Application of fractional differential equation to interpret the dynamics of dissolved heavy-metal uptake in streams at a wide range of scales

Fractional differential equations (FDEs) provide promising models to simulate non-Fickian transport in heterogeneous systems such as geological media, but the FDEs have seldom been used to model reactive transport across a wide range of spatial scales. To fill the knowledge gap, this study proposed a fractional-order advection-dispersion-reaction (fADR) model to quantify the dynamics of dissolved heavy metals, such as manganese (Mn), moving in streams with oxidative precipitation and sorption in storage zones. The fADR model was applied to fit the Mn concentration profiles documented in the literature at the hyporheic flow path scale (0.30m in length), the reach scale (100-3000m), and the basin scale ( ∼ 20000 m) at Pinal Creek, Arizona. Numerical results showed that compared with standard transport models such as the OTIS model and the single-rate mass transfer model, the fADR model can better capture the observed plumes at all scales. This is because the space fractional-diffusive term in the fADR model can capture super-diffusive jumps of dissolved metals driven by turbulence at the bedform scale and flooding in a decade-long time scale. Meanwhile, the time fractional derivative term in the fADR model describes complex solute retention due to multiple-rate mass exchange between the mobile zone (stream or the hyporheic flux) and various storage regimes with different properties (such as streambed sediments and stagnant portions in the hyporheic zone). This does not rely on the equilibrium chemistry condition for solutes in storage zones assumed by standard hyporheic-exchange models. In addition, the first-order reaction in the fADR model can efficiently characterize the mass decline of Mn downstream resulting from enhanced Mn oxidation (such as oxidation of MN(II) to +3 or +4 oxidation states) due to the input of streamflow with increased pH and dissolved oxygen and/or groundwater recharge with high dissolved metals into the hyporheic zone. The decoupled super-diffusion and retention for Mn can exhibit scale-dependent behaviors due to the evolution of driving mechanisms, which can be characterized by the parsimonious, phenomenological FDE by adjusting the indexes. Therefore, the application of FDEs helps us to interpret the physical and geochemical processes in streams across scales.