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Introduction Many geotechnical engineering problems include the flow of water around a cut-off, which is installed to reduce the up-lift pressure on the dam and reduce the possibility of internal erosion along the soilstructure contact. This example will illustrate the ability of SEEP/W to simulate the flow regime of a soil, where a cut-off exists below the upstream side of an embankment.

Numerical Simulation Figure 1 shows the problem configuration, where an embankment is used to retain water on the left side of the domain. Water can seep into the ground from the reservoir retained on the left and eventually exit on the ground surface beyond the downstream toe. The cut-off is modelled using interface elements, as shown in Figure 2. Interface elements are created with the Draw Mesh Properties command. Both the ‘Generate mesh along line’ option and the ‘Generate interface elements’ option must be selected. The thickness of the interface elements is set to 1m, but is arbitrary in this case. Material properties can be assigned to interface elements the same as ordinary elements. Materials are assigned by clicking near the Line on both sides.

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Figure 1. Problem configuration.

Figure 2. Interface elements used to represent cut-off.

As this is a steady-state analysis and the soil will remain saturated, the Saturated Only material model is used. For this example, the soil is considered to be isotropic and homogenous, with a saturated hydraulic conductivity of 0.001 ft/sec (≈ 3 x 10-2 cm/sec). The intention in this example is to prevent water from seeping through the cut-off; that is, it is perfectly impermeable to water flow. In SEEP/W, this can be achieved by assigning a material to the interface elements with a Material Model set to ‘none’ (Figure 3).

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Figure 3. Material model for perfectly impermeable flow.

Setting the Material Model to ‘none’ is a flag in the computer code to treat these elements as missing. Physically, a missing element is analogous to a perfectly impermeable element (if the element is not present, no flow can cross the element). When the mesh is displayed, elements with a Material Model set to ‘none’ are shown as light grey. When the mesh is not displayed the selected material color is shown (compare Figure 1 with Figure 2). A total head boundary condition of 60 ft was applied to the surface of the soil under the reservoir to represent a pressure head of 20 ft. The soil surface on the downstream side of the embankment was given a boundary condition of total head equals 40 ft so that the surface remains saturated, or the pressure head remains at 0 ft.

Results and Discussion Total head contours are equivalent to equipotential lines; that is, the energy potential along a contour is constant. Figure 4 shows the resulting total head contours at 2-foot intervals.

Figure 4. Resulting total head contours and flow paths.

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Flow paths were added to the solution by using the Draw Flow Paths command (Figure 4). A flow path is a line that a droplet of water will follow from where it enters the domain to where it exits the domain. Notice that the Flow Paths cross the equipotential lines at right angles, as in flow-net construction. This condition only occurs if the soil is homogeneous and isotropic. The results look like a flow net, but this is not a true flow net. The position of the flow paths was manually selected to make it look like a flow net. In a true flow net the amount of water flow in each flow channel must be the same. Manually, positioning the flow paths does not ensure that this is exactly true. While SEEP/W does not create a technical true flow-net, combining the total head contours with flow paths makes it possible to create a reasonable resemblance of a flow-net, which can be useful for interpretation and presentation purposes. The blue shading above the mesh domain in Figure 4 is a reflection of the pressure along the top surface of the mesh. Notice the sharp drop across the cut-off. Another way of portraying the effect of the cutoff on the pressure regime is to draw a graph along the surface of the soil under the reservoir and embankment (Figure 5). Once again, notice the sharp pressure drop across the cut-off at the 50-foot mark, where the cut-off is located. Uplift pressure 1,300

Pore-Water Pressure (psf)

1,200 1,100 1,000 900 800 700 600 500 400

0

20

40

60

80

100

120

X (ft)

Figure 5. Uplift pressure at the base of the reservoir and embankment.

The amount of seepage can also be determined using a special graphing technique. Figure 6 shows the inflow to the domain at each node along the region edge under the reservoir. The same line can then be used to create a graph of water rate versus time (Figure 7).

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0.0004

Water Rate (ft³/sec)

0.00035

0.0003

0.00025

0.0002

0.00015

0.0001

5e-005 0

10

20

30

40

50

X (ft)

Figure 6. Nodal water rate values at base of reservoir.

0.0004

Water Rate (ft³/sec)

0.00035

0.0003

0.00025

0.0002

0.00015

0.0001

5e-005 -1

0

1

X (ft)

Figure 7. Nodal flow values versus time.

By selecting the “sum (Y) vs. Average (X)” option in the Draw Graph window (Figure 8), the resulting water rate graph versus time window is created (Figure 9). Hovering over the data point then gives the total inflow, which in this case is approximately 5.63x10-3 ft3/sec. This is a positive value, signifying flow is moving into the system. Selecting the downstream ground surface region edge gives the same value but with a negative sign indicating flow out of the same.

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Figure 8. Summation of nodal flow values.

0.0062

Water Rate (ft³/sec)

0.006

0.0058

0.0056

0.0054

0.0052

0.005 -1

0

1

X (ft)

Figure 9. Total water flow into the domain under the reservoir.

Another way to view the flow regime is to look at flow vectors (Figure 10). The vectors give a sense of direction and relative rates. The longest vectors are at the lower tip of the cut-off, indicating that this is the point where the flow is the fastest. Another point of concentration is the right-lower corner of the embankment. In the lower left and right corners of the domain, the vectors are so short they are not visible at the displayed scale. This is an indication that the flow in these corners is nearly stagnant.

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Figure 10. Flow vectors added to the results window.

It is worth noting that the pore-water pressure distribution in this case is not influenced by the hydraulic conductivity. Changing the hydraulic conductivity will not change the position of the total head contours with all else being equal. The quantity of the seepage however will be directly proportional to the hydraulic conductivity, even though the pressure distribution does not change. This is also evident from examining the governing partial differential equation, which for steady-state conditions is: 𝐾𝑥

∂ 2ℎ ∂𝑥2

+ 𝐾𝑦

∂ 2ℎ ∂𝑦2

=0

Equation 1

where 𝐾𝑥 and 𝐾𝑦 is the hydraulic conductivity in the x- and y-direction and ℎ is the hydraulic head. For isotropic, homogeneous conditions, 𝐾𝑥 equals 𝐾𝑦. We can then write the equation as: 𝐾

∂ 2ℎ ∂𝑥2

+𝐾

∂ 2ℎ ∂𝑦2

=0

Equation 2

Now we can divide both sides of the equation by k, and then the result is: ∂ 2ℎ ∂𝑥2

+

∂ 2ℎ ∂𝑦2

=0

Equation 3

This is the Laplacian equation, which describes the pressure distribution; it does not include the hydraulic conductivity.

Summary and Conclusions In this example, a steady-state water transfer analysis was developed to simulate the flow regime of a simple cut-off under an embankment. The example highlights the use of flow paths and total head 7

contours to create an image that resembles a flow net. Remember, the flow paths are not flow lines. The pore-water pressure distribution could also be replicated using a different hydraulic conductivity, with a change in the water rates.

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