Global-Search Distributed-Gauss-Newton Optimization Method and Its Integration With the Randomized-Maximum-Likelihood Method for Uncertainty Quantification of Reservoir Performance
- Chaohui Chen (Shell International Exploration and Production Company) | Guohua Gao (Shell Global Solutions US Incorporated) | Ruijian Li (Shell Exploration and Production Company) | Richard Cao (Shell Exploration and Production Company) | Tianhong Chen (Shell Exploration and Production Company) | Jeroen C. Vink (Shell Global Solutions International) | Paul Gelderblom (Shell Global Solutions International)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- October 2018
- Document Type
- Journal Paper
- 1,496 - 1,517
- 2018.Society of Petroleum Engineers
- Uncertainty Quantification, Randomized Maximum Likelihood, Distributed Gauss-Newton, Unconventional reservoir, History Matching
- 1 in the last 30 days
- 150 since 2007
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Although it is possible to apply traditional optimization algorithms together with the randomized-maximum-likelihood (RML) method to generate multiple conditional realizations, the computation cost is high. This paper presents a novel method to enhance the global-search capability of the distributed-Gauss-Newton (DGN) optimization method and integrates it with the RML method to generate multiple realizations conditioned to production data synchronously.
RML generates samples from an approximate posterior by minimizing a large ensemble of perturbed objective functions in which the observed data and prior mean values of uncertain model parameters have been perturbed with Gaussian noise. Rather than performing these minimizations in isolation using large sets of simulations to evaluate the finite-difference approximations of the gradients used to optimize each perturbed realization, we use a concurrent implementation in which simulation results are shared among different minimization tasks whenever these results are helping to converge to the global minimum of a specific minimization task. To improve sharing of results, we relax the accuracy of the finite-difference approximations for the gradients with more widely spaced simulation results. To avoid trapping in local optima, a novel method to enhance the global-search capability of the DGN algorithm is developed and integrated seamlessly with the RML formulation. In this way, we can improve the quality of RML conditional realizations that sample the approximate posterior.
The proposed work flow is first validated with a toy problem and then applied to a real-field unconventional asset. Numerical results indicate that the new method is very efficient compared with traditional methods. Hundreds of data-conditioned realizations can be generated in parallel within 20 to 40 iterations. The computational cost (central-processing-unit usage) is reduced significantly compared with the traditional RML approach.
The real-field case studies involve a history-matching study to generate history-matched realizations with the proposed method and an uncertainty quantification of production forecasting using those conditioned models. All conditioned models generate production forecasts that are consistent with real-production data in both the history-matching period and the blind-test period. Therefore, the new approach can enhance the confidence level of the estimated-ultimate-recovery (EUR) assessment using production-forecasting results generated from all conditional realizations, resulting in significant business impact.
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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.
Chen, C., Gao, G., Gelderblom, P. et al. 2016a. Integration of Cumulative-Distribution-Function Mapping with Principal-Component Analysis for the History Matching of Channelized Reservoirs. SPE Res Eval & Eng 19 (2): 278–293. SPE-170636-PA. https://doi.org/10.2118/170636-PA.
Chen, C., Gao, G., Ramirez, B. A. et al. 2016b. Assisted History Matching of Channelized Models Using Pluri-Principal Component Analysis. SPE J. 21 (5): 1793–1812. SPE-173192-PA. https://doi.org/10.2118/173192-PA.
Chen, C., Jin, L., Gao, G. et al. 2012. Assisted History Matching Using Three Derivative-Free Optimization Algorithms. Presented at the SPE Europec/EAGE Annual Conference, Copenhagen, Denmark, 4–7 June. SPE-154112-MS. https://doi.org/10.2118/154112-MS.
Chen, C., Li, R., Gao, G. et al. 2016c. EUR Assessment of Unconventional Assets Using Parallelized History Matching Workflow Together With RML Method. Presented at the Unconventional Resources Technology Conference, San Antonio, Texas, 1–3 August. URTEC-2429986-MS.
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. Computat. Geosci. 17 (4): 689–703. https://doi.org/10.1007/s10596-013-9351-5.
Chen, Y. and Oliver, D. S. 2014. History Matching of the Norne Full-Field Model with an Iterative Ensemble Smoother. SPE Res Eval & Eng 17 (2): 244–256. SPE-164902-PA. https://doi.org/10.2118/164902-PA.
Davis, M. W. 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.
Efendiev, Y., Datta-Gupta, A., Ginting, V. et al. 2005. An Efficient Two-Stage Markov Chain Monte Carlo Method for Dynamic Data Integration. Water Resour. Res. 41 (12). https://doi.org/10.1029/2004WR003764.
