Percolation cluster statistics and conductivity semi-variograms

Cluster statistics of percolation theory have been shown to generate expres sions for the distribution of hydraulic conductivity values in accord with field studies. Percolation theory yields directly the smallest possible gen eralized resistance value, R-c, for which a continuous path through an infi nite heterogeneous system can avoid all larger resistances. R-c, defines an infinite system hydraulic conductivity. Cluster statistics generate the nu mber of clusters of resistors of a given size with a given R, for which a c ontinuous path through the cluster can avoid resistances larger than R. The probability that a volume of size x(3) 'falls on' a particular cluster giv es the probability that volume has a characteristic resistance, R. Determin ing the semi-variogram of the hydraulic conductivity is now elementary; it is necessary only to determine whether translation h of the center of the v olume x(3) removes it from the cluster in question. If the cluster is large r than (x + h)(3), then, on the average, the same cluster resistance R will control K. Otherwise, the value of K at x + h will be uncorrelated with it s value at x. The condition is then expressed as an integral related to the one, which gives the distribution of K. Then an integral over the derived distribution of K gives the variogram. Results obtained are that the variog ram should be similar to either the exponential or Gaussian forms typically in use, if K is a power law function of random variables (as in Poiseuille 's Law), or more closely related to the spherical approximation if K is an exponential function of random variables.