How To Draw Flow Nets For Dams

2.3 Drawing a Flow Cyberspace for Flow Beneath an Impermeable Dam

Consider the steps for cartoon a menstruation cyberspace through the homogeneous porous sand under an impermeable, concrete dam that is keyed into sand as shown in Figure v. In the field, the porous geologic textile below the dam extends a long distance in the upgradient and downgradient directions, merely only a portion of information technology is illustrated here. The dam is 21 meters broad in the management perpendicular to the figure. The h2o level in the reservoir contained by the dam is 10 meters in a higher place the surface of the low hydraulic conductivity material below the aquifer which is used every bit a datum. The water level in the reservoir beneath the dam is 4 meters lower than the h2o behind the dam and the water beneath the dam runs off downstream.

Figure showing a concrete dam keyed into soil

Figure 5 – A concrete dam keyed into soil.

We begin constructing a flow internet 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 causeless to prevent menses from crossing those boundaries and so are labeled as no-menstruation boundaries (Figure 6). Any user-friendly level tin can be used as a datum for a flow net. In this case the horizontal boulder surface below the dam provides a convenient reference for head measurements. An open body of water is hydrostatic, so the hydraulic caput on the sand at the bottom of the reservoir is equal to the elevation of the reservoir h2o. So, these locations are constant-head boundaries with a caput 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 and so they must be fatigued far enough from the dam and so that no significant leakage occurs betwixt the reservoirs and the underlying sand at the distant ends of the organisation. The highest rate of seepage into the sand will be immediately up gradient of the dam with seepage decreasing with altitude upwards gradient. If, afterwards amalgam a flow cyberspace, it appears that the diagram is not broad enough, it can be redrawn with greater lateral extent from the dam until an acceptable menstruation internet is obtained. Using knowledge of Darcy'southward Police force and the fact that flow is parallel to no-menstruum boundaries, one menstruum path tin be drawn forth the concrete dam from the upgradient to the downgradient reservoir (Figure 6).

Figure showing steps 1 to 3 of graphical construction of flow net

Figure 6 – Step ane – Draw the organisation to scale (no-flow boundaries are indicated past grayness zones), Footstep two – Describe equipotential lines to coincide with head boundaries (blackness lines), Footstep 3 – Depict flow lines to coincide with no-catamenia boundaries (blue arrow following the no-flow purlieus of the dam.

The next step is to envision how water is probable to motion through the system and sketch some flow lines (Figure vii). The menstruum lines should be drawn perpendicular to the constant-head boundaries. Exercise not exist concerned if your offset try at sketching flow lines is not right because errors in drawing the flow lines will bear witness upward as the equipotential lines are drawn, and tin can be corrected past erasing and redrawing until the catamenia cyberspace is correct. The first sketching of flow lines simply gets the process started.

Figure showing step 4 of graphical construction of flow net

Figure 7 – Pace four: Describe menstruum lines along paths where you envision groundwater flowing (blue arrows), ensuring they are perpendicular to equipotential lines on the boundaries. Do not be concerned well-nigh getting them right initially. Cartoon 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-caput boundary at the downstream reservoir. The equipotential lines demand to be drawn perpendicular to both the no-menstruation boundaries and the menses 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. Cartoon a flow net by mitt is a trial-and-error process because the equipotential lines and flow lines are adapted until curvilinear squares are formed. It is useful to sketch round shapes within and touching the boundaries of the space formed past the equipotential lines and flow lines. If the shapes are not circular, every bit in the first endeavor to draw a period net shown in Figure 8, so the lines should be adjusted.

Figure showing step 5 of graphical construction of flow net

Figure 8 – Step 5: Draw equipotential lines between lines drawn at abiding head boundaries (black lines), ensuring they are perpendicular to no-menstruation boundaries, perpendicular to flow lines and attempting to form curvilinear squares. Drawing a flow net by hand is a trial-and-error procedure because the equipotential lines and flow lines are adapted until curvilinear squares are formed. It is useful to sketch round shapes inside and touching the boundaries of the infinite formed by the equipotential lines and flow lines. If the shapes are not circular, as in this beginning attempt to describe the period cyberspace, then the lines should be adjusted.

