Integration of Support Vector Regression With Distributed Gauss-Newton Optimization Method and Its Applications to the Uncertainty Assessment of Unconventional Assets

Guo, Zhenyu; Chen, Chaohui; Gao, Guohua; Cao, Richard; Li, Ruijian; Liu, Hope

doi:10.2118/191373-PA

Integration of Support Vector Regression With Distributed Gauss-Newton Optimization Method and Its Applications to the Uncertainty Assessment of Unconventional Assets

Authors: Zhenyu Guo (University of Tulsa) | Chaohui Chen (Shell Exploration and Production Company Incorporated) | Guohua Gao (Shell Global Solutions US Incorporated) | Richard Cao (Shell Exploration and Production Company Incorporated) | Ruijian Li (Shell Exploration and Production Company Incorporated) | Hope Liu (Shell Exploration and Production Company Incorporated)
DOI: https://doi.org/10.2118/191373-PA
Document ID: SPE-191373-PA
Publisher: Society of Petroleum Engineers
Source: SPE Reservoir Evaluation & Engineering
Volume: 21
Issue: 04
Publication Date: November 2018

Document Type: Journal Paper
Pages: 1,007 - 1,026
Language: English
ISSN: 1094-6470
Copyright: 2018.Society of Petroleum Engineers
Keywords: machine learning, Unconventional, distributed computation, EUR assessment
Downloads: 5 in the last 30 days
: 145 since 2007

Show more detail: View rights & permissions

SPE Member Price:	USD 12.00
SPE Non-Member Price:	USD 35.00

Summary

Reservoir model parameters generally have very large uncertainty ranges, and need to be calibrated by history matching (HM) available production data. Properly assessing the uncertainty of production forecasts (e.g., with an ensemble of calibrated models that are conditioned to production data) has a direct impact on business decision making. It requires performing numerous reservoir simulations on a distributed computing environment. Because of the current low-oil-price environment, it is demanding to reduce the computational cost of generating multiple realizations of history-matched models without compromising forecasting quality. To solve this challenge, a novel and more efficient optimization method (referred to as SVR-DGN) is proposed in this paper, by replacing the less accurate linear proxy of the distributed Gauss-Newton (DGN) optimization method (referred to as L-DGN) with a more accurate response-surface model of support vector regression (SVR).

Resembling L-DGN, the proposed SVR-DGN optimization method can be applied to find multiple local minima of the objective function in parallel. In each iteration, SVR-DGN proposes an ensemble of search points or reservoir-simulation models, and the flow responses of these reservoir models are simulated on high-performance-computing (HPC) clusters concurrently. All successfully simulated cases are recorded in a training data set. Then, an SVR proxy is constructed for each simulated response using all training data points available in the training data set. Finally, the sensitivity matrix at any point can be calculated analytically by differentiating the SVR models. SVR-DGN computes more-accurate sensitivity matrices, proposes better search points, and converges faster than L-DGN.

The quality of the SVR proxy is validated with a toy problem. The proposed method is applied to a real field HM example of a Permian liquid-rich shale reservoir. The uncertain parameters include reservoir static properties, hydraulic-fracture properties, and parameters defining relative permeability curves. The performance of the proposed SVR-DGN optimization method is compared with the L-DGN optimizer and the hybrid Gauss-Newton with a direct-pattern-search (GN-DPS) optimizer, using the same real field example. Our numerical tests indicate that the SVR-DGN optimizer can find better solutions with smaller values of the objective function and with a less computational cost (approximately one-third of L-DGN and 1/30 of GN-DPS). Finally, the proposed method is applied to generate multiple conditional realizations for the uncertainty quantification of production forecasts.

File Size

1 MB

Number of Pages

References

Amudo, C., May, R. S., Graf, T. et al. 2008. Experimental Design and Response Surface Models as a Basis for Stochastic History Match—A Case Study on Complex Reservoirs in the Niger Delta. Presented at the International Petroleum Technology Conference, Kuala Lumpur, 3–5 December. SPE-23665-MS. https://doi.org/10.2118/23665-MS.

Bazargan, H., Christie, M., Elsheikh, A. H. et al. 2015. Surrogate Accelerated Sampling of Reservoir Models With Complex Structures Using Sparse Polynomial Chaos Expansion. Advances in Water Resources 86 (Part B): 385–399. https://doi.org/10.1016/j.advwatres.2015.09.009.

Chen, C., Jin, L., Gao, G. et al. 2012. Assisted History Matching Using Three Derivative-Free Optimization Algorithms. In Proc., SPE Europec/EAGE Annual Conference, Copenhagen, Denmark, 4–7 June. SPE-154112-MS. https://doi.org/10.2118/154112-MS.

Chen, B. and Reynolds, A. C. 2016. Ensemble-Based Optimization of the Water-Alternating-Gas-Injection Process. SPE J. 21 (3): 786–798. SPE-173217-PA. https://doi.org/10.2118/173217-PA.

