vapor pressure of water) is reached. Alternatively, air will come out of the solution – or indeed ingress from the free water surface if this is open to the atmosphere. The pressure will not then be vapor pressure but simply atmospheric.
We have shown in Chapter 1 that once voids open, a unique relationship exists between the degree of saturation Sw and the pore pressures pw (see Figure 1.6). Using this relation, which can be expressed by formula or simply a graph, we can modify the equation used in Section 2.2 to deal with the problem of partial saturation without introducing any additional variables assuming that the air throughout is at constant (atmospheric) pressure. Note that both phenomena of densification and dilation will be familiar to anybody taking a walk on a sandy beach after the tide has receded leaving the sand semi‐saturated. First, one can note how when the foot is placed on the damp sand, the material appears to dry in the vicinity of the applied pressure. This obviously is the dilation effect. However, if the pressure is not removed but reapplied several times, the sand “densifies” and becomes quickly almost fluid. Clearly, liquefaction has occurred. It is surprising how much one can learn by keeping one’s eyes open!
The presence of negative water pressures will, of course, increase the strength of the soil and thus have a beneficial effect. This is particularly true above the free water surface or the so‐called phreatic line. Usually, one is tempted to assume simply a zero pressure throughout that zone but for non‐cohesive materials, this means almost instantaneous failure under any dynamic load. The presence of negative pressure in the pores assures some cohesion (of the same kind which allows castles to be built on the beach provided that the sand is damp). This cohesion is essential to assure the structural integrity of many embankments and dams.
2.3.2 The Modification of Equations Necessary for Partially Saturated Conditions
The necessary modification of Equations (2.20) and (2.21) will be derived below, noting that generally we shall consider partial saturation only in the slower phenomena for which u–p approximation is permissible.
Before proceeding, we must note that the effective stress definition is modified and the effective pressure now becomes (viz Section 1.3.3)
with the effective stress still defined by (2.1).
Equation (2.20) remains unaltered in form whether or not the material is saturated but the overall density ρ is slightly different now. Thus in place of (2.12), we can write
neglecting the weight of air. The correction is obviously small and its effect insignificant.
However, (2.21) will now appear in a modified form which we shall derive here.
First, the water momentum equilibrium, Equation (2.13), will be considered. We note that its form remains unchanged but with the variable p being replaced by pw. We thus have
(2.26a)
(2.26b)
As before, we have neglected the relative acceleration of the fluid to the solid.
Equation (2.14), defining the permeabilities, remains unchanged as
(2.27a)
(2.27b)
However, in general, only scalar, i.e. isotropic, permeability will be used here
(2.28a)
(2.28b)
where I is the identity matrix. The value of k is, however, dependent strongly on Sw and we note that:
(2.29)
Such typical dependence is again shown in Figure 1.6.
Finally, the conservation Equation (2.16) has to be restructured, though the reader will recognize similarities.
The mass balance will once again consider the divergence of fluid flow wi,i to be augmented by terms previously derived (and some additional ones). These are
1 Increased pore volume due to change of strain assuming no change of saturation: δijdεij = dεii
2 An additional volume stored by compression of the fluid due to fluid pressure increase: nSwdpw/Kf
3 Change of volume of the solid phase due to fluid pressure increase: (1 − n)χwdpw/Ks
4 Change of volume of solid phase due to change of intergranular contact stress: −KT/Ks(dεii + χwdpw/Ks)
5 And a new term taking into account the change of saturation: ndSw
Adding to the above, as in Section 2.2, the terms involving density changes, on thermal expansion, the conservation equation now becomes:
(2.30a)
