# Transport of Sediment Mixtures in Steady Flow with an Extra Contribution of Their Finest Fractions: Laboratory Tests and Modeling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experiment Gdańsk, 2021

- a 0.73 m long outflow section, with a bottom inlet supplying the channel with water;
- starting section with a length with a length of 2.5 m;
- a cuvette with the input sediment, i.e., a depression along the entire channel, 1 m long an 0.08 m high;
- the test section 2.5 m long;
- sediment trap, in the form of another cuvette, i.e., a depression channel 0.73 m long and 0.08 m high, with a bottom inlet and outlet.

- Filling the cuvette with the water-saturated sediment. Placing the wet sediment in the cuvette prevented formation of air bubbles in the deposited mixture;
- Taking a sample from the container filled with sediment prepared for testing;
- Leveling the sediment to the edge of the cuvette and removing the remains of sediment from the vicinity of the cuvette;
- Filling the channel up to the water depth H = 0.05 m;
- Performing a test for a given flow (Times for individual tests are given in Table 1;
- Measurements of the velocity over the bottom using the Prandtl tube (0.1 m above the lower edge of the cuvette, in the period between the flow stabilization and the development of wrinkles);
- Water removal from the channel after the test is completed;
- Collection of sediment from the trap and control area;
- Measurement of sediment volume captured in the sand trap;
- Documentation of bathymetry in a cuvette by taking photos, performing a bathymetric measurement;
- Collecting sediment samples for sieve analysis;
- Removal of sediment from cuvette and its cleaning.

^{2}/s] is the sediment transport rate;$\text{}d={d}_{r}$ is the representative diameter for sediment mixture in the dense layer, assumed as ${d}_{r}={d}_{50}$; $s={\mathsf{\rho}}_{\mathrm{s}}/\mathsf{\rho}$ is the relative density; ${\mathsf{\rho}}_{s}$ is the density of sediments; $g$ is the acceleration due to gravity and ρ is water density, while non-dimensional shear stress ${\theta}_{*}^{\prime}$ is defined as:

#### 2.2. Experiment by Elhakeem and Imran (2012) [53]

^{3}. The experiments were conducted for flows ranging from 9 to 15 L/s at the average velocities from 0.429 to 9.676 m/s. The duration of the tests ranged from 1800 to 3600 s. The inclination of channel was from 0.0037 to 0.0062, and the water depths were from 0.096 to 0.134 m. A total of 32 runs were conducted under the equilibrium condition within the lower regime, primarily dunes. During the run, coarse material (${d}_{i}$ > 2.87 mm) was trapped at the downstream end of the flume (Figure 6), collected manually from the coarse sediment trap, and fed into the sediment feeder at the upstream section of the flume.

## 3. Theoretical Investigations

#### 3.1. Transport Model for Non-Uniform Sediment

#### 3.2. Model Results for Uniform Sediment

#### 3.3. Model Results for Non-Uniform Sediments

## 4. Discussion

#### 4.1. Non-Uniform Sediment Transport with Limited Availability of Very Fine Fractions

#### 4.2. Comparison Calculations with Measuremends

#### 4.2.1. Experiment Gdańsk 2021

#### 4.2.2. Experiment by Elhakeem and Imran [53]

## 5. Conclusions

- Transport calculations conducted by the presented model separately for all sediment fractions in mixture including the mutual interactions between them have shown:
- due to assumed strong interactions between the sediment fractions in the moving layer of densely concentrated sediments all the fractions are characterized by the same velocity and concentration vertical profiles;
- in the contact layer vertical profiles of velocities and concentrations vary for individual fractions due to turbulent water pulsations and chaotic collisions of grains;
- the agreement between the calculated transport and measurements was achieved within plus/minus a factor of two of the measurements;
- calculations of the granulometric distributions of sediment from the trap conducted using presented model have shown good agreement with the measurements (plus/minus a factor of two of the measurements);
- results confirm that the mechanism of interactions between fractions, and in particular the influence of finer grains in the mixture on the increase in transport of coarser fractions, is well described by the model.

