SEARCH

SEARCH BY CITATION

Keywords:

  • water retention;
  • capillary water;
  • bound water;
  • van Genuchten model;
  • clay

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

[1] We provide a unified model for the soil-water retention function, including the effect of bound and capillary waters for all types of soils, including clayey media. The model combines a CEC-normalized isotherm describing the sorption of the bound water (and the filling of the trapped porosity) and the van Genuchten model to describe the capillary water sorption retention but ignore capillary condensation. For the CEC-normalized isotherm, we tested both the BET and Freundlich isotherms, and we found that the Freundlich is more suitable than the BET isotherm in fitting the data. It is also easier to combine the Freundlich isotherm with the van Genuchten model. The new model accounts for (1) the different types of clay minerals, (2) the different types of ions sorbed in the Stern layer and on the basal planes of 2:1 clays, and (3) the pore size distribution. The model is validated with different data sets, including mixtures of kaolinite and bentonite. The model parameters include two exponents (the pore size exponent of the van Genuchten model and the exponent of the Freundlich isotherm), the capillary entry pressure, and two critical water contents. The first critical water content is the water content at saturation (porosity), and the second is the maximum water content associated with adsorption forces, including the trapped nonbound water.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

[2] In clayey soils and rocks, there are two types of water, bound and capillary. Some of the water molecules correspond to hygroscopic or hydration water covering the mineral surface (bound water). The second type of water is the capillary (free) water in the pores, which can include drainable and nondrainable waters. The distinction between these two waters can be made on the basis of the main force acting on the water molecule (capillary forces versus adsorption forces). Many attempts have been made to develop a consistent soil-water retention curve working at a broad range of water saturations, from full saturation to oven dryness, including properties like water condensation [see Ross et al., 1991; Rossi and Nimmo, 1994; Tuller et al., 1999; Or and Tuller, 1999; Tuller and Or, 2003, 2005; and recently Lebeau and Konrad, 2010]. Based on film water in triangular pore geometries, some recent works [Tuller et al., 1999; Or and Tuller, 1999; Tuller and Or, 2003, 2005] have realistically described capillary condensation mechanisms at very dry or high matric suction conditions. One of the remaining challenges is to accurately describe bound water in clay-rich materials and to combine an adequate isotherm for the bound water with a capillary water retention curve.

[3] Recently, Woodruff and Revil [2011] have introduced a normalized clay-water isotherm model based on Brunauer, Emmett, and Teller (BET) theory describing the sorption and desorption of bound water in mixtures of clays, sand-clay mixtures, and shales. We will see, however, that the BET isotherm is not suitable for use with a capillary water retention curve because it does not predict a finite amount of bound water at a very high relative humidity. Calvet and Prost [1971] observed that the amount of adsorbed water decreases as the quantity of exchangeable cations decreases and they suggested that at low relative humidity, the sorption of the water molecules are mostly associated with the sorbed cations [see also Laird, 1999, and Montes-Hernandez et al., 2003]. This implies in turn that the influence of the cations sorbed on the mineral surface needs to be accounted for in such models.

[4] Following Krushin [2005] and Woodruff and Revil [2011] normalized the clay-water sorption isotherms (sorption and desorption) of clayey materials by their cation exchange capacity (CEC), accounting for a correction factor depending on the type of counterion sorbed on the mineral surface. With such a normalization, all the data collapse into two master curves, one for sorption and one for desorption. The model of Woodruff and Revil [2011] was, however, not considering the capillary water.

[5] We propose below an extension of the model of Woodruff and Revil [2011] to include the capillary water by combining the CEC-normalized isotherm with the van Genuchten model [van Genuchten, 1980] for capillary water to describe the complete water retention curve from saturation to oven dryness. Our model accounts for the pore size distribution, the clay mineralogy, and the type of counterions that are present on the mineral surface. Such an approach also follows a similar line of thinking as the modeling effort described recently in Lebeau and Konrad [2010] by combining bound and capillary water into a single closed-form mathematical expression [see also Tuller et al., 1999; Or and Tuller, 1999; Tuller and Or, 2003, 2005]. Rather than using the normalized BET isotherm proposed by Woodruff and Revil [2011], we show that the empirical Freundlich isotherm is more suited to be combined with the van Genuchten retention model.

2. Three Types of Water

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

[6] In a water-saturated porous material, there are actually three types of pore water (see Figure 1): (1) a bound water content, inline image corresponding to the sorbed (bound) pore water (in red in Figure 1); (2) the capillary water content, inline image (in blue in Figure 1a) that can be drained; and (3) the water content that is not bound and not drainable, inline image (in yellow in Figure 1a). So the water content, inline image, can be written as

  • display math(1)
image

Figure 1. Three types of pore waters. (a) In a water-saturated porous material, the connected porosity comprises a bound water porosity, a drainable porosity, and a nondrainable and nonbound liquid water porosity. (b) Below the residual water saturation, there is a regime controlled by film flow from the pockets of residual nonbound water molecules in order to compensate for the desorption of the bound water molecules from the mineral surface.

