On Advective Transport in Fractal Permeability and velocity Fields

Abstract

We consider advective transport in a steady state random velocity field with homogeneous increments. Such a field is self‐affine with a power law dyadic semivariogram γ(s) proportional to d2ω, where d is distance and ω is a Hurst coefficient. It is characterized by a fractal dimension D = E + 1 − ω, where E is topological dimension. As the mean and variance of such a field are undefined, we condition them on measurement at some point x0. We then introduce a tracer at another point y0 and invoke elementary theoretical considerations to demonstrate that its conditional mean dispersion is local at all times. Its conditional mean concentration and variance are given explicitly by well‐established expressions which, however, have not been previously recognized as being valid in fractal fields. Once the conditional mean travel distance s of the tracer becomes large compared to the distance between y0 and x0, the corresponding dispersion and dispersivity tensors grow in proportion to s1+2ω, where 0 < ω < 1. This supralinear rate of growth is consistent with that exhibited by apparent longitudinal dispersivities obtained by standard methods of interpretation from tracer behavior observed in a variety of geologic media under varied flow and transport regimes. Filtering out modes from the fractal velocity field with correlation scales larger than some s0 allows an asymptotic transport regime to develop when s ≫ s0. The corresponding asymptotic dispersivities grow in proportion to s02ω, when 0 < ω ≤ ½. 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 gives rise to a self‐affine velocity field, while s is sufficiently small to insure that the variance of the log permeabilities, which grows as a power of s, remains nominally less than one. An analysis of published apparent longitudinal dispersivity data in light of the above theoretical results supports my earlier conclusion that when one juxtaposes data from a large number of generally dissimilar geologic media from a variety of locales, one observes a tendency toward self‐affine behavior with a Hurst coefficient ω ≃ 0.25. At any given locale such media may or may not exhibit fractal behavior; if they do, ω may or may not be close to 0.25.