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asterisk Baseline EndFraction identical-to upper C Subscript s Baseline plus StartFraction italic n upper S Subscript w Baseline Over upper K Subscript f Baseline EndFraction plus StartFraction left-parenthesis alpha minus n right-parenthesis chi Subscript w Baseline Over upper K Subscript s Baseline EndFraction"/>

      which, of course, must be identical with (2.17) when S_w = 1 and χ_w = 1, i.e. when we have full saturation. The above modification is mainly due to an additional term to those defining the increased storage in (2.17). This term is due to the changes in the degree of saturation and is simply:

      (2.31)

      but here we introduce a new parameter C_S defined as

      (2.32)

      The final elimination of w in a manner identical to that used when deriving (2.21) gives (neglecting density variation):

      (2.33a)

      or

(2.33b)

      The small changes required here in the solution process are such that we found it useful to construct our computer program for the partially saturated form, with the fully saturated form being a special case.

      In the time‐stepping computation, we still always assume that the parameters S_w, k_w, and C_s change slowly and hence we will compute these at the start of the time interval keeping them subsequently constant.

      Previously, we mentioned several typical cases where pressure can become negative and hence saturation drops below unity. One frequently encountered example is that of the flow occurring in the capillary zone during steady‐state seepage. The solution to the problem can, of course, be obtained from the general equations simply by neglecting all acceleration and fixing the solid displacements at zero (or constant) values.

If we consider a typical dam or a water‐retaining embankment shown in Figure 2.3, we note that, on all the surfaces exposed to air, we have apparently incompatible boundary conditions. These are:

      Clearly, both conditions cannot be simultaneously satisfied and it is readily concluded that only the second is true above the area where the flow emerges. Of course, when the flow leaves the free surface, the reverse is true.

Computation will easily show that negative pressures develop near the surface and that, therefore, a partially saturated zone with very low permeability must exist. The result of such a computation is shown in Figure 2.3 and indeed it will be found that very little flow occurs above the zero‐pressure contour. This contour is, in fact, the well‐known Phreatic line and the partially saturated material procedure has indeed been used frequently purely as a numerical device for its determination (see Desai 1977a, 1977b; Desai and Li 1983 etc.). Another example is given in Figure 2.4. Here a numerical solution of Zienkiewicz et al. (1990b) is given for a problem for which experimental data are available from Liakopoulos (1965).

Figure 2.3 A partially saturated dam. Initial steady‐state solution. Only saturation (a) and pressure contours (b) are shown. Contour interval in (b) is 75 kPa. The Phreatic line is the boundary of the fully saturated zone in (a)

Figure 2.4 Test example of partially saturated flow experiment by Liakopoulos (1965). (a) Configuration of test (uniform inflow interrupted at t = 0); (b) pressures with – – –, computed; ––––––, recorded; (c) data (linear elastic analysis, E = 3000 kPa).

      Source: From Liakopoulos (1965)

      In the practical code used for earthquake analysis, we shall use this partially saturated flow to calculate a wide range of soil mechanics phenomena. However, for completeness in Section 2.4, we shall show how the effects of air movement can be incorporated into the analysis.

2.4 Partially Saturated Behavior with Air Flow Considered (p_a ≥ 0)

      2.4.1 The Governing Equations Including Air Flow

      This part of the chapter is introduced for completeness – though the effects of the air pressure are insignificant in most problems. However, in some cases of consolidation and confined materials, the air pressures play an important role and it is useful to have means for their prediction. Further, the procedures introduced are readily applicable to other pore–fluid mixtures. For instance, the simultaneous presence of water and oil is important in some areas of geomechanics and coupled problems are of importance in the treatment of hydrocarbon reservoirs. The procedures used in the analysis follow precisely the same lines as introduced here.

      In particular, the treatment following the physical approach used in this chapter has been introduced by Simoni and Schrefler (1991), Li et al. (1990)

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