A new finite difference model is developed to characterize solute transport in a fractured medium. In the model, the dispersivity of individual fractures is employed as a variable increasing with travel distance from a source. To demonstrate the applicability of the model, the effect of fracture characteristics on flow and solute transport in a fractured medium is simulated. The fracture characteristics are described by a power law fracture length distribution and fracture spatial density, which are characterized by length exponent (a) and proximity to percolation threshold (Deltarho), respectively. The results show that neither the average linear velocity nor the breakthrough time of solutes change significantly with increasing Deltarho. For constant Deltarho, as a becomes larger, the average linear velocity decreases while the breakthrough time increases. The megascopic dispersivity increases slightly at first, and then decreases afterwards with increasing Deltarho.