Computational Geomechanics. Manuel Pastor

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Over partial-differential t EndFraction plus rho psi ModifyingAbove r With ampersand c period dotab semicolon Subscript normal i slash normal i Baseline minus i Subscript normal i slash normal i Baseline minus rho g Subscript i Baseline equals rho upper G Subscript i"/>

where is the local value of the velocity field of the π phase in a fixed point in space, i is the flux vector associated with Ψ, g the external supply of Ψ, and G is the net production of Ψ. The relevant thermodynamic properties Ψ are mass, momentum, energy, and entropy. The values assumed by i, g, and G are given in Table 2.2 (Hassanizadeh and Gray 1980, 1990; Schrefler 1995). The constituents are assumed to be microscopically nonpolar; hence, the angular momentum balance equation has been omitted. This equation shows, however, that the stress tensor is symmetric.

Table 2.2 Thermodynamic properties for the microscopic mass balance equations.

      Sources: Adapted from Hassanizadeh and Gray (1980, 1990), and Schrefler (1995).

Quantity	ψ	i	g	G
Mass	1	0	0	0
Momentum		t m	g	0
Energy				0
Entropy	Λ	Φ	S	φ

      2.5.3 Macroscopic Balance Equations

For isothermal conditions, as here assumed, the macroscopic balance equations for mass, linear momentum, and angular momentum are then obtained by systematically applying the averaging procedures to the microscopic balance Equation (2.60) as outlined in Hassanizadeh and Gray (1979a, 1979b, 1980). The balance equations have here been specialized for a deforming porous material, where the flow of water and of moist air (mixture of dryair and vapor) is taking place (see Schrefler 1995).

      The local thermodynamic equilibrium hypothesis is assumed to hold because the time scale of the modeled phenomena is substantially larger than the relaxation time required reaching equilibrium locally. The temperatures of each constituent in a generic point are hence equal. Further, the constituents are assumed to be immiscible and chemically nonreacting. All fluids are assumed to be in contact with the solid phase. As throughout this book, stress is defined as tension positive for the solid phase, while pore pressure is defined as compressively positive for the fluids.

      In the averaging procedure, the volume fractions η^π appear which are identified as follows: for solid phase η^s = 1 − n, for water η^w = nS_w, and for air η^a = nS_a.

      The averaged macroscopic mass balance equations are given next. For the solid phase, this equation reads

(2.61)

      where is the mass averaged solid phase velocity and ρ^π is the intrinsic phase averaged density. The intrinsic phase averaged density ρ^π is the density of the π phase averaged over the part of the control volume (Representative Elementary Volume, REV) occupied by the π phase. The phase averaged density ρ_π, on the contrary, is the density of the π phase averaged over the total control volume. The relationship between the two densities is given by

      (2.62)

      For water, the averaged macroscopic mass balance equation reads

(2.63)

where is the quantity of water per unit time and volume, lost through evaporation and v^w the mass averaged water velocity.

      For air, this equation reads

      (2.64)

      where

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