3 Darcian Head Profiles

Let us again consider a steady-state scenario shown in Figure 11a. In this case, the water level in both vessels remain fixed such that either side of the cylinder of porous media is bounded by a constant hydraulic head, and the difference in hydraulic head drives the flow (that is, water flows from a region of higher potential energy to a region of lower potential energy). In accordance with Darcy’s law, there is a hydraulic gradient in the porous medium, and the water is flowing from left to right, as expressed by the hydraulic gradient, which is equal to -0.4 m/m, as shown on the head profile (Figure 11b).

Figure 11 – Demonstration of steady-state horizontal flow and associated horizontal gradient; water flows left to right in accordance with the direction of decreasing potential energy, as expressed by the hydraulic head profile: a) experimental setup and b) hydraulic head profile (Cohen and Cherry, 2020).

Note that the hydraulic head within each column of water is uniform with depth as shown in Figure 12 and as described previously. The hydraulic head at each point of measurement in the piezometer is equal to the elevation of the top of the water column in the piezometer. In theory, there is negligible (and perhaps immeasurable) vertical hydraulic head gradient in each vessel, because there is also vertical flow in the vessel, but the differences in head throughout the vessel are insignificant given the slow flow rate and minimal friction imparted by the sides of the vessels. Thus, for all practical purposes, the head profile can be considered uniform in each water column.

This experimental setup provides an opportunity to introduce some simple hydraulic modeling concepts, depicted as the equivalent hydraulic model (Figure 12c).

Figure 12 – Relationship between piezometer measurements, hydraulic head distribution, and associated boundary conditions of the experimental setup. This is a steady-state scenario for a system with constant head boundaries (Cohen and Cherry, 2020).

Figure 13a shows the same experimental setup, except the cylinder of sand is vertical. As such, plotting hydraulic head as a function of elevation is appropriate (Figure 13b). Hydraulic head decreases with decreasing elevation. As a result, the hydraulic gradient is positive (+0.4), whereas the gradient was negative (-0.4) in the horizontal setup (Figure 11). This difference in sign is simply an artifact of using elevation as the spatial coordinate; Darcy’s law still applies, and water flows downward in the direction of decreasing hydraulic head.

Figure 13 – Demonstration of vertical flow and associated head profile: The head profile is defined by water levels measured in piezometers and their respective points of measurement, which are the open ends of each piezometer in the sand column. Water flows downward, from higher head to lower head (Cohen and Cherry, 2020).

Figure 14 is a modified version of the multi-piezometer experimental setup shown in Figure 13b. In the case of Figure 14, the piezometers are “nested” in that they are closely spaced but measure the head at different depths because the open end of each piezometer (the point of measurement) is located at different elevations.

Figure 14 – Head profile using nested piezometers. Even though the piezometers are positioned close to one another and their respective water levels decrease towards the right, the measurements are actually representative of the vertical head profile, because all flow is vertical owing to the geometry of the cylinder, and the vertical position of each point of measurement is different (Cohen and Cherry, 2020).

Figure 15 compares the head profiles measured with the piezometers in Figure 13b to the nested piezometer configuration (Figure 14). The head profiles are identical, because the points of measurements and the boundary conditions are identical.

Figure 15 – Comparison of head profile using different piezometer orientations: a) experimental setup; b) boundary conditions; and, c) head profile. The head profiles are identical, because the points of measurement and the boundary conditions are identical (Cohen and Cherry, 2020).

Figure 16 illustrates vertical hydraulic gradients for downward and upward one-dimensional flow. The relationship between hydraulic gradient and hydraulic conductivity, K, (−dh/dL 1/K) still holds; only the orientation has changed.

Figure 16 – Vertical head profiles for downward and upward flow scenarios. The head profile defines the direction of flow, and changes in slope are indicative of variable hydraulic conductivity (Cohen and Cherry, 2020).