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### 2.2. Transport of liquid water

The continuity equation for liquid water is given by [14] and [15]:

 (10)

where ρL is the mass of liquid water per unit volume of liquid phase and vL is the velocity of water. The bulk equation is obtained by averaging Eq. (10) over the REV:

 (11)

This integral can be divided in two parts:

 (12)

Using the definition of the volumetric phase average (Eq. (7)), one can write:

 (13)

The average of the time derivative gives [12] and [13]:

 (14)

where SoLG is the surface of the liquid/gas interface, SoLS is the surface of the liquid/solid interface, u is the velocity of the interface, nLG is a unit vector pointing outward the liquid phase at the liquid/gas interface, and nLS is a unit vector pointing outward the liquid phase at the liquid/solid interface. Because it is assumed that the deformations of the solid matrix could be neglected, the last integral on the right-hand side of Eq. (14) can be dropped, which leaves:

 (15)

The average of the divergence in Eq. (13) is given by [12] and [13]:

At the solid/liquid interface, it is assumed that the liquid velocity is zero (the so-called no-slip condition of fluid mechanics [16]). Hence, the last integral on the right-hand side of Eq. (16) can be neglected, which leaves:

 (17)

Substituting Eqs. Figs. (15) and (17) in Eq. (13), one finds:

 (18)

According to Whitaker [15], the integral in Eq. (18) corresponds to the rate of vaporization per unit volume of the liquid phase at the liquid/gas interface and is denoted by . In addition, the average value corresponds to the density of the liquid ρL, which can be assumed constant. Eq. (18) can thus be simplified:

 (19)

The next step consists of determining the average value of the liquid velocity. The starting point is the Darcy equation [17]:

 (20)

The quantity is the bulk velocity of the liquid, K is the permeability of the material, μ is the viscosity of the fluid, P is the pressure on the liquid, and g is the gravitational acceleration. Darcy originally derived this equation to describe the transport of water through the material at the macroscopic scale. Whitaker [15] showed that in materials having very small pores, the capillary forces are dominant:

 (21)

The quantity pc is the capillary pressure and k is the permeability of the liquid-filled pore space.

Eq. (21) is based on the assumptions that gravity effects are negligible and that the pressure is uniform throughout the liquid and gaseous phases. It should also be emphasized that the validity of the equation also rests on the hypothesis that the liquid phase is continuous. The latter assumption will be further discussed in the last section of this report.

The bulk velocity of the liquid can be related to its intrinsic average counterpart through:

 (22)

Substituting Eqs. Figs. (21) and (22) into Eq. (19) gives:

 (23)

Since = ƒ(θL) [15], the chain rule allows to write:

 (24)

The substitution of Eq. (24) in Eq. (23) gives:

 (25)

Let

 (26)

Eq. (25) is now expressed as a function of a single field quantity θL to give a complete description of liquid water transport:

 (27)

Since Eq. (27) is expressed in the form of a diffusion equation, DL can be assimilated to a water diffusion coefficient. However, it should be emphasized that the movement of liquid water considered in this section arises by capillary suction. It is not, per se, a diffusive phenomenon.

With the definition of DL given in Eq. (26), combined with Eq. (24), the velocity of the liquid phase (Eq. (21)) can now be written as:

 (28)

Next: Transport of water vapor Up: Water transport Previous: General considerations