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Bentonite is an important material for low-permeability engineering systems used in dams and hazardous waste facilities. While models to characterize the hydraulic conductivity of bentonite have been developed in previous studies, these models were not applicable to various density conditions for Na- and Ca-bentonite. In this study, we present a new model for estimating the hydraulic conductivity of bentonite applicable to a wide range of density conditions for Na- and Ca-bentonite. In order to consider flow paths in compacted bentonite, a lamination structure of montmorillonite stacks was assumed. Our hydraulic model discriminated interlayer pores and other pores by applying a method for estimating the probability of connected pores and hydraulic coefficients governed by the plane Poiseuille flow equation. The model was consistent with the experimental data investigated in previous studies on the hydraulic conductivity of Na-bentonite and was in good agreement with the data of Ca-bentonite in the range of an effective montmorillonite density (ρem) at 500 kg/m3 ≤ ρem ≤ 1,500 kg/m3 and 600 kg/m3 ≤ ρem, respectively. However, some experimental values on Ca-bentonite under low-density conditions were far higher than the model results in the case where Ca-bentonite forms flow paths of relatively large pores.

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  • A model for estimating the hydraulic conductivity of bentonite was proposed.

  • The model discriminated interlayer pores from other pores in the bentonite.

  • The relation between a wide range of effective montmorillonite densities and basal spacings for Na-montmorillonite was proposed.

bentonite, hydraulic conductivity, hydraulic model, montmorillonite
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Low-permeability clay soil is a key material for water control systems, and hydraulic conductivity is a significant factor to evaluate the system. Low hydraulic conductivity soils ensure the safety of drinking water in the natural system (Nahin et al. 2019) and are used for groundwater control in engineering systems such as dams and hazardous waste facilities (Gueddouda et al. 2016; Park & Oh 2018; Santos & Esquivel 2018). Bentonite, which is a representative material for engineering systems, uses as clay liners and backfills for hazardous waste facilities and cutoff walls for earth dams, and is a suitable material to prevent groundwater flow and migration of hazardous materials (i.e., heavy metals and radionuclides) (Meer & Benson 2007; Malusis et al. 2011). Montmorillonite, a major component of bentonite clay, plays a role in this function via a filling of the bentonite pores through its swelling property. Compacted bentonite and sand-bentonite mixtures are used in engineering systems when suitable for the performance required (Lee & Shackelford 2005; Gueddouda et al. 2016).

Many experiments for determining the hydraulic conductivities of bentonite and montmorillonite have been conducted for using as an engineered barrier in radioactive waste disposal facilities (e.g., Suzuki et al. 1992; Matsumoto et al. 1997; Maeda et al. 1998). Some experiments have revealed that compacted bentonite and montmorillonite saturated with water are composed of stacking laminations of several montmorillonite particles (Pusch et al. 1990). Other studies have found that the water flowing near the montmorillonite surface has a higher viscosity than bulk water due to an effect of the surface charge of montmorillonite (Low 1976; Maeda et al. 1998; Ichikawa et al. 1999). Based on these experimental results, models for estimating the hydraulic conductivity of bentonite have been proposed. Komine (2008) considered two montmorillonite particles as a base unit and proposed a flow of water through their interval; however, they did not distinguish the interlayer pores of the montmorillonite from the other pores (hereinafter defined as external pores) and neglected to perform any quantitative analysis of the viscosity change of water. Since the width of the interlayer pore differs from that of the external pore, the hydraulic conductivities in the interlayer and the external pore also differ. The same problems arose when the Kozeny–Carman relation was applied to bentonite and montmorillonite (Ren et al. 2016; Kobayashi et al. 2017; Kohno 2021). Tanaka et al. (2009) proposed a model for hydraulic conductivity estimation that distinguished between interlayer pores and external pores. Their model, however, focused on high-effective montmorillonite density (the montmorillonite weight divided by the sum of the volumes of pores and montmorillonite) conditions and thus could not be applied under low-density conditions.

