# Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

^{1}

^{2}

^{3}

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^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{−3}] is the material density, the symbol ${\partial}^{\gamma}/\partial {t}^{\gamma}$ represents the Caputo time-fractional derivative with order $\gamma $ [dimensionless] (0 < γ ≤ 1), $\alpha $ [dimensionless] (1 < $\alpha $ ≤ 2) is the index of the (positive) Riemann–Liouville space-fractional derivative, V [LT

^{−1}] is the average flow velocity, D [L

^{α}T

^{−1}] is the effective dispersion coefficient, and ${\beta}^{*}$ = 1 [T

^{γ−1}] is used here for unit conversion (so that velocity V can have the commonly used dimension). Here, the Riemann–Liouville space-fractional derivative is needed since the corresponding FPDE on a bounded domain governs a well-defined stochastic process (while the bounded value problem for the FPDE with the Caputo space-fractional derivative generates negative solutions) [12,13]. Second, the space-nonlocal FADEs with a space-fractional derivative can capture super-diffusion (which represents fast displacement with the plume variance growing faster than linear in time) driven by hydrologic mechanisms including local-scale river turbulence, large-scale flooding, and other preferential flow paths, even though super-diffusion has been consistently ignored by current time-nonlocal transport models in hydrology [14]. Notably, the FADE (1) uses the one-side space-fractional derivative because the fast displacement for pollutant particles is usually one dimensional (from upstream to downstream) in geomedia (meaning that the two-side FPDEs are not appropriate for modeling typical hydrologic processes). Third, the time-nonlocal FADEs with a time-fractional derivative can simulate sub-diffusion due to chemical/physical sorption/desorption and/or retention of pollutants in geomedia [15].

## 2. Hierarchical Method Using Multi-Level Fractional-Derivative Models

#### 2.1. Multi-Scale Modeling, Anomalous Transport, and Classical Models

^{−1}~10

^{0}m); (b) the reach scale (10

^{1}~10

^{3}m), where most field tests were conducted; and (c) the watershed scale (>10

^{3}m), which is critical for aquatic ecosystems.

_{s}[ML

^{−3}] denote the chemical concentration in the main channel and the storage zone, respectively; $a$ [T

^{−1}] is the rate constant for mass exchange between stream and the storage zone; A and A

_{S}[L

^{2}] are the stream and storage-zone cross-sectional areas, respectively; and V and D are the same as those in the FADE (1) (the medium here is the open channel). The finite-size, single storage zone can be separated into the streambed and the hyporheic zone by adding one more governing equation in (2). The TSM (2) is the best-known phenomenological model for the stream-aquifer system. Implementation software and variants of the TSM (1) include the popular software OTIS/OTEQ [33,34] and the geochemical submodels MINTEQ and MINEQL [35,36]. If $a=0$, then there is no mass exchange or storage zone, and the TSM (2) reduces to the classical advection-dispersion equation (ADE) for conservative solutes moving in a homogeneous system.

#### 2.2. Development of FADEs for Multi-Scaling Transport in the River Corridor

#### 2.2.1. Geomorphologic Unit Scale: Fixed-Index FADE for Stable Anomalous Dynamics

^{γ−1}] is the capacity coefficient describing the mass ratio between the adsorbed and mobile pollutants in equilibrium; the symbol ${\partial}^{\gamma ,{\lambda}_{t}}/\partial {t}^{\gamma ,{\lambda}_{t}}$ represents the Caputo type, tempered, time-fractional derivative ${\partial}^{\gamma ,{\lambda}_{t}}f\left(t\right)/\partial {t}^{\gamma ,{\lambda}_{t}}={e}^{-{\lambda}_{t}t}{\partial}^{\gamma}\left[{e}^{{\lambda}_{t}t}f\left(t\right)\right]/\partial {t}^{\gamma}$ with the temporal truncation parameter ${\lambda}_{t}$ [T

^{−1}] (whose inverse defines the maximum residence time) [40]; ${\lambda}_{x}$ [L

