Introduction

Bentonite is recognized generally as an ideal buffer/backfill material for artificial barriers in deep geological repositories for nuclear waste due to its low hydraulic conductivity, high swelling characteristics, and good self-sealing capacity (Karnland et al. Reference Karnland, Olsson, Nilsson and Sellin2007; Dohrmann et al. Reference Dohrmann, Kaufhold, Lundqvist, Bergaya and Lagaly2013; Sellin and Leupin Reference Sellin and Leupin2013; Sun et al. Reference Sun, Zhang, Li and Zhang2015; Chen et al. Reference Chen, Zhu, Ye, Cui and Chen2016; Kaufhold and Dohrmann Reference Kaufhold and Dohrmann2016). Bentonite can fill cracks between bentonite and the surrounding crystalline rock to form an enclosed environment, effectively restricting nuclide migration. During the operation of a repository, however, groundwater solution may infiltrate into and thereby affect the swelling properties of the buffer/backfill material. The effect of saline groundwater on the swelling characteristics of bentonite is expressed mainly in two aspects. First of all, the salinity of groundwater has a strong inhibitory effect on swelling potential (Villar Reference Villar2006; Siddiqua et al. Reference Siddiqua, Blatz and Siemens2011). This relationship can be explained by DDL theory as the spacing between clay particles decreases with an increase in ion concentration in the pore water. Secondly, the variation of cations adsorbed by bentonite also affects swelling characteristics (Segad et al. Reference Segad, Bo and Cabane2012a, Reference Segad, Hanski, Olsson, Ruokolainen, Åkesson and Bob; Liu Reference Liu2013).

Montmorillonite is regarded as the only component in bentonite responsible for water uptake and swelling processes. A plate-like montmorillonite layer is ~0.96 nm thick. Montmorillonite plates often combine in a face-to-face orientation (Saiyouri et al. Reference Saiyouri, Tessier and Hicher2004). The face of the montmorillonite layer has a permanent negative charge resulting from isomorphous substitutions of divalent cations (e.g. Mg²⁺ and Fe²⁺) for trivalent Al³⁺ in the octahedral sheet and trivalent Al³⁺ for tetravalent Si⁴⁺ in the tetrahedral sheet. This charge is compensated by the preferential adsorption of cations on the layer surface, such as Na⁺, K⁺, Ca²⁺, or Mg²⁺, which commonly, are hydrated and exchangeable. If the cations in salt solutions are significantly different from the exchange population present on the exchange sites, a cation exchange reaction will occur.

The swelling phase of compacted bentonite specimens may be divided into two dominant processes: crystalline swelling and osmotic swelling (Liu Reference Liu2013). In crystalline swelling, water molecules are inserted between the unit layers caused by hydration of the interlayer cations (Holmboe et al. Reference Holmboe, Wold and Jonsson2010; Siddiqua et al. Reference Siddiqua, Blatz and Siemens2011; Ferrage Reference Ferrage2016; Birgersson Reference Birgersson2017). Four discrete layers of water molecules exist in a Na-montmorillonite, while less than 3 layers are present in other cationic montmorillonites (Yong Reference Yong1999; Liu Reference Liu2013). Upon further water infiltration, the distance between montmorillonite layers increases gradually, the hydrated cations deviate from the outer montmorillonite surfaces into the interparticle pore space, and the DDLs begin to form on the clay surface. This phase is called the osmotic swelling phase. Ca-montmorillonite particles may divide into stacks of 2–20 unit layers depending on the pore water salinity and other experimental conditions. Na-montmorillonite can divide into smaller stacks of 1–5 unit layers (Cadene et al. Reference Cadene, Durand-Vidal, Turq and Brendle2005; Segad et al. Reference Segad, Bo and Cabane2012a, Reference Segad, Hanski, Olsson, Ruokolainen, Åkesson and Bob), suggesting a higher swelling potential for Na-montmorillonite than its calcium counterpart. Thus, the ion-exchange reaction between cations existing in groundwater and that adsorbed on exchange positions on the layer surface may also change the swelling characteristics greatly (Schramm and Kwak Reference Schramm and Kwak1982a, Reference Schramm and Kwakb; Segad et al., Reference Segad, Bo and Cabane2012a, Reference Segad, Hanski, Olsson, Ruokolainen, Åkesson and Bob).

