FLAC3D Theory and Background • Fluid-Mechanical Interaction

# Unsteady Groundwater Flow in a Confined Layer

Note

To view this project in FLAC3D, use the menu command . The project’s main data files are shown at the end of this example.

A long embankment of width \(L\) = 100 m rests on a shallow saturated layer of soil. The width
(\(L\)) of the embankment is large in comparison to the layer thickness, and its permeability is
negligible when compared to the permeability, \(k\) = 10^{-12} m^{2}/(Pa sec), of the
soil. The Biot modulus for the soil is measured to be \(M\) = 10 GPa. Initial steady-state conditions
are reached in the homogeneous layer. The purpose is to study the pore-pressure change in the layer as the
water level is raised instantaneously upstream by an amount \(H_0\) = 2 m. This corresponds to a
pore-pressure rise of \(p_1 = H_0 {\rho}_w g\) (with the water density \({\rho}_w\) = 1000 kg/m^{3} and acceleration of gravity \(g\) = 10 m/s^{2}) at the upstream side of the embankment.
Figure 1 shows the geometry of the problem:

The flow in the layer may be assumed to be one-dimensional. The model has width \(L\). The excess pore pressure, \(p\), initially zero, is raised suddenly to the value \(p_1\) at one end of the model. The corresponding analytical solution has the form

where the \(z\)-axis is running along the embankment width and has its origin at the upstream side, \(\hat{p} = {p\over p_1}\), \(\hat{z} = {z \over {L}}\), \(\hat{t} = c t/L^2\), and \(c = M k\).

In the FLAC3D model, the layer is defined as a column of 25 zones. The excess pore pressure is fixed at the value of 2 × 10^{4} Pa at the face located at \(z\) = 0, and at zero at the face located at \(z\) = 100 m. The model grid is shown in Figure 2.

The analytical solution is programmed as a FISH function for direct comparison to the numerical results at selected fluid-flow times corresponding to \(\hat{t}\) = 0.05, 0.1, 0.2 and 1.0. The analytical and numerical pore-pressure results for these times are stored in tables.

UnsteadyGroundwaterFlowConfinedLayer.dat
contains the FLAC3D data file for this problem, using the implicit servo to automatically switch from explicit
to implicit and increase the implicit timestep. UnsteadyGroundwaterFlowConfinedLayer-Steady.dat
contains the data file using the `zone fluid steady-state`

command to immediately find the steady state solution.

The comparison of analytical and numerical excess pore pressures at four fluid-flow times for the transient solution is shown in Figure 3.
Normalized excess pore pressure (\(p/p_1\)) is plotted versus normalized distance (\(z/L\)) in
the figure, where Tables 2, 4, and 6 contain the analytical solution for excess pore pressures, and Tables 1, 3, and 5 contain the FLAC3D solutions. The four flow times are 5 × 10^{4}, 10^{5}, 2 × 10^{5}, and 10^{6} seconds. Steady-state conditions are reached by the last time considered. The difference between analytical and numerical pore pressures at steady state is less than 0.2%.

break

Data File

**UnsteadyGroundwaterFlowConfinedLayer.dat**

```
model new
model title 'Unsteady groundwater flow in a confined layer: Implicit Method'
zone create brick size 1 1 25 point 1 (10 0 0) point 2 (0 10 0) ...
point 3 (0 0 100)
zone face skin
; --- fluid flow model ---
model configure fluid-flow
zone fluid property mobility-coefficient 1e-12 biot-modulus 1e10
zone face apply pore-pressure 2e4 range group 'Bottom'
zone face apply pore-pressure 0 range group 'Top'
; --- settings ---
zone results pore-pressure on
; Switch to implicit servo and increase dt automatically.
zone fluid implicit servo on
model solve-fluid time 5e4
model results export 'confinedlayer-005'
model solve-fluid time 5e4
model results export 'confinedlayer-010'
model solve-fluid time 10e4
model results export 'confinedlayer-020'
model solve-fluid time 80e4
model results export 'confinedlayer-100'
model save 'confinedlayer'
```

**SteadyState.dat**

```
model new
model title 'Unsteady groundwater flow in a confined layer: Implicit Method'
zone create brick size 1 1 25 point 1 (10 0 0) point 2 (0 10 0) ...
point 3 (0 0 100)
zone face skin
; --- fluid flow model ---
model configure fluid-flow
zone fluid property mobility-coefficient 1e-12
zone face apply pore-pressure 2e4 range group 'Bottom'
zone face apply pore-pressure 0 range group 'Top'
; --- settings ---
zone fluid steady-state
;
model save 'confinedlayer-steadystate'
```

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