Download figure to PowerPoint

[7] We assume from now the sum of the bound water plus the trapped water is equal to inline image, the water content ascribed to the adsorption force. Therefore,

  • display math(2)

where inline image denotes the suction potential.

[8] Assuming that the sorbed water corresponds strictly speaking to a maximum of p-layers of water molecules covering the surface of the minerals (assuming there are no hydrophobic minerals), we have,

  • display math(3)

where S denotes the surface area between the solid phase and the connected pore space, V is the total volume of the porous material, d denotes the thickness of a water molecule (0.29 nm), inline image represents the specific surface area (surface per unit mass of grains), and inline image denotes the mass density of the grains (typically 2650 kg m−3). The value of p can be determined though an analysis comparing the strength of the capillary force with the strength of the adsorption force (p is likely to be between 2 and 4, as discussed below). Some simple calculations for bentonite show that half of the pore water can be bound water. Consequently, in clayey materials, due to the strong value of the specific surface area, the volumetric fraction of bound water can be very substantial.

[9] Three points will need to be discussed more, later in this paper:

[10] 1. The idea that the surface is uniformly covered by layers of water molecules is a simplification of reality in which an important fraction of the water molecules can be partly clustered around the cations sorbed in the Stern layer [see Woodruff and Revil, 2011, Figure 2b]. Our model will account explicitly for the amount of bound water sorbed on the counterions sorbed on the mineral surface.

image

Figure 2. Hydration of the clay mineral surface. (a) Specific surface area as a function of CEC (in meq g−1 with 1 meq g−1 = 96,320 C kg−1 in SI units). The two lines correspond to 1 to 3 elementary charges per unit surface area. Modified from Revil et al. [2013]. (b) Hydrated cations like sodium are sorbed on the surface of the clay minerals (both the basal surfaces and edges). Such sorption partly controlled the amount of sorbed water molecules on the surface of the mineral.

Download figure to PowerPoint

[11] 2. The first two layers of bound water molecules have distinct properties (dielectric constant and viscosity) that are different from the bulk water. However, the third layer of water molecules is known to have properties that are similar to the bulk pore water [Israelashvili, 1992].

[12] 3. The capillary force dominates the adsorption forces only at very high relative humidities (>0.98). This implies that the bound water, from a force standpoint, includes more than two layers of water molecules.

[13] The situation for saturations below the irreducible water saturation is depicted in Figure 1b. The pockets of residual water (bound water excluded) would serve as source terms to generate flow along the surface of the minerals where the bound water molecules get desorbed when the relative humidity becomes lower and lower. If the kinetics of desorption is slow enough, the desaturation occurs first by emptying the pockets of residual (nonbound) water. When these pockets are totally empty, desorption occurs from the layer of remaining bound water molecules. During sorption, at the opposite, there is no physical mechanism to have film flow from the mineral surface to these pockets. Therefore the saturation/desaturation curves are expected to exhibit a hysteresis. We do not consider, in this model development, any spontaneous pore filling or emptying mechanisms (i.e., pore snap off) [see Tuller et al., 1999; Or and Tuller, 1999; Tuller and Or, 2003, 2005; Haas and Revil, 2009].

3. Water Sorption Desorption Isotherms

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

3.1. Extended van Genuchten Model

[14] To continue, we first need a model to describe the capillary pressure curve. We chose, in the following, the van Genuchten capillary pressure model described by the following function for the total water content θ [van Genuchten, 1980]:

  • display math(4)

(constrained form of van Genuchten's model) where inline image was originally described as the residual water content. In the current model, inline image corresponds to the volumetric water content ascribed to adsorptive forces, considering also the water molecules in the trapped porosity. The water content θs = ϕ denotes the water content at saturation (porosity), 1/α denotes the capillary entry pressure, and ψ represents the suction pressure (in m), which is defined as negative in soil physics. The exponent, n, called the pore size exponent, is a structural parameter used to characterize the effect of the pore size distribution (similar to the pore size index, λ, in the Brooks and Corey model). Therefore, equation (2) can be written as

  • display math(5)

[15] We choose the van Genuchten retention model, because according to Nimmo [1991] and Ross et al. [1991], among many other works, the van Genuchten model successfully describes the water retention curve at high and medium values of the water content. However, it may fail to describe the water retention curve at low values of water content. In our model, the low values of water content are partially explained by the sorption-desorption of the bound water, and the redistribution of the water molecules from the trapped porosity by film flow, as shown in Figure 1b. Other models could be used as well to describe the capillary pressure curve, for example the model developed by Assouline et al. [1998].

[16] The suction pressure inline image (in Pa) (here defined as a positive value) can be related to the relative humidity, hr, by the Kelvin equation [e.g., Or and Wraight, 2000; Lu and Likos, 2004],

  • display math(6)

where inline image (m3 mol−1) is the molar volume of water (1.8016 × 10−5 m3 Mol−1), R = 8.31432 J K−1 mol−1 is the universal gas constant, T is the absolute temperature (in Kelvin), and inline image ≈ 137 × 106 Pa at 25°C. Connecting the two potentials, as done in equation (6), should only be considered as an approximation. Indeed, the adsorption isotherm we used describes adsorption to flat surfaces. In contrast, the Kelvin equation describes capillary condensation corresponding to the filling of pores at a given relative humidity due to the effect of water surface curvature on potential.

