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Diffusion in one‐dimensional multifractal porous media
Author(s) -
Lovejoy S.,
Schertzer D.,
Silas P.
Publication year - 1998
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/1998wr900007
Subject(s) - scaling , random walk , exponent , multifractal system , porous medium , statistical physics , diffusion , anomalous diffusion , phase transition , singularity , physics , condensed matter physics , mathematics , materials science , fractal , porosity , mathematical analysis , thermodynamics , geometry , statistics , computer science , philosophy , linguistics , composite material , innovation diffusion , knowledge management
We examine the scaling properties of one‐dimensional random walks on media with multifractal diffusivities, which is a simple model for transport in scaling porous media. We find both theoretically and numerically that the anomalous scaling exponent of the walk is d w =2 + K (− 1) where K (− 1) is the scaling exponent of the reciprocal spatially averaged (“dressed”) resistance to diffusion. Since K (− 1) > 0, the walk is subdiffusive; the walkers are effectively trapped in a hierarchy of barriers. The trapping is dominated by contributions from a specific order of singularity associated with a phase transition between anomalous and normal diffusion. We discuss the implications for transport in porous media.