- The experimental investigations on transport of sediment mixture with large amount of very fine non-cohesive fractions resulted in proposed modification by inclusion of possible deficit of very small fractions in the active layer of the bottom. The compatibility of the transport calculation results for all fractions with measurements Gdańsk 2021 is within plus/minus a factor of two of the measurements. This confirms the lack of full availability of very fine fractions in the bottom and be the reason for a significant reduction in transport of those fractions.
- Comparison of the calculations by presented modified model of grain size distributions with measurements Gdańsk 2021 shows consistency with the experimental results within plus/minus a factor of two of the measurements.
- The presented study provides verified three-layer model which enables the proper description of sediment transport and grain size distributions of transported fractions in steady flow for any bed sediment mixtures, including poorly and well sorted grains with large amount of very fine non-cohesive fractions.
- The present study provides a useful engineering tool for prediction of transport in steady flow of sediment mixtures with various non-cohesive fractions including very fine and fine. Calculations are possible with just a few measurable properties of particles and water. Parameters do not need tuning against experiments.
- The next step of model development will be the extension to transport modeling in steady flow of sand mixtures with cohesive admixtures. The authors look forward to work also on model extension to predict sediment transport under highly transient (e.g., dam-break) flows.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The approximation of transport measurements (Gdańsk 2021 experiments) and the mean values of measurement tests by curve with a coefficient of determination ${R}^{2}=0.8878$.

**Figure 4.**Grain size distribution of quartz sand in the trap at the flow (

**a**) 8 L/s and (

**b**) 14 L/s in the Gdańsk 2021 experiment.

**Figure 5.**Exemplary vertical profile: (

**a**) vertical velocity profile for the measured $u\left(z\right)$ and the logarithmic profile with the depth average shear velocity $\overline{{u}_{f*}^{\prime}}$; (

**b**) the shear velocities taken from $u\left(z\right)$ measurements in comparison with the dept average value $\overline{{u}_{f*}^{\prime}}$.

**Figure 6.**Experimental setup for steady flow measurements by Elhakeem and Imran (2012) [53].

**Figure 7.**Particle size distribution determined for various mixtures caught in a trap in the experiments by [53], tests 1–8 for the initial particle size distribution (

**a**) M1, (

**b**) M2, (

**c**) M3, (

**d**) M4.

**Figure 8.**Vertical structure of: (

**a**) sediment transport profile with velocity and concentration of the i-th fraction of sediment; (

**b**) shear stress profile; after Kaczmarek et al. [50] with modification for non-uniform sediments.

**Figure 9.**Calculated concentrations: (

**a**) in the contact layer, (

**b**) in grain collision sublayer and calculated velocities: (

**c**) in the contact layer; (

**d**) in grain collision sublayer for homogeneous sediments $\left({d}_{r}={d}_{i}\right)$ and $u{\prime}_{f*}=0.042\text{}\mathrm{m}/\mathrm{s}$.

**Figure 10.**Calculation results ($u{\prime}_{f*}=0.042\mathsf{m}\mathsf{/}\mathsf{s})$ of concentration in vertical profiles for heterogeneous sediments with the representative diameter ${d}_{r}$ in: (

**a**) the contact layer, (

**b**) grain collision sublayer and of velocity vertical profile in: (

**c**) the contact layer; (

**d**) grain collision sublayer.

**Figure 11.**Exemplary result of calculations: the influence of fine fractions on transport of coarse fractions for $u{\prime}_{f*}=0.042\text{}\mathrm{m}/\mathrm{s}$.

**Figure 12.**Model results of the erosion thickness $\Delta {h}_{i}$ of the i-th sediment fraction caused by transport ${q}_{i}$ and the reduced thickness $\Delta {h}_{i}^{n}$ caused by ${q}_{i}^{n}$ due to limited availability of very fine fractions.