Chen, C., Li, R., Gao, G. et al. 2016. EUR Assessment of Unconventional Assets Using Parallelized History Matching Workflow Together With the RML Method. Presented at the Unconventional Resources Technology Conference, San Antonio, Texas, 1–3 August. URTeC-2429986-MS. https://doi.org/10.15530/URTeC-2016-2429986.

Chen, B., He, J., Wen, X. et al. 2017a. Uncertainty Quantification and Value of Information Assessment Using Proxies and Markov Chain Monte Carlo Method for a Pilot Project. Journal of Petroleum Science and Engineering 157: 328–339. https://doi.org/10.1016/j.petrol.2017.07.039.

Chen, C., Gao, G., Li, R. et al. 2017b. Integration of Distributed Gauss-Newton With Randomized Maximum Likelihood Method for Uncertainty Quantification of Reservoir Performance. In Proc., SPE Reservoir Simulation Conference, Montgomery, Texas, 20–22 February. SPE-182639-MS. https://doi.org/10.2118/182639-MS.

Chitralekha, S. B., Trivedi, J. J., and Shah, S. 2010. Application of the Ensemble Kalman Filter for Characterization and History Matching of Unconventional Oil Reservoirs. Presented at the Canadian Unconventional Resources and International Petroleum Conference, Calgary, 19–21 October. SPE-137480-MS. https://doi.org/10.2118/137480-MS.

Cortes, C. and Vapnik, V. 1995. Support-Vector Networks. Machine Learning 20 (3): 273–297.

De Brabanter, K. 2011. Least-Squares Support Vector Regression With Applications to Large-Scale Data: A Statistical Approach. Faculty of Engineering, KU Leuven, Katholieke Universiteit Leuven.

Dyn, N. 1987. Interpolation of Scattered Data by Radial Functions. In Topics in Multivariate Approximation, eds. C. K. Chui, L. L. Schumaker, and F. I. Utreras, Chapter 5, 47–61. Orlando, Florida: Academic Press, Inc. https://doi.org/10.1016/B978-0-12-174585-1.50009-9.

Elsheikh, A. H., Hoteit, I., and Wheeler, M. F. 2014. Efficient Bayesian Inference of Subsurface Flow Models Using Nested Sampling and Sparse Polynomial Chaos Surrogates. Comput. Method Appl. Mech. Eng. 269: 515–537. https://doi.org/10.1016/j.cma.2013.11.001.

Emerick, A. A. and Reynolds, A. C. 2012. History Matching Time-Lapse Seismic Data Using the Ensemble Kalman Filter With Multiple Data Assimilations. Computational Geosciences 16 (3): 639–659. https://doi.org/10.1007/s10596-012-9275-5.

Emerick, A. A. and Reynolds, A. C. 2013a. Ensemble Smoother With Multiple Data Assimilations. Computers & Geosciences 55: 3–15. https://doi.org/10.1016/j.cageo.2012.03.011.

Emerick, A. A. and Reynolds, A. C. 2013b. History-Matching Production and Seismic Data in a Real Field Case Using the Ensemble Smoother With Multiple Data Assimilation. In Proc., SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18–20 February. SPE-163675-MS. https://doi.org/10.2118/163675-MS.

Evensen, G. 1994. Sequential Data Assimilation With a Nonlinear Quasi-Geostrophic Model Using Monte Carlo Methods To Forecast Error Statistics. Journal of Geophysical Research 99 (C5): 10143–10162. https://doi.org/10.1029/94JC00572.

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. Journal of Complexity 27 (1): 111–125. https://doi.org/10.1016/j.jco.2010.09.001.

Gao, G. and Reynolds, A. C. 2004. An Improved Implementation of the LBFGS Algorithm for Automatic History Matching. In Proc., SPE Annual Technical Conference and Exhibition, Houston, 26–29 September. SPE-90058-MS. https://doi.org/10.2118/90058-MS.

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., Alpak, F. O. et al. 2015. An Efficient Optimization Work Flow for Field-Scale In-Situ Upgrading Developments. SPE J. 20 (4): 701–715. SPE-163634-PA. https://doi.org/10.2118/163634-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. 2017a. Distributed Gauss-Newton Method for History Matching Problems With Multiple Best Matches. Computational Geosciences 21 (5–6): 1325–1342. https://doi.org/10.1007/s10596-017-9657-9.

Gao, G. Jiang, H., Van Hagan, P. et al. 2017b. A Gauss-Newton Trust Region Solver for Large-Scale History Matching Problems. In Proc., SPE Reservoir Simulation Conference, Montgomery, Texas, 20–22 February. SPE-182602-MS. https://doi.org/10.2118/182602-MS.

