Power‐law velocity distributions in fracture networks: Numerical evidence and implications for tracer transport

Abstract

Velocity distributions in two‐ and three‐dimensional networks of discrete fractures are studied through numerical simulations. The distribution of 1/v , where v is the velocity along particle trajectories, is closely approximated by a power law (Pareto) distribution over a wide range of velocities. For the conditions studied, the power law exponents are in the range 1.1–1.8, and generally increase with increasing fracture density. The same is true for the quantity 1/bv , which is related to retention properties of the rock; b is the fracture half‐aperture. Using a stochastic Lagrangian methodology and statistical limit theorems applicable to power‐law variables, it is shown that the distributions of residence times for conservative and reacting tracers are related to one‐sided stable distributions. These results are incompatible with the classical advection dispersion equation and underscore the need for alternative modeling approaches.