Introduction

The sealing issue

The long-term containment function of an underground radioactive waste repository will require the implementation of sealing devices in the boreholes, the galleries, and the access shafts in order to minimize the possible water flow along those structures and to delay the migration of radionuclides towards the biosphere. A solution which is often contemplated consists of emplacing a swelling core, confined by some rigid blocks at both ends (Arnold et al. 2011). Bentonite is often favoured as an active component of the core material because it has a strong swelling potential in order to clog all the voids, as well as a very low hydraulic conductivity. Moreover, when confined, it can develop a swelling pressure capable of closing the fractures (at least partially) generated by the excavation of the openings. In order to study the feasibility of such sealing systems, numerous laboratory experiments have been performed, as well as in situ experiments conducted in Underground Research Laboratories (URL), such as TSX (Tunnel Sealing Experiment) (Martino et al. 2008) and ESP (Enhanced Sealing Project) (Dixon et al. 2014) at Atomic Energy of Canada Limited (AECL)’s URL in Canada, FEBEX (Full-scale Engineered Barriers Experiment) at Grimsel, Switzerland (Huertas et al. 2000), EB (Engineered Barrier Emplacement Experiment) at the Mont Terri, Switzerland (Villar et al. 2014), RESEAL (Repository Sealing) at Hades URL, Belgium (van Geet et al. 2007), and KEY at the Laboratoire souterrain de Meuse Haute Marne (LSMHM), France (Barnichon and Deleruyelle 2009). In France, the Institut de Radioprotection et de Sûreté Nucléaire (IRSN) has conducted the SEALEX (Sealing Performance Experiment) project in the Tournemire URL, in order to test various technical parameters that could influence global hydraulic and mechanical performances of a seal and to provide IRSN with feedback and knowledge on the key parameters (Barnichon and Deleruyelle 2009).

In the SEALEX project, a series of sealing configurations were emplaced in large-diameter (60 cm) boreholes, excavated in argillite (Barnichon et al. 2011). The concept for each test is based on:

  • A confined swelling clay-based core emplaced at the borehole end;

  • The presence of an annular void required for emplacement (radial void equal to 2 cm);

  • A system to impose a constant fluid pressure (or to measure water flux in a later stage), based on porous geotextile membranes at both ends of the clay-based core;

  • Systems to ensure a watertight seal at both ends: an inflatable packer at the back of the borehole, and a downstream lid with inflatable O-rings coupled with resin at the top part of the core;

  • A confinement system, based on a rigid central tube, an outer rigid plate, and anchor bolts, in order to ensure a volume controlled condition (along the axial direction) during the tests;

  • An intra-core instrumentation of hydro-mechanical (HM) characteristics based on wireless instrumentation.

After installation, all tests start with a flooding phase designed to fill all the technological voids with water as fast as possible (i.e. mostly the annular void and the spacing between blocks). It is followed by a saturation phase, in which a 1 m water head is imposed at both ends of the core allowing the clay-based core to reach full saturation. Then, the axial confining pressure is removed in order to mimic the hypothetical failure of confinement blocks in real repositories for some disruptive scenarios. Finally, a performance test phase is performed after reaching the steady state at the end of the previous phase.

The efficiency of a swelling core depends primarily on the hydro-mechanical properties of the material. For this purpose, bentonite-based materials are favoured, for which the potential swelling pressure is strongly correlated with the initial dry density of the montmorillonite fraction. For the SEALEX project, a 70/30 mixture of MX80 bentonite/sand, compacted at different dry densities, has been chosen, targeting a 4-MPa swelling pressure. Several laboratory tests have been carried out to finalize specifications of this material, and also to provide input data required for modelling.

All the tests are isothermal (temperature close to constant), and thus, thermal or thermally coupled processes are considered to have negligible effects on the system behaviour. Therefore, dominantly hydro-mechanical processes in unsaturated as well as saturated regimes can be expected in these experiments. However, chemical effects cannot be ruled out a priori. For example, exfoliation at the edges of the sand–bentonite core, driven by chemical processes relating to the osmotic potential of the bentonite (Pusch et al. 1990), can lead to the formation of a bentonite-based gel. This has been observed on the bentonite studied here in specifically designed experiments (Saba 2014). With progressive saturation and swelling of the core, the gel can be re-compacted.

Specifically, the following processes are potentially involved in the in situ experiments:

  1. 1.

    Resaturation of a pre-compacted clay-based core under free and confined conditions, leading to complex load/suction paths;

  2. 2.

    Development of a high swelling pressure;

  3. 3.

    Continuous evolution of the dry density of the material, with significant impact on the hydro-mechanical properties;

  4. 4.

    Influence of the voids surrounding the core.

Experiments environment in the Tournemire URL

Two particular in situ experiments performed within the SEALEX project are considered in this paper: the WT-1 experiment which aimed at quantifying the hydro-mechanical behaviour of the argillite, without the effects of the bentonite–sand core, and the PT-A1 experiment which aimed at studying the fully coupled hydro-mechanical behaviour of the bentonite–sand plug, surrounded by the argillite.

Both experiments are located in boreholes, drilled from the existing 2008 gallery in the Tournemire URL as shown in Fig. 1. The WT-1 and PT-A1 (with bentonite/sand cylinder) have similar orientations, though WT-1 is 3.75 m long and PT-A1 is 5 m long.

