Advertisement

Multiscale Stochastic Representations Using Polynomial Chaos Expansions with Gaussian Process Coefficients

  • Charanraj Thimmisetty
  • Fred Aminzadeh
  • Kelly Rose
  • Roger GhanemEmail author
Original Article
  • 802 Downloads

Abstract

We address an important component of risk mitigation for ultra-deep sea drilling in the Gulf of Mexico (GoM), namely the probabilistic characterization of fluid fluxes at the seafloor from future drilling operations. In the process, we develop a stochastic representation of functions defined on a high-dimensional space conditional on their marginal statistics and their global correlation structure. The representation leverages a particular structure of the functional dependence of interest which exhibits scale separation. Specifically, we construct a polynomial chaos representation for scalar quantities of interest whose coefficients are themselves random. The intrinsic randomness of the polynomial chaos expansion (PCE) reflects local uncertainty and captures dependence on a subset of the parameters, while randomness in the PCE coefficients captures a global structure of the uncertainty and dependence on the remaining parameters in the high-dimensional space. This construction is demonstrated by predicting wellbore signatures in the GoM where a 120-dimensional table is populated at several thousand wellbore locations throughout the GoM. Physics-based models of multiphase flow in porous media are used to calculate the PCE representations at the sites where data is available. In this context, random parameters describing the subsurface define the parameter set with respect to which PCE is constructed. A Gaussian process in parameter space is then developed for each coefficient in these representations. The combined probabilistic representation permits the delineation of separate stochastic influences on predictions of interest.

Notes

Acknowledgments

The assistance of Dr. Nima Jabbari and Dr. Arman Khodabakhsh Nejad with the PIPESIM simulations is gratefully acknowledged, as is the assistance of Ms. Corinne Disenhof in formatting the BOEM data to make it spatially referenced. This technical effort was performed in support of the National Energy Technology Laboratory’s ongoing research under Section 999 of the Energy Policy Act of 2005. This publication was prepared in part as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, expresses or implies, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference therein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed therein do not necessarily state or reflect those of the United States Government or any agency thereof.

