Skip to main content

Advertisement

SpringerLink
Log in

Probabilistic Analysis of a Rock Salt Cavern with Application to Energy Storage Systems

  • Original Paper
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

This study focuses on the failure probability of storing renewable energy in the form of hydrogen or compressed air in rock salt caverns. The validation of the short- and long-term integrity and stability of rock salt cavern is a prerequisite in their design process. The present paper provides a reliability-based analysis of a typical renewable energy storage cavern in rock salt. An elasto-viscoplastic creep constitutive model is implemented into a numerical model of rock salt cavern to assess its behavior under different operation conditions. Sensitivity measures of different variables involved in the mechanical response of cavern are computed by elementary effect global sensitivity method. Subset simulation methodology is conducted to measure the failure probability of the system with a low computational cost. This methodology is further validated by a comparison with a Monte Carlo-based probabilistic analysis. The propagation of parameter uncertainties and the failure probability against different failure criteria are evaluated by utilizing a Monte Carlo-based analysis. In this stage, the original finite element model is substituted by a surrogate model to further reduce the computational effort. Finally, a reliability analysis approach is employed to obtain the minimum admissible internal pressure in a cavern.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

Abbreviations

\(\alpha\) :

Hardening parameter (\(\hbox {MPa}^{2-n}\))

\(\beta\) :

Constitutive model parameter (−)

\(\beta _1\) :

Constitutive model parameter (\(\hbox {MPa}^{-1}\))

\(\beta _{\mathrm{r}}\) :

Reliability index (−)

\(\gamma _{\mathrm{cr}}\) :

Density of cap rock (\(\hbox {kN/m}^3\))

\(\gamma _{\mathrm{rs}}\) :

Density of rock salt (\(\hbox {kN/m}^3\))

\(\gamma\) :

Ultimate parameter (−)

\(\dot{{\varepsilon }}^{\mathrm{el}}_{ij}, \dot{{\varepsilon }}_{ij}\) :

Tensor of elastic and total strain rate (\(\hbox {s}^{-1}\))

\(\dot{{\varepsilon }}^{\mathrm{vp}}_{ij}\) :

Tensor of viscoplastic strain rate (\(\hbox {s}^{-1}\))

\(\dot{{\varepsilon }}^{\mathrm{cr}}_{ij}\) :

Tensor of creep strain rate (\(\hbox {s}^{-1}\))

\(\eta\) :

Hardening parameter (−)

\(\overline{\eta ^*_{\mathrm{M}}}\) :

Maxwell coefficient (MPa s)

\(\theta\) :

Lode’s angle (\({}^{\circ }\))

\(\sigma _{ij}\) :

Stress tensor (MPa)

\(\mu\) :

Fluidity parameter (\(\hbox {s}^{-1}\))

\(\mu ^*\) :

EE index (−)

\(\nu\) :

Poisson ratio (−)

\(\xi\) :

Accumulated viscoplastic strain (−)

\(a_1\) :

Constitutive model parameter (\(\hbox {MPa}^{2-n}\))

b :

Constitutive model parameter (−)

\(\hbox {COV}_{P_{\mathrm{F}}}\) :

COV of the failure probability (−)

\(\hbox {DF}\) :

Indicator of dilatant behavior (−)

\(\hbox {DF}_{u,p}\) :

Tolerable and present value of DF (−)

E :

Elastic modulus (MPa)

\(\hbox {EE}_i\) :

Elementary effect of ith parameter (−)

F :

Failure event (−)

\(F^{\mathrm{vp}}\) :

Yield surface (\(\hbox {MPa}^2\))

\(F_0^{\mathrm{vp}}\) :

Normalizing factor (\(\hbox {MPa}^2\))

\(G_{x}\) :

Performance function (−)

\(G_{s,v,l}\) :

Performance function against dilatancy, volume loss, admissible minimum \(P_i\) (−)

\(I_1\) :

First invariant of the deviatoric stress (MPa)

\(J_2\) :

Second invariant of the deviatoric stress (\(\hbox {MPa}^2\))

k :

Number of input parameters in GSA (−)

l :

Number of intermediate levels in SubSim (−)

\(m_0\) :

Constitutive model parameter (\(\hbox {MPa}^{-1}\))

n :

Transition parameter (−)

N :

Flow rule exponent (−)

p :

