Page 52 - Numerical Analysis and Modelling in Geomechanics

COMPRESSED AIR TUNNELLING 33
with complex boundary conditions. In general, the soil is inhomogeneous and
anisotropic. The permeability varies with the degree of saturation and both air
and soil are compressible. Furthermore, the process of lowering the groundwater
is time-dependent. In addition, air and water are miscible fluids. The physical
complexity of the problem makes analysis of the flow of air through soils during
compressed air tunnelling extremely difficult. 11 Due to these complexities and
the uncertainty regarding several details of the flow, simplifying assumptions
must be made in numerical modelling of the problem.

Assumptions for the numerical model

The air and water permeability of the soil is usually assumed to be constant
within a subregion of the ground and so the entire region can be divided into
subregions with constant permeabilities. At least two subregions will exist: a
saturated subregion below the deformed groundwater surface and an unsaturated
region above it. Also different soil layers with different permeabilities can be
considered as subregions within these two main subregions. The air permeability
of the unsaturated region above the groundwater level will be much higher than
that of the saturated subregion below the groundwater level.
Although the whole process of lowering the groundwater level is time-
dependent, only the final steady-state condition after establishment of the
deformed groundwater profile is considered in the numerical model. Darcy’s law
is used as the flow law, but it is only valid for laminar flow conditions, which
excludes large hydraulic gradients and large soil particles. The analysis is
performed for air and water under isothermal conditions.
Air and water are assumed to be immiscible fluids. This implies that the soil
mass is subdivided into a fully saturated zone below the deformed water table
and an unsaturated zone above it. A free boundary will exist that separates the two
zones (see Figure 2.3). The shape and position of this boundary are a priori
unknown and should be determined as part of the solution.

Governing equations
The formulation of the problem is based on the continuity equation, the equation
of motion (Darcy’s law) and Boyle’s law as the equation of state for ideal gases.
For the specific case of steady-state flow and in the absence of sources and sinks,
the equation of continuity as an expression of the mass conservation can be
written as: 12
(2.3)

Darcy’s law relates the flow rate to the driving potential as: