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Using Local Second Gradient Model and Shear Strain Localisation to Model the Excavation Damaged Zone in Unsaturated Claystone

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Abstract

The drilling of galleries induces damage propagation in the surrounding medium and creates, around them, the excavation damaged zone (EDZ). The prediction of the extension and fracture structure of this zone remains a major issue, especially in the context of underground nuclear waste storage. Experimental studies on geomaterials indicate that localised deformation in shear band mode usually appears prior to fractures. Thus, the excavation damaged zone can be modelled by considering the development of shear strain localisation bands. In the classical finite element framework, strain localisation suffers a mesh-dependency problem. Therefore, an enhanced model with a regularisation method is required to correctly model the strain localisation behaviour. Among the existing methods, we choose the coupled local second gradient model. We extend it to unsaturated conditions and we include the solid grain compressibility. Furthermore, air ventilation inside underground galleries engenders a rock–atmosphere interaction that could influence the damaged zone. This interaction has to be investigated in order to predict the damaged zone behaviour. Finally, a hydro-mechanical modelling of a gallery excavation in claystone is presented and leads to a fairly good representation of the EDZ. The main objectives of this study are to model the fractures by considering shear strain localisation bands, and to investigate if an isotropic model accurately reproduces the in situ measurements. The numerical results provide information about the damaged zone extension, structure and behaviour that are in very good agreement with in situ measurements and observations. For instance, the strain localisation bands that develop in chevron pattern during the excavation and rock desaturation, due to air ventilation, are observed close to the gallery.

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Abbreviations

\(a^t\) :

Current configuration of quantity \(a\) at a given time \(t\)

\({\dot{a}}\) :

Time variation of quantity \(a\)

\({\rm da}\) :

Time variation of quantity \(a\) between two given times for iterative procedures

\(a^*\) :

Virtual quantity \(a\)

\(\alpha \) :

Deviatoric strain increment

\(\beta _{\rm c}\) :

Cohesion parameter for suction influence

\(\beta _{\rm E}\) :

Young’s modulus parameter for suction influence

\(\Gamma \) :

Porous material boundary

\(\Gamma _\sigma \) :

Part of the porous material boundary on which \(\overline{t}_i\) and \(\overline{T}_i\) are applied

\(\Gamma _q\) :

Part of the porous material boundary on which \(\bar{q}\) is prescribed

\(\delta _{ij}\) :

Kronecker symbol

\(\Delta _{{\rm R/S/W}}\) :

Non-equilibrium forces of balance equations for iterative procedures

\(\epsilon _{ij}\) :

Total strain field

\(\epsilon ^{\rm p}_{ij}\) :

Plastic strain field

\({\hat{\epsilon }}_{ij}\) :

Deviatoric total strain field

\({\hat{\epsilon }}^{\rm p}_{ij}\) :

Deviatoric plastic strain field

\(\epsilon _{\rm v}\) :

Volumetric strain

\(\epsilon _{\rm eq}\) :

Von Mises’ equivalent deviatoric total strain (total deviatoric strain)

\(\epsilon ^{\rm p}_{\rm eq}\) :

Von Mises’ equivalent deviatoric plastic strain

\(\theta \) :

Tangential direction

\(\lambda _{ij}\) :

Lagrange multipliers field

\(\mu _{\rm w}\) :

Water dynamic viscosity

\(\nu \) :

Drained Poisson’s ratio

\(\rho \) :

Density

\(\rho _{\rm s}\) :

Solid grain density

\(\rho _{\rm w}\) :

Water density

\(\sigma _{ij}\) :

Cauchy total stress field

\(\sigma' _{ij}\) :

Bishop’s effective stress field

\(\hat{\sigma }_{ij}\) :

Deviatoric stress field

\(\sigma'\) :

Bishop’s mean effective stress

\(\sigma _{x/y/z,0}\) :

Initial principal total stresses

\(\Sigma _{ijk}\) :

Double stress dual of the virtual micro second gradient field

\(\tilde{\Sigma }_{ijk}\) :

Jaumann double stress rate

\(\tau _{ij}\) :

Additional stress field associated to the microstructure (microstress)

\(\upsilon _{ij}\) :

Microkinematic gradient field

\(\phi _{\rm c}\) :