References

  1. 1.
    B.M. Adams, L.E. Bauman, W.J. Bohnhoff, K.R. Dalbey, M.S. Ebeida, J.P. Eddy, M.S. Eldred, P.D. Hough, K.T. Hu, J.D. Jakeman, J.A. Stephens, L.P. Swiler, D.M. Vigil, T.M. Wildey, Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.0 User’s Manual, Sandia Technical Report SAND2014-4633 (2006)Google Scholar
  2. 2.
    K. Anjyo, J. Lewis, RBF interpolation and Gaussian process regression through an RKHS formulation. J. Math. Ind. 3(6), 63–71 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    BOEM: 2010 Atlas of Gulf of Mexico gas and oil sands data, accessed 2012. https://www.data.boem.gov/GGStudies/Files/2010 (2012)
  4. 4.
    BOEM: Borehole data, Gulf of Mexico, accessed 2012. https://www.data.boem.gov/Well/Borehole/Default.aspx (2012)
  5. 5.
    L.A. Burke, S.A. Kinney, R.F. Dubiel, J.K. Pitman, Regional map of the 0.70 psi/ft pressure gradient and development of the regional geopressure-gradient model for the onshore and offshore Gulf of Mexico basin, USA: Gulf Coast Association of Geological Societies Journal, 1, 97–106 (2012)Google Scholar
  6. 6.
    Z. Chen, G. Huan, Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, PA (2006)Google Scholar
  7. 7.
    N. Cressie, The origins of kriging. Math. Geol. 22(3), 239–252 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    B. Dindoruk, P.G. Christman et al., PVT properties and viscosity correlations for Gulf of Mexico oils. SPE Reserv. Eval. Eng. 7(06), 427–437 (2004)Google Scholar
  9. 9.
    D.B. Dunson, N. Pillai, J.H. Park, Bayesian density regression. J. R. Stat. Soc. Ser. B (Stat Methodol.) 69(2), 163–183 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    R.G. Ghanem, P.D. Spanos, Vol. 41. Stochastic finite elements: a spectral approach (Springer, Berlin, 1991)CrossRefGoogle Scholar
  11. 11.
    A.A. Giunta, L.P. Swiler, S.L. Brown, M.S. Eldred, M.D. Richards, E.C. Cyr, in The surfpack software library for surrogate modeling of sparse irregularly spaced multidimensional data. 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, AIAA, (2006), pp. 1708–1736Google Scholar
  12. 12.
    T. Gneiting, M. Genton, P. Guttorp, Geostatistical space-time models, stationarity, separability and full symmetry. In Statistical Methods for Spatio-Temporal Systems, 151–175, ed. by B. Finkenstadt, L. Held, V. Isham, Taylor and Francis, Boca Raton, (2007)Google Scholar
  13. 13.
    A. Hagedorn, B. Kermit, Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits. J. Petrol. Tech. 17(4), 475–484 (1965)CrossRefGoogle Scholar
  14. 14.
    E. Isaaks, R. Srivastava. Applied geostatistics (Oxford University, London, 2011)Google Scholar
  15. 15.
    D. Krige, A statistical approach to some basic mine valuation problems on the witwatersrand. J. Chem. Metall. Min. Soc. 52, 119– 139 (1951)Google Scholar
  16. 16.
    N.S.N. Lam, Spatial interpolation methods: a review. The American Cartographer. 10(2), 129–150 (1983)CrossRefGoogle Scholar
  17. 17.
    G. Matheron, Principles of geostatistics. Econ. Geol. 58(8), 1246–1266 (1963)CrossRefGoogle Scholar
  18. 18.
    M.K. McNutt, R. Camilli, T.J. Crone, G.D. Guthrie, P.A. Hsieh, T.B. Ryerson, O. Savas, F. Shaffer, Review of flow rate estimates of the deepwater horizon oil spill. Proc. Natl. Acad. Sci. 109(50), 20,260–20,267 (2012)CrossRefGoogle Scholar
  19. 19.
    P. Müller, A. Erkanli, M. West, Bayesian curve fitting using multivariate normal mixtures. Biometrika. 83(1), 67–79 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, E. Duchesnay, Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12, 2825–2830 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    C. Rasmussen, C. Williams. Gaussian processes for machine learning (MIT press, Cambridge, 2006)zbMATHGoogle Scholar
  22. 22.
    S. Sakamoto, R. Ghanem, Simulation of multi-dimensional non-gaussian non-stationary random fields. Probab. Eng. Mech. 17(2), 167–176 (2002)CrossRefGoogle Scholar
  23. 23.
    ECLIPSE 2009 User’s Manual, Schlumberger Corporation (2009)Google Scholar
  24. 24.
    PIPESIM Version 2011 User’s Manual, Schlumberger Corporation (2011)Google Scholar
  25. 25.
    M. Shinozuka, G. Deodatis, Simulation of stochastic processes by spectral representation. Appl. Mech. Rev. 44(4), 191–204 (1991)MathSciNetCrossRefGoogle Scholar
  26. 26.
    C. Soize, R. Ghanem, Data-driven probability concentration and sampling on manifold. J. Comput. Phys. 321, 242–258 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    C. Thimmisetty, R. Ghanem, J. White, X. Chen, High-dimensional intrinsic interpolation using Gaussian process regression and diffusion maps. Math. Geosci. (2017)  https://doi.org/10.1007/s11004-017-9705-y
  28. 28.
    J. Vogel, Inflow performance relationships for solution-gas drive wells. J. Petrol. Tech. 20(1), 83–92 (1968)CrossRefGoogle Scholar
  29. 29.
    M. Voltz, R. Webster, A comparison of kriging, cubic splines and classification for predicting soil properties from sample information. J. Soil Sci. 41(3), 473–490 (1990)CrossRefGoogle Scholar
  30. 30.
    G. Wahba, Spline models for observational data, vol. 59 Siam (1990)Google Scholar
  31. 31.
    D. Xiu, G.E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetCrossRefGoogle Scholar
  32. 32.
    F. Yamazaki, M. Shinozuka, Digital generation of non-gaussian stochastic fields. J. Eng. Mech. 114(7), 1183–1197 (1988)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermoreUSA
  2. 2.University of Southern CaliforniaLos AngelesUSA
  3. 3.National Energy Technology LaboratoryAlbanyUSA

Personalised recommendations