Optimization Algorithms Based on Combining FD Approximations and Stochastic Gradients Compared with Methods Based Only on a Stochastic Gradient

Authors
Xia Yan (PetroChina Coalbed Methane Company Limited) | Albert C. Reynolds (University of Tulsa)
DOI
http://dx.doi.org/10.2118/163613-PA
Document ID
SPE-163613-PA
Publisher
Society of Petroleum Engineers
Source
SPE Journal
Volume
19
Issue
05
Publication Date
October 2014
Document Type
Journal Paper
Pages
873 - 890
Language
English
ISSN
1086-055X
Copyright
2014.Society of Petroleum Engineers
Disciplines
5.5.8 History Matching, 1.7.5 Well Control, 5.5 Reservoir Simulation
Keywords
reservoir description and dynamics, stochastic gradient, finite difference
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Summary

Optimization algorithms that incorporate a stochastic gradient [such as simultaneous-perturbation stochastic approximation (SPSA), simplex, EnOpt) are easy to implement in conjunction with any reservoir simulator. However, for realistic problems, a stochastic gradient provides only a rough approximation of the true gradient, and, in particular, the angle between a stochastic gradient and the associated true gradient is typically far from zero even though a properly computed stochastic gradient usually represents an uphill direction. This paper develops a more robust optimization procedure by replacing the components of largest magnitude of the stochastic gradient with a finite-difference (FD) approximation of the pertinent partial derivatives. In essence, the objective of the method is to determine which components of the unknown true gradient are most important and then replace the corresponding components of the stochastic gradient with more-accurate FD approximations. This modified gradient can then be used in a gradient-based optimization algorithm to find the minimum or maximum of a given cost function. Our focus application is the estimation of optimal well controls, but it is clear that the method could also be used for other applications, including history matching.

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References

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.

Brouwer, D. and Jansen, J. 2004. Dynamic Optimization of Waterflooding With Smart Wells Using Optimal Control Theory. SPE J. 9 (4): 391–402. SPE-78278-PA. http://dx.doi.org/10.2118/78278-PA.

Chen, C., Li, G., and Reynolds, A.C. 2010. Closed-Loop Reservoir Management on the Brugge Test Case. Computat. Geosci. 14 (4): 691–703. http://dx.doi.org/10.1007/s10596-010-9181-7.

Chen, C., Li, G., and Reynolds, A. C. 2011. Robust Constrained Optimization of Short and Long-Term NPV for Closed-Loop Reservoir Management. Presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 21–23 February. SPE-141314-MS. http://dx.doi.org/10.2118/141314-MS.

Chen, Y. and Oliver, D. 2009. Ensemble-Based Closed-Loop Optimization Applied to Brugge Field. Presented at SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 2–4 February. SPE-118926-MS. http://dx.doi.org/10.2118/118926-MS.  

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.

Conn, A. R., Scheinberg, K., and Vicente, L. N. 2009. Introduction to Derivative Free Optimization. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics. 

Custódio, A. L. and Vicente, L. N. 2007. Using Sampling and Simplex Derivatives in Pattern Search Methods. SIAM J. Optimiz. 18 (2): 537–555. http://dx.doi.org/10.1137/050646706.

Custódio, A. L., Rocha, H., and Vicente, L. N. 2008. Incorporating Minimum Frobenius Norm Models in Direct Search. Technical Report No. 08–51, Department of Mathematics, University of Coimbra, Coimbra, Portugal (2008).

de Montleau, P., Cominelli, A., Neylong, K., et al. 2006. Production Optimization under Constraints Using Adjoint Gradients. Oral presentation given at the 10th European Conference on the Mathematics of Oil Recovery, Amsterdam, The Netherlands, 4–7 September.

Do, S. 2012. Application SPSA-Type Algorithms to Production Optimization. PhD dissertation, the University of Tulsa, Tulsa, Oklahoma (2012).

Do, S. and Reynolds, A. C. 2013. Theoretical Connections Between Optimization Algorithms Based On an Approximate Gradient. Computat. Geosci. 17 (6): 959–973. http://dx.doi.org/10.1007/s10596-013-9368-9.

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. http://dx.doi.org/10.2118/90058-PA.

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.

Jansen, J., Brouwer, D., Naevdal, G., et al. 2005. Closed-Loop Reservoir Management, First Break 23 (1): 43–48. http://dx.doi.org/10.3997/1365-2397.2005002.

Kraaijevanger, J. F. B. M., Egberts, P. J. P., Valstar, J. R., et al. 2007. Optimal Waterflood Design Using the Adjoint Method. Presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 26–28 February. SPE-105764-MS. http://dx.doi.org/10.2118/105764-MS.

Li, G. and Reynolds, A. C. 2009. Iterative Ensemble Kalman Filters for Data Assimilation. SPE J. 14 (3): 496–505. SPE-109808-PA. http://dx.doi.org/10.2118/109808-PA.

Li, G. and Reynolds, A. C. 2011. Uncertainty Quantification of Reservoir Performance Predictions Using a Stochastic Optimization Algorithm. Computat. Geosci. 15 (3): 451–462. http://dx.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. http://dx.doi.org/10.2118/87336-PA.

