Heterogeneous media consisting of segregated flow regions are fractionalorder systems, where the regionalscale anomalous diffusion can be described by the fractional derivative model (FDM). The standard FDM, however, first, cannot characterize the Darcyscale 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. Latetime 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 latetime 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 Darcyscale dispersion follows the tempered stable law, and the model parameters might be related to the soil size and flow conditions.
Geological formations usually exhibit multiscale physical and/or chemical heterogeneity, which can lead to the space and/or time nonlocal dependency for solute transport (see the extensive review by Zhang et al. [
Two major knowledge gaps, however, remain for the FDM. First, while the FDM has been applied by hydrologists to simulate contaminant transport through regionalscale heterogeneous porous and fractured media for more than a decade [
This study attempts to fill the above knowledge gaps. We apply a tempered fractional derivative model (TFDM), which is a generalization of the standard FDM, to capture bromide transport through laboratory repacked sand. This way, the applicability of the fractionalorder partial differential equation on Darcyscale dispersion can be tested reliably. Hence the first knowledge gap can be filled. The combined study of laboratory experiments and stochastic analysis may also reveal the trend of major transport parameters varying with sand properties. Such trend might lead to the answer regarding the second knowledge gap.
Laboratory experiments of solute transport through sand columns were conducted extensively by the hydrology community to explore the dynamics of dissolved solutes. For example, recent experiments [
The rest of the paper is organized as follows. In Section
We conducted laboratory experiments to measure the breakthrough curve (BTC) of one conservative tracer (bromide, as NaBr) in various sand packs. A glass column with internal dimensions of 150 mm (length) × 15.9 mm (diameter) was packed with glass beads or silica sand (Figure
(a) Photograph of the experimental setup. (b) The medium: silica sand (left) and glass beads (right).
Three different types of column experiments were conducted. For the first type of experiment (denoted by Run 1 in the following), the column was filled with uniform glass beads with an average diameter of 0.4 mm (Figure
For the second type of experiments (denoted by Run 2), two different sizes of glass beads were well mixed and packed, forming networks with mobile and relatively immobile domains. The first group of these experiments included glass beads with average diameters of 1 and 0.2 mm, while the second group included glass beads with average diameters of 0.4 and 0.2 mm.
For the third type of experiments (denoted by Run 3), silica sand with a specific particle size distribution was used to represent “natural” soil, where the irregular shape of the sand may affect the interconnected pores and the corresponding tracer dynamics. The overall particle size distribution was obtained by combining the following size fractions obtained by sieving: 0.85~1.0 mm (representing the coarse sand), 0.35~0.425 mm (medium sand), and 0.15~0.25 mm (fine sand), respectively. For description simplicity, in the following we denote the three size fractions by 1, 0.4, and 0.2 mm size silica sand, respectively, corresponding to the glass beads with similar diameters used in the second type of experiment. The sand was then cleaned with acid before packing.
After the column was packed, the following steps were performed to obtain BTCs. A fivepoint calibration of the bromide ion selective electrode (ISE) (Orion) was performed. The potential of standard solutions was measured from the lowest to the highest bromide concentration. Deionized water (DI) was run through the column for at least 2 hours prior to tracer injection to establish the flow domain. The pulse of bromide (of volume 10 mL) was then injected into the column (oriented horizontally) at a concentration of 1 mol/L, and discrete samples were collected from the outlet using a fraction collector (Teledyne ISCO). The sampling interval at early and late times was shortened to better record the tails of the BTC, known to be critical signals of nonFickian transport. The bromide concentration was measured in all collected samples using the calibrated ISE probe with a detection limit of 10^{−5} mol/L.
The flow velocity was adjusted during the experiment using a rotary peristaltic pump and controller (Cole Palmer Masterflex) (Figure
Figure
BTC for the first type of experiment (i.e., Run 1 where the column filled with homogeneous glass beads) (a). (b) is the loglog plot of (a). The parameters shown in (b) are those fitted by the tempered fractional derivative model (
A slight latetime tail in the BTC (Figure
The BTC measured in Run 2 contains a much heavier latetime BTC tail than the “homogeneous” case (Figure
The measured BTCs for the laboratory experiment Run 2 (i.e., the column filled with glass beads) (a) and Run 3 (i.e., the column filled with silica sand) (b). In the legend shown in (a), “
Note that the BTC tail declines faster for a higher fluid velocity, as shown by the glass beads with a 0.4 + 0.2 mm mixture (Figure
The observed BTC for Run 3 also contains an apparent latetime tail (Figure
The latetime tail in the BTC shrinks with the increase of fluid velocity (Figure
The laboratory observed bromide BTCs (symbols) versus the bestfit or predicted solutions using the TFDM model (
Nonlocal transport theories were developed recently to capture nonFickian diffusion, as reviewed extensively by Haggerty et al. [
Comparison of the TFDM (2) and the standard CTRW model reviewed by Berkowitz et al. [
Comparison  TFDM ( 
The standard CTRW [ 

