# Advanced Numerical Modeling of Sediment Transport in Gravel-Bed Rivers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Sediment Transport and Bed Variation Module

- The horizontal surface is unchanged z
_{d}= 0, so sediment sorting occurs only in the active layer (1) and the upmost sublayer (2); i.e., the bed consists of only two layers. - The bed sediment is moving under two forms: infiltration or bed load.
- The flow and sediment transport are one-dimensional.

_{b}= q

_{b}(x,t) = total specific volumetric bed load discharge. Its fraction q

_{b,j}is determined by experiment or semi-experiment when considering non-equilibrium bed load [21]. In this study, the fractional bed load rate is calculated while using the bed-load equation proposed by Wilcock and Crowe [22]. Further, Equation (1) can also be rewritten as:

_{a}= p

_{a}(x,t) = the average porosity of active layer; p

_{s}= p

_{s}(x,t) = the average porosity of sublayer layer (Figure 3). If the porosity of layers is constant, the porosity source term S

_{p}= 0 and Equation (3) becomes the Exner equation. Therefore, Equation (3) can be considered as the expansion of the Exner equation, when considering the variation of porosity in space and time.

_{a,j}= the size fraction of active layer; and, β

_{b,j}= the size fraction j of sediment transport in the form of bed load. The function S

_{F,j}expresses the exchange process of the active layer and sublayers of the size fraction j. Determining the S

_{F,j}value can be considered as the quantification of the bottom infiltration process and Section 2.2 provides a detailed description of this function.

_{s,j}is the size fraction j of sediment in the subsurface layer. As can be seen in Equations (3)–(6), the bed porosity variation and the sediment exchange between two neighbor bed layers are considered. Furthermore, the experimental studies show that the bed porosity depends on two main parameters: grain size distribution and compaction degree of bed material. Section 2.3 describes details regarding the determination of porosity.

#### 2.2. Infiltration Process

_{m}and fine sediment d

_{m}can be used to evaluate the depth of the infiltration process. For example, according to [23], if the ratio (D

_{m}/d

_{m}) of river bed less than 10, then fine sediment is almost impossible to infill through the void space of the gravel bed. If this ratio is between 10 and 30, then fine sediments will be deposited in the void space of the top gravel-bed layer and create a sand seal. If 30 < D

_{m}/d

_{m}< 70, some fine sediments will be clogged at the void of the upper layer, but they will not clearly create a sand seal. When the ratio D

_{m}/d

_{m}greater than 70, all of the fine sediment will settle down to the bottom of the gravel bed. These critical values may be changed when the bed material consists of almost uniform spherical particles for fine and coarse sediments. In other words, the infiltration process depends on the characteristic parameters of the bottom (including the size distribution as well as the shape of the sediment particles). Additionally, this process depends on the near bottom hydraulic conditions, i.e., near-bed flow water and flow through the void space of the gravel bed. Cui et al. [3] proposed a theoretical model, in which the fraction of fine sediment decreases exponentially along with the depth of the gravel bed. Wooster et al. [24] and Gibson et al. [25] also used this hypothesis with laboratory observation data to develop their models, in which the fine reduction sediment is determined as a function of gravel depth.

_{F,j}depends on sediment discharge of size class j, the time variation of bed porosity and active layer thickness. However, effects of sediments in the stratum layer was not yet considered in this equation.

_{i}= the mass of a particle i; ${\overrightarrow{\mathrm{u}}}_{\mathrm{i}}$ = the velocity of a particle; $\overrightarrow{\mathrm{g}}$ = gravity acceleration; ${\overrightarrow{\mathrm{f}}}_{\mathrm{i},\mathrm{k}}$ = interaction force between particle i and particle k (contact force); ${\overrightarrow{\mathrm{f}}}_{\mathrm{i},\mathrm{f}}$ = interaction force between the particle i and the fluid; I = moment of inertia; ${\overrightarrow{\mathsf{\omega}}}_{\mathrm{i}}$ = angular velocity; d

_{i}= diameter of the grain i; ${\overrightarrow{\mathrm{n}}}_{\mathrm{i},\mathrm{k}}$ = directional contact = vector connecting the center of grains i and k.

