Efficient parameter estimation for a methane hydrate model with active subspaces

  • Mario Teixeira ParenteEmail author
  • Steven Mattis
  • Shubhangi Gupta
  • Christian Deusner
  • Barbara Wohlmuth
Original Paper


Methane gas hydrates have increasingly become a topic of interest because of their potential as a future energy resource. There are significant economical and environmental risks associated with extraction from hydrate reservoirs, so a variety of multiphysics models have been developed to analyze prospective risks and benefits. These models generally have a large number of empirical parameters which are not known a priori. Traditional optimization-based parameter estimation frameworks may be ill-posed or computationally prohibitive. Bayesian inference methods have increasingly been found effective for estimating parameters in complex geophysical systems. These methods often are not viable in cases of computationally expensive models and high-dimensional parameter spaces. Recently, methods have been developed to effectively reduce the dimension of Bayesian inverse problems by identifying low-dimensional structures that are most informed by data. Active subspaces is one of the most generally applicable methods of performing this dimension reduction. In this paper, Bayesian inference of the parameters of a state-of-the-art mathematical model for methane hydrates based on experimental data from a triaxial compression test with gas hydrate-bearing sand is performed in an efficient way by utilizing active subspaces. Active subspaces are used to identify low-dimensional structure in the parameter space which is exploited by generating a cheap regression-based surrogate model and implementing a modified Markov chain Monte Carlo algorithm. Posterior densities having means that match the experimental data are approximated in a computationally efficient way.


Constitutive modeling Soil plasticity Bayesian inversion Dimension reduction 

Mathematics Subject Classification (2010)

62-07 65C20 68U20 


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Funding information

Financial support for BW, SM, and MTP was provided by the German Research Foundation (DFG, Project WO 671/11-1). The work of SG and CD was further funded by the German Federal Ministries of Economy (BMWi) and Education and Research (BMBF) through the SUGAR project (grant nos. 03SX250, 03SX320A, and 03G0856A), and the EU-FP7 project MIDAS (grant agreement no. 603418).


