K is never the property of a "point", but is always an average over some representative elementary volume (REV). When K is a function of position within an aquifer, i.e. K = K(x,y,z) the aquifer is considered to be heterogeneous. Heterogeneity can abrupt, trending, or "uniform". In the last case K varies locally, but the statistical distribution of K is similar in different portions of the aquifer. In some cases we can use an "equivalent homogeneous medium" approximation, replacing the spatially variable K with some type of equivalent average K. This is the type of assumption we are using to analyze our permeameter experiments and it is also an assumption invoked by Darcy in his original experiment. The types of averages are defined in the text: the arithmetic mean, the harmonic mean and the geometric mean. The harmonic mean is always less than the arithmetic mean, and the geometric mean is between these. The geometric mean has been suggested as the best estimate of an equivalent average K for uniformly heterogeneous materials. The geometric mean is also often close to the K value determined from the arithmetic mean of log transformed Ks.
There is a second type of variability of K. K may have a directional nature. For example, if there is sedimentary layering or a preferred set of joints or bedding plane fractures, an aquifer may have a higher equivalent average K parallel to the beds or fractures than perpendicular to them. We refer to this directional character as anisotropy and represent the Ks in 3 directions as Kx, Ky, and Kz. In a 3D version of Darcy's law, we must then use the appropriate K for the direction of the gradient and direction of flow. We need to compute the specific discharges in each direction (qx, qy, and qz) separately, using the hydraulic gradient and K in the appropriate direction, then add the q vectors to obtain the resultant overall magnitude and direction of flow.
In the most general case, there are actually 9 components of the K "tensor". However, if we are careful to choose the coordinate axes of our flow problem to correspond to the principal directions of anisotropy, only 3 of these (Kxx, Kyy, and Kzz) will be nonzero. We often abbreviate these three principal components of the anisotropy tensor as Kx, Ky, and Kz. When you see a single subscript for the K, you can assume that the principal directions of anisotropy are aligned with the coordinate axes of the flow problem.
In some cases we may be able to account for local heterogeneity, such as layering, by using an average effective hydraulic conductivity that varies depending on the direction of flow. Thus, we replace a heterogeneous but isotropic system with an equivalent one that is treated as homogeneous but anisotropic and K is a tensor rather than a scalar. For the case of perfect layering, the effective K for flow parallel to the layers can be computed as the arithmetic mean of the layer Ks. The effective K for flow perpendicular to the layers can be computed as the harmonic mean of the layer Ks.
III. Refraction of flow lines (and equipotentials) across abrupt boundaries
Refer to the figure below from the text by Freeze and Cherry. The red dashed line is a, the blue solid line that runs along the interface between the two formations is b, and the green line is c.
By continuity, the flow between two adjacent streamlines in the upper unit (Q1) must equal the flow between two adjacent streamlines in the lower unit (Q2).
By Darcy's law, Q1 = -K1 aw (D h)/dl1 and Q2 = -K2 cw (D h)/dl2 where w is the width of the system in the direction perpendicular to the section shown and D h is the head change from one equipotential (dashed line) to the next. Note that aw is the area through which flow occurs in the upper unit, cw is the area through which flow occurs in the lower unit. Setting these two flows equal to each other and eliminating common terms, we get
K1 a/dl1 = K2 c/dl2
Note that tan q1 = dl1/a and tan q2 = dl2/c
So K1/tan q1 = K2/tan q2
And K1/K2 = tan q1/tan q2