Eide, A. L., Holden, L., Reiso, E. et al. 1994. Automatic History Matching by Use of Response Surface and Experimental Design. Oral presentation given at the 4th European Conference on the Mathematics of Oil Recovery, Roros, Norway, 7–10 June.
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. 2012. Combining the Ensemble Kalman Filter With Markov-Chain Monte Carlo for Improved History Matching and Uncertainty Characterization. SPE J. 17 (2): 418–440. SPE-141336-PA. https://doi.org/10.2118/141336-PA.
Emerick, A. A. and Reynolds, A. C. 2013a. Investigation on the Sampling Performance of Ensemble-Based Methods with a Simple Reservoir Model. Computat. Geosci. 17 (2): 325–350. https://doi.org/10.1007/s10596-012-9333-z.
Emerick, A. A. and Reynolds, A. C. 2013b. Ensemble Smoother with Multiple Data Assimilation. Comput. Geosci. 55 (June): 3–15. https://doi.org/10.1016/j.cageo.2012.03.011.
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. https://doi.org/10.1029/94JC00572.
Evensen, G. 2007. Data Assimilation: The Ensemble Kalman Filter. Berlin: Springer.
Ferreira, O. P., Gonçalves, M. L. N., and Oliveira, P. R. 2011. Local Convergence Analysis of the Gauss-Newton Method Under a Majorant Condition. J. Complexity 27 (1): 111–125. https://doi.org/10.1016/j.jco.2010.09.001.
Furrer, R. and Bengtsson, T. 2007. Estimation of High-Dimensional Prior and Posterior Covariance Matrices in Kalman Filter Variants. J. Multivariate Anal. 98 (2): 227–255. https://doi.org/10.1016/j.jmva.2006.08.003.
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 (56): 1325–1342. https://doi.org/10.1007/s10596-017-9657-9.
Gao, G., Zafari, M., and Reynolds, A. C. 2006. Quantifying Uncertainty for the PUNQ-S3 Problem in a Bayesian Setting With RML and EnKF. SPE J. 11 (4): 506–515. SPE-93324-PA. https://doi.org/10.2118/93324-PA.
Gonçalves, 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.
Goodwin, N. 2015. Bridging the Gap Between Deterministic and Probabilistic Uncertainty Quantification Using Advanced Proxy Based Methods. Presented at the SPE Reservoir Simulation Symposium, Houston, 23–25 February. SPE-173301-MS. https://doi.org/10.2118/173301-MS.
Hooke, R. and Jeeves, T. A. 1961. “Direct Search” Solution of Numerical and Statistical Problems. J. ACM 8 (2): 212–229. https://doi.org/10.1145/321062.321069.
Houtekamer, P. L. and Mitchell, H. L. 2001. A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation. Mon. Weather Rev. 129 (1): 123–137. https://doi.org/10.1175/1520-0493(2001)129%3C0123:ASEKFF%3E2.0.CO;2.
Ingber, I. 1993. Simulated Annealing: Practice Versus Theory. Math. Comput. Model. 18 (11): 29–57. https://doi.org/10.1016/0895-7177(93)90204-C.
Kitanidis, P. K. 1995. Quasi-Linear Geostatistical Theory for Inversing. Water Resour. Res. 31 (10): 2411–2419. https://doi.org/10.1029/95WR01945.
Li, G. and Reynolds, A. C. 2011. Uncertainty Quantification of Reservoir Performance Predictions Using a Stochastic Optimization Algorithm. Computat. Geosci. 15 (3): 451–462. https://doi.org/10.1007/s10596-010-9214-2.
Li, R., Reynolds, A. C., and Oliver, D. S. 2003. History Matching of Three-Phase Flow Production Data. SPE J. 8 (4): 328–340. SPE-87336-PA. https://doi.org/10.2118/87336-PA.
Liu, J. S. 2008. Monte Carlo Strategies in Scientific Computing. New York City: Springer-Verlag New York.
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.
Maucec, M., Douma, S. G., Hohl, D. et al. 2007. Streamline-Based History Matching and Uncertainty—Markov-Chain Monte Carlo Study of an Offshore Turbidite Oil Field. Presented at the SPE Annual Technical Conference and Exhibition, Anaheim, California, 11–14 November. SPE-109943-MS. https://doi.org/10.2118/109943-MS.
Meyn, S. P. and Tweedie, R. L. 1993. Markov Chains and Stochastic Stability. London: Springer-Verlag.
