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Homogeneous chaos basis adaptation for design optimization under uncertainty: Application to the oil well placement problem

Published online by Cambridge University Press:  03 August 2017

Charanraj Thimmisetty ,
Panagiotis Tsilifis  and
Roger Ghanem
Charanraj Thimmisetty
Affiliation:
Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, California, USA
Panagiotis Tsilifis
Affiliation:
Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, California, USA
Roger Ghanem*
Affiliation:
Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, California, USA
*
Reprint requests to: Roger Ghanem, Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA. E-mail: ghanem@usc.edu
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Abstract

A new method is proposed for efficient optimization under uncertainty that addresses the curse of dimensionality as it pertains to the evaluation of probabilistic objectives and constraints. A basis adaptation strategy previously introduced by the authors is integrated into a design optimization framework that construes the optimization cost function as the quantity of interest and computes stochastic adapted bases as functions of design space parameters. With these adapted bases, the stochastic integrations at each design point are evaluated as low-dimensional integrals (mostly one dimensional). The proposed approach is demonstrated on a well-placement problem where the uncertainty is in the form of a stochastic process describing the permeability of the subsurface. An analysis of the method is carried out to better understand the effect of design parameters on the smoothness of the adaptation isometry.

Keywords

Adapted BasesKarhunen–Loeve ExpansionOptimization Under UncertaintyOptimal Well PlacementPolynomial Chaos Expansions
Type
Special Issue Articles
Information
AI EDAM , Volume 31 , Special Issue 3: Uncertainty Quantification for Engineering Design , August 2017 , pp. 265 - 276
Copyright
Copyright © Cambridge University Press 2017 

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References

REFERENCES

Bangerth, B., Klie, H., Wheeler, M.F., Stoffa, P.L., & Sen, M.K. (2006). On optimization algorithms for the reservoir oil well placement problem. Computational Geosciences 10, 303–319.Google Scholar
Beckner, B.L., & Song, X. (1995). Field development planning using simulated annealing-optimal economic well scheduling and placement. Proc. SPE Annual Technical Conf. Exhibition. Richardson, TX: Society of Petroleum Engineers.Google Scholar
Ben-Tal, A., El Ghaoui, L., & Nemirovski, A. (2009). Robust Optimization. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Bilionis, I., & Zabaras, N. (2013). Solution of inverse problems with limited forward solver evaluations: a bayesian perspective. Inverse Problems 30, 015004.Google Scholar
Bottou, L. (2010). Large-scale machine learning with stochastic gradient descent. Proc. COMPSTAT'2010. Heidelberg, Germany: Physica-Verlag.Google Scholar
Byrd, R.H., Lu, P., Nocedal, J., & Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing 16, 1190–1208.Google Scholar
Cameron, R., & Martin, W. (1947). The orthogonal development of nonlinear functionals in series of Fourier-hermite functionals. Annals of Mathematics 48, 385–392.Google Scholar
Chen, Z., Huan, G., & Ma, Y. (2006). Computational Methods for Multiphase Flows in Porous Media, Vol. 2. Philadelphia, PA: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Christie, M.A., & Blunt, M.J. (2001). Tenth SPE comparative solution project: a comparison of upscaling techniques. Proc. SPE Reservoir Evaluation & Engineering, Vol. 4. Richardson, TX: Society of Petroleum Engineers.Google Scholar
Corey, A.T. (1994). Mechanics of Immiscible Fluids in Porous Media. Highlands Ranch, CO: Water Resources Publications.Google Scholar
Eldred, M.S. (2009). Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design. Proc. 50th AIAA, ASME, ASCE, AHS, ASC Structures, Structural Dynamics, and Materials and Co-Located Conf., Palm Beach, CA.Google Scholar
Ghanem, R., & Spanos, P. (1991). Stochastic Finite Elements: A Spectral Approach. New York: Springer–Verlag.Google Scholar
Güyagüler, B. (2002). Optimization of well placement and assessment of uncertainty. PhD thesis. Stanford University.Google Scholar
Güyagüler, B., & Horne, R.N. (2001). Uncertainty assessment of well placement optimization. Proc. SPE Annual Technical Conf. Exhibition. Richardson, TX: Society of Petroleum Engineers.Google Scholar
Heyman, D.P., & Sobel, M.J. (2003). Stochastic Models in Operations Research: Stochastic Optimization, Vol. 2. North Chelmsford, MA: Courier Corporation.Google Scholar
Janson, S. (1999). Gaussian Hilbert spaces. Cambridge: Cambridge University Press.Google Scholar
Keshavarzzadeh, V., Meidani, H., & Tortorelli, D.A. (2016). Gradient based design optimization under uncertainty via stochastic expansion methods. Computer Methods in Applied Mechanics and Engineering 306, 47–76.Google Scholar
Kleywegt, A.J., Shapiro, A., & Homem-de Mello, T. (2002). The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization 12, 479–502.Google Scholar
Le Maître, O.P., Reagan, M.T., Najm, H.N., Ghanem, R.G., & Knio, O.M. (2002). A stochastic projection method for fluid flow: II. Random process. Journal of Computational Physics 181, 9–44.CrossRefGoogle Scholar
Peaceman, D.W. (2000). Fundamentals of Numerical Reservoir Simulation, Vol. 6. New York: Elsevier.Google Scholar
Reagan, M.T., Najm, H.N., Ghanem, R.G., & Knio, O.M. (2003). Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection. Combustion and Flame 132, 545–555.Google Scholar
Rian, D.T., & Hage, A. (1994). Automatic optimization of well locations in a north sea fractured chalk reservoir using a front tracking reservoir simulator. Proc. Int. Petroleum Conf. Exhibition of Mexico. Richardson, TX: Society of Petroleum Engineers.Google Scholar
Robbins, H., & Monro, S. (1951). A stochastic approximation method. Annals of Mathematical Statistics 22(3), 400–407.Google Scholar
Scott, D.W. (2015). Multivariate Density Estimation: Theory, Practice, and Visualization. Hoboken, NJ: Wiley.Google Scholar
Spall, J.C. (1992). Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on Automatic Control 37, 332–341.Google Scholar
Stone, H.L. (1970). Probability model for estimating three-phase relative permeability. Journal of Petroleum Technology 22(2), 214–218.Google Scholar
Stone, H.L. (1973). Estimation of three-phase relative permeability and residual oil data. Journal of Petroleum Technology 12(4).Google Scholar
Tipireddy, R., & Ghanem, R.G. (2014). Basis adaptation in homogeneous chaos spaces. Journal of Computational Physics 259, 304–317.Google Scholar
Tsilifis, P., & Ghanem, R. (2016). Reduced Wiener chaos representation of random fields via basis adaptation and projection. Journal of Computational Physics. Advance online publication.Google Scholar
Tsilifis, P., Ghanem, R., & Hajali, P. (2017). Efficient Bayesian experimentation using an expected information gain lower bound. SIAM/ASA Journal on Uncertainty Quantification 5(1), 30–62.Google Scholar
Yeten, B., Durlofsky, L.J., & Aziz, K. (2002). Optimization of nonconventional well type, location and trajectory. Proc. SPE Annual Technical Conf. Exhibition. Richardson, TX: Society of Petroleum Engineers.Google Scholar