Fig. 1
figure 1

Location of the WT-1 and PT-A1 experiments in Tournemire URL, and location of measuring pore pressure chambers GSM270_P1 and GSM270_P2

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Initial pore pressure

Prior to, during, and after the excavation of the SEALEX boreholes, the evolution of the pore pressure (absolute pressure) in the argillite around the 2008 gallery was recorded at distances of 3 and 6 m from the gallery wall, in horizontal and vertical directions. The configuration is shown in Fig. 2, and the corresponding pressure evolution over the 4-year period is displayed in Fig. 3. Starting from the value of injection pressure (400 kPa), the pressure profiles in different directions and distances may be described as follows:

Fig. 2
figure 2

Locations of the sensors in a vertical cross section

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Fig. 3
figure 3

Time evolution of pore pressure around the gallery (vertical lines indicate the start and end of excavation of PT-A1 borehole) : sensors labelled GSMx_P1 are located at 6 m from the gallery wall, and sensors labelled GSMx_P2 are located at 3 m from the gallery wall

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  • The downward vertical chambers (GSM180_P1 and GSM180_P2) decreased rapidly to reach a value close to atmospheric pressure at the end of 2013.

  • The upward vertical chambers (GSM0_P1 and GSM0_P2) decreased rapidly to reach a value close to 200 kPa at the end of 2013.

  • The horizontal chambers (GSM270_P1 and GSM270_P2) increased rapidly, and then:

    • The chamber closer to the gallery (GSM270_P2, at 3 m radial distance) almost stabilized at 600 kPa, and then started to decrease from the end of 2010.

    • The chamber farther from the gallery (GSM270_P1, at 6 m radial distance) increased up to 1000 kPa in March 2012. It then started to decrease slightly.

  • The pressure in both horizontal chambers responded to the excavation of borehole PT-A1 (located 2 m radially from the chambers), which took place between 02 May until 16 May 2012, and which induced a temporary increase of pressure in these chambers that lasted several months.

The pressure data show very different trends in the horizontal plane compared to the vertical plane. In the horizontal plane, pressures are around or above hydrostatic, continuing to rise at 6 m radial distance, but falling to around 200 kPa at 3 m radial distance. This sort of pressure distribution is expected when there is anisotropy in the mechanical properties of the rock. It is also noticeable that the oscillations due to the relative humidity in the gallery can be seen in the data from the horizontal plane. Pressures in the vertical plane are more difficult to understand. A radial pattern of desaturation might be expected, and hence, pressures dependent on the radial distance from the centre of the tunnel would be expected. However, the data show little pressure dependence on the radial distance, and instead pressure above the tunnel is around 100 kPa higher than pressure below the tunnel.

The relative humidity and temperature recorded continuously in the gallery exhibit seasonal cycles.

Rock properties

The Tournemire argillite is made of clay (mica, kaolinite, illite), carbonate (calcite) and quartz (Boisson et al. 2001). The material has a sub-horizontal bedding structure, thus transverse isotropic mechanical and hydraulic properties, which are different along directions parallel or perpendicular to the bedding planes. The main properties of the argillite, derived from Daupley (1997), Boisson et al. (2001) and Valès et al. (2004), are summarized in the Table 1.

Table 1 Tournemire argillite main properties
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The in situ principal stresses are as follows (Rejeb and Tijani 2003):

  • σh = 2.1 ± 1 MPa oriented N 72° ± 15° E; it has 10° pending with horizontal direction

  • σv = 3.8 ± 0.4 MPa oriented N 72°E; it has 10° pending with vertical direction

  • σH = 4 ± 2 MPa oriented N 162° ± 15° E

Modelling teams

The DECOVALEX project (DEvelopment of COupled process models and their VALidation against EXperiment) was initiated to support the development of computer codes and to compare model calculations with results from field and laboratory experiments. In the latest phase of the project (2012–2015), one of the tasks consisted of the analysis of the SEALEX project.

In a companion paper (Millard et al. 2016), the first two steps of the task, consisting of calibrating the bentonite–sand mixture model parameters on the basis of laboratory tests and in modelling a 1/10th scale mock-up representative of the in situ SEALEX experiments, have been described in detail. In the present paper, the two other steps of the task devoted to the comparative modelling of in situ tests are presented. Some specific aspects are also treated in other companion papers (Thatcher and Bond 2016a, b).

The participating modelling teams and the corresponding funding organisations are:

  1. 1.

    CNSC (Canadian Nuclear Safety Commission, Canada)

  2. 2.

    IRSN (Institut de Radioprotection et de Sûreté Nucléaire, France);

  3. 3.

    Quintessa on behalf of RWM (Radioactive Waste Management, United Kingdom);

  4. 4.

    UoE (University of Edinburgh) on behalf of RWM (Radioactive Waste Management, United Kingdom);

  5. 5.

    UGN (Institute of Geonics CAS) on behalf of RAWRA (RadioActive Waste Repository Authority, Czech Republic);

  6. 6.

    UFZ (Helmholtz Centre for Environmental Research, Germany).

The different codes used by the teams are listed in Table 2. Most of the codes are finite element codes, and only QPac, the code used by Quintessa, is based on finite volumes for hydraulics, and mixed elements for mechanics (Quintessa 2013).