^{−1}] is the truncation parameter in space; $\overline{V}$ and $\overline{D}$ denote the spatially averaged velocity and dispersion coefficient, respectively; $z$ stands for the vertical direction (pointing to the hyporheic zone); and ${K}_{r}$ and ${K}_{r}^{*}$ [T

^{−1}] denote the reaction rate for pollutants in the open channel and the storage zone, respectively. Equation (3a) shows that the solute concentration change is due to the advective flux, the super-diffusive flux, and chemical reactions in the mobile phase, since particles embedded in the immobile phase cannot move. Equation (3b) implies that the immobile phase concentration is related to the historical mobile concentration (at the same location) filtered by the memory function, as well as the mass loss due to reactions.

#### 2.2.2. Reach Scale: Variable-Index FADE for Evolution of Anomalous Transport in a Non-Stationary System

#### 2.2.3. Watershed Scale: Distributed-Order FADE for Combing Anomalous Transport in Sub-Basins

## 3. Applications

#### 3.1. Application 1: Bedload Transport along Riverbed

^{γ−1}, and $\overline{D}=0.00421\left(\pm 0.0003\right)$ m

^{2}/s, where the symbol “$\pm $” denotes the 90% confidence interval.

^{−1}] is used for unit conversion), $D\left(x\right)=4.1\left(\pm 0.022\right)+0.0064\left(\pm 0.0002\right)x/{L}_{0}$ ft

^{α}/h, $\gamma =0.80\left(\pm 0.017\right)$, and $\beta =0.30\left(\pm 0.0149\right)$ h

^{γ−1}.

#### 3.2. Application 2: Heavy Metal Moving in a Stream Varying from Geomorphologic Unit Scale to Watershed Scale

## 4. Discussion

#### 4.1. FADE Applicability in Capturing Anoamlous Scaling in Rivers

#### 4.2. Fractional Index within a Single Scale: When Will it Reach Stable?

^{−1}~10

^{0}m), a single hydrofacies in aquifers (10

^{−1}~10

^{2}m), or the same type of soil in the vadose zone (10

^{−1}~10

^{1}m) may be captured by the FADE (7) with a stable index. This conclusion explains why the single-index FADE (7a) fits contaminant transport at the well-known MADE aquifer, Mississippi, U.S. (adjacent to fluvial-deltaic deposits of the Gulf of Mexico Basin) and the Cape Cod aquifer, Massachusetts, U.S. (consisting of glacio-fluvial outwash sediments) well [11]. The observed tracer plumes (at the MADE and Cape Cod sites) extended 100~200 m downgradient, which was on the scale of interconnected coarse sand/gravel hydrofacies, and therefore, according to the above-mentioned conclusion, anomalous transport at these two fluvial aquifers can be characterized by the FADE (7a) with a stable index, proving the finding in Zhang et al. [11].

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Application 1: (

**a**) The measured (symbols, from Martin et al. [46]) versus the best-fit snapshots for bedload moving along a fixed gravel bed. (

**b**) is the log-log plot of (

**a**) to show the tail.

**Figure 2.**Application 1: Modeled (red lines, using the variable-index fractional advection-dispersion equation (FADE) (4)) versus measured (circles, by Sayre and Hubbell [47]) snapshots of bed sediment at the North Loup River, Nebraska at ten sampling cycles (

**a**–

**j**). For comparison purposes, the results of the FADE (4) with a fixed index (dotted lines) and the original mode (without considering anomalous diffusion) from Sayre and Hubbell [47] (dashed lines) are also shown.

**Figure 3.**Application 2: (

**a**) three scales at the Pinal Creek Basin, Arizona (modified from Puckett et al. [1]). (

**b**–

**d**): The measured (symbols) vs. modeled (lines) BTCs using the FADE (7) for the dissolved Mn transport at all three scales in the stream.

**Figure 4.**Finite elements for a laboratory-scale fractured medium, which has a dimension of 10 × 3 m (length × width).