The number of levels of the grid space in Morris method (−)

\(P_{\mathrm{i}}\) :

Internal pressure (MPa)

\(P_{\mathrm{F}}\) :

Probability of failure (−)

r :

Number of trajectories in Morris method (−)

\(R^2\) :

Coefficient of determination (−)

\(s_{ij}\) :

\(=\sigma _{ij}-I_1/3\) (MPa)

\(s_m\) :

System realization based on mth set of input data (\(m=1,\ldots ,Z_s\)) (−)

\(\hbox {VL}_{u, p}\) :

Tolerable and current volume convergence (−)

\(y_j\) :

Intermediate failure threshold (−)

\(Z_{\mathrm{s}}\) :

Number of model evaluations in each intermediate level of SubSim (−)

References

  • Ahmed A, Soubra AH (2012) Extension of subset simulation approach for uncertainty propagation and global sensitivity analysis. Georisk Assess Manag Risk Eng Syst Geohazards 6:162–176

    Article  Google Scholar 

  • Au SK, Beck JL (2001) Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 16:263–277

    Article  Google Scholar 

  • Au SK, Wang Y (2014) Subset simulation. Wiley, New York

    Book  Google Scholar 

  • Bérest P, Brouard B (2003) Safety of salt caverns used for underground storage. Oil Gas Sci Technol 58:361–384

    Article  Google Scholar 

  • Bérest P, Brouard B, Karimi-Jafari M, Sambeek LV (2007) Transient behavior of salt caverns, interpretation of mechanical integrity tests. Int J Rock Mech Min Sci 44(5):767–786

    Article  Google Scholar 

  • Bräuer V, für Geowissenschaften und Rohstoffe B, (2011) Geotechnical Exploration of the Gorleben Salt Dome. Description of the Gorleben site, BGR

  • Buljak V (2010) Proper orthogonal decomposition and radial basis functions algorithm for diagnostic procedure based on inverse analysis. FME Trans 38:129–136

    Google Scholar 

  • Cacuci DG (2003) Sensitivity & uncertainty analysis volume 1: theory. CRC Press, Boca Raton

    Book  Google Scholar 

  • Chen J, Ren S, Yang C, Jiang D, Li L (2013) Self-healing characteristics of damaged rock salt under different healing conditions. Materials 6:3438–3450

    Article  Google Scholar 

  • Cristescu N, Hunsche U (1998) Time effects in rock mechanics. Wiley, New York

    Google Scholar 

  • Desai C, Zhang D (1987) Viscoplastic model for geologic material with generalized flow rule. Int J numer Anal Met 11:603–627

    Article  Google Scholar 

  • Einstein HH, Baecher GB (1983) Probabilistic and statistical methods in engineering geology. Rock Mech Rock Eng 16(1):39–72

    Article  Google Scholar 

  • Filimonov LY, Lavrov VA, Shafarenko MY, Shkuratnik LV (2001) Memory effects in rock salt under triaxial stress state and their use for stress measurement in a rock mass. Rock Mech Rock Eng 34(4):275–291

    Article  Google Scholar 

  • Fuenkajorn K, Sriapai T, Samsri P (2012) Effects of loading rate on strength and deformability of Maha Sarakham salt. Eng Geol 135–136:10–23

    Article  Google Scholar 

  • Günther RM, Salzer K, Popp T, Lüdeling C (2015) Steady-state creep of rock salt: improved approaches for lab determination and modelling. Rock Mech Rock Eng 48(6):2603–2613

    Article  Google Scholar 

  • Guo Y, Yang C, Mao H (2012) Mechanical properties of jintan rock salt under complex stress paths. Int J Rock Mech Min 56:54–61

    Google Scholar 

  • Haldar A, Mahadevan S (2000) Probability, reliability, and statistical methods in engineering design. Wiley, New York

    Google Scholar 

  • Hansen FD, Mellegard KD, Senseny PE (1984) Elasticity and strength of ten natural rock salts. In: Hardy HRJ, Langer M (eds) Proceedings of the first conference the mechanical behavior of salt, pp 53–70

  • Hasofer AM, Lind NC (1974) Exact and invariant second-moment code format. J Eng Mech Div 100:111–121

    Google Scholar 

  • Hastings WK (1970) Monte carlo sampling methods using markov chains and their applications. Biometrika 57:97–109

