Robust Uncertainty Quantification Through Integration of Distributed-Gauss-Newton Optimization With a Gaussian Mixture Model and Parallelized Sampling Algorithms
- Guohua Gao (Shell Global Solutions (US)) | Jeroen C. Vink (Shell Global Solutions International) | Chaohui Chen (Shell International Exploration and Production) | Mariela Araujo (Shell Global Solutions (US)) | Benjamin A. Ramirez (Shell International Exploration and Production) | James W. Jennings (Shell International Exploration and Production) | Yaakoub El Khamra (Shell Global Solutions (US)) | Joel Ita (Shell Global Solutions (US))
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- November 2019
- Document Type
- Journal Paper
- 1,481 - 1,500
- 2019.Society of Petroleum Engineers
- history matching, distributed Gauss-Newton optimization method, uncertainty quantification, Gaussian mixture model, acceptance-rejection algorithm
- 42 in the last 30 days
- 110 since 2007
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Uncertainty quantification of production forecasts is crucially important for business planning of hydrocarbon-field developments. This is still a very challenging task, especially when subsurface uncertainties must be conditioned to production data. Many different approaches have been proposed, each with their strengths and weaknesses. In this work, we develop a robust uncertainty-quantification work flow by seamless integration of a distributed-Gauss-Newton (GN) (DGN) optimization method with a Gaussian mixture model (GMM) and parallelized sampling algorithms. Results are compared with those obtained from other approaches.
Multiple local maximum-a-posteriori (MAP) estimates are determined with the local-search DGN optimization method. A GMM is constructed to approximate the posterior probability-density function (PDF) by reusing simulation results generated during the DGN minimization process. The traditional acceptance/rejection (AR) algorithm is parallelized and applied to improve the quality of GMM samples by rejecting unqualified samples. AR-GMM samples are independent, identically distributed samples that can be directly used for uncertainty quantification of model parameters and production forecasts.
The proposed method is first validated with 1D nonlinear synthetic problems with multiple MAP points. The AR-GMM samples are better than the original GMM samples. The method is then tested with a synthetic history-matching problem using the SPE01 reservoir model (Odeh 1981; Islam and Sepehrnoori 2013) with eight uncertain parameters. The proposed method generates conditional samples that are better than or equivalent to those generated by other methods, such as Markov-chain Monte Carlo (MCMC) and global-search DGN combined with the randomized-maximum-likelihood (RML) approach, but have a much lower computational cost (by a factor of five to 100). Finally, it is applied to a real-field reservoir model with synthetic data, with 235 uncertain parameters. AGMM with 27 Gaussian components is constructed to approximate the actual posterior PDF. There are 105 AR-GMM samples accepted from the 1,000 original GMM samples, and they are used to quantify the uncertainty of production forecasts. The proposed method is further validated by the fact that production forecasts for all AR-GMM samples are quite consistent with the production data observed after the history-matching period.
The newly proposed approach for history matching and uncertainty quantification is quite efficient and robust. The DGN optimization method can efficiently identify multiple local MAP points in parallel. The GMM yields proposal candidates with sufficiently high acceptance ratios for the AR algorithm. Parallelization makes the AR algorithm much more efficient, which further enhances the efficiency of the integrated work flow.
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Aanonsen, S. I., Naevdal, 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. https://doi.org/10.2118/117274-PA.
Alabert, F. 1987. The Practice of Fast Conditional Simulations Through the LU Decomposition of the Covariance Matrix. Math Geol 19 (5): 369–386. https://doi.org/10.1007/BF00897191.
Bocquet, M. and Sakov, P. 2014. An Iterative Ensemble Kalman Smoother. Q J R Meteorol Soc 140 (682): 1521–1535. https://doi.org/10.1002/qj.2236.
Chen, C., Gao, G., Li, R. et al. 2018. Global-Search Distributed-Gauss-Newton Optimization Method and Its Integration With the Randomized-Maximum-Likelihood Method for Uncertainty Quantification of Reservoir Performance. SPE J. 23 (5): 1496–1517. SPE-182639-PA. https://doi.org/10.2118/182639-PA.
Chen, C., Gao, G., Ramirez, B. A. et al. 2016. Assisted History Matching of Channelized Models by Use of Pluri-Principal-Component Analysis. SPE J. 21 (5): 1793–1812. SPE-173192-PA. https://doi.org/10.2118/173192-PA.
Chen, Y. and Oliver, D. S. 2010. Cross-Covariances and Localization for EnKF in Multiphase Flow Data Assimilation. Computat Geosci 14 (4): 579–601. https://doi.org/10.1007/s10596-009-9174-6.