[17] Note that in this paper, we will not account for the effect of suction on the porosity change. For high porosity materials, we expect, however, that suction would change the effective stress of the material (see Revil and Mahardika [2013] for a general theory) and therefore the porosity, which could be model through a constitutive equation between the porosity and the effective stress [e.g., Revil et al., 2002].

3.2. The CEC Normalized Isotherm Equations

[18] The second contribution associated with the bound water is determined using the approach developed by Woodruff and Revil [2011]. The relationship between the water content (volume of water per volume of material) and the gravimetric water content, inline image, ascribed to adsorptive forces (mass of water ascribed to adsorptive forces per unit mass of solid) is given by

  • display math(7)

where inline image and inline image denote the mass densities of the water and solid phases, respectively. Using equations (6) and (7), the data can be either displayed in ( inline image, hr), ( inline image ψ), (θ, hr), or (θ, ψ) diagrams.

[19] In the approach of Woodruff and Revil [2011], the relationship between the gravimetric water content and the relative humidity is given by the BET isotherm [Brunauer et al., 1938],

  • display math(8)

where inline image represents the gravimetric water content (mass of water per unit mass of grains) when the first monolayer is fully saturated and C denotes the BET constant, which is related to the enthalpy of sorption for the first molecular layer of water. The BET isotherm can be rewritten explicitly as a function between the gravimetric water content and the relative humidity:

  • display math(9)

[20] To avoid having an exceedingly high sorption water content at high humidity (which creates a problem when we want to combine the isotherm with a capillary retention curve at high water content), we propose to use the Freundlich isotherm as an alternative to the BET equation [Freundlich, 1909]:

  • display math(10)

where k and m (both dimensionless) are the Freundlich adsorption isotherm coefficients corresponding to the adsorption capacity and the adsorption strength, respectively.

[21] Woodruff and Revil [2011] normalized the BET isotherm by the CEC (expressed in meq g−1 or C kg−1 with 1 meq g−1 = 96,320 C kg−1 and 1 meq g−1 = 1 Mol kg−1), which represents the number of active sites of the mineral surface per unit mass of grains. The CEC of various clay minerals is shown in Figure 2a, and the sorption of hydrated cations in Figure 2b. The CEC is grossly proportional to the specific surface area. The ratio between the CEC and the specific surface area corresponds to the charge per unit surface area of the clay minerals. The active sites on the mineral surface correspond to the silanol and aluminol surface groups at the edges of the clay crystals or to the basal planes where isomorphic substitutions exist in the crystalline framework (especially for smectite). The idea to scale the water sorption/desorption isotherms by the CEC is based on the observation that the scaled isotherms, inline image, fall under two trends (one for sorption and one for desorption), whatever the type of clay (see Figure 3, Krushin [2005], and Woodruff and Revil [2011]). Normalizing the isotherms by the CEC yields the following scaled BET and Freundlich isotherm (see Appendix A):

  • display math(11)
  • display math(12)

where we have used the following normalized parameters inline image, inline image for the BET equation, and inline image for the Freundlich isotherm. The units of inline image and inline image are both in C kg−1 (or meq g−1 with 1 meq g−1 = 96,320 C kg−1 in Si units).

image

Figure 3. Comparison between the water desorption isotherm for smectite and the Georgia kaolinite, which is characterized by an unusually high CEC for a kaolinite. (a) Unscaled isotherms. (b) Scaled isotherms using the cation exchange capacity. The first layer is believed to saturate the surface of the clay minerals at a relative humidity of 0.47 and the second layer is formed at a relative humidity of 0.65. (c and d) Same work with the data from Hatch et al. [2012]. The lines are just guides for the eyes. The discrepancy between the normalized curves above 70% relative humidity could be due to water condensation.

Download figure to PowerPoint

[22] As an example, we show in Figure 4 how the normalized isotherms inline image for illite and kaolinite can be fitted with the BET and Freundlich normalized isotherms, equations (11) and (12). For the BET isotherm, the best fit of the data yields inline image = 4.2 × 105 C kg−1 and inline image = 23.3 × 10−7 kg C−1. We will see later that such a high value for inline image (equivalent to 12 water molecules per site for the first hydration layer) may be due to the presence of Ca2+, which is indeed observed by Hatch et al. [2012]. The best fit of the data with the Freundlich isotherm yields inline image = 1.6 × 105 C kg−1 (1.7 meq g−1) and m = 2.3 ± 0.2. Note that the Freundlich isotherm provides a substantially better fit of the data for illite and kaolinite by comparison with the BET sorption isotherm (compare Figures 4a and 4b).

image

Figure 4. Comparison between the normalized BET and Freundlich sorption isotherms for illite and kaolinites. (a) Fit of the normalized data (normalization is done with the CEC) using the normalized BET sorption isotherm (data from Hatch et al. [2012]). The best fit of the data yields inline image = 4.2 × 105 C kg−1 and inline image = 23.3 × 10−7 kg C−1. (b) Fit of the same data with the Freundlich normalized sorption isotherm. The best fit yields inline image = 1.6 × 105 C kg−1 (1.7 meq g−1) and m = 2.3 ± 0.2. Note that the Freundlich isotherm fits the data better than the BET isotherm.

