Environmental Engineering Reference
In-Depth Information
The ensuing mathematical model is governed by the following equations, which
are the same as for the consolidation problem (whether isothermal or not):
- a linear momentum balance equation for water (fluids), for the solid skeleton
and for the whole multiphase medium (mixture);
- a mass balance equation for each constituent;
- an energy balance equation for the mixture solid plus fluid(s).
When dealing with isothermal problems, the last equation can be dropped. In the
case of geothermal reservoirs, the system may transform into a three-phase one due
to the presence of a gaseous phase (steam).
Each one of these equations for the generic constituent contains, in its general
form, terms representing the effects of all the fields present. In real applications
there is an obvious necessity to reduce the complexity of the model by introducing
simplifying assumptions, and partial models are often used in practice. Of course the
admissibility and accuracy of such simplifications have to be carefully assessed to
obtain meaningful numerical models and reliable forecasts.
Mathematical models based on the definition of Terzaghi's effective stress,
usually applied for the analysis of subsidence due to water extraction, are
subdivided into two categories. In the first one a simultaneous solution is sought for
the linear momentum balance of the two- or three-phase system and the mass
balance equation of the fluids present. Fluid overpressures and solid skeleton
deformations are usually assumed to be unknowns following Biot's theory [BIO
41], which is a combination of the above-mentioned equations. The second
formulation generally neglects the coupling between the fluid and solid phase,
firstly calculating the water pressure by solving the fluid mass balance equation (in
two or three dimensions). Subsidence is then calculated using a structural code
following the hypothesis of one- (or three-)dimensional deformation. Deformation
effects in the mass balance equation of the fluids are sometimes introduced by
assuming only a vertical displacement of the layers [COR 84].
Here we summarize the governing equations of Biot's theory, which will be used
in the following applications. For a detailed discussion about these equations, the
interested reader is referred to [LEW 98]. The numerical model for a fully saturated
soil comprises:
- the linear momentum balance equation for the two-phase system, where
convective and inertial terms have been disregarded;
T
[5.1]
L
σ
+
ρ
g
=
0