  1. 1.
    Andrade, J.E., Chen, Q., Le, P.H., Avila, C.F., Evans, T.M.: On the rheology of dilative granular media: bridging solid- and fluid-like behavior. Journal of the Mechanics and Physics of Solids 60(6), 1122–1136 (2012)CrossRefGoogle Scholar
  2. 2.
    Bastian, P., Heimann, F., Marnach, Ś.: Generic implementation of finite element methods in the Distributed and Unified Numerics Environment (DUNE). Kybernetika 46(2), 294–315 (2010)Google Scholar
  3. 3.
    Beven, K., Freer, J.: Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. J. Hydrol. 249(1–4), 11–29 (2001)CrossRefGoogle Scholar
  4. 4.
    Brooks, S., Gelman, A., Jones, G., Meng, X.L.: Handbook of Markov Chain Monte Carlo. CRC press, Boca Raton (2011)Google Scholar
  5. 5.
    Bui-Thanh, T., Burstedde, C., Ghattas, O., Martin, J., Stadler, G., Wilcox, L.C.: Extreme-scale UQ for Bayesian inverse problems governed by pdes. In: Proceedings of the international conference on high performance computing, networking, storage and analysis, p. 3. IEEE Computer Society Press (2012)Google Scholar
  6. 6.
    Bui-Thanh, T., Girolami, M.: Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte Carlo. Inverse Problems 30(11), 114,014,23 (2014)CrossRefGoogle Scholar
  7. 7.
    Butler, T., Jakeman, J., Wildey, T.: Combining push-forward measures and Bayes’ rule to construct consistent solutions to stochastic inverse problems. SIAM J. Sci. Comput. 40(2), A984–A1011 (2018)CrossRefGoogle Scholar
  8. 8.
    Choi, J., Dai, S., Cha, J., Seol, Y.: Laboratory formation of noncementing hydrates in sandy sediments. Geochem. Geophys. Geosyst. 15(4), 1648–1656 (2014)CrossRefGoogle Scholar
  9. 9.
    Constantine, P., Gleich, D.: Computing active subspaces with Monte Carlo. arXiv:1408.0545 (2014)
  10. 10.
    Constantine, P.G.: Active subspaces, SIAM spotlights, vol. 2. Society for industrial and applied mathematics (SIAM), Philadelphia, PA. Emerging ideas for dimension reduction in parameter studies (2015)Google Scholar
  11. 11.
    Constantine, P.G., Diaz, P.: Global sensitivity metrics from active subspaces. Reliability Engineering & System Safety 162, 1–13 (2017)CrossRefGoogle Scholar
  12. 12.
    Constantine, P.G., Dow, E., Wang, Q.: Active subspace methods in theory and practice: applications to kriging surfaces. SIAM J. Sci. Comput. 36(4), A1500–A1524 (2014)CrossRefGoogle Scholar
  13. 13.
    Constantine, P.G., Kent, C., Bui-Thanh, T.: Accelerating Markov chain Monte Carlo with active subspaces. SIAM J. Sci. Comput. 38(5), A2779–A2805 (2016)CrossRefGoogle Scholar
  14. 14.
    Cortesi, A., Constantine, P., Magin, T.E., Congedo, P.M.: Forward and backward uncertainty quantification with active subspaces: application to hypersonic flows around a cylinder. Research report RR-9097, INRIA Bordeaux, équipe CARDAMOM. (2017)
  15. 15.
    Cui, T., Law, K.J.H., Marzouk, Y.M.: Dimension-independent likelihood-informed MCMC. J. Comput. Phys. 304, 109–137 (2016)CrossRefGoogle Scholar
  16. 16.
    Dawe, R.A., Thomas, S.: A large potential methane source—natural gas hydrates. Energy sources. Part A: recovery utilization, and environmental effects 29(3), 217–229 (2007)Google Scholar
  17. 17.
    Dedner, A., Flemisch, B., Klöfkorn, R.: Advances in DUNE: proceedings of the DUNE: user meeting, held in October 6Th–8Th 2010 in Stuttgart, Germany. SpringerLink: Bücher. Springer, Berlin (2012)Google Scholar
  18. 18.
    Deusner, C., Bigalke, N., Kossel, E., Haeckel, M.: Methane production from gas hydrate deposits through injection of supercritical CO2. Energies 5(7), 2112 (2012)CrossRefGoogle Scholar
  19. 19.
    Freer, J., Beven, K.: Bayesian estimation of uncertainty in runoff prediction and the value of data: an applicaiton of the GLUE approach. Water Resour. Res. 32(7), 2161–2173 (1996)CrossRefGoogle Scholar