The previously proposed models also have the problem of not employing generalized equations for application to a wide range of effective montmorillonite densities. The interlayer pore width of montmorillonite changes in accordance with variations in the interlayer cation, water content, and ionic strength of the solution (Foster et al. 1954; Norrish 1954; Fink & Thomas 1964; Fukushima 1984; Zhang & Low 1989). The relation between the interlayer pore width and water content has been parameterized by the effective montmorillonite density of Na- and Ca-montmorillonite. Although the relations under high-effective montmorillonite density conditions (over 800 kg/m3) have been obtained (Kozaki et al. 1998; Suzuki et al. 2001; Wang et al. 2021), those under low-effective montmorillonite density conditions have been insufficient. The relation under low-effective montmorillonite density conditions is significant because sand-bentonite mixtures containing 10–30% bentonite have the potential to be used in low-level radioactive waste disposal facilities and other engineering systems. The difference between Na- and Ca-bentonite is important to evaluate the hydraulic performance of the clay liner used in hazardous waste facilities because their permeabilities are quite different when the bentonite content is low (Maeda et al. 1998). However, the estimation method of the hydraulic conductivities considering the water flow paths in bentonite for both Na- and Ca-type of various bentonite content is insufficient.

The objectives of this study are to evaluate the relation among the interlayer pore, external pore width, and effective montmorillonite density, and to develop a model to estimate the hydraulic conductivity of both Na- and Ca-bentonite under the wide range of effective montmorillonite densities based on this relation.

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A schematic image of a montmorillonite particle, stack, interlayer pore, and external pore is shown in Figure 1. The porosities of these two pores are calculated as outlined below, assuming that the montmorillonite particle shape is cylindrical. A volume of the stack is expressed as follows:
(1)
where V is the volume of the stack (m3), R is the radius of the montmorillonite particle (m), n is the number of montmorillonite particles in the stack (–), t is the thickness of the montmorillonite particle (m), and d is the basal spacing (m).
Figure 1
Schematic image of a montmorillonite particle, stack, interlayer pore, and external pore.
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Schematic image of a montmorillonite particle, stack, interlayer pore, and external pore.

Figure 1
Schematic image of a montmorillonite particle, stack, interlayer pore, and external pore.
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Schematic image of a montmorillonite particle, stack, interlayer pore, and external pore.

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The weight of the montmorillonite particle is expressed as follows:
(2)
where m is the weight of the montmorillonite particle (kg), and ρm is the particle density of the montmorillonite (kg/m3).
Assuming that the compacted montmorillonite specimen is a cube of the length H (m), the weight of the montmorillonite in the specimen is expressed as follows:
(3)
where Mm is the weight of the montmorillonite in the specimen (kg), ρem is the effective montmorillonite density (representing bulk density when the specimen is composed of only montmorillonite) (kg/m3), and H is the length of the specimen (m).
The number of stacks in the specimen is expressed as follows:
(4)
where N is the number of the stacks in the specimen (–).
The total volume of the stacks in the specimen (Vt), the total volume of the external pores in the specimen (Vext), and the total volume of the interlayer pores in the specimen (Vint) are expressed by the following equations.
(5)
(6)
(7)
An external pore porosity (εext) and an interlayer pore porosity (εint) are expressed as follows:
(8)
(9)

The number of particles in a single stack and the basal spacing are necessary to solve Equations (8) and (9). X-ray diffraction (XRD) revealed the relations between the water content and basal spacing for a Na-montmorillonite suspension (Foster et al. 1954; Norrish 1954; Fink & Thomas 1964; Zhang & Low 1989) and the relations between the effective montmorillonite density (ρem) and basal spacing (d) for a compacted Na-montmorillonite (Kozaki et al. 1998; Suzuki et al. 2001; Holmboe et al. 2012). Kozaki et al. (1998) observed the basal spacing of a three-water-layer hydrate state (3WH state, d = 1.88 nm) and a two-water-layer hydrate state (2WH state, d = 1.56 nm) for compacted bentonite. The reported values of the effective montmorillonite density (ρem) corresponding to the 3WH state alone were 1,000 kg/m3 < ρem ≤ 1,300 kg/m3, to a mixed state of the 3WH state and 2WH state were 1,300 kg/m3 < ρem < 1,600 kg/m3, and to the 2WH state alone were 1,600 kg/m3 ≤ ρem. Additionally, Suzuki et al. (2001) observed the basal spacing of a mixed state of d = 3.52 nm and a 3WH state. The reported ρem values corresponding to the mixed state were 800 kg/m3 ≤ ρem ≤ 1,000 kg/m3. While these former reports determined the hydrate states by assessing whether the diffraction peaks of the basal spacing appeared, Holmboe et al. (2012) determined the hydrate states based on the peak ratio using data fitting to the diffraction peaks. While the hydration states at 1,300 kg/m3 < ρem were in agreement with the reports of Kozaki et al. (1998) and Suzuki et al. (2001), those at ρem ≤ 1,300 kg/m3 were not. Holmboe et al. (2012) determined that the hydration states at ρem ≤ 1,300 kg/m3 were a mixed state of crystalline swelling (stepwise expansion) and osmotic swelling (continuous expansion), and did not observe a clear diffraction peak resulting from the 3WH state at ρem = 1,000 kg/m3. In contrast, Kozaki et al. (1998) and Suzuki et al. (2001) observed this peak at the same density. Holmboe et al. (2012) explained this difference by the effects of the water content without using any profile fitting or the particle size. The degree of difference of the basal spacing reported by Kozaki et al. (1998) and Holmboe et al. (2012), however, was only about 0.1 nm, and a value assumed not to be significantly different in the estimation of the hydraulic conductivity because the change in the hydraulic conductivities would be order of 3% based on our proposed model.

