The modelling teams used the above geometric information to build a conceptual model of the HE-D experiment. The test history shown below was shared by the teams for consistency purposes (considering 6/04/2004 as day 0):
-
Day −29: Excavation: borehole diameter = 0.30 m.
-
Day −28: Heaters emplaced and pressurized (1 MPa).
-
Day 0: First heating phase: 325 W/heater.
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Day 91: Second heating phase: 975 W/heater.
-
Day 339: Cooling phase begins.
-
Day 518: End of test.
Conceptualization of THM processes in the HE-D experiment
THM processes in the context of geological disposal were discussed by Tsang (1987) who used an interaction diagram similar to the one shown in Fig. 10 in order to discuss the coupling of THM processes. Following that discussion, the international DECOVALEX project was started in 1991 and has been continuing for many phases up to the present one. Coupled THM models were developed, calibrated, validated and improved throughout the different phases of DECOVALEX.
The conceptualization of the HE-D experiment follows many similar coupled processes shown in Fig. 10. The fundamental assumption is the idealization of Opalinus Clay as a porous medium. Opalinus Clay at the Mont Terri URL is essentially saturated with water in the pores although de-saturation due to ventilation effects can occur around galleries that have been left open for prolonged periods of time. In the case of the HE-D experiment, the excavation of the heating tunnel was performed in a period of approximately 7 days. Lining, tubing and pressurization followed immediately after excavation (Fig. 5, left diagram). The borehole was drilled with compressed air and the borehole diameter is relatively small (30 cm), in order to minimize the formation of microcracks along the walls. The heaters were also installed in a rock mass sufficiently far from the existing niche (Fig. 1), and the rock mass around the heaters is saturated as evidenced by positive pore water pressure measurements (Zhang et al. 2007). Due to the above considerations, it could be assumed that during the HE-D experiment the Opalinus Clay mass affected by heating would be saturated. When heat is supplied to the Opalinus Clay, the THM processes that take place can be conceptualized as illustrated in Fig. 11. Focusing on a Representative Elementary Volume (REV) that contains both the water and solid phases, the following processes are assumed to occur:
-
Near the heat source the temperature increases. The heat dissipates away from the heat source by conduction in the solid matrix, and by both conduction and convection in the pore water. However, due to the very low permeability of the Opalinus Clay (in the range of 10−19–10−20 m2, see Table 3), heat convection is negligible.
-
The temperature increase causes thermal expansion of both the solid matrix and the pore water. This expansion is constrained by the existing in situ stress in the Opalinus Clay formation; therefore, thermal stresses will develop in the solid matrix. The thermal expansion coefficient of the pore water is much higher than the one of the solid matrix. Therefore, the solid matrix inhibits the expansion of the water, resulting in an increase in the pore water pressure. Zones closer to the heat source would experience higher pressure increases.
-
The increase in pressure in the vicinity of the heat source results in a hydraulic gradient that triggers pore water flow away from the heat source. That flow will gradually dissipate the increase in pore pressure in the vicinity of the heat source.
Mathematical models of THM processes for saturated isotropic porous media have been developed in the past (e.g. Nguyen and Selvadurai 1995). Opalinus Clay would exhibit the same coupled THM processes; however, the bedded nature of the clay suggested that anisotropic effects could be significant. Gens et al. (2007) analysed the HE-D experiment as follows. The Opalinus Clay is a stiff layered Mesozoic clay of marine origin. When saturated stiff clays are subjected to thermal loading, they develop a strong pore pressure response. In turn, the generated pore pressures will impact on subsequent THM behaviour. The performance of the HE-D experiment provided the opportunity to observe in situ the development of coupled THM behaviour in this type of material. The intensity of cross-coupling between the different aspects of the problem turned out to be very variable and could be organized in a hierarchical manner:
-
The strongest coupling is found from thermal to hydraulic and mechanical behaviour. The pore pressure generation is primarily controlled by temperature increase and the largest contributor to deformation and displacements is thermal expansion.
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Significant but more moderate effects are identified from the coupling of hydraulic to mechanical behaviour. The dissipation of pore pressures induces additional displacements and strains, but because of the high clay stiffness, are smaller than thermally induced deformations.
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In principle, mechanical damage could impact the hydraulic results because of the development of a higher permeability due to material damage. The damaged zone, however, appears to be very limited in this case and the effects appear to be insignificant.
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There is no noticeable coupling from hydraulic to thermal behaviour. Practically, all heat transport is by conduction and the thermal conductivity of the material does not change, as the material remains saturated throughout.