Table 2 Computer codes
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In situ test WT-1

The in situ test WT-1 was used as test case for the modelling of the hydraulic behaviour of the rock surrounding the bentonite–sand plug. In this section, the test is first described, then the modelling of the test is presented, and finally the predicted results are compared to the measurements.

Description of the WT-1 test

The in situ WT-1 experiment aimed at quantifying the water mass ‘losses’ that occured within the host rock and around the downstream lid system, without the effects of the bentonite–sand core. Compared to the full SEALEX test design (e.g. PT-A1 test), the WT-1 test had a purposefully oversimplified design (see Fig. 4) that considered only the downstream lid with its water sealing system (inflated O-rings and injected epoxy resin), where water mass ‘losses’ resulting from a full test outer boundary condition would take place.

Fig. 4
figure 4

A schematic description of the WT-1 experiment

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The excavation of the borehole was completed in March 2011. It started on March 1, 2011 and ended on March 4, 2011. The advancement rate of the excavation front was around 1 m per day. After excavation was completed, the borehole head was closed with a plastic sheet to limit desaturation of the borehole walls. From the March 29 to 31, 2011 (i.e. about 25 days after the end of excavation), the experimental device was introduced into the borehole, the packers were inflated, and some resin was injected to ensure the tightness of the device. Finally, hydration supplied by an outside water tank (as shown in Fig. 1) was started on the April 7, 2011 following the strategies of the two phases described below:

  • First, a back pressure of 2 bars (i.e. 20 m water head) was applied in the tank, which allowed filling the void between the rock and the device in 2 h.

  • Second, the back pressure was removed, and the upper level of the water in the tank was constantly maintained at 1 m above the axis of the borehole (i.e. 1 m water head).

The final position of the set-up is as follows:

  • The 60-cm-diameter borehole is open from its head at the gallery for a distance of 335 cm. Accordingly, relative humidity in this length of the borehole is assumed to be the same as in the gallery;

  • At the far end of the borehole, there is a 5-cm gap between the rock and the experimental set-up that is filled by water. Adjacent to this gap, the borehole is lined for a 35 cm length with a steel cylinder and injected resin (condition with zero normal flow). As an approximation, the borehole can be considered as closed in the last 40 cm (i.e. disregarding the presence of the water filled gap).

The injected water mass was constantly recorded. No leak was visually observed. First the flooding of the cavity (nominal thickness of 5 cm) took place in 2 h, during which a mass of 16600 g of water was injected. This water mass corresponds closely to the theoretical water mass required to fill the initial voids in the test. Approximately 1370 g more water was injected than expected, which is thought to have been absorbed by the host rock itself during the initial 2-h flooding phase.

The evolution of the injected water mass with time showed a different behaviour in the first 3 months from that during the next 10 months in the experiment, which was attributed to experimental errors, and corrected by adding 200 g of water to fit the initial trend to the final one. On the basis of this correction, the ‘Reference’ curve (i.e. as expected to be simulated by modelling) was considered as the one composed of the first three months trend and of the corrected one. It is shown in Fig. 5.

Fig. 5
figure 5

Reference curve (i.e. corrected) of water mass injected after initial flooding

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Discussion of the test result

If the initial three months period is temporarily ignored, the remaining data follow a roughly linear trend, with a change in slope after one year. The reason for this change is not clear. It was interpreted by Quintessa as a change in the EDZ permeability as any fractures heal (Thatcher and Bond 2016a). Subtracting the two linear trends from the injected water data reveals an oscillation with a period of one year, reflecting the oscillating relative humidity in the experimental gallery (Fig. 6). This demonstrates that the pressure gradient causing flow into the argillite is driven by the difference in pressure between the applied head and the suction on the borehole walls induced by the relative humidity.

Fig. 6
figure 6

Relative humidity in the tunnel compared to the difference between the injected water mass and the linear trend lines

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The modelling of the WT-1 test was done by some teams, using on the one hand either a pure hydraulic or a fully coupled hydro-mechanical formulation, and on the other hand two-dimensional or three-dimensional models. From the calibration of the argillite hydraulic parameters on the test result, the teams obtained large differences on parameters such as the intrinsic permeability, ranging from 10−21 m2 to 2. 10−19 m2. The discrimination between the different approaches would require additional test measurements. Nevertheless, in view of the WT-1 results, most of the teams concluded that the argillite was unlikely to be a major hydraulic sink on the timescale of the PT-A1 experiment.

In situ PT-A1 test

The main objective of the PT-A1 test was to study the fully coupled hydro-mechanical behaviour of the bentonite–sand plug, surrounded by the argillite. In this section, the test is first described, then the modelling of the test is presented, with some emphasis on the technological void modelling, and finally the predicted results are compared to the measurements.

Description of the PT-A1 test

The in situ PT-A1 test aimed at quantifying the evolution of the hydro-mechanical characteristics (swelling pressure, pore pressure, water saturation) within the bentonite–sand as the core resaturated after emplacement in the borehole.