Physical theory behind the model  Scaling limit of CTRW  The general master equation 
Form of the memory function 


Number and value of parameters to capture subdiffusion  3 ( 
4 ( 
Mechanism for subdiffusion  Diffusion  Slow advection 
Modeling superdiffusion  Applicable with 
N/A 
Modeling the mixed diffusion  Applicable with 
N/A 
Multidimensional extension  Multiscaling index [ 
No multiscaling index 
Spatial variability of transport 


Molecular diffusion, however, may also cause the subdiffusive effect, as suggested by the physical process of multiple rate mass transfer [
The tempered stable model proposed by Meerschaert et al. [
In our representation, we propose the following tempered fractional derivative model, or TFDM, by generalizing the current time fractional derivative models [
Model (
The two fractional derivative terms in model (
The model in (
Model (
For Run 2 using 1 + 0.2 mm glass beads, we first fit the BTC using the TFDM (
For Run 2 using 0.4 + 0.2 mm glass beads, the bestfit parameters for the BTC with the slowest flow velocity
The same conclusion is found for Run 3 with silica sand (Figures
Laboratory experiments show that the subdiffusion increases by increasing the range of the sand size distribution, especially when the column is filled with glass beads. A wider sand size distribution tends to enhance subdiffusion, because broader distributions of particle diameter more readily form immobile regions. Bromide transport through silica sand is not as sensitive to the size distribution as the glass beads, which might be due to either the strong influence of the irregular shape of silica sand on the mass exchange between mobile and relatively immobile regions and/or the relatively large flow rate required for the laboratory experiments that counterbalances the size effect. Future studies are needed to explore further the influence of sand shape and low flow rate on subdiffusion.
Water flow rates across the sand column also affect subdiffusion. As the flow rate in nonaggregated material increases, the percentage of the total domain dominated by diffusive transport decreases. Hence the increase of fluid velocity likely decreases the contribution of diffusion to the arrival times of solute particles. In particular, if the observation period is short, the latetime subdiffusive behavior affected by the flow rate may not be detected. Hence the total experimental period should be as long as possible, to identify the full behavior of subdiffusion at late times.
The TFDM parameters can efficiently capture the subtle variation of subdiffusion. For example, the temporal scale index
In addition, the standard FDM tends to overestimate the latetime BTC tail (see, e.g., the dashed line in Figure
Finally, the bestfit space scale index
To further understand the subdiffusive process, we simulate again the measured BTCs by assuming that the observed subdiffusion is driven by slow advection. A time fractional derivative model can be built to describe an advectiondominated subdiffusion. Assuming a CTRW with independent jump sizes and waiting times (or the elapsed time during two subsequent jumps), the corresponding scaling limit is
Applications show that model (
Bestfit of BTCs (solid lines) using model (
Further applications show that model (
It is possible to extend model (
The TFDM model (
The TFDM (
One of the major limitations of the TFDM (
The fractional engine is a promising tool to capture anomalous dispersion in heterogeneous media, but major challenges do exist, including the Darcyscale fractional dispersion and the influence of medium heterogeneity. In this study, laboratory experiments were combined with stochastic model analysis to explore the applicability of the fractional engine in capturing Darcyscale dispersion in sand columns filled with various materials and to explore the potential link between medium properties and model parameters. The following four main conclusions are drawn.
The tempered fractional derivative model can capture subdiffusion at the Darcyscale. The physical model distinguishes solute status, contains the least number of parameters, and can be extended conveniently to capture advanced transport processes. Most importantly, the TFDM can characterize the transient decline rate of the latetime BTC, probably due to the finite distribution of particle waiting times, while the standard FDM tends to overestimate the latetime tail of BTC.
Both the sand particle size distribution and the fluid velocity can affect the Darcyscale subdiffusion. All of the measured BTCs of bromide contain an apparent latetime tail, which is heavier for a wider particle size distribution of sand or a smaller fluid velocity. These two properties can enhance the relative contribution of diffusion to the latetime arrivals of solute particles. Hence both medium properties and flow conditions can affect subdiffusion, which is consistent with the conclusion of Berkowitz and Scher [
Diffusioncontrolled subdiffusion is possible. In heterogeneous or even homogeneous sand columns, subdiffusive transport due to molecular diffusion occurs even for a large Peclet number. The relatively immobile regions formed during soil repacking cause diffusioncontrolled subdiffusion, following the physical process of multirate mass transfer. The diffusioncontrolled subdiffusion can be apparent in undisturbed soils, where the immobile zones are almost inevitable and the corresponding mass transfer rate varies significantly in space.
There are serious limitations in applying the slowadvectiondominated subdiffusive model, such as model (
This work was supported by the Desert Research Institute (DRI) and the National Science Foundation (NSF) under Grant no. DMS1025417. Part of the experiments was conducted by Wallace Atterberry supported by the first author. This paper does not necessary reflect the views of NSF or DRI. The authors thank three anonymous reviewers for helpful comments that improved the paper.