On swelling behaviours of a bentonite under different water contents

In this study, the pressure measured on compacted bentonites when they are wetted under a deformation confined condition is defined as the apparent swelling pressure (p_s); here ‘apparent’ emphasises that p_s may be different from the pressure between layers of montmorillonite in bentonites during the same wetting process. The effect of initial gravimetric water content (w_i) on p_s is an important issue in the design and construction of the buffer and backfill materials of a geological disposal system in high-level radioactive waste disposal projects. Some studies have been conducted to observe changes of p_s for specimens with different w_i conditions; however, a consensus has not been achieved. Fig. 1 summarises some representative data of past studies on this issue, where the p_s after reaching an equilibrium state by wetting, hereafter termed the equilibrium swelling pressure (p_eq), is plotted against the specimen dry density (ρ_d) or w_i. Some data show that p_eq decreases with an increase in w_i (Suzuki & Fujita, 1999; Sugiura et al., 2009; Cui & Si, 2014), whereas some other data show an insignificant effect of w_i (e.g. Tanai et al., 2010). Some studies have also shown the effect of w_i is not apparent or is unclear, partially due to data scattering (e.g. Komine & Ogata, 1994; Villar & Lloret, 2008). It also seems that for Kunigel V1, a Japanese bentonite, the w_i effect is minor for relatively loose specimens, but becomes significant as ρ_d increases (Suzuki & Fujita, 1999; Tanai et al., 2010).

In contrast, the swelling behaviours of compacted bentonites during wetting are essentially a result of swelling of montmorillonite and suction change in the bentonite. Several models have been proposed to simulate/predict the swelling behaviours of compacted bentonites based on traditional constitutive models and the microscopic behaviours of montmorillonite. Some have focused on the final equilibrium states (e.g. Komine & Ogata, 2004), some dealt with pure montmorillonite (e.g. Likos & Wayllace, 2010) and some others have addressed behaviours in the full process of wetting (e.g. Gens & Alonso, 1992; Alonso and Gens, 1999; Kyokawa, 2021). These models could predict some behaviours of bentonites well; however, predictions of the w_i effect on p_s do not seem to have been well achieved quantitatively (e.g. Lloret et al., 2003; Kyokawa, 2021).

With the research background given above, in the present study a series of swelling pressure tests was conducted with a wide w_i and dry density (ρ_d) ranges to reveal experimentally the w_i effect on p_s behaviours. In addition, a series of X-ray diffraction (XRD) tests to examine swelling behaviours of montmorillonite with different gravimetric water content (w), and a series of compaction tests to find the stress–strain behaviours under different w_i were also conducted. Discussions were extended to include the w_i effect on p_s, pore water density in a bentonite, swelling types of montmorillonite for various w_i and the relation between compaction curves and ρ_d–p_eq curves.

The commercial bentonite Kunigel V1 (K_V1), a candidate material for use in the Japanese geological disposal project, was used for all tests conducted in this study. The physical properties of K_V1 are summarised in Table 1. Note that all properties were obtained on the same batch of K_V1 as used in this study, although some were reported by previous studies (Wang et al., 2020a, 2021; Shirakawabe et al., 2021). It is expected that Ca²⁺ can be extracted from some soluble minerals of K_V1 so that the summation of extractable cations is higher than the cation exchange capacity (CEC) (Shirakawabe et al., 2022). The water retention characteristics of K_V1 powder are shown in Fig. 2; these were obtained by curing initially oven-dried K_V1 in environments at constant relative humidity (RH) until no apparent mass change (∼1·0–1·5 months). The humidity of less than 100% was controlled by saturated chemical solutions (e.g. Wang et al., 2020b). Distilled water was used to keep the RH at 100%, in which the curing period was arbitrarily determined. The total suction of all cured powder was measured by a WP4 dew-point potentiometer. Note that the K_V1 for the data in Fig. 2 is from a different batch wherein the water retention characteristics may be slightly different.

Three test programmes – the swelling pressure test (p_s test), XRD test and compaction tests (p_c tests) – were conducted in this study. Table 2 provides general information regarding these tests and some details have been summarised in the tables in the Appendix 1.

For the p_s tests, five w_i cases with representative w_i values of 0, 7, 18, 20 and 33% were prepared. Material with w_i = 0% was obtained by oven drying at 110°C for about 24 h; w_i = 7% was obtained in the laboratory environment. The other three cases were prepared by direct water and KV_1 mixing. To prepare these materials, 50 g K_V1 was first spread on a plastic plate and a predetermined amount of distilled water was sprayed on them. Then, they were mixed uniformly and cured in sealed containers for at least 1 month. Intuitively, the distribution of water molecules would differ somewhat depending on the material preparation methods – that is, directly mixing with liquid water or curing by water vapour. The former method was selected because it is much easier to prepare a wide range of w_i by this method, and it is closer to the condition when manufacturing buffer blocks in geological disposal. In addition, Wang et al. (2020a) observed that, regardless of direct absorption of liquid water or water vapour, the relationships between w and the basal spacing of montmorillonite in K_V1 are very consistent in the w range of 10–21%. This fact suggests that preparation methods may not affect the swelling of montmorillonite or the distribution of water molecules significantly. Fig. 3(a) shows materials in three w_i cases. For the w_i = 18% and 20% cases, powder was loosely aggregated during mixing and these aggregates were crushed during specimen preparation. For the w_i = 33% case, all materials stuck together to form a block. After curing, the block was ground through a 1 mm length square mesh for specimen preparation.

