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
5.1.4 Monitoring and Control, 6 Reservoir Description and Dynamics, 5.1 Design and Optimization, 5 Production and Operations, 6.8 Fundamental Research in Reservoir Description and Dynamics
Keywords
stochastic gradient, reservoir description and dynamics, 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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