176 Computational Modelling in Hydraulic and Coastal Engineering
normal derivatives are properly modified, to conform to non-uniform grids,
or the domain is extended to a node outside the flow domain, where the φ
values are extrapolated from the interior of the flow domain.
Example 7.1
The problem to be dealt with involves groundwater flow below a dam
with a sheet pile. The solution domain comprises of an aquifer of
known geometry, and the goal is to simulate the distribution of the
velocity potential φ(x, y) or the piezometric head, h. That information can help to estimate the seepage losses (water escaping under the
dam) and to assess the danger of downstream soil destabilization (piping). Also, since the zone under the dam usually is not perfectly sealed
and is semi-permeable, the flow under the dam needs to be investigated
(Figure 7.3).
The mathematical model consists of the Laplace equation in its
finite differences form (Equation 7.27). The discretization of the flow
domain is done using a rectangular grid. Boundaries of constant pressure head are the soil surface upstream and downstream of the dam.
Impermeable boundaries are the dam base (if sealed), the aquifer bed
φ i,s
φ i,j
Free surface
Δx
Δx
Δs
h i
Reference datum
i
j
x
Figure 7.2 Discretization of the free surface boundary.
normal derivatives are properly modified, to conform to non-uniform grids,
or the domain is extended to a node outside the flow domain, where the φ
values are extrapolated from the interior of the flow domain.
Example 7.1
The problem to be dealt with involves groundwater flow below a dam
with a sheet pile. The solution domain comprises of an aquifer of
known geometry, and the goal is to simulate the distribution of the
velocity potential φ(x, y) or the piezometric head, h. That information can help to estimate the seepage losses (water escaping under the
dam) and to assess the danger of downstream soil destabilization (piping). Also, since the zone under the dam usually is not perfectly sealed
and is semi-permeable, the flow under the dam needs to be investigated
(Figure 7.3).
The mathematical model consists of the Laplace equation in its
finite differences form (Equation 7.27). The discretization of the flow
domain is done using a rectangular grid. Boundaries of constant pressure head are the soil surface upstream and downstream of the dam.
Impermeable boundaries are the dam base (if sealed), the aquifer bed
φ i,s
φ i,j
Free surface
Δx
Δx
Δs
h i
Reference datum
i
j
x
Figure 7.2 Discretization of the free surface boundary.