alpha ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i plus upper S Subscript w Baseline StartFraction n Over upper K Subscript f Baseline EndFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript w plus StartFraction alpha minus n Over upper K Subscript s Baseline EndFraction chi Subscript w Baseline ModifyingAbove p With ampersand c period dotab semicolon Subscript w plus n ModifyingAbove upper S With ampersand c period dotab semicolon Subscript w plus italic n upper S Subscript w Baseline StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript w Baseline Over rho Subscript w Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 2nd Row identical-to w Subscript i comma i Baseline plus alpha ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i Baseline plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript w Baseline Over upper Q Superscript asterisk Baseline EndFraction plus italic n upper S Subscript w Baseline StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript w Baseline Over rho Subscript w Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0 EndLayout"/>
or
(2.30b)
Now, however, Q* is different from that given in Equation 2.17 and we have in its place
(2.30c)
which, of course, must be identical with (2.17) when Sw = 1 and χw = 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 CS 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 Sw, kw, and Cs 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).
