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Micro-Macro Analysis and Phenomenological Modelling of Salt Viscous Damage and Application to Salt Caverns

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Abstract

This paper aims to gain fundamental understanding of the microscopic mechanisms that control the transition between secondary and tertiary creep around salt caverns in typical geological storage conditions. We use a self-consistent inclusion-matrix model to homogenize the viscoplastic deformation of halite polycrystals and predict the number of broken grains in a Representative Elementary Volume of salt. We use this micro-macro modeling framework to simulate creep tests under various axial stresses, which gives us the critical viscoplastic strain at which grain breakage (i.e., tertiary creep) is expected to occur. The comparison of simulation results for short-term and long-term creep indicates that the initiation of tertiary creep depends on the stress and the viscoplastic strain. We use the critical viscoplastic deformation as a yield criterion to control the transition between secondary and tertiary creep in a phenomenological viscoplastic model, which we implement into the Finite Element Method program POROFIS. We model a 850-m-deep salt cavern of irregular shape, in axis-symmetric conditions. Simulations of cavern depressurization indicate that a strain-dependent damage evolution law is more suitable than a stress-dependent damage evolution law, because it avoids high damage concentrations and allows capturing the formation of a damaged zone around the cavity. The modeling framework explained in this paper is expected to provide new insights to link grain breakage to phenomenological damage variables used in Continuum Damage Mechanics.

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Abbreviations

\(\varvec{m},\varvec{n}\) :

Unit sliding vector and unit normal vector in global coordinate

\(\varvec{M},\varvec{N}\) :

Unit sliding vector and unit normal vector in local coordinate

\(\varvec{a^l},\varvec{A^l}\) :

\(l\mathrm{th}\) slip system in global and local coordinates

\(\varvec{\dot{\varepsilon }}^{e}, \varvec{\dot{\varepsilon }}^{vp}\) :

Elastic strain rate, viscoplastic strain rate

\(\gamma _0,\gamma\) :

Reference strain rate, viscoplastic strain of a grain

\(\varvec{P}\) :

Projection tensor

\(\Psi , \theta , \Phi\) :

Angles representing the grain orientation

\(\tau _0, \tau\) :

Reference shear stress, local shear stress of a grain

\(\varvec{\sigma }, \varvec{\varepsilon }\) :

Microscopic stress and strain tensors of a grain

\(\varvec{\overline{\sigma }}, \varvec{\overline{\varepsilon }}\) :

Macroscopic stress and strain tensors of a matrix

\(h^l\) :

Local shear stress-dependent sign coefficient

p :

Probability of the occurrence of a specific grain orientation

\(A, B, n, n_1, n_2\) :

Material constants

\(\varvec{L^*}\) :

Hill’s tensor

\(\kappa , \widetilde{\kappa }\) :

Bulk modulus, effective bulk modulus

\(\mu , \widetilde{\mu }\) :

Shear modulus, effective shear modulus

\(\nu , \widetilde{\nu }\) :

Poisson’s ratio, effective Poisson’s ratio

\(N, N_b, N_g\) :

Number of total grains, number of broken grains, number of good grains

\(D_m, D_M\) :

Damage variable in the homogenization model and the phenomenological model

\(\sigma _T\) :

Mono-crystal tensile strength

R :

Universal gas constant

Q :

Activation energy for the slip mechanism

T :

Absolute temperature

\(\varvec{s},J_2\) :

Deviatoric stress tensor, second deviatoric stress invariant

\(\sigma _e\) :

Equivalent von Mises stress

\(C, \xi , \varphi\) :

Damage accumulation parameters during the tertiary creep

\(\varkappa\) :

Ductility parameter

\(\varvec{C_0},\varvec{C_D}\) :

Stiffness of the intact material, stiffness of the damaged material

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Acknowledgments

Financial support for this research was provided by the School of Civil and Environmental Engineering at the Georgia Institute of Technology, and by the National Science Foundation (Grant No. CMMI-1362004/1361996).

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Drive, Atlanta, GA, 30332, USA

    Cheng Zhu & Chloé Arson

  2. Laboratoire Navier/CERMES, Paris-Est University, 6 & 8 Avenue Blaise Pascal, Cité Descartes, Champs sur Marnes, 77455, Marnela Vallée Cedex 2, France

    Ahmad Pouya

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Correspondence to Cheng Zhu.

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Zhu, C., Pouya, A. & Arson, C. Micro-Macro Analysis and Phenomenological Modelling of Salt Viscous Damage and Application to Salt Caverns. Rock Mech Rock Eng 48, 2567–2580 (2015). https://doi.org/10.1007/s00603-015-0832-9

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