Nonetheless, flow nets can still be graphically constructed if the manner in which the hydraulic conductivity varies with direction is the same everywhere in the flow system (that is, if the hydraulic conductivity is “homogeneously anisotropic”). Often, both equipotential lines and flow lines need to be erased and redrawn repeatedly before achieving curvilinear squares with equipotential lines meeting the water table at right angles and at an elevation equal to their value. Even after adjustment, a hand drawn flow net is only an approximate solution to the flow equations. For the purpose of this book, a fairly precise flow net is shown as Figure Box 4-5. 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.
The act of drawing the flow net must be done for a homogeneous and isotropic system; however, there is a procedure for scaling an anisotropic system so that an isotropic flow net can be drawn and then transformed into an anisotropic flow net. However, for cases in which the hydraulic conductivity is non-homogeneous (i.e., heterogeneous), constructing a flow net requires a numerical method using a computer. 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). In this case the horizontal bedrock surface below the dam provides a convenient reference for head measurements.
BOX 4 – Drawing a Flow Net For an Unconfined System With a Water Table Boundary
As such, a grid obtained by drawing a series of equipotential lines is called a flow net. The flow net is an important tool in analysing two-dimensional irrotational flow problems. The construction of a groundwater flow net begins with the definition of the flow domain and the boundary conditions along the domain boundary. Two common boundary conditions are (1) constant hydraulic head along the boundary and (2) no flow across the boundary. After the boundary conditions are specified, the flow net is constructed by following an iterative, step-by-step procedure. The procedure for constructing a graphical flow net does not accommodate boundaries with a defined flux other than zero.
- 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.
- We assume the pressure is atmospheric in the drain (that is, the water flowing to the drain discharges at the end of the drain without backing up water in the drain).
- Nonetheless, these computer-generated equipotential lines and flow lines form a bona fide flow net, because they satisfy the groundwater flow equation.
- In that case, the flow lines and equipotential lines of a flow net will not meet at right angles.
- The difference between the value of head on adjacent equipotential lines is called the contour interval.
5 Drawing a Flow Net for an Unconfined System with a Water Table Boundary
The heterogeneous case has a lower-K zone within the aquifer in Figure 17a and a higher-K zone in Figure 17b. Anisotropy can occur in a horizontal flow net as well as in a vertical one. Anisotropy in the horizontal plane is generally the result of a directional fabric in the material such as fracture planes.
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.
In contrast to the previous section, material at the ground surface is impermeable, and earth material is brought in from nearby to construct a dam, so in this case water flows through the dam instead of under the dam. The dam surface draw flow nets is sealed to prevent infiltration of water into the dam structure. There is no flow across flow lines so a water table without recharge can be viewed as a no flow boundary of unknown position until after the flow net is drawn.
BOXES
This method involves using equations for boundary conditions and solutions of Laplace equations obtained by mathematical procedures. Boundary equations are known only for relatively simple cases, but due to the mathematical calculations present, this method could make simple cases difficult to solve. Gradient for any flow field is given by h/l, where h is the head lost in that field and l is the length of the field.
There can be any number of flow lines that a fluid particle can take within a soil mass. If equipotential lines are drawn at equal intervals, it means that the headloss between any two equipotential lines is the same. A system of flow lines and equipotential lines constitutes the flow net. Travel time of a packet of water traversing a flow line is indicated with blue numbers next to the arrow heads of the flow lines. The travel time is 3.1 years from entry to exit all packets traversing system with completely open boundary (Figure 16a).
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 Box 5-3 – A groundwater flow system beneath an irrigated field with long, parallel drains. Which is about 125 oil drums full of water each day, and would take about 100 days to fill an Olympic-size swimming pool. As noted earlier, it is important to recognize that the volumetric flow rate determined from a flow net is an approximate value. The second flow net pictured here (modified from Ferris, et al., 1962) shows a flow net being used to analyze map-view flow (invariant in the vertical direction), rather than a cross-section. Note that this problem has symmetry, and only the left or right portions of it needed to have been done. To create a flow net to a point sink (a singularity), there must be a recharge boundary nearby to provide water and allow a steady-state flowfield to develop.
1 What is Graphical Construction Of a Flow Net?
Figure 9 – Creating shapes with a constant aspect ratio is a requirement when drawing a flow net. Sketching a circle within the shapes can help discern whether the shapes are curvilinear squares. 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. Figure Box 4-4 – The shapes are curvilinear squares if circles fit approximately within them, but some flow nets may include partial flow tubes as shown here by the narrow flow tube at the bottom of the flow net. Since the head drops are uniform by construction, the gradient is inversely proportional to the size of the blocks. Big blocks mean there is a low gradient, and therefore low discharge (hydraulic conductivity is assumed constant here).
- Two types of boundary conditions are used in graphical construction of two-dimensional, steady-state flow nets.
- Atmospheric pressure is used as the zero-reference point for quantifying pressure so, at the drain, the pressure is zero and the hydraulic head is equal to the elevation.
- 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.
- Figure 13 – Click here to go to Box 4 which describes the procedure for drawing a flow net with a water table boundary.
BOX 5 – Drawing Flow Nets for Anisotropic Systems
Unconfined groundwater systems have a water table boundary which requires special consideration when drawing a flow net because the location of the water table boundary is not known until after the flow net construction is completed. Such systems may also have a seepage face, where groundwater seeps out along a sloping section of ground surface. The position of the water table and the length of the seepage face need to be adjusted along with the flow and equipotential lines while drawing the flow net.
7 Flow Nets Provide Insight into Groundwater Flow
The horizontal hydraulic conductivity, Kx, is 0.16 m/d and the vertical hydraulic conductivity, Ky, is 0.01 m/d. The ground surface elevation is 0.6 m above bedrock, and the centers of the 0.1 m-diameter circular drains are 0.2 m above bedrock (so the bottom of each drain is at 0.15 m and the top is at 0.25 m). Now, suppose the field is flooded to a water elevation 0.8 m above bedrock. To understand the rationale behind the geometric transformation, it is useful to consider how hydraulic conductivity varies with direction.
Begin the steps for drawing a flow net as described in section 2.3 and shown in Figure Box 4-1. The position of the water table is not known until the flow system is revealed by following the rules for drawing a flow net, so the initial sketch indicates this uncertainty by using a dashed line with “? Irregular points (also called singularities) in the flow field occur when streamlines have kinks in them (the derivative doesn’t exist at a point). An equivalent amount of flow is passing through each streamtube (defined by two adjacent blue lines in diagram), therefore narrow streamtubes are located where there is more flow.