  20. 20.
    Grey, Z.J., Constantine, P.G.: Active subspaces of airfoil shape parameterizations. arXiv:1702.02909 (2017)
  21. 21.
    Gupta, S., Deusner, C., Haeckel, M., Helmig, R., Wohlmuth, B.: Testing a thermo-chemo-hydro-geomechanical model for gas hydrate bearing sediments using triaxial compression lab experiments. Geochem. Geophys. Geosyst. 18(9), 3419–3437 (2017)CrossRefGoogle Scholar
  22. 22.
    Gupta, S., Helmig, R., Wohlmuth, B.: Non-isothermal, multi-phase, multi-component flows through deformable methane hydrate reservoirs. Comput. Geosci. 19(5), 1063–1088 (2015)CrossRefGoogle Scholar
  23. 23.
    Haario, H., Laine, M., Mira, A., Saksman, E.: DRAM: efficient adaptive MCMC. Stat. Comput. 16 (4), 339–354 (2006)CrossRefGoogle Scholar
  24. 24.
    Hager, C., Wohlmuth, B.: Nonlinear complementarity functions for plasticity problems with frictional contact. Comput. Methods Appl. Mech. Eng. 198(41), 3411–3427 (2009). CrossRefGoogle Scholar
  25. 25.
    Hager, C., Wohlmuth, B.: Semismooth newton methods for variational problems with inequality constraints. GAMM Mitteilungen 33, 8–24 (2010)CrossRefGoogle Scholar
  26. 26.
    Hairer, M., Stuart, A.M., Vollmer, S.J.: Spectral gaps for a metropolis–hastings algorithm in infinite dimensions. Ann. Appl. Probab. 24(6), 2455–2490 (2014). CrossRefGoogle Scholar
  27. 27.
    Holodnak, J.T., Ipsen, I.C.F., Smith, R.C.: A probabilistic subspace bound with application to active subspaces ArXiv e-prints (2018)Google Scholar
  28. 28.
    Huang, J., Griffiths, D.V.: Return mapping algorithms and stress predictors for failure analysis in geomechanics. J. Eng. Mech. 135(4), 276–284 (2009). CrossRefGoogle Scholar
  29. 29.
    Hyodo, M., Li, Y., Yoneda, J., Nakata, Y., Yoshimoto, N., Nishimura, A.: Effects of dissociation on the shear strength and deformation behavior of methane hydrate-bearing sediments. Mar. Pet. Geol. 51, 52–62 (2014)CrossRefGoogle Scholar
  30. 30.
    Hyodo, M., Nakata, Y., Yoshimoto, N., Ebinuma, T.: Basic research on the mechanical behaviour of methane hydrate sediments mixture. Soils. Found. 45(1), 75–85 (2005)Google Scholar
  31. 31.
    Jefferson, J.L., Gilbert, J.M., Constantine, P.G., Maxwell, R.M.: Reprint of: Active subspaces for sensitivity analysis and dimension reduction of an integrated hydrologic model. Computers & Geosciences 90, 78–89 (2016)CrossRefGoogle Scholar
  32. 32.
    Jirasek, M., Bazant, Z.: Inelastic analysis of structures. Wiley, London (2002)Google Scholar
  33. 33.
    Kaipio, J., Somersalo, E.: Statistical and computational inverse Problems, vol. 160. Springer Science & Business Media, Berlin (2006)Google Scholar
  34. 34.
    Kimoto, S., Oka, F., Fushita, T.: A chemo-thermo-mechanically coupled analysis of ground deformation induced by gas hydrate dissociation. Int. J. Mech. Sci. 52(2), 365–376 (2010)CrossRefGoogle Scholar
  35. 35.
    Klar, A., Soga, K., NG, Y.A.: Coupled deformation-flow analysis for methane hydrate extraction. Geotechnique 60(10), 765–776 (2010)CrossRefGoogle Scholar
  36. 36.
    Klar, A., Uchida, S., Soga, K., Yamamoto, K.: Explicitly coupled thermal flow mechanical formulation for gas-hydrate sediments. SPE J. 18, 196–206 (2013)CrossRefGoogle Scholar
  37. 37.
    Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 481–492. University of California Press, Berkeley (1951)Google Scholar
  38. 38.
    Lee, J.Y., Francisca, F.M., Santamarina, J.C., Ruppel, C.: Parametric study of the physical properties of hydrate-bearing sand, silt, and clay sediments: 2. Small-strain mechanical properties. J. Geophys. Res. 115(B11), 11p (2010)Google Scholar
  39. 39.
    Lee, J.Y., Yun, T.S., Santamarina, J.C., Ruppel, C.: Observations related to tetrahydrofuran and methane hydrates for laboratory studies of hydrate bearing sediments. Geochem. Geophys. Geosyst. 8(6), Q06003 (2007)CrossRefGoogle Scholar