The schematic layout of the experiment is shown in Fig. 7. The swelling bentonite–sand core (MX80 bentonite based) is confined between two fixed stainless steel packers, which provide both hydraulic boundary and mechanical stiffness. The upstream packer is emplaced at 5 m from the main borehole head. The chamber left at the end of the borehole is filled up with an injected grouting. The downstream plate is retained by a confining system (referred to as a confining plug) maintaining an approximately constant volume condition. Note that in the real test configuration, contrary to the schematic description in Fig. 7, the upstream and downstream filters are connected to the same tank, located in the gallery. The packers have a series of three rubber inflatable cushions (O-ring) all around and conduits for the hydration tubings. Once inflated with water, the O-rings close the 20-mm-wide gap between the plate and the surrounding rock. To improve the watertightness of the system, the annular space between the three O-rings is filled up with resin (Sikadur 52).

Fig. 7
figure 7

A schematic description of the PT-A1 experiment (including a detail of the three rubber inflatable O-rings)

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The PT-A1 experiment is located in a borehole, drilled from the gallery as already shown in Fig. 1. The excavation of the PT-A1 borehole was performed in 2012. It started on May 2, 2012 and ended on May 16, 2012. The work was interrupted twice, firstly between the May 4, 2012 and the May 9, 2012, and secondly between the May 10, 2012 and the May 14, 2012. Therefore, the effective time necessary for the excavation was 6 days, which corresponds to an advancement rate of the excavation front of around 0.77 m per day. After excavation was completed, the borehole head was closed with a plastic sheet to limit desaturation of the borehole walls. From June 12, 2012 to July 5, 2012 (i.e. about 26 days after the end of excavation), the experimental device was introduced into the borehole, the packers were inflated. Finally, hydration supplied by a water tank in the access gallery was started on the September 25, 2012 following the strategies of the two phases described below:

  • First, a back pressure of 2 bars (i.e. 20 m water head) was applied in the tank, which allowed filling the void between the rock and the device in 2 h, while a temporary venting system enabled air evacuation during flooding of the system. An amount of 80 l of water was injected, which corresponds first to the technological void which has a volume of about 44 l, and also to the various existing gaps:

  • Gap between the downstream lid and the rock (about 33 l),

  • Gaps between the bentonite blocks,

  • Gaps within the blocks, around the sensors.

  • Second, the back pressure was removed, and the upper level of the water in the tank was constantly maintained at 1 m above the axis of the borehole (i.e. 1 m water head).

The bentonite–sand core was made of 8 monolithic pre-compacted discs, of 0.56 m diameter and 0.15 m thickness. During installation, the volume and mass of each bentonite disc have been measured. Using these data together with the fabrication water content (11%), the initial dry density of the bentonite–sand core material has been estimated as 1.94 Mg/m3, and the initial degree of saturation as 0.73. The estimated dry density after filling of the technological gap is 1.67 Mg/m3. The assembled core of 8 discs was emplaced in the borehole (0.6 m diameter), which led to a non-uniform gap around the core (as illustrated in Fig. 12), thus introducing three-dimensional features into the problem.

The instrumentation of the test was designed to try to catch these features. The core was instrumented by means of wireless sensors, positioned as illustrated in Fig. 8. During the test, problems arose with several sensors, and only a limited number of sensors could supply useful measurements.

Fig. 8
figure 8

Relative humidity sensors and stress pressure cells. Left core side view, upstream on the left. Right section C, the blue zone being the water-filled technological void

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A first group of eight sensors, shown in black in Fig. 8, was used to record the relative humidity at various locations in the core. Only four of them supplied relevant measurements, which are plotted in Fig. 9. A difference is observed in the experiment between the sensors RH 22-1 and RH 22-2 which are located one above the other, and for which, therefore, different hydration histories might be expected. On the contrary, the sensors RH 52-1 and RH 52-2, which are located at symmetric horizontal locations, should have given approximatively the same answer, but this was not found experimentally.

Fig. 9
figure 9

Relative humidity measurements

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The axial stress was recorded by the two pressure cells, shown in red in Fig. 8, located at the ends of the core. The corresponding measurements are plotted in Fig. 10. The axial stresses differ by a factor greater than 2. Among the reasons that could explain this difference are the friction forces at the bottom of the core, as well as the heterogeneity of the bentonite.

Fig. 10
figure 10

Axial stress measurements

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Radial stresses were also measured in one given section (section C) at three locations, as shown in Fig. 8. The corresponding measurements are plotted in Fig. 11. Unexpectedly, the measurements at symmetric locations S 60-1 and S 60-3 are different. This might be due to some asymmetry when emplacing the bentonite–sand core in the borehole.

Fig. 11
figure 11

Radial stress measurements

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The injected water mass was also continuously recorded. No leak was detected. The evolution of the cumulative injected water mass, after the flooding of the cavity, is displayed in Fig. 12. The difference in the trend, compared to the WT-1 experimental result (Fig. 5), can be explained by the presence of entrapped air within the hydration lines, which was discovered after the test, and which prevented further water intake.

Fig. 12
figure 12

Evolution of cumulative injected water mass after the flooding of the cavity

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Bentonite–sand mixture modelling

The simulation of the hydro-mechanical response of the sealing core material in a sealing experiment is a really challenging task. Indeed, because of the presence of a technological void between the bentonite–sand core and the surrounding rock, the core material is subjected to a complex load–suction path. It is first subjected to a constrained swelling under hydration in the axial direction, but to a free swelling in the radial direction. Then, when the gap is closed, the swelling occurs under constrained volume in both directions, but with a reduced hydration from the lateral side. Moreover, some gel is formed at the bentonite surface (Saba 2014). The hydro-mechanical properties of this gel are evolving during the closure of the gap by the swelling bentonite.

