# Validation of an Experimental Procedure to Determine Bedload Transport Rates in Steep Channels with Coarse Sediment

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{50}(mm) < 28.6) and mild slopes (~0% < So < 2%) [9,10,11]. On the other hand, steep rough-bedded channels constitute the main component of mountainous drainage systems. These channels are the principal source of sediment to milder slope downstream channels [12]. Thus, sediment transport in steep channels influences processes like landscape morphology evolution, sediment routing during hydrological events, and flow hydraulics in river systems. To determine the development of these processes, understanding and knowledge of bedload transport in steep channels must be improved. Sediment transport in steeper channels with coarser material is a complex process. The continuously changing driving forces, basin sediment production, and riverbed conditions, such as high gradients and elements of roughness, result in a high uncertainty in the quantification of sediment transport rates [1,13,14]. The uncertainty reported can reach some orders of magnitude [12,15,16]. The presence of large diameter particles in the stream causes changes in flow hydraulics (e.g., velocity profile) and turbulence intensity, hydraulic jumps, areas of flow acceleration and deceleration, and high spatial variability of boundary shear stress [17,18]. Some equations have been proposed to describe bedload transport rates, movement thresholds, and channel roughness of steep channels with coarse sediment [19]. Smart [11] developed aA relation for slopes ranging from 3% to 20% was developed by [11]. Moreover, the performance of this model was found to be better than other similar models in the case of high slopes [20]. More recent studies [21], Yager et al. [22] have considered the typical features of steep channels such as wide grain size distribution ranging from large almost immobile boulders to finer more mobile sediments, the effective stress that causes sediment movement (which is smaller than the total shear stress), and the reduced amount of mobile sediment. However, the variation and randomness of parameters like grain size distribution, spatial distribution of immobile grains, and the streambed conditions, need to be analyzed as fundamental parameters for bedload transport [22]. Additionally, Juez et al. [23] found that better predictions of bedload transport rates are obtained when gravity projections, which become more significant due to high slopes, are considered in bedload transport models. Therefore, as the first approach to developing a consistent procedure to capture the high variability present in natural rivers, this work presents the validation of an experimental methodology to determine bedload transport rates for steep slopes (3% to 5%) using sediments with both uniform sizes and with a grain size distribution.

## 2. Background

_{o}is the boundary shear stress, ρ

_{s}the sediment density, ρ the fluid density, d

_{s}the sediment diameter, g the gravitational acceleration, s the sediments relative density and

_{μ}the fluid viscosity. The first term represents a stability parameter, τ*, to analyze threshold conditions. This parameter represents the dimensionless boundary shear stress [5].

_{c}, defines the boundary shear stress for which motion begins. In other words, sediment transport starts when τ* > τ*

_{c}. The critical Shields parameter (τ*

_{c}) is a function of the shear Reynolds number or particle Reynolds number (R

_{ep}).

_{sh}, defined as V

_{sh}= √(τ

_{o}/ρ). Expressing the above function in terms of shear velocity the following final function results.

_{s}− ρ)/ρ, and q* is the dimensionless bedload transport rate, also known as the Einstein bedload number. It is expressed as:

_{s}is the volumetric bedload transport rate per unit width (m

^{3}/s/m or m

^{2}/s) and g

_{o}represents the gravity vector projection (g

_{o}= cos

^{2}(φ)) [23]. Angle φ corresponds to the inclination of the bed with respect to the horizontal. This projection has been defined by Juez et al. [41] to include effects of steep slopes on pressure distribution and friction, considering that the gravity vector projection has improved the prediction capacity of the sediment transport models with respect to the estimations using gravity vector directly [2,20].

## 3. Experimental Procedure

^{3}) of different diameters. For each simulation, the channel gradient was set first. The sediment was placed at a total length of about 5 m at the beginning of each run. A first layer of immobile sediment, of a thickness from 2 to 3 d

_{s,}was located on the bottom of the channel. Over this immobile layer, a layer of mobile sediment was placed, with a thickness of from 3 to 5 d

