (2): 97–106. SPE-71203-PA. http://dx.doi.org/10.2118/71203-PA.
Computer Modelling Group (CMG). 2013. IMEX Three-Phase, Black-Oil Reservoir Simulator. Calgary: Computer Modelling Group.
Evensen, G. 1994. Sequential Data Assimilation with a Nonlinear Quasi-Geostrophic Model Using Monte Carlo Methods to Forecast Error Statistics. J. Geophys. Res. 99 (C5): 10143–10162. http://dx.doi.org/10.1029/94JC00572.
Gang, T. 2006. Data Integration and Model Improvement for Naturally Fractured Reservoirs. PhD dissertation, University of Tulsa, Tulsa.
Gu, Y. 2006. History Matching Production Data Using the Ensemble Kalman Filter. PhD dissertation, University of Oklahoma, Norman, Oklahoma
Gu, Y. and Oliver, D. S. 2007. An Iterative Ensemble Kalman Filter for Multiphase Fluid Flow Data Assimilation. SPE J. 12 (4): 438–446. SPE-108438-PA. http://dx.doi.org/10.2118/108438-PA.
Houtekamer, P. L. and Mitchell, H. L. 1998. Data Assimilation Using An Ensemble Kalman Filter Technique. Mon. Weather Rev. 126 (3): 796–811. http://dx.doi.org/10.1175/1520-0493(1998)126%3C0796:DAUAEK%3E2.0.CO;2.
Hustad, O. S. and Browning, D. J. 2010. A Fully Coupled Three-Phase Model for Capillary Pressure and Relative Permeability for Implicit Compositional Reservoir Simulation. SPE J. 15 (4): 1003–1019. SPE-125429-PA. http://dx.doi.org/10.2118/125429-PA.
Juanes, R. and Blunt, M. J. 2007. Impact of Viscous Fingering on the Prediction of Optimum WAG Ratio. SPE J. 12 (4): 486–495. SPE-99721-PA. http://dx.doi.org/10.2118/99721-PA.
Kerig, P. D. and Watson, A. T. 1987. A New Algorithm for Estimating Relative Permeabilities from Displacement Experiments. SPE Res Eng 2 (1):103–112. SPE-14476-PA.
Larsen, J. A. and Skauge, A. 1998. Methodology for Numerical Simulation with Cycle-Dependent Relative Permeabilities. SPE J. 3 (2): 163–173. SPE-38456-PA. http://dx.doi.org/10.2118/38456-PA.
Liu, N. and Oliver, D. S. 2005. Critical Evaluation of the Ensemble Kalman Filter on History Matching of Geologic Facies. SPE Res Eval & Eng 8 (6): 470–477. SPE-92867-PA. http://dx.doi.org/10.2118/92867-PA.
Lorentzen, R. J., Nævdal, G., Vàlles, B. et al. 2005. Analysis of the Ensemble Kalman Filter for Estimation of Permeability and Porosity in Reservoir Models. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, 9–12 October. SPE-96375-MS. http://dx.doi.org/10.2118/96375-MS.
Oliver, D. S. and Chen, Y. 2011. Recent Progress on Reservoir History Matching: A Review. Computat. Geosci. 15 (1): 185–221. http://dx.doi.org/10.1007/s10596-010-9194-2.
Oliver, D. S., Reynolds, A. C. and Liu, N. 2008. Inverse Theory for Petroleum Reservoir Characterization and History Matching, first edition. Cambridge, UK: Cambridge University Press.
Parrish, D. R. 1966. Flooding Process for Recovery of Oil. US Patent No. 3,244,228.
Reynolds, A. C., Li, R. and Oliver, D. S. 2004. Simultaneous Estimation of Absolute and Relative Permeability by Automatic History Matching of Three-Phase Flow Production Data. J Can Pet Technol 43 (3): 37–46. PETSOC-04-03-03. http://dx.doi.org/10.2118/04-03-03.