A specimen for the p_s test was prepared by placing predetermined material into a ring combination as shown in Fig. 3(b), where a specimen ring with an inner diameter of 28 mm and thickness of 2 mm was at the bottom to hold the specimen. The material was statically compacted by a compaction block on a jack frame for 2 min. Then the specimen surfaces were trimmed so that they were the same thickness as the specimen ring. The specimen together with the specimen ring was confined into the swelling pressure apparatus shown in Fig. 3(c), where a pressure sensor with a capacity of 7 MPa was installed. The above preparation took ∼5 min for each specimen and the time that the specimen surfaces were directly exposed to the atmosphere was less than 2 min, which ensured negligible moisture loss. Distilled water was supplied from the top and pore air could exhaust from the air outlets at the bottom. Water was filled into the circular water groove and transported to the specimen through a membrane filter (permeability of the membrane filter was ∼10⁻⁷ m/s (Wang et al., 2017)) between the specimen and the top plate. The above testing techniques have been validated to be reliable by Wang et al. (2021, 2022).

As shown in Table 2, the initial dry density (ρ_di) of prepared specimens ranges from 1·1 Mg/m³ to 1·9 Mg/m³. As w_i increases, particularly for the case of w_i = 33%, it becomes difficult to compact the material into a relatively dense state in 2 min, thus the ρ_di range becomes narrow. In total, nine apparatuses were employed in this programme, and tests conducted simultaneously were put into the same group in Tables 3–7 (see Appendix 1). The actual w_i values of the prepared specimens shown in these tables were measured using K_V1 powder immediately after specimen preparation.

For the XRD test series, two types were conducted (Table 2): the XRD test designating tests on specimens initially prepared with different values of w_i, and the XRD–p_s test corresponding to specimens with w_i of either 0% (i.e. 110°C oven dry) or 7% (i.e. laboratory environment), but they were wetted in a deformation confined condition. For the XRD test, K_V1 was directly mixed with distilled water for different w_i conditions and filled into specimen rings with arbitrary densities. The specimens were sealed and cured for at least 1 month to achieve water uniformity before XRD scanning. Twenty specimens with w_i range from 29·4% to 103% were prepared (Table 8, Appendix 1). Specimens in XRD-p_s tests were prepared by static compaction in the same manner as that described in Fig. 3(b). Then they were installed in an XRD swelling pressure cell (Fig. 4(a)) for XRD scanning while being wetted. The cell can sufficiently restrict specimen deformation and measure p_s during wetting, while in this study, the final XRD profiles obtained after releasing p_s at the end of the wetting only were depicted. Wang et al. (2021) described details of this cell and typical XRD profiles during wetting. Fifteen specimens with a final water content (w_f) range from 25·8% to 64·8% were tested (Table 9, Appendix 1).

An X-ray diffractometer (RINT-TTR III, Rigaku Corp.) with a parallel Cu Kα beam (source power: 50 kV and 300 mA) was used (Fig. 4(b)). Scanning angle (2θ) ranged from 1° to 10° with a scanning speed of 1°/min. During XRD scanning, the specimen surface was covered by a polyester film 6 μm thick to avoid water evaporation. By observing XRD peak positions of montmorillonite in KV_1, the basal spacing of montmorillonite (d₀₀₁) can be obtained with Bragg's law (i.e. the equation in Fig. 4(c), where λ is the wavelength of Cu Kα  = 0·15418 nm, and n = 1).

In the p_c test programme, eight specimens with four w_i cases (Table 2) were prepared. The w_i = 0% material was obtained by oven drying and the w_i = 8% material was from the laboratory environment. For the w_i = 20% case, the material was prepared by curing in the environment with relative humidity (RH) of ∼98% controlled by saturated potassium sulfate (K₂SO₄) solution. The actual w_i difference between two specimens in the w_i = 20% case is due to the difference in the curing period (i.e. about 1 month difference). For the w_i = 25% case, the material was cured in a RH = ∼100% environment controlled by distilled water at room temperature of ∼23°C.

