Computational Geomechanics. Manuel Pastor

      The linear momentum balance equation for the solid phase is obtained in a similar way, taking into account Equations (2.68) instead of (2.67).

By summing this momentum balance equation with Equation (2.76) written for water and air and by taking into account the definition of total stress (2.69), assuming continuity of stress at the fluid‐solid interfaces and by introducing the averaged density of the multiphase medium

      (2.77)

      we obtain the linear momentum balance equation for the whole multiphase medium

(2.78)

      The mass balance equations are derived next.

The macroscopic mass balance equation for the solid phase (2.61), after differentiation and dividing by ρ^s is obtained as

(2.79)

This equation is used in the subsequent mass balance equations to eliminate the material time derivative of the porosity. For incompressible grains, as assumed here, . For compressible grains, see Equation (2.89) and related remarks.

The mass balance equation for water (2.63) is transformed as follows. First in Equation (2.75), the material time derivative of the water density with respect to the moving solid phase and the relative velocity v^ws are introduced. Then the derivatives are carried out, the quantity of water lost through evaporation is neglected and the material time derivative of the porosity is expressed through Equation (2.79), yielding

(2.80)

      The mass balance equation for air is derived in a similar way

      (2.81)

      To obtain the equations of Section 2.4.2, further simplifications are needed, which are introduced next.

An updated Lagrangian framework is used where the reference configuration is the last converged configuration of the solid phase. Further, the strain increments within each time step are small. Because of this, we can neglect the convective terms in all the balance equations. Neglecting in the linear momentum balance Equation (2.78) further the relative accelerations of the fluid phases with respect to the solid phase yields the equilibrium Equation (2.34a)

      (2.82)

      The linear momentum balance equation for fluids (2.76) by omitting all acceleration terms, as in Section 2.2.2, can be written for water

      (2.83)

      where

      and

(2.84)

      and for air

      (2.85)