Computational Geosciences

, Volume 15, Issue 3, pp 565–577

An integrative approach to robust design and probabilistic risk assessment for CO2 storage in geological formations

  • Sergey Oladyshkin
  • Holger Class
  • Rainer Helmig
  • Wolfgang Nowak
Original Paper

Abstract

CO2 storage in geological formations is currently being discussed intensively as a technology with a high potential for mitigating CO2 emissions. However, any large-scale application requires a thorough analysis of the potential risks. Current numerical simulation models are too expensive for probabilistic risk analysis or stochastic approaches based on a brute-force approach of repeated simulation. Even single deterministic simulations may require parallel high-performance computing. The multiphase flow processes involved are too non-linear for quasi-linear error propagation and other simplified stochastic tools. As an alternative approach, we propose a massive stochastic model reduction based on the probabilistic collocation method. The model response is projected onto a higher-order orthogonal basis of polynomials to approximate dependence on uncertain parameters (porosity, permeability, etc.) and design parameters (injection rate, depth, etc.). This allows for a non-linear propagation of model uncertainty affecting the predicted risk, ensures fast computation, and provides a powerful tool for combining design variables and uncertain variables into one approach based on an integrative response surface. Thus, the design task of finding optimal injection regimes explicitly includes uncertainty, which leads to robust designs with a minimum failure probability. We validate our proposed stochastic approach by Monte Carlo simulation using a common 3D benchmark problem (Class et al., Comput Geosci 13:451–467, 2009). A reasonable compromise between computational efforts and precision was reached already with second-order polynomials. In our case study, the proposed approach yields a significant computational speed-up by a factor of 100 compared with the Monte Carlo evaluation. We demonstrate that, due to the non-linearity of the flow and transport processes during CO2 injection, including uncertainty in the analysis leads to a systematic and significant shift of the predicted leakage rates toward higher values compared with deterministic simulations, affecting both risk estimates and the design of injection scenarios.

Keywords

Polynomial chaos CO2 storage Multiphase flow Porous media Risk assessment Uncertainty Integrative response surfaces 