After the material had been prepared, it was filled into the apparatus shown in Fig. 5, and vertical loading was applied stepwise by adjusting the pressure in the air cylinder. The applied vertical compaction load (p_c) was monitored by a load cell and the vertical deformation was measured by a displacement transducer. During the loading, the water path was closed and the air path was open. For most specimens, p_c was initialised from 0·05 MPa and p_c was doubled in the next step until p_c reached ∼15 MPa, after which unloading to ∼0·01 MPa was applied stepwise. The stress increment of each step in the main loading (or unloading) was sometimes quadrupled (or quartered) in w_i = 24·7% cases to reduce the duration of the test. In some tests, instant unloading and reloading during the loading process was applied to confirm the elastic behaviour. The rest time at each step of the main loading and unloading was about 24–36 h until no apparent deformation was observed. The final specimen thicknesses are ∼10 mm for w_i = 0% and 8% cases, ∼9 mm for the case of w_i = 20% and ∼7 mm for the case of w_i = 25%. It seems that the change in water content for most specimens during the tests can be ignored, except for the w_i = 20% case, in which one specimen showed 4% reduced water and another one could not be measured. Nevertheless, the w_i values are used to indicate the specimen conditions in this study.

Typical p_s time histories of specimens in five w_i cases of p_s tests are shown in Fig. 6. The final dry density (ρ_df) values of specimens shown on each of the p_s curves were calculated with consideration of possible deformation during the tests (details are explained in Appendix 2). For the cases of w_i = 0% and 7%, the p_s–time histories have a similar character: p_s increases non-monotonically with a p_s drop, which has also been observed by many other researchers (Villar & Lloret, 2008; Lee et al., 2012; Chen et al., 2019; Tanaka & Watanabe, 2019). The drop seems to first increase and then reduce or vanish as ρ_df increases and the increase of w_i also seems to weaken the drop, which has often been observed for K_V1 in past studies (Komine & Ogata, 1994; Suzuki and Fujita, 1999). For the cases of w_i = 18% and 20%, the p_s drop still exists weakly for relatively loose specimens, but it almost disappears when w_i reaches 33% in Fig. 6(e). Fig. 6(f) shows two examples of p_s curves for specimens with different w_i at relatively dense and relatively loose conditions. For the relatively loose case (i.e. ρ_df = ∼1·3 Mg/m³), they all have a slightly increasing trend from ∼10 h until the end of the tests, although this trend seems not to be affected greatly by w_i. For the relatively dense case (i.e. ρ_df = ∼1·6 Mg/m³), the increasing trend is visibly weakened compared to the observed maximum p_s value. For even denser specimens (e.g. ρ_df = ∼1·7 Mg/m³), it seems that a p_s equilibrium is reached faster for initially drier specimens (i.e. cases of w_i = 0% and 7%) compared to wetter cases. The increasing trend seems also to be affected by specimen thickness based on tests by Wang et al. (2022), which found that the increasing trend lasts longer on a normalised time scale for thicker specimens regardless of the specimen density. The trend of increase may be a result of fine adjustment of the internal (i.e. inter-particle) stress, which seems to take longer in relatively loose, initially wet, or large specimens.

Aggregates and individual crystalline particles in K_V1 powder can be directly observed from the scanning electron microscopy (SEM) images in Fig. 7. The aggregates seem to be wrapped by montmorillonite particles; however, considering that the montmorillonite content (C_m) of K_V1 is only 53%, it is also expected that non-expansive particles may be cocooned in aggregates, as illustrated schematically in Fig. 8(a). The p_s drop observed for relatively low w_i cases (i.e. w_i = 0% and 7%) would be induced by particle movement. Owing to the development of p_s, highly stressed individual particles, aggregates and dispersed particles from aggregates may move to less stressed locations, resulting in tentative p_s reduction as illustrated in Fig. 8(b). In contrast, for higher w_i cases (e.g. w_i = 18% and 20%), the original aggregates may partially disperse and form larger aggregates with others (Figs 8(c) and 3(a)), through which the drop in p_s disappears for relatively dense specimens because inter-particle stress distribution may be relatively uniform. For even higher w_i cases (e.g. w_i = 33%), aggregates and individual particles may initially form even larger aggregates (Figs 8(d) and 3(a)), and thus no p_s drop can be observed.

Figure 9 shows the relation between ρ_df and p_eq for all w_i cases, where the final measured p_s at the end of the wetting was assigned to p_eq. The overall trend suggests that w_i has no apparent effect on p_eq, except for the data in the w_i = 18% case with ρ_df > 1·7 Mg/m³, as indicated by the encircling dashed line. These data are apparently smaller than those in the cases of w_i = 0% and 7%, which may be explained by measurement scattering, although another possible reason is discussed in the section headed ‘Relation between swelling and compaction of K_V1’. A semi-log scale figure is depicted in the insert, in which the data seem to be able to divided into three linear zones from ρ_df = 1·45 Mg/m³ and 1·72 Mg/m³.

It has often been observed in past studies for compacted bentonite that the calculated degree of saturation (S^*_r) after water absorption can exceed 100% (e.g. Pusch et al., 1990; Villar, 2002). This is also the case for specimens in p_s tests, as shown in Fig. 10(a), where S^*_r was calculated by equation (1).

