2.3 Drawing a Flow Net for Flow Beneath an Impermeable Dam

Consider the steps for drawing a flow net through the homogeneous porous sand under an impermeable, concrete dam that is keyed into sand as shown in Figure 5. In the field, the porous geologic material below the dam extends a long distance in the upgradient and downgradient directions, but only a portion of it is illustrated here. The dam is 21 meters wide in the direction perpendicular to the figure. The water level in the reservoir contained by the dam is 10 meters above the surface of the low hydraulic conductivity material below the aquifer which is used as a datum. The water level in the reservoir below the dam is 4 meters lower than the water behind the dam and the water below the dam runs off downstream.

Figure 5 – A concrete dam keyed into soil.

We begin constructing a flow net by drawing the outline of the flow system to scale and labeling all boundary conditions (Figure 6). The low permeability concrete dam and layer underlying the sand are assumed to prevent flow from crossing those boundaries and so are labeled as no-flow boundaries (Figure 6). Any convenient level can be used as a datum for a flow net. In this case the horizontal bedrock surface below the dam provides a convenient reference for head measurements. An open body of water is hydrostatic, so the hydraulic head on the sand at the bottom of the reservoir is equal to the elevation of the reservoir water. So, these locations are constant-head boundaries with a head of 10 m on the ground surface upgradient of the dam and a head of 6 meters on the downgradient side (Figure 6). The lateral portions of the aquifer are not bounded so they must be drawn far enough from the dam so that no significant leakage occurs between the reservoirs and the underlying sand at the distant ends of the system. The highest rate of seepage into the sand will be immediately up gradient of the dam with seepage decreasing with distance up gradient. If, after constructing a flow net, it appears that the diagram is not wide enough, it can be redrawn with greater lateral extent from the dam until an acceptable flow net is obtained. Using knowledge of Darcy’s Law and the fact that flow is parallel to no-flow boundaries, one flow path can be drawn along the concrete dam from the upgradient to the downgradient reservoir (Figure 6).

Figure 6 – Step 1 – Draw the system to scale (no-flow boundaries are indicated by gray zones), Step 2 – Draw equipotential lines to coincide with head boundaries (black lines), Step 3 – Draw flow lines to coincide with no-flow boundaries (blue arrow following the no-flow boundary of the dam.

The next step is to envision how water is likely to move through the system and sketch some flow lines (Figure 7). The flow lines should be drawn perpendicular to the constant-head boundaries. Do not be concerned if your first attempt at sketching flow lines is not correct because errors in drawing the flow lines will show up as the equipotential lines are drawn, and can be corrected by erasing and redrawing until the flow net is correct. The first sketching of flow lines simply gets the process started.

Figure 7 – Step 4: Draw flow lines along paths where you envision groundwater flowing (blue arrows), ensuring they are perpendicular to equipotential lines on the boundaries. Do not be concerned about getting them right initially. Drawing flow nets is a trial-and-error process. As experience with flow nets grows, intuition improves and it becomes easier to place flow lines in nearly the right position on the first attempt.

Next, draw equipotential lines to show how hydraulic head varies from the constant-head boundary at the upstream reservoir to the constant-head boundary at the downstream reservoir. The equipotential lines need to be drawn perpendicular to both the no-flow boundaries and the flow lines. The equipotential lines and flow lines should intersect to form shapes with a constant aspect ratio, preferably “curvilinear squares”, quadrilaterals with curved sides and having an aspect ratio close to 1. Drawing a flow net by hand is a trial-and-error process because the equipotential lines and flow lines are adjusted until curvilinear squares are formed. It is useful to sketch round shapes within and touching the boundaries of the space formed by the equipotential lines and flow lines. If the shapes are not circular, as in the first attempt to draw a flow net shown in Figure 8, then the lines should be adjusted.

