# 5.4 Spatial and Directional Variation of Hydraulic Conductivity

As discussed previously in this book, the transmission properties of earth materials are influenced by the processes that are active when the material is formed or deposited, as well as the changes in the material that occur during post-formation processes. Both sets of processes influence the direction and magnitude of hydraulic conductivity at each location throughout the material. In the general case, hydraulic conductivity varies with direction, and when aligned with the principal coordinate axes, it is represented as *K*_{x}, *K*_{y} and *K*_{z}.

If formation of the earth material results in uniform pore sizes that are well connected in all directions at a given location (e.g., *K*_{x} = *K*_{y} = *K*_{z}) the hydraulic conductivity at that location is *isotropic*. In a material with isotropic hydraulic conductivity, water passes through a material with equal ease regardless of the direction that the water is forced through the material. That is, given the same gradient in any direction, the resulting flow rate is the same. Figure 33a provides a schematic of an isotropic porous medium where *K* is the same in every direction at every location. In contrast, if the pore structure and connections resulted in a setting where hydraulic conductivity is lower in one direction than another, the porous medium is *anisotropic*, for example, where *K*_{x} = *K*_{y}, but *K*_{z} has a smaller value as shown in Figure 33b. The medium shown in Figure 33b could be the result of compacting the medium shown in Figure 33a, or from a depositional process that oriented non-spherical grains in a preferred direction. Anisotropy can also involve different values of *K* in all of the principal directions *K*_{x} ≠ *K*_{y} ≠ *K*_{z}. In most sediments and sedimentary rocks, the deposition process produces micro layering resulting in horizontal hydraulic conductivities being about the same, *K*_{x} = *K*_{y}, and vertical values, *K*_{z}, that are smaller. This results in isotropic conditions in the horizontal plane (map view) with anisotropic conditions in a cross-sectional representation as shown in Figure 33b.

When hydraulic conductivity distributions are observed at the broader scale of a geologic formation (not at only one location), two general conditions can be found in four combinations as illustrated in Figure 34. The formation can be either homogeneous or *heterogeneous*, and it can be either isotropic or anisotropic:

- If all locations have the same value of
*K*_{x},*K*_{y},*K*_{z}, then the hydraulic conductivity distribution is*isotropic*and*homogeneous*(Figure 34a). - If the same anisotropic conditions are present at all locations within a formation, then the hydraulic conductivity distribution is
*anisotropic*and*homogeneous*(Figure 34b). - If the hydraulic conductivity is isotropic at all locations, but the isotropic values differ from location to location, then the hydraulic conductivity distribution is
*isotropic*and*heterogeneous*(Figure 34c). - If at different locations within the formation, different sets of anisotropic conditions are present, then the hydraulic conductivity distribution is
*anisotropic*and*heterogeneous*(Figure 34d).

The three-dimensional components of hydraulic conductivity can be used to derive the hydraulic conductivity, *K*_{s}, associated with a particular groundwater flow path (*q*) constructed within the groundwater system. For example, if the spatial hydraulic conductivity conditions (*K*_{x}, *K*_{y}, *K*_{z}) are known, the hydraulic conductivity associated with a flow line at some angle to the coordinate system can be established using a hydraulic conductivity ellipsoid (Freeze and Cherry, 1979, Chapter 2). In two dimensions an ellipse can be constructed using the square root of *K*_{max} and *K*_{min} values as shown in Figure 35 and expressed in Equations 34 and 35.

(34) |

Equation 34 relates the hydraulic conductivity, *K*_{s}, in any angular direction, *θ*, to the components *K*_{x} and *K*_{z}. Equation 34 can be written in rectangular coordinates by setting *x* = *r* cos* θ* and *z* = *r* sin *θ* resulting in Equation 35.

(35) |

Equation 35 is the equation of an ellipse. Major axes are the square root of *K*_{x} and *K*_{z} as shown in Figure 35d (Freeze and Cherry, 1979). As indicated in Figure 35b and d, the hydraulic conductivity in any direction, *K*_{s}, in an isotropic or anisotropic medium can be determined graphically if *K*_{x} and *K*_{z} are known. Although the *x*, *y*, and *z* axes are shown here in their classic orientation, the axes may be oriented in any direction within a formation as shown in Figure 36.