  40. 40.
    Leube, P.C., Geiges, A., Nowak, W.: Bayesian assessment of the expected data impact on prediction confidence in optimal sampling design. Water Resources Research 48(2), W02501 (2012)CrossRefGoogle Scholar
  41. 41.
    Lukaczyk, T., Palacios, F., Alonso, J.J., Constantine, P.: Active subspaces for shape optimization. In: Proceedings of the 10th AIAA multidisciplinary design optimization conference, pp. 1–18 (2014)Google Scholar
  42. 42.
    Martin, J., Wilcox, L.C., Burstedde, C., Ghattas, O.: A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion. SIAM J. Sci. Comput. 34(3), A1460–A1487 (2012)CrossRefGoogle Scholar
  43. 43.
    Masui, A., Haneda, H., Ogata, Y., Aoki, K.: Effects of methane hydrate formation on shear strength of synthetic methane hydrate sediments. The Fifteenth International Offshore and Polar Engineering Conference 8, 364–369 (2005)Google Scholar
  44. 44.
    Miyazaki, K., Masui, A., Sakamoto, Y., Aoki, K., Tenma, N., Yamaguchi, T.: Triaxial compressive properties of artificial methane-hydrate-bearing sediment. Journal of Geophysical Research: Solid Earth 116(B06102), (2011)Google Scholar
  45. 45.
    Miyazaki, K., Masui, A., Tenma, N., Ogata, Y., Aoki, K., Yamaguchi, T., Sakamoto, Y.: Study on mechanical behavior for methane hydrate sediment based on constant strain-rate test and unloading-reloading test under triaxial compression. International Journal of Offshore and Polar Engineering 20(1), 61–67 (2010)Google Scholar
  46. 46.
    Moridis, G.J., Collett, T.S., Boswell, R., Kurihara, M., Reagan, M.T., Koh, C., Sloan, E.D.: Toward production from gas hydrates: current status, assessment of resources, and simulation-based evaluation of technology and potential. SPE Reserv. Eval. Eng. 12, 745–771 (2009)CrossRefGoogle Scholar
  47. 47.
    Moridis, G.J., Collett, T.S., Pooladi-Darvish, M., Hancock, S., Santamarina, C., Boswell, R., Kneafsey, T., Rutqvist, J., Kowalsky, M.B., et al., Reagan M.T.: Challenges, uncertainities and issues facing gas production from gas hydrate deposits. SPE Reserv. Eval. Eng. 14, 76–112 (2011)Google Scholar
  48. 48.
    Nowak, W., de Barros, F.P.J., Rubin, Y.: Bayesian geostatistical design: task-driven optimal site investigation when the geostatistical model is uncertain. Water Resources Research 46(3), W03535 (2010)CrossRefGoogle Scholar
  49. 49.
    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: Machine learning in python. J. Mach. Learn. Res. 12, 2825–2830 (2011)Google Scholar
  50. 50.
    Piñero, E., Marquardt, M., Hensen, C., Haeckel, M., Wallmann, K.: Estimation of the global inventory of methane hydrates in marine sediments using transfer functions. Biogeosciences 10(2), 959–975 (2013)CrossRefGoogle Scholar
  51. 51.
    Pinkert, S.: The lack of true cohesion in hydrate-bearing sands. Granul. Matter 19(3), 57 (2017)CrossRefGoogle Scholar
  52. 52.
    Pinkert, S., Grozic, J.L.H.: Prediction of the mechanical response of hydrate-bearing sands. J. Geophys. Res. Solid Earth 119(6), 4695–4707 (2014)CrossRefGoogle Scholar
  53. 53.
    Pinkert, S., Grozic, J.L.H., Priest, J.A.: Strain-softening model for hydrate-bearing sands. International Journal of Geomechanics 15(6), 04015, 007 (2015)CrossRefGoogle Scholar
  54. 54.
    Priest, J.A., Rees, E.V.L., Clayton, C.R.I.: Influence of gas hydrate morphology on the seismic velocities of sands. J. Geophys. Res. Solid Earth 114(B11), B11205 (2009)CrossRefGoogle Scholar
  55. 55.
    Rutqvist, J.: Status of the TOUGH-FLAC simulator and recent applications related to coupled fluid flow and crustal deformations. Computers &, Geosciences 37, 739–750 (2011)CrossRefGoogle Scholar
  56. 56.
