Applications of the Cluster Statistics

This chapter presents a conceptually straightforward treatment of spatial correlations of “random” heterogeneous media, but does not intend to capture at this point even a majority of the actual behavior. A great deal of work still needs to be done since what has been accomplished so far neglects the expected geological complications due to patterns of deposition (on a wide range of scales), dewatering, alteration, deformation, and fracture. A fundamental point of this chapter will be that, even if a medium itself does not exhibit correlations, the transport properties of this medium will be correlated over distances which can be very large. In fact a simple physical result emerges, namely that the length scale of correlations in the measurement of a conduction process is directly proportional to the size of the volume of measurement (Hunt [1], given in Sect. 9.3 here), known in the hydrologic community as the “support” volume. This result is observed over 3–4 orders of magnitude of length, i.e., over 10+ orders of magnitude of the volume [2]. Although there is no proof yet that the percolation theoretical prediction is at the root of this experimental result, it is certainly a viable candidate.