Due to the effects of salt solution on the swelling of bentonite being very complex, few methods can predict accurately the swelling characteristics of bentonite in a salt solution. The DDL model was found to be suitable in predicting osmotic swelling of compacted bentonites in a dilute salt solution (Tripathy et al. Reference Tripathy, Sridharan and Schanz2004; Xiang et al. Reference Xiang, Xu and Jiang2014). However, this model has limitations in estimating crystalline swelling and is not suitable for specimens in concentrated solution (Yong Reference Yong1999; Liu Reference Liu2013). Rao and Thyagaraj (Reference Rao and Thyagaraj2007) pointed out that an increase in salt concentration of pore fluid leads to an increase in osmotic suction, which acts as an additive effective stress component leading to a decline in swelling potential. Based on fractal analyses of bentonite micro-structures, the effective stress incorporated with osmotic suction was expressed by Xu et al. (Reference Xu, Xiang, Jiang, Chen and Chu2014), and the correlation of montmorillonite void ratio (e _m) with effective stress (p _e) conforms to a fractal model with $e_m=K{p_e}^{D_s-3}$ , where K is the swelling coefficient of montmorillonite and D _s is the surface fractal dimension. This model can be used to describe the swelling behavior affected by the salinity of pore water. The effect of a salt solution incorporating ion-exchange reactions has not been well studied, however. Especially, a comprehensive theoretical description of the influence of swelling characteristics of compacted bentonite in salt solution is lacking.

The objectives of the present work were: (1) to investigate the influence factors of different salt solutions on swelling strain; (2) to calculate the effect of exchange reactions on swelling strain; and (3) to predict the swelling strains of compacted bentonite in a concentrated salt solution.

Predictive Model

As mentioned above, the correlation of montmorillonite void ratio to effective stress was satisfied by the swelling fractal model. The effective stress on bentonite particles is defined generally as the net interaction force between soil particles, and in the case of montmorillonite swelling, effective stress can be regarded as the interaction force between montmorillonite layers. When compacted specimens are in contact with distilled water, the effective stress is balanced by the vertical pressure, σ, applied in swelling tests, without the osmotic suction of pore water. While in a salt solution, the effective stress, p _e, of compacted bentonite exposed to osmotic suction could be represented as (Xu et al. Reference Xu, Xiang, Jiang, Chen and Chu2014; Xiang et al. Reference Xiang, Xu, Xie and Fang2016):

(1) $p_e=\sigma +\pi {\left(\frac{\sigma }{\pi }\right)}^{D_s-2}$

where σ is vertical overburden pressure applied on the specimen, π is osmotic suction of pore water and can be expressed as (Rao and Thyagaraj Reference Rao and Thyagaraj2007):

(2) $\pi =\mathrm{iM}RT$

where M is the molar concentration of the salt solution, R is the universal gas constant, T is absolute temperature in Kelvin, and i is the Van’t Hoff factor and equals to 2 and 3 for the sodium chloride and calcium chloride solutions, respectively (Rao and Shivananda Reference Rao and Shivananda2005). The CaCl₂ solution, therefore, has 1.5 times the osmotic suction of the NaCl solution at the same molar concentration.

By incorporating osmotic suction in effective stress, the fractal model provides an effective method to express the effect of pore-water salinity on swelling characteristics of compacted bentonite. The swelling coefficient of montmorillonite, K, is an inherent parameter of montmorillonite, and is independent of the salinity of pore water. As mentioned above, in addition to pore- water salinity, the main factors influencing the swelling behavior of montmorillonite are the surface charge density and the cations adsorbed on exchange positions on the layer surface. Due to the fact that the salt solution usually does not participate in isomorphous substitutions, the surface charge density is basically unchanged. However, cations on the exchange positions would be exchanged via the ion-exchange reaction. The influence of changes in exchangeable cations on swelling characteristics is reasonably described by the swelling coefficient K (Xiang et al. Reference Xiang, Xu and Jiang2014).

The DDL model was suitable in predicting osmotic swelling of bentonites in distilled water (Liu Reference Liu2013). Thus, the coefficient K can be estimated on the assumption that the fractal equation should be equivalent to the DDL model in the case of osmotic swelling in distilled water. Rearranging the fractal model gives

(3) $K=e_m{p_e}^{3-D_s}$

According to the DDL model, the effective stress, p _e, and the montmorillonite void ratio, e _m, can be calculated by the average distance, d _m, between the 2:1 montmorillonite layers (Fig. 1), i.e. the distance from the top of a montmorillonite layer to the bottom of the one directly above it. Osmotic swelling is assumed generally to follow the completion of crystalline swelling. Thus, the distance d should be larger when crystalline swelling has fully taken place. Osmotic swelling was assumed to develop to such an extent that the montmorillonite particles cannot divide into small stacks (Fig. 1). The separation distance between the montmorillonite particles was given by h, the particles consist of n _s, the number of montmorillonite layers, and the unit montmorillonite layers were separated by an interlayer distance d _m. The value of e _m for montmorillonite is then related to h as (Liu Reference Liu2013):

