Box 5 Drawing Flow Nets for Anisotropic Systems Graphical Construction of Groundwater Flow Nets

Mathematically, the process of constructing a flow net consists of contouring the two harmonic or analytic functions of potential and stream function. These functions both satisfy the Laplace equation and the contour lines represent lines of constant head (equipotentials) and lines tangent to flowpaths (streamlines). Together, the potential function and the stream function form the complex potential, where the potential is the real part, and the stream function is the imaginary part. The geometric transformation from an anisotropic system to an isotropic system can be viewed as transforming the hydraulic conductivity ellipse into a circle.

When the water table is a flow line, equipotential lines meet the water table at right angles. 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. 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. To illustrate inclusion of a water table in a flow net, consider flow through an earthen dam with a hydraulic conductivity of 0.2 m/d resting on an impermeable base (Figure Box 4-1).

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 video illustrates the process of drawing a groundwater flow net below a dam from a headwater reservoir on the left to a tailwater reservoir on the right. A cutoff wall protruding below the base of the dam increases the flow path length, decreasing the flow velocity, thus reducing the potential for the groundwater flow to erode the porous material at the toe of the dam. (13 minutes)A discussion of the material in this video is provided in Section 2.6 “The “Hear See Do” of Flow Nets” of the Groundwater Project book “Graphical Construction of Groundwater Flow Nets”.

• This video illustrates the process of drawing a groundwater flow net below a dam from a headwater reservoir on the left to a tailwater reservoir on the right.
• For an anisotropic system in a plan view, it is necessary to know the principal directions and align the x–y coordinate system to these directions.
• Big blocks mean there is a low gradient, and therefore low discharge (hydraulic conductivity is assumed constant here).
• The next step is to envision how water is likely to move through the system and sketch some flow lines (Figure 7).
• A flow net is a graphical representation of two-dimensional steady-state groundwater flow through aquifers.
• If Ky is less than Kx, the circle will be larger than the original ellipse (and circumscribe it), whereas if Ky is greater than Kx, the circle will be smaller than the original ellipse and the ellipse will circumscribe the circle.

The groundwater flow equation is based on Darcy’s Law and conservation of mass. The groundwater flow equation is derived and discussed in another Groundwater Project book (Woessner and Poeter, 2020). A brief overview of Darcy’s Law, specific discharge, average linear groundwater velocity, and groundwater travel time are provided in Box 2 because the concepts are central to the material presented in this book. In that case, the flow lines and equipotential lines of a flow net will not meet at right angles.

Exit Gradient (iexit)

Computer processing could also be used with this method to get a rapid solution. To provide additional practice in visualizing flow nets, this book uses the online computer software, TopoDrive, which simulates groundwater flow in a topographically-driven flow system. The total head drop, H, is estimated as 0.6 m (that is, the difference between the 0.8 m head at the ground surface and the average 0.2 m head along the drain). The average head along the drain is estimated as 0.2 m because the head at the top of the drain is 0.25 m, the center is 0.2 m, and the bottom is 0.1 m. Figure Box 5-5 – Geometric transformation of the (a) anisotropic system on the left to an (b) isotropic system on the right by transforming the vertical axis. The transformed geometry on the right is 4 times taller and as wide as the left one.

In Figure 16b, the outlet is restricted to only a portion of the right side. For groundwater to exit the aquifer, flow tubes narrow near the outlet opening. The narrowing of flow tubes is accompanied by an increase in hydraulic gradient (equipotential lines closer together) in accordance with Darcy’s Law. Hydraulic head contours range from 10 to 21 meters and are labeled in black.

• A flow net can also be constructed for two-dimensional flow in a plan view.
• Figure Box 5-2 – Transformation of an anisotropic hydraulic conductivity ellipse (center) into an isotropic ellipse (circle) by either transforming the x-axis (left) or the y-axis (right).
• Note that this problem has symmetry, and only the left or right portions of it needed to have been done.
• Because the same flow pattern repeats itself in alternating mirror images throughout the field, only a small portion of the drained field needs to be drawn to develop a flow net (Figure Box 5-4).
• This method involves the construction of a scaled model for studying a flow problem.

These types of points often do make other types of solutions (especially numeric) to these problems difficult, while the simple graphical technique handles them nicely. With this, it is understood that the flow net gives a pictorial representation of the path taken by a flow particle and the head variation along the path. Stay informed about new book releases, events, and ways to participate in the Groundwater Project. The flow needs to be doubled to account for drainage from both sides of the drain, so approximately 2.4 liters per minute. Either transform results in an acceptable isotropic geometry for the system as shown in Figure Box 5-2.

For example, in an aquifer with homogeneous and isotropic hydraulic conductivity, computer-generated equipotential lines cross flow lines at right angles. Numerically generated flow nets are usually used to display flow patterns rather than to compute flow rates, because flow rates are calculated by numerically solving the groundwater flow equations. Two requirements need to be kept in mind when drawing the equipotential and flow lines in order to obtain an accurate solution to the groundwater flow equation. First, the equipotential lines and the flow lines need to intersect at right angles.