_{i}= stiffness of grain i; δ

_{i,k}= The characteristic of the contact and displacement (also called the length of the springs in the two directions above); α

_{i}= damping coefficient; and, Δu

_{i}= relative velocity of grain at the moment of collision. Additionally, following Coulomb, the product of the coefficient of friction μ and the orthogonal force component determines the value of tangential friction. In the nonlinear contact force, the Hertz–Mindlin model, which is the tangential force component, will increase until the ratio (f

^{(}

^{τ}

^{)}/f

^{(n)}) reaching a value of μ, and it keeps the maximum value until the particles are no longer in contact with each other. A detail of the force models, as well as the method for determining the relevant coefficients, can be found in [28].

#### 2.3. Porosity Estimation

^{(D)}), where p

^{(D}

^{)}is porosity of coarse sediment. Correspondingly, the relative ratio of coarse sediment volume to the total volume of bed sediment is determined as:

^{(d)}and f

^{(d)}for fine sediment, we have f

^{(d)}= (1 − f

^{(D)}) and obtained the correlation between porosity of bed and porosity of coarse sediment and the relative proportions of fine sediment, as follows:

^{(D)}× p

^{(d)}). Substituting this value into Equation (13), we obtain the relationship between the bottom porosity and the characteristics of fine sediment for this case, as follows:

_{m}

_{in}when the fraction of fine sediment is approximately equal to the porosity of the coarse sediment. Kamann et al. [33] has improved Equation (15) and then proposed the same expression to calculate the bed porosity:

^{(d)}

_{min}is a fine size fraction when the bed porosity reaches the minimum value.

_{1}> d

_{2}> ... > d

_{j}> … > d

_{n}, initial specific volume V

_{1}, V

_{2}, …, V

_{j}, …, V

_{n}, and size fraction β

_{1}, β

_{2}, …, β

_{j}, …, β

_{n}.

_{j}and N

_{j}are calculated using the critical size ratio (d/D = 0.154) and comparing with the other diameters.

_{j}) grain based on M

_{j}, N

_{j}(see Figure 5). The middle zone is the controlling mixture of grain size jth (occupation), the first zone, and the last zone is presented filling mechanism. Particles in the first zone can fill in the structure component with diameter jth without disturbing in structure, and diameter jth is also filled in last zone structure.

_{j}, which consists of three parts (Figure 5).

## 3. Results and Discussions

#### 3.1. Infiltration of Fine Sediments into Gravel-Bed

^{3}). We refer to fine sediment as sand. The simulations are carried out for a bed with uniform sand and gravel (Case 1) and a graded sand-gravel-bed (Case 2). The sediment bed layer is 0.1 m thick with the following characteristics (Table 1):

- Case-1: uniform gravel size D = 10 mm and uniform sand with a size d defined from the following size ratios:Critical ratio for tetrahedral packing d/D = 0.154;One and halftime of the critical ratio for tetrahedral packing d/D = 0.231; and,Critical ratio for cubical packing d/D = 0.414.
- Case-2: Multi size fractions of sand and gravel bed:Sand with a mean diameter of d
_{m}= 0.26 mm and standard deviation of σ^{(d)}= 1.94; and,Gravel with a mean diameter of D_{m}= 7.1 mm and a standard deviation of σ^{(D)}= 1.35.

^{(D)}= 0.454. The fine sediment was fed over the bed surface from the 20,000th step. The simulation process was stopped when the void space in the bed surface has been clogged or the bed has been fulfilled with fine sediments. Figure 6 presents the structure of the gravel-bed, as well as the vertical distribution of fine sediments with three different sizes at the end of simulation time. The simulated results are qualitatively comparable to the experimental results that were conducted by [37,38] for the idealized packing of the purely bimodal bed sediments. For the case d/D = 0.154 (Figure 6a), it can be observed that the fine sediments mostly filled into the void space and were then trapped there. However, while increasing the size ratio to d/D = 0.231, the fine sediment infiltrated into only a part of the void space (Figure 6b). Figure 6c shows the results for the case d/D = 0.414, where the initial void space was smaller than the diameters of fine sediment. The fine sediment was trapped in the bed surface and could not move down into the gravel bed. As expected, the calculated results confirm that the size ratio has a significant effect on the infiltration process, as well as the vertical distribution of the fine sediments.

