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Dispersion in Fractal Fracture Networks
Author(s) -
Ross Benjamin
Publication year - 1986
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/wr022i005p00823
Subject(s) - dispersion (optics) , exponent , standard deviation , diffusion , position (finance) , fracture (geology) , fractal , particle (ecology) , constant (computer programming) , scale (ratio) , random walk , mathematics , statistical physics , physics , mathematical analysis , mechanics , statistics , geotechnical engineering , geology , optics , thermodynamics , philosophy , linguistics , oceanography , finance , quantum mechanics , computer science , economics , programming language
The solute transport equation describes dispersion as a diffusion‐like process characterized by a constant dispersivity, implying that small pulses of contaminant grow as the square root of distance traveled. Measured dispersivities tend to be proportional to the scale of the measurement, suggesting that the pulses grow in linear proportion to distance. An exponent r , defined by the relationship d ∼ L r where d is the standard deviation of position of a contaminant particle and L is the distance traversed, may be used to characterize dispersion. It takes a value ½ in the first case and 1 in the second. The movement of a contaminant particle is analyzed here as a random walk through a fracture network composed of fissures of varying transmissivity. If large‐transmissivity fractures are sufficiently few, then one calculates r = ½. If large‐transmissivity fractures are numerous, one obtains 2 r = q + 1, where q is an exponent that describes the relationship between fracture spacing and transmissivity. Inserting the reasonable value q = 1 gives r = 1. This theory thus provides a physical basis for the observed scale‐dependence of dispersivity.