This study simulates the effect of fracture characteristics on permeability, average linear velocity, breakthrough time, and megascopic dispersivity of a fractured medium. The authors use a power law for fracture length distribution and a fractional Brownian motion for hydraulic fracture aperture spatial distribution, which can be characterized by a and H, respectively. A new finite difference model is developed for solute transport that considers advection, adsorption, first-order decay, and scale-dependent dispersivity of a single fracture using a numerical dispersion term caused by finite difference approximation. The ranges of 1.4 less than or equal to a less than or equal to 2.2 and 0.1 less than or equal to H less than or equal to 0.9 are considered The results show that the permeability is not very affected by a, but increases slightly with increasing H. The average linear velocity decreases with increasing a, but increases with increasing H. Both the breakthrough time and the megascopic dispersivity increase as a becomes larger but decrease as H becomes larger Finally, the megascopic dispersivity is proportional to the linear size of a medium by the power of 8, which increases from approximately 0.65 to 0.85 with increasing a, which shows the analogy of the fractured medium to a highly heterogeneous porous medium.