A dual-porosity model for the study of chemical effects on the swelling behaviour of MX-80 bentonite
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Abstract
Significant chemical influence on the swelling potential of MX-80 bentonite was observed during swelling tests where specimens were hydrated with highly concentrated brine. The maximum swelling pressure for specimens hydrated with brine was about 30% of the maximum swelling pressure for the same specimens hydrated with de-ionized water. The maximum swelling pressure was attained within tens of hours of brine infiltration and further decreased by half within a year. A fully coupled hydro–mechanical–chemical (HMC) dual-porosity model is proposed in this paper to interpret the swelling behaviour of MX-80 when infiltrated with brine. The dependence of hydraulic and mechanical properties on such factors as porosity, salinity and water content was investigated. A nonlinear elastic constitutive model was proposed to correlate the swelling pressure with the variation in the microporosity. The chemical effects on the mechanical behaviour were coupled at the micropore level. A number of relationships have been developed for MX-80, i.e. micropore permeability as a function of void ratio, water retention characteristics of micropores and macropores, micropore dependence on water content and the diffusion coefficients of the two types of pore structure. The proposed model was successful in reproducing both quantitatively and qualitatively the experimental results from two sets of infiltration experiments on compacted MX-80 bentonite.
Keywords
Bentonite Brine Coupled processes Dual porosity Salinity Swelling1 Introduction
Bentonite has been widely considered as seal materials in nuclear waste disposal facilities [1, 13, 15, 43, 44, 53]. Bentonite is primarily composed of montmorillonite minerals. Montmorillonite is characterized with high specific surface area, high swelling potential and a strong tendency to bound water. These characteristics could ensure a very low permeability and high sealing capacity of the bentonite when wetted. Deep geological disposal is being considered in Canada and many other countries for the long-term management of nuclear waste. Deep geological repositories (DGR) for nuclear waste rely on a system of engineered and natural barriers to contain and isolate the waste. Bentonite-based buffer and seal engineered barrier systems (EBS) are major components of this multi-barrier system. Once emplaced between the waste containers and the host rock, the swelling potential of the bentonite upon saturation with porewater from the host rock is relied on in order to seal gaps and fissures that may exist in the EBS itself and/or the surrounding host rock in the vicinity of the waste emplacement areas. The chemical characteristics of the pore fluid, i.e. pH, salinity, cation types, have been shown to affect the swelling behaviour of the bentonite [13, 21, 32, 49]. In Canada, the porewater in deep geological formations is characterized by high salinity. For example, the total dissolved solids (TDS) of the porewater at the Bruce site, Ontario, where a DGR for low- and intermediate-level wastes is being proposed, are up to 450 g/L at depths of more than five hundred metres. The highly concentrated brine in the host rock would seep into the EBS, triggering complex hydraulic (H)–mechanical (M)–chemical (C) processes that could affect the swelling potential of the bentonite. These HMC processes are coupled and are far more complex than the swelling behaviour of bentonite infiltrated with de-ionized (DI) water. Herbert et al. [18] found that brine with elevated salt concentration in contact with MX-80 bentonite can decrease its swelling pressure to a steady-state value that might take years to be reached. Swelling pressure tests were conducted at Queen’s University, Canada, in collaboration with the CNSC [42]. The results showed that the swelling pressure of MX-80 bentonite hydrated with DI water reached a plateau in a monotonous manner after approximately 1 month. On the other hand, for the same specimens hydrated with brine, the swelling pressure increased to a maximum value within tens of hours but reduced slowly to an equilibrium state after approximately 1 year. The final values of the swelling pressure of specimens hydrated with brine are approximately one order of magnitude smaller compared to the DI case. In order to interpret the observed chemical effects on the swelling behaviour of bentonite, the authors have developed in this paper a fully coupled HMC model that takes into account the dual-porosity structure of the bentonite. Although most of the assumptions used in this work might be applicable to other types of bentonite, the focus is on MX-80, a sodium bentonite.
