Box 5 – Drawing Flow Nets for Anisotropic Systems

Hydraulic conductivity in an anisotropic aquifer varies with direction. In that case, the flow lines and equipotential lines of a flow net will not meet at right angles. 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”). Under this circumstance, a flow net can be graphically constructed by (1) transforming the geometry of the system to an isotropic system, (2) drawing a flow net for the isotropic system, and (3) transforming the flow net back to the original anisotropic system.

To understand the rationale behind the geometric transformation, it is useful to consider how hydraulic conductivity varies with direction. If we extract a core sample from a porous medium along a given direction and measure the hydraulic conductivity along the longitudinal axis of the core, we obtain the directional hydraulic conductivity (K_d) in that direction. If hydraulic conductivity is isotropic, then K_d is the same in all directions. If hydraulic conductivity is anisotropic, then K_d varies with direction. The direction in which K_d attains its maximum value is known as the maximum principal direction. The K_d in this direction is denoted as K_max. Perpendicular to maximum principal direction is the minimum principal direction, along which K_d attains its minimum value, denoted as K_min. Development and explanation of the hydraulic conductivity ellipse is provided in Groundwater Project book (Woessner and Poeter, 2020).

A polar-coordinates plot is a useful way to illustrate K_d. In such a plot, the square root of K_d is plotted in all directions as distance from the origin. In the isotropic case, the result is a circle. In the anisotropic case, the result is an ellipse, known as the hydraulic conductivity ellipse (Figure Box 5-1a). The major and minor axes of the ellipse are aligned respectively in the maximum and minimum principal directions of K_d. In the general case, the ellipse can be in any orientation. Figure Box 5-1b, shows the case in which the ellipse’s major and minor axes are aligned with a rectangular (x–y) coordinate system. In this case, K_max and K_min can be written as K_x and K_y (if K_max is associated with the x direction) or K_y and K_x (if K_max is associated with the y direction). It is for this setting that we will illustrate the geometric transformation procedure below.

Figure Box 5-1 – Hydraulic conductivity ellipse (a) in general orientation and (b) with major and minor axes aligned with the rectangular (x–y) coordinate system.

The geometric transformation from an anisotropic system to an isotropic system can be viewed as transforming the hydraulic conductivity ellipse into a circle. This can be done by transforming either the y axis or the x axis. 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.

(Box 5-1)

If K_y is less than K_x, the circle will be larger than the original ellipse (and circumscribe it), whereas if K_y is greater than K_x, the circle will be smaller than the original ellipse and the ellipse will circumscribe the circle.

Alternatively, we could multiply the x-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, X is defined by Equation Box 5-2.

(Box 5-2)

Either transform results in an acceptable isotropic geometry for the system as shown in Figure Box 5-2.

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).

An example is provided to make the process clear. Suppose you want to draw a flow net for an irrigated field with many parallel drains as shown in Figure Box 5-3. In this system, the drains are 100 m long. The horizontal hydraulic conductivity, K_x, is 0.16 m/d and the vertical hydraulic conductivity, K_y, 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). If the bedrock is the datum, then its elevation is 0.0 m. Now, suppose the field is flooded to a water elevation 0.8 m above bedrock.

Figure Box 5-3 – A groundwater flow system beneath an irrigated field with long, parallel drains.

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).

Figure Box 5-4 Only a small portion of the field with parallel drains needs to be drawn to develop a flow net. A groundwater divide (no-flow boundary) occurs halfway between two adjacent drains. In addition, flow to the drain from the left is a mirror image of flow from the right. This symmetry enables us to consider the flow to the drain from only one side. The flow net developed for the above figure will repeat across the many drains of the field in alternating mirror images.

The horizontal hydraulic conductivity is 16 times higher than the vertical, K_x = 16 K_y, so the ratio of the semi-axes of the ellipse is = = 4. To convert the anisotropic system to an isotropic system, we can stretch the system vertically by a factor of 4. We leave the x-coordinates as they are and multiply the y-coordinates by 4. The ground surface is at 0.6 m so it is increased to 2.4 m; the top of the drain is at 0.25 m so it is increased to 1.0 m; the bottom is at 0.15 m so it is increased to 0.6 m, while the bedrock remains at 0 m (4*0 = 0). The transformed system has the same width, but is 4 times taller than the anisotropic system. The shape of the circular drain becomes an oval (Figure Box 5-5).

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. The shape of the circular drain becomes an oval.

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. 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. Consequently, the hydraulic head at the top of the drain is 0.25 m, at the midpoint it is 0.2 m and at the bottom it is 0.15 m. We expect water to flow from the ground surface to the drain, so we sketch in flow lines and equipotential lines and continue to adjust until they form curvilinear squares (Figure Box 5-6).

Figure Box 5-6 – An isotropic flow net is drawn in the transformed isotropic system (b) on the right.

Next, we transform the section back to its original coordinate system (Figure Box 5-7). This is done by shifting each x and y coordinate where the equipotential lines and flow lines intersect, dividing the y coordinate by 4, and plotting the intersection point at the original x location and the reduced y location on the original system geometry. Once the intersection points are plotted, the lines can be connected. The transformed equipotential lines and flow lines do not meet at right angles and the intersecting lines do not form squares in the anisotropic system geometry (Figure Box 5-7).

Figure Box 5-7 – The (b) isotropic flow net on the right is transformed back to the (a) anisotropic geometry on the left.

Finally, we estimate the volumetric flow through the system. To accomplish this, we use the formula presented here as Equation Box 5-3. It is discussed and derived in Box 4 of this book.

(Box 5-3)

where:

When Equation Box 5-3 is applied to an anisotropic system, an equivalent hydraulic conductivity is used to account for the differing values in the horizontal and vertical direction. Equivalent hydraulic conductivity for an anisotropic system is calculated as shown in Equation Box 5-4.

(Box 5-4)

Using Equation Box 5-4, the equivalent hydraulic conductivity for the flow net shown in Figure Box 5-7a is:

= = 

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.

The total flow to one side of the drain is calculated using Equation Box 5-3 with an equivalent hydraulic conductivity of 0.04 m/d, a total head drop H of 0.6 m, 5 flow tubes (n_f), 7 head drops (n_d), and a length of drain, w, of 100 m:

∼ ∼

The flow needs to be doubled to account for drainage from both sides of the drain, so approximately 2.4 liters per minute.

Discharge from both sides to the drain = Q_{total-both-sides} = 2Q_total ~

A 0.3 m diameter pipe can transport 50 liters per minute without backing up so we made a reasonable assumption when we decided that the drain would be at atmospheric pressure.

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. The process of creating the flow net is similar. 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 directions and align the x–y coordinate system to these directions. The geometric transformation can then be carried out for flow net construction.

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