The understanding of solute transport in rock fractures is of great importance in many engineering fields. In this study, two groups of experiments on artificial and natural single fractures with different fracture apertures and roughness were conducted to investigate the changes of solute transport regimes. The time fractional advection–dispersion equation (tFADE) as a promising model was applied to describe the anomalous transport. The performance of the classical advection-dispersion equation (ADE) and tFADE was compared according to the fitting precision of breakthrough curves (BTCs). The responses of the fitting parameters in the tFADE to the experimental conditions were also discussed. The results indicated that the non-Fickian transport more likely occurred in the short distance transport, and the larger Peclet number ( Pe ) led to the increase of the exponent of the power-law function in the phase of concentration decline. The tFADE was superior to ADE in capturing the non-Fickian transport especially the tailing behavior. The fractional order of time α in the tFADE was the key parameter to describe the anomalous transport process, and its responding mechanisms of were revealed: the best-fit α decreased with the increase of flow velocity and the decrease of the fracture aperture. The roughness of the single fracture which leads to a complex flow field had a significant effect on the best-fit α . The findings of this study can help for better understanding the effectiveness and physical significance of the tFADE.