_{s}. A schematic of the flume and sediment configuration is shown in Figure 2. Once uniform flow was established, the simulation time began. Sediment transported before this is not considered for the transport rate calculation. No sediment was fed at the inlet of the simulation zone because the transport rates obtained from a set of calibration experiments showed no dependence on sediment feed rate. Each run was considered complete when the total mobile sediment layer was transported. At the end of the channel, transported sediment was collected in a sediment trap. For each simulation, discharge, flow depth, and water temperature were measured. An experimental configuration consisted of a discharge, a channel slope, and a sediment type. Three experiments were performed for each configuration with different running times to verify the validity of the transport rates obtained. Water temperature was measured at the beginning and at the end of each simulation. The profile length required to reach uniform flow conditions was calculated for each experiment. The test section was located 3.5 m downstream from the inlet weir. The maximum length, based on flow and geometry conditions, that is needed to reach normal depth (from critical to normal flow depth in the downstream direction and vice versa in the upstream direction) is 3 m. The distance from the end of the test section to the outlet was 3.5 m. Thus, normal depth was ensured by placing the test zone downstream far enough to reach uniform conditions for steep slopes and upstream enough for mild slopes. The flow started in a hydraulic jump downstream of the inlet weir and ended in a free fall. Normal depth was verified at the beginning, at the middle and at the end of the test section. Sediment collected in each run was dried and weighed to determine the transport rate by dividing the weight of the transported sediment by the collection time.

_{50}for this grain size distribution is 53.10 mm. In the present study d

_{84}is considered as the characteristic diameter with d

_{84}= 21.70 mm.

## 4. Data Analysis

^{3}/s), Y flow depth (m), Rh hydraulic radius, V mean flow velocity (m/s), Q* dimensionless discharge, Y* dimensionless flow depth, Rh* dimensionless hydraulic radius, and V* dimensionless mean flow velocity.

## 5. Results and Discussion

_{s}) has been selected, since as slope increases it decreases, meaning particle diameter has a higher influence on flow and sediment transport mechanics [52,53]. Slope has mainly been neglected in the study of hydraulic processes under the assumption of mild slopes. Even though in bedload sediment transport it is indirectly included for the calculation of other parameters such as bed shear stress, in this case, due to its high values, slope becomes an important parameter that must be considered directly as an independent variable.

_{s}) in the denominator in Equation (12). If it is inverted and put in the numerator it becomes a comparable parameter to bedload layer thickness. An important difference is that Equation (12) considers slope as an independent variable and Equation (7) does not. From the correlation of between slope and bedload transport rate, it has been shown that slope, due to its high values, has an important impact on transport rates. Therefore, its inclusion in Equation (12) represents an improvement over Equation (7).

_{s}= 10 mm and 15 mm (Figure 4b and Figure 5b, respectively), and to [7,44,45] for d

_{s}= 25 mm (Figure 6b). For the highest slope (5%) experimental rates are similar to [6,46] for d

_{s}= 10 mm and 15 mm (Figure 4c and Figure 5c, respectively), [48] for d

_{s}= 15 mm (Figure 5c), and [11,48,49] for d

_{s}= 25 mm (Figure 6c).

_{s}= 25 mm) and Case D (Granulometry) have similar behavior. Since characteristic diameter d

_{84}(20.8 mm) was used to put variables in dimensionless form for Case D, this supports the definition of d

_{84}as the characteristic diameter for sediment mixtures [54,55,56]. Even though the grain size distribution is built from the same uniform diameters, this fact could lead to the idea that for steep slopes the size distribution may have less influence on the transport process, and can be represented with a single particle size [54].

## 6. Conclusions

_{84}has been found to be appropriate. However, more data are needed to affirm this supposition because the grain size distribution used here was built with just three different particle diameters and they were the same as those used for the uniform size experiments.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Schematic of the simulation configuration, (

**b**) test zone prepared for a simulation, (

**c**) test zone during a simulation, and (

**d**) sediment trap with the sediment collected in a simulation.

**Figure 4.**Experimental and calculated transport rates for uniform grain sizes of (

**a**) d

_{s}= 10 mm So = 3%, (

**b**) d

_{s}= 10 mm So = 4%, and (

**c**) d

_{s}= 10 mm So =5%.

**Figure 5.**Experimental and calculated transport rates for uniform grain sizes of (

**a**) d

_{s}= 15 mm So = 3%, (

**b**) d

_{s}= 15 mm So = 4%, and (

**c**) d

_{s}= 15 mm So = 5%.

