Transient Closure of a Cylindrical Hole in a Salt Formation Considered as a Norton–Hoff Medium

  • Sivaprasath Manivannan
  • Pierre BérestEmail author
Technical Note


This paper focuses on the mechanical behavior of boreholes or elongated cylindrical salt caverns. A typical problem is as follows: a single cavern is leached out from a salt formation whose extension, when compared to the dimensions of the cavern, is so large that the salt formation can be considered as an infinite medium. For such a problem, closed-form solutions are available for the simple case of an idealized spherical or cylindrical cavern when steady state has been reached (Van Sambeek et al. 1993; Wang et al. 2015; Van Sambeek and DiRienzo 2016; Cornet et al. 2017). The objective of this paper was to prove that, in the case of an idealized cylindrical cavern, simple equations govern the transient behavior of the cavern.

Laboratory Creep Tests

A large amount of literature is dedicated to the mechanical behavior of salt. A typical creep test can be described as follows (Fig. 1, left). At t = 0, a constant uniaxial load, \(\sigma\)


Salt caverns Cavern creep closure Closed-form solutions Reverse creep Transient creep Non-linear viscoelasticity Norton–Hoff constitutive law 

List of Symbols


Internal radius of the hollow cylinder


Parameter of the Norton–Hoff power law


External radius of the hollow cylinder


Young’s modulus


Second invariant of the deviatoric stress tensor


Exponent of the Norton–Hoff power law


Internal pressure at r = a

\({P_\infty }\)

External pressure at r = b


Parameter of the Norton-Hoff power law

\(r,\varphi ,z\)

Cylindrical coordinates

\({s_{{\text{rr}}}},{s_{\varphi \varphi }},{s_{{\text{zz}}}}\)

Radial, tangential and axial deviatoric stresses

\(S={\text{ }}{s_{rr}}\)

Radial deviatoric stress


Steady-state radial deviatoric stress




Absolute temperature


Period of a pressure cycle

\(\dot {u}\)

Radial displacement rate


Change in internal volume of the cylinder


Ratio (b/a) between external and internal radii of the cylinder


Wellbore overall compressibility coefficient


Open hole compressibility coefficient


Casing compressibility coefficient


Brine compressibility coefficient

\({\sigma _{{\text{rr}}}},{\sigma _{\varphi \varphi }},{\sigma _{{\text{zz}}}}\)

Radial, tangential and axial stresses

\({\sigma _0}\)

Reference stress



\(\varepsilon _{{{\text{rh}}}}^{{{\text{tr}}}}\)

Rheological transient strain

\(\varepsilon _{{}}^{{{\text{el}}}}\)

Elastic strain

\(\varepsilon _{{}}^{{{\text{ss}}}}\)

Steady-state strain


Internal parameter of the Munson–Dawson constitutive law


Poisson’s ratio



This work was performed in the framework of Sivaprasath Manivannan’s PhD Thesis supported by Total SA. Pierre Bérest’s contribution was funded partially by the French Agence Nationale de la Recherche (ANR) in the framework of the FluidStory Project devoted to storage of O2 and CO2 in salt caverns. This project includes researchers from Armines, Areva-H2Gen, BRGM, Brouard Consulting, Geostock, Geogreen and Ecole Polytechnique. Reviewers’ and Editor comments were quite helpful.


