8.2 Determining Groundwater Flow Directions

When evaluating the direction of groundwater flow, the first step is to plot the head data on a map or cross section, then create contours of equal head, i.e., equipotential lines, as shown in Figure 64 and Figure 65. Representations in cross-sectional views are created using axes that are equally scaled, x and z. In geologic investigations cross-sectional representations are often shown with the vertical scale exaggerated (e.g., horizontal to vertical rations of 1 to 10, 1 to 100, etc.); this allows small changes in features to be visible. However, cross sections without vertical exaggeration are required when attempting to interpret groundwater flow directions. Maps are already scaled equally in the x and y direction and are most commonly used to represent two-dimensional groundwater flow patterns. An example of the process used to construct equipotential lines is illustrated in Figure 66.

Figure showing the construction of equipotential lines using measurements from three wells
Figure 66 – Example of constructing equipotential lines using measurements from three wells (blue circles). The three values of head are shown as large black values 102, 93, and 79. The space between wells is divided into segments to linearly interpolate the values of head between the wells. Then dashed black equipotential lines are drawn connecting equal values of interpolated heads. If the material has isotropic and homogeneous hydraulic conductivity, then groundwater flow (large blue arrow) will occur in the direction of the maximum gradient (from high values to low values), which is at right angles (yellow boxes) to the equipotential lines.

Alternatively, equipotential lines can be drawn using computer software to interpolate sparse data points to a regular grid (e.g., the public domain, open-source, geographical information system QGIS; the commonly-used, commercial gridding and contouring program SURFER©; or code in Python© or Matlab©). These interpolated values are then contoured. There are many interpolation methods that are not discussed here, but often their name makes it easy to envision their procedure (e.g., closest point; nearest neighbor; triangulation, inverse-distance weighting of surrounding data; kriging which includes functions describing continuity of trends in the coordinate directions; trend surface polynomials).

One challenge with automated interpolation is that the conditions at the boundaries of the data are usually poorly represented because the programs infer data trends beyond the measured data field (extrapolation). Many practitioners prefer to use digitized versions of their own hand drawn contour maps over computer generated versions because this allows geologic insight to be incorporated into the interpretation and often allows boundary conditions to be represented in a more realistic manner. This is particularly so when the number of data points (monitoring wells) are relatively few and when they are not uniformly distributed within the study area. With any method of interpolation and contouring, results must be scrutinized to see if they make hydrogeologic sense. This process is referred to as a sensibility analysis. Ask the question, “What would this look like if it was hand contoured?” Some interpolation methods yield poor representations of a system. When the output doesn’t make sense, data quality and interpolation methods need to be reviewed, and in some cases additional field data collection proposed.

Gradient and Flow Directions in Isotropic Material

The head distribution and interpolated equipotential lines can be used to calculate a hydraulic gradient at any location within the flow field. A gradient is a vector, that is it has both magnitude and direction. The maximum gradient is used to determine the direction of groundwater flow and is referred to as –grad h, with the negative sign indicating that groundwater flow is from high head to low head (Figure 67a). In Figure 67b the gradient (h2h1)/L1 is greater than the gradient (h2h1)/L2, and there is no direction with a larger gradient than (h2h1)/L1. In this case, –grad h is defined over the distance L1. For isotropic and homogeneous material, the path of groundwater flow is parallel to –grad h and flow lines are constructed at right angles to equipotential lines as illustrated in Figure 68.

Figure showing the relationship of the hydraulic gradient vector to mapped head
Figure 67 – The relationship of the hydraulic gradient vector -grad h to mapped head (dashed blue equipotential contours) a) –grad h for a map view of a head field where h8 > h3; and, b) –grad h for a flow field interpreted from head measurements at three wells (blue dots). Head values are the large values adjacent to the well locations. –grad h is defined by the maximum hydraulic gradient.
Water table map of an isotropic and homogeneous unconfined aquifer.
Figure 68 – Water table map of an isotropic and homogeneous unconfined aquifer. Head measurements are plotted at well locations (black dots). The surveyed river stages (blue triangles) also represent head values as groundwater is discharging to the river and the water table is connected to the river stage. Heads are contoured using a 10 m interval creating the equipotential lines shown as dashed black lines. Blue arrows represent flow lines constructed at right angles (indicated by red squares) to the equipotential lines.

Flow Directions in Anisotropic Materials

As stated previously in Sections 4 and 5, most sediments and sedimentary rocks are deposited such that their hydraulic conductivity is similar in all directions of the horizontal plane (Kx and Ky). However, those same depositional process create anisotropy in the vertical direction due to layering of the sediments, thus vertical hydraulic conductivity, Kz, is commonly orders of magnitude less than horizontal hydraulic conductivity (Kx and Ky).

When the two-dimensional hydraulic conductivity distribution is anisotropic and homogeneous (KxKy, or KxKz) groundwater flow is influenced by the maximum component of hydraulic conductivity. If the component of maximum hydraulic conductivity does not parallel –grad h, then flow lines cross equipotential lines at an angle other than a right angle.