As any studies have hardly shown the relation between the basal spacing and the effective montmorillonite density at 800 kg/m3 < ρem (Holmboe et al. 2012), this relation was calculated from the relation between the water content (w) and the basal spacing under an assumed condition of water saturation.
(10)
where w is the water content (–), Mw is the weight of water in the specimen, and ρw is the water density (kg/m3).
The relation between the basal spacing and the effective montmorillonite density calculated from Equation (10) using the previous experimental water content data (Foster et al. 1954; Norrish 1954; Fink & Thomas 1964; Zhang & Low 1989; Holmboe et al. 2012) is shown in Figure 2. The osmotic swelling occurred at ρem ≤ 900 kg/m3 and the crystalline swelling occurred at 700 kg/m3 ≤ ρem. The density conditions of the osmotic and crystalline swelling determined by the montmorillonite suspension were consistent with the results for the compacted montmorillonite, whose density was 800 kg/m3 ≤ ρem ≤ 1,000 kg/m3 with a mixed state of osmotic and crystalline swelling (Kozaki et al. 1998; Suzuki et al. 2001). It is an important point that Figure 2 shows a continuous decreasing of basal spacing as increasing density at ρem < 1,000 kg/m3. Figure 2 also shows that d values for the crystalline swelling, d = 1.88 nm, 1.56 nm, and mixed state of them, are consistent with the results for the compacted montmorillonite, whose density is 1,000 kg/m3 < ρem (Kozaki et al. 1998). The values of the effective montmorillonite density (ρem) corresponding to the basal spacing affected by the osmotic swelling alone were determined as ρem < 700 kg/m3, those affected by the mixed state of osmotic and crystalline swelling were determined as 700 kg/m3 ≤ ρem < 1,000 kg/m3, and those affected by the crystalline swelling alone were determined as 1,000 kg/m3 ≤ ρem, for Na-montmorillonite. A regression curve of the basal spacing for osmotic swelling is determined by the following equations.
(11)
Figure 2
Relation between the effective montmorillonite density and basal spacing for Na-montmorillonite.
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Relation between the effective montmorillonite density and basal spacing for Na-montmorillonite.

Figure 2
Relation between the effective montmorillonite density and basal spacing for Na-montmorillonite.
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Relation between the effective montmorillonite density and basal spacing for Na-montmorillonite.

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In the case of the result of the d-value calculated by using Equation (11) less than or equal to 3.2 nm.
(12)

The basal spacing jumped from 1.88 to 3.2–4.1 nm (Norrish 1954; Fukushima 1984; Zhang & Low 1989; Suzuki et al. 2001). We assumed that the basal spacing jumped from 1.88 to 3.2 nm (Zhang & Low 1989), and that the basal spacing was osmotic swelling as determined by Equations (11) and (12). As noted above, the difference in the basal spacing of 0.1 nm assumed not to be significantly different in the estimation of the hydraulic conductivity.

Although the swelling behavior of Ca-montmorillonite is the same as that of Na-montmorillonite at 1,000 kg/m3 ≤ ρem, the 3WH state supplants osmotic swelling as the dominant state at lower-density conditions (Fukushima 1984; Matusewicz et al. 2013). Therefore, the basal spacing of crystalline swelling is reasonable under all density conditions for Ca-montmorillonite.

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X-ray CT observations of compacted montmorillonite showed that the montmorillonite particles were oriented perpendicularly to the direction of compaction (Takahashi & Tachi 2019). Diffusion tests using tritiated water (HTO) also revealed that the effective diffusivities for compacted montmorillonite changed according to the direction of diffusion (axial and perpendicular directions to compaction) (Sato & Suzuki 2003; Suzuki et al. 2004). The effective diffusivity is expressed as follows (Tachi & Yotsuji 2014):
(13)
(14)
where Dei is the effective diffusivity of species i (m2/s), εtotal is the total porosity (–), Dwi is the diffusivity of species i in bulk water (m2/s), G is a geometric factor (–), δg is the geometric constrictivity (–), δel is the electric constrictivity (–), and τ is the tortuosity (–).
The geometric constrictivity and electric constrictivity depend on the pore shape and the electrostatic interactions with a negatively charged clay surface, respectively. The tortuosity represents an actual diffusion path length relative to the specimen length. From the viewpoint of the diffusional direction, the pore shape and the electrostatic interactions do not affect the differences in the effective diffusivities of HTO. Therefore, the effective diffusivity of HTO is affected by only the diffusion path length. Figure 3 shows a schematic image of the diffusion path lengths when the stacks laminate with an average angle θ. The symbols θa,y and θp,x represent the lamination angles of the stack in axial and perpendicular directions to compaction, respectively (x and y are any positive integer). The average angles of θa,y and θp,x are equal to θ. La,y and Lp,x represent the diffusion path lengths in the axial and perpendicular directions to compaction per stack, respectively. The total lengths of La,y and Lp,x are equal to La and Lp, respectively. Assuming this laminated condition, the diffusion path lengths for the axial and perpendicular directions to compaction are given as follows:
(15)
(16)
where La is the diffusion path length axial to the compaction (m) and Lp is the diffusion path length perpendicular to the compaction (m).
Figure 3
Schematic image of path lengths.
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Schematic image of path lengths.

Figure 3
Schematic image of path lengths.
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Schematic image of path lengths.

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The relation between the effective diffusivity ratio of the diffusional direction of HTO and the average laminated angle of stacks is calculated from Equations (13)–(16).
(17)
where DeHTO,p and DeHTO,a are the effective diffusivity of HTO for the perpendicular and axial directions to compaction, respectively (m2/s), and τp and τa represent the tortuosity perpendicular to and along the axial direction of compaction, respectively (–).
Figure 4 shows the relation between the effective montmorillonite density and the average laminated angles of the stacks. The average laminated angles of the stacks were calculated from the effective diffusivities from previous studies (Sato & Suzuki 2003; Suzuki et al. 2004). The effective montmorillonite density increases as the average laminated angle tends to decrease. The average laminated angles at arbitrary densities are obtained by least-squares regression (Equation (18)). Since the range of effective montmorillonite densities of the effective diffusivity is narrow due to the limited data, the average laminated angles out of this range are obtained by extrapolation of the regression line.
(18)
(19)
where Rdet2 is the coefficient of determination, RSS is the residual sum of squares, and TSS is the total sum of squares.
Figure 4
Relation between the effective montmorillonite density and the averaged laminated angle of stacks calculated from the effective diffusivity of Sato & Suzuki (2003) and Suzuki et al. (2004).
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Relation between the effective montmorillonite density and the averaged laminated angle of stacks calculated from the effective diffusivity of Sato & Suzuki (2003) and Suzuki et al. (2004).

Figure 4
Relation between the effective montmorillonite density and the averaged laminated angle of stacks calculated from the effective diffusivity of Sato & Suzuki (2003) and Suzuki et al. (2004).
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Relation between the effective montmorillonite density and the averaged laminated angle of stacks calculated from the effective diffusivity of Sato & Suzuki (2003) and Suzuki et al. (2004).

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The coefficient of determination (R2 = 0.58) is reasonably good. The average laminated angles of the stacks calculated from Sato & Suzuki (2003) are lower values than those calculated from Suzuki et al. (2004). High ionic strength solutions (0.55 and 1.1 M) were used in diffusion tests by Sato & Suzuki (2003), while pure water was used in them by Suzuki et al. (2004). It is possible that the difference of ionic strength affects the average laminated angles of the stacks, but we cannot explain moreover about this because the usable data are limited.

On the other hand, the effective diffusivity of HTO showed no changes in response to shifts in the diffusional directions for the bentonite and sand-bentonite mixtures because of containing minerals other than montmorillonite (Sato & Suzuki 2003). An average laminated angle of 45° was therefore selected for this study, as the montmorillonite particles do not appear to orient in a particular direction in the bentonite and sand-bentonite mixtures.