The classical framework of multi-phase flow in deformable porous media has been used by all the teams to model the hydro-mechanical processes taking place in the bentonite–sand core. The general formulation of multi-phase flow including liquid water, vapour, and dry air (Olivella et al. 1994, 1995) has been used only by IRSN, while the other teams adopted the well-known Richards equation (Richards 1931) based on assumptions of constant atmospheric gas pressure and negligible mass of vapour in front of the mass of liquid water. Apart from this difference in mass balance equations, the main differences between the theoretical formulations used by the teams came essentially from the hydro-mechanical constitutive equations of the bentonite–sand mixture. The corresponding equations have been detailed in Millard et al. (2016). Here, the focus is on two salient features which are the water retention curve, and the mechanical constitutive equations.

Water retention curve

The dry density of the bentonite–sand mixture has a crucial influence on the hydraulic properties of the material, such as the retention curve (Man and Martino 2009). As an example, the water retention curves determined for the same bentonite–sand mixture but compacted at different dry densities, as reported by Baumgartner (2006), are shown in Fig. 13. On this figure, the different curves correspond to different effective Montmorillonite dry density (EMDD) of the mixture, defined by:

$${\text{EMDD}} = \frac{{f_{\text{m}} .f_{\text{c}} .\rho_{\text{d}} }}{{\left[ {1 - \left( {\frac{{\left( { 1 { - }f_{\text{c}} } \right)\rho_{d} }}{{G_{a} .\rho_{w} }}} \right) - \left( {\frac{{\left( {1 - f_{m} } \right)f_{c} .\rho_{d} }}{{G_{n} .\rho_{w} }}} \right)} \right]}}$$

where f m is the mass fraction of montmorillonite in clay fraction, fc the mass fraction of clay in dry solid, \(\rho_{d}\) the dry bulk density, \(\rho_{w}\) the water density, \(G_{a}\) the relative density of aggregate component (specific gravity), and \(G_{n}\) the relative density of non-montmorillonite clay component.

Fig. 13
figure 13

Water retention curve of bentonite as a function of dry density (from Baumgartner 2006)

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In the same manner as for the argillite, some teams chose to idealize the water retention curve by the Van Genuchten expression (Van Genuchten 1980). UFZ used the classical expression without modification, whereas other teams preferred to adapt it to account for its dependency on the dry density of the material (Man and Martino 2009). CNSC used the following equation:

$$S_{e} = \left[ {1 + \left( {\alpha_{\text{EMDD}} h} \right)^{N} } \right]^{ - m}$$

where h is the pressure head, and \(\alpha_{\text{EMDD}}\) is a coefficient which depends on the effective Montmorillonite dry density EMDD of the bentonite–sand mixture (\(F_{\text{corr}}\) being a calibration parameter):

$$\alpha_{\text{EMDD}} = \left( {0.0705 \times \left( {EMDD} \right)^{2} - 0.3065 \times \left( {EMDD} \right) + 0.313} \right) \times F_{\text{Corr}}$$

IRSN let the parameters P0 and λ 0 depend on porosity variation \(\vartheta_{0} - \vartheta ,\) where \(\vartheta_{0}\) is the initial porosity, a and b are calibration parameters:

$$P_{0} = P_{0} e^{{a\left( {\vartheta_{0} - \vartheta } \right)}} \quad \lambda_{0} = \lambda_{0} e^{{b\left( {\vartheta_{0} - \vartheta } \right)}}$$

In order to reflect the influence of the double structure porosity of the bentonite, as accounted for in the BExM model (Navarro et al. 2015), Quintessa used a water retention curve relating the suction under free swelling conditions \(s_{free}\) to the water content by mass \(\omega\), and the dry density:

$$s_{free} = \alpha { \exp }\left( {\beta \rho_{d} } \right) + \gamma { \exp }\left( {\varepsilon \omega } \right)$$

where \(\alpha , \beta ,\gamma ,\varepsilon\) are constants calibrated against the data. The suction is explicitly linked to the mean stress (\(p\)) through the following equation, suggested by Dueck (2004):

$$s = {\text{s}}_{\text{free}} - p$$

For UoE, the water retention curve was fitted with an empirical relationship derived by Bond and Benbow (2009), giving also the free suction:

$$s_{\text{free}} = M_{b} \left( {{ \exp }\left( {a - b\omega } \right) + { \exp }\left( {c - d\omega } \right)} \right)$$

where \(M_{b}\) is the mass fraction of bentonite, and a, b, c, and d are material parameters. Stress-dependent water retention properties are determined in the same manner as Quintessa through the Dueck (2004) suction–mean stress relationship.