**Figure 13.**Comparison of sediment transport calculations with modification due to very fine fraction deficit, with calculations without modification, and with calculations for uniform sediments (${d}_{r}={d}_{i}=0.125\text{}\mathrm{mm}$ and d

_{r}= d

_{i}= 0.35 mm).

**Figure 14.**Comparison of grain size distribution calculations by presented model with modification for limited availability of fine fractions with measurements Gdańsk 2021.

**Figure 15.**Transport calculations for Gdańsk 2021 data, approximated by linear curve $y=ax$ with a coefficient of determination ${R}^{2}$.

**Figure 16.**Transport calculations for the data by [53] approximated by curve $y=ax$ with a coefficient of determination ${R}^{2}$.

**Figure 17.**Comparison of grain size distribution calculations with measurements by [53] for: initial grain size distribution: (

**a**) M1, (

**b**) M2, (

**c**) M3 and (

**d**) M4.

TR Test | Flow Rate [L/s] | $\overline{\mathit{u}}$ -Depth Averaged [m/s] | Friction | Rep. of Tests [-] | Test Time [s] | Sediment Transport Maximum Mean Minimum [m ^{3}/ms] | Fractions d _{90}/d_{50}/d_{10}[mm] | $\mathit{R}\mathit{e}=\frac{\overline{\mathit{u}}\mathit{H}}{\mathit{\nu}}$ [-] | |
---|---|---|---|---|---|---|---|---|---|

u_{f*}[m/s] | θ* [-] | ||||||||

Inputsand | 0.23/0.22/0.14 | ||||||||

TR_0_7 | 7.0 | 0.5000 | 0.0097 | 0.2587 | 2 | 3600 | 8.00∙10^{−9} | 0.23/0.14/0.03 | 13,462 |

5.50∙10^{−9} | |||||||||

3.00∙10^{−9} | |||||||||

TR_0_8 | 8.0 | 0.5715 | 0.0158 | 0.0690 | 4 | 3600 | 1.90∙10^{−8} | 0.23/0.21/0.11 | 15,385 |

6.60∙10^{−7} | |||||||||

3.43∙10^{−6} | |||||||||

TR_0_9 | 9.0 | 0.6429 | 0.0177 | 0.0862 | 3 | 3600 | 7.59∙10^{−7} | 0.24/0.21/0.12 | 17,308 |

6.84∙10^{−7} | |||||||||

5.61∙10^{−7} | |||||||||

TR_0_10 | 10.0 | 0.7143 | 0.0185 | 0.0949 | 3 | 3600 | 7.95∙10^{−7} | 0.24/0.21/0.11 | 19,231 |

7.38∙10^{−7} | |||||||||

6.80∙10^{−7} | |||||||||

TR_0_11 | 11.0 | 0.7857 | 0.0194 | 0.1035 | 3 | 3600 | 1.16∙10^{−6} | 0.24/0.22/0.12 | 21,154 |

1.08∙10^{−6} | |||||||||

1.00∙10^{−6} | |||||||||

TR_0_12 | 12.0 | 0.8571 | 0.0201 | 0.1121 | 3 | 3600 | 1.52∙10^{−6} | 0.26/0.24/0.13 | 23,077 |

1.29∙10^{−6} | |||||||||

1.06∙10^{−6} | |||||||||

TR_0_13 | 13.0 | 0.9285 | 0.0244 | 0.1638 | 3 | 1800 | 6.77∙10^{−6} | 0.24/0.22/0.12 | 25,000 |

4.86∙10^{−6} | |||||||||

2.94∙10^{−6} | |||||||||

TR_0_14 | 14.0 | 1.0000 | 0.0273 | 0.2070 | 3 | 900 | 1.33∙10^{−5} | 0.26/0.24/0.14 | 26,923 |

9.60∙10^{−6} | |||||||||

5.91∙10^{−6} | |||||||||

TR_0_15 | 15.0 | 1.0714 | 0.0306 | 0.2587 | 3 | 900 | 1.80∙10^{−5} | 0.25/0.22/0.11 | 28,846 |

1.44∙10^{−5} | |||||||||

1.07∙10^{−5} |

**Table 2.**Main parameters and transport results by Elhakeem and Imran (2012) [53].

1M1 | 2M1 | 3M1 | 4M1 | 5M1 | 6M1 | 7M1 | 8M1 | |

${\theta}_{*}^{\prime}$[-] | 0.058 | 0.066 | 0.069 | 0.078 | 0.085 | 0.091 | 0.100 | 0.114 |

$q\left[\mathrm{g}/\mathrm{m}/\mathrm{s}\right]$ | 4.24 | 14.37 | 18.98 | 27.05 | 47.89 | 57.81 | 103.63 | 123.23 |

${d}_{50}$[mm] | 2.03 | 1.95 | 1.98 | 2.04 | 2.26 | 2.49 | 2.66 | 2.50 |

1M2 | 2M2 | 3M2 | 4M2 | 5M2 | 6M2 | 7M2 | 8M2 | |

${\theta}_{*}^{\prime}$[-] | 0.058 | 0.072 | 0.092 | 0.101 | 0.113 | 0.121 | 0.131 | 0.144 |

$q\left[\mathrm{g}/\mathrm{m}/\mathrm{s}\right]$ | 3.15 | 13.76 | 47.04 | 51.75 | 72.92 | 133.87 | 123.73 | 179.11 |

${d}_{50}$[mm] | 1.67 | 1.70 | 1.65 | 1.65 | 1.53 | 1.92 | 1.89 | 2.13 |

1M3 | 2M3 | 3M3 | 4M3 | 5M3 | 6M3 | 7M3 | 8M3 | |

${\theta}_{*}^{\prime}$[-] | 0.073 | 0.082 | 0.087 | 0.098 | 0.106 | 0.116 | 0.126 | 0.141 |

$q\left[\mathrm{g}/\mathrm{m}/\mathrm{s}\right]$ | 8.52 | 22.48 | 26.52 | 34.82 | 60.46 | 69.63 | 137.55 | 151.90 |

${d}_{50}$[mm] | 1.36 | 1.18 | 1.36 | 4.27 | 4.45 | 4.94 | 5.04 | 5.45 |

1M4 | 2M4 | 3M4 | 4M4 | 5M4 | 6M4 | 7M4 | 8M4 | |

${\theta}_{*}^{\prime}$[-] | 0.060 | 0.076 | 0.077 | 0.084 | 0.095 | 0.110 | 0.131 | 0.156 |

$q\left[\mathrm{g}/\mathrm{m}/\mathrm{s}\right]$ | 5.93 | 24.42 | 20.20 | 27.36 | 54.77 | 67.70 | 108.36 | 214.55 |

${d}_{50}$[mm] | 1.40 | 1.58 | 1.45 | 1.52 | 1.36 | 1.47 | 1.54 | 1.42 |

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**MDPI and ACS Style**

Zawisza, J.; Radosz, I.; Biegowski, J.; Kaczmarek, L.M.
Transport of Sediment Mixtures in Steady Flow with an Extra Contribution of Their Finest Fractions: Laboratory Tests and Modeling. *Water* **2023**, *15*, 832.
https://doi.org/10.3390/w15050832

**AMA Style**

Zawisza J, Radosz I, Biegowski J, Kaczmarek LM.
Transport of Sediment Mixtures in Steady Flow with an Extra Contribution of Their Finest Fractions: Laboratory Tests and Modeling. *Water*. 2023; 15(5):832.
https://doi.org/10.3390/w15050832

**Chicago/Turabian Style**

Zawisza, Jerzy, Iwona Radosz, Jarosław Biegowski, and Leszek M. Kaczmarek.
2023. "Transport of Sediment Mixtures in Steady Flow with an Extra Contribution of Their Finest Fractions: Laboratory Tests and Modeling" *Water* 15, no. 5: 832.
https://doi.org/10.3390/w15050832