Here, S_r is the theoretical degree of saturation; ρ_pw is the average pore water density in compacted bentonite; ρ_w is the free water density ( = 1·0 Mg/m³); and ρ_d is the specimen dry density. In Fig. 10(a), S^*_r was calculated by assuming ρ_pw is equal to ρ_w = 1 Mg/m³, and assigning the w_f and ρ_df of p_s test specimens to w and ρ_d, respectively.

It is shown that S^*_r values are mostly higher than 100% for w_i = 0%, 7% and 18% cases, close to 100% for the w_i = 20% case on average and close to 90% for the w_i = 33% case. Although there is significant data scatter, it seems that the maximum S^*_r is larger for a lower w_i case under the same ρ_df condition up to 1·7 Mg/m³, and it also increases with an increase in ρ_df for the w_i = 0% case up to 1·7 Mg/m³. Two possible reasons resulting in S^*_r > 100% are considered: measurement errors and the possibility of ρ_pw > 1 Mg/m³. Uncertainty of S^*_r induced by measurement errors can be estimated from equation (2) by taking the partial derivative of equation (1).

It has been estimated by Wang et al. (2021) that ∂G_s = ±0·03 and ∂w =  ±1·5%, and ∂ρ_d should be in a range from −0·02 to +0·01 Mg/m³ as estimated in Appendix 2. Then an upper boundary of ∂S_r + 100% as a function of ρ_df can be obtained by assigning ∂w = +1·5%, ∂ρ_d = +0·01 Mg/m³ and ∂G_s = −0·03 as indicated by the dashed line in Fig. 10(a). Note that in the calculation of ∂S_r by equation (2), ρ_pw = ρ_w = 1 Mg/m³ was assumed and w can be obtained from equation (1) for a given ρ_d by setting S_r = 100%. It can be seen that many data distribute above the upper boundary, implying that ρ_pw > 1 Mg/m³ should be a fact.

It is highly likely that the water density near negatively charged plates can be much higher than 1 Mg/m³ based on either numerical simulations or experimental results (Moore & Reynolds, 1997; Israelachvili, 2011; Zhang & Lu, 2018; Gonella et al., 2021). This is believed to be induced by interactions between water molecules, counterions and charged plates, especially in a very narrow space (Israelachvili, 2011; Gonella et al., 2021). Jacinto et al. (2012) estimated that the water density in the interlayer space of montmorillonite of a compacted bentonite (i.e. MX-80) may be 1·32 Mg/m³, 1·17 Mg/m³ and 1·16 Mg/m³ for one-, two- and three-layer hydration states, respectively. They also estimated that ρ_pw for a MX-80 specimen with ρ_d of 1·60 Mg/m³ was 1·09 Mg/m³. Intuitively, if the water density in the interlayer space of montmorillonite increases as the amount of water in the space reduces, S^*_r should increase as ρ_d increases; however, the correlation between S^*_r and ρ_d seems very poor, both in Fig. 10(a) and according to other studies (e.g. Villar, 2002).

The poor correlation between S^*_r and ρ_df may suggest that water molecules could not occupy all positions that can theoretically hold them because, for instance, the wetting duration was inadequate, pore air was trapped and the driving forces to arrange the water molecules in an ideal state is too weak due to the effects of existing substances. In terms of the duration of wetting, Wang et al. (2021) conducted similar p_s tests on K_V1 with w_i of 7–9% and a wetting duration of ∼20 h. They found that S^*_r values were mostly under 100%, much smaller than those of the w_i = 7% case in this study with a wetting duration of ∼40 h. To further confirm the effect of wetting duration, three wetting durations (40 h, 65 h and 110 h) were applied to specimens in the w_i = 0% case; these indicate, as shown in Fig. 10(b), that a wetting duration longer than 40 h does not seem to be able to further increase S^*_r significantly. However, the decrease of S^*_r with an increasing w_i in Fig. 10(a) suggests that, with a relatively high w_i, pore air may be easily trapped because of low air permeability, and/or the initially existing water molecules may be obstacles for an ideal water molecule arrangement.

Pore water types and their corresponding densities of bentonites have been estimated based on data such as total and external specific surface area, occupation areas and percentages of hydrated water molecules, montmorillonite basal spacing, and so on (Cases et al., 1992; Fernández et al., 2004; Jacinto et al., 2012). A similar calculation was not conducted in the present study since there are not sufficient reliable experimental data of K_V1 for such a calculation. However, even with such a calculation result, directly measured ρ_pw data are crucial for calculation method validation. The highly scattered data in Fig. 10 suggest directly measuring ρ_pw of saturated bentonites is still a technical challenge. Therefore, in this study, it is roughly concluded that the ρ_pw of K_V1 in the tested range may be 1·1–1·2 Mg/m³ considering the difference between an envelope of S^*_r and the upper boundary of ∂S_r + 100% in Fig. 10(a).