As Quintessa, UGN used a relation between the water content, the suction and the dry density:

$$\omega = \left\{ {\begin{array}{*{20}c} {\omega \left( s \right) = a \cdot \ln \left( s \right) + b {\text {for}}\, \omega \left( s \right) < \omega_{ \hbox{max} } } \\ {\omega_{ \hbox{max} }\quad {\text{elsewhere}}} \\ \end{array} } \right.$$

This relation is constructed for a given dry density, by considering a simple bilinear representation of the experimental data in the ln(s) -\(\omega\) plane. The dry density enters the expression as:

$$\omega_{ \hbox{max} } = \frac{{\rho_{s} - \rho_{d} }}{{\rho_{s} }}\frac{{\rho_{w} }}{{\rho_{d} }} + \omega_{\text{res}}$$

where \(\rho_{s}\) is the density of solid, \(\omega_{ \hbox{max} }\) and \(\omega_{res}\) are the maximal and residual mass water content, respectively. The value \(\omega_{\text{res}}\) could be related to the dry density of the bentonite component, but it was found sufficient to keep a constant value. The effective water saturation is obtained from the water content through:

$$S_{e} = \frac{{\omega - \omega_{\text{res}} }}{{\omega_{ \hbox{max} } - \omega_{\text{res}} }}$$

Mechanical constitutive equations

Since the bentonite–sand mixture may exhibit an elasto-plastic response under some stress–suction paths, as observed in some laboratory experiments (Wang et al. 2012; Saba 2014), Quintessa and IRSN chose to use an elasto-plastic model, whereas the other teams preferred to use an elastic material model, including nonlinear effects. Except IRSN and UoE, all the teams had to incorporate an additional swelling strain to be able to reproduce the swelling behaviour of the bentonite–sand mixture.

IRSN used the well-known mechanical elasto-plastic Barcelona Basic Model (Alonso et al. 1990), with variable volumetric elastic properties with respect to mean net stress p and suction s:

$$d\varepsilon_{v}^{e} = \frac{{\kappa_{i} \left( s \right)}}{1 + e} \frac{dp}{p} + \frac{{\kappa_{s} \left( p \right)}}{1 + e} \frac{ds}{s + 0.1}$$

In this expression, κi and κs are elastic stiffness parameters for changes in net mean stress and suction, respectively, and e is the void ratio. The elastic stiffness parameter for changes in mean net stress depends on suction as:

$$\kappa_{i} \left( s \right) = \kappa_{i0} \left( {1 + \alpha_{i} s} \right)$$

where \(\kappa_{i0}\) is the elastic stiffness parameter in saturated conditions and \(\alpha_{i}\) a model parameter. The elastic stiffness parameter for suction changes depends in turn on mean stress as:

$$\kappa_{s} \left( p \right) = \kappa_{s0} \left( {1 + \alpha_{s} { \ln }\frac{p}{{p_{\text{ref}} }}} \right)$$

where \(\kappa_{s0}\), \(\alpha_{s}\), and \(p_{\text{ref}}\) are model parameters.

Quintessa proposed an original approach based on the Modified Cam Clay (MCC) model, with some small modifications. The elastic bulk modulus is given by:

$$K = K_{0} - \kappa p^{\prime}$$

and the plastic yield surface is given by:

$$\left[ {\frac{q}{M}} \right]^{2} +\, p^{\prime} \left( {p^{\prime} - p_{c} } \right) = 0$$

where \(K\) is the bulk modulus, \(p^{\prime}\) is effective mean stress, \(q\) is deviatoric stress, \(p_{c}\) is the pre-consolidation pressure, and \(K_{0}\), \(v_{\kappa }\), \(\kappa ,\) \(M\) are all constant parameters. The virgin consolidation line is derived from an original relationship, called ‘Internal Limit Curve’ (ILC), which is based on relationships derived from published data (Wang et al. 2013a, b) and on a conceptual understanding of energy balances in the bentonite. It is used to parametrize both the plastic failure of the material and the water retention curve. It relates the mean stress (or swelling pressure) to the dry density (or water content):

$$p = \tilde{\alpha }*exp\left( {\tilde{\beta }\rho_{d} } \right)$$

where \(\tilde{\alpha }\) and \(\tilde{\beta }\) are model parameters.

UoE used a nonlinear elastic strain-dependent model, where the elastic modulus is updated as a function of the volumetric strain:

$$E_{t} = E_{t = 0} \left( {1 - \left( {\zeta \mathop \sum \limits_{t = 1}^{t - 1} \varepsilon_{v} } \right)^{d} } \right)$$

where \(E_{t = 0}\) is the reference starting Young’s modulus, \(\zeta\) is a hardening or softening factor depending on the strain history and direction, \(\varepsilon_{v}\) is the total volumetric strain, and \(d\) is a fitting parameter. This function leads to an increase in elastic modulus with a decrease in sample volume i.e. strain hardening, and vice versa (strain softening). The initial void ratio of the sample is used as a control to switch between the softening and hardening factors in the calculation of the Young’s modulus. This allows strain softening and hardening to be parameterized separately.

UGN used for the mechanical constitutive equations an incremental elasticity model, with a constant Poisson’s ratio and a variable elasticity modulus, which depends on two parameters \(\xi\) and \(\eta\), and represents a function of dry density:

$$E = \xi { \exp }\left( {\eta \frac{{\rho_{d} }}{{\rho_{w} }}} \right)$$

Finally, UFZ and CNSC chose an isotropic linear elastic model, with constant parameters.

PT-A1 model set-ups

One main difficulty was to build a conceptual model consistent with the results of both PT-A1 and WT-1 tests. Some teams chose to use an axisymmetric or a bidimensional model, thus supplying an average response for the radial stresses evolution in the bentonite–sand core, while some other teams preferred a three-dimensional model. Concerning the surrounding argillite, some teams considered a coupled hydro-mechanical model, whereas some others did not model it, assuming a specific boundary condition at the wall of the borehole, on the basis of the WT-1 results.