Adjust the position of menstruation lines and equipotential lines until a circle fills the space between the lines fairly well equally in Effigy nine. If an oval is needed to fill the space and then it is not a curvilinear square. A slight misfit of the circles is non of import. In order to make a deviation to the estimation of catamenia through the arrangement, the misfits need to be large enough such that it is necessary to add or delete period lines or equipotential lines in order to obtain the near-curvilinear squares because achieving the proper ratio of the number of period lines and equipotential lines is primal to cartoon a valid flow net. The number of flow lines is the same in Figure 8 and Effigy nine, but the number of equipotential lines differ indicating the redrawing was necessary to obtain a menstruum net that tin can be used to calculate menses through the system.

Figure showing flow net after final adjustment of flow lines and equipotential lines

Figure 9 – Creating shapes with a abiding aspect ratio is a requirement when cartoon a menses cyberspace. The all-time way to reach that is past drawing curvilinear squares. Sketching a circumvolve within the shapes can aid discern whether the shapes are curvilinear squares. A slight misfit is not of import. The misfits demand to be large plenty such that information technology is necessary to add or delete menstruation 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 adequate form, the next footstep is to summate the values of the equipotential lines and label them. The equipotential lines represent hydraulic heads within the organisation between the boundary heads. The difference betwixt the value of head on adjacent equipotential lines is called the profile interval. This interval is abiding for the entire catamenia net. To determine the magnitude of the contour interval, beginning make up one's mind the total head drop across the flow net, H, and split that past the number of head drops, north _d, in the flow internet as shown in Equation i.

contour interval = $\displaystyle \frac{H}{n_{d}}$

(1)

where:

contour interval	=	head difference between side by side equipotential lines (Fifty)
H	=	total head drop across the flow net domain (Fifty)
n _d	=	number of head drops in the flow net (dimensionless)

The total caput driblet beyond the organisation is:

H = 10 m – half dozen yard = four thousand

A caput drop is represented past the zone between adjacent equipotential lines. The number of caput drops is not capricious. Information technology is determined by drawing a flow net while adhering to the rules regarding the placement of equipotential lines. In the menses net for flow under the concrete dam in that location are fourteen head drops (due north _d =14) every bit shown in Figure 10.

Figure showing the number of head drops in a flow net

Effigy ten – The advisable number of head drops (spaces between equipotential lines) and flow tubes (spaces between menses lines) are adamant past following the rules for drawing a flow net. This menses net has 14 head drops and 3 menstruum tubes.

This catamenia cyberspace drawing began with 2 internal period lines, creating three flow tubes beneath the dam. A valid flow cyberspace can be fatigued showtime with one, ten, or whatsoever number of flow tubes, equally long as the appropriate number of equipotential lines are added to form shapes of constant attribute 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 period tubes to the number of head drops that is of import. The ratio of flow tubes to caput drops for the flow internet of Figure ten is iii/14 = 0.214. If the menstruum net is drawn with ii flow tubes, ix 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 caput drops will produce curvilinear squares, for a ratio of 5/28 = 0.217. These small-scale differences in the ratio of flow tubes to caput drops illustrate that drawing a flow internet with paper and pencil yields an estimate solution.

The contour interval for the flow net is:

contour interval = $\displaystyle \frac{H}{n_{d}}$ = $\displaystyle \frac{4\ \textup{m}}{14}$ = 0.2857 m

The labeled equipotential lines are shown in Figure 11. The menstruation net does non provide precision to the 3 pregnant figures shown in the profile labels in the diagram. Three meaning figures are shown, non because the system is known to high precision, merely to adequately illustrate the difference in caput between adjacent contour lines.

Figure showing step 6 of graphical construction of flow net

Effigy 11 – Stride vi: Calculate the equipotential line contour interval and characterization the equipotential lines. In this instance the contour interval is ~0.29 meters.

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

Source: https://books.gw-project.org/graphical-construction-of-groundwater-flow-nets/chapter/drawing-a-flow-net-for-flow-beneath-an-impermeable-dam/

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