Modelling the hydromechanical behaviour of expansive granular mixtures upon hydration

Bentonite pellet-powder mixtures are candidate sealing materials in radioactive waste disposal concepts. The mixture is installed in galleries in dry state as a granular material. The material is progressively hydrated by the pore water of the host rock and becomes homogeneous. Before homogenisation, the granular structure controls the material behaviour. In the present work, a modelling approach able to address particular features of pellet-powder mixtures is introduced. Two domains are considered: i) granular, and ii) homogeneous. The material behaviour before homogenisation is studied through Discrete Element Method (DEM) simulations. Constitutive laws for the granular state are proposed from DEM results. The behaviour of the homogenised material is described by a modified Barcelona Basic Model (BBM). Transition from granular to homogeneous states depends on suction and relative volume fractions of pellets and powder. Swelling pressure tests performed in the laboratory are satisfactorily simulated using this approach.


Introduction
Bentonite pellet-powder mixtures are considered as candidate sealing materials in radioactive waste disposal concepts [1][2][3][4][5]. These materials are characterised by a low permeability, good radionuclide retention capacity, and ability to swell upon hydration.
Pellet-powder mixtures are installed in the galleries in dry state as a granular assembly. The granular material undergoes hydration by the pore water of the host rock and progressively becomes homogeneous. Before homogenisation, the mechanical behaviour of the material is controlled by its granular nature.
The present work aims at proposing a modelling approach accounting for particular features of pelletpowder mixtures. Two domains are considered: i) granular, controlled by interactions between pellets with powder in free-swelling conditions in inter-pellet voids, and ii) homogeneous. The behaviour in granular state is studied using Discrete Element Method (DEM). A modified Barcelona Basic Model (BBM) [6] is used to describe the behaviour of the homogenised mixture. Transition from granular to homogeneous states is described by criteria depending on suction and volume fractions of pellets and powder.
First, swelling pressure tests performed in the laboratory on three pellet-powder mixtures are presented. Then, DEM simulations of periodic pellet assemblies are presented. Model equations proposed for granular and homogeneous states are then introduced. Finally, numerical simulations of swelling pressure tests are performed using the proposed model.

Swelling pressure tests 2.1 Material
Three MX80 pellet-powder mixtures are used to perform swelling pressure tests. Pellets are subspherical. Their diameter is 7 mm and initial suction (s 0 ) and dry density are 89 MPa and 1.91 Mg/m 3 respectively [7]. Powder is made of crushed pellets. The reference mixture has pellet-powder proportions of 70/30. All mixtures have the same mass of pellets but various powder contents. Mixtures are referred to as 70/30, 70/15 and 70/0. Volume fraction of pellets (Φ 1 ) and powder (Φ 2 ) are defined as the ratio of total volume of pellets/powder to total volume of the sample. Material properties are presented in Table 1.

Experimental results
Results are presented in Figure 1. It is considered that, owing to the powder density in 70/30 test, the influence of granular structure is not significant. Conversely, powder is considered to be close to free-swelling conditions in 70/15 test. The granular structure controls the macroscopic response at suction higher than 4 MPa. At lower suction homogenises and the swelling behaviour is controlled by granular structure.

Overview
DEM simulations of pellet assemblies have been performed in Ref. [10]. It was suggested that the intrinsic

DEM simulations of pellet assemblies have been
It was suggested that the intrinsic behaviour of pellet assemblies large granular assemblies than in simulations of swelling pressure tests. In this respect, it is proposed to constitutive laws for pellet assemblies from DEM simulations of large assemblies. Isotropic compressions of pellet assemblies are performed using DEM. Pellets are modelled as spheres of constant diameter a = 7.53 mm to keep identical mass and density as real pellets. The same numerical method as in Ref [11] is used. The same numerical samples as in Ref. [10] are used: periodic cubic samples composed of 4000 beads, with varying initial value of Values of Φ 1 (0) considered in this study are: 0.638; 0.628; 0.595; 0.577. The initial number of coordination in these samples are respectively: 6.0 The total volumetric strain of the pellet assemblies (ε V ) is characterised as a function of a dimensionless pressure parameter (m g ). Contacts are elastic perfectly plastic, using Hertz's law in elasticity and pellet strength as the elastic limit of contacts.