Oliver, D. S., Reynolds, A. C., and Liu, N. 2008. Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge, UK: Cambridge University Press, Cambridge.

Patelli, E. and Pradlwarter, H. 2010. MC Gradient Estimation in High Dimensions. Int. J. Numer. Meth. Eng. 81 (2): 172–188. http://dx.doi.org/10.1002/nme.2687.

Pellissetti, M. F., Pradlwarter, H. J., and Schueller, G. I. 2005. Relative Importance of Uncertain Parameters in Aerospace Applications. Oral presentation given at the III European Conference on Computational Mechanics, Lisbon, Portugal, 5–9 June.

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. http://dx.doi.org/10.2118/119094-PA.

Pradlwarter, H. 2005. Relative Importance of Uncertain Structural Parameters. Oral presentation given at the European Conference on Spacecraft Structures, Materials and Mechanical Testing, Noordwijk, The Netherlands, 10–12 May.

Pradlwarter, H. 2007a. Relative Importance of Uncertain Structural Parameters. Part I: Algorithm. Comput. Mech. 40 (4): 627–635. http://dx.doi.org/10.1007/s00466-006-0127-9.

Pradlwarter, H. 2007b. Relative Importance of Uncertain Structural Parameters. Part II: Applications. Comput. Mech. 40 (4): 637–649. http://dx.doi.org/10.1007/s00466-006-0128-8.

Reynolds, A. C., He, N., and Oliver, D. S. 1999. Reducing Uncertainty in Geostatistical Description with Well-Testing Pressure Data. In Reservoir Characterization: Recent Advances, ed. R. A. Schatzinger and J. F. Jordan, Chap. 10, 149–162. Tulsa, Oklahoma: American Association of Petroleum Geologists.

Reynolds, A. C., Zafari, M., and Li, G. 2006. Iterative Forms of the Ensemble Kalman Filter. Oral presentation given at the 10th European Conference on the Mathematics of Oil Recovery, Amsterdam, The Netherlands, 4–7 September.

Rodrigues, J. R. P. 2006. Calculating Derivatives for Automatic History Matching. Computat. Geosci. 10 (1): 119–136. http://dx.doi.org/10.1007/s10596-005-9013-3.

Sarma, P., Chen, W., Durlofsky, L., et al. 2008. Production Optimization With Adjoint Models Under Nonlinear Control-State Path Inequality Constraints. SPE Res Eval & Eng 11 (2): 326–339. SPE-99959-PA. http://dx.doi.org/10.2118/99959-PA.

Sarma, P., Durlofsky, L. J., Aziz, K., et al. 2007. A New Approach to Automatic History Matching Using Kernel PCA. Presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 26–28 February. SPE-106176-MS. http://dx.doi.org/10.2118/106176-MS.

Spall, J. C. 1992. Multivariate Stochastic Approximation Using a Simultaneous Perturbation Gradient Approximation. IEEE Trans. Automat. Contr. 37 (3): 332–341. http://dx.doi.org/10.1109/9.119632.

Spall, J. C. 1998. Implementation of the Simultaneous Perturbation Algorithm for Stochastic Optimization. IEEE Trans. Aero. Elec. Sys. 34 (3): 817–823. http://dx.doi.org/10.1109/7.705889.

Spall, J. C. 2001. Accelerated Second-Order Stochastic Optimization Using Only Function Measurements. Oral presentation given at the 2001 Winter Simulation Conference.

van Essen, G., den Hof, P. V., and Jansen, J. 2011. Hierarchical Long-Term and Short-Term Production Optimization. SPE J. 16 (1): 191–199. SPE-124332-PA. http://dx.doi.org/10.2118/124332-PA.

van Essen, G., Zandvliet, M., den Hof, P. V., et al. 2009. Robust Waterflooding Optimization of Multiple Geological Scenarios. SPE J. 14 (1): 202–210. SPE-102913-PA. http://dx.doi.org/10.2118/102913-PA.

Wu, Z., Reynolds, A. C., and Oliver, D. S. 1999. Conditioning Geostatistical Models to Two-Phase Production Data. SPE J. 3 (2): 142–155. SPE-56855-PA. http://dx.doi.org/10.2118/56855-PA.

Yan, X. and Reynolds, A. C.  2013. An Optimization Algorithm Based on Combining FD Approximations and Stochastic Gradients. Presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18–20 February. SPE-163613-MS. http://dx.doi.org/10.2118/163613-MS.

Zakirov, I. S., Aanonsen, S. I., Zakirov, E. S., et al. 1996. Optimizating Reservoir Performance by Automatic Allocation of Well Rates. Oral presentation given at the 5th European Conference on the Mathematics of Oil Recovery, Leoben, Austria, 3–6 September.

Zhang, F. and Reynolds, A. C. 2002. Optimization algorithms for automatic history matching of production data. Oral presentation given at the 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, 3–6 September.

Zhao, H., Chen, C., Do, S., et al. 2011. Maximization of a Dynamic Quadratic Interpolation Model for Production Optimization. SPE J. 18 (6): 1012–1025. SPE-141317-PA. http://dx.doi.org/10.2118/141317-PA.