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Stochastic transport theories rely on the assumptions that the principles of stationarity and ergodicity are satisfied. These assumptions imply that the pattern of heterogeneity can be viewed as being spatially periodic, thus yielding a finite correlation scale. We examine a type of heterogeneity which does not satisfy these principles: self‐similar, or fractal heterogeneity, which exhibits a pattern (over a large range of scales) that is independent of the scale of observation and thus will possess a very large correlation scale. We review the basic concepts of fractal geometry and develop scaling relationships for fractal travel distance versus scale of observation. We then develop Lagrangian models for dispersion in a single fractal streamtube and for a set (or bundle) of fractal streamtubes. The results are compared to classical one‐dimensional advection‐dispersion theory, existing fractal and stochastic theories, and to a summary of field‐measured dispersivity data. For the single streamtube model, field‐measured dispersivity is proportional to the straight‐line travel distance raised to a power of D ‐ 1, where D is the fractal dimension. For the set of fractal streamtubes, field‐measured dispersivity is proportional to the straight‐line travel distance raised to a power of 2 D ‐ 1. The analytical expressions are verified and elaborated by a fractal random walk computer model. The computer model is used to show the self‐similar nature of heterogeneity that is independent of the scale of observation. Comparisons of the field dispersivity data with our fractal models indicate that most tracer tests have been performed in media that approximate a set of fractal streamtubes.
Wheatcraft et al. (1988) studied this question.