Dimensionless pressure parameter
In real pellet assemblies, upon mechanical loading depends on suction because this latter affects pellet stiffness [ phenomenon, a dimensionless pressure parameter, used in DEM to account for both mean stress and pellet stiffness variations. In this respect, characterising the volumetric behaviour under determine the material response under both mean stress and pellet stiffness variations.
The mean stress in a granular assembly, written as follows [11,12]: Where V is the volume of the granular assembly, is the number of contacts in the assembly, mean value for all contacts, r is the distance between two pellets in contact and F is the contact force, given by Hertz's law: where E and ν are the pellet Young modulus ratio, and δ is the normal deflection at contact between two pellets.
From equations (1) and (2) it appears that volume changes in the granular assembly can be related to the following ratio: 1 which introduces the aforementioned of pellet assemblies is better addressed in assemblies than in simulations of swelling In this respect, it is proposed to determine constitutive laws for pellet assemblies from DEM simulations of large assemblies.
of pellet assemblies are Pellets are modelled as spheres mm to keep identical mass and density as real pellets. The same numerical method ] is used. The same numerical samples as in ] are used: periodic cubic samples composed of 4000 beads, with varying initial value of Φ 1 (Φ 1 (0) ). considered in this study are: 0.638; 0.628; 0.595; 0.577. The initial number of coordination in these samples are respectively: 6.0; 5.8; 4.4; 4.4 [11].
volumetric strain of the pellet assemblies ) is characterised as a function of a dimensionless Contacts are elastic perfectly plastic, using Hertz's law in elasticity and pellet strength

Dimensionless pressure parameter
In real pellet assemblies, the macroscopic response upon mechanical loading depends on suction because this latter affects pellet stiffness [7]. To consider this phenomenon, a dimensionless pressure parameter, m g , is n DEM to account for both mean stress and pellet stiffness variations. In this respect, characterising the volumetric behaviour under m g variations allows to determine the material response under both mean stress and pellet stiffness variations.
The mean stress in a granular assembly, p, can be (1) is the volume of the granular assembly, N c is the number of contacts in the assembly, < > is the is the distance between two is the contact force, given by (2) are the pellet Young modulus and Poisson deflection at contact between equations (1) and (2) it appears that volume changes in the granular assembly can be related to the (3) hich introduces the aforementioned m g .

Note concerning pellet swelling
In real pellet assemblies, particle swelling occurs along with stiffness variation [7]. Since this latter is addressed by m g , only the influence of variation of be characterised in the model. It is however not addressed using DEM. Indeed, considering particle swelling (variation of a) at constant constant N c in these conditions, it is easily shown equations (1), (2), and (3), that: where ε V1 is the pellet volumetric strain.

Hydromechanical behaviour of a pellet
In DEM, each particle has the same behaviour as a pellet. The model proposed in Ref. [7] is used to describe the pellet behaviour [10]. The main model equations are summarised as follows: where R is the elastic limit of the pellet, effective mean stress applied to the pellet, are model parameters [7]. Figure 2 presents the evolution of ε V as a function of in DEM simulations for samples of various bilinear relationship is obtained. The slope change occurs at a threshold value of m g , denoted by appears to depend on Φ 1 (0) (Figure 3). Equality corresponds to the occurrence of contact plasticity in the granular assembly. Figure 2. Evolution of ε V as a function of m g swelling swelling occurs along . Since this latter is addressed , only the influence of variation of a on ε V needs to be characterised in the model. It is however not considering particle ) at constant m g , assuming , it is easily shown from (4) is the pellet volumetric strain.

Hydromechanical behaviour of a pellet
has the same behaviour as a ] is used to describe The main model equations are (7) is the elastic limit of the pellet, p 1 ' is the effective mean stress applied to the pellet, α m , β m and C as a function of m g for samples of various Φ 1 (0) . A bilinear relationship is obtained. The slope change , denoted by m g * , which Equality m g = m g * corresponds to the occurrence of contact plasticity in the

General remark
The proposed model is based on hardening plasticity. It accounts for two Plasticity and hardening are common to both granular and homogeneous domains. Elastic laws are different depending on the state of the material. described in the following subse

Granular-homogeneous transition
In the present model, the main hypothesis is the consideration of two distinct domains. The material is either considered granular or homogeneous. In the granular domain, interactions between pellets control the mechanical behaviour of the assembly and DEM results are used to determine constitutive laws. In the homogeneous domain, the mixture is considered to have homogenised and its behaviour is described using a modified version of the BBM [ granular materials, it is considered that coarse particles control the mechanical behaviour if the proportion of small particles is smaller than a threshold value. In this case, it is not relevant for characterising the transition since volumetric proportion of powder remains nearly constant upon hydration.
Experimental results in literature [ bentonite materials undergo a significant fabric rearrangement at low suction. considered that below a threshold suction the material behaviour is no longer controlled by the granular structure.
Results of swelling pressure tests.
DEM results are interpreted as follows: * 2 * (8) are model parameters (see Table   powder mixtures The proposed model is based on hardening elastoplasticity. It accounts for two-stress variables, p and s. Plasticity and hardening are common to both granular and homogeneous domains. Elastic laws are different depending on the state of the material. The model is described in the following subsections.

