Modelling of oedometer tests on pellet-powder bentonite mixtures to support mock-up test analysis

Bentonite mixtures of MX-80 (80% of high-density pellets and 20% of bentonite powder on a mass basis) have been recently proposed as a candidate material for sealing deep geological disposals of high-level radioactive waste. A loading/unloading oedometer test at constant water content has been performed on this mixture, which has been modelled using the finite element Code_Bright. The constitutive model used to represent the mechanical response is the Barcelona Expansive Model (BExM), since a multimodal pore size distribution characterises the pore network of the mixture. During compression at constant water content, an increase in the degree of saturation and a consequent reduction of suction is induced. Consequently, two competing effects occur at different pore-size scales: (a) compression due to mean net stress increase; and (b) expansion on induced suction reduction that mainly affects the micro-porosity level inside aggregates. A sensitivity analysis has been performed to explore these effects, in which the elastic compressibility parameter at the micro-porosity scale for changes in mean effective stress plays an important role.


Introduction
Heterogeneous mixtures of high-density bentonite have been suggested as a suitable sealing material for nuclear disposals of high-level radioactive waste. These mixtures display a high swelling capacity, low permeability and high radionuclide migration retardation properties [1,2]. Furthermore, bentonite-based mixtures can efficiently decrease the gaps between the rock and the seal due to operational advantages in terms of emplacement [3][4][5].
The French Institute for Radiological Protection and Nuclear Safety (IRSN) has launched VSEAL project, which relies on a series of in situ experiments performed in IRSN's Underground Research Laboratory (Tournemire, France). These experiments aim at studying the long-term hydration process of an engineered barrier composed of a binary mixture of MX-80 bentonite, as well as the impact of gas migration at different hydraulic states. The binary mixture is constituted by high-density bentonite pellets (80% mass ratio) and bentonite powder. Based on the design of the in situ experiments, a laboratory mock-up test (1/10th scale) has also been launched. The small-scale mock-up test (100 mm in diameter and 350 mm high) presents independent top (fast injection) and radial (slow injection) water systems to mimic real hydraulic conditions. Within this context, several loading/unloading oedometer tests at different hydraulic states have been carried out to support the mock-up test and understand the mechanical response of the mixture. This paper focuses on numerical modelling using the finite element program Code_Bright [6] of a specific loading/unloading oedometer test performed under constant water content conditions. In particular, the Barcelona Expansive Model (BExM) [7,8] has been used to describe the mechanical behaviour of the double porosity mixture (micro and macro-porosity). BExM allows the distinction of the deformational response of these two porosity levels, which are affected by mean net stress and suction changes (macro-porosity) and mean effective stress changes (micro-porosity). Therefore, on loading at constant water content, two competing effects at different scales may occur (compression response on stress increase, and expansion on suction decrease associated with the degree of saturation increase). Within this context, the elastic stiffness parameter at the microporosity level for changes in mean effective stress displays an essential role by inducing the expansion of the micro-pore volume on compression and reducing the macro-pore volume. A sensitivity analysis is performed in the paper to investigate the effect of this microporosity elastic stiffness parameter on the global mechanical response of the mixture.

