Péclet-Number-Dependent Longitudinal Dispersion in Discrete Fracture Networks

Abstract

Dispersion in fractured media impacts many environmental and geomechanical practices. It is mainly controlled by the structure of fracture networks and the Péclet number $(Pe)$, but predicting it remains challenging. In this study, numerous three-dimensional stochastic discrete fracture networks (DFNs) were generated, where the density, size, and orientation vary significantly. The aperture and conductivity are proportional to the size, following power-laws. Through flow and transport simulation, we evaluated the longitudinal dispersion coefficients $\left(D_L\right)$. We found that, as density increases, the tortuosity decreases and the first passage time distributions approximate bell-shaped curves more closely, which suggests, but does not fully guarantee, that an asymptotic dispersion regime may emerge for denser DFNs, as solute particles traverse more fractures and the macroscopic inter-fracture mixing is more homogeneous. We then determined the $D_L$ values for DFNs in which the time evolution of the variance of particle displacements becomes linear and hence asymptotic. The results show that both $Pe$ and fracture density affect $D_L$, but the former has a much stronger influence than the latter. A new Péclet number $\left(P{e}^c\right)$ was recalculated for all DFNs, where the characteristic length scale accounts for the influence of large fractures. Dimensionless $D_L$ values show a unique power-law relationship with high $P{e}^c$ values. Furthermore, when advection dominates, the dimensionless $D_L$ can be described by a universal finite-size scaling function depending on fracture density and domain sizes. The findings of this study enhance the understanding of transport in fracture networks and imply the potential for predicting $D_L$ in a broad range of scenarios using statistics on fracture parameters obtainable at the field scale.