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The interstitial space, or the pores, may, in a practical situation, be filled with two or more fluids. We shall, in this section, consider only two fluids with the degree of saturation by each fluid being defined by the proportion of the total pore volume n (porosity) occupied by each fluid. In the context of soil behavior discussed in this book, the fluids will invariably be water and air, respectively. Thus, we shall refer to only two saturation degrees, S_w that for water and S_a that for air, but the discussion will be valid for any two fluids.

It is clear that if both fluids fill the pores completely, we shall always have

(1.18)

Clearly, this relation will be valid for any other pair of fluids, e.g. oil and water and indeed the treatment described here is valid for any fluid conditions.

The two fluids may well present different areas of contact with the solid grains of the material in the manner illustrated in Figure 1.5a and b. The average pressure reducing the interstitial contact and relevant to the definition of effective stress found in the previous section (Equations (1.16) and (1.17)) can thus be taken as

(1.19)

where the coefficients χ_w and χ_a refer to water and air, respectively, and are such that

(1.20)

The individual pressures p_w and p_a are again referring to water and air and their difference, i.e.

(1.21)

is dependent on the magnitude of surface tension or capillarity and on the degree of saturation (p_c is often referred to, therefore, as capillary pressure).

Figure 1.5 Two fluids in pores of a granular solid (water and air). (a) Air bubble not wetting solid surface (effective pressure p = p_w); (b) Both fluids in contact with solid surfaces (effective pressure p = χ_w p_w + χ_a p_a).

Depending on the nature of the material surface, the contact surface may take on the shapes shown in Figure 1.5 with

(1.22a)

and

(1.22b)

Occasionally, the contact of one of the phases and the solid may disappear entirely as shown in Figure 1.5a giving isolated air bubbles and making in this limit

(1.23)

In many situations, in soil mechanics, it is sufficient to take χ equal to the respective degrees of saturation (Lewis and Schrefler 1982; Nuth and Laloui 2008).

Whatever the nature of the contact, we shall find, neglecting the hysteresis during the wetting and drying cycles, that a unique relationship between p_c and the saturation S_w can be written, i.e.

(1.24)

Indeed, the degree of saturation will similarly affect flow parameters such as the permeability k to which we shall make reference in the next chapter, giving

(1.25)

Many studies of such relationship are reported in the literature (Liakopoulos 1965; Neuman 2017; Van Genuchten et al. 1977; Narasimhan and Witherspoon 1978; Safai and Pinder 1979; Lloret and Alonso 1980; Bear et al. 1984; Alonso et al. 1987). Figure 1.6 shows a typical relationship.

Figure 1.6 Typical relations between pore pressure head, h_w = p_w/χ_w, saturation, S_w, and relative permeability, k_r = k_w (S_w)/k_w(1) (Safai and Pinder 1979). Note that relative permeability decreases very rapidly as saturation decreases.

Source: From Safai and Pinder (1979).

The concepts of dealing with the effects of multiple pore pressure by introducing an average pressure and using the standard definition of effective stress (1.19, 1.16, and 1.17) were first introduced by Bishop (1959). Certainly, the arguments for thus extending the original concepts are less clear than is the case when only a single fluid is present. However, the results obtained by this extension are quite accurate. We shall, therefore, use such a definition in the study of partially saturated media.

In many cases occurring in practice, the air pressure is close to zero (atmospheric datum) as the pores are interconnected. Alternatively, negative air pressure occurs as cavitation starts and here the datum is the vapor pressure of water. In either case, the effect of p_a can be easily neglected as the water pressure simply becomes negative from Equation (1.24). Such negative pressures are responsible for the development of certain cohesion by the soil and are essential in the study of free surface conditions occurring in embankments, as we shall see later.