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Based on the previous studies (Komine 2008; Iida et al. 2011), hydraulic conductivities were modeled, assuming that each stack was a parallel plate. Some studies (Bolt 1956; Sidharan & Choudhury 2008) obtained the relation between the pore width of parallel clay plates and the void ratio, but they did so without assuming different widths for the external and interlayer pores. In the present study, we estimated the hydraulic conductivities of water flowing through both interlayer pores and external pores, with the assumption that the pore widths differed. As mentioned above, the external pore width is calculated from the stacks of n layers of particles. The relation between the void ratio contributed by the external pores and the external pore width is given as follows:
(20)
where eext is the void ratio contributed by the external pores (–), Sext is a specific surface area of the stack (m2/kg), and Aext is a half distance of the width of the external pore (m).
The interlayer pore width of the osmotic swelling is calculated from a regression curve (Equations (11) and (12)) in Figure 2 plotting the relation between the effective montmorillonite density and basal spacing. The basal spacing of the mixed state of the osmotic and crystalline swelling was at 700 kg/m3 ≤ ρem < 1,000 kg/m3 for Na-montmorillonite. As the abundance ratio of the number of stacks attributed to the osmotic or crystalline swelling was difficult to evaluate precisely, we assumed that the osmotic swelling stacks had an abundance ratio of 100% at ρem = 700 kg/m3 and 0% at ρem = 1,000 kg/m3, and vice versa for the crystalline swelling. The abundance ratio of the stacks between these densities was calculated from linear interpolation. The basal spacing of the mixed state consisting of the 3WH and 2WH states was estimated at 1,300 kg/m3 < ρem < 1,600 kg/m3 for Na-montmorillonite and Ca-montmorillonite. We assumed that the stacks had an abundance ratio of 100% at ρem = 1,300 kg/m3 and 0% at ρem = 1,600 kg/m3 for the 3WH state, and vice versa for the 2WH state. Between these densities, the abundance ratio of the stacks was calculated in the same manner used for Na-montmorillonite at 700 kg/m3 ≤ ρem < 1,000 kg/m3. The external pore width was calculated by Equations (1), (11), (12), and (20) using the calculation parameters as shown in Table 1. The external pore width decreases as the effective montmorillonite density increases, and also as the number of laminations in the stack (n) decreases (Figure 5). If the number of laminations exceeds 4 for Na-montmorillonite, the external pore width at ρem = 1,000 kg/m3 is larger than that at ρem = 700 kg/m3, based on the assumed ratio of the osmotic swelling to the 3WH states. Having a large external pore width implies that the specimen has enough space for montmorillonite swelling. Since only crystalline swelling was observed at ρem = 1,000 kg/m3 (Kozaki et al. 1998), it would be unreasonable to assume that the external pore width at ρem = 1,000 kg/m3 is larger than that at ρem = 700 kg/m3. In contrast to the unreasonable behavior observed for Na-montmorillonite with an n over 3, the external pore width in the case of Ca-montmorillonite decreases monotonically as the effective montmorillonite density increases in every lamination number. This difference between Na-montmorillonite and Ca-montmorillonite was due to the difference in their swelling behavior. Although Na-montmorillonite could form the osmotic state (the d-value was over 3.2 nm) under any effective montmorillonite density condition other than 1,000 kg/m3 ≤ ρem, Ca-montmorillonite could swell only up to the 3WH state (the d-value was 1.88 nm) under the density condition of ρem < 1,000 kg/m3. Therefore, we determined that the number of laminations for Na-montmorillonite was 2 and 3, while that for Ca-montmorillonite was an arbitrary number above 2.
Table 1

Properties of montmorillonite particle

Diameter of montmorillonite particle (2R) 3.91 × 10−7 [m] (Suzuki et al. 2012) 
Thickness of montmorillonite particle (t) 9.55 × 10−10 [m] (Tournassat et al. 2003) 
Particle density of montmorillonite (ρm) 2,770 (kg/m3) (Komine 2008) 
Diameter of montmorillonite particle (2R) 3.91 × 10−7 [m] (Suzuki et al. 2012) 
Thickness of montmorillonite particle (t) 9.55 × 10−10 [m] (Tournassat et al. 2003) 
Particle density of montmorillonite (ρm) 2,770 (kg/m3) (Komine 2008) 
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Figure 5
Relation between effective montmorillonite density and external pore width (n represents the number of laminations in a single stack).
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Relation between effective montmorillonite density and external pore width (n represents the number of laminations in a single stack).

Figure 5
Relation between effective montmorillonite density and external pore width (n represents the number of laminations in a single stack).
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Relation between effective montmorillonite density and external pore width (n represents the number of laminations in a single stack).