Typical XRD profiles obtained in XRD tests and XRD-p_s tests are shown in Fig. 11, where profile intensity was normalised with respect to the peak intensity at about 2θ = ∼4·6° that corresponds to d₀₀₁ of ∼1·9 nm. With a relatively high w (e.g. 72·4–102·8%) in Fig. 11(a), profile peaks near 2θ = ∼2° with shoulders at 2θ = 3–4° are observed. The major peaks and shoulders designate (001) and (002) surface reflections of montmorillonite, respectively (Viani et al., 1983). As w decreases, the normalised intensity of the peak near 2θ = ∼2° weakens (Fig. 11(b)) and it vanished visually at about w = 34·6%. Results in Fig. 11(c) were obtained from XRD_p_s tests, where w_f values indicated in the figure were obtained after wetting. Results both in the XRD tests and XRD_p_s tests indicate that, in a w range of 36–65%, peaks near 2θ = ∼2° and at 2θ = ∼4·6° coexist.

The relations between w and d₀₀₁ obtained in this study are summarised in Fig. 12 together with data from Wang et al. (2020a). The stepwise increases of d₀₀₁ from 1·0 nm to 1·25 nm, 1·55 nm to 1·9 nm were designated as the ‘crystalline swelling’ state (e.g. Norrish, 1954; Meleshyn & Bunnenberg, 2005), of which the swelling of montmorillonite in this stage is expected to be driven mainly by the hydration force of counterions in the montmorillonite (e.g. Israelachvili, 2011; Zhang & Lu, 2018). It can be seen that the differences between two successive steps are about 0·25–0·35 nm, which is comparable to the size of a water molecule; d₀₀₁ = 1·0 nm, 1·25 nm, 1·55 nm and 1·9 nm are often said to correspond to zero-layer (0w), one-layer (1w), two-layer (2w) and three-layer (3w) hydration states (L), respectively (Foster et al., 1954; Fink & Thomas, 1964; Watanabe & Sato, 1988; Sato et al., 1992; Morodome & Kawamura, 2009; Wang et al., 2020a). The big d₀₀₁ gap between 1·9 and 4 nm was named the ‘forbidden basal spacing’ by Ravina & Low (1977), where an XRD peak could not be observed. Meleshyn & Bunnenberg (2005) explained the mechanism of the forbidden basal spacing, based on Monte Carlo simulations, that at d₀₀₁ = 1·9 nm, a special chain structure may form between counterions, water molecule and the montmorillonite surface in the interlayer space of Na type montmorillonite, such that water molecules cannot further penetrate into the interlayer space but accumulate around montmorillonites. Until the accumulated water approaches a certain level, the chain is broken and water enters the interlayer again associated with a jump of d₀₀₁ to 4·0 nm. It seems that d₀₀₁ = 4 nm is still a step until w exceeds ∼57% and then d₀₀₁ monotonically increases with w, as has often been observed by past studies (Foster et al., 1954; Norrish, 1954; Fink & Thomas, 1964; Fink et al., 1968; Zhang & Low, 1989). Swelling from d₀₀₁ = 4 nm was often called ‘osmotic swelling’ and interpreted by the diffuse double layer theory, although some argued its applicability to montmorillonites (e.g. Pashley, 1981; Viani et al., 1983; Lubetkin et al., 1984; Gonella et al., 2021).

As observed in Fig. 12, there is a crystalline and osmotic swelling mixed zone (termed the mixed swelling zone hereinafter) in a w range from 36 to 65%. Supposing that w_f values of saturated specimens in p_s tests are in this range, ρ_df of these specimens should be in the range from 1·05 to 1·46 Mg/m³ with the assumption of ρ_pw = 1·1 Mg/m³ according to equation (1). This range changes to 1·08–1·49 Mg/m³ for the ρ_pw = 1·15 Mg/m³ case (note that the maximum S^*_r in Fig. 10(a) is about 115% for specimens with ρ_df up to 1·50 Mg/m³, which is the reason to assume ρ_pw = 1·15 Mg/m³). These two ρ_df ranges are very consistent with the ρ_df range of specimens in XRD_p_s tests estimated from p_eq values (Table 9), which is 1·11–1·43 Mg/m³. By taking the average of these three ranges, it may be said that in the ρ_df range of 1·08–1·46 Mg/m³, crystalline and osmotic swellings co-exist and drive the development of p_s of compacted K_V1 specimens during wetting. In contrast, the crystalline swelling should govern p_s behaviours of compacted K_V1 for the ρ_df > 1·46 Mg/m³ condition. The above conclusions seem also to be supported by the relation between ρ_df and p_eq in Fig. 9, where the line slope changes in the semi-log scale just at ρ_df = 1·45 Mg/m³. Note that the montmorillonite break-up and force between montmorillonite particle issues, which have often been mentioned to explain the swelling behaviours of bentonites (e.g. Stepkowska, 1990; Villar, 2002; Baeyens & Bradbury, 2003; Suzuki et al., 2005; Laird, 2006), were not considered in this study mainly because clear evidence, to the best of the present authors’ knowledge, has not been found to show their occurrence as major issues in the scope of this study.

The equilibrated ρ_d after each loading or unloading step in p_c tests is plotted against p_c in Fig. 13 with a semi-logarithmic scale. It is very clear that the loading curves become steeper as w_i increases, while unloading and reloading curves seem to follow the same gentle slope, except for the w_i = 24·7% case. The steeper loading curves suggest that stiffness reduction is associated with an increasing w_i and the constant gentle slope of unloading and reloading curves imply very small elastic deformation.

The p_c loading curves with similar w_i in Figs 13(a), 13(b) and 13(d) were averaged and replotted to Fig. 14, where p_eq data obtained from p_s tests were added. The p_eq data are categorised into three clusters: cluster 1 are data for specimens with w_f in between 24·7% and 36%; cluster 2 are data for w_f < 24·7%; cluster 3 are data for w_f > 36%. The lower side boundary of mixed swelling zone in Fig. 12, 36%, is selected as one threshold. Another threshold value, 24·7%, is from w of K_V1 cured under RH = ∼100% (used for the specimen preparation in p_c tests), which seems to be the boundary where 3w starts to be fully achieved, as shown in Fig. 12. With the above categorisation, it can be seen that these three data clusters roughly correspond to the three linear zones visually separated in the insert of Fig. 9. It is also very interesting to see that data cluster 1 distributes closely along the compaction loading curve (p_c line) with w_i = 24·7%. In contrast, cluster 2 exhibits a trend to move to the upper side of the w_i = 24·7% p_c line as p_eq increases, and cluster 3 apparently departs from the w_i = 24·7% p_c line as p_eq decreases.

It is clear that the L of montmorillonite is 3w for a w value between 24·7 and 36% (Fig. 12). It is assumed herein that L of specimens in cluster 1 of Fig. 14 also stayed at 3w as the majority under the pressure of p_eq, and 3w was also the majority for the w_i = 24·7% specimen in the p_c test in the p_eq range of the cluster 1. This assumption is supported by results of Takahashi et al. (2015), which measured d₀₀₁ and p_eq of a pure montmorillonite while wetting it with water. They found that under p_eq of 3 MPa, 3w could fully develop; under p_eq = 6 MPa, a mixture of 2w and 3w occurred; and under p_eq = 13 MPa, only 2w developed. The maximum p_eq of data cluster 1 is about 4·7 MPa, indicating the assumption is reasonable. With this assumption, compaction and swelling behaviours of the tested bentonite may be illustrated by Fig. 15, which indicates that, in the w range discussed, the states of a specimen with an initial L of 0w, 1w or 2w in the p_s test would finally reach the p_c line of specimen with an initial L of 3w, although p_s curves during wetting may vary depending on the initial L values. The mechanism may be explained by Fig. 16, that is, the arrangement of particles and aggregates of specimens may be different depending on their initial L values, while the wetting process of the p_s test is similar to the compaction process of the p_c test. In the p_c test, the ‘void’ of a specimen is reduced by compaction loading, while in the p_s test, it is reduced by the size increase of the montmorillonite particles, where the interlayer space of montmorillonite has to be excluded from the ‘void’.

With the conceptual ideas shown in Figs 15 and 16, the following question arises: how would p_eq evolve for a specimen with initial L of 3w? It would be zero according to the concepts in Fig. 16 because a constant L = 3w is assumed; however, apparently this is wrong as the specimen still has swelling potential by wetting. To answer this question, the mechanism of forbidden basal spacing (Meleshyn & Bunnenberg, 2005) from L = 3w at d₀₀₁ = 1·9 nm to the osmotic swelling starting from d₀₀₁ = ∼4·0 nm may be borrowed. Since d₀₀₁ = 1·9 nm and d₀₀₁ = 4·0 nm were observed at a w range of 36–65% in Fig. 12, the accumulation–breakthrough process of the forbidden basal spacing may be illustrated by Fig. 17(a). With this mechanism, it may be said that p_eq measured for an initial L = 3w specimen during wetting is the result of swelling to d₀₀₁ = ∼ 4·0 nm of some montmorillonites located in areas that have accumulated more water. In other words, although the average w is smaller than 36%, d₀₀₁ = ∼ 4·0 nm still develops locally. Because it is partially developed, it may be expected that p_eq will not be able to reach the p_c line, as illustrated by p_s4 in Fig. 15.