Governing equations
From the conceptual understanding of the physical phenomena described above, the governing equations of the models are developed by considering a Representative Elementary Volume (REV) as illustrated in Fig. 11, and invoking the basic principles of conservation of mass, heat and momentum. Additionally, constitutive relationships that are specific to the material thermal, hydraulic and mechanical behaviour must also be assumed. The governing equations are very similar for all teams, since they were based on the same basic principles as discussed in the previous section. For example, the governing equations formulated by the CNSC consist of five coupled partial differential equations with temperature (T), pore pressure (p) and displacement vector (u
i
) as primary unknowns:
$$\frac{\partial }{{\partial x_{i} }}\left( {\kappa_{ij} \frac{\partial T}{{\partial x_{j} }}} \right) + q = \rho C\frac{\partial T}{\partial t}$$
(1)
$$\begin{aligned} & \frac{\partial }{{\partial x_{i} }}\left( {\frac{{\rho_{\text{w}} k_{ij} }}{\mu }\left( {\frac{\partial p}{{\partial x_{j} }} + \rho_{\text{w}} g \, j} \right)} \right) + \rho_{\text{w}} \left[ {\frac{ - n}{{K_{\text{w}} }} + \frac{n - \alpha }{{K_{\text{s}} }}} \right]\frac{\partial p}{\partial t} \\ & \quad + \rho_{\text{w}} \alpha \frac{\partial }{\partial t}\left( {\frac{{\partial u_{\text{k}} }}{{\partial x_{\text{k}} }}} \right) + \rho_{\text{w}} \left( {\left( {1 - \alpha } \right)\beta - n\beta_{\text{w}} - \left( {1 - n} \right)\beta_{\text{s}} } \right)\frac{\partial T}{\partial t} = 0 \\ \end{aligned}$$
(2)
$$\frac{1}{2}C_{ijkl} \left( {\frac{{\partial^{2} u_{k} }}{{\partial x_{j} \partial x_{k} }} + \frac{{\partial^{2} u_{l} }}{{\partial x_{i} \partial x_{l} }}} \right) + \alpha \frac{{\partial p_{f} }}{{\partial x_{i} }} = 0$$
(3)
A summary of the meaning and hypotheses used for each equation is given as follows:
-
(a)
Equation of heat conservation:
Equation (1) is derived from the consideration of conservation of heat. In addition, it is assumed that:
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Local thermodynamic equilibrium exists between the liquid and solid; therefore, the temperature is the same in the two phases at a point around a REV.
-
Heat conduction is the dominant mechanism of heat transport, and convection due to pore water flow could be neglected due to low permeability of OPA, resulting in low pore water velocity.
In Eq. (1), κ
ij
is the thermal conductivity tensor (W/m/°C), ρ is the density of the bulk medium (kg/m3), C is the bulk specific heat of the medium (J/kg/°C) and q accounts for distributed heat generation in the poroelastic medium (W/m3).
-
(b)
Equation of conservation of mass for the pore water:
Equation (2) is derived from considerations of mass conservation of pore water. Equation (2) has four terms, and the assumptions underlying each one are as follows:
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The first term of the equation results from a generalization of Darcy’s law of non-isothermal pore water flow in a saturated porous media. In this term, k
ij
is the saturated permeability tensor (m2), μ (kg/m/s) is the viscosity of water, and ρ
w is the density of water, which are both functions of temperature.
-
The second term is a storage term coming from considerations of compressibility of the water and the solid phase, where n is porosity, 1/K
w is the coefficient of compressibility of water (Pa−1) and 1/K
s is the coefficient of compressibility of the solid phase (Pa−1).
-
The third term is a storage term originating from the consideration of compressibility of the porous skeleton.
-
The fourth term represents water flow due to the difference in thermal expansion between the water and the solid material, where β
w, β
s and β (1/°C) are the coefficient of volumetric thermal expansion of the water, the solid material and the solid skeleton, respectively.
-
(c)
Equation of momentum conservation:
Equation (3) is derived from the consideration of momentum conservation of the porous medium. Since inertial effects are neglected, Eq. (3) reduces to the equation of equilibrium of the porous medium. The major assumption in Eq. (3) is that Biot’s effective stress principle is valid. Therefore, external loads will result in a total stress σ in the porous medium, which would result in an effective stress σ′ in the solid skeleton and a pressure in the pore water according to the equation:
$$\sigma = \sigma^{\prime} + \alpha p$$
(4)
where α is Biot’s coefficient of poroelasticity.
