# Two-Dimensional Numerical Simulation Study on Bed-Load Transport in the Fluctuating Backwater Area: A Case-Study Reservoir in China

^{*}

## Abstract

**:**

## 1. Introduction

_{4}from the water surface were detected in the fluctuating backwater area, likely due to a shallower water column and abundant organic matter [17]. The construction of hydro-junctions in the upper river will lead to variations of the incoming water and sediment conditions, and in turn, changes to the deposition and erosion processes operating in the fluctuating backwater reach of the downstream reservoir. Lu developed a 2D mathematical model using the boundary-fitting orthogonal curvilinear coordinate system, which was employed to predict the space-time changes of sedimentation in the Chongqing reach, part of the fluctuating backwater reach of the TGR [18].

## 2. Mathematical Model

#### 2.1. Shallow-Water Equations

**U**(x, t) = vector of conserved variables; and

**u**in the x-direction and y-direction, respectively.

**S**(

**U**) in Equation (1) can include several physical phenomena (i.e., wind, Coriolis force, outfall etc.). Here, only bed slope

**S**(

_{b}**U**) and fraction effects

**S**(

_{f}**U**) are considered. Therefore, the source term is

**S**(

_{b}**U**) and

**S**(

_{f}**U**) can be expressed as

_{fx}and τ

_{fy}= components of the shear stress τ of the river bed in the x-direction and y-direction, respectively, and according to the Manning formula:

_{75}/D

_{25}, which is called the non-uniformity coefficient of river sediment distribution and has an effect on roughness; κ = 0.4, termed the Carmen coefficient; R

_{b}= hydraulic radius corresponding to river bed resistance; K

_{s}= comprehensive roughness of bed sediment, representing the average or median sediments; and B = the degree of influence of the non-uniformity coefficient, calculated by:

**U**(x, t = 0) and all boundary conditions to fully describe the fluctuating backwater area of the reservoir.

#### 2.2. Movable Bed Model

**g**= (g

_{b}_{bx}, g

_{by}) = non-uniform bed-load discharge. Bed-load discharge is often formulated semi-empirically, written as per Sharmov [29], in which velocity is the main variable.

_{c}= incipient velocity, which is approximately equal to the point velocity acting directly on the sediment. Many formula for calculating the incipient velocity U

_{c}ignoring viscous force exist. The basic form can be defined as

_{s}and γ = sediment and water densities, respectively; and K = several variations of the coefficient by different formulations, as illustrated in Table 1 [29]. In this work, K = 1.437 was adopted as the mean value of several incipient velocity formulas.

## 3. Numerical Model

_{i}= the area of element i; and ${\mathit{F}}_{\mathit{n}}(\mathit{U})=\mathit{E}(\mathit{U})\mathrm{cos}\theta +\mathit{G}(\mathit{U})\mathrm{sin}\theta $ = normal flux, which takes the normal direction outside the boundary as the positive direction and the angle between n and x as $\theta $.

^{n}

^{+1}is expressed as

_{ni}= upwind bed-load transport contribution of each neighboring element, across each element edge. The schematic diagram of the element is shown (Figure 1a).

_{n}of the sediment transport rate in the normal direction of the two neighboring elements.

_{bn}> 0, the upstream of element 1 is located upstream of element 2; If g

_{bn}< 0, the result is the opposite Figure 1b. Secondly, after the upstream and downstream elements are identified, Equation (22) is used to calculate the sediment transport rate of the element interface (g

_{n}):

_{1n}/g

_{2n}, calculated by:

