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Keywords:

  • percolation theory; solute transport; scaling

Abstract

We present a fundamental theory of solute dispersion in porous using (i) critical path analysis and cluster statistics of percolation theory far from the percolation threshold and (ii) the tortuosity and structure of large clusters near the percolation threshold. We use the simplest possible model of porous media, with a single length scale of heterogeneity in which the statistics of local conductances are uncorrelated. This combination of percolation-based techniques allows comprehensive investigation and predictions concerning the process of dispersion. Our predictions, which ignore molecular diffusion and make minimal use of unknown parameters, account for results obtained in a comprehensive set of nearly 1100 experiments performed on systems ranging in size from centimeters to 100 km. The success of our simple treatment overturns many existing notions about transport in porous media, such as (1) multiscale heterogeneity must be accounted for in predictions (single scale is sufficient), (2) geologic correlations are of great importance (the randomness of percolation theory is more appropriate for prediction than the most complicated models in other frameworks), (3) geologic complexity is more important than statistical physics (exactly the reverse), (4) knowledge of the subsurface is more important than knowledge of the initial conditions of the plume (the latter is critical, the former may be virtually irrelevant), (5) diffusion is dominant over advection (diffusion appears seldom to be relevant at all), (6) fracture networks are fundamentally different, and more complex, than porous media (the two are mostly equivalent), (7) the fractal structure of the medium is relevant to power-law behavior of the dispersion (in fact, at short times it is the heterogeneity of the medium, while at long times it is the fractal structure of the critical paths), and (8) there is a relation between an increase in dispersion with scale and a similar increase in the hydraulic conductivity (in fact the present model is consistent with both a diminishing hydraulic conductivity and a diminishing solute velocity with increasing spatial scale). © 2009 Wiley Periodicals, Inc. Complexity, 16,43–55, 2010