Computational Geomechanics. Manuel Pastor

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)

(2.24)

      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

(2.25)

      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 p_w. 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 S_w 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 w_i,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)

      or

      (2.30b)

Now, however, Q^* is different from that given in Equation 2.17 and we have in its place

      (2.30c)