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

  • Charanraj Thimmisetty (a1), Panagiotis Tsilifis (a2) and Roger Ghanem (a2)

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.

Corresponding author
Reprint requests to: Roger Ghanem, Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA. E-mail:
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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, 303319.
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.
Ben-Tal, A., El Ghaoui, L., & Nemirovski, A. (2009). Robust Optimization. Princeton, NJ: Princeton University Press.
Bilionis, I., & Zabaras, N. (2013). Solution of inverse problems with limited forward solver evaluations: a bayesian perspective. Inverse Problems 30, 015004.
Bottou, L. (2010). Large-scale machine learning with stochastic gradient descent. Proc. COMPSTAT'2010. Heidelberg, Germany: Physica-Verlag.
Byrd, R.H., Lu, P., Nocedal, J., & Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing 16, 11901208.
Cameron, R., & Martin, W. (1947). The orthogonal development of nonlinear functionals in series of Fourier-hermite functionals. Annals of Mathematics 48, 385392.
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.
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.
Corey, A.T. (1994). Mechanics of Immiscible Fluids in Porous Media. Highlands Ranch, CO: Water Resources Publications.
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.
Ghanem, R., & Spanos, P. (1991). Stochastic Finite Elements: A Spectral Approach. New York: Springer–Verlag.
Güyagüler, B. (2002). Optimization of well placement and assessment of uncertainty. PhD thesis. Stanford University.
Güyagüler, B., & Horne, R.N. (2001). Uncertainty assessment of well placement optimization. Proc. SPE Annual Technical Conf. Exhibition. Richardson, TX: Society of Petroleum Engineers.
Heyman, D.P., & Sobel, M.J. (2003). Stochastic Models in Operations Research: Stochastic Optimization, Vol. 2. North Chelmsford, MA: Courier Corporation.
Janson, S. (1999). Gaussian Hilbert spaces. Cambridge: Cambridge University Press.
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, 4776.
Kleywegt, A.J., Shapiro, A., & Homem-de Mello, T. (2002). The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization 12, 479502.
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, 944.
Peaceman, D.W. (2000). Fundamentals of Numerical Reservoir Simulation, Vol. 6. New York: Elsevier.
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, 545555.
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.
Robbins, H., & Monro, S. (1951). A stochastic approximation method. Annals of Mathematical Statistics 22(3), 400407.
Scott, D.W. (2015). Multivariate Density Estimation: Theory, Practice, and Visualization. Hoboken, NJ: Wiley.
Spall, J.C. (1992). Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on Automatic Control 37, 332341.
Stone, H.L. (1970). Probability model for estimating three-phase relative permeability. Journal of Petroleum Technology 22(2), 214218.
Stone, H.L. (1973). Estimation of three-phase relative permeability and residual oil data. Journal of Petroleum Technology 12(4).
Tipireddy, R., & Ghanem, R.G. (2014). Basis adaptation in homogeneous chaos spaces. Journal of Computational Physics 259, 304317.
Tsilifis, P., & Ghanem, R. (2016). Reduced Wiener chaos representation of random fields via basis adaptation and projection. Journal of Computational Physics. Advance online publication.
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), 3062.
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.
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