# 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.

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 (*h*_{2}–*h*_{1})/*L*_{1} is greater than the gradient (*h*_{2}–*h*_{1})/*L*_{2}, and there is no direction with a larger gradient than (*h*_{2}–*h*_{1})/*L*_{1}. In this case, –*grad h* is defined over the distance *L*_{1}. 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.

# 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 (*K*_{x} and *K*_{y}). However, those same depositional process create anisotropy in the vertical direction due to layering of the sediments, thus vertical hydraulic conductivity, *K*_{z}, is commonly orders of magnitude less than horizontal hydraulic conductivity (*K*_{x} and *K*_{y}).

When the two-dimensional hydraulic conductivity distribution is anisotropic and homogeneous (*K*_{x} ≠ *K*_{y}, or *K*_{x} ≠ *K*_{z}) 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/*K*_{max}^{0.5} and 1/*K*_{min}^{0.5}, 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 *K*_{x} is greater than *K*_{z}.

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

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 (*Q*_{1} = *Q*_{2}) 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.

(91) |

where:

K_{1} |
= | hydraulic conductivity of layer 1 (L/T) |

K_{2} |
= | hydraulic conductivity of layer 2 (L/T) |

θ_{1} |
= | angle the flow line makes with a perpendicular to the boundary in stratum 1 (degrees) |

θ_{2} |
= | 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.