Another difficulty consisted in modelling the hydro-mechanical behaviour of the technological void.

Quintessa modelled only the bentonite using an axisymmetric representation. The presence of the technological void was modelled by imposing a hydrostatic pressure condition together with an inflow volume limited to the void volume. In addition, they treated the contact between the bentonite–sand core and the rock by a specified stress which increased as the bentonite expanded radially and had a very large value when the bentonite had filled the void.

UoE used an axisymmetric model, but handled the technological void differently. A moving mesh was used to record when the void became filled and to switch on or off the boundary conditions as a function of the nodal displacements. For the hydraulic process, the lateral boundary condition was a prescribed pressure to represent a water-filled void until the nodal radial displacement reached the limit of the technological void space. Once a node was calculated to have reached a displacement equal to the filling of the void, the hydraulic boundary condition at the bentonite edge was removed in order to account for the lack of hydraulic contribution of the host rock (i.e. became a no-flow boundary), and the mechanical free boundary was changed to a zero displacement boundary.

UFZ used an axisymmetric model to investigate the fluid flow and axial stresses, and a bidimensional cross-section model to calculate the radial stresses (Fig. 14). They distinguished two steps in the modelling process, a free swelling step associated with an open gap over a given period of 17 days, followed by a constrained swelling step corresponding to a closed gap. In the axisymmetric model, the calculation started with a free radial displacement and a prescribed water pressure at the lateral boundary and then changed to zero radial displacement and zero water flux when the gap was closed. In the cross-section model, the gap was modelled by finite elements with a high intrinsic permeability (5 10−5 m2) and a low Young’s modulus (9.8 10−3 MPa) during the first step, and with the bentonite properties after the gap closure.

Fig. 14
figure 14

UFZ models used for the PT-A1 test. Calculation steps and boundary conditions of axisymmetric and cross-section models

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The three other teams (CNSC, IRSN, and UGN) used a three-dimensional representation of the PT-A1 test. Only one half of the domain was considered for symmetry reasons. IRSN and UGN included in their model the hydration systems (downstream and upstream lids), the concrete block between the host rock and upstream lid, the bentonite–sand core, the annular gap, and the surrounding host rock (Fig. 15), whereas CNSC chose to represent only the bentonite core, but including the interfaces between the bentonite discs (Fig. 16).

Fig. 15
figure 15

IRSN model used for the PT-A1 test

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Fig. 16
figure 16

CNSC model used for the PT-A1 test

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For IRSN, the gap was modelled as a material with high porosity. Permeability was taken as a high value for saturated conditions (Table 3). The permeability was assumed to be dependent on porosity. A function with very low capillary pressure was considered for the retention curve (Table 3). This implies that saturation takes place sharply as capillary pressure vanishes. The dependence of the retention curve on density or porosity has been taken into account. Relative permeability was calculated with a power function of degree of saturation.

Table 3 Gap parameters used by IRSN
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The mechanical response of the gap was achieved using two different stiffness values depending on the opening. When the gap is open, the elastic modulus is very low. On the contrary, when it is closed the elastic modulus is 100 MPa (i.e. high stiffness) (Table 3). In this way, relative displacement of the air gap boundaries (nodes) stops when closure takes place because closure leads to stiffness increase.

In the same spirit, UGN treated the void as a saturated porous material with a very low initial dry density, which gradually increased as the gap closed. For the considered applications, it is important that the gap remains fully saturated before closing and serves as a transport zone for the water. After closing, the mechanical response of the gap remains active and it is very close to the mechanical response of the plug. The gap closing moment was identified as the moment when the plug swelled over the full gap (nearly touching the borehole wall). From that moment, the flow in the gap was prevented, which means a direct interconnection of the flow models in the plug and the host rock. This interconnection used the COMSOL extrusion coupling operator.

In addition to the filling of the gap with water and its closure due to swelling of the sand–bentonite plug, CNSC also modelled the flow between the sand–bentonite discs along the interface between discs. The gap was simulated as a very compressible saturated material with unit porosity. The sand–bentonite disc-to-disc interface was simulated as a fracture with unit porosity, and an initial permeability of 10-14 m2. The permeability was set to be variable as a function of swelling pressure development, reaching a permeability of 10-21 m2 at full swelling pressure.

Concerning the boundary conditions, at both ends of the core the axial displacement was restrained, and a zero water pressure was prescribed. On the lateral surface of the domain, which represents the internal surface of the excavated borehole, a no-flow and a zero displacement boundary conditions were assigned, neglecting the effect of the rock.

PT-A1 model results

In the following, the relevant quantities measured during the test are compared to the model calculations. The values of relative humidity are first shown in Figs. 17 and 18, corresponding to sensors located, respectively, at 22 cm and 52 cm from the outer end of the bentonite core. The fact that sensors RH 22-1 and RH 22-2 responded in a different way could be reproduced by some three-dimensional models, as is the case for UGN and IRSN, whereas CNSC found about the same response for the two sensors. The amplitude of the difference was slightly underestimated by UGN but well reproduced by IRSN. On the contrary, by studying only half of the core, these models could not predict the non-symmetric responses of the sensors RH 52-1 and RH 52-2. Most of the models reproduced correctly the response of the sensor RH 52-1, with some differences at the very beginning, but the overall hydration kinetics was rather well captured in particular by the three-dimensional models.

