Computational Geomechanics. Manuel Pastor

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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)

      with p_s = p_w S_w + p_a S_a in the case of thermodynamic equilibrium or for incompressible solid grains (2.50).

The sum of (2.67), written for air and water and of (2.68), gives the total stress σ, acting on a unit area of the volume fraction mixture

(2.69)

      This is the form of the effective stress (2.51), also called generalized Bishop stress (Nuth and Laloui 2008), employed in the following, as already explained.

Moist air in the pore system is assumed to be a perfect mixture of two ideal gases, dry air and water vapour, with π = ga and π = gw, respectively. The equation of a perfect gas is hence valid

(2.70)

      where M_π is the molar mass of constituent π, R the universal gas constant, and θ the common absolute temperature. Further, Dalton’s law applies and yields the molar mass of moisture

(2.71)

Water is usually present in the pores as a condensed liquid, separated from its vapor by a concave meniscus because of surface tension. The capillary pressure is defined as p_c = p_g − p_w, see Equation (2.43).

      The momentum exchange term of the linear momentum balance equation for fluids has the form

      (2.72)

      where v^πs

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