Computational Geomechanics. Manuel Pastor

Computational Geomechanics - Manuel Pastor


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      or

      (2.42b)italic nabla Superscript upper T Baseline bold v plus alpha upper S Subscript a Baseline bold m Superscript upper T Baseline ModifyingAbove bold-italic epsilon With ampersand c period dotab semicolon plus upper S Subscript a Baseline StartFraction n Over upper K Subscript a Baseline EndFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript a Baseline plus StartFraction alpha minus n Over upper K Subscript s Baseline EndFraction chi Subscript a Baseline ModifyingAbove p With ampersand c period dotab semicolon Subscript a Baseline plus n ModifyingAbove upper S With ampersand c period dotab semicolon Subscript a Baseline plus italic n upper S Subscript a Baseline StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript a Baseline Over rho Subscript a Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0

      Now, in addition to the solid phase displacement ui(u), we have to consider the water pressure pw and the air pressure pa as independent variables.

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

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

      (2.44)upper S Subscript w Baseline plus upper S Subscript a Baseline equals 1 and chi Subscript w Baseline plus chi Subscript a Baseline equals 1

      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.

      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.

      has to be equal to one, which results in

      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 ps, 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)StartFraction p Subscript s Baseline Over theta Subscript s Baseline EndFraction minus StartFraction p Subscript w Baseline Over theta Subscript w Baseline EndFraction equals minus k overbar ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i

      where k overbar is a coefficient and θs and θw 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

      where χw 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 As = As(ρs, εij, θs, Sw) that