# 4.1 General Considerations

Let us first consider a simple experimental setup to illustrate some of the most fundamental aspects of equipotential contours and associated flow directions (for an isotropic medium). As shown in Figure 17, piezometers are inserted to various depths on either side of a horizontal cylinder filled with water-saturated, porous medium through which water flows under laminar conditions (in other words, water is flowing according to Darcy’s law). Since the water level in a piezometer represents the hydraulic head *at the point of measurement* (the open end at the bottom of each piezometer), the hydraulic head at all three points on the left side (*x*_{1}=10 cm) is equal to 20 cm (*h*_{1}=20 cm). Similarly, the hydraulic head at all three points on the right side (*x*_{2}=40 cm) is 14 cm (*h*_{2}=14 cm). As shown in Figure 17b, an *equipotential contour connects points of equal hydraulic head*. Stated differently, *the hydraulic head at all points along an equipotential contour are equal*. Accordingly, there is no gradient along the equipotential contour and the *hydraulic gradient vector* must be orthogonal (i.e., at 90°) to the contour line. In other words, the flow direction is orthogonal to equipotential contours for isotropic conditions.

The measurements at each location show that the hydraulic gradient in the *z* direction is zero. That is, hydraulic head does not change in the z direction (Δ*h**/**Δ**z**=0*), which is consistent with the fact that water cannot flow in the vertical direction due to the presence of the impermeable sides (“no-flow boundaries”) of the horizontal cylinder. The absence of a vertical gradient adjacent to the boundaries is expressed by equipotential lines that are perpendicular to the no-flow boundaries.

Figure 17 also shows the horizontal hydraulic head profile (horizontal hydraulic gradient). Note that hydraulic gradient can be measured using the head profile and by using the equipotential contours.

If the difference in head between the sets of piezometers was less, the hydraulic gradient would be smaller (hydraulic head profile would not be as steep and spacing between the 2-cm equipotential contour lines would be larger).

# Example Problem 3

a) Draw equipotential lines in the sand at 1 ft intervals.

b) To what level will water rise in the hypothetical piezometer?

Click here for solution to Example Problem 3

Having now established the basic elements of equipotential contours for one-dimensional flow and the associated groundwater flow direction, consider the experimental setup and corresponding hydraulic representation of two-dimensional flow in Figure 18. The overall flow geometry is as we would expect given that, based on the heads at the constant head boundaries, flow is left to right, and there needs to be at least some vertical component of flow owing to the geometry of the enclosed system and the location of the boundary conditions. As before, the equipotential contours meet the no-flow boundaries at right-angles. In addition, the equipotential contour lines at the constant head boundaries are defined by the elevation of the water body at those boundaries, which was illustrated in Figure 12. Note also that because the porous medium is isotropic, the flow lines meet the equipotential contours at right angles (90°).