Chen, Y. and Oliver, D. S. 2012. Ensemble Randomized Maximum Likelihood Method as an Iterative Ensemble Smoother. Math Geosci 44 (1): 1–26. https://doi.org/10.1007/s11004-011-9376-z.
Chen, Y. and Oliver, D. S. 2013. Levenberg-Marquardt Forms of the Iterative Ensemble Smoother for Efficient History Matching and Uncertainty Quantification. Comput Geosci 17 (4): 689–703. https://doi.org/10.1007/s10596-013-9351-5.
Christie, M., Demyanov, V., and Erbsa, D. 2006. Uncertainty Quantification for Porous Media Flows. J Comput Phys 217 (1): 143–158. https://doi.org/10.1016/j.jcp.2006.01.026.
Chu, L., Reynolds, A. C., and Oliver, D. S. 1995. Computation of Sensitivity Coefficients for Conditioning the Permeability Field to Well-Test Data. In Situ 19 (2): 179–223.
Davis, M. 1987. Production of Conditional Simulations via the LU Decomposition of the Covariance Matrix. Math Geol 19 (2): 91–98. https://doi.org/10.1007/BF00898189.
De Paola, G., Torrado, R. R., and Embid, S. 2014. New Methodology for the Generation of the Structural and Petrophysical Conceptual Model With Limited Information. Oral presentation given at the AAPG International Conference and Exhibition, Istanbul, Turkey, 14–17 September.
Ehrendorfer, M. 2007. A Review of Issues in Ensemble-Based Kalman Filtering. Meteorol J 16 (6): 795–818. https://doi.org/10.1127/0941-2948/2007/0256.
Elsheikh, A. H., Wheeler, M. F., and Hoteit, I. 2013. Clustered Iterative Stochastic Ensemble Method for Multi-Modal Calibration of Subsurface Flow Models. J Hydrol 491 (29 May): 40–55. https://doi.org/10.1016/j.jhydrol.2013.03.037.
Emerick, A. A. and Reynolds, A. C. 2010. EnKF-MCMC. Presented at the SPE EUROPEC/EAGE Annual Conference and Exhibition, Barcelona, Spain, 14–17 June. SPE-131375-MS. https://doi.org/10.2118/131375-MS.
Emerick, A. A. and Reynolds, A. C. 2011. Combining Sensitivities and Prior Information for Covariance Localization in the Ensemble Kalman Filter for Petroleum Reservoir Applications. Computat Geosci 15 (2): 251–269. https://doi.org/10.1007/s10596-010-9198-y.
Emerick, A. A. and Reynolds, A. C. 2013. Ensemble Smoother With Multiple Data Assimilation. Comput Geosci 55 (June): 3–15. https://doi.org/10.1016/j.cageo.2012.03.011.
Evensen, G. 2007. Data Assimilation: The Ensemble Kalman Filter. New York City: Springer.
Evensen, G. 2009. The Ensemble Kalman Filter for Combined State and Parameter Estimation. IEEE Control Syst Mag N Y 29 (3): 83–104. https://doi.org/10.1109/MCS.2009.932223.
Ferreira, O. P., Gonc¸alves, M. L. N., and Oliveira, P. R. 2011. Local Convergence Analysis of the Gauss–Newton Method Under a Majorant Condition. J Complex 27 (1): 111–125. https://doi.org/10.1016/j.jco.2010.09.001.
Fletcher, R. 1987. Practical Methods of Optimization, 2nd edition. New York: John Wiley & Sons.
Gao, G. and Reynolds, A. C. 2006. An Improved Implementation of the LBFGS Algorithm for Automatic History Matching. SPE J. 11 (1): 5–17. SPE-90058-PA. https://doi.org/10.2118/90058-PA.
Gao, G., Vink, J. C., Chen, C. et al. 2016a. A Parallelized and Hybrid Data-Integration Algorithm for History Matching of Geologically Complex Reservoirs. SPE J. 21 (6): 2155–2174. SPE-175039-PA. https://doi.org/10.2118/175039-PA.
Gao, G., Vink, J. C., Chen, C. et al. 2016b. Uncertainty Quantification for History Matching Problems With Multiple Best Matches Using a Distributed Gauss-Newton Method. Presented at the SPE Annual Technical Conference and Exhibition, Dubai, 26–28 September. SPE-181611-MS. https://doi.org/10.2118/181611-MS.
Gao, G., Vink, J. C., Chen, C. et al. 2017. Distributed Gauss-Newton Optimization Method for History Matching Problems With Multiple Best Matches. Computat Geosci 21 (5–6): 1325–1342. https://doi.org/10.1007/s10596-017-9657-9.
