An efficient approximation of non-Fickian transport using a time-fractional transient storage model

Abstract

The classical transient storage (TS) model is widely used to describe a non-Fickian solute transport process induced by the solute mass exchange between the main channel and the immobile zone. It has been shown that the single rate TS model tends to underestimate the slower exchange that occurs in a deeper or longer hyporheic flow path. This long-term retention can be better described by the fractional mobile/immobile model (FMIM). However, in a real-world application, this method usually overestimates the late time concentrations in a breakthrough curve (BTC), which can be better described by the tempered-time-fractional model (TTFM). In this study, we introduced a fractional-in-time derivative TS model (FTTS), which can describe broad waiting times in a particle motion process. First, a fully-implicit numerical scheme was applied to solve the FTTS and the method was validated by comparing its results with the analytical solutions of the classical advection-dispersion model (ADE) and the FMIM. Then, the FTTS was applied to fit the synthetic data generated by the STAMMT-L and field tracer experiment data. Further, a variance-based global sensitivity analysis was performed to assess the influence rank of the parameters to the heavy tail of the BTCs. The results indicated that the FTTS could fit the BTCs generated by the ADE and FMIM well. In synthetic cases, the FTTS could reproduce different heavy tailing BTCs accurately. In addition, the FTTS could well describe tracer data in natural streams and performed better than the TTFM. The sensitivity analysis indicated that both the fractional-in-time term and the TS term in the FTTS had important impacts on the tail of the BTC, which could make the FTTS more flexible for describing the tailing phenomenon in a BTC.