Heterogeneous media consisting of segregated flow regions are fractional-order systems, where the regional-scale anomalous diffusion can be described by the fractional derivative model (FDM). The standard FDM, however, first, cannot characterize the Darcy-scale dispersion through repacked sand columns, and second, the link between medium properties and model parameters remains unknown. To fill these two knowledge gaps, this study applies a tempered fractional derivative model (TFDM) to capture bromide transport through laboratory repacked sand. Column transport experiments are conducted first, where glass beads and silica sand with different diameters are repacked individually. Late-time tails are observed in the breakthrough curves (BTC) of bromide even in relatively homogeneous glass beads. The TFDM can capture the observed subdiffusion, especially the late-time BTC with a transient declining rate. Results also show that both the size distribution of repacked sand and the magnitude of fluid velocity can affect subdiffusion. In particular, a wider sand size distribution or a smaller flow rate can enhance the subdiffusion, leading to a smaller time index and a higher truncation parameter in the TFDM. Therefore, the Darcy-scale dispersion follows the tempered stable law, and the model parameters might be related to the soil size and flow conditions.
Modeling ramp-hold indentation measurements based on Kelvin–Voigt fractional derivative model