Download figure to PowerPoint

[23] In Figure 5, we use the data of Figure 4, but we represent them in terms of the water content and capillary pressure in a (ψ, θ) diagram. We see very clearly that the Freundlich isotherm implies that the water content reaches an asymptotic limit at low capillary pressures. The departure from the trends for a capillary pressure below 50 MPa may indicate the effect of water condensation in the porous material.

image

Figure 5. Representation of the volumetric content of the adsorbed water as a function of the capillary pressure (data from Hatch et al. [2012]). For a capillary pressure smaller than 50 MPa, the departure from the trend predicted by the Freundlich isotherm (the plain and dashed lines) denotes the effect of the capillary water. We use the same parameters as in Figure 4b.

Download figure to PowerPoint

[24] Using equations (7), (11), and (12), we write the two isotherms for the water content:

  • display math(13)
  • display math(14)

[25] The charge per unit pore volume [e.g., Revil, 2012], defined by

  • display math(15)

which corresponds to a parameter that is used in a lot of petrophysical models of rock properties (see recently Revil [2012, 2013] for the complex conductivity of siliciclastic sediments, and Revil and Mahardika [2013] for their seismoelectric properties). Equations (13) and (14) can also be rearranged to get the water saturation inline image ascribed to adsorption forces (with inline image the connected porosity):

  • display math(16)
  • display math(17)

[26] In all cases (gravimetric water content, water content, saturation), the isotherms can be written as

  • display math(18)
  • display math(19)
  • display math(20)

where inline image denotes the scaled isotherm. This function does not depend on the cation exchange capacity, but just on the type of sorption mechanism on the surface of the mineral. We will see that different types of clay minerals may be characterized by different functions, inline image. This is probably because for illite and kaolinite, most of the sorption occurs on the crystalline edges, while for smectite, most of the sorption takes place on the basal planes. The normalized isotherms are given by

  • display math(21)
  • display math(22)

where inline image. The scaled isotherm is characterized by a single parameter (b or m).

[27] The first generalization is to extend the previous model to a set of different mineral phases. We will use the subscript/superscript j to describe the mineral phase j and we will assume that the water sorption on silica and carbonate minerals can be neglected, because their surface area would, in general, be much smaller than the surface area associated with the clay minerals. A generalization to a polymineralic soil or clay mixture is

  • display math(23)
  • display math(24)
  • display math(25)

where the sum is extended to the N mineral phases (typically, kaolinite, illite, and smectite), inline imageis the mass fraction of mineral j in the soil, inline image denotes the cation exchange capacity of phase j, inline image will be either given by the BET normalized scaled isotherms or the Freundlich scaled isotherm, and inline image is a constant (in isothermal conditions). According to Caroll [1959], the CEC of kaolinite is the range 0.03–0.10 meq g−1, the CEC of chlorite and illite are in the range 0.10–0.40 meq g−1, the CEC of smectite is in the range 0.70–1.5 meq g−1, and the CEC of vermiculite is in the range 1.0–2.0 meq g−1 (see Table 1).

Table 1. Sorption Properties of the Different Clay Minerals Using the Freundlich Isotherm
Clay jCECj (meq g−1) inline image(meq g−1)m (–)
Kaolinite0.03–0.101.62
Illite0.10–0.401.62
Smectite0.70–1.52.81
Vermiculite1.0–2.02.81

[28] Two other effects need to be accounted for. The first is related to the type of cations populating the surface of the clay minerals, while the second is related to the difference between sorption and desorption.

3.3. Type of Cation and Hysteresis

[29] According to Laird [1999], the water molecules are mostly located around the cation sites located on the mineral surface (see Figure 2b). This explains why the cation exchange capacity, rather than the specific surface area, is a better parameter to normalize water sorption isotherms and therefore why the mass fraction rather than the surface fraction is a better way to scale the isotherms for a polymineralic assemblage. Because the cation exchange capacity depends on the type of ion used, we need to find a way to scale the cation exchange capacity with the type of ions. Woodruff and Revil [2011] proposed that the measured cation exchange capacity for a cationic species i and a mineral phase j, is scaled to the absolute cation exchange capacity of the phase, CECj (measured using a cation that has a very strong affinity with the clay surface like ammonium or cobalt), according to

  • display math(26)

where inline image (therefore inline image) denotes a scaling parameter that is dependent on the type of cation populating the mineral surface. Woodruff and Revil [2011] found δ(Li) = 0.4, δ(Na) = 0.5, δ(Mg) = 0.6, and δ(Ca) = 0.8. We will provide some different values later on. In summary, the CECj in equations (23) and (24) should be replaced by the scaled cation exchange capacity inline image. Then the two parameters entering the sorption isotherm need to be scaled with the cation-dependent effective cation exchange capacity.