Material description and oedometer test
The investigated material is MX-80 bentonite mixture with a mass base proportion of 80% high-density pellets and 20% powder. MX-80 bentonite presents a high E3S Web of Conferences 195, 04004 (2020) E-UNSAT 2020 https://doi.org/10.1051/e3sconf/202019504004 montmorillonite content (80%), liquid limit 420 -560%, plastic limit 62%, and density of particles 2.77 Mg/m 3 [9]. High-density bentonite pellets (approximately 7 mm in diameter) have been produced by compaction to a dry density ȡ d = 1.99 Mg/m 3 . Bentonite powder has been fabricated by crushing pellets. The dry density of the powder is around 1.10 Mg/m 3 . The mixture has been prepared at a dry density ȡ d =1.49 Mg/m 3 and water content w = 8.5% to 9.8% corresponding to a void ratio e = 0.859 (porosity φ = 0.462) and degree of saturation Sr = 0.27 to 0.32. Figure 1 presents the mixture and the two components (pellets and powder). The mixture is characterised by a multi-porosity network that evolves during the mechanical test (loading/unloading). Figure 2 shows the pore size distributions (PSDs) obtained by mercury intrusion porosimetry of the pellet and the powder at their initial state. Bimodal distributions are observed for both components. The pellet displays dominant modes of entrance pore size at approximately 20 nm (micro-pores inside aggregates) and between 10 μm and 20 μm (macro-pores between aggregates and fissures) [3,10,11]. The powder shows two dominant pore size families at 20 nm (inside aggregates) and between 170 μm and 190 μm (between aggregates). Consequently, the mixture displays a multi-modal PSD with micro-porosity (intraaggregate pores in powder and pellets), macro-porosity (inter-aggregate pores in powder and pellets), and larger macro-pores between the shielding skeleton of the pellets and the powder (inter-grain porosity considered hereafter as part of the macro-porosity). These PSDs are used to estimate the initial micro-porosity = 0.193 and the macro-porosity = 0.269.
The loading/unloading paths have been performed with the oedometer cell and lever mechanism presented in Figure 3. The diameter of the ring is 50 mm with a height of 20 mm. The cell avoids water loss while keeping air pressure under atmospheric conditions. The test has been carried out with a mixture at a constant water content of around 9.8% and with an initial total suction around 85 MPa measured with a dew-point mirror psychrometer. Table 1 shows the stress paths followed on step loading/unloading, as well as the elapsed time on each step. The elapsed time is required for the modelling stage.

Macro-porosity
Microporosity   Figure 4 plots the vertical stress steps applied against the elapsed time. The evolution of (total) porosity at the end of each step is also included. These time-evolution plots will be used when plotting the numerical results. More conventional compressibility curves on loading/unloading and in terms of void ratio are shown in Figure 5. A pre-consolidation (vertical) stress under partial saturation slightly higher than 1 MPa can be obtained from the plotted results

