ON ADVECTIVE TRANSPORT IN FRACTAL PERMEABILITY AND VELOCITY-FIELDS

We consider advective transport in a steady state random velocity fiel d with homogeneous increments. Such a field is self-affine with a powe r law dyadic semivariogram gamma(s) proportional to d(2 omega), where d is distance and omega is a Hurst coefficient. It is characterized by a fractal dimension D = E + 1 - omega, where E is topological dimensi on. As the mean and variance of such a field are undefined, we conditi on them on measurement at some point x(0). We then introduce a tracer at another point y(0) and invoice elementary theoretical consideration s to demonstrate that its conditional mean dispersion is local at all times. Its conditional mean concentration and variance are given expli citly by well-established expressions which, however, have not been pr eviously recognized as being valid in fractal fields. Once the conditi onal mean travel distance s of the tracer becomes large compared to th e distance between y(0) and x(0), the corresponding dispersion and dis persivity tensors grow in proportion to s(1+2 omega), where 0 < omega < 1. This supralinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained by standard methods df interpretation from tracer behavior observed in a variety of geolog ic media under varied flow and transport regimes. Filtering out modes from the fractal velocity held with correlation scales larger than som e s(0) allows an asymptotic transport regime to develop when s >> s(0) . The corresponding asymptotic dispersivities grow in proportion to s( 0)(2 omega), when 0 < omega less than or equal to 1/2. This linear to sublinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained from calibrated numerical models in a variety of media. A self-affine natural log permeability field g ives rise to a self-affine velocity field, while s is sufficiently sma ll to insure that the variance of the log permeabilities, which grows as a power of s, remains nominally less than one. An analysis of publi shed apparent longitudinal dispersivity data in light of the above the oretical results supports my earlier conclusion that when one juxtapos es data from a large number of generally dissimilar geologic media fro m a variety of locales, one observes a tendency toward self-affine beh avior with a Hurst coefficient omega similar or equal to 0.25. At any given locale such media may or may not exhibit fractal behavior; if th ey do, omega may or may not be close to 0.25.