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The hydraulic conductivities of compacted montmorillonite and bentonite are estimated by assuming a plane Poiseuille flow in the external and interlayer pores. Hydraulic paths are presumed to compose complicatedly by the coupling of the external and interlayer pores. Based on the probability of connected pores at arbitrary adjacent cross-sections (Childs & Collis-George 1950; Marshall 1958; Millington & Quirk 1961), this study follows the method of Millington & Quirk (1961) and uses coefficients of a plane Poiseuille flow (Komine 2008) (M&Q model). The relation between the hydraulic conductivity and probability of connected pores at arbitrary adjacent cross-sections is given as follows:
(21)
where kM&Q is the hydraulic conductivity (m/s), g is the gravitational acceleration (m/s2), μ is the viscosity of water (Pa·s), Xj,k is the smaller width of Xj and Xk (m), and f(Xj) and f(Xk) are probabilities of the presence of the pore widths of Xj or Xk in the arbitrary cross-sections (–).
An apparent hydraulic gradient is calculated by dividing a pressure difference between the upper and lower ends of the specimen by the specimen length. Since the hydraulic paths are longer than the specimen length, as mentioned before, however, the actual hydraulic gradient in the pores is smaller than the apparent hydraulic gradient. A relation between the apparent hydraulic gradient (Iapp) and the actual hydraulic gradient in the pores (I) can be expressed using the average laminated angle of the stacks (θ) (Tanaka et al. 2009):
(22)
where ∂p/∂z is the pressure gradient in the flow direction (Pa/m), and Δp is the pressure difference between the upper and lower ends of the specimen (Pa).
As Equation (21) does not consider the difference between the actual hydraulic gradient and the apparent hydraulic gradient in the pores, the apparent hydraulic conductivity for the specimen is expressed as follows:
(23)
where A is the half distance of pore width (m); the subscripts 2WH, 3WH, and osm stand for the interlayer pores of the 2WH state, 3WH state, and osmotic swelling state, respectively; and the subscript ext stands for the external pore.

To clarify the effect of the pore connections, we examine extreme cases where the external pores and interlayer pores are assumed to be independent of each other (parallel model) and the pore connections are assumed to be completely random (series model). The apparent hydraulic conductivities for the parallel and series model are expressed as follows:

Parallel model
(24)
Series model
(25)
where εi is the porosity contributed by pore i (–).
The viscosity of water in the pores affected by the surface charge of montmorillonite is expressed as follows (Andrade & Dodd 1946):
(26)
where μ(x) is the viscosity of water at distance x from the montmorillonite surface, μ0 is the viscosity coefficient of the bulk water (Pa·s), fve is the viscoelectric constant (m2/V2), and Ψ(x) is an electric potential at distance x from the montmorillonite surface (V).
The Gouy-Chapman diffuse double layer theory is applied to calculate the electric potential of near the montmorillonite surface. Since the ion distribution in the diffuse double layer obeys the Poisson–Boltzmann equation, a relation between the electric potential and ion distribution is expressed as follows:
(27)
where ε0 is the vacuum permittivity (C/V/m), εw is a relative permittivity of solution (–), zi is a valence of species i (–), e is the elementary charge (C), cb,i is a number concentration of species i in the bulk solution (1/m3), kB is the Boltzmann constant (J/K), and T is the absolute temperature (K).
Equation (27) is solved by taking a surface charge density and the potential gradient of the mid-plane between two parallel plates (Verwey & Overbeek 1948; Tachi & Yotsuji 2014).
(28)
where σ0 is the surface charge density of montmorillonite (C/m2), CEC is a cation exchange capacity (C/kg), and S is the specific surface area of montmorillonite (m2/kg)].

For the sake of simplicity, the number concentration in the bulk solution is calculated by assuming a 1:1 electrolyte solution concentration ratio obtained based on the ionic strength of equilibrium concentration with respect to Kunipia F (Na-montmorillonite; solid-to-liquid ratio: 250 g/l (Oda & Shibata 1999)), a refined bentonite commercially available from Kunimine Industries Co. Ltd, Japan, using PHREEQC (Parkhurst & Appelo 2013). Table 2 shows the parameters used for solving Equations (26)–(28).