## 4. Numerical Simulation

#### 4.1. Study Case Description

^{2}, and the length of the river is 18.22 km [35]. The upstream bed has a steep slope with an average of 13.2% and the main channel of the upstream is narrower at approximately 8 m. The bed slope of the downstream reach is gentle, with an average of 4.5%. The main channel upstream is wider at approximately 25 m. The Longxi River basin experiences frequent seismic activity, which has resulted in large changes in topography, with the overall terrain showing higher elevations in the north moving to lower elevations in the south. After the Wenchuan earthquake, a large amount of loose material generated by the earthquake was accumulated in the basin. Due to the steepness of the ditch bed and huge variability in terrain between the hillsides, these loose sediments have large potential energy, which objectively increases the possibility of the movement of loose sediments and facilitates the transport of loose partials into the downstream river. The upstream and downstream reaches of the Longxi River experienced different morphological changes due to the earthquake [36]. The field investigation found more boulders and less sedimentation in the upstream reach with finer sediment particles, whereas a large degree of siltation occurred in the downstream reach of the Longxi River (Figure 2c,d). Therefore, it is of practical significance to study the Longxi River (Zipingpu Reservoir tail) for the study of the characteristics of bed-load sediment transport in the fluctuating backwater area.

^{3}s

^{−1}, 669 m

^{3}s

^{−1}, and 795 m

^{3}s

^{−1}, respectively. In the present work, the typical flood processes of 10- and 50-year frequencies were modeled (Figure 3a). In addition, the grain size distribution of the numerical simulation was scaled with the size of the physical model in Figure 3b. In addition, the average and median particle sizes were D

_{m}= 37.01 mm and D

_{50}= 23.64 mm, respectively.

_{i}= the point value i of physical variables at element faces j; x

_{ij}= element averaged values i of physical variables at element faces j; and A

_{ij}= the element j area. Both configurations are simulated on a 2D triangular unstructured mesh composed of 20,583 elements and 10,556 nodes. A detailed view of the mesh is shown in Figure 4.

#### 4.2. Validation Test

^{3}s

^{−1}was introduced into the system as the inlet boundary condition and two water levels of 890 m and 885 m were imposed as the outside boundaries. As illustrated in Figure 6, the average simulated water level was consistent with the measured value of the physical model, indicating that the hydraulic numerical model is believable.

^{3}. During the test, all sediments laid in the upstream reach were transported downstream by water flow, and no sediment remained above section CS33 (Figure 5e). Figure 7 shows a comparison of the numerical simulation results and physical model measurement as the morphologies of two randomly-chosen cross sections of the sediment of the above two schemes. The comparison illustrates that the simulation results are consistent with the results of the experimental test and that a certain minor error of sedimentation thickness exists of an approximate average 0.15–0.24 m. The numerical sediment transport model can therefore be regarded as credibly validated by the physical model.

#### 4.3. Evaluation of Morphological Bed Changes in the Fluctuating Backwater Area

#### 4.3.1. Type 1: Reservoir Storage

^{3}s

^{−1}as the steady flood discharge of a 10-year frequency in the Longxi River. The downstream elevation is set to vary uniformly with time within 870 m to 890 m. The upstream sediment is set as the discharge of bed-load. To show the overall trend and local characteristics, the reservoir sedimentation plane morphology is displayed with the variation in the contour of the bed elevation of the river-bed.

#### 4.3.2. Type 2: Reservoir Drawdown

^{3}s

^{−1}as a steady mean flood discharge of Longxi River. The downstream elevation was set to uniformly vary with time within a range of an initial 890 m to 870 m. The upstream sediment load was set to zero.

#### 4.3.3. Type 3: Continuous Flood Process

## 5. Conclusions

- Sediment transport is by far the more uncertain process, which is the most significant innovation of this study. This present study shows the implementation that the river channel sedimentation morphology is changed by the change of water level in the downstream reach and provides evidence for the effect of incidental events on river bed morphology, which increases the factors driving the change of river bed morphology and challenges the traditional theory that the shape of a river channel is mainly determined by the upstream water and sediment and the physical boundary conditions of the river channel, rather than random events.
- The sedimentation in the fluctuating backwater area is mainly deposited in the main channel and the difference in elevation of the river-bed between the beach and channel decreases with time. In the river bend, the sedimentation is mainly concentrated on the convex bank, readily resulting in the growth of a convex bