Computational Geomechanics. Manuel Pastor

Computational Geomechanics

Computational Geomechanics - Manuel Pastor

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rho Subscript f Baseline left-bracket ModifyingAbove bold w With ampersand c period dotab semicolon plus bold w nabla Superscript upper T Baseline bold w right-bracket slash n With bar plus rho Subscript f Baseline bold b equals 0"/>

      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)k Subscript italic i j Baseline upper R Subscript j Baseline equals w Subscript i

      (2.14b)bold k upper R equals bold w

      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 k equals k prime slash rho prime Subscript f Baseline g prime where rho prime Subscript f and g′ are the fluid density and gravitational acceleration at which the permeability is measured.

      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 KT is the average bulk modulus of the solid skeleton and εii 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)w Subscript i comma i Baseline plus ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i Baseline plus StartFraction n ModifyingAbove p With ampersand c period dotab semicolon Over upper K Subscript f Baseline EndFraction plus StartFraction left-parenthesis 1 minus n right-parenthesis ModifyingAbove p With ampersand c period dotab semicolon Over upper K Subscript upper S Baseline EndFraction minus StartFraction upper K Subscript upper T Baseline Over upper K Subscript upper S Baseline EndFraction left-parenthesis ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i Baseline plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Over upper K Subscript upper S Baseline EndFraction right-parenthesis plus n StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript f Baseline Over rho Subscript f Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0

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

      (2.16a)w Subscript i comma i Baseline plus alpha ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i Baseline plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Over upper Q EndFraction plus ModifyingBelow n StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript f Baseline Over rho Subscript f Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline With bar equals 0

      or in vectorial form

      (2.16b)italic nabla Superscript normal upper T Baseline bold w plus alpha bold m Superscript upper T Baseline ModifyingAbove bold epsilon With ampersand c period dotab semicolon plus StartFraction ModifyingAbove p With ampersand c period dotab semicolon Over upper Q EndFraction plus ModifyingBelow n StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript f Baseline Over rho Subscript f Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline With bar equals 0

      where

      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

      and

bold u equals bold u overbar o n normal upper Gamma equals normal upper Gamma Subscript u

      Further

      (2.19a)StartLayout 1st Row normal upper Gamma equals normal upper Gamma Subscript p Baseline union normal upper Gamma Subscript w Baseline 2nd Row p equals p overbar o n normal upper Gamma equals normal upper Gamma Subscript p EndLayout

      and

      (2.19b)bold n Superscript normal upper T Baseline bold w equals w Subscript <hr><noindex><a href=Скачать книгу