Hydraulic–mechanical (HM)- or thermal–hydraulic–mechanical (THM)-coupled models incorporating dual-porosity structures have been developed and successfully validated against experimental observations in a number of pioneering studies [3, 12, 16, 17, 19, 37, 46, 47, 50, 55]. Momentum and mass conservation equations for each pore structure are required for the simulation of water flow and vapour transport processes [37]. Dual-porosity-based hydraulic flow and chemical transport models were reported with respect to unsaturated bentonite soils [31, 33, 41, 54]. Recently, a HMC-coupled dual-porosity model was developed by Musso et al. [34] to address the volume change of expansive soil under cycling of chemical permeation. This approach, although only addressing the fully saturated situation, provides an excellent framework to interpret the chemical influences on the swelling behaviour of bentonite [8]. In the present study, the authors developed a HMC-coupled model taking into account the dual-porosity structure of bentonite. The model was based on the dual-porosity HMC-coupled model from Musso et al. [33], but it was extended to address a full spectrum of hydraulic flow in both unsaturated and saturated states. Two types of pore structures were considered: the micropores within the clay aggregates and the macropores between those aggregates. The model considers porewater flow and solute transport in both pore structures and hydraulic and chemical exchange between the two structures through a semi-permeable membrane. This paper is structured as follows: (1) characterization of hydraulic and transport properties of bentonite; (2) derivation of governing HMC equations; (3) development of finite element (FE) models; and (4) mathematical simulation of swelling tests.
1.1 Characterization of hydraulic and transport properties of bentonite
1.1.1 Double-porosity in bentonite
When water is added to an unsaturated bentonite sample, it infiltrates first into the macropores and then into the micropores and nanopores, resulting in an expansion of the aggregate. This process is involved with a subdivision of the stacks, as well as an increase in the thickness of the diffuse double layer (DDL) [48]. The DDL is generally in the scale of tens of nm to a few micrometres. The proportion of macropores in the size range of 2–300 μm reduces significantly, while micropores in the size range of 0.01–2 μm increase in a wetting cycle with DI water. Manca et al. [29] studied the microstructural evolution of bentonite and noticed a steady and quasi-linear increase in micropore volume with decreasing suction in the wetting path. Under confined conditions, the tendency for the aggregate to expand results in the development of a swelling pressure.
The mathematical model that is described in this paper is based on the multiple porosity concept as previously discussed. We did not take into account the nanopore and only consider the micropore structure within an aggregate and the macropore structure between the aggregates. This simplification seems to be sufficient in order to explain the main phenomena associated with the swelling of compacted bentonite [17]. Microstructural morphological studies on bentonite under wetting paths indicate a gradual reduction of the macropores and steady cumulative growth in the abundance of micropores [29]. As shown in Fig. 2c, when full saturation occurs with DI water, the macropores would practically disappear. However, when the bentonite is saturated with brine, the reduction of macropores is much less pronounced with increasing salinity in the pore fluid, as demonstrated in Fig. 2a, b.
1.2 Hydraulic conductivity of a dual-porosity medium infiltrated with brine
We have \(\frac{\gamma }{\mu } = 7.6 \times 10^{6}\) 1/m s. The specific surface area S in unit volume of typical MX-80 can be calculated from specific surface area (SSA) value of 561 m^{2}/g [5], which leads to \(S = {\text{SSA}}*\gamma_{s} \times 10^{6} = 1.51 \times 10^{9}\) 1/m.
This C_{kc} value could be much larger than the well-recognized value of 5 for packed sandy soils as indicated by Carrier III [7]. This is likely due to the thick electric double layer of clay particles, that greatly hinders the transmissivity of compacted clay, through electric–hydraulic-coupled effects [25] and thus increases the C_{kc} factor. It is shown by the above analysis that the Kozeny–Carman equation is suitable for permeability estimation regarding the specific case of MX-80 bentonite hydrated with DI water.
Note that w^{m} is distinct from the total water content w and is governed by the osmotic suction of chemical solution.