    Article  Google Scholar 

  • Heusermann S, Rolfs O, Schmidt U (2003) Nonlinear finite element analysis of solution mined storage caverns in rock salt using the lubby2 constitutive model. Comput Struct J 81:629–638

    Article  Google Scholar 

  • Homma T, Saltelli A (1996) Importance measure in global sensitivity analysis of nonlinear models. Reliab Eng Syst Saf 52(12):1–17

    Article  Google Scholar 

  • Hou Z (2003) Mechanical and hydraulic behavior of rock salt in the excavation disturbed zone around underground facilities. Int J Rock Mech Min Sci 40:725–738

    Article  Google Scholar 

  • Hunsche U, Hampel A (1997) Rock salt—the mechanical properties of the host rock material for radio active waste. Eng Geol 52:271–291

    Article  Google Scholar 

  • Hunsche U, Hampel A (1999) Rock salt–the mechanical properties of the host rock material for radio active waste repository. Eng Geol 52:271–291

    Article  Google Scholar 

  • Khaledi K, Miro S, König M, Schanz T (2014) Robust and reliable metamodels for mechanized tunnel simulations. Comput Geotech 61:1–12

    Article  Google Scholar 

  • Khaledi K, Mahmoudi E, Datcheva M, König D, Schanz T (2016a) Sensitivity analysis and parameter identification of a time dependent constitutive model for rock salt. J Comput Appl Math 293:128–138

    Article  Google Scholar 

  • Khaledi K, Mahmoudi E, Datcheva M, Schanz T (2016b) Analysis of compressed air storage caverns in rock salt considering thermo-mechanical cyclic loading. Environ Earth Sci 75(15):1–17

    Article  Google Scholar 

  • Khaledi K, Mahmoudi E, Datcheva M, Schanz T (2016c) Stability and serviceability of underground energy storage caverns in rock salt subjected to mechanical cyclic loading. Int J Rock Mech Min Sci 86:115–131

    Google Scholar 

  • Lj Ma, Xy Liu, Fang Q, Hf Xu, Hm Xia, Eb Li, Sg Yang, Wp Li (2012) A new elasto-viscoplastic damage model combined with the generalized hoek-brown failure criterion for bedded rock salt and its application. Rock Mech Rock Eng 46(1):53–66

    Google Scholar 

  • Mahmoudi E, Khaledi K, König D, Schanz T (2015) Numerical simulation of deep and shallow energy storage systems in rock salt. In: Schanz T, Hettler A (eds) Aktuelle Forschung in der Bodenmechanik 2015, Springer, Berlin, Heidelberg, pp 69–83

    Google Scholar 

  • Mahmoudi E, Khaledi K, von Blumenthal A, König D, Schanz T (2016) Concept for an integral approach to explore the behavior of rock salt caverns under thermo-mechanical cyclic loading in energy storage systems. Environ Earth Sci 75(14):1–19

    Article  Google Scholar 

  • Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1092

    Article  Google Scholar 

  • Minkley M, Muehlbauer J (2007) Constitutive models to describe the mechanical behavior of salt rocks and the imbedded weakness planes. In: 6th Conference on the mechanical behavior of salt—SALTMECH6. Hannover, Germany, pp 119–127

  • Miro S, Hartmann D, Schanz T (2014) Global sensitivity analysis for subsoil parameter estimation in mechanized tunneling. Comput Geotech 56:80–88

    Article  Google Scholar 

  • Miro S, König M, Hartmann D, Schanz T (2015) A probabilistic analysis of subsoil parameters uncertainty impacts on tunnel-induced ground movements with a back-analysis study. Comput Geotech 68:38–53

    Article  Google Scholar 

  • Mollon G, Dias D, Soubra AH (2013) Range of the safe retaining pressures of a pressurized tunnel face by a probabilistic approach. J Geotech Geoenviron Eng 139:1954–1967

    Article  Google Scholar 

  • Morris MD (1991) Technometrics, factorial sampling plans for preliminary computational experiments. Taylor & Francis Ltd, London

    Google Scholar 

  • Nazary S, Mirzabozorg H, Noorzad A (2013) Modeling time-dependent behavior of gas caverns in rock salt considering creep, dilatancy and failure. Tunn Undergr Space Technol 33:171–185

