Theoretical head variograms for steady flow in statistically homogeneous aquifers

Flow through two‐dimensional porous formations is modeled as a random process using an approximate formulation of flow physics in order to obtain an expression for the head variogram. (The variogram is related to the variance of the head difference between two points separated by a vector h and has leapt into prominence owing to its central role in kriging.) The model domain investigated here is unbounded, composed of arbitrarily placed circular blocks of random, statistically homogeneous dimension and transmissivity, these two properties being independently distributed. The model medium can represent only transmissivity autocorrelation functions which decline monotonically from 1 to 0. Steady, uniform average flow is investigated. Flow physics are simplified by ignoring any differences among blocks of the same size and transmissivity and then using their random setting to ease spatial integrations involved in head statistics. The head variogram proves significantly anisotropic, with heads differing more widely on average for a fixed lag parallel to the head gradient than perpendicular to it. Field practice presently ignores anisotropy; the implications are examined for a hypothetical case and indicate that kriged standard deviation is overestimated perpendicular to the mean flow and underestimated parallel to it. For lag magnitude greater than the largest block diameter the variogram is logarithmic in the lag magnitude and proportional to the head gradient squared. This leads to a technique to reduce data scatter in empirically fitted variograms. Once an empirical head variogram is selected, the inverse problem may be solved for a particularly simple medium structure.