Graphical Solution

The downstream end of the water table should meet the dam surface at an elevation higher than the surface of the downstream reservoir (forming a seepage face as shown in Figure Box 4-2), and at a slope equal to the slope of the dam face. In the same way that the initial position of the water table is unknown until after the flow net is drawn, the length of the seepage face is unknown until after a valid flow net is drawn. Hydraulic head along a seepage face is equal to the elevation of the ground surface because the gage pressure along the seepage face is zero. Unlike the water table, the location of the seepage face boundary is known because it will be on the downgradient face of the dam, only its length is unknown before sketching the flow net.

A flow net can also be constructed for two-dimensional flow in a plan view. An important assumption for graphical construction of a flow net in a plan view is the absence of areally distributed recharge, such as infiltration of precipitation to the flow system. Figure 4 illustrates a plan view of a flow net between a lake and pond in an area constrained by bedrock. If the rate of recharge is trivial relative to the volumetric lateral flow from the lake to the pond, then the flow net is a sufficiently accurate for evaluating the groundwater system. Figure Box 5-2 – Transformation of an anisotropic hydraulic conductivity ellipse (center) into an isotropic ellipse (circle) by either transforming the x-axis (left) or the y-axis (right). Therefore, the use of electrical models for solving complex fluid flow problems is common.

A flow net is a graphical representation of two-dimensional steady-state groundwater flow through aquifers. Figure 13 – Click here to go to Box 4 which describes the procedure for drawing a flow net with a water table boundary. Figure Box 5-7 – The (b) isotropic flow net on the right is transformed back to the (a) anisotropic geometry on the left. Figure Box 5-6 – An isotropic flow net is drawn in the transformed isotropic system (b) on the right. For example, the best-known theoretical solution was given by Kozeny for flow through an earthen dam with a filter drain at the base towards the downstream side.

8 Drawing a Flow Net For a System With Anisotropic Hydraulic Conductivity

In an electrical model, voltage represents the total head, current to velocity and conductivity to permeability. Ohm’s law is analogous to Darcy’s law, therefore measuring voltage can locate equipotential lines. Figure Box 4-1 – Step 1 – Draw the system to scale, Step 2 – Draw equipotential lines to coincide with head boundaries, Step 3 – Draw flow lines to coincide with no-flow boundaries. 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 an acceptable flow net has been drawn.

2 Drawing a Flow Net for a Homogeneous Isotropic System

When transforming the y axis, we multiply the y-coordinates of the ellipse by the ratio . That is, any point (x,y) in the original coordinate system will be moved to a point (x,Y) in the transformed coordinate system where, Y is defined in Equation Box 5-1. This method involves the construction of a scaled model for studying a flow problem. Piezometer tubes can be used for the determination of the heads at various points. Figure Box 4-6 – Some flow nets may include partial flow tubes as shown here by the narrow flow tube at the bottom of the flow net. Although, for expediency, we have drawn the flow lines in the correct position here, it is likely the first attempt to draw flow lines will require adjustment when drawing a flow net.

However, the principal directions for flow in the plan view might not be as obvious as for flow in a vertical cross section (as above example). The principal directions in a vertical cross section are often (but not always) taken to be horizontal and vertical because many subsurface settings consist of horizontal layers. By contrast, the principal directions for flow in a plan view are generally not in east-west/north-south directions. For an anisotropic system in a plan view, it is necessary to know the principal draw flow nets directions and align the x–y coordinate system to these directions. The geometric transformation can then be carried out for flow net construction. A flow line is a line along which a fluid particle flows within a flow line.

1 What is Graphical Construction of a Flow Net?

Because the water pressure is equal to the atmospheric pressure at the water table, the equipotential lines need to intersect the water table at the elevation equal to the value of the equipotential line label. A key difference between graphical versus numerical construction of a flow net is that the graphical method requires creating both equipotential lines and flow lines, whereas the numerical method does not. Groundwater professionals commonly use a groundwater model to compute hydraulic head, then later use a flow path tracking model (also known as a particle tracking model) to compute flow lines.

Perpendicular to maximum principal direction is the minimum principal direction, along which Kd attains its minimum value, denoted as Kmin. Development and explanation of the hydraulic conductivity ellipse is provided in Groundwater Project book (Woessner and Poeter, 2020). Drawing a flow net is a trial-and-error process because equipotential and flow lines are adjusted until curvilinear squares are formed. The added complication when drawing an unconfined flow net is that the position of the upper boundary (the water table) and length of the seepage face are also adjusted while working to create curvilinear squares.

A flow net is drawn in the transformed section (Figure Box 5-6) according to the steps of flow net construction under isotropic conditions as described in section 2.2 of this book. We know the hydraulic head at the ground surface is equal to the elevation of the ponded water (0.8 m). 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). Hydraulic head is the sum of pressure head in terms of a height of a column of water and elevation.