Table 2

Parameters for the calculation of hydraulic conductivity

Density of water (ρw) 997 (kg/m3) 
Gravitational acceleration (g) 9.81 (m/s2) 
Viscosity of bulk solution (μ) 8.94 × 10−4 (Pa·s) 
Vacuum permittivity (ε0) 8.85 × 10−12 (C/V/m) 
Relative permittivity of solution (εw) 78.5 
Elementary charge (e) 1.60 × 10−19 (C) 
Boltzmann constant (kB) 1.38 × 10−23 (J/K) 
Viscoelectric constant (fve) 1.02 × 10−15 (m2/V2) (Lyklema & Overbeek 1961) 
CEC of montmorillonite (CEC) 1.11 × 102 (meq/100 g) (Suzuki et al. 2012) 
Ionic strength of solution 0.03 (mol/l) 
Density of water (ρw) 997 (kg/m3) 
Gravitational acceleration (g) 9.81 (m/s2) 
Viscosity of bulk solution (μ) 8.94 × 10−4 (Pa·s) 
Vacuum permittivity (ε0) 8.85 × 10−12 (C/V/m) 
Relative permittivity of solution (εw) 78.5 
Elementary charge (e) 1.60 × 10−19 (C) 
Boltzmann constant (kB) 1.38 × 10−23 (J/K) 
Viscoelectric constant (fve) 1.02 × 10−15 (m2/V2) (Lyklema & Overbeek 1961) 
CEC of montmorillonite (CEC) 1.11 × 102 (meq/100 g) (Suzuki et al. 2012) 
Ionic strength of solution 0.03 (mol/l) 
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The hydraulic conductivities vary with the changes in the ionic strength of the solution permeating the specimen caused by the changing viscosity of the water and the basal spacing (Norrish 1954). Although the ionic strength is a significant factor for the hydraulic conductivity, this paper only considers the basic case of permeation by pure water. More cases will have to be considered to clarify the effects of the ionic strength on the hydraulic conductivity.

Figure 6 shows the relation of the effective montmorillonite density and hydraulic conductivity, which is the comparison of this study with experimental data of previous studies (Suzuki et al. 1992; Matsumoto et al. 1997; Maeda et al. 1998; Hasegawa 2004; Butsuda et al. 2006; Ito et al. 2010a, 2010b). For Na-montmorillonite, the M&Q model of n = 2 and n = 3 is in good agreement with the experimental data of previous studies. For Na-bentonite and sand-bentonite mixtures, the M&Q model of n = 3 is consistent with the scattered experimental data at 500 kg/m3 ≤ ρem ≤ 1,500 kg/m3. For Ca-bentonite and sand-bentonite mixtures, the M&Q model of n = 3–20 is in agreement with the experimental data.
Figure 6
Relation between the effective montmorillonite density and hydraulic conductivity.
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Relation between the effective montmorillonite density and hydraulic conductivity.

Figure 6
Relation between the effective montmorillonite density and hydraulic conductivity.
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Relation between the effective montmorillonite density and hydraulic conductivity.

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The M&Q model is compared with the parallel model and the series model. The results of the M&Q model are similar to those of Na-montmorillonite and Na-bentonite (including sand-bentonite mixtures), with the exception of the densities, namely, 700 kg/m3 < ρem ≤ 1,300 kg/m3. The porosity ratio of the external pores in these densities is higher than that in other densities (Figure 7), increases sharply in the range of the mixed state of the osmotic swelling state and the 3WH state (700 kg/m3 < ρem < 1,000 kg/m3), and decreases sharply in the range of the 3WH state only (1,000 kg/m3 ≤ ρem ≤ 1,300 kg/m3). The increase of the external pores can be explained by the effect causing the decrease of the interlayer pore width (which changes from the osmotic swelling state to the 3WH state) and is larger than the effect causing the increase in the number of stacks. The decrease of the ratio of the external pores is only affected by the increase of the number of stacks (density) because the interlayer pore width does not change at 1,000 kg/m3 ≤ ρem ≤ 1,300 kg/m3. Since the parallel model considers the hydraulic paths flowing through the external pores and the interlayer pores independently, the hydraulic conductivity is controlled by the large pores, that is, the external pores. In the series model, in contrast, the hydraulic conductivity is strongly affected by the small pores, that is, the interlayer pores with the 3WH state. The M&Q model is an intermediate result between the parallel model and the series model at 700 kg/m3 < ρem ≤ 1,300 kg/m3. The M&Q model reflects the permeability of dominant flow paths and reproduces the scattered experimental data at 500 kg/m3 ≤ ρem ≤ 1,500 kg/m3 for Na-bentonite. On the other hand, this model estimates far higher and far lower than the experimental data at ρem < 500 kg/m3 and 1,500 kg/m3 < ρem, respectively. The reason for differences between the experimental data and the model is not clear; however, it is possible that the flow paths are kept in the external pore under high-density conditions, and are prevented by particle-selves, which meant generating dead-ends, under low-density conditions.
Figure 7
Relation between the porosity ratio and effective montmorillonite density (n = 3).
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Relation between the porosity ratio and effective montmorillonite density (n = 3).