  1. Bérest P, Brouard B, Gharbi H (2015) Rheological and geometrical reverse creep in salt caverns. In: Roberts L, Mellegard K, Hansen F (eds) Proc. 8th conf. mechanical behavior of Salt VIII. Taylor and Francis Group, London, pp 199–208CrossRefGoogle Scholar
  2. Bérest P, Brouard B, Karimi-Jafari M (2017) Geometrical versus rheological transient creep closure in a salt cavern. CR Acad Sci Paris Méca 345:735–741. CrossRefGoogle Scholar
  3. Breunese JN, van Eijs RMHE, De Meer S, Kroon IC (2003) Observation and prediction of the relation between salt creep and land subsidence in solution mining: the Barradeel case. In: Proc. SMRI fall technical conference, Chester, UK, pp 38–57Google Scholar
  4. Brouard B (1998) Sur le comportement des cavités salines, étude théorique et experimentation in situ. PhD Thesis, Ecole Polytechnique, Palaiseau, France, 253 pGoogle Scholar
  5. Brouard B, Bérest P, Karimi-Jafari M, Rokahr RB, Staudtmeister K, Zander-Schiebenhöfer D, Fourmaintraux D, de Laguérie P, You T (2006) Salt-cavern abandonment field test in Carresse. Research report n°2006-01 prepared for the SMRI. Clarks Summit, Pennsylvania, 96 pGoogle Scholar
  6. Brouard B, Bérest P, de Greef V, Béraud JF, Lheur C, Hertz E (2013) Creep closure rate of a shallow salt cavern at Gellenoncourt, France. Int J Rock Mech Min Sci 62:42–50CrossRefGoogle Scholar
  7. Brouard Consulting RESPEC (2013) Analysis of cavern MB#1 Moss Bluff blowout data. Research Report n°2013-01 prepared for the SMRI. Clarks Summit, Pennsylvania, p 197Google Scholar
  8. Cornet JS, Dabrowski M, Schmid DW (2017) Long-term cavity closure in non-linear rocks. Geophys J Int 210:1231–1243. CrossRefGoogle Scholar
  9. Denzau H, Rudolph F (1997) Field test for determining the convergence of a gas storage cavern under load conditions frequently changing between maximum and minimum pressure and is finite element modelling. In: Proc. SMRI spring meeting, Cracow, Poland, pp 71–84Google Scholar
  10. Eickemeier R, Paar WA, Wallner M (2002) Assessment of subsidence and cavern convergence with respect to brine field enlargement. In: Proc. SMRI spring technical conference, Banff, Alberta, Canada, pp 161–178Google Scholar
  11. Hugout B (1988) Mechanical behavior of salt cavities -in situ tests- model for calculating the cavity volume evolution. In: Hardy RH Jr and Langer M (eds) Proc. 2nd Conf. Mech. Beh. of Salt, Hannover, September 1984. Trans Tech Pub., Clausthal-Zellerfeld, pp 291–310Google Scholar
  12. Karimi-Jafari M, Bérest P, Brouard B (2006) Transient behavior of salt caverns. In: Proc. SMRI fall meeting, Rapid City, South Dakota, pp 253–270Google Scholar
  13. Lestringant C, Bérest P, Brouard B (2010) Thermo-mechanical effects in compressed air storage (CAES). In: Proc. SMRI fall technical conference, Leipzig, Germany, pp 29–44Google Scholar
  14. Munson DE, Dawson PR (1984) Salt constitutive modeling using mechanism maps. In: Reginald Hardy H, Jr, Langer M (eds) Proc. 1st Conf. Mech. Beh. of Salt. Trans Tech Pub, Clausthal-Zellerfeld, pp 717–737Google Scholar
  15. Pouya A (1991) Comportement rhéologique du sel gemme. Application à l’étude des excavations souterraines. PhD Thesis, Ecole Nationale des Ponts et Chaussées, October 18, 1991Google Scholar
  16. Preece DS (1987) Borehole creep closure measurements and numerical computations at the Big Hill, Texas storage site. In: Proc. 6th Int. congress on rock mechanics, vol 1, pp 219–224Google Scholar
  17. Urai J, Spiers CJ (2007) The effect of grain boundary water on deformation mechanisms and rheology of rocksalt during long-term deformation. In: Wallner M, Lux KH, Minkley W, Hardy RH Jr (eds) Proc. 6th Conf. Mech. Beh. of Salt. Taylor & Francis, London, pp 149–158Google Scholar
  18. Van Sambeek LL, DiRienzo AL (2016) Analytical solutions for stress distributions and creep closure around open-holes or caverns using multilinear segmented creep laws. In: Proc. SMRI fall meeting, Salzburg, Austria, pp 225–238Google Scholar
  19. Van Sambeek L, Fossum A, Callahan G, Ratigan J (1993) Salt mechanics: empirical and theoretical developments. In: Kakihana H, Reginald Hardy Jr H, Hoshi T, Toyokura K (eds), Proc. 7th Symp. on Salt, vol. I. Elsevier, Amsterdam pp 127–134Google Scholar
  20. Wang L, Bérest P, Brouard B (2015) Mechanical behavior of salt caverns: closed-form solutions vs numerical computations. Rock Mech Rock Eng 48(6):2369–2382CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mécanique des SolidesEcole PolytechniquePalaiseauFrance

Personalised recommendations