[30] The second point is related to the observation that water sorption in clayey soils exhibits hysteresis. Hysteresis can be due to a number of mechanisms. One of them is likely that the mechanisms described in Figure 1b are not reversible. Indeed, film flow from the trapped water pockets occurs during desaturation but not during saturation.

[31] If the two previous effects are accounted for, we obtain the following hysteretic function:

  • display math(27)
  • display math(28)

where bw denotes a parameter associated with the amount of water trapped in the pore space during desorption (bw is on the order of 100 mg of water per gram of dry clay according to Woodruff and Revil [2011]) and inline image denotes the value of inline image for sorption.

[32] We now need to find the characteristics of the adsorption for the different clay minerals. For the sorption of a monolayer, a bilayer, or three layers of water molecules on the mineral surface, we have

  • display math(29a)
  • display math(29b)
  • display math(29c)
  • display math(30)

for the BET and Freundlich isotherms, respectively, inline image, and p represents the number of hydration layers. Newman [1983] found that a complete monolayer of water is obtained at a mean relative humidity of 0.47, using a database of 62 smectitic soils from the UK. Taking inline image yields C = 1.27. Using this value, the relative humidity for the saturation of the second layer is inline image = 0.65, in agreement with Figure 3.

[33] For kaolinite and illite, we come back to Figure 4a. The parameter C is determined by inline image = 50. Therefore, the BET isotherm yields inline image = 0.12 (monolayer), and inline image 0.51 (bilayer). If we use the Freundlich isotherm instead, and if we use inline image = 23.3 × 10−7 kg C−1, and inline image = 1.6 × 105 C kg−1 (see Figure 4), we obtain inline image = 0.10 (monolayer), and inline image 0.51 (bilayer).

[34] As discussed below in section 6, the capillary force dominates the sorption forces in clayey materials at very high relative humidity (around 0.98–0.99). Using the Freundlich isotherm with m =2.3, inline image = 23.3 × 10−7 kg C−1, and inline image = 1.6 × 105 C kg−1, such a high relative humidity corresponds to two to three layers of water molecules. At the opposite, the BET isotherm (with C = 50) predicts that the third layer is formed at a low relative humidity of 0.74.

[35] The normalized parameter inline image is defined by the mass of water per unit mass of grains divided by the CEC, which is itself the charge per unit mass of grains. Consequently, inline image is defined as the mass of water ( inline image) per unit charge on the surface of the mineral ( inline image), and is given by

  • display math(31)

where e and N are the elementary charge (1.6 × 10−19 C) and the Avogadro number (6.02 × 1023 Mol−1), respectively, inline image denotes the number of water molecule per site (for a monolayer), Zi is the valence of i, NS represents the number of sites on the mineral surface, and Mw is the molar mass of water in kg Mol−1 (18.02 × 10−3 kg Mol−1). Equation (31) simplifies to

  • display math(32)

[36] For sodium (Ζ(Na) = 1), Michot et al. [2005] found 4 ± 1 water molecules per Na+ site (for the first layer) for four synthetic Na saponites (a trioctahedral mineral belonging to the smectite group). Using inline image(Na+) = 4 yields inline image = 7.5 × 10−7 kg C−1. With C = 1.27, we obtain inline image = 2.8 × 105 C kg−1 for sorption (2.9 meq g−1). These values can be compared in turn to the values given by Woodruff and Revil [2011]: inline image = 0.08 g meq−1 (8.3 × 10−7 kg C−1) and inline image = 6.9 meq g−1 (6.6 × 106 C kg−1) using 1 meq g−1 = 96,320 C kg−1. Note that for smectites, like Ca-vermiculite, Ca2+ in the interlayer porosity is characterized by 8 water molecules in total (n(Ca2+) = 8), and therefore forms a Ca(H2O)82+ solvation complex, consistent with X-ray diffraction studies [Sposito et al., 1999].

3.4. Unified Model

[37] In summary, our generalized isotherm, inline image, takes the form given by equations (2), (5), and (24). The unified capillary and sorption isotherms are defined by six parameters: (1) the porosity inline image (water content at saturation), (2) the CEC, (3) the type of cation defining inline image, (4) the BET or Freunlich isotherm coefficient (b or m), and finally, (5) and (6) two parameters of the van Genuchten model (α and n). The value of inline image is kept constant. The chart to determine the water sorption isotherm from the material properties and the environmental conditions is shown in Figure 6. One remaining issue is if we can combine either the BET or Freundlich sorption isotherms with the van Genuchten model. It will be shown that the BET isotherm leads to unrealistic water contents at high relative humidity. Therefore, in general, it cannot be combined easily with the van Genuchten model. In the following, we use a combination of the Freundlich and van Genuchten models and we will show that this leads to a reasonable unified water retention model.

image

Figure 6. Forward and inverse modeling chart. This chart shows how the material properties and the environmental variables (relative humidity and temperature) are used to determine the water sorption isotherm. Measuring a sorption isotherm and knowing the environmental conditions, one can retrieve the properties of the material. The nonlinear optimization is done with the least square criterion using the Gauss-Newton method.