Figure 8 – Step 5: Draw equipotential lines between lines drawn at constant head boundaries (black lines), ensuring they are perpendicular to no-flow boundaries, perpendicular to flow lines and attempting to form curvilinear squares. Drawing a flow net by hand is a trial-and-error process because the equipotential lines and flow lines are adjusted until curvilinear squares are formed. It is useful to sketch round shapes within and touching the boundaries of the space formed by the equipotential lines and flow lines. If the shapes are not circular, as in this first attempt to draw the flow net, then the lines should be adjusted.

Adjust the position of flow lines and equipotential lines until a circle fills the space between the lines fairly well as in Figure 9. If an oval is needed to fill the space then it is not a curvilinear square. A slight misfit of the circles is not important. In order to make a difference to the estimation of flow through the system, the misfits need to be large enough such that it is necessary to add or delete flow lines or equipotential lines in order to obtain the near-curvilinear squares because achieving the proper ratio of the number of flow lines and equipotential lines is key to drawing a valid flow net. The number of flow lines is the same in Figure 8 and Figure 9, but the number of equipotential lines differ indicating the redrawing was necessary to obtain a flow net that can be used to calculate flow through the system.

Figure 9 – Creating shapes with a constant aspect ratio is a requirement when drawing a flow net. The best way to achieve that is by drawing curvilinear squares. Sketching a circle within the shapes can help discern whether the shapes are curvilinear squares. A slight misfit is not important. The misfits need to be large enough such that it is necessary to add or delete flow or equipotential lines in order to obtain the near-curvilinear squares as in the transition from the previous figure to this figure.

Once the flow net has an acceptable form, the next step is to calculate the values of the equipotential lines and label them. The equipotential lines represent hydraulic heads within the system between the boundary heads. The difference between the value of head on adjacent equipotential lines is called the contour interval. This interval is constant for the entire flow net. To determine the magnitude of the contour interval, first determine the total head drop across the flow net, H, and divide that by the number of head drops, n_d, in the flow net as shown in Equation 1.

contour interval =

(1)

where:

The total head drop across the system is:

H = 10 m – 6 m = 4 m

A head drop is represented by the zone between adjacent equipotential lines. The number of head drops is not arbitrary. It is determined by drawing a flow net while adhering to the rules regarding the placement of equipotential lines. In the flow net for flow under the concrete dam there are fourteen head drops (n_d =14) as shown in Figure 10.

Figure 10 – The appropriate number of head drops (spaces between equipotential lines) and flow tubes (spaces between flow lines) are determined by following the rules for drawing a flow net. This flow net has 14 head drops and 3 flow tubes.

This flow net drawing began with two internal flow lines, creating three flow tubes beneath the dam. A valid flow net can be drawn beginning with one, ten, or any number of flow tubes, as long as the appropriate number of equipotential lines are added to form shapes of constant aspect ratio, preferably curvilinear squares. No matter how many flow lines are drawn, the process of creating shapes of constant aspect ratio will yield approximately the same ratio of the number of flow tubes to the number of head drops. It is the ratio of the number of flow tubes to the number of head drops that is important. The ratio of flow tubes to head drops for the flow net of Figure 10 is 3/14 = 0.214. If the flow net is drawn with 2 flow tubes, 9 head drops will be needed to create curvilinear squares, for a ratio of 2/9 = 0.2222. If 5 flow tubes are used then 23 head drops will produce curvilinear squares, for a ratio of 5/28 = 0.217. These small differences in the ratio of flow tubes to head drops illustrate that drawing a flow net with paper and pencil yields an approximate solution.

The contour interval for the flow net is:

contour interval = = = 0.2857 m

The labeled equipotential lines are shown in Figure 11. The flow net does not provide precision to the 3 significant figures shown in the contour labels in the diagram. Three significant figures are shown, not because the system is known to high precision, but to adequately illustrate the difference in head between adjacent contour lines.

Figure 11 – Step 6: Calculate the equipotential line contour interval and label the equipotential lines. In this case the contour interval is ~0.29 meters.

It is useful to remember that the approximate solution provided by a hand drawn flow net is sufficient for practical applications because the error is slight compared with the uncertainty associated with assuming the material is homogeneous and isotropic, and with estimating the value of hydraulic conductivity.