    Santamarina, J.C., Ruppel, C.: The impact of hydrate saturation on the mechanical, electrical, and thermal properties of hydrate-bearing sand, silts, and clay. Geophysical Characterization of Ga Hydrates. Geophys Dev. Ser 14, 373–384 (2010)Google Scholar
  57. 57.
    Simo, J., Hughes, T.: Computational inelasticity. Interdisciplinary applied mathematics. Springer, New york (2006)Google Scholar
  58. 58.
    Sloan, E.D.: Gas hydrates: review of physical/chemical properties. Energ. Fuel. 12, 191–196 (1998)CrossRefGoogle Scholar
  59. 59.
    de Souza Neto, E., Peric, D., Owen, D.: Computational methods for plasticity: theory and applications. Wiley, New York (2011)Google Scholar
  60. 60.
    Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numerica 19, 451–559 (2010)CrossRefGoogle Scholar
  61. 61.
    Sultan, N., Cochonat, P., Canals, M., Cattaneo, A., Dennielou, B., Haflidason, H., Laberg, J.S., Long, D., Mienert, J., Trincardi, F., Urgeles, R: Triggering mechanisms of slope instability processes and sediment failures on continental margins: a geotechnical approach. Mar. Geol. 213(1-4), 291–321 (2004)CrossRefGoogle Scholar
  62. 62.
    Sultan, N., Cochonat, P., Foucher, J.P., Mienert, J.: Effect of gas hydrates melting on sea floor slope instability. Mar. Geol. 213(1), 379–401 (2004)CrossRefGoogle Scholar
  63. 63.
    Troldborg, M., Nowak, W., Tuxen, N., Bjerg, P.L., Helmig, R., Binning, P.J.: Uncertainty evaluation of mass discharge estimates from a contaminated site using a fully Bayesian framework. Water Resources Research 46(12), W12552 (2010)CrossRefGoogle Scholar
  64. 64.
    Uchida, S., Soga, K., Yamamoto, K.: Critical state soil constitutive model for methane hydrate soil. Journal of Geophysical Research: Solid Earth 117, B03209 (2012)CrossRefGoogle Scholar
  65. 65.
    Vollmer, S.J.: Dimension-independent mcmc sampling for inverse problems with non-gaussian priors. SIAM/ASA Journal on Uncertainty Quantification 3(1), 535–561 (2015)CrossRefGoogle Scholar
  66. 66.
    Vrugt, J., ter Braak, C., Gupta, H., Robinson, B.: Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling? Stoch. Env. Res. Risk A. 23(7), 1011–1026 (2008)CrossRefGoogle Scholar
  67. 67.
    Vrugt, J.A., Ter Braak, C., Diks, C., Robinson, B.A., Hyman, J.M., Higdon, D.: Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling. International Journal of Nonlinear Sciences and Numerical Simulation 10(3), 273–290 (2009)CrossRefGoogle Scholar
  68. 68.
    Waite, W.F., Santamarina, J.C., Cortes, D.D., Dugan, B., Espinoza, D.N., Germaine, J., Jang, J., Jung, J.W., Kneafsey, T.J., Shin, H., Soga, K., Winters, W.J., Yun, T.S.: Physical properties of hydrate-bearing sediments. Reviews of Geophysics 47(4), RG4003 (2009)CrossRefGoogle Scholar
  69. 69.
    Wood, D.: Soil behaviour and critical state soil mechanics. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  70. 70.
    Xuerui, G., Marcelo, S.: A geomechanical model for gas hydrate-bearing sediments. Environmental Geotechnics 4(2), 143–156 (2017)CrossRefGoogle Scholar
  71. 71.
    Yun, T.S., Santamarina, J.C., Ruppel, C.: Mechanical properties of sand, silt, and clay containing tetrahydrofuran hydrate. J. Geophys. Res. 112(B04), 106 (2007)Google Scholar
  72. 72.
    Zienkiewicz, O., Taylor, R.: The finite element method for solid and structural mechanics. The finite element method elsevier science (2013)Google Scholar

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© Springer Nature Switzerland AG 2018
corrected publication September/2018

Authors and Affiliations

  1. 1.Chair for Numerical MathematicsTechnical University MunichGarching bei MünchenGermany
  2. 2.Department of MathematicsUniversity of BergenBergenNorway
  3. 3.GEOMAR Helmholtz Centre for Ocean Research KielKielGermany

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