1.3 Chemical transport properties: tortuosity and diffusion
1.4 WRC for MX-80 bentonite
Soil suction is defined as the energy required for extracting a unit volume of water from a soil in order to overcome retention mechanisms that exist in that soil [14]. For the current problem under study, three distinct water retention mechanisms are considered: water adsorption, capillary retention and osmosis. The total suction (S_{u}) is equal to the sum of matric suction (p_{m}) and osmotic suction (π), according to the definition of soil suction by the International Society of Soil Science and others [23, 52]. Water adsorption upon clay minerals at high suction levels and capillary retention at lower suction levels [10, 48] are responsible for matric suction. Water adsorption is mainly governed by the physicochemistry of clay minerals and takes place in the intra-aggregate micropores in terms of electric double layers between the clay stacks [48]. Experiments have shown that the total void ratio or dry density has no effect on the WRC of MX-80 bentonite at suction levels higher than 10 MPa, suggesting that the predominant mechanism of water retention is adsorption in that high suction range. The second mechanism of retention is mainly attributable to capillarity effects in the inter-aggregate macropores when the soil approaches a saturated state, e.g. suction becomes less than the air entry value (AEV). For the purpose of modelling water flow in MX-80 within a dual-porosity framework, we need relationships for the WRC in both types of pores that take into consideration the above three types of water retention mechanisms. These relationships are derived as follows.
1.4.1 Macropore WRC
- (1)
Adsorption and capillarity are the respective mechanisms for water retention in the micropore and macropore; osmosis is not considered;
- (2)
the AEV α could be expressed as a function of macropore void ratio e_{M}, \(\alpha = 0.2/e_{\text{M}}\) (MPa);
- (3)
the micropore void ratio (e_{m}) could be expressed as a function of water ratio (ratio of volume of water over solid volume) e_{w}, \(e_{\text{m}} = 0.48\left( {e_{\text{w}} } \right)^{2} + 0.1e_{\text{w}} + 0.31\);
- (4)
the water ratio is dependent on the gravimetric water content, e_{w} = G_{s}w.
1.4.2 Micropore WRC
1.5 Swelling pressure at various dry densities and salinities
1.6 HMC-coupled model for MX-80 bentonite
1.6.1 Key assumptions
- (a)
Dual-porosity structure
- (b)
Chemical-dependent WRC
- (c)
Water content-dependent microporosity
- (d)
Membrane effect
Semi-permeable membrane effect is prominent for bentonite in contact with saline water [21, 25, 27]. The membrane has a filtering effect: it delays movement of solutes with respect to the flow of water into the micropores.
1.7 Hydraulic flow equations
In the saturated state, p represents the pore pressure measurable by pore pressure transducer. The relationship between different suction and pore pressure parameters is as follows. Let S_{u} be the total suction, then S_{u} = p_{m} + π. The matric suction p_{m} = u_{a} − p, with u_{a} the air pressure, considered as atmospheric (0), and p the water pressure, then p = − p_{m}. Then, the net pore pressure P is the negative of total suction as P = − S_{u} = p − π
1.7.1 Dual-porosity flow equations
Based on Eq. (27) for a single-porosity medium, we can derive the flow equations for a dual-porosity medium as follows, where the superscripts m and M for a parameter, respectively, qualify that parameter as associated with the micropore or the macropore.
1.8 Solute transport equations
By neglecting the compressibility of solid matrix and pore fluid, the variation of porosity is assumed to be dependent only on volumetric deformation due to variation of pore volume.
According to Eq. (11), the apparent diffusion coefficient D_{a}/D_{0} = 0.05–0.075 when n = 0.41 falls exactly on the regressed line in Fig. 4. Therefore, the calibrated membrane coefficient reasonably reflects the constricting effect of the diffuse layer at the clay–water interface. It also suggests the viability of the above form of expression Eq. (44).
Our numerical simulation of brine–bentonite interaction leads to the following set of parameters, i.e. L_{c} = 10 μm, H = 0.004 (B_1.6 sample) for salinity C^{m} in the unit of molar/m^{3}.
1.9 Constitutive relationship for mechanical behaviour
In this study, the mass of 1 L model water was measured as 1223.1 g. The total dissolved salt (TDS) of the Queen’s University’s model water (MW) supernatant was 328.9 g/L. In this case, the mass of DI water (894.2 g) was calculated as 73% of the mass of 1 L of MW2 supernatant. Therefore, X = 0.73 is justifiable for saline water of \(\rho_{l}\) = 1223 kg/m^{3}. Then, \(e_{\text{m}} = \frac{{w\rho_{\text{s}} }}{{X\rho_{l} }} = \frac{2.75}{0.73*1.223}w = 3.08\;{\text{w}}\).
