Page 54 - Numerical Analysis and Modelling in Geomechanics

COMPRESSED AIR TUNNELLING 35

(2.7)

Boundary conditions

Typical boundary conditions for this problem can be boundaries of constant air
pressure, Φ=constant, or impermeable boundaries, ∂Φ/∂n=0. Figure 2.3 shows
the generalised geometry of a tunnel driven under compressed air passing
underneath a road, illustrating the typical boundary conditions and the original
and the deformed groundwater profiles.

Numerical solution

On the basis of the above assumptions, a numerical model has been developed
using the finite element method to simulate the problem on the basis of
isothermal, steady-state potential flow and applied to the problem of compressed
air tunnelling. The model can predict the distribution of the air pressure in the
ground due to the application of, or change in, the tunnel pressure.
The shape and position of the free boundary, separating the saturated and
unsaturated zones, are a priori unknown and should be determined as part of the
solution. Initially, a position is assumed for the deformed groundwater surface.
The problem is then analysed using the assumed boundary and the pore-air
pressures are calculated at a number of points (finite element nodes) on or very
close to this boundary. The calculated pore-air pressures at various points on the
boundary are then compared with the corresponding hydrostatic water pressures.
If the pore-air and pore-water pressures are equal (within an acceptable degree of
accuracy) on all the nodes on the assumed ground water level, the assumed
boundary is accepted as the final position of the deformed groundwater table
under the equilibrium conditions. Otherwise, the assumed boundary is updated
and the process is repeated until the condition of equal air and water pressures on
all nodes on the free boundary is satisfied. In this way, the shape and position of
the deformed groundwater surface are determined in an iterative process, as the
location of points in the ground at which the air pressure balances the water
pressure.
Figure 2.4 shows a typical output of the two-dimensional model showing
contours of pore-air pressure heads. Figure 2.5 shows a typical output of the
three-dimensional model showing the deformed shape and position of the
groundwater table. The model calculates the pore-air pressure distribution in the
soil medium, the zone of influence of the air flow, the direction and velocity of
the flow of air at every point in this zone, and the final location of the
groundwater surface after the application of compressed air. The cumulative air
loss from the tunnel can be calculated from this information. 1, 11