Box 7  Transformation for 2-D Flow in an Anisotropic Medium

William W. Woessner; Eileen P. Poeter

Box 7 Transformation for 2-D Flow in an Anisotropic Medium

A second graphical method for determining flow directions in anisotropic materials involves transforming one axis of the map or cross section to account for the anisotropy, plotting the head values then drawing equipotential lines and flow lines as if the system were isotropic. Subsequently, the entire drawing is transformed back to the original scale. The transformation involves either stretching the drawing in the direction of the minimum hydraulic conductivity or compressing the drawing in the direction of the maximum hydraulic conductivity. The most convenient way to explain the process is to orient the x and z axes parallel to K_max and K_min of the field system, respectively.

Consider the system shown in Figure Box 7-1, where a water body with an elevation of 8 meters overlies the ground surface at an elevation of 6 meters. A 1-meter diameter drain with a head of 0 (at atmospheric pressure) is centered 2 meters above the bedrock such that the bottom of the drain is 1.5 meters above bedrock. All boundaries other than the ponded water and the drain are no-flow. The horizontal hydraulic conductivity, K_x, is 0.16 m/day and the vertical hydraulic conductivity, K_z, is 0.01 m/day.

Figure showing use of axis transformation to determine flow lines in an anisotropic homogeneous system. — **Figure Box 7-1 –** Example of using the axis transformation to determine flow lines in an anisotropic homogeneous system. a) The problem domain is a sand represented in cross section with no vertical exaggeration where saturated flow occurs through anisotropic material from a pond bottom (the pond surface is 8 m above the underlying low permeability bed and the bottom of the pond is 6 m above the bed). There is a 1 m diameter drain pipe at atmospheric pressure on the left boundary. All boundaries other than the pond and the drain are no-flow boundaries. The hydraulic conductivity ellipse has a ratio of the square root of K_x to that square root of K_z, which is 4. b) Transformation of the z axis and drawing of flow lines at right angles to the equipotential lines. c) The final flow lines at the original scale (z axis is returned to the original scale). (all after Freeze and Cherry, 1979)

To generate the anisotropic flow field the plot scale is transformed into X’ and Z’ based on the hydraulic conductivity ellipse. The scale of the x axis remains the same, X’ = x, and the z-axis is transformed as Z’ = z(K_x^0.5 / K_z^0.5). Substituting the values for K_x and K_z results in:

$\displaystyle Z'=z\left ( \frac{0.16^{0.5}}{0.1^{0.5}} \right )=z\left ( \frac{0.4}{0.1} \right )=4z$

So, the z-axis is expanded by a factor of 4 as shown in Figure Box 7-1b. If the head values were not plotted on the pre-transformed scaled section, then the head data are plotted at the x,Z‘ location with their Z’ four times their z value (red dots and values on Figure Box 7-1b). Next, the head data are contoured (dashed black lines) and flow lines are drawn perpendicular to the equipotential lines (solid blue arrows). Finally, the Z’ location of every point in the image is transformed back to the original scale as shown in Figure Box 7-1c to obtain flowlines for the anisotropic system. After the reverse transformation, the flow lines are no longer perpendicular to the equipotential lines. The transformation is only valid when the original cross-sectional representation is plotted without vertical exaggeration.

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