Compression friction angle

\(\phi _{{\rm c},0}\) :

Initial compression friction angle

\(\phi _{\rm c,f}\) :

Final compression friction angle

\(1/\chi _{\rm w}\) :

Water compressibility

\(\psi \) :

Dilatancy angle

\(\omega _{ij}\) :

Spin tensor

\(\Omega \) :

Porous material volume

\({\rm I}_\sigma \) :

First stress invariant

\({\rm II}_{\hat{\sigma }}\) :

Second deviatoric stress invariant

\(b\) :

Biot’s coefficient

\(B_{\phi }\) :

Friction angle hardening coefficient

\(B_{\rm c}\) :

Cohesion softening coefficient

\(c\) :

Cohesion

\(c_0\) :

Initial saturated cohesion

\(c_{\rm f}\) :

Final saturated cohesion

\(c_{\rm sat}\) :

Saturated cohesion

\(D\) :

Second gradient elastic modulus

\(Du_i\) :

Normal derivative of the displacement field

\(E\) :

Drained Young’s modulus

\(E_{\rm sat}\) :

Saturated Young’s modulus

\(F_{ij}\) :

Macrodeformation gradient field

\(h_{ijk}\) :

Micro second gradient field

\(K\) :

Drained bulk modulus of the poroelastic material

\(k_{ij}\) :

Intrinsic water permeability tensor

\(k_{\rm r,w}\) :

Relative water permeability

\(K_{\rm s}\) :

Isotropic bulk modulus of the solid grains

\(m\) :

Yield surface parameter

\(M\) :

van Genuchten coefficient

\(M_{\rm v}\) :

Molar mass of water vapour

\(M_{\rm w}\) :

Water mass

\(m_{{\rm w},i}\) :

Water mass flow

\(n\) :

Porosity

\(N\) :

van Genuchten coefficient

\(p_{\rm c}\) :

Capillary pressure (matric suction)

\(p_{\rm g}\) :

Gas pressure

\(P_{\rm r}\) :

van Genuchten air entry pressure

\(p_{\rm v}\) :

Partial pressure of water vapour

\(p_{{\rm v},0}\) :

Pressure of saturated water vapour

\(p_{\rm w}\) :

Pore water pressure

\(q\) :

Deviatoric stress

\(\bar{q}\) :

Input water mass per unit area

\(Q\) :

Water sink term

\(q_{{\rm w},i}\) :

Average speed of water relative to the solid grains

\(R\) :

Gas constant

\(R_{\rm c}\) :

Uniaxial compression strength

\(R_{\rm c,sat}\) :

Saturated uniaxial compression strength

RH:

Air relative humidity

\(S_{\max }\) :

Maximum water degree of saturation

\(S_{\rm res}\) :

Residual water degree of saturation

\(S_{\rm{r,w}}\) :

Water degree of saturation

\(T\) :

Absolute temperature

\(\overline{t}_i\) :

Classical external traction force per unit area

\(\overline{T}_i\) :

Additional external double force per unit area

\(u_i\) :

Displacement field

\(u_{\rm r}\) :

Radial displacement

\(u_{\perp ,i}\) :

Normal vector of the displacement field

\(W_{\rm int}\) :

Internal work

\(W_{\rm ext}\) :

External work

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Acknowledgments

The authors would like to thank the Andra for the availability of the results of experimental measurements performed in its URL and the FRIA-F.R.S.-FNRS, the National Funds of Scientific Research in Belgium, for their financial support.

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  1. Département ArGEnCo, Université de Liège, Chemin des Chevreuils 1, 4000, Liège, Belgium

    Benoît Pardoen, Séverine Levasseur & Frédéric Collin

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  1. Benoît Pardoen
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  2. Séverine Levasseur
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  3. Frédéric Collin
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Correspondence to Benoît Pardoen.

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B. Pardoen: FRIA, F.R.S.-FNRS scholarship holder.

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Pardoen, B., Levasseur, S. & Collin, F. Using Local Second Gradient Model and Shear Strain Localisation to Model the Excavation Damaged Zone in Unsaturated Claystone. Rock Mech Rock Eng 48, 691–714 (2015). https://doi.org/10.1007/s00603-014-0580-2

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