This parameter can be a viable measure to evaluate and validate the assumption of the micropore constitutive equation. Manca et al. [29] conducted MIP test to determine the macropore (r > 1 μm) void ratio for bentonite under permeation of 4 M saline at e^{M} = 0.35 for e_{0} = 0.78, corresponding to the portion of microporosity \(f_{\text{m}} = 0.55\). The osmotic suction of 4 M NaCl solution is reported as \(\pi = 22.5 {\text{MPa}}\) [26]. This leads to a calculated w = 0.16 from the WRC (Fig. 6). Then the micropore void ratio is given as e_{m} = 2.9*0.16 = 0.464, where the ratio 2.9 is the average of 2.75 and 3.08 for DI and brine solution, respectively. This corresponds to the portion of microporosity f_{m} = 0.59 and is close to the experimental results of f_{m} = 0.55 (7% variance error). Therefore, the proposed model is an acceptable approximate that can correlate the microporosity with water content.
Using this formula, we can analyse the double-porosity proportion by the water content and suction level for any bentonite samples. For instance, the initial condition of the as-compacted bentonite specimen (by Queen’s University) is w_{0} = 0.11, \(\rho_{\text{d}}\) = 1.6 g/cm^{3}, which indicates \(e_{\text{m}} = 0.3\). and \(f_{\text{m}} = 0.42\). When saturated with brine, the osmotic suction level of 40 MPa corresponds to a gravimetric water content of 0.15. At a dry density of 1 g/cm^{3}, this results in a micropore void ratio \(e_{\text{m}} = 3.08\;{\text{w}}\) and the portion of microporosity \(f_{\text{m}} = 0.643\), which is much less than 1.0, suggesting a significant portion of macropore even at full saturation.
1.10 Finite element model for constant volume swelling test
Model input parameters
Variable | Unit | Value/expression | |
---|---|---|---|
Validation A (Posiva data) | Validation B (Queen’s U data) | ||
Initial conditions | |||
G _{s} | 2.75 | 2.75 | |
ρ _{d} | 1.7 | 1.6 | |
e _{0} | G_{s}/1.7–1 | G_{s}/1.6–1 | |
w _{0} | 0.05 | 0.11 | |
e _{m} | Equation (6) | Equation (6) | |
Saturated permeability | |||
\(k_{\text{s}}^{\text{M}}\) | m/s | 4*k^{m} | |
\(k_{\text{s}}^{\text{M}}\) | m/s | 2*Equation (5) | Equation (5) |
Relative permeability | |||
\(k_{\text{r}}^{\text{m}}\) | S _{e} ^{3} | 1 | |
\(k_{\text{r}}^{\text{m}}\) | (0.8)^{6} | 1 | |
Chemo-osmosis | |||
ω | – | 0.97–0.98 | |
k _{c} | Pa m^{3}/mol | – | 8.83E3 |
Mass exchange | |||
\(\bar{\alpha }\) | m/Pa s | 2* k^{m}*ρ_{l} | 10*k^{m}*ρ_{l} |
D _{e} | m^{2}/s | – | Equation (12) |
L _{c} | m | – | 1E−5 |
\(\bar{D}\) | m/s | – | Equation (50) |
H | m^{3}/mol | – | 0.004–0.005 |
\(Q^{\text{M}}\) | m/s | – | Equation (45) |
Solute transport | |||
D _{0} | m^{2}/s | – | 1E–9 |
τ ^{m} | – | – | D_{0}/Eq. (12) |
τ ^{M} | – | – | 1 |
Water retention curve | |||
\(S_{\text{u}}^{\text{m}}\) | Pa | From Eq. (14) | From Eq. (14) |
\(S_{\text{ut}}\) | Pa | 2800 | Equation (15) |
\(S_{\text{u}}^{\text{M}}\) | Pa | From Eq. (13) | From Eq. (13) |
Mechanical property | |||
K | MPa | 36.4 | 19.3 (DI) 19.3 (MW) |
Boundary condition | |||
\({\mathbb{L}}\) | m/s | 0.5E7*k^{M} | 0.5E7*k^{M} |
P _{in} | MPa | 2 | 0.015 |
C _{in} | M | 0 | 4.6 |
Dimension | |||
Radius | mm | 25 | 19 |
Height | mm | 65 | 12 |
1.11 Simulation of Posiva test
Due to the very low salinity of the permeant, the modelling ignored the effect of pore fluid chemistry and considered mainly the interaction between macropores and micropores. As shown in Table 1, most of the model parameters are the same as what we derived theoretically for MX-80 bentonite. These include the micropore void ratio and the water retention curves for both the micropore and macropore. Only minor adjustments were made with respect to the permeability and unsaturated state variables
1.12 Simulation of Queen’s University tests
1.12.1 Experimental results of temporal variation of swelling pressure
Experimental conditions for Queen’s University dataset
Sample ID | Soil composition | Dry density (g/cm^{3}) | Permeant |