Guo, Z., Chen, C., Gao, G. et al. 2017a. EUR Assessment of Unconventional Assets Using Machine Learning and Distributed Computing Techniques. Presented at the SPE/AAPG/SEG Unconventional Resources Technology Conference, Austin, Texas, 24–26 July. URTEC-2659996-MS. https://doi.org/10.15530/URTEC-2017-2659996.
Guo, Z., Chen, C., Gao, G. et al. 2017b. Applying Support Vector Regression to Reduce the Effect of Numerical Noise and Enhance the Performance of History Matching. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 9–11 October. SPE-187430-MS. https://doi.org/10.2118/187430-MS.
Gonc¸alves, M. L. N. 2013. Local Convergence of the Gauss–Newton Method for Injective-Overdetermined Systems of Equations Under a Majorant Condition. Comput Math Appl 66 (4): 490–499. https://doi.org/10.1016/j.camwa.2013.05.019.
Grana, D., Fjeldstad, T., and Omer, H. 2017. Bayesian Gaussian Mixture Linear Inversion in Geophysical Inverse Problems. Math Geosci 49 (4): 493–525. https://doi.org/10.1007/s11004-016-9671-9.
He, J., Reynolds, A. C., Tanaka, S. et al. 2018. Calibrating Global Uncertainties to Local Data: Is the Learning Being Over-Generalized? Presented at the SPE Annual Technical Conference and Exhibition, Dallas, 23–26 September. SPE-191480-MS. https://doi.org/10.2118/191480-MS.
Islam, A. W. and Sepehrnoori, K. 2013. A Review of SPE’s Comparative Solution Projects (CSPs). J. Pet. Sci. Res. 2 (4): 167–180.
Kitanidis, P. K. 1995. Quasi-Linear Geostatistical Theory for Inversing. Water Resour Res 31 (10): 2411–2419. https://doi.org/10.1029/95WR01945.
Kourounis, D., Durlofsky, L. J., Jansen, J. D. et al. 2014. Adjoint Formulation and Constraint Handling for Gradient-Based Optimization of Compositional Reservoir Flow. Comput Geosci 18 (2): 117–137. https://doi.org/10.1007/s10596-013-9385-8.
Li, G. and Reynolds, A. C. 2009. Iterative Ensemble Kalman Filters for Data Assimilation. SPE J. 14 (3): 496–505. SPE-109808-PA. https://doi.org/10.2118/109808-PA.
Li, W., Zhang, D., and Lin, G. 2015. A Surrogate-Based Adaptive Sampling Approach for History Matching and Uncertainty Quantification. Presented at the SPE Reservoir Simulation Symposium, Houston, 23–25 February. SPE-173298-MS. https://doi.org/10.2118/173298-MS.
Link, W. A. and Eaton, M. J. 2011. On Thinning of Chains in MCMC. Methods Ecol Evol 3 (1): 112–115. https://doi.org/10.1111/j.2041-210X.2011.00131.x.
Liu, J. S. 1996. Metropolized Independent Sampling With Comparison to Rejection Sampling and Importance Sampling. Stat Comput 6 (2): 113–119. https://doi.org/10.1007/BF00162521.
Liu, N. and Oliver, D. S. 2003. Evaluation of Monte Carlo Methods for Assessing Uncertainty. SPE J. 8 (2): 188–195. SPE-84936-PA. https://doi.org/10.2118/84936-PA.
Luo, X., Stordal, A. S., Lorentzen, R. J. et al. 2015. Iterative Ensemble Smoother as an Approximate Solution to a Regularized Minimum-Average-Cost Problem: Theory and Applications. SPE J. 20 (5): 962–982. SPE-176023-PA. https://doi.org/10.2118/176023-PA.
Mohammadi, B. 2015. Ensemble Kalman Filters and Geometric Characterization of Sensitivity Spaces for Uncertainty Quantification in Optimization. Comput Methods Appl Mech Eng 290 (15 June): 228–249. https://doi.org/10.1016/j.cma.2015.03.006.
Odeh, A. 1981. Comparison of Solutions to a Three Dimensional Black-Oil Reservoir Simulation Problem (includes associated paper 9741). J Pet Technol 33 (1): 13–25. SPE-9723-PA. https://doi.org/10.2118/9723-PA.
Oliver, D. S. 1996. Multiple Realization of the Permeability Field From Well-Test Data. SPE J. 1 (2):145–155. SPE-27970-PA. https://doi.org/10.2118/27970-PA.
Oliver, D. S. 2017. Metropolized Randomized Maximum Likelihood for Improved Sampling From Multimodal Distributions. SIAM/ASA J. Uncertainty Quantification 5 (1): 259–277. https://doi.org/10.1137/15M1033320.
