Computational Geomechanics. Manuel Pastor

Скачать книгу

the mass balance equations for both water and air have to be written. These are derived in a manner identical to that used for Equation (2.30). Thus, for water, we have

(2.41a)

      (2.41b)

      and for air

      (2.42a)

      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

Скачать книгу