#### 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]:

- (4)

(constrained form of van Genuchten's model) where was originally described as the residual water content. In the current model, corresponds to the volumetric water content ascribed to adsorptive forces, considering also the water molecules in the trapped porosity. The water content *θ*_{s} = *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

- (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 (in Pa) (here defined as a positive value) can be related to the relative humidity, *h*_{r}, by the Kelvin equation [e.g., *Or and Wraight*, 2000; *Lu* *and* *Likos*, 2004],

- (6)

where (m^{3} mol^{−1}) is the molar volume of water (1.8016 × 10^{−5} m^{3} Mol^{−1}), *R* = 8.31432 J K^{−1} mol^{−1} is the universal gas constant, *T* is the absolute temperature (in Kelvin), and ≈ 137 × 10^{6} 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, , ascribed to adsorptive forces (mass of water ascribed to adsorptive forces per unit mass of solid) is given by

- (7)

where and denote the mass densities of the water and solid phases, respectively. Using equations (6) and (7), the data can be either displayed in ( , *h*_{r}), ( *ψ*), (*θ*, *h*_{r}), 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],

- (8)

where 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:

- (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]:

- (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, , 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):

- (11)

- (12)

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

[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.

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

- (13)

- (14)

[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

- (23)

- (24)

- (25)

where the sum is extended to the *N* mineral phases (typically, kaolinite, illite, and smectite), is the mass fraction of mineral *j* in the soil, denotes the cation exchange capacity of phase *j*, will be either given by the BET normalized scaled isotherms or the Freundlich scaled isotherm, and 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 IsothermClay *j* | CEC_{j} (meq g^{−1}) | (meq g^{−1}) | *m* (–) |
---|

Kaolinite | 0.03–0.10 | 1.6 | 2 |

Illite | 0.10–0.40 | 1.6 | 2 |

Smectite | 0.70–1.5 | 2.8 | 1 |

Vermiculite | 1.0–2.0 | 2.8 | 1 |

[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, CEC_{j} (measured using a cation that has a very strong affinity with the clay surface like ammonium or cobalt), according to

- (26)

where (therefore ) 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 CEC_{j} in equations (23) and (24) should be replaced by the scaled cation exchange capacity . 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.

[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, = 23.3 × 10^{−7} kg C^{−1}, and = 1.6 × 10^{5} 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.