Goncalves, M. L. N. 2013. Local Convergence of the Gauss–Newton Method for Injective-Overdetermined Systems of Equations Under a Majorant Condition. Computers & Mathematics With Applications 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-173303-MS. https://doi.org/10.2118/173303-MS.

Guo, Z., Reynolds, A. C., and Zhao, H. 2017a. A Physics-Based Data-Driven Model for History-Matching, Prediction and Characterization of Waterflooding Performance. In Proc., SPE Reservoir Simulation Conference,Montgomery, Texas, 20–22 February. SPE-182660-MS. https://doi.org/10.2118/182660-MS.

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.

Hastings, W. K. 1970. Monte Carlo Sampling Methods Using Markov Chains and Their Applications. Biometrika 57 (1): 97–109. https://doi.org/10.2307/2334940.

He, J., Xie, J., Wen, X-H. et al. 2016. An Alternative Proxy for History Matching Using Proxy-for-Data Approach and Reduced Order Modeling. Journal of Petroleum Science and Engineering 146: 392–399. https://doi.org/10.1016/j.petrol.2016.05.026.

Kahrobaei, S., Van Essen, G., Van Doren, J. et al. 2013. Adjoint-Based History Matching of Structural Models Using Production and Time-Lapse Seismic Data. In Proc., SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18–20 February. SPE-163586-MS. https://doi.org/10.2118/163586-MS.

Kitanidis, P. K. 1995. Quasi-Linear Geostatistical Theory for Inversing. Water Resources 31 (10): 2411–2419. https://doi.org/10.1029/95WR01945.

Le, D. H. and Reynolds, A. C. 2015. An Adaptive Ensemble Smoother for Assisted History Matching. TUPREP Research Report, The University of Tulsa.

Le, D. H., Emerick, A. A., and Reynolds, A. C. 2015a. An Adaptive Ensemble Smoother With Multiple Data Assimilation for Assisted History Matching. In Proc., SPE Reservoir Simulation Symposium, Houston, 23–25 February. SPE-173214-MS. https://doi.org/10.2118/173214-MS.

Le, D. H., Younis, R., and Reynolds, A. C. 2015b. A History Matching Procedure for Non-Gaussian Facies Based on ES-MDA. In Proc., SPE Reservoir Simulation Symposium, Houston, 23–25 February. SPE-173233-MS. https://doi.org/10.2118/173233-MS.

Leeuwen, V., Evensen, P. J., and Geir. 1996. Data Assimilation and Inverse Methods in Terms of a Probabilistic Formulation. Monthly Weather Review 124 (12): 2898–2913. https://doi.org/10.1175/1520-0493(1996)124<2898.DAAIMI>2.0.CO:2.

Li, H., Sarma, P., and Zhang, D. A. 2011. Comparative Study of the Probabilistic-Collocation and Experimental Design Methods for Petroleum-Reservoir Uncertainty Quantification. SPE J. 16 (3): 429–439. SPE-140738-PA. https://doi.org/10.2118/140738-PA.

Li, R., Reynolds, A. C., and Oliver, D. S. 2003. Sensitivity Coefficients for Three-Phase Flow History Matching. J Can Pet Technol 42 (4): 70–77. PETSOC-03-04-04. https://doi.org/10.2118/03-04-04.

Liang, F. and Lin, G. 2014. Simulated Stochastic Approximation Annealing for Global Optimization With a Square-Root Cooling Schedule. Journal of the American Statistical Association 109 (506): 847–863. https://doi.org/10.1080/01621459.2013.872993.

Liu, D. C. and Nocedal, J. 1989. On the Limited Memory BFGS Method for Large-Scale Optimization. Mathematical Programming 45 (1–3): 503–528. https://doi.org/10.1007/BF01589116.

Liu, N. and Oliver, D. S. 2003. Evaluation of Monte Carlo Methods for Assessing Uncertainty. SPE J. 8 (02): 188–195. SPE-84936-PA. https://doi.org/10.2118/84936-PA.

Mercer, J. 1909. Functions of Positive and Negative Type, and Their Connection With the Theory of Integral Equations. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 209, 415–446.

Micchelli, C. A. 1984. Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions. In Approximation Theory and Spline Functions, pp. 143–145. Netherlands: Springer.

Nævdal, G., Mannseth, T., and Vefring, E. H. 2002. Near-Well Reservoir Monitoring Through Ensemble Kalman Filter. In Proc., SPE/DOE Improved Oil Recovery Symposium, Tulsa, 13–17 April. SPE-75235-MS. https://doi.org/10.2118/75235-MS.

Nash, S. G. and Sofer, A. 1996. Linear and Nonlinear Programming. Blacklick, Ohio: McGraw-Hill Science/Engineering/Math.