homogeneous transition
In the present model, the main hypothesis is the consideration of two distinct domains. The material is either considered granular or homogeneous. In the granular domain, interactions between pellets control the mechanical behaviour of the assembly and DEM results are used to determine constitutive laws. In the homogeneous domain, the mixture is considered to have homogenised and its behaviour is described using a modified version of the BBM [6]. In non-reactive ranular materials, it is considered that coarse particles behaviour if the proportion of small particles is smaller than a threshold value. In this case, it is not relevant for characterising the transition n of powder remains nearly Experimental results in literature [13] suggest that bentonite materials undergo a significant fabric rearrangement at low suction. As a consequence, it is considered that below a threshold suction, denoted by s * , the material behaviour is no longer controlled by the The main hypothesis associated to the granular domain is that powder is in free-swelling conditions ( Figure 1). It is considered that this condition is no longer verified if the volume fraction of powder in interpellet voids (Φ mat ) reached a threshold value, denoted by Φ mat * . Transition from granular to homogeneous states is considered irreversible. Because of lack of data, parameters s * and Φ mat * are estimated. The following values are proposed: s * = 3 MPa [7,13] and Φ mat * = Φ 1 (i.e. powder in the inter-pellet voids reaches the same volume fraction as pellets in the total volume). Note that it is a first estimation and that pellets are likely not to totally control the macroscopic response until one of these criteria is verified. Conversely, influence of the granular structure is likely not to be immediately totally lost as soon as a criterion is verified.

Granular material
Elastic volumetric strain ε V e is considered a function of m g and ε V1 . Since both depend on p and s, dε V e is written as: where, from equation (8), According to hypotheses discussed in Ref. [7], pellets, and in this case powder grains, are considered elastic, composed of micropores [14] only, and fullysaturated. In the granular domain, ε V1 is a function of p and s, and powder volumetric strain, ε V2 , is only a function of s: where, in the granular domain, And Values of model parameters are summarised in Table 2.

Homogenised material
The initial BBM [6] is adapted to account for more than one porosity. Three void ratios are considered: i) macrostructural void ratio (e M ), associated to the volume of voids between pellets and powder grains (Ω vM ), ii) pellet and iii) powder grain microstructural void ratios (e m1 and e m2 ), associated to intra-pellet and intra-powder grain volume of voids (Ω v1 and Ω v2 ). Respectively denoting by Ω s1 and Ω s2 the volume of solid in pellets and powder, void ratios are defined as: Total void ratio, e, is written as: Dimensionless stiffness parameters for the macrostructure, κ and κ s , and for the microstructure, κ m , describe the elastic (superscript e) increment of void ratios of the different levels of structure as functions of the stress variables of the model: where p atm is the atmospheric pressure and p 2 ' is the powder effective mean stress. In the present work, hydraulic equilibrium is assumed between each level of structure for simplicity. In the homogeneous domain, powder is assumed to be as affected as pellets by mechanical loadings. Effective stresses applied to both pellets and powder are thus identical. Thus, p 1 ' = p 2 ' and de m1 = de m2 .
Consequently, from equation (17), denoting by e 0 the initial void ratio:

Plasticity and hardening
Plasticity and hardening are addressed as in the original BBM [6]. The most important equations are summarised hereafter. For further details, interested readers may directly refer to Ref. [6]. The yield surface of the model is describe with q the deviatoric stress, M the slope of the state line, p 0 the preconsolidation stress, function of With p c a reference stress and λ dimensionless stiffness parameter, function of where λ(0), r, and β M are model parameters.
Hardening law is written: where dε V p is the increment of total plastic strain, calculated from the consistency condition accounting for an associated flow rule. Note that, since microstructure is considered elastic, dε V p 5 Simulations of laboratory tests

Method
The model is used to simulate swelling pressure tests. s is decreased from s = s 0 to s = 0.1 MPa. taken equal to 0 for simplicity. Gravity is not considered. No volumetric strain is allowed.

