Computational Geomechanics. Manuel Pastor

Computational Geomechanics

alt="v Subscript i Baseline equals k Subscript italic a i j Baseline left-parenthesis minus p Subscript a Sub Subscript comma j Subscript Baseline plus rho Superscript a Baseline b Subscript j Baseline right-parenthesis"/>

where

eta Superscript a Baseline v Subscript i Superscript italic a s Baseline equals v Subscript i

and

(2.86) k Subscript a Baseline Subscript italic i j Baseline equals StartFraction upper K Subscript italic i j Baseline upper K Superscript italic r a Baseline Over mu Superscript a Baseline EndFraction

The phase densities appearing in Sections 2.2–2.4 are intrinsic phase averaged densities as indicated above.

The mass balance equation for water is obtained from Equation (2.80), taking into account the reference system chosen, dividing by ρ^w, developing the divergence term of the relative velocity and neglecting the gradient of water density. This yields

(2.87)

where the first of Equation (2.84) has been taken into account. This coincides with Equation (2.41a) for incompressible grains (α = 1) except for the source term and the second‐order term due to the change in fluid density. This last one could be introduced in the constitutive relationship (2.75).

Similarly, the mass balance equation for air becomes

(2.88)

where again the first of Equation (2.86) has been taken into account and the gradient of water density has been neglected. Similar remarks as for the water mass balance equation apply. In particular, the constitutive relationships for moist air, Equations (2.70) and (2.71), have been used.

Finally, if, for the solid phase, the following constitutive relationship is used (viz. Lewis and Schrefler 1998)

(2.89)

where K_s is the bulk modulus of the grain material, then the mass balance equations are obtained in the same form as in Section 2.4 (with χ_w = S_w), though this is not in agreement with what was assumed here for the effective stress.

2.5.6 Nomenclature for Section 2.5

As this section does not follow the notations use of the book, we summarize below for purposes of nomenclature:

aπmass averaged acceleration of π phaseaπsacceleration relative to the solidbexternal momentum supply StartFraction upper D Over italic upper D t EndFraction material time derivativeEspecific intrinsic energyFπdeformation gradient tensorfπdifferentiable functionf/i∂f/∂xif,i∂f/∂Xigexternal momentum supply related to gravitational effectsGnet production of thermodynamic property Ψhintrinsic heat sourceiflux vector associated with thermodynamic property Ψ k overbar constitutive coefficientKintrinsic permeability tensorKrπrelative permeability of π phasekπdynamic permeability tensorKwwater bulk modulus ModifyingAbove m With ampersand c period dotab semicolon mass rate of water evaporationMπmolar mass of constituent πnporositypccapillary pressurepgadry air pressure pgwvapor pressurepaair pressurepssolid pressurepwpressure of liquid waterpπmacroscopic pressure of the π phase ModifyingAbove bold r With ampersand c period dotab semicolon local value of the velocity fieldRuniversal gas constantRconstitutive tensorRπrigid body rotation tensorSintrinsic entropy sourceSwdegree of water saturationSadegree of air saturationtcurrent timet mmicroscopic stress tensortπpartial stress tensor ModifyingAbove bold t With ampersand c period circ semicolon πexchange of momentum due to mechanical interaction of the π phase with other phases ModifyingAbove bold u With ampersand c period dotab semicolon mass averaged solid phase velocityUπright stretch tensorvπsvelocity of the π phase with respect to the solid phasevwmass averaged water velocityvamass averaged air velocityvvolume averaged water relative velocitywvolume averaged air relative velocityVπleft stretch tensorxπmaterial pointXπreference configurationεlinear strain tensorΦentropy fluxφincrease of entropyηπvolume fraction of the π phaseχwBishop parameterλspecific entropyμπdynamic viscosityθπabsolute temperature of constituent πρaveraged density of the multi‐phase mediumρπintrinsic phase averaged density of the π phaseρπphase averaged density of the π phaseρmicroscopic densityσ ′effective stress tensorΨgeneric thermodynamic property or variable

Superscripts or subscripts

ga =dry airgw =vapora =airw =waters =solid

Computational Geomechanics. Manuel Pastor