---|---|---|---|
B_1.61_DI | Bentonite MX-80 | 1.61 | DI water |
B_1.60_MW | Bentonite MX-80 | 1.60 | Model water (brine) |
B_1.42_MW | Bentonite MX-80 | 1.42 | Model water (brine) |
SB_1.8_MW | Bentonite 70% Sand 30% | 1.80 | Model water (brine) |
Pore fluid chemistry has a significant impact on the swelling of bentonite as shown in Fig. 15. With the same dry density, DI water infiltration results in a peak swelling pressure about three times higher than the MW case. Within 500 h, no obvious variation in swelling pressure for the case of DI water can be noticed from the test data. After 500 h, the swelling pressure gradually decreases and reached a stable value by 2000 h for DI water, while for MW, the swelling pressure took much longer to stabilize after 8000 h of continuous permeation. The swelling pressure for DI water decreases by approximately 10% from the peak, whereas for MW, a decrease of 80–90% from the peak was observed. Sand–bentonite mixtures (SB) show a very similar swelling behaviour with bentonite with the same value in EMDD. Our modelling efforts will focus on reproducing the gradual decrease in swelling pressure under hydration of brine water.
1.12.2 Modelled temporal variation of swelling pressure
Figure 18 gives a brief illustration of the overall HMC-coupled processes occurring in MX-80 bentonite under permeation of brine. It is shown that the micropore water content w^{m} varies significantly with time at different locations. At 100 h, the modelled w^{m} reaches a peak value at regions close to the outlet boundary, where e^{M} remains in a low level before the chemical front passes that point. In the beginning, the macropore becomes saturated quickly, while chemical transport therein is delayed compared to the water movement due to chemical exchange with the micropore. In regions further away from the inlet, water would be preferentially absorbed into the micropore because of the lack of osmotic suction gradient. The swelling pressure peak may be correlated with the chemical front in the macropore. The extremely concentrated brine poses a very high osmotic suction on the clay aggregate that draws water from the micropore, causing a decrease of the microporosity and then a decline of the swelling pressure. This observation corresponds well to the conceptual model and seems well justified.
2 Conclusion
In this study, a HMC-coupled model was developed for compacted unsaturated bentonite. The model is based on a dual-porosity framework and considers the effect of salinity on various hydraulic and mechanical properties. A series of relationships for HMC properties have been proposed for MX-80 bentonite. Effective approaches to estimate the WRC and hydraulic conductivity for each porosity component were proposed and verified. The coupled HMC model was used to simulate laboratory swelling experiments on compacted MX-80 bentonite specimens infiltrated with DI water and brine, respectively. A good agreement between the model results and the experimental data, both in trends and absolute values, suggest that the main processes have been captured.
Many of the parameters representive of the hydraulic–chemical characteristics of the bentonite are derived by best-fit correlations of available data from the literature combined with the ones determined from laboratory tests at Queen’s University. Despite the variability of the data sources, these empirical correlations when used as input to the proposed model are able to reproduce the main processes found from swelling tests performed at two different laboratories. Further in-depth microporosimetric studies, as part of CNSC’s ongoing regulatory research, are being conducted at Queen’s University in order to verify, calibrate and further refine the current model and improve our understanding of the effects of salinity on the swelling potential of bentonite.
Notes
References
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