Model parameters
Parameters related to the granular domain are obtained from Ref. [7] or from DEM simulation results. Parameters related to the transition from granular to homogeneous states are estimated as described in Parameters related to the homogenised domain are conveniently estimated.
In bentonite materials, the swelling potential and apparent preconsolidation stress increase with increasing dry density. To account for these features, it is proposed to estimate p 0 (0) from results of oedometer tests performed on MX80 materials, available ( Figure 4). The following relationship is proposed:

Simulations of laboratory tests
The model is used to simulate swelling pressure tests.
MPa. Gas pressure is taken equal to 0 for simplicity. Gravity is not considered. anular domain are obtained DEM simulation results. Parameters related to the transition from granular to are estimated as described in 4.2. Parameters related to the homogenised domain are In bentonite materials, the swelling potential and apparent preconsolidation stress increase with increasing dry density. To account for these features, it is proposed from results of oedometer tests performed on MX80 materials, available in Ref. [15][16][17] . The following relationship is proposed: where p 0 * and n p are parameters deduced from experimental results. To allow the swelling potential to increase with increasing dry density while avoiding to fit an additional parameter, it is proposed to allow according to the following relationship: where e m0 is the initial value of dense materials, as e 0 approaches κ m and a higher swelling potential would be computed.

Results and discussion
Using a single set of parameters (Table 2) and initial material properties (Table 1), the three swelling pressure tests are simulated. Results are in Figure 5.
Although some differences between experimental and model results are obtained, the main trends of the swelling pressure evolution are satisfactorily reproduced. In particular, the homogenisation of the 70/30 material a high suction, the response of the 70/15 material controlled by the pellet assembly until low suction, and final swelling pressures for all materials are close to experiments.
are parameters deduced from Saturated preconsolidation stress as a function of aterials. Results from Marcial [15], Guerra et al. [17]. Dashed line = 80 MPa and n p = 3.7 To allow the swelling potential to increase with increasing dry density while avoiding to fit an additional parameter, it is proposed to allow κ s to vary from 0 to κ m according to the following relationship: is the initial value of e m . In this respect, in approaches e m0 , κ s would approach and a higher swelling potential would be computed. This approach is not directly supported by experimental t is suggested that other methods allowing κ s could be suitable. κ m and κ are thus the only elastic stiffness parameters to estimate mixtures have the same values of = 0.45. e 0 is respectively equal to 1.64, 1.17, 0.847 for the 70/0, 70/15, and 70/30

Results and discussion
Using a single set of parameters (Table 2) and initial material properties (Table 1), the three swelling pressure tests are simulated. Results are compared to experiments Although some differences between experimental and model results are obtained, the main trends of the swelling pressure evolution are satisfactorily reproduced. In particular, the homogenisation of the 70/30 material at high suction, the response of the 70/15 material controlled by the pellet assembly until low suction, and final swelling pressures for all materials are close to  In spite of these simplifications, the conceptual approach proposed in the present work appears to provide interesting results regarding the modelling of pellet-powder mixtures, where powder can be in free swelling conditions in granular state and homogenise upon hydration.

Plasticity and hardening
Some hypotheses have been considered in the Hydraulic equilibrium between pellets and powder can induce a lower suction in pellets in the model compared to experiment, and thus a computed homogenisation at higher imposed s. Parameters in the granular domain are constant in the from Figure 2, a dependency of f εa and Also, it is considered 1, which is exact only if all contacts are The computed response can thus be characterised in the granular domain. imensionless stiffness parameters in the homogenised considered constant, and their values are estimated without direct experimental . It is however worth mentioning that κ, λ(0), r are close to values proposed in Ref. [5] to model Finally, a more accurate description of the transition from granular to homogeneous states is an interesting perspective of the , the conceptual approach proposed in the present work appears to the modelling of powder mixtures, where powder can be in freeswelling conditions in granular state and the material can

Conclusion
A conceptual approach was proposed to model the hydromechanical behaviour of bentonite pellet mixtures. Constitutive laws and parameters related to the granular state are obtained through DEM simulations. The behaviour of the homogeneous material is by a modified version of the BBM.
The proposed approach is able to address features of these materials such as powder in free swelling conditions, mechanical behaviour controlled by pellet assemblies in loose-powder materials, transit homogeneous state, increase of swelling potential and preconsolidation stress with increasing dry density.
The material behaviour is simplified and improvement of the approach can be proposed in future works. However, in spite of these simplificati swelling pressure tests performed in the laboratory on pellet-powder mixtures with various powder contents were satisfactorily simulated using a single set of parameters, which none of the traditional modelling framework based only on a continuu achieve.