Fig. 17
figure 17

Comparison between measured and predicted relative humidities at 22 cm

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Fig. 18
figure 18

Comparison between measured and predicted relative humidities at 52 cm

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It is interesting to note, in Figs. 17 and 18, that some teams predicted first a decrease of the relative humidity, while others predicted a continuous increase with time. The decrease can be explained by the fact that because of the expansion of the outer part of the bentonite under hydration, the inner core also slightly expands, and its porosity increases, thus causing a drop in suction. This effect can be reduced by incorporating a special material in the gap (accounting for the gel formation), as done for example by UGN, which brings some additional radial stiffness to the bentonite.

The comparison of axial stresses is shown in Fig. 19. The test measurements show a significant difference between the two axial stresses as measured at the ends of the bentonite–sand mixture. Among the reasons that could explain this difference are the friction forces at the bottom of the core, as well as the heterogeneity of the bentonite. For a large number of predictions, there is no difference for the two axial stresses, whereas for IRSN the difference does exist but it is much smaller than in the experiment. Most probably, this can be explained by the differences in the gap modelling. Nevertheless, it can be noted that the predicted axial stresses lie within the range of the test measurements. The decreases of stress which are predicted by Quintessa in the early days and around 200 days are attributed to plastic collapse, as in laboratory tests modelling (Millard et al. 2016). These features are also very sensitive to the model parametrization.

Fig. 19
figure 19

Comparison between measured and predicted axial stresses

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The radial stresses are compared in Fig. 20. This feature could not be reproduced, even by a three-dimensional analysis. Such models, as used by IRSN and UGN, gave, however, a rather good prediction of the stresses S 60-1 or S 60-2.

Fig. 20
figure 20

Comparison between measured and predicted radial stresses

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As can be expected, compared to the three-dimensional models, an axisymmetric model leads to a lower predicted average radial stress during the transient period corresponding to the gap closure. The difficulties in the interpretation of the experimental results are reflected in the overall scatter of the predictions, which in turn is mostly due to the differences in assumptions about the non-uniformity of the gap, and its hydro-mechanical behaviour. In particular, it can be noticed that the various gap modelling assumptions result in significant differences in the build-up time of the radial stresses when the gap is closing. The early development of radial stresses measured seems to confirm the formation of a gel at the bentonite surface, which enabled the transmission of radial stresses between the plug and the surrounding rock. Teams that represented the gap with a hydro-mechanical material that evolved with gap closure were able to match this gradual increase in radial stress e.g. IRSN, UGN, and UFZ. However, teams that treated the gap simply as a void calculated a later onset of radial stress development e.g. UoE, Quintessa.

Finally, the evolution of the injected water mass is compared in Fig. 21. The differences which can be observed are attributable to the various approaches followed by the teams to account for different initial voids (technological void, interfaces, etc.), and for their initial flooding at the start of water injection. In particular, Quintessa took into account as starting value, the measured water mass required to fill all the voids, whereas other teams considered only the water mass necessary to fill the gap volume.

Fig. 21
figure 21

Comparison between measured and predicted injected water mass

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Unaware of the presence of entrapped air within the hydration lines, the models could not reproduce the trend of the experimental curve, i.e. a first increase of injected water followed by a decrease. Quintessa even predicted, at the beginning, an opposite trend, due to water squeezing faster than the water uptake by the bentonite. They found that this behaviour was very sensitive to the hydro-mechanical parametrization of their model.

Conclusion

The SEALEX experiments performed in the Tournemire underground research laboratory have been selected for a comparative modelling exercise of a sealing system, in the framework of the DECOVALEX-2015 project. The analysis of the in situ included the identification and discussion of possible processes that may influence the test results and particularly the bentonite–sand behaviour. The modelling work carried out allowed for the assessment of the relevance of a number of processes which must be properly addressed:

  • The dry density of the bentonite–sand mixture and its evolution with time has a crucial influence on the hydro-mechanical behaviour of the material. As a consequence, a full characterization of the material requires performing laboratory tests at different dry densities.

  • A correct evaluation of the dependency of the water permeability on both the saturation degree and the bentonite dry density (or porosity) was found to be necessary.

  • An accurate representation of the retention curve is also necessary, and in particular its dependence on the dry density (or porosity) of the bentonite. This does reflect the double porosity character of this material, the evolution of which under load and suction changes must be incorporated in one way or another in the material behaviour law. In particular, a single water retention test under constant volume is not sufficient to characterize the retention curve, and a water retention test under free swelling should be preferred.

  • For the mechanical aspects, laboratory tests on bentonite–sand mixture clearly show an elasto-plastic response under certain stress–suction paths (Millard et al. 2016). Although many teams obtained satisfactory results in the simulation of the PT-A1 in situ test while using an elastic (nonlinear) behaviour, it seems that a realistic predictive material modelling should incorporate such features.

  • Because of the large swelling capacity of the bentonite, an additional swelling strain has to be incorporated in many classical models.

  • The technological void plays an important role on the short-term behaviour of the sealing device. With regards to this particular point, an accurate modelling of this phase will require additional research on the exfoliation of the core under hydration at its outer boundaries, which creates a bentonite-based gel and on its progressive hardening under compaction.