(4) $h=n_s{\mathrm{te}}_m-\left(n_s-1\right)d_m$

where t is the thickness of a montmorillonite layer and d _m should be taken as a weighted average of separation distance because it depends on the types of exchangeable cations, and is defined as

(5) $d_m=\frac{1}{\mathrm{CEC}}{\sum }_i\mathrm{EX}{C}_i{d}_i$

where CEC is the cation exchange capacity of bentonite and EXC _i is the exchange capacity of the ith exchangeable cation, and d _i is the fixed distance between unit montmorillonite layers when only the ith exchangeable cation is present. It is approximated by d _i = 1.2, 0.6, and 0.9 nm for Na-, K-, and Ca- (or Mg-) montmorillonites, respectively (Yong Reference Yong1999; Liu Reference Liu2013).

Fig. 1 A schematic representation of osmotic swelling with a diffuse double-layer existing between montmorillonite particles (Liu Reference Liu2013)

The montmorillonite void ratio e _m can be expressed as:

(6) $e_m=\frac{d-R_{\mathrm{ion}}}{t+R_{\mathrm{ion}}}$

where d is half of the average distance between the montmorillonite layers and R _ion is the radius of unhydrated exchangeable cations and is used as the weighted average to take into account the demixing of cations

(7) $R_{\mathrm{ion}}=\frac{1}{\mathrm{CEC}}{\sum }_i\mathrm{EX}{C}_i{R}_i$

where R _i is radius of ith exchangeable cation.

Combining Eqs. 4 and 6, if osmotic swelling takes place, indicates that h should be larger than d _m and the distance d should satisfy:

(8) $d>\frac{t+R_{\mathrm{ion}}}{t}{d}_m+R_{\mathrm{ion}}$

Once the value of d _m is obtained, K can be calculated from Eq. 3 if p _e is known. The value of p _e is given by the DDL theory (Bolt Reference Bolt1956) as:

(9) $p_e=f_r-f_a$

where f _r is the repulsive force and f _a is the Van der Waals attractive force, which is expressed by the Hamaker’s equation (Casimir and Polder Reference Casimir and Polder1948; Liu and Neretnieks Reference Liu and Neretnieks2008):

(10) $f_a=\frac{A_h}{6\pi }\left(\frac{1}{d^3}-\frac{2}{{\left(d+t\right)}^3}+\frac{1}{{\left(d+2t\right)}^3}\right)$

where A _h is the Hamaker constant (J).

Repulsive pressure f _r is given by the Gouy-Chapman theory (Bolt Reference Bolt1956; Iwata et al. Reference Iwata, Tabuchi and Warkentin1988)

(11) $f_r=2n_{0}kT\left(coshu-1\right)$

where n ₀ is the concentration of ions in pore water, k, is the Boltzmann’s constant, T is the absolute temperature, u is the electrostatic potential at the mid-plane between parallel montmorillonite layers, and is described as (Iwata et al. Reference Iwata, Tabuchi and Warkentin1988; Liu Reference Liu2013):

(12) $u={sinh}^{-1}\left(2sinh{u}_m+\left(4/\mathrm{\kappa d}\right)sinh\left(u_h/2\right)\right)$

where the two scaled potentials on the right-hand side are for the isolated particles and can be written explicitly as (Iwata et al. Reference Iwata, Tabuchi and Warkentin1988; Liu and Neretnieks Reference Liu and Neretnieks2008):

(13) $u_m=4tan{h}^{-1}\left(exp\left(-\mathrm{\kappa d}/2\right)tanh\left(z/4\right)\right)$

and

(14) $u_h=4tan{h}^{-1}\left(exp\left(-\mathrm{\kappa d}\right)tanh\left(z/4\right)\right)$

In these expressions, κ is the reciprocal Debye length (Iwata et al. Reference Iwata, Tabuchi and Warkentin1988):

(15) $\kappa =\sqrt{\frac{2n_0{\nu }^{2}e{\text{'}}^2}{\epsilon kT}}$

z is the scaled surface-potential of the isolated particle, and is described as (Iwata et al. Reference Iwata, Tabuchi and Warkentin1988; Liu Reference Liu2013):

(16) $z=2{sinh}^{-1}\left(96.5\times \frac{\mathrm{CEC}}{S}\sqrt{\frac{1}{8\epsilon n_{0}kT}}\right)$

where e′ is the charge on the electron, ε is the static permittivity of pore water, S is the total specific surface area of bentonite, and ν is the average valence of exchangeable cations. In the case involving a mixture of several types of exchangeable cation, a simple method to apply the DDL model in the evaluation of p _e is to define υ as the weighted average valency of exchangeable cations,

