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(phi) of a given fraction phi 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(phi). The expected value and variance of t(phi) are available explicitly if the aperture and length distribution have finite variance. The expected t(phi) 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(phi) 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.