Download figure to PowerPoint

4. Comparison With Experimental Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

4.1. End-Member Clay Minerals

[38] First, we test our model on bentonite and kaolinite before we test the model on a mixture of the two materials. Figure 7 shows the fit of the sorption data for bentonite and kaolinite. The Georgia kaolinite has an unusually high CEC for a kaolinite. It is in the range 0.20–0.25 meq g−1, according to the laboratory measurements made by Cason and Reed [1977]. BET isotherm represents the bentonite data well, while the Freundlich isotherm better fits kaolinite data; however, the BET isotherm also fits the kaolinite data as well (not shown here). In Figure 8, we fit the van Genuchten model to the experimental data for the bentonite and kaolinite. The good fit between the van Genuchten model and the data indicates that the van Genuchten provides a good capillary retention model for these clays.

image

Figure 7. Sorption isotherms for bentonite and kaolinite. (a) Sorption isotherm of a bentonite (porosity 0.82) using the BET isotherm. (b) Sorption isotherm for a kaolinite (porosity 0.58) using the Freundlich isotherm. We obtain m = 1.20 ± 0.05, so very close to a linear adsorption isotherm. We use inline image = 2.8 × 105 C kg−1 for bentonite and inline image = 1.6×105 C kg−1 for kaolinite (see Table 1). The uncertainty on the model parameters is determined from the residual normal distributions approximated by the Student's t distribution and using two standard deviations (95% confidence interval).

Download figure to PowerPoint

image

Figure 8. Capillary pressure curve for bentonite and the Georgia kaolinite. (a) Fit of the backfill data from Engelhardt et al. [2003] using the van Genuchten model (the backfill is composed of 30% in weight sodium bentonite from Wyoming and 70% crushed diorite, porosity 0.48). (b) Fit of the Georgia kaolinite data from Likos and Lu [2003] (filter paper and humidity chamber) and Lu and Kaya [2013] (DC method) using the van Genuchten model (porosity 0.58). See Figure 7 for the confidence intervals.

Download figure to PowerPoint

4.2. Complete Sorption Isotherms for the End-Members

[39] As mentioned above, the BET sorption isotherm leads to unrealistic water contents at high relative humidity (Figure 4a). The unified isotherm is therefore based on combining the Freundlich sorption isotherm with the van Genuchten capillary model. This combination yields the following equation for sorption:

  • display math(33)

where,

  • display math(34)
  • display math(35)

[40] The water content, inline image, denotes the maximum water content ascribed to adsorption forces and is defined by

  • display math(36)

or when inline image. Such limit is well defined for the Freundlich isotherm, but not for the BET isotherm as mentioned above. Equation (33) ensures a continuous function of the water content with the suction as well as its derivatives, as discussed by Lebeau and Konrad [2010] and shown in Figure 9. In its simplest form, the model can be fitted to experimental data with five parameters. This model includes two critical water content parameters ( inline image, the porosity) and the maximum water content ascribed to adsorption forces but including also the nonbound and nondrainable water ( inline image), the capillary entry pressure (1/α), the pore size parameter n, and the sorption parameter m.

image

Figure 9. Unified capillary pressure curve combining the capillary pressure and the Freundlich sorption isotherm. Values of the model parameters: n = 1.9, m = 0.3, inline image = 0.6, inline image = 0.22, and α = 10−5 Pa−1. In our model, the capillary pressure curve is a composite of two curves, one describing the behavior of the capillary water and one describing the behavior the adsorbed water, including the effect of the trapped water.

Download figure to PowerPoint

[41] In Figures 10 and 11, we use the data from Likos and Lu [2001] for smectite and kaolinite, respectively. These data are fitted with equations (33) and (34). The optimization is done with the least square method without regularization. The unknown parameters are α, m, n, inline image, and inline image. The water content, inline image, is used to compute the CEC using equation (35). In this exercise, m should be considered to be an empirical parameter as the optimized values for kaolinite and smectite are below 1. Equation (33) is able to fit the data. The inverted effective CEC is 0.75 meq g−1 for smectite (consistent with the presence of sodium, we used inline image = 2.8 × 105 C kg−1) and 0.38 meq g−1 for kaolinite (with inline image = 1.6 × 105 C kg−1 for kaolinite). The CEC of kaolinite is somewhat high with respect to the range reported in Table 1, which indicates that this kaolinite may contain some mixed layer clay minerals.

image

Figure 10. Unified sorption isotherm applied to smectite. The experimental data are from Likos and Lu [2001]. We used inline image = 2.8 × 105 C kg−1. The inverse of the capillary entry pressure is α = (3.5 ± 0.6) × 10−6 Pa−1. See Figure 7 for the confidence intervals.

