Transport and retention in fractured rock: Consequences of a power-law distribution for fracture lengths

A probabilistic model for the transport of a reacting species in fractured rock is presented. Particles are transported by advection through a series of n rock fractures, and also diffuse and react chemically in the surrounding porous medium. The fracture attributes are unobserved with predefined statistical distribution. The time of arrival t_φ of a given fraction φ of an initial solute pulse, a key quantity used in a variety of applications, is related to the statistics for fracture apertures and lengths. A classification scheme is developed for the large n asymptotics of t_φ. The expected value and variance of t_φ are available explicitly if the aperture and length distribution have finite variance. The expected t_φ is infinite, and its probability distribution is related to asymmetrical Levy distributions in the case of a power-law distribution for lengths. The most probable time of arrival is proposed as a robust alternative to the expected value. A scaling transition in the most probable t_φ versus n is found as the power-law exponent changes. These results suggest that risks associated with migrating contaminants may be misrepresented by conventional stochastic analyses.