Oliver, D. S. and Alfonzo, M. 2018. Calibration of Imperfect Models to Biased Observations. Computat Geosci 22 (1): 145–161. https://doi.org/10.1007/s10596-017-9678-4.
Oliver, D. S., Reynolds, A. C., and Liu, N. 2008. Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge, UK: Cambridge University Press.
Reynolds, A. C. 2014. My Decade-Long Journey Through the Field of Ensemble-Based Data Assimilation. Presented at the Ninth International EnKF Workshop, Bergen, Norway, 23–25 June.
Robert, C. P. and Casella, G. 2010. Introducing Monte Carlo Methods With R. New York City: Springer Science+Business Media.
Sakov, P., Oliver, D. S., and Bertino, L. 2012. An Iterative EnKF for Strongly Nonlinear Systems. Mon Weather Rev 140 (6): 1988–2004. https://doi.org/10.1175/MWR-D-11-00176.1.
Sargsyan, K., Huan, X., and Najm, H. N. In press. Embedded Model Error Representation for Bayesian Model Calibration. arXiv preprint arXiv:1801.06768 (submitted 19 January 2018).
Sebacher, B., Stordal, A., and Hanea, R. 2016. Complex Geology Estimation Using the Iterative Adaptive Gaussian Mixture Filter. Computat Geosci 20 (1): 133–148. https://doi.org/10.1007/s10596-015-9553-0.
Smith, K. W. 2007. Cluster Ensemble Kalman Filter. Tellus A 59 (5): 749–757. https://doi.org/10.1111/j.1600-0870.2007.00246.x.
Sondergaard, T. and Lermusiaux, P. F. J. 2013a. Data Assimilation With Gaussian Mixture Models Using the Dynamically Orthogonal Field Equations. Part I: Theory and Scheme. Mon Weather Rev 141 (6): 1737–1760. https://doi.org/10.1175/MWR-D-11-00295.1.
Sondergaard, T. and Lermusiaux, P. F. J. 2013b. Data Assimilation With Gaussian Mixture Models Using the Dynamically Orthogonal Field Equations. Part II: Applications. Mon Weather Rev 141 (6): 1737–1760. https://doi.org/10.1175/MWR-D-11-00296.1.
Stordal, A. S. 2015. Iterative Bayesian Inversion With Gaussian Mixtures: Finite Sample Implementation and Large Sample Asymptotics. Computat Geosci 19 (1): 1–15. https://doi.org/10.1007/s10596-014-9444-9.
Stordal, A. S., Valestrand, R., Karlsen, H. A. et al. 2012. Comparing the Adaptive Gaussian Mixture Filter With the Ensemble Kalman Filter on Synthetic Reservoir Models. Computat Geosci 16 (2): 467–482. https://doi.org/10.1007/s10596-011-9262-2.
Sun, W., Vink, J. C., and Gao, G. 2017. A Practical Method to Mitigate Spurious Uncertainty Reduction in History Matching Workflows With Imperfect Reservoir Model. Presented at the SPE Reservoir Simulation Conference, Montgomery, Texas, 20–22 February. SPE-182599-MS. https://doi.org/10.2118/182599-MS.
Tarantola, A. 2005. Inverse Problem Theory and Methods for Model Parameter Estimation. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics.
Valestrand, R., Naevdal, G., Shafieirad, A. et al. 2012. Refined Adaptive Gaussian Mixture Filter—Application on a Real Field Case. Presented at the SPE Europec/EAGE Annual Conference, Copenhagen, Denmark, 4–7 June. SPE-154479-MS. https://doi.org/10.2118/154479-MS.
Vink, J. C., Gao, G., and Chen, C. 2015. Bayesian-Style History Matching: Another Way to Underestimate Forecast Uncertainty? Presented at the SPE Annual Technical Conference and Exhibition, Houston, 28–30 September. SPE-175121-MS. https://doi.org/10.2118/175121-MS.
von Neumann, J. 1951. Various Techniques Used in Connection With Random Digits. J. Res. Nat. Bur. Stand. Appl. Math. Series 12: 36–38.
Wild, S. M. 2009. Derivative Free Optimization Algorithms for Computationally Expensive Function. PhD dissertation, Cornell University, Ithaca, New York (January 2009).
Yu, G., Sapiro, G., and Mallat, S. 2012. Solving Inverse Problems With Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity. IEEE Trans Image Process 21 (5): 2481–2499. https://doi.org/10.1109/TIP.2011.2176743.
Zhou, W. and Zhang, L. 2010. Global Convergence of a Regularized Factorized Quasi-Newton Method for Nonlinear Least Squares Problems. Comput. Appl. Math. 29 (2): 195–214, 2010. https://doi.org/10.1590/S1807-03022010000200006.