Poro-elasticity

Abstract The Canadian Nuclear Safety Commission (CNSC) used the finite element code FRACON to perform blind predictions of the FEBEX heater experiment. The FRACON code numerically solves the extended equations of Biot s poro-elasticity. The rock was assumed to be linearly elastic, however, the poro-elastic coefficients of variably saturated bentonite were expressed as functions of net stress and void ratio using the state surface equation obtained from suction-controlled oedometer tests. In this paper, we will summarize our approach and predictive results for the Thermo-Hydro-Mechanical response of the bentonite. It is shown that the model correctly predicts drying of the bentonite near the heaters and re-saturation near the rock interface. The evolution of temperature and the heater thermal output were reasonably well predicted by the model. The trends in the total stresses developed in the bentonite were also correctly predicted, however the absolute values were underestimated probably due to the neglect of pore pressure build-up in the rock mass. 

In order to predict the T-H-M response of the bentonite, a coupled T-H-M transient analysis was performed with the Finite Element Code FRACON. The governing equations incorporated in the FRACON code were derived from an extension of Biot s (1941) theory of poro-elasticity to include the T-H-M behaviour of the unsaturated FEBEX bentonite. The model formulation(Nguyen, Selvadurai and Armand, 2003) resulted in three governing equations where the primary unknowns are temperature, the displacement vector and the pore fluid pressure, as follows ... 

Equation (1) is the equation of conservation of heat, where heat conduction is assumed to be the only mechanism of heat transport. In this equation, K j is the thermal conductivity tensor (W/m / C), p is the density of the bulk medium (kg/m ), C is the bulk specific heat of the medium (J/kg/ °C) and q accounts for distributed heat generation in the poro-elastic medium (W/m ). 

The FEBEX T-H-M experiment is a valuable and important project which should lead to an improvement in the understanding of the behaviour of the bentonite barrier around heat-emitting Nuclear Fuel Waste(NFW) containers. Such large field experiments should always be undertaken with the simultaneous development of constitutive and computational models to interpret the experiments. The FEBEX bentonite possesses strong nonlinear behaviour in the unsaturated state. In order to simulate that behaviour, we have adopted a nonlinear poro-elastic approach. In this approach, the coefficients of the poroelastic equations are assumed to be functions of suction and the void ratio. These functions are derived from the state-surface equation which has been experimentally obtained from suction-controlled oedometric tests performed by the Spanish research organizations UPC and CIEMAT. 

The equations of poro-elasticity were solved with the finite element code FRACON. The FRACON code was used to predict the in-situ T-H-M experiment at the FEBEX gallery in Grimsel, Switzerland. The FRACON correctly predicted that the bentonite would resaturate from the rock interface. Near the heater, it also correctly predicted that initial drying will take place, followed by a slow resaturation. The model correctly predicted that at the end of 1000 days of heating, resaturation of the bentonite was still... 

The effects of temperature can be prevalent for some formation rocks and in-situ conditions. Temperature-induced changes in pore pressure and rock matrix stress (thermo-poro-elasticity model) can be modelled using Equations 10 to 12. In addition, to take into account the chemical potential effects, gradient of rock water potential instead of pore pressure gradient is used in Equation 11. 

Wong, H., Morvan, M.,DeleruyeUe, F. and Leo, C.J. Analytical study of mine closure behaviour in a poro-elastic medium. Computers and Geotechnics, 2008, 35(5), pp645-654. 

In poro-elastic models, where fluid storage and resistance to flow in pores within the polymer are modeled (Mazzoldi et al. 1999)... 

Example 3 illustrates the use of thermodynamics principles in formulating constitutive equations for a poro-viscoelastic medium. The ultimate purpose here is also to develop solutions for a long horizontally aligned tunnel with a circular cross-section embedded in a poro-viscoelastic massif. The setting of the problem is similar to Example 2 discussed above except that the spherical cavity is replaced by a long lined tunnel (Dufour et al. 2009). We start by restricting to small strain problems where the strain tensor of a viscoelastic material can be decomposed into an elastic part (denoted by superscript e ) and a viscoelastic part (superscript V ) ... 

---