Download figure to PowerPoint

image

Figure 11. Unified sorption isotherm applied to kaolinite. The experimental data are from Likos and Lu [2001]. We used inline image = 1.6 × 105 C kg−1. The inverse of the capillary entry pressure is α = (2.1 ± 0.9) × 10−5 Pa−1. See Figure 7 for the confidence intervals. Note that the CEC seems pretty high with respect to the range of values provided in Table 1 for kaolinite.

Download figure to PowerPoint

4.3. Mixtures

[42] In Figure 12, we show that the desorption isotherm of a mixture can be predicted from the sorption isotherms of the two end-members using the weighted average justified above in our model. We use mixtures of kaolinite and smectite to demonstrate the model. First we fitted the two end-members separately (kaolinite and smectite), and then we used a simple mixing formula employing these two end-members and their mass fractions. This process is used to predict the isotherms for various mixtures of kaolinite and smectite. Subsequently, we obtained an excellent agreement between the model prediction and the experimental data, as predicted by equation (23).

image

Figure 12. Prediction of desorption isotherms using the end-member desorption isotherms (data from Likos and Lu [2003]). The isotherms of the kaolinite and bentonite mixes are predicted from the end-member isotherms and the mass fraction of kaolinite and bentonite in the mixture. These results are in excellent agreement with the predictions of our model.

Download figure to PowerPoint

4.4. Effect of the Type of Cation

[43] Now, we re-evaluate the data by Montes-Hernandez et al. [2003], who studied the water sorption on various homoionic MX-80 bentonites. The CEC of the MX-80 bentonite is in the range of 0.70–0.80 meq g−1 [Koch, 2008]. We will use a mean of 0.75 meq g−1 in the following analysis. The normalization should be done with inline image. The normalized isotherm can be written as

  • display math(37)

which we can be written as

  • display math(38)

[44] For this analysis, we use sodium as a reference, with δ(Na) = 0.5. Then we can use the δ(Na) to find the value of δ for the other ions by adjusting the data in such a way that they all fall on the same trend. We then determine the following scaling factors for taking into account the effect of the ion: δ(Li) = 0.4, δ(Mg) = 0.8, and δ(Ca) = 0.8, and δ(K) = 0.4 (see Figure 13). Note that the value of δ(Mg) is slightly higher than the one reported by Woodruff and Revil [2011] (δ(Mg) = 0.6). Fitting the data of Figure 13, we obtain inline image = 0.277 ± 0.011. Taking a CEC of 0.75 meq g−1 yields inline image= 2.7 ± 0.1 meq g−1 for smectite (see Table 1). Therefore, for a given type of ion (i) and clay (j), the Freundlich isotherm can be written as

  • display math(39)

where the values of inline image are reported in Table 1 for the main types of clay minerals.

image

Figure 13. Normalized sorption isotherms for monoionic bentonite (data from Montes-Hernandez et al. [2003]). The normalization is done with the parameter δi that is characteristic of the type of ion sorbed on the mineral surface or located into the interlayer porosity. See Figure 7 for the confidence intervals. The effect of the sorbed cation shows how the hydration of the sorbed cations is of fundamental importance to the understanding of the amount of hydration water that covers the surface of a clay mineral.

Download figure to PowerPoint

4.5. Further Validations

[45] To evaluate the applicability of our model for different types of soils, we tested the model against five other soils from the database investigated by Lebeau and Konrad [2010]. We were especially interested in the clayey sands and loams, including the sandy loam of Pachepsky et al. [1984], the Adelato loam [Mualem, 1976], the Gilat loam [Mualem, 1976], and the Shonai sand [Mehta et al., 1994]. The water retention curve fits for the data are shown in Figures 14 and 15.

image

Figure 14. Prediction of the complete capillary pressure curve for four soil samples (data from Mualem [1976] and Pachepsky et al. [1984]). The solid line corresponds to the fit of the model with the model parameters given on the graphs. See Figure 7 for the confidence intervals.

Download figure to PowerPoint

image

Figure 15. Prediction of the complete capillary pressure curve for the Shonai sand (data from Metha et al. [1994]). The solid line corresponds to the best fit of the model with the model parameters given on the graph. See Figure 7 for the confidence intervals.

Download figure to PowerPoint

[46] For desorption, the relationship between inline image and the CEC is given by

  • display math(40)

where bw is the excess mass of water per gram of dry clay (see equation (28) above), and inline image≈ 1. The CEC are provided in Table 2. Inverting equation (40) yields

  • display math(41)
Table 2. Inverted Properties of Five Soil Samples and the Computation of the CEC Using Equation (40) and inline image= 2.8 meq g−1
Soil inline image (–)α (Pa−1)nm inline imageCEC (meq g−1)
Sandy loam0.010±0.009(1.2±0.2)×10−41.5±0.10.4±0.30.42±0.010.059±0.012
Pachapa loam0.073±0.008(6.3±0.4)×10−52.2±0.10.25±0.10.46±0.010.123±0.008
Adelanto loam0.11±0.02(4.0±0.8)×10−51.6±0.10.47±0.120.44±0.010.089±0.010
Gilat loam0.09±0.009(2.1±0.3)×10−42.0±0.20.12±0.070.45±0.01 
Shonai sand0.045±0.004(4.1±0.1)×10−45.3±0.90.03±0.010.42±0.01 

[47] According to Feigenbaum et al. [1991], the CEC of the Gilat soil is 0.096 ± 0.008 meq g−1 (Na+), in agreement with the CEC predicted from our model (see Table 2, 0.089 ± 0.010 meq g−1). Using bw ≈ 30 mg per gram of dry clay, inline image≈ 1, and the values reported in Table 2 for the Gilat soil yields 0.086 meq g−1 (Na+). This gives a value for bw that is not too far from the value suggested by Woodruff and Revil [2011] (100 mg per gram of dry clay).

5. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

[48] In the literature, capillary forces have been proposed to dominate over the adsorption force at quite high relative humidity. For instance, according to Silva and Grifoll [2007], adsorption should give rise to multilayer films with thicknesses in the order of nanometers, and capillary forces should prevail at relative humidities in the range of 0.98 to 0.99. In the case of a monodisperse close packing of spherical particles, Tokunaga [2009] showed that capillary forces should dominate at a matric potential of approximately −26 γ d, where γ represents the surface tension between the vapor and liquid water phases, and d denotes the particle diameter. For kaolinite, with a diameter in the order of 2 μm, capillary forces should prevail at a matric potential of approximately −9.45 × 105 Pa (or a relative humidity of 99.3%). For illite and smectite, the relative humidity at which the capillary forces dominate would be even higher. Therefore, the water film on the clay surfaces may form several additional layers past the second hydration layer at a relative humidity of about 0.50. That said, these additional layers of water molecules will have properties that are very similar to the properties of the bulk water. Indeed, past the second hydration layer, the water molecules have the same viscosity and dielectric constant as the free water molecules [e.g., Israelashvili, 1992].

[49] A second point that will need to be explored is the expansion of the present model to compute the relative permeability for both capillary and film flow, and to compute the streaming current density associated with his flow. Such electrokinetic effects provide a powerful method to monitor pore water flow as discussed recently by Revil and Mahardika [2013]. In parallel, it will be interesting to monitor very low water contents using complex conductivity measurements expanding the model developed recently by Revil [2013] to very low saturations.

6. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

[50] Based on this work, the following conclusions have been reached.

[51] (1) The normalization by the CEC of the sorption/desorption isotherm for single minerals leads to two master curves for sorption and desorption. We have extended the approach developed by Woodruff and Revil [2011] to include the Freundlich isotherm. This isotherm is better suited to be combined with the van Genuchten capillary pressure model to provide a unified model of various types of soils.

[52] (2) For a mixture of various minerals, the sorption/desorption isotherms are given by the weighted sum of the sorption isotherms of the different minerals using the mass fraction of each mineral as weighting factor. This procedure is both justified from a theoretical point of view and conforms to the experimental data for mixtures of kaolinite and smectite.

[53] (3) We provide a way to compute a unified isotherm and capillary pressure curve combining the van Genuchten model and the CEC-normalized Freundlich model for a mixture of clay minerals and nonclay minerals. This new model includes two exponents, m and n, the capillary entry pressure (1/α), and two critical water content parameters, one being the water content at saturation ( inline image, the porosity) and the second being the maximum water content inline image ascribed to adsorption forces, including the trapped nonbound water. The maximum water content inline image can be related to the cation exchange capacity of the material.

[54] (4) The new water retention model was tested through a few data sets of soils, indicating it is capable of describing water isotherms for clayey media.

Appendix: A

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

Normalized Sorption Isotherms

[55] Normalizing the sorption isotherms with the cation exchange capacity yields

  • display math(A1)

where inline image. The units of inline image and inline image are both in C kg−1 (or meq g−1). Note that 1 meq g−1 = 96,320 C kg−1. After straightforward algebraic manipulations, we obtain

  • display math(A2)

[56] Defining the following sorption parameter inline image, we obtain

  • display math(A3)

and inline image is in C kg−1 (or meq g−1). In addition, we can easily show that inline image. After some algebraic manipulations, and using inline image (dimensionless), we obtain the following nonnormalized and normalized sorption isotherms

  • display math(A4)
  • display math(A5)

[57] The normalization of the Freundlich isotherm is straighforward. We start the normalization with the isotherm inline image. Next the normalization is completed by replacing k by inline image, where inline image denotes a constant. The normalized isotherm can be written as

  • display math(A6)

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References

[58] We thank the National Science Foundation (NSF award CMMI 0926276) for financial support. A. Revil thanks also the Organics, Clays, Sands and Shales (OCLASSH) Consortium at the Colorado School of Mines. We warmly thank Marc Lebeau and Jean-Marie Konrad for sharing their database with us. We thank the Associate Editor, Denis O'Carroll, and three anonymous referees for their very constructive comments.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Three Types of Water
  5. 3. Water Sorption Desorption Isotherms
  6. 4. Comparison With Experimental Data
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix: A
  10. Acknowledgments
  11. References