Mohamed, L., Christie, M., Demyanov, V. et al. 2010. Application of Particle Swarms for History Matching in the Brugge Reservoir. Presented at the SPE Annual Technical Conference and Exhibition, Florence, Italy, 19–22 September. SPE-135264-MS. https://doi.org/10.2118/135264-MS.
Oliver, D. S. 1996. On Conditional Simulation to Inaccurate Data. Math. Geol. 28 (6): 811–817. https://doi.org/10.1007/BF02066348.
Oliver, D. S. 2015. Metropolized Randomized Maximum Likelihood for Sampling from Multimodal Distributions. ArXiv, https://arxiv.org/pdf/1507.08563v2.pdf.
Oliver, D. S. and Chen, Y. 2011. Recent Progress on Reservoir History Matching: A Review. Computat. Geosci. 15 (1): 185–211. https://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. Cambridge, UK: Cambridge University Press.
Peters, L., Arts, R., Brouwer, G. et al. 2010. Results of the Brugge Benchmark Study for Flooding Optimization and History Matching. SPE Res Eval & Eng 13 (3): 391–405. SPE-119094-PA. https://doi.org/10.2118/119094-PA.
Powell, M. J. 2006. The NEWUOA Software for Unconstrained Optimization Without Derivatives in Large-Scale Nonlinear Optimization. In Large-Scale Nonlinear Optimization, ed. G. Di Pillo and M. Roma, 255–297. Boston: Springer.
Powell, M. J. 2007. Developments of NEWUOA for Unconstrained Minimization Without Derivatives. Technical Report DAMTP NA2004/08, Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge, UK.
Reynolds, A. C. 2014. My Decade-Long Journey Through the Field of Ensemble-Based Data Assimilation. Oral presentation given at the Ninth International EnKF Workshop, Bergen, Norway, 23–25 June.
Reynolds, A. C., He, N., and Oliver, D. S. 1999. Reducing Uncertainty in Geostatistical Description with Well-Testing Pressure Data. In AAPG Memoir 71: Reservoir Characterization–Recent Advances, ed. R. A. Schatzinger and J. F. Jordan, Chap. 10, 149–162. Tulsa: American Association of Petroleum Geologists.
Shirangi, M. G. and Durlofsky, L. 2015. Closed-Loop Field Development Optimization Under Uncertainty. Presented at the SPE Reservoir Simulation Symposium, Houston, 23–25 February. SPE-173219-MS. https://doi.org/10.2118/173219-MS.
Shirangi, M. G. and Emerick, A. A. 2016. An Improved TSVD-based Levenberg-Marquardt Algorithm for History Matching and Comparison with Gauss-Newton. J. Pet. Sci. Eng. 143 (July): 258–271. https://doi.org/10.1016/j.petrol.2016.02.026.
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.
Stordal, A. S. and Elsheikh, A. H. 2015. Iterative Ensemble Smoothers in the Annealed Importance Sampling Framework. Adv. Water Resour. 86A (December): 231–239. https://doi.org/10.1016/j.advwatres.2015.09.030.
Stordal, A. S., Valestrand, R., Karlsen, H. A. et al. 2011. 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.
Tarantola, A. 2005. Inverse Problem Theory and Methods for Model Parameter Estimation. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics.
Valestrand, R., Nævdal, 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.
van Leeuwen., P. J. 1999. Comment on “Data Assimilation Using an Ensemble Kalman Filter Technique.” Mon. Weather Rev. 127 (6): 1374–1377. https://doi.org/10.1175/1520-0493(1999)127%3C1374:CODAUA%3E2.0.CO;2.
Vink, J. C., Gao, G., and Chen, C. 2015. Bayesian Style History Matching: Another Way to Under-Estimate 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.
Yeh, T., Uvieghara, T., Jennings, J. W. et al. 2016. A Practical Workflow for Probabilistic History Matching and Forecast Uncertainty Quantification: Application to a Deepwater West Africa Reservoir. Presented at the SPE Annual Technical Conference and Exhibition, Dubai, 26–28 September. SPE-181639-MS. https://doi.org/10.2118/181639-MS.
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): 382–391. SPE-95750-PA. https://doi.org/10.2118/95750-PA.
Zhang, F. and Reynolds, A. C. 2002. Optimization Algorithms for Automatic History Matching of Production Data. Oral presentation given at ECMOR VIII–8th European Conference on Mathematics of Oil Recovery, Freiberg, Germany, 3–6 September.
Zhao, H., Li, G., Reynolds, A. C. et al. 2013. Large-Scale History Matching With Quadratic Interpolation Models. Computat. Geosci. 17 (1): 117–138. https://doi.org/10.1007/s10596-012-9320-4.