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s Baseline equals sigma prime Subscript italic i j Baseline minus left-parenthesis 1 minus n right-parenthesis p Subscript s Baseline delta Subscript italic i j"/>

      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 is the velocity of the π phase relative to the solid.

      It is assumed that R^π is invertible, its inverse being and k^π is defined by the following relation:

(2.73)

where μ_π is the dynamic viscosity with dimensions [mass] [length]⁻¹ [time]⁻¹, K the intrinsic permeability [length]² and T the temperature above some datum. The permeability of (2.73) is the dynamic permeability [length]³ [mass]⁻¹ [time]; to obtain the soil mechanics permeability k′ [length]/[time], Equation (2.73) has to be multiplied by the specific weight of water γ_w of dimensions [mass] [length]⁻² [time]⁻². In the case of more than one fluid flowing, the intrinsic permeability is modified as

(2.74)

      where K^rπ is the relative permeability, a function of the degree of saturation. For the water density, the following holds:

(2.75)

      where K_w is the bulk modulus of water.

      2.5.5 General Field Equations

      The macroscopic balance laws are now transformed and the constitutive equations introduced, to obtain the general field equations.

The linear momentum balance equation for the fluid phases is obtained first. In Equation (2.65), the fluid acceleration is expressed, taking into account Equation (2.59), and introducing the relative fluid acceleration a^πs. Further, Equations (2.67) and (2.68) are introduced. The terms dependent on the gradient of the fluid velocity and the effects of phase change are neglected and a vector identity for the divergence of the stress tensor is used. Finally, Equations (2.73) and (2.74) are included, yielding

(2.76)Скачать книгу