Computational Geomechanics. Manuel Pastor

Computational Geomechanics - Manuel Pastor


Скачать книгу
and Schrefler and Zhan (1993) for the flow of water with air.

      The alternative approach of using the mixture theory in these problems was outlined by Li and Zienkiewicz (1992) and Schrefler (1995).

      Some simple considerations will allow the basic equations for the dynamics of the soil containing two pore fluids to be derived. They have been solved by Schrefler and Scotta (2001) and an example will be shown in Section 8.5.

      2.4.2 The Governing Equation

      The dynamics of the total mixture can, just as in Section 2.3, be written in precisely the same form as that for a single fluid phase (see (2.11)). For completeness, we repeat that equation here (now, however, a priori omitting the small convective terms)

      (2.34b)bold upper S Superscript upper T Baseline bold sigma minus rho ModifyingAbove bold u With two-dots plus rho bold b equals 0

      (2.35)rho equals italic n upper S Subscript w Baseline rho Subscript w Baseline plus n left-parenthesis 1 minus upper S Subscript w Baseline right-parenthesis rho Subscript a Baseline plus left-parenthesis 1 minus n right-parenthesis rho Subscript s

      noting that

upper S Subscript a Baseline equals 1 minus upper S Subscript w

      For the flow of water and air, we can write the Darcy equations separately, noting that

      (2.37a)k Subscript w Baseline upper R Subscript i Superscript w Baseline equals w Subscript i

      (2.37b)k Subscript w Baseline bold upper R Superscript normal w Baseline equals bold w

      (2.38a)k Subscript a Baseline upper R Subscript i Superscript a Baseline equals v Subscript i

      (2.38b)k Subscript a Baseline bold upper R Superscript a Baseline equals bold v

      Here we introduced appropriate terms for coefficients of permeability for water and air, while assuming isotropy. A new variable v now defines the air velocity.

      The approximate momentum conservation Equation (see 2.13) can be rewritten in a similar manner using isotropy but omitting acceleration terms for simplicity. We therefore have for water

      (2.39a)w Subscript i Baseline equals k Subscript w Baseline left-parenthesis minus p Subscript w comma i Baseline plus rho Subscript w Baseline b Subscript i Baseline right-parenthesis

      (2.39b)bold w equals k Subscript w Baseline left-parenthesis minus nabla p Subscript w Baseline plus rho Subscript w Baseline bold b right-parenthesis

      and for air

      (2.40a)v Subscript i Baseline equals k Subscript normal a Baseline left-parenthesis minus p Subscript normal a comma i Baseline plus rho Subscript normal a Baseline b Subscript i Baseline right-parenthesis

      (2.40b)bold v equals k Subscript normal a Baseline left-parenthesis minus nabla p Subscript normal a Baseline plus rho Subscript normal a Baseline bold b right-parenthesis

      Finally, the mass balance equations for both water and air have to be written. These are derived in a manner identical to that used for Equation (2.30). Thus, for water, we have

      (2.41b)italic nabla Superscript upper T Baseline bold w plus alpha upper S Subscript w Baseline m Superscript upper T Baseline ModifyingAbove bold-italic epsilon With ampersand c period dotab semicolon plus upper S Subscript w Baseline StartFraction n Over upper K Subscript w Baseline EndFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript w Baseline plus StartFraction alpha minus n Over upper K Subscript s Baseline EndFraction chi Subscript w Baseline ModifyingAbove p With ampersand c period dotab semicolon Subscript w Baseline plus n ModifyingAbove upper S With ampersand c period dotab semicolon Subscript w Baseline plus italic n upper S Subscript w Baseline StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript w Baseline Over rho Subscript w Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon Subscript 0 Baseline equals 0

      and for air

      (2.42a)v Subscript i comma i Baseline plus alpha upper S Subscript a Baseline ModifyingAbove epsilon With ampersand c period dotab semicolon Subscript italic i i Baseline plus upper S Subscript a Baseline StartFraction n Over upper K Subscript w Baseline EndFraction ModifyingAbove p With ampersand c period dotab semicolon Subscript a Baseline plus StartFraction alpha minus n Over upper K Subscript s Baseline EndFraction chi Subscript a Baseline ModifyingAbove p With ampersand c period dotab semicolon Subscript a Baseline plus n ModifyingAbove upper S With ampersand c period dotab semicolon Subscript a Baseline plus italic n upper S Subscript a Baseline StartFraction ModifyingAbove rho With ampersand c period dotab semicolon Subscript a Baseline Over rho Subscript a Baseline EndFraction plus ModifyingAbove s With ampersand c period dotab semicolon <hr><noindex><a href=Скачать книгу