Computational Geomechanics. Manuel Pastor

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 v^a is the mass averaged air velocity.

      The linear momentum balance equation for the fluid phases is

(2.65)

      where t^π is the partial stress tensor, ρ_πb^π the external momentum supply due to gravity, ρ_πa^π the volume density of the inertia force, the sum of the momentum supply due to averaged mass supply, and the intrinsic momentum supply due to a change of density and referred to the deviation of the velocity of constituent π from its mass averaged velocity, and accounts for exchange of momentum due to mechanical interaction with other phases. is assumed to be different from zero only for fluid phases. For the solid phase, the linear momentum balance equation is hence

      (2.66)

      The average angular momentum balance equation shows that for nonpolar media, the partial stress tensor is symmetric t^π _ji = t^π _tj at the macroscopic level also and the sum of the coupling vectors of angular momentum between the phases vanishes.

      2.5.4 Constitutive Equations

      Constitutive models are selected here which are based on quantities currently measurable in laboratory or field experiments and which have been extensively validated. Most of them have been obtained from entropy inequality; see Hassanizadeh and Gray (1980, 1990).

      It can be shown that the stress tensor in the fluid is

(2.67)

      where p^π is the fluid pressure, and in the solid phase is

(2.68)