Some insight on Barcelona Expansive Model (BExM)
The Barcelona Expansive Model (BExM) [7] is a mechanical constitutive model particularly suitable for geomaterials with bimodal pore-size distribution (i.e., macro and micro-porosity).
The model assumes that for the macrostructural level, both elastic and plastic strains can develop as a result of mean net stress and suction changes. The behaviour of the macrostructural level is defined by the Barcelona Basic Model (BBM) [12]. A fundamental assumption of the framework is that the microstructural behaviour is independent of the macrostructural state and only responds to changes in suction and stresses at the local microstructural level.
The increment of the total elastic volumetric strain of the medium is computed as the sum of the increments of macro and micro elastic deformations [7]: where ݀ߝ ௩ is the increment of the volumetric elastic strain; ݀ߝ ௩ ெ the increment of the volumetric elastic strain related to changes in the macro-porosity; and ݀ߝ ௩ ெ the increment of the volumetric elastic strain related to changes in the micro-porosity.
The increment of the macrostructural volumetric elastic strain ݀ߝ ௩ ெ is expressed as a function of the increments of mean net stress and suction as where ‫‬ refers to the mean net stress, ‫ݏ‬ is the (total) suction, ‫‬ ௧ the atmospheric pressure, ݁ ெ the macrostructural void ratio, ߢ ெ the elastic compressibility at the macro-porosity level for changes in mean stress, and ߢ ௦ ெ the elastic macro compressibility for changes in suction. The initial ݁ ெ can be estimated based on the (total) initial void ratio e and the initial microstructural void ratio ݁ ெ which is determined based on the initial microporosity ߶ ெ [13].
The increment of the microstructural strain ݀ߝ ௩ ெ is assumed to be volumetric, nonlinear elastic and proportional to the increment of the mean effective stress as where ሺ‫‬ ‫ݏ‬ሻ is the mean effective stress, ݁ ெ the microstructural void ratio, and ߢ ெ the elastic compressibility at the micro level for changes in mean effective stress.
The parameters of the mechanical model and their values are summarised in Table 2. As observed, BExM involves a number of parameters associated with both elastic and plastic behaviour. Some parameters have been obtained from experimental results on the mixture, such as the saturated elastoplastic compressibility parameter ߣሺͲሻ (in terms of void ratio), and the saturated pre-consolidation (yield) stress ‫‬ ‫כ‬ [13]. Parameters not directly obtained from experimental results were taken from [14].
However, the emphasis has been placed on the previously described elastic volumetric behaviour. As will be later discussed, the modelled response is highly sensitive to the elastic compressibility parameter at the micro-porosity scale ߢ ெ .  Regarding the hydraulic behaviour, the van Genuchten equation [15] has been used to describe the water retention curve of the mixture where S e is the effective degree of saturation, S r the degree of saturation, S rl the residual degree of saturation, S ls the maximum degree of saturation under saturated conditions, and P and ɉ are material parameters. The water retention curve will be used to compute the changes in (total) suction during loading/unloading, based on S r changes at constant water content induced on compression (loading) and expansion (unloading). The hydraulic behaviour in BExM is complemented with an equation for the intrinsic permeability evolution, which is an exponential function of the macro-porosity defined as where ߶ ெ is the macro-porosity; b a parameter and k 0 the reference intrinsic permeability at a reference macroporosity ߶ ெ The saturated permeability was measured in the laboratory, and a value of k 0 = 1.17x10 -20 m 2 for ߶ ெ ൌ0.269 was considered by [6].
The dependency of liquid relative permeability on the degree of saturation is expressed as where n is a material parameter, and S e the effective degree of saturation previously defined in equation (4).
The hydraulic parameters used in the simulations are summarised in Table 3.  Figure 6a shows the model geometry with a plane of symmetry. Figure 6b displays the boundary conditions, in which horizontal displacements are restricted. The model is applied under plane strain conditions with normal (out-of-plane direction) displacements not allowed.  Table 2). ] Fig. 6. (a) Model geometry under plane strain conditions. (b) Mechanical and hydraulic boundary conditions.   Figure 9 plots the time evolution of the computed changes in (total) suction associated with the changes in the degrees of saturation. On loading to vertical stress of 5 MPa, the suction reduces to around 22 -28 MPa (s = 85MPa is the value at the initial state). On unloading to vertical stress 0.1 MPa, the suction reaches values around 30 -46 MPa.    (Figures 8 and 9). For 0.08, the total porosity decreases to 0.33 at 5 MPa starting E3S Web of Conferences 195, 04004 (2020) E-UNSAT 2020 https://doi.org/10.1051/e3sconf/202019504004 from φ = 0.462 at the initial state. Interestingly, this decrease in total porosity on loading is accompanied by an increase in micro-porosity. In fact, two competing effects are occurring at the microstructural level. The increase in vertical stress tends to induce compression in the micropore volume, while (total) suction decrease leads to aggregate swelling and the corresponding increase in micro-porosity. It seems that κ micro plays a fundamental role since lower values reduce aggregate swelling and allow more compression of the macropores and the associated decrease in total porosity.

Concluding remarks
This paper presents experimental and modelling results of an oedometer test performed under constant water content conditions on a binary mixture of MX-80 (bentonite pellets and powder). The study addresses the effect of the micro-porosity compressibility κ micro on the global volume change behaviour of the mixture. Modelling results show that total porosity decreases inducing the increase of the degree of saturation and the decrease of suction. The sensitivity analysis demonstrates that changing values of κ micro affects the mechanical behaviour of the binary mixture significantly. For lower values of κ micro , the material shows larger compressibility resulting in a higher decrease in total porosity. These results show how crucial it is to choose a reliable value when modelling experimental mock-up tests.