The other parameters of Eq. (3) are C
ijkl
, the stiffness tensor that relates stress to strain and β, the coefficient of volumetric thermal expansion of the solid matrix. All teams involved with this work assumed the rock to be elastic; therefore, the stiffness tensor was derived from Hooke’s law. However, to take into account the anisotropy induced by bedding, most teams used Hooke’s law for a transversely isotropic material, with five independent elastic constants as detailed in the next section.
Numerical models
The governing equations, with their corresponding boundary and initial conditions, were numerically solved with different computer codes that use various numerical methods. BGR/UFZ, ENSI used the finite element method, with the OpenGeosys computer code (Kolditz et al. 2014). CNSC and JAEA also used the finite element method, with the COMSOL (COMSOL AB 2013) and THAMES codes, respectively. CAS used a self-developed numerical code EPCA3D, which is a combination of multiple techniques and theories, such as finite element method, cellular automaton, elasto-plastic theory and principle of statistics (Pan et al. 2009a, b; Pan and Feng 2013). CNWRA used the Finite Difference code FLAC and set up their model for Step 1a with the following conditions: a 2D-axisymmetric model centred on the heater borehole axis of a 8 m radius and a 28 m length; assumed water-saturated conditions; ambient conditions on exterior boundaries; symmetry conditions on the borehole axis; applied heat load and pressure on the heated borehole section; and material parameter values based on previous reports on HE-D modelling. KAERI used the Finite Difference code FLAC3D. LBNL used TOUGH-FLAC, based on linking the TOUGH2 multiphase flow simulator (Integral Finite Difference) with the FLAC3D geomechanical code (Finite Difference Method) (Rutqvist et al. 2002, 2014). The combined TOUGH2 and FLAC3D codes (TOUGH-FLAC) were applied for simulating sequentially coupled multiphase fluid flow and heat transport coupled with geomechanical issues (stress and deformation) (Rutqvist et al. 2002; Rutqvist 2011). Except for CNWRA, who used an axisymmetric model, all teams used a 3D representation of the geometry. The main features and assumptions of the numerical models are summarized in Table 2. It is worth noting that all teams assumed a poroelastic behaviour for the rock mass, except for LBNL who used a Mohr–Coulomb criterion to determine the onset of potential plastic deformations. Typical geometries and meshes are shown in Fig. 12. The main input parameters used by the teams are summarized in Table 3. As shown in Tables 2 and 3, most teams considered the anisotropy of the THM properties of the Opalinus Clay.
Table 2 Main features and assumptions of numerical models
Table 3 Main Opalinus Clay parameters considered by the teams
The initial conditions for the boundary value problems are presented in Table 4. The initial pore water pressure at Mont Terri is approximately 2 MPa. The lower value used by most of the teams reflects the fact that the HE-D area was subjected to drainage well before the start of the test through the nearby galleries (Gallery 98 and MI niche).
Table 4 Initial conditions considered by the teams
The main boundary conditions are:
Most teams applied heat fluxes to the heaters, but some of them applied temperature directly. Those applying heat fluxes considered the heat loss mentioned in the previous section (by including all elements in borehole BHE D0 or by estimation). Hydraulic boundary conditions at the heater surface were either drained or undrained.
External boundary conditions (surrounding galleries) were taken into account in different ways. In general, the teams were required to consider atmospheric pressure, stress relaxation and varying temperature at those boundaries.
Verification with analytical solutions
Verification of numerical results with analytical solutions provides confidence in the adequacy of the numerical model (in terms of mesh size, time step and assumed boundary conditions) and its implementation into the computer code. The CAS, CNSC and BGR/UFZ teams compared their numerical results with analytical solutions proposed by Booker and Savvidou (1985) for a point heat source in an infinite isotropic porous medium (Wang et al. 2016). Analytical solutions for transversely isotropic media (such as could be assumed for Opalinus Clay) do not presently exist. However, the temperature solution from Booker and Savvidou (1985) could be transformed in order to take this transverse anisotropy into account (Garitte et al. 2012; Nguyen et al. 2016).
A comparison of the modelling results with the analytical solution, using the input parameters in Table 5, is presented in Fig. 13. The modelling results compare satisfactorily with the analytical solution providing confidence in the model implementation. Some slight discrepancy is, however, noticed. This is attributable to mesh discretization, but mainly to boundary effects, since the analytical solution is based on an infinite medium, while the numerical models were developed for finite domains surrounding the point heat source. The research teams have tested those two factors by refining the mesh and extending the boundaries of the numerical models farther from the heat source. The results presented in Fig. 13 are the final ones where a visually satisfactory agreement is found between the numerical and analytical solutions.
Table 5 Input parameters for the verification with analytical solution