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Abstract

Summary

Uncertainty quantification workflows which map input uncertainties to output uncertainties are an important part of many reservoir simulation applications. Examples are estimation of prediction uncertainties including history data, reserves estimation, or optimization under uncertainty. Over the last few years, proxy modeling techniques have turned out to be essential for improving the efficiency of the employed methods, with one example being stochastic sampling techniques like Markov Chain Monte Carlo. An established proxy model family is the generalized Polynomial Chaos Expansion (gPCE). Some applications to reservoir simulation exist - also in the context of history matching.

In this work we focus on a non-intrusive, efficient way to construct such gPCE representations for multi-dimensional input uncertainties: sparse-grid spectral projection – also known as Smolyak grid. In standard approaches, which derive their multi-dimensional cubature rules from full tensor-products of one-dimensional rules, the number of required points grows exponentially with the input dimension. This causes a significant computational effort since each point represents a forward simulation. On the other hand, sparse-grid techniques used in this work show polynomial scaling behavior by “thinning out” the full tensor-product. However, in order to avoid possible numerical instabilities of this technique a design of simulation cases needs to be applied. Practical consequences and guidelines for real applications will be described in detail.

The numerical framework for constructing gPCEs is applied to uncertainty quantification workflows in reservoir simulation. We investigate performance criteria for representing key performance indicators using gPCEs, e.g., cumulative properties like oil production total or transient properties such as bottom hole pressure. Practical application designs are derived and described.

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2014-09-08
2024-04-26

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References

  1. Alkhatib, A., & King, P.
    (2014). Robust quantification of parametric uncertainty for surfactant-polymer flooding. Computational Geosciences, 18(1), 77–101. doi:10.1007/s10596‑013‑9384‑9
     https://doi.org/10.1007/s10596-013-9384-9 [Google Scholar]
  2. Bazargan, H., Christie, M., & Tchelepi, H.
    (2013). Efficient Markov chain Monte Carlo sampling using polynomial chaos expansion. Proceedings of the SPE Reservoir Simulation Symposium, Feb 18–20, The Woodlands, TX, USA. doi:10.2118/163663‑MS
     https://doi.org/10.2118/163663-MS [Google Scholar]
  3. Blatman, G., & Sudret, B.
    (2011). Adaptive sparse polynomial chaos expansion based on least angle regression. Journal of Computational Physics, 230(6), 2345–2367. doi:10.1016/j.jcp.2010.12.021
     https://doi.org/10.1016/j.jcp.2010.12.021 [Google Scholar]
  4. Bungartz, H.-J., & Griebel, M.
    (2004, 5). Sparse grids. Acta Numerica, 13, 147–269. doi:10.1017/S0962492904000182
     https://doi.org/10.1017/S0962492904000182 [Google Scholar]
  5. Elman, H., & Liao, Q.
    (2013). Reduced basis collocation methods for partial differential equations with random coefficients. SIAM/ASA Journal on Uncertainty Quantification, 1(1), 192–217. doi:10.1137/120881841
     https://doi.org/10.1137/120881841 [Google Scholar]
  6. Gerritsma, M., van der Steen, J.-B., Vos, P., & Karniadakis, G.
    (2010). Time-dependent generalized polynomial chaos. Journal of Computational Physics, 229(22), 8333–8363. doi:10.1016/j.jcp.2010.07.020
     https://doi.org/10.1016/j.jcp.2010.07.020 [Google Scholar]
  7. Gerstner, T., & Griebel, M.
    (1998). Numerical integration using sparse grids. Numerical Algorithms, 18(3–4), 209–232. doi:10.1023/A%3A1019129717644
     https://doi.org/10.1023/A%3A1019129717644 [Google Scholar]
  8. (2003). Dimension-adaptive tensor-product quadrature. Computing, 71(1), 65–87. doi:10.1007/s00607‑003‑0015‑5
     https://doi.org/10.1007/s00607-003-0015-5 [Google Scholar]
  9. Jaynes, E. T.
    (2003). Probability theory - the logic of science. Cambridge, UK: Cambridge University Press.
     [Google Scholar]
  10. Keese, A., & Matthies, H. G.
    (2003). Numerical methods and Smolyak quadrature for nonlinear stochastic partial differential equations. Informatikbericht, Institut für Wissenschaftliches Rechnen, Technische Universität Braunschweig. Retrieved from http://www.digibib.tu-bs.de/?docid=00001471
     [Google Scholar]
  11. Le Maître, O. P., & Knio, O. M.
    (2010). Spectral methods for uncertainty quantification with applications to computational fluid dynamics. Springer. doi:10.1007/978‑90‑481‑3520‑2
     https://doi.org/10.1007/978-90-481-3520-2 [Google Scholar]
  12. Nobile, F., Tempone, R., & Webster, C.
    (2008, January). A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM Journal on Numerical Analysis, 46(5), 2309–2345. doi:10.1137/060663660
     https://doi.org/10.1137/060663660 [Google Scholar]
  13. Pajonk, O., Rosić, B. V., & Matthies, H. G.
    (2013). Sampling-free linear Bayesian updating of model state and parameters using a square root approach. Computers & Geosciences, 55, 70–83. doi:10.1016/j.cageo.2012.05.017
     https://doi.org/10.1016/j.cageo.2012.05.017 [Google Scholar]
  14. Saad, G., & Ghanem, R.
    (2009). Characterization of reservoir simulation models using a polynomial chaos-based ensemble Kalman filter. Water Resources Research, 45, W04417. doi:10.1029/2008WR007148
     https://doi.org/10.1029/2008WR007148 [Google Scholar]
  15. Smolyak, S.
    (1963). Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Mathematics Doklady, 240–243.
     [Google Scholar]
  16. Sochala, P., & Le Maître, O.
    (2013). Polynomial chaos expansion for subsurface flows with uncertain soil parameters. Advances in Water Resources, 62, Part A, 139–154. doi:10.1016/j.advwatres.2013.10.003
     https://doi.org/10.1016/j.advwatres.2013.10.003 [Google Scholar]
  17. Tompkins, M., & Prange, M.
    (2012). Efficient estimation of polynomial chaos proxies using generalized sparse quadrature. Proceedings of the SEG Annual Meeting. Las Vegas: Society of Exploration Geophysicists. doi:10.1190/segam2012‑0811.1
     https://doi.org/10.1190/segam2012-0811.1 [Google Scholar]
  18. Xiu, D., & Karniadakis, G. E.
    (2002). The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing, 24(2), 619–644. doi:10.1137/S1064827501387826
     https://doi.org/10.1137/S1064827501387826 [Google Scholar]
  19. Zander, E.
    (2014, May 19). SGLib - A Matlab/Octave toolbox for stochastic Galerkin methods. doi:10.5281/zenodo.9966
     https://doi.org/10.5281/zenodo.9966 [Google Scholar]
  20. Zeng, L., & Zhang, D.
    (2010). A stochastic collocation based Kalman filter for data assimilation. Computational Geosciences, 14(4), 721–744. doi:10.1007/s10596‑010‑9183‑5
     https://doi.org/10.1007/s10596-010-9183-5 [Google Scholar]
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Evaluation of Non-intrusive Generalized Polynomial Chaos Expansion in the Context of Reservoir Simulation
Conference Proceedings 2014, 1 (2014); https://doi.org/10.3997/2214-4609.20141866
/content/papers/10.3997/2214-4609.20141866
/content/papers/10.3997/2214-4609.20141866
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