**Figure 5.**Fracture: The best-fit (lines, using the FADE (7)) versus the Monte Carlo BTCs (symbols) of dissolved pollutants moving at different travel distances along the saturated fracture-matrix medium plotted in Figure 4.

**Figure 6.**Fracture: The change of the time index γ in the FADE (7a) with the travel distance for fitting the Breakthrough curves (BTCs) shown in Figure 5. The dashed line shows the asymptote. (

**b**) is the semi-log plot of (

**a**).

**Table 1.**Properties and standard models for pollutants moving at three representative scales in streams.

Properties | Geomorphologic Unit Scale | Reach Scale | Watershed Scale |
---|---|---|---|

Hydrologic/biogeochemical factors on pollutant dynamics | Geomorphology; Turbulence; Local-scale mass exchange between channel and riverbed due to hydrologic & biogeochemical uptake | Variation in hydrologic dynamics & system properties; Broad biogeochemical functions | Climate change (including extreme rainfall events); Sub-watershed properties; Long-term land use/land cover change |

Anomalous transport properties | Super-diffusion due to hydro-function (turbulence); Sub-diffusion due to physical/biogeochemical functions | Non-stationary evolution of residence times and/or super-diffusion; Heavy-tailed residence times and strong uptake/retention | Mixing of anomalous diffusion from sub-basins; Long-term competition between fast jumps (due to flooding) and retention |

Standard models for pollutant transport at each scale | Physically based models, such as the Advective Pumping Model (APM) | Phenomenological models: Advection-dispersion equation; Time nonlocal or spatiotemporally nonlocal models | River continuum model (i.e., for DOM) [27]; Integrate reach-scale model; Fractal topography model; Pulse-Shunt model [28] |

Standard models’ limitation in modeling pollutant dynamics | They cannot well capture local-scale super-diffusion due to turbulence | They cannot capture non-stationary, scale dependent anomalous dispersion at the scale of 10^{1}~10^{3} m | They cannot capture mixed non-stationary anomalous transport in complex river networks |

**Table 2.**The best-fit parameters in the FADE (7a) for the snapshots shown in Figure 5. In the legend, “RMSE” denotes the root mean square error which evaluates the fitting.

Travel Distance (m) | V (m/d) | D (m^{2}/d) | γ [–] | RMSE |

0.02 | 1.035 | 0.005 | 0.83 | 0.0212 |

0.04 | 1.035 | 0.005 | 0.83 | 0.0234 |

0.08 | 1.035 | 0.005 | 0.83 | 0.0215 |

0.10 | 1.035 | 0.005 | 0.83 | 0.0243 |

0.40 | 1.035 | 0.050 | 0.84 | 0.0251 |

0.80 | 1.035 | 0.120 | 0.84 | 0.0231 |

1.00 | 1.035 | 0.120 | 0.84 | 0.0221 |

4.00 | 1.035 | 0.250 | 0.91 | 0.0243 |

8.00 | 1.035 | 0.250 | 0.93 | 0.0254 |

10.00 | 1.035 | 0.250 | 0.93 | 0.0206 |

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**MDPI and ACS Style**

Zhang, Y.; Zhou, D.; Wei, W.; Frame, J.M.; Sun, H.; Sun, A.Y.; Chen, X.
Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications. *Mathematics* **2021**, *9*, 790.
https://doi.org/10.3390/math9070790

**AMA Style**

Zhang Y, Zhou D, Wei W, Frame JM, Sun H, Sun AY, Chen X.
Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications. *Mathematics*. 2021; 9(7):790.
https://doi.org/10.3390/math9070790

**Chicago/Turabian Style**

Zhang, Yong, Dongbao Zhou, Wei Wei, Jonathan M. Frame, Hongguang Sun, Alexander Y. Sun, and Xingyuan Chen.
2021. "Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications" *Mathematics* 9, no. 7: 790.
https://doi.org/10.3390/math9070790