Computational Geomechanics. Manuel Pastor

      In the above, we consider only the balance of the fluid momentum and R represents the viscous drag forces which, assuming the Darcy seepage law, can be written as

      (2.14a)

      (2.14b)

      Note that the underlined terms in (2.13) represent again the convective fluid acceleration and are generally small. Also note that, throughout this book, the permeability k is used with dimensions of [length]³·[time]/[mass] which is different from the usual soil mechanics convention k′ which has the dimension of velocity, i.e. [length]/[time]. Their values are related by where and g′ are the fluid density and gravitational acceleration at which the permeability is measured.

The final equation is one accounting for the mass balance of the flow. Here we balance the flow divergence w_ii by the augmented storage in the pores of a unit volume of soil occurring in time dt. This storage is composed of several components given below in order of importance:

      1 the increased volume due to a change in strain, i.e.: δijdεij = dεii = mTdε

      2 the additional volume stored by compression of void fluid due to fluid pressure increase: ndp/Kf

      3 the additional volume stored by the compression of grains by the fluid pressure increase: (1 − n)dp/KSand

      4 the change in volume of the solid phase due to a change in the intergranular effective contact stress .

      Here K_T is the average bulk modulus of the solid skeleton and ε_ii the total volumetric strain.

      Adding all the above contributions together with a source term and a second‐order term due to the change in fluid density in the process, we can finally write the flow conservation equation

      (2.15)

      This can be rewritten using the definition of α given in Equation (1.15b) as

      (2.16a)

      or in vectorial form

      (2.16b)

      where

(2.17)

      In (2.16), the last two (underlined) terms are those corresponding to a change of density and rate of volume expansion of the solid in the case of thermal changes and are negligible in general. We shall omit them from further consideration here.

      Equations (2.11), (2.13), and (2.16) together with appropriate constitutive relations specified in the manner of (2.3) define the behavior of the solid together with its pore pressure in both static and dynamic conditions. The unknown variables in this system are:

       The pressure of fluid (water), p ≡ pw

       The velocities of fluid flow wi or w

       The displacements of the solid matrix ui or u.

      The boundary condition imposed on these variables will complete the problem. These boundary conditions are:

      1 For the total momentum balance on the part of the boundary Γt, we specify the total traction ti(t) (or in terms of the total stress σij nj (σ⋅ G) with ni being the ith component of the normal at the boundary and G is the appropriate vectorial equivalence) while for Γu, the displacement ui(u), is given.

      2 For the fluid phase, again the boundary is divided into two parts Γp on which the values of p are specified and Γw where the normal outflow wn is prescribed (for instance, a zero value for the normal outward velocity on an impermeable boundary).

      Summarizing, for the overall assembly, we can thus write

(2.18)

      and

      Further

      (2.19a)

      and

      (2.19b)

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