Okano, H. and Koda, M. 2003. An Optimization Algorithm Based on Stochastic Sensitivity Analysis for Noisy Landscapes. Reliability Engineering and System Safety 79 (2): 245–252. https://doi.org/10.1016/S0951-8320(02)00236-3.

Oliver, D. S. 1996. On Conditional Simulation to Inaccurate Data. Math. Geology 28: 811–817.

Oliver, D. S. 2015. Metropolized Randomized Maximum Likelihood for Sampling From Multimodal Distributions. https://doi.org/10.1137/15M1033320. ArXiv e-prints, https://arxiv.org/pdf/1507.08563v2.pdf.

Platt, J. 1998. SequentialMinimal Optimization: A Fast Algorithmfor Training Support VectorMachines. Microsoft Research. TechReport:MSR-TR-98-14.

Powell, M. J. D. 2004. Least Frobenius Norm Updating of Quadratic Models That Satisfy Interpolation Conditions. Math. Program. B 100 (1): 183–215. https://doi.org/10.1007/s10107-003-0490-7.

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. https://doi.org/10.2118/04-03-03.

Schaaf, T., Coureaud, B., and Labat,N. 2008. Using ExperimentalDesign, Assisted HistoryMatching Tools and Bayesian Framework to Get Probabilistic Production Forecasts. Presented at the Europec/EAGE Conference and Exhibition, Rome, 9–12 June. SPE-113498-MS. https://doi.org/10.2118/113498-MS.

Smola, A. J. and Scho¨lkopf, B. 2004. A Tutorial on Support Vector Regression. Statistics and Computing 14 (3): 199–222.

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-182602-MS. https://doi.org/10.2118/182602-MS.

Suykens, J. A. and Vandewalle, J. 1999. Least Squares Support Vector Machine Classifiers. Neural Processing Letters 9 (3): 293–300. https://doi.org/10.1023/A:1018628609742.

Suykens, J. A., De Brabanter, J., Lukas, L. et al. 2002. Weighted Least Squares Support Vector Machines: Robustness and Sparse Approximation. Neurocomputing 48 (1–4): 85–105. https://doi.org/10.1016/S0925-2312(01)00644-0.

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, 20–30 September. SPE-175121-MS. https://dx.doi.org/10.2118/175121-MS.

Wantawin, M., Yu, W., and Sepehrnoori, K. 2017. An Iterative Response-Surface Methodology by Use of High-Degree-Polynomial Proxy Models for Integrated History Matching and Probabilistic Forecasting Applied to Shale-Gas Reservoirs. SPE J. 22 (6): 2012–2031. SPE-187938-PA. https://doi.org/10.2118/187938-PA.

Zhao, H., Li, G., and Reynolds, A. C. 2013. Large-Scale History Matching With Quadratic Interpolation Models. Computational Geosciences 17 (1): 117–138. https://doi.org/10.1007/s10596-012-9320-4.

Zhao, Y., Forouzanfar, F., and Reynolds, A. C. 2017. History Matching of Multi-Facies Channelized Reservoirs Using ES-MDA With Common Basis DCT. Computational Geosciences 21 (5–6): 1343–1364. https://doi.org/10.1007/s10596-016-9604-1.

Zhu, C., Byrd, R. H., Lu, P. et al. 1997. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN Routines for Large-Scale Bound Constrained Optimization. ACM Trans. on Mathematical Software 23 (4): 550–560. https://doi.org/10.1145/279232.279236.

Zhu, D. and Okuno, R. 2015. Analysis of Narrow-Boiling Behavior for Thermal Compositional Simulation. Presented at the SPE Reservoir Simulation Symposium, Houston, 23–25 February. SPE-173234-MS. https://doi.org/10.2118/173234-MS.

Zhu, D., Eghbali, S., Shekhar, C. et al. 2017. A Unified Algorithm for Phase-Stability/Split Calculation for Multiphase Isobaric-Isothermal Flash. SPE J. 23 (2): 498–521. SPE-175060-PA. https://doi.org/10.2118/175060-PA.

Zhou, W. and Zhang, L. 2010. Global Convergence of a Regularized Factorized Quasi-Newton Method for Nonlinear Least Squares Problems. Computational & Applied Mathematics. 29 (2). https://doi.org/10.1590/S1807-03022010000200006.


	PetroWiki was initially created from the seven volume Petroleum Engineering Handbook (PEH) published by the Society of Petroleum Engineers (SPE).


	The SEG Wiki is a useful collection of information for working geophysicists, educators, and students in the field of geophysics. The initial content has been derived from : Robert E. Sheriff's Encyclopedic Dictionary of Applied Geophysics, fourth edition.

Integration of Support Vector Regression With Distributed Gauss-Newton Optimization Method and Its Applications to the Uncertainty Assessment of Unconventional Assets

Summary

References

You may also be interested in…

EUR Assessment of Unconventional Assets Using Machine Learning and Distributed Computing Techniques

Other Resources