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References

  1. 1.
    Augustin, F., Gilg, A., Paffrath, M., Rentrop, P., Wever, U.: Polynomial chaos for the approximation of uncertainties: chances and limits. Eur. J. Appl. Math. 19, 149–190 (2008)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Askey, R., Wilson, J.: Some basic hypergeometric polynomials that generalize Jacobi polynomials. Memoirs of the American Mathematical Society, p. 319. AMS, Providence (1985)Google Scholar
  3. 3.
    Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klofkorn, R., Kornhu R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. II. Implementation and tests in DUNE. Comput. 82(2–3), 121–138 (2008)MATHGoogle Scholar
  4. 4.
    Birkholzer, J.T., Zhou, Q., Tsang, Ch.-F.: Large-scale impact of CO2 storage in deep saline aquifers: A sensitivity study on pressure response in stratified systems. Int. J. Greenhouse Gas Control 3, 181–194 (2009)CrossRefGoogle Scholar
  5. 5.
    Class, H., Ebigbo, A., Helmig, R., Dahle, H., Nordbotten, J.N., Celia, M.A., Audigane, P., Darcis., M., Ennis-King, J., Fan, Y., Flemisch, B., Gasda, S., Jin, M., Krug, S., Labregere, D., Naderi, A., Pawar, R.J., Sbai, A., Sunil, G.T., Trenty, L., Wei, L.: A benchmark-study on problems related to CO2 storage in geologic formations. Comput. Geosci. 13, 451–467 (2009)CrossRefGoogle Scholar
  6. 6.
    Cortis, A., Oldenburg, C., Benson, S.M.: The role of optimality in characterizing CO2 seepage from geologic carbon sequestration sites. Int. J. Greenhouse Gas Control 2, 640–652 (2008)CrossRefGoogle Scholar
  7. 7.
    Ebigbo, A., Class, H., Helmig, R.: CO2 leakage through an abandoned well: problem-oriented benchmarks. Comput. Geosci. 11(2), 103–115 (2007)MATHCrossRefGoogle Scholar
  8. 8.
    Flemisch, B., Fritz, J., Helmig, R., Niessner, J., Wohlmuth, B.: DUMUX: a multi-scale multi-physics toolbox for flow and transport processes in porous media. In: Ibrahimbegovic, A., Dias, F. (eds.) ECCO3MAS Thematic Conference on Multi-scale Computational Methods for Solids and Fluids, Cachan, France, 28–30 November 2007Google Scholar
  9. 9.
    Ghomain, Y., Pope, G.A., Sepehrnoori, K.: Development of a response surface based model for minimum miscibility pressure (MMP) correlation of CO2 flooding, paper SPE 116719. In: 2008 SPE Annual Technical Conference and Exhibition, Denver, CO, 21–24 September 2008. doi:10.2118/116719-MS
  10. 10.
    Jakeman, J.D., Roberts, S.G.: Stochastic Galerkin and collocation methods for quantifying uncertainty in differential equations: a review. Aust. N.Z. Ind. Appl. Math. J. 50, C815–C830 (2008)MathSciNetGoogle Scholar
  11. 11.
    Jaynes, E.T.: On the rationale of maximum-entropy methods. Proc. IEEE 106, 939–952 (1982)CrossRefGoogle Scholar
  12. 12.
    IPCC: Special report on carbon dioxide capture and storage. Technical Report, Intergovernmental Panel on Climate Change (IPCC), prepared by Working Group III. Cambridge University Press, Cambridge (2005)Google Scholar
  13. 13.
    Isukapalli, S.S., Roy, A., Georgopoulos, P.G.: Stochastic response surface methods (SRSMs) for uncertainty propagation: Application to environmental and biological systems. Risk Anal. 18(3), 351–363 (1998)CrossRefGoogle Scholar
  14. 14.
    Hansson, A., Bryngelsson, M.: Expert opinions on carbon dioxide capture and storage—a framing of uncertainties and possibilities. Energy Policy 37, 2273–2282 (2009)CrossRefGoogle Scholar
  15. 15.
    Huang, S., Mahadevan, S., Rebba, R.: Collocation-based stochastic finite element analysis for random field problems. Probabilist. Eng. Mech. 22, 194–205 (2007)CrossRefGoogle Scholar
  16. 16.
    Foo, J., Wan, X., Karniadakis, G.E.: The multi-element probabilistic collocation method (ME-PCM): error analysis and applications. J. Comput. Phys. 227, 9572–9595 (2008)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kopp, A., Class, H., Helmig, H.: Investigations on CO2 storage capacity in saline aquifers—part 1: dimensional analysis of flow processes and reservoir characteristics. Int. J. Greenhouse Gas Control 3, 263–276 (2009)CrossRefGoogle Scholar
  18. 18.
    Lacroix, S., Vassilevski, Y., Wheeler, M.F., Wheeler, J.: Iterative solvers of the implicit parallel accurate reservoir simulator (IPARS). Numer. Linear Algebra Appl. 4, 537–549 (2001)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Li, H., Zhan, D.: Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resour. Res. 43, 44–48 (2009)Google Scholar
  20. 20.
    Lin, G., Tartakovsky, A.M.: An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media. Water Resour. Res. 32, 712–722 (2009)Google Scholar
  21. 21.
    Maltz, F.H., Hitzl, D.L.: Variance reduction in Monte Carlo computations using multi-dimensional hermite polynomials. J. Comput. Phys. 2, 345–376 (1979)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nordbotten, J., Celia, M., Bachu, M.: Injection and storage of CO2 in deep saline aquifers: analytical solution for CO2 plume evolution during injection. Transp. Porous Media 58(3), 339–360 (2005)CrossRefGoogle Scholar
  23. 23.
    Nordbotten, J.M., Kavetski, D., Celia, M.A., Bachu, S.: A semi-analytical model estimating leakage associated with CO2 storage in large-scale multi-layered geological systems with multiple leaky wells. Environ. Sci. Technol. 43(3), 743–749 (2009)CrossRefGoogle Scholar
  24. 24.
    Oladyshkin, S., Class, H., Helmig, R., Nowak, W.: Datadriven robust design and probabilistic risk assessment: application to underground carbon dioxide storage. Abstract #H41L-03 presented at 2010 Fall Meeting, AGU, San Francisco, California, 13–17 Dec (2010)Google Scholar
  25. 25.
    Shi, L., Yang, J., Zhang, D., Li, H.: Probabilistic collocation method for unconfined flow in heterogeneous media. J. Hydrol. 365, 4–10 (2009)CrossRefGoogle Scholar
  26. 26.
    Villadsen, J., Michelsen, M.L.: Solution of Differential Equation Models by Polynomial Approximation, p. 446. Prentice-Hall, Englewood Cliffs (1978)MATHGoogle Scholar
  27. 27.
    Webster, M., Tatang, M.A., Mcrae, G.J.: Application of the probabilistic collocation method for an uncertainty analysis of a simple ocean model. MIT Joint Program on the Science and Policy of Global Change Report Series No. 4. Massachusetts Institute of Technology, Cambridge (1996)Google Scholar
  28. 28.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Wildenborg, A.F.B., Leijnse, A.L., Kreft, E., Nepveu, M.N., Obdam, A.N.M., Orlic, B., Wipfler, E.L., van der Grift, B., van Kesteren, W., Gaus, I., Czernichowski-Lauriol, I., Torfs, P., Wójcik., R.: Risk assessment methodology for CO2 storage—the scenario approach. In: Benson, S.M. (ed.) The CO2 Capture and Storage Project for Carbon Dioxide Storage in Deep Geological Formations for Climate Change Mitigation, Ch. 33., pp. 1293–1316. Elsevier, Amsterdam (2005)CrossRefGoogle Scholar
  30. 30.
    Woodbury, A.D., Ulrych, T.J.: Minimum relative entropy: forward probabilistic modeling. Water Resour. Res. 29(8), 2847–2860 (1993)CrossRefGoogle Scholar
  31. 31.
    Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Sergey Oladyshkin
    • 1
  • Holger Class
    • 1
  • Rainer Helmig
    • 1
  • Wolfgang Nowak
    • 1
  1. 1.SRC Simulation Technology, Institute of Hydraulic Engineering (LH2)University of StuttgartStuttgartGermany

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