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

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.