Figure 7
Relation between the porosity ratio and effective montmorillonite density (n = 3).
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Relation between the porosity ratio and effective montmorillonite density (n = 3).

Close modal

For Ca-bentonite (including sand-bentonite mixtures), the hydraulic conductivity of the series model markedly differs from that of the experimental results at ρem < 1,000 kg/m3, because the model result represents a harmonic mean of the external pores and the 3WH state interlayer pores. This implies that the main flow path of Ca-bentonite at ρem < 1,000 kg/m3 is the external pores.

The scattered data on the hydraulic conductivities of Ca-bentonite (including sand-bentonite mixtures) at 600 kg/m3 ≤ ρem indicate that the aggregation of montmorillonite particles occurs due to difference in the strength of matrix suction at compaction (Sakita et al. 2020). Many small pores are distributed in the external pores because montmorillonite particles are likely to be dispersed in between sand grain skeletons when the degree of saturation is high at compaction (see the schematic image in Figure 7, from Sakita et al. 2020). In contrast, relatively large pores are generated in the external pores because the montmorillonite particles aggregate around the sand grains when the degree of saturation is low at compaction. Ca-montmorillonite swells only up to the 3WH state in the case when the effective montmorillonite density is low. When the aggregation of montmorillonite particles around the sand grains at the low-initial degree of saturation, pores between the sand grains are not filled by Ca-montmorillonite and are created relatively large one in them. Although the M&Q models from n = 3–20 are in agreement with the experimental results showing >80% saturation at compaction, they are inconsistent with the <80% saturation as shown in Figure 8. The M&Q model appears to enable the estimation of the hydraulic conductivity for Ca-bentonite when the degree of saturation at compaction is high, as the model assumes a homogeneous distribution of montmorillonite particles. The M&Q model, on the other hand, does not apply when the degree of saturation at compaction is low, as the distribution of montmorillonite particles is heterogeneous due to the generated flow paths of the relatively large pores. The heterogeneous condition is a unique condition when using a Ca-bentonite-sand mixture at the low-initial degree of saturation. Therefore, the proposed model can be applied to most cases with permeation through pure water or a low-ionic strength solution.
Figure 8
Relation between the effective montmorillonite density of Ca-type bentonite mixed soil and hydraulic conductivity (degree of initial saturation, Sakita et al. 2020).
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Relation between the effective montmorillonite density of Ca-type bentonite mixed soil and hydraulic conductivity (degree of initial saturation, Sakita et al. 2020).

Figure 8
Relation between the effective montmorillonite density of Ca-type bentonite mixed soil and hydraulic conductivity (degree of initial saturation, Sakita et al. 2020).
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Relation between the effective montmorillonite density of Ca-type bentonite mixed soil and hydraulic conductivity (degree of initial saturation, Sakita et al. 2020).

Close modal
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We proposed a model to demonstrate the relation between the hydraulic conductivity of bentonite and an effective montmorillonite density. Importantly, the model distinguishes between the interlayer pores of montmorillonite and the other pores (external pores), because their widths differ. We calculated the interlayer pore width from a proposed equation showing the relation between the basal spacing of montmorillonite and the effective montmorillonite density. The number of montmorillonite particle laminations in the stack was also significant as a variable required for the calculation of the external pore width. The proposed model was sufficient to estimate the hydraulic conductivity of bentonite and reproduced the difference between low-permeability and high-permeability at low-density conditions for Na- and Ca-bentonite, respectively. As the proposed model assumed a homogeneous distribution of montmorillonite particles in the specimen, however, care had to be taken to disallow application to Ca-bentonite when the saturation at compaction was low, as relatively large pores were generated due to the aggregation of the montmorillonite particles near the sand grains. The heterogeneous conditions were created by aggregation and the little swelling of Ca-montmorillonite, while the large swelling of Na-montmorillonite ensured the homogeneous condition and kept low hydraulic conductivity at low-density conditions without considering the effect of the initial degree of saturation.

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All relevant data are included in the paper or its Supplementary Information.

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The authors declare there is no conflict.

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