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      or

      (2.42b)

      Now, in addition to the solid phase displacement u_i(u), we have to consider the water pressure p_w and the air pressure p_a as independent variables.

      However, we note that now (see (1.21))

(2.43)

      and that the relation between p_c and S_W is unique and of the type shown in Figure 1.6. p_c now defines S_W and from the fact that

      (2.44)

      air saturation can also be found.

      We have now the complete equation system necessary for dealing with the flow of air and water (or any other two fluids) coupled with the solid phase deformation.

2.5 Alternative Derivation of the Governing Equation (of Sections 2.2–2.4) Based on the Hybrid Mixture Theory

      It has already been indicated in Section 2.1 that the governing equations can also be derived using mixture theories. The classical mixture theories (viz. Green 1969; Morland 1972; Bowen 1980, 1982) start from the macro‐mechanical level, i.e. the level of interest for our computations, while the so‐called hybrid mixture theories (viz. Whitaker 1977; Hassanizadeh and Gray 1979a, 1979b, 1980) start from micro‐mechanical level. The equations at the macro‐mechanical level are then obtained by spatial averaging procedures. Further, there exists a macroscopic thermodynamical approach to Biot’s theory proposed by Coussy (1995). All these theories lead to a similar form of the balance equations. This was in particular shown by de Boer et al. (1991) for the mixture theory, the hybrid mixture theories and the classical Biot’s theory.

The theories differ in the constitutive equations, usually obtained from the entropy inequality. This is here shown in particular for the effective stress principle because this was extensively discussed in Chapter 1. Let’s consider first the fully saturated case. Runesson (1978) shows, for instance, that the principle of effective stress follows from the mixture theory under the assumption of incompressible grains. This means that α in Equation (1.16a) or (2.1a)

(2.45a)

      has to be equal to one, which results in

(2.45b)

      Only in this case, the two formulations given by Biot’s theory and the mixture theory coincide.

      In the hybrid mixture theories, the concept of the solid pressure p_s, i.e. the pressure acting on the solid grains is introduced (viz. Hassanizadeh and Gray 1990). From the application of the second principle of thermodynamics, it appears that under conditions close to thermodynamic equilibrium

      (2.46)

      where is a coefficient and θ_s and θ_w the absolute temperatures of solid and water, respectively. Under the assumption of local thermodynamic equilibrium, where temperatures are equal and rate terms vanish, it follows that

(2.47)

and the effective stress principle in the form of (2.45b) can be derived following Hassanizadeh and Gray (1990). Equation (2.47) holds also under nonequilibrium conditions; it has however to be assumed that the solid grains remain incompressible, the same assumption as with the mixture theory.

Let us consider now the partially saturated case. There are again several conflicting expressions in literature. The first expression for partially saturated soils was developed by Bishop (1959) and may be written as follows, see Equations (2.36), (2.24), and (1.19)

(2.48)

      where χ_w is the Bishop parameter, usually a function of the degree of saturation, see (1.22a) and (1.22b). The same expression, but with Bishop’s parameter equal to the water degree of saturation was derived by Lewis and Schrefler (1982, 1987) using volume averaging. Hassanizadeh and Gray (1990) find for the partially saturated case under the assumption of the following form of the Helmholtz free energy for the solid A^s = A^s(ρ^s, ε_ij, θ^s, S_w) that

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