**Figure 6.**Experimental and calculated transport rates for uniform grain sizes of (

**a**) d

_{s}= 25 mm So = 3%, (

**b**) d

_{s}= 25 mm So = 4%, and (

**c**) d

_{s}= 25 mm So = 5%.

**Figure 7.**Experimental and calculated transport rates for grain size distribution (Case D) with (

**a**) So = 3%, (

**b**) So = 4%, and (

**c**) So = 5%.

**Figure 8.**Variation of dimensionless transport rates with dimensionless discharge, particle size (d

_{s}) and slope (So).

**Figure 9.**Comparison between predictions obtained using the corrected gravity (CG) and using gravity vector directly (G) for d

_{s}= 10 mm (Case A), d

_{s}= 15 mm (Case B), d

_{s}= 25 mm (Case C).

**Figure 10.**Comparison between measured and calculated bedload transport rates within the range of an order of magnitude of accuracy for d

_{s}= 10 mm (Case A), d

_{s}= 15 mm (Case B), d

_{s}= 25 mm (Case C).

Author | Equation | Range |
---|---|---|

Meyer-Peter (1949, 1951) [44,45]
| ${q}^{*}={\left(4{\tau}_{*}-0.188\right)}^{3/2}$ | $1.25<s<4.2$ ${d}_{s}={d}_{50}$ |

Einstein (1942) [6]
| ${q}^{*}=2.1ex{p}^{\left(-0.391\frac{1}{{\tau}_{*}}\right)}$ | $1.25<s<4.25$ $0.315<{d}_{s}<28.6\text{}\mathrm{mm}$ ${q}^{*}<0.4$ ${d}_{s}\approx {d}_{35}-\text{}{d}_{45}$ |

Wong & Parker (2006) [46]
| ${q}^{*}=4.93{\left({\mathsf{\tau}}_{*}-0.047\right)}^{1.6}$ ${q}^{*}=3.97{\left({\mathsf{\tau}}_{*}-0.0495\right)}^{3/2}$ | $\mathrm{s}=2.55$ |

Ashida & Michiue (1972) [47]
| ${q}^{*}=17\left({\tau}_{*}-{\tau}_{*c}\right)\left(\sqrt{{\tau}_{*}}-\sqrt{{\tau}_{*c}}\right)$ | ${\tau}_{*c}=0.05$ |

Meyer-Peter & Müller (1948) [7]
| ${q}^{*}=8{\left({\tau}_{*}-{\tau}_{*c}\right)}^{3/2}$ | ${\tau}_{*c}=0.047$ |

Yalin (1963) [48]
| ${q}_{s}=0.635s{\tau}_{*}{}^{\frac{1}{2}}\left(1-\frac{ln\left(1-as\right)}{as}\right)$ $a=2.45{\left(R+1\right)}^{0.4}{\tau}_{*}{{}_{c}}^{0.5}$ $s=\frac{{\tau}_{*}-{\tau}_{*c}}{{\tau}_{*c}}\text{}$ | $0.8\text{}\mathrm{mm}{d}_{s}28.6\text{}\mathrm{mm}$ |

Parker (1979) [49]
| ${q}^{*}=11.2\frac{{\left({\mathsf{\tau}}_{*}-0.03\right)}^{4.5}}{{\tau}_{*}{}^{3}}$ | Shield’s numbers occurring in gavel bed rivers |

Cheng (2002) [50]
| ${q}^{*}=13{\tau}_{*}{}^{3/2}exp\left(-\frac{0.05}{{\tau}_{*}{}^{3/2}}\right)$ | $\mathrm{s}=2.69-2.53$ $Q=0.093-1.119\text{}f{t}^{3}/s$ ${S}_{0}=0.73-1.2\text{}\%$ ${d}_{s}=0.068-0.27\text{}ft$ |

Nielsen (1992) [51]
| ${q}_{s}={C}_{s}{\delta}_{s}{V}_{s}$ ${C}_{s}=0.65$ $\frac{{V}_{s}}{{V}_{*}}=4.8$ $\frac{{d}_{s}}{{\delta}_{s}}=2.5\left({\tau}_{*}-{\tau}_{c}\right)$ | |

