Typically the spreading length-scale in a diffusion process increases in time as ℓ∼tn,n=1/2. In the presence of spatial heterogeneities, however, so-called anomalous diffusion can occur, here the time exponent n≠1/2. The objective of this paper is to present a numerical simulation that directly demonstrates the link between spacial heterogeneity and anomalous diffusion. The simulation is of the infiltration of a fluid into a horizontal unit area Hele-Shaw cell in which the permeability at specified locations, laid out as a Sierpinski fractal carpet pattern, can differ from the nominal value K = 1. When there is no permeability contrast, the fluid infiltration has a diffusion-like behavior, i.e., the filled plan-form area increases in time as F=Atn,n=1/2. When there is a permeability contrast, however, although the evolution of infiltration still follows a power law, anomalous behavior is observed; subdiffusive (n<1/2) when K < 1 and superdiffusive (n>1/2) when K > 1. These anomalous behaviors persist, even when the permeability contrast is only imposed over the largest sized fractal pattern element. But, if the pattern over which the permeability contrast is imposed has a subdomain length scale (obtained by filling in the largest sized element in the Sierpinski carpet with the smaller sized elements), normal diffusion n=1/2 behavior is recovered. In the special case that the imposed permeability in the pattern elements is K蠐1, an approximate model, directly relating the coefficient and exponent in the infiltration power law to the porosity and fractal dimension of the carpet pattern, is derived and validated.
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© 2015. American Geophysical Union. All Rights Reserved.
- anomalous diffusion
- direct simulation
- fractal carpet
- spacial heterogeneities