Dispersivity computations are developed for heterogeneous aquifers which exhibit structured variability across a wide range of scales. Multiscale, fractional Gaussian and self-similar random field models are used to describe the statistical structure of hydraulic conductivity variations. The concept of a relative dispersivity (Arij) is introduced, following a similar concept introduced earlier in the context of turbulent diffusion. The relative dispersivity depends on the plume size and is influenced only by velocity variations at scales smaller than the plume scale. In contrast, the dispersivity associated with the ensemble average concentration (Aij) is influenced by all scales of velocity variations, including very large scales. In fractional Gaussian media, previous analyses predict that Aij grows as a power function of the plume displacement. In contrast, the longitudinal relative dispersivity approaches a constant value, and the transverse components gradually approach zero. In fractional Guassian and self-similar media the longitudinal dispersivity is shown to be a power function of the transverse second moment of the plume. The initial plume dimensions have a strong influence on dispersivity values for plumes in media with structured variability across a wide range of scales.

A generalization of self-similar random field models is presented to accommodate the widely observed feature of statistical anisotropy in ln K variability. A practical illustration of the results is presented based on ln K variability data at the Cape Cod site. Two-scale exponential and self-similar models are fit to the ln K variograms, and dispersivity computations are developed for two plumes with very different initial dimensions. At a displacement of about 1 km, a longitudinal relative dispersivity (Ar11) value of 30 m is estimated for the sewage plume with initial dimensions of 500¿500¿10 m, while the estimate corresponding to the tracer test plume with initial dimensions of 5¿5¿1 m is about 1--2 m. The corresponding ensemble longitudinal dispersivity (A11) is about 90 m. Relative dispersivities are more appropriate for characterizing the dilution and spreading at individual heterogeneous aquifers. Dispersivities associated with the ensemble average concentration will tend to overestimate the degree of dilution and spreading in an aquifer, and this error can be very large in media with a widerange of scales of heterogeneity. ¿ American Geophysical Union 1995.