When the p_eq of p_s test specimens exceeds 4·9 MPa, all p_eq data are from w_i = 0% and 7% cases (Fig. 9) and belong to cluster 2 (Fig. 14). From this pressure magnitude, it may be expected that L = 3w could not fully develop and L = 2w remains at the end of wetting based on the results of Takahashi et al. (2015) introduced above. Maybe for this reason, p_eq data start to depart from the L = 3w p_c line and approach the L = 2w p_c line (w_i = 21·4%). Although the data for specimens with ρ_df >1·7 Mg/m³ are insufficient to give a detailed discussion, the possible reason for the relatively lower p_eq values from the case of w_i = 18% is interpreted herein. It is supposed that there is also a p_c line for the L = 2w and 3w mixed state, as indicated by the grey dashed line in the insert of Fig. 14. With a similar idea as that shown for L = 3w case in Fig. 15, p_s test specimens with L = 0w and 1w and ρ_df > 1·7 Mg/m³ may also finally reach the grey dashed line by wetting with a final L of mixed 2w and 3w. The particle level mechanism is depicted in Fig. 17(b), borrowing the idea of the accumulation–breakthrough process in Fig. 17(a); that is, in order to change from L = 2w to 3w, water molecules need to accumulate to a certain level outside the montmorillonite. Cases et al. (1992) found from water adsorption and desorption isotherm data that water also accumulates at the external surface of montmorillonite first and then enters the interlayer space during wetting for L changed from 0w to 1w, which is also indirect evidence of the accumulation–breakthrough process assumed herein. In contrast, for specimens with initial L of 2w in p_s tests (i.e. w_i = 18% case), as illustrated in Fig. 17(c), montmorillonites changed to L = 3w can only contribute p_eq where the magnitude is lower than the cases with initial L = 0w and 1w. It is worth mentioning that the L values illustrated above and in Fig. 17 only represent the main possible conditions of hydrated water, because different L states would commonly coexist in most conditions (e.g. Cases et al., 1992; Holmboe et al., 2012).

For data p_eq cluster 3 in Fig. 14, apparently, osmotic swelling is involved in the final state of the p_s test specimens, so these data may imply another p_c line covering osmotic swelling. Nevertheless, it can be roughly summarised that w_i may have few effects on p_eq in the p_s tests if the initial L of montmorillonite is at a lower state than its final L state, whereas if the initial L is on the same or a similar level as the final L, p_eq may be smaller.

Three test programmes – the swelling pressure test (p_s test), XRD test and compaction tests (p_c test) – were conducted on a Japanese bentonite (i.e. K_V1) to understand its behaviours at different initial water contents (w_i). In the  p_s tests, apparent swelling pressure (p_s) of K_V1 was measured under five w_i cases (i.e. 0, 7, 18, 20 and 33%) and in an initial dry density range (ρ_di) from 1·1 to 1·9 Mg/m³. In the X-ray diffraction tests, K_V1 specimens with a water content (w) range from 25 to 103% were scanned to confirm the basal spacing of montmorillonite (d₀₀₁). In the p_c tests, K_V1 for four w_i cases (i.e. 0, 8, 20 and 25%) was compacted to about 15 MPa. The conclusions are drawn as follows.

• (a)

  The relations between final dry density (ρ_df) and equilibrium swelling pressure (p_eq) for different w_i cases in p_s tests seem to follow the same curve except a few data in the w_i = 18% case with ρ_df > 1·7 Mg/m³. The result suggests that w_i will have a minor effect on p_eq in certain w_i and ρ_df ranges.

• (b)

  The calculated degree of saturation (S^*_r) by assuming the average pore water density (ρ_pw) = 1·0 Mg/m³ is much higher than 100% for most specimens after wetting in p_s tests. The calculation uncertainty of S^*_r induced by measurement errors and the fact that ρ_pw > 1·0 Mg/m³ are expected to be the reasons for this. The ρ_pw in the conditions tested is estimated to be 1·1–1·2 Mg/m³.

• (c)

  Results of XRD tests imply that in a w range of 36–65%, crystalline swelling at d₀₀₁ = 1·9 nm and osmotic swelling with d₀₀₁ ≥ ∼4·0 nm co-exist. This w range corresponds to a ρ_df range of roughly 1·08–1·46 Mg/m³ for K_V1 specimens in p_s tests. This ρ_df range suggests boundaries of swelling types of montmorillonite governing p_eq of K_V1; in other words, crystalline swelling governs p_eq for ρ_df > 1·46 Mg/m³, osmotic swelling governs p_eq for ρ_df < 1·05 Mg/m³ and both types of swelling are involved between 1·46 and 1·05 Mg/m³.

• (d)

  The relation between ρ_df and p_eq of specimens with w_f value between 24·7% and 36% in p_s tests seems to follow well a compaction curve (p_c line) of specimens with w_i = 24·7% obtained in p_c tests. Since w between 24·7% and 36% corresponds to the three-layer (3w) hydration state (L) of montmorillonite in K_V1, the consistency of data obtained from p_s and p_c tests is interpreted as follows: if the initial condition L of p_s test specimens is lower than 3w – that is, L = zero-layer (0w), one-layer (1w) or two-layer (2w) hydration state – p_eq would reach the p_c line of a specimen with L = 3w. This interpretation implies a possible mechanism – namely, the ‘void’ reduction process – because the swelling of montmorillonite by wetting in the p_s test may be similar to the ‘void’ decreasing process through applying load in the compaction test, where the interlayer space of montmorillonite has to be excluded from the ‘void’. With the above interpretation and mechanism, the minor effect of w_i on p_eq in relatively low w_i and ρ_d ranges and the apparent effect in relatively high w_i and ρ_d ranges may be explained.

