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 [formula presented] 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 [formula presented] of a given fraction [formula presented] 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 [formula presented] asymptotics of [formula presented] The expected value and variance of [formula presented] are available explicitly if the aperture and length distribution have finite variance. The expected [formula presented] 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 [formula presented] versus [formula presented] is found as the power-law exponent changes. These results suggest that risks associated with migrating contaminants may be misrepresented by conventional stochastic analyses.