(17) $\nu =\frac{1}{\mathrm{CEC}}{\sum }_i\mathrm{EX}{C}_i{\nu }_i$

It is important to note that predicting swelling characteristics of compacted bentonite using this method is based on the temperature, which is assumed to be static and uniform throughout the swelling system. In a real bentonite barrier of deep geological repositories, temperatures will be higher and there will be a temperature gradient (Åkesson et al. Reference Åkesson, Jacinto, Gatabin, Sanchez and Ledesma2009; Zheng et al. Reference Zheng, Rutqvist, Birkholzer and Liu2015; Villar et al. Reference Villar, Martín, Romero, Iglesias and Gutiérrez-Rodrigo2016). In such circumstances, the proposed method may be of doubtful utility for estimating swelling characteristics.

Prediction and Comparisons

Before prediction of the swelling deformation of bentonite in various salt solutions, the swelling coefficient K should be calculated. As mentioned above, the K value can be calculated by the DDL model in the case of osmotic swelling, where the distance d should satisfy Eq. 8. The radius of typical exchangeable cations is listed in Table 3; combined with the exchangeable cation compositions of bentonite specimens swollen in various different solutions (Table 2), the value of d of each specimen should be >1.2 nm. In addition, previous research indicated that larger values of d would appear in the bentonite specimens with higher exchangeable Na⁺, where osmotic swelling has been fully developed (Saiyouri et al. Reference Saiyouri, Tessier and Hicher2004; Ferrage et al. Reference Ferrage, Lanson, Sakharov, Geoffroy, Jacquot and Drits2007; Liu Reference Liu2013). The values of d from different specimens varied, therefore, according to the exchangeable cation composition (Table 2). The physical parameters (Table 4), taken from the literature (Liu Reference Liu2013; Xiang et al. Reference Xiang, Xu and Jiang2014), were then used to calculate the swelling coefficient, K, (Table 5), which indicated clearly that K increased directly with the exchangeable Na⁺ content.

Table 3 Non-hydrated radius of typical ions (Liu Reference Liu2013)

Table 4 Physical parameters for determination of K

Table 5 Variables used in the calculation of the swelling coefficient, K, for different exchangeable cation compositions of compacted bentonite

d is half of the average distance between the montmorillonite layers, e _m is the montmorillonite void ratio, u _h and u _m are the two scaled potentials on the right-hand side are for the isolated particles, u is the electrostatic potential at the mid-plane between parallel montmorillonite layers, f _r is the repulsive force, f _a is the Van der Waals attractive force, and p _e is the effective stress

Once the swelling coefficient K is obtained, the swelling deformation of specimens in salt solution can be predicted by the fractal model. The relationship between ε_max of bentonite and e _m at full saturation can be written as (Xu, Matsuoka and Sun Reference Xu, Matsuoka and Sun2003)

(19) ${\epsilon }_{\mathrm{sm}\mathrm{ax}}=\left(\frac{C_m{\rho }_b/{\rho }_m{e}_m+1}{{\rho }_b}{\rho }_{d0}-1\right)\times 100\%$

where C _m is the montmorillonite content in bentonite, ρ_b is the specific gravity of bentonite, ρ_m is the specific gravity of montmorillonite, and ρ_do is the initial dry density.

The measured montmorillonite void ratio can be obtained from Eq. 19. The montmorillonite void ratio of specimens with 1.7 g/cm³ initial dry density was estimated by the fractal model (Table 6), as the swelling coefficient, K, was obtained, and the effective stress, p _e, was calculated incorporating the osmotic suction of the inundating solution. Good agreement was found between the predicted montmorillonite void ratio and the measured values. Experimental data of GMZ bentonite in distilled water and NaCl solution from Ye et al. (Reference Ye, Zhang, Chen, Chen, Wang and Cui2014) were also plotted for comparison (Fig. 5). Predicted values agreed well with experimental results for the relationship between the montmorillonite void ratio and the applied load.

Table 6 Comparison of predicted montmorillonite void ratio with measured values (ρ_d = 1.7 g/cm³)

The swelling strain of bentonite in salt solution can also be predicted (Fig. 6). The estimations of maximum swelling strain also fitted well with experimental data. Predicted values indicated that under the same vertical pressure, montmorillonite void ratio was independent of the initial dry density of compacted bentonite. The initial dry density had a significant effect on maximum swelling strain.

Fig. 5 Comparison of measured montmorillonite void ratio with the predicted void ratio as a function of the vertical pressure on bentonite immersed in distilled water and NaCl solution

Fig. 6 Comparison between estimation of the maximum swelling strain and experimental data from specimens in various solutions