Can a Time Fractional‐Derivative Model Capture Scale‐Dependent Dispersion in Saturated Soils?

Abstract

Time nonlocal transport models such as the time fractional advection-dispersion equation (t-fADE) were proposed to capture well-documented non-Fickian dynamics for conservative solutes transport in heterogeneous media, with the underlying assumption that the time nonlocality (which means that the current concentration change is affected by previous concentration load) embedded in the physical models can release the effective dispersion coefficient from scale dependency. This assumption, however, has never been systematically examined using real data. This study fills this historical knowledge gap by capturing non-Fickian transport (likely due to solute retention) documented in the literature (Huang et al. 1995) and observed in our laboratory from small to intermediate spatial scale using the promising, tempered t-fADE model. Fitting exercises show that the effective dispersion coefficient in the t-fADE, although differing subtly from the dispersion coefficient in the standard advection-dispersion equation, increases nonlinearly with the travel distance (varying from 0.5 to 12 m) for both heterogeneous and macroscopically homogeneous sand columns. Further analysis reveals that, while solute retention in relatively immobile zones can be efficiently captured by the time nonlocal parameters in the t-fADE, the motion-independent solute movement in the mobile zone is affected by the spatial evolution of local velocities in the host medium, resulting in a scale-dependent dispersion coefficient. The same result may be found for the other standard time nonlocal transport models that separate solute retention and jumps (i.e., displacement). Therefore, the t-fADE with a constant dispersion coefficient cannot capture scale-dependent dispersion in saturated porous media, challenging the application for stochastic hydrogeology methods in quantifying real-world, preasymptotic transport. Hence improvements on time nonlocal models using, for example, the novel subordination approach are necessary to incorporate the spatial evolution of local velocities without adding cumbersome parameters.

Nonlocal transport models for capturing solute transport in one‐dimensional sand columns: Model review, applicability, limitations and improvement

“…The ST-fADE (7) also has the best-fit dispersion coefficient/dispersivity increasing in space. by Huang et al (1995) and evaluated by Garrard et al (2017). After crossing this zone, the local dispersion coefficient and dispersivity tend to decline (Figure 2).…”

“…This section expands the work in Garrard et al (2017) by using a more reliable dataset and checking the feasibility of the nonlocal transport models reviewed above.…”

“…Most of the measured tracer BTCs in Huang et al (1995), however, were incomplete with apparent missing data especially at the late time, resulting in high uncertainty in quantifying the scale-dependent dispersion at the relatively larger spatial/temporal scale. Additional tests with reliable and complete datasets, therefore, are needed to systematically validate the finding in Garrard et al (2017). Most importantly, can the other nonlocal transport models, especially the ST-fADE and the CTRW framework, which were assumed to capture unsolved heterogeneity in the velocity field, capture the scale-dependent dispersion observed in the lab?…”

Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions

“…[ 1 , 2 , 3 , 4 ]. In general, the fractional advection-dispersion models with the Caputo fractional derivative [ 5 ] may match the real observation better than the classical advection-dispersion models [ 6 , 7 ], which were applied to describe the transport of chemical pollutants in shale gas exploitation [ 8 ]. For example, the time- [ 9 ] and space- [ 10 ] fractional advection-dispersion models have been verified to be able to capture some non-Fickian transport.…”

“…For example, the time- [ 9 ] and space- [ 10 ] fractional advection-dispersion models have been verified to be able to capture some non-Fickian transport. The tempered advection-dispersion models, such as the promising models, can capture the scale-dependent dispersion (see [ 5 ]) and predict the truncated power-law breakthrough curves very nicely (see [ 11 ]). In fact, the spatial evolution of the conservative solute molecules with the complex distribution is closely related to the physical and chemical interactions between them and the porous media [ 12 , 13 , 14 ].…”