_{m}= 0.26 mm and standard deviation σ

^{(d)}= 1.94. The gravel mixture has a median grain size of D

_{m}= 7.1 mm and the end of simulation time σ

^{(D)}=1.35. Random packing during the first 41,000 time steps with porosity p

^{(D)}= 0.407 generated the gravel-bed. The infiltration process was stopped when the top gravel layer has been filled. Figure 7a shows the bed materials along the flume centerline at the end of the simulation, where the gradation of saturated gravels with percolated fines can be seen. Furthermore, as the pores on the surface layer were filled with fine sediment, fine particles could not infiltrate through this layer and saturate the subsurface pore space. Figure 7b compares the mean values of the simulated fine fractions along the depth, with the observations that were conducted by Gibson et al. [25].

#### 3.2. Bed Form Movement and Porosity Variation

^{3}/s/m, a constant water elevation of 10 m long the channel, and the initial bed level defined, as follows:

_{1}= 1 mm; d

_{2}= 7 mm), which was then used to compose a desired size gradation. The initial size fractions are (βa

_{,1}= 0.75 and βa,

_{2}= 0.25) in the active layer and (β

_{S,1}= 0.65 and β

_{S,2}= 0.35) in the stratum layer.

#### 3.3. Sulaiman’s Experiment

_{50}= 15 mm, and the fine fraction ranges from 0.5 mm to 4.75 mm with d

_{0}= 2 mm. The sediments were mixed and then thoroughly homogenized. Experiments have been carried out for two situations. In Run-1, no sediment was supplied at the inlet; coarse sediment did not move actively, and only fine sediment was removed from the bed. In Run-2, an amount of the fine sediment fraction was continuously fed from the upstream of the flume, and these fine sediments could be deposited into the coarse bed or transported downstream. The condition of the riverbed at the end of Run-1 was used as an initial condition of Run-2. Cumulative time steps for Run-1 are 20, 65, 130, and 250 min and for Run-2 were 30, 50, 66, and 82 min. The total duration of Run-1 and Run-2 was 332 min. Table 2 summarizes the experimental conditions. Water depth h and velocity v are the initial average water depth and velocity in the uniform region. More details of the experiments description can be seen in [16].

_{coarse}/d

_{fine}= 7.5, the minimum porosity is equal to 0.195. At the end of the simulation, the fine sediment wholly filled in the void structure of gravel in the middle of the flume. In the erosion case, most of the fine particles are removed from gravel upstream (x = 4.2 m) and the porosity is reached the maximum value of 0.4. Afterward, the porosity decreased to a minimum value of 0.195 when the fine fraction increased from 0.22 to 0.30. In the deposition case, in contrast to the erosion case, the fine fraction is greater than 0.30, and the porosity is proportional to a fine fraction. The increase in porosity in the deposition case is less than that in the erosion case.

#### 3.4. SAFL’s Experiment

_{90}and the coefficient of active layer thickness n

_{active}= 2. The roughness height also varies over time and space, as defined by the product of D

_{90}and roughness coefficient of n

_{k}= 1.8.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The structure diagram of the vertical cross-section is based on the multi-layers model. (1) = active layer; (2) and (3) = substrate layers; E

_{a}and E

_{m}= Active layer thickness and active-stratum layer thickness; z

_{b}= bed elevation; z

_{c}and z

_{d}= substrate elevations.

**Figure 4.**Schematic structure of bottom sediments as a function dependent on the magnitude of the fine component sediment.

**Figure 6.**The structure of bed gravel and fine sediment distribution depends on the size ratio (d/D).

**Left**: Fine sediments distribution at a cross-section along the channel at the final computational step with d/D = 0.154 (