This work was performed as a part of the activities of the Research Institute of Sustainable Future Society, Waseda University, and supported by the Ministry of Economy, Trade and Industry (METI) of Japan, Waseda University Grant for Special Research Projects (project numbers 2021C-231 and 2021R-034) and Taisei Foundation. All XRD tests were conducted and technically supported by the MCCL, Waseda University (Izutani et al., 2016). The authors express their deep gratitude to all those mentioned above.

See Tables 3–9.

This section presents a way of determining ρ_df and its uncertainty for p_s test specimens. First, ρ_di and ρ_df of a specimen are calculated based on equations (3)–(6).

In equations (3)–(6), M_s, M_wi and M_wf are dry mass, wet mass before a test and wet mess after testing the specimen, respectively. V_i and ΔV are the initial volume and volume change (set swelling as positive) of the specimen, respectively. A balance with a solution of 0·1 mg and micrometers calibrated to 1 μm was used for the mass and length measurements. V_i is assumed to be equal to the inner volume of the specimen ring, which would be reasonable because the specimen surfaces were carefully trimmed before a test and radial deformation of the stainless specimen ring was expected to be negligible.

M_s can be calculated by either equation (4) or equation (5). The difference in magnitude of M_s (ΔM_s = M_s of equation (5) − M_s of equation (4)) for all tested specimens is summarised in Fig. 18 plotted against the M_s calculated by equation (4). M_wf in equation (5) would be slightly underestimated due to material loss when removing membrane filters from a specimen at the end of the test. For this reason, most ΔM_s values are negative. For some relatively dense specimens, bentonite was sometimes squeezed out of the specimen ring locally after wetting. For those specimens, the material that had been squeezed out was removed when measuring M_wf, and therefore the ΔM_s values as indicated by grey symbols are apparently larger. In addition, the ΔM_s values of data in the dotted line areas are apparently larger than the others, which may have been induced by w_i measurement error. The measured w_i was 35·8% for these data (Table 7), whereas their w_i values would be close to 33% based on extra confirmation. In the present study, M_s calculated by equation (4) was adopted in principle to obtain ρ_df, while for the data in gray and data in the dotted line areas, equation (5) was used for M_s calculation. The uncertainty of M_s would range from −0·015 to +0·004 g from Fig. 18.

There are two major sources for ΔV in equation (6): membrane filter compression and apparatus expansion, both of which are induced by development of p_s and cause vertical swelling deformation of the specimen. The former was estimated by measuring the change in thickness of the membrane filter before and after a test, and the latter was obtained by measuring the change in thickness of the apparatus immediately before wetting and before termination of the test. Fig. 19 depicts the relation between apparatus expansion and p_eq, where an increase of the apparatus expansion with an increasing p_eq is observed. Data scattering is expected to be induced mainly by the measurement error. Under the same p_eq, the measurement error seems to be in a range of ±0·01 mm at the maximum, which corresponds to an uncertainty of ΔV of ±6 mm³. Measurement error for membrane filter thickness changes could not be estimated because the filters were used repeatedly in the tests. However, the error should be smaller than that for apparatus expansion because the filter thickness measurement was easier. Herein, it was assumed to be ±0·005 mm based on measurements in Wang et al. (2021) and the corresponding uncertainty of ΔV is ±3 mm³.

The uncertainty of ρ_df (∂ρ_df) can be calculated by equation (7), which is the partial derivative of equation (6)

With ∂(ΔV) =  ±9 mm³ and ∂M_s = $−0⋅015+0⋅004g$⁠, and assuming (V_i +  ΔV) is equal to the volume of a standard specimen (28 mm dia. and 2 mm thick), ∂ρ_df roughly ranges from −0·02 to +0·01 Mg/m³ for a ρ_df range of 1·0–1·8 Mg/m³.

C_m

montmorillonite content

d₀₀₁

basal spacing of montmorillonite

G_s

specific gravity

L

hydration states of interlayer water of montmorillonite ( = 0–3w).

p_c

applied vertical compaction load

p_eq

equilibrium swelling pressure on p_s time history

p_s

apparent swelling pressure of compacted bentonite during wetting

S_r

degree of saturation

S^*_r

calculated degree of saturation with assumption of ρ_pw = ρ_w = 1 Mg/m³

S_ri and S_rf

initial and final degrees of saturation, respectively

w

gravimetric water content

w_i

initial gravimetric water content

w_f

final gravimetric water content

θ

half the angle between incident and diffracted X-ray beams of the diffractometer

θ_pk

diffraction angle of a mineral, which is equal to θ at peak position of X-ray scanned profile

λ

wavelength of incident X-ray beams ( = 0·15418 nm for Cu Kα source)

ρ_d

dry density

ρ_df

final dry density

ρ_di

initial dry density

ρ_pw

average pore water density in compacted bentonite

ρ_w

free water density