Computational Geomechanics. Manuel Pastor
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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)
However, just as in Equation (2.25), we have to write
(2.35)
noting that
For definition of effective stress, we again use (2.24) now, however, without equating the air pressure to zero, i.e. writing
For the flow of water and air, we can write the Darcy equations separately, noting that
(2.37a)
(2.37b)
for water as in (2.27) and for the flow of air:
(2.38a)
(2.38b)
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)
(2.39b)
and for air
(2.40a)
(2.40b)
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)
and for air
(2.42a)