Smart (1984) [11]
| for uniform sediment${q}^{*}=4.2\text{}{S}^{0.6}\text{}C\text{}{\tau}_{*}{}^{0.5}\left({\tau}_{*}-{\tau}_{*c}\right)$ for sediment mixtures${q}^{*}=4\left[{\left(\frac{{d}_{90}}{{d}_{30}}\right)}^{0.2}\text{}{S}^{0.6}\text{}C\text{}{\tau}_{*}{}^{0.5}\left({\tau}_{*}-{\tau}_{*c}\right)\right]$ $C=\frac{\text{}mean\text{}flow\text{}velocity}{bed\text{}shear\text{}velocity}\text{}$ | ${S}_{0}=3-20\text{}\%$ ${d}_{s}=2-10.5\text{}\mathrm{mm}$ ${\rho}_{s}=2670-2680\text{}\mathrm{kg}/{\mathrm{m}}^{3}\text{}$ |

Case | Particle Diameter d _{s} (mm) | Discharge Q (l/s) | Channel Slope So (%) | Flow Depth Y (m) | Froude Number Fr | Reynolds Number Re | Relative Submenrgence Y/d _{s} |
---|---|---|---|---|---|---|---|

A | 10 | 4.02–23.16 | 3.0–5.0 | 0.020–0.065 | 0.99–1.51 | 7400–74,000 | 2.00–6.50 |

B | 15 | 5.06–23.16 | 3.0–5.0 | 0.035–0.075 | 0.40–1.22 | 7800–74,000 | 2.33–5.00 |

C | 25 | 5.34–23.16 | 3.0–5.0 | 0.035–0.090 | 0.42–0.93 | 8100–74,000 | 1.4–3.60 |

D | d_{84} = 21.70 | 6.45–23.16 | 3.0–5.0 | 0.035–0.085 | 0.51–1.09 | 9900–74,000 | 1.57–3.82 |

Dimensionless Parameter | Symbol | Correlation Coefficient |
---|---|---|

discharge | Q* | 0.94–0.98 |

flow depth | Y* | 0.93–0.97 |

mean flow velocity | V* | 0.92–0.97 |

hydraulic radius | Rh* | 0.92–0.97 |

Slope | So | 0.92–0.99 |

Definition | Case A | Case B | Case C | Case D | |
---|---|---|---|---|---|

W | Coefficient | 1.966 × 10^{−}^{27} | 2.144 × 10^{−}^{77} | 3.608 × 10^{−}^{22} | 1.669 × 10^{−}^{12} |

x | Dimensionless discharge exponent | 6.230 | 17.730 | 7.006 | 4.134 |

y | Dimensionless flow depth exponent | −8.137 | −32.177 | −9.468 | −3.542 |

z | Slope exponent | 0.419 | 0.986 | 1.401 | 0.746 |

Statistic | Case A | Case B | Case C | Case D |
---|---|---|---|---|

Multiple correlation coefficient | 0.98 | 0.97 | 0.98 | 0.98 |

Determination coefficient R^{2} | 0.95 | 0.94 | 0.96 | 0.95 |

Adjusted R^{2} | 0.95 | 0.94 | 0.96 | 0.95 |

Typical error | 0.26 | 0.28 | 0.25 | 0.21 |

Observations | 35 | 35 | 35 | 35 |

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**MDPI and ACS Style**

Carrillo, V.; Petrie, J.; Timbe, L.; Pacheco, E.; Astudillo, W.; Padilla, C.; Cisneros, F.
Validation of an Experimental Procedure to Determine Bedload Transport Rates in Steep Channels with Coarse Sediment. *Water* **2021**, *13*, 672.
https://doi.org/10.3390/w13050672

**AMA Style**

Carrillo V, Petrie J, Timbe L, Pacheco E, Astudillo W, Padilla C, Cisneros F.
Validation of an Experimental Procedure to Determine Bedload Transport Rates in Steep Channels with Coarse Sediment. *Water*. 2021; 13(5):672.
https://doi.org/10.3390/w13050672

**Chicago/Turabian Style**

Carrillo, Veronica, John Petrie, Luis Timbe, Esteban Pacheco, Washington Astudillo, Carlos Padilla, and Felipe Cisneros.
2021. "Validation of an Experimental Procedure to Determine Bedload Transport Rates in Steep Channels with Coarse Sediment" *Water* 13, no. 5: 672.
https://doi.org/10.3390/w13050672