The flow directions in anisotropic settings are dependent on the magnitude of the component hydraulic conductivities illustrated by using an inverse hydraulic-conductivity tensor ellipse (Figure 69). Once –grad h is identified for a set of equipotential lines and the inverse hydraulic-conductivity tensor ellipse is constructed with axes of 1/Kmax0.5 and 1/Kmin0.5, and the anisotropic flow direction at a given location can be determined by graphical construction (Liakopoulos, 1965). The hydraulic-conductivity tensor ellipse described in Section 5.4 (Figures 34 and 35) is used to illustrate the directional nature of K whereas the inverse hydraulic conductivity tensor ellipse is used to determine groundwater flow directions.

Figure 69 presents the steps used to construct flowlines in an anisotropic setting where Kx is greater than Kz.

Figure showing the determinaton of flow direction in a cross section of anisotropic homogeneous material
Figure 69 – Determining flow direction in a cross section of anisotropic homogeneous material is illustrated for a case where Kx > Kz: a) head values are obtained (black dots are heads from wells) and equipotential lines (dashed lines) are constructed; b) then an inverse hydraulic-conductivity tensor ellipse is generated for site conditions; and, c) the ellipse is placed with its center on an equipotential line at the location where the flow direction is to be determined. The ellipse axes are aligned with the direction of maximum and minimum K on the map and cross section. A line representing –grad h (red solid arrow) is drawn from the ellipse center (at a right angle [red square] to the equipotential line) to the ellipse perimeter. A tangent to the ellipse at the point where –grad h intercepts its perimeter is constructed (black solid line). Then the anisotropic flowline (blue dashed arrow) is constructed from the center of the ellipse to the tangent line crossing it at a right angle (red square). The flow path is traced through the system by plotting additional inverse hydraulic-conductivity tensor ellipses where the dashed blue flow line crosses the next equipotential line. d) The inverse hydraulic-conductivity tensor ellipse is a circle under isotropic and homogeneous conditions. This results in flow lines being parallel to –grad h at all locations.

Figure 70 illustrates how flow direction is impacted by the orientation of the equipotential lines in an anisotropic media.

Figure showing the use of the hydraulic conductivity tensor ellipse graphical method to determine flow directions
Figure 70 – Examples of using the inverse hydraulic-conductivity tensor ellipse graphical method to determine flow directions in four examples of equipotential line orientations in the xz field. Values of Kx are greater than Kz, The method and symbols shown here are defined in Figure 69. a) Two non-parallel or non-vertical orientations of equipotential lines with the anisotropic flow path indicated by the dashed blue arrow. b) Equipotential lines oriented parallel to the Kx axis (upper diagram) and Kz axis (lower diagram). For the cases shown in (b), the –grad h (red arrow) and anisotropic flow line (dashed blue arrow) are identical.

A second graphical approach for representing flow in an anisotropic system is presented in Box 7. Click here to read Box 7 about a graphical method of axis transformation to draw two-dimensional flow in an anisotropic, homogeneous medium.

Flow Directions at Interfaces of Differing Hydraulic Conductivity

Groundwater flow is also impacted by contrasts in hydraulic conductivity as flow passes from one isotropic homogeneous region to another (Figure 71). If a flow line intersects the hydraulic conductivity boundary at an angle other than 90°, it is refracted into the adjacent formation. The magnitude of refraction relates to the ratio of the hydraulic conductivity values as shown in Equation 91. The refraction occurs because discharge through in the area between two flow lines, the flow tube or stream tube, must be the same on both sides of the interface (Q1 = Q2) under steady state conditions and Darcy’s Law must be honored on both sides of the interface. Given that, the equipotential lines must be closer together (larger hydraulic gradient) and the stream tubes must be wider (flow through a larger cross-sectional area) in the lower hydraulic conductivity side of the interface than on the higher hydraulic conductivity side. In addition to the flow line refraction, the equipotential lines also refract as they cross the hydraulic conductivity boundary.

Figure showing the refraction of flow lines and equipotential lines at the interface between layers of different hydraulic conductivities
Figure 71 – Examples of flow lines and equipotential lines refracting at the interface between layers of different hydraulic conductivities. Gray and yellow areas are saturated, isotropic, homogeneous materials of different hydraulic conductivity. a) Dashed lines are equipotential lines decreasing from upper left to the lower right. In the gray area K1 = 55 m/d and in the yellow area K2 = 120 m/d. The flow lines (blue arrows) intersect the boundary at 45°. They are refracted into the yellow area at an angle of 65.4°. The equipotential field is also refracted. b) Flow line refraction in a sequence of isotropic, homogeneous aquifers with K2 (yellow) 10 times K1 (gray).
\displaystyle \frac{K_1}{K_2}=\frac{{\tan{\ \theta}}_1}{{\tan{\ \theta}}_2} (91)

where:

K1 = hydraulic conductivity of layer 1 (L/T)
K2 = hydraulic conductivity of layer 2 (L/T)
θ1 = angle the flow line makes with a perpendicular to the boundary in stratum 1 (degrees)
θ2 = angle the flow line makes with a perpendicular to the boundary in stratum 2 (degrees)

The derivation of the tangent law is presented in Box 8. Click here to read Box 8 which presents the derivation of the tangent law.

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Hydrogeologic Properties of Earth Materials and Principles of Groundwater Flow Copyright © 2020 by The Authors. All Rights Reserved.