    Article  Google Scholar 

  • Nguyen MD, Braham S, Durup JG (1993) Surface subsidence over deep solution mined storage cavern field. In: International conference on case histories in geotechnical engineering

  • Olivella S, Gens A (2002) A constitutive model for crushed salt. Int J Numer Anal Methods Geomech 26:719–746

    Article  Google Scholar 

  • Olivella S, Gens A, Carrera J, Alonso E (1996) Numerical formulation for a simulator (Code-Bright) for the coupled analysis of saline media. Eng Comput 13:87–112

    Article  Google Scholar 

  • Perzyna P (1966) Fundamental problems in viscoplasticity. Adv Appl Mech 9:243–377

    Article  Google Scholar 

  • Phoon KK (2008) Reliability-based design in geotechnical engineering: computations and applications. CRC Press, Boca Raton

    Google Scholar 

  • Phoon KK, Ching J (2014) Risk and reliability in geotechnical engineering. CRC Press, Boca Raton

    Google Scholar 

  • Ratigan J, Yogt T (1993) Lpg storage at mont belvieu, texas: a case history. SPE Adv Technol Ser 1:204–211

    Article  Google Scholar 

  • Ratigan WR, Hannum WD (1980) Mechanical behavior of new mexico rock salt in triaxial compression up to 200 c. J Geophys Res 85:891–900

    Article  Google Scholar 

  • Roberts LA, Buchholz SA, Mellegard KD, Düsterloh U (2015) Cyclic loading effects on the creep and dilation of salt rock. Rock Mech Rock Eng 48(6):2581–2590

    Article  Google Scholar 

  • Saltelli A, Andres T, Ratto M (2008) Global sensitivity analysis. The primer. Wiley, New York

    Google Scholar 

  • Sane S, Desai C, Jenson J, Contractor D, Carlson A, Clark P (2008) Disturbed state constitutive modeling of two pleistocene tills. Quat Sci Rev 27:267–283

    Article  Google Scholar 

  • Santoso A, Phoon K, Quek S (2011) Modified metropolishastings algorithm with reduced chain correlation for efficient subset simulation. Probab Eng Mech 26:331–341

    Article  Google Scholar 

  • Schulze O (2007) Investigation on damage and healing of rock salt. In: 6th Conference on the mechanical behavior of salt—SALTMECH6, Hannover, Germany, 22–25 May, pp 33–43

  • Sobol’ IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55:271–280

    Article  Google Scholar 

  • Tang WH, Yucemen MS, Ang AHS (1976) Probability based short-term design of slope. Can Geotech J 13:201–215

    Article  Google Scholar 

  • Wainwright HM, Finsterle F, Jung Y, Zhou Q, Birkholzer JT (2014) Making sense of global sensitivity analyses. Comput Geosci 65:84–94

    Article  Google Scholar 

  • Wang L, Bérest P, Brouard B (2015) Mechanical behavior of salt caverns: closed-form solutions vs numerical computations. Rock Mech Rock Eng 48(6):2369–2382

    Article  Google Scholar 

  • Wang Y (2011) Reliability based design of spread foundations by monte carlo simulations. Geotechnics 61:677–685

    Google Scholar 

Download references

Acknowledgments

This work was performed in the frame of the project ANGUS+ funded by the Federal Ministry of Education and Research (BMBF) under Grant No. 03EK3022C. The authors are grateful for this support.

Author information

Authors and Affiliations

  1. Chair of Foundation Engineering, Soil and Rock Mechanics, Ruhr-Universität Bochum, Universitätsstr. 150, 44780, Bochum, Germany

    Elham Mahmoudi, Kavan Khaledi, Diethard König & Tom Schanz

  2. Chair of Computing in Engineering, Ruhr-Universität Bochum, Universitätsstr. 150, 44780, Bochum, Germany

    Shorash Miro

Authors
  1. Elham Mahmoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kavan Khaledi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shorash Miro
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Diethard König
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Tom Schanz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elham Mahmoudi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mahmoudi, E., Khaledi, K., Miro, S. et al. Probabilistic Analysis of a Rock Salt Cavern with Application to Energy Storage Systems. Rock Mech Rock Eng 50, 139–157 (2017). https://doi.org/10.1007/s00603-016-1105-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00603-016-1105-y

Keywords

Search

Navigation