**a**), d/D = 0.231(

**b**), and d/D = 0.414 (

**c**);

**Right**: Fine fraction variation in gravel depth with different size ratio.

**Figure 7.**(

**a**)—Structure of bed sediments at a cross-section along the channel at the final computational step (

**b**)—Fine fraction distribution at the end of simulation; and, (

**c**)—Bed porosity distribution along the depth at the beginning and the end of the computation time.

**Figure 8.**The performance of bed-porosity variation model in different times (

**a**)—Bed elevation; (

**b**)—Porosity of active-layer; (

**c**)—Coarse size fraction of active-layer; and, (

**d**)—Fine size fraction of active-layer.

**Figure 9.**Comparison between the conventional model and the new model at the final time step (

**a**)—Porosity of the active-layer; and, (

**b**)—Bed elevation.

**Figure 10.**Schematic drawing of experimental channel and apparatus [16].

**Figure 11.**Bed variation for surface layer in comparison with observations from flume measurements [16]: (

**a**) bed elevation—erosion case; (

**b**) bed elevation—deposition case (

**c**) Fine fraction variation in erosion and deposition cases; and, (

**d**) Porosity variation in erosion and deposition cases.

**Figure 12.**Flume set up for St. Anthony Falls Laboratory (SAFL) downstream fining experiments [15].

**Figure 13.**Simulated results in comparison with possible observations and Cui’s study [15]: (

**a**) bed and water elevations obtained from constant porosity model; (

**b**) bed and water elevations obtained from both constant and variable porosity models; (

**c**) Sand fraction obtained from variable porosity model; and, (

**d**) Calculated porosity variation.

Density of Sphere (kg/m^{3}) | Density of Water (kg/m^{3}) | Young’s Modulus (Pa) | Poisson Ratio | Friction Between Grains | Coefficient of Restitution |
---|---|---|---|---|---|

2700 | 1000 | 5.0 × 10^{6} | 0.45 | 0.5 | 0.4 |

**Table 2.**Experimental conditions for two runs, where: q

_{w}—water discharge; q

_{s}—sediment discharge; Fr—Froude number; τ—non-dimensional bed shear stress [16].

Exp. | q_{w} (m^{2}/s) | q_{s} × 10^{−6} (m^{2}/s) | h (m) | v (m/s) | Fr | τ (Fine) | τ (Coarse) |
---|---|---|---|---|---|---|---|

Run-1 Run-2 | 0.034 0.034 | 0 31.8 | 0.039 0.045 | 0.879 0.754 | 1.428 1.133 | 0.178 0.203 | 0.026 0.030 |

Erosion | Deposition | |||||
---|---|---|---|---|---|---|

Variation | Constant | Sulaiman | Variation | Constant | Sulaiman | |

Bed Elevation | ||||||

R | 0.99510 | 0.99442 | 0.99412 | 0.99451 | 0.99538 | 0.99465 |

RMSE | 0.00585 | 0.00631 | 0.00560 | 0.00347 | 0.00490 | 0.00414 |

MAE | 0.00451 | 0.00546 | 0.00442 | 0.00275 | 0.00424 | 0.00343 |

Fine Fraction | ||||||

R | 0.98936 | 0.98953 | 0.99124 | 0.96269 | 0.98205 | 0.96953 |

RMSE | 0.16423 | 0.17410 | 0.26149 | 0.07897 | 0.09265 | 0.07297 |

MAE | 0.12371 | 0.12518 | 0.18397 | 0.05929 | 0.08579 | 0.05088 |

**Table 4.**Parameters for SAFL Downstream Fining Experiments; where: q

_{w}—water discharge, q

_{s}—sediment discharge, ξ

_{d}—downstream water depth, S

_{0}—flume slope, fs—sand fraction [15].

Exp. | q_{w} (m^{2}/s) | q_{s} (m^{2}/s) | ξ_{d} (m) | S_{0} (%) | fs % | Time (h) |
---|---|---|---|---|---|---|

Run 1 | 0.163 | 2.37 × 10^{−4} | 0.4 | 0.20 | 33 | 2, 8, 16.83 |

2 h | 8 h | 16 h | |||||||
---|---|---|---|---|---|---|---|---|---|

R | RMSE | MAE | R | RMSE | MAE | R | RMSE | MAE | |

Variation | 0.987 | 0.019 | 0.010 | 0.998 | 0.016 | 0.013 | 0.989 | 0.033 | 0.027 |

Constant | 0.980 | 0.023 | 0.011 | 0.996 | 0.024 | 0.018 | 0.988 | 0.034 | 0.032 |

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**MDPI and ACS Style**

Bui, V.H.; Bui, M.D.; Rutschmann, P.
Advanced Numerical Modeling of Sediment Transport in Gravel-Bed Rivers. *Water* **2019**, *11*, 550.
https://doi.org/10.3390/w11030550

**AMA Style**

Bui VH, Bui MD, Rutschmann P.
Advanced Numerical Modeling of Sediment Transport in Gravel-Bed Rivers. *Water*. 2019; 11(3):550.
https://doi.org/10.3390/w11030550

**Chicago/Turabian Style**

Bui, Van Hieu, Minh Duc Bui, and Peter Rutschmann.
2019. "Advanced Numerical Modeling of Sediment Transport in Gravel-Bed Rivers" *Water* 11, no. 3: 550.
https://doi.org/10.3390/w11030550