Rommelse, J. 2009. Data Assimilation in Reservoir Management. PhD dissertation, Technical University of Delft, Delft, The Netherlands.
Shahverdi, H. 2012. Characterization of Three-Phase Flow and WAG Injection in Oil Reservoirs. PhD dissertation, Heriot-Watt University, Edinburgh, Scotland.
Shahverdi, H. and Sohrabi, M. 2013. An Improved Three-Phase Relative Permeability and Hysteresis Model for the Simulation of A Water-Alternating-Gas Injection. SPE J. 18 (5): 841–850. SPE-152218-PA. http://dx.doi.org/10.2118/152218-PA.
Shahverdi, H. and Sohrabi, M. 2015. Modeling of Cyclic Hysteresis of Three-Phase Relative Permeability during Water-Alternating-Gas Injection. SPE J. 20 (1): 35–48. SPE-166526-PA. http://dx.doi.org/10.2118/166526-PA.
Shahverdi, H., Sohrabi, M., Fatemi, S. M. et al. 2013. Investigation of Two-Phase and Three-Phase Relative Permeability in Different Rocks with Similar Pore Size Distribution. Presented at the SPE/EUROPEC/EAGE Annual Conference and Exhibition, London, 10–13 June. SPE-164789-MS. http://dx.doi.org/10.2118/164789-MS.
Skjervheim, J., Evensen, G., Aanonsen, S. I. et al. 2007. Incorporating 4D Seismic Data in Reservoir Simulation Models Using Ensemble Kalman Filter. SPE J. 12 (3): 282–292. SPE-95789-PA. http://dx.doi.org/10.2118/95789-PA.
Stenger, B. A., Al Kendi, S. A., Al Katheeri, A. B. et al. 2011. Interpretation of Immiscible WAG Repeat Pressure-Falloff Tests. SPE Res Eval & Eng 14 (6): 687–701. SPE-137062-PA. http://dx.doi.org/10.2118/137062-PA.
Stone, H. L. 1970. Probability Model for Estimating Three-Phase Relative Permeability. J Can Pet Technol 23 (2): 214–218. SPE-2116-PA. http://dx.doi.org/10.2118/2116-PA.
Stone, H. L. 1973. Estimation of Three-Phase Relative Permeability and Residual Oil Data. J Can Pet Technol 12 (4): 53–61. PETSOC-73-04-06. http://dx.doi.org/10.2118/73-04-06.
Wang, Y., Li, G. and Reynolds, A. C. 2010. Estimation of Depths of Fluid Contacts by History Matching Using Iterative Ensemble-Kalman Smoothers. SPE J. 15 (2): 509–525. SPE-119056-PA. http://dx.doi.org/10.2118/119056-PA.
Zafari, M. and Reynolds, A. C. 2007. Assessing the Uncertainty in Reservoir Description and Performance Predictions with the Ensemble Kalman Filter. SPE J. 12 (3): 378–387. SPE-95750-PA. http://dx.doi.org/10.2118/95750-PA.
Zhang, Y. and Yang, D. Y. 2014. Estimation of Relative Permeability and Capillary Pressure for Tight Formations by Assimilating Field Production Data. Inverse Probl. Sci. Eng. 22 (7): 1150–1175. http://dx.doi.org/10.1080/17415977.2013.856899.
Zheng, S. and Yang, D. 2013. Pressure Maintenance and Improving Oil Recovery by Means of Immiscible Water-Alternating-CO2 Processes in Thin Heavy-Oil Reservoirs. SPE Res Eval & Eng 16 (1): 60–71. SPE-157719-PA. http://dx.doi.org/10.2118/157719-PA.
Zuo, L., Chen, Y., Dengen, Z. et al. 2014. Three-Phase Relative Permeability Modeling in the Simulation of WAG Injection. SPE Res Eval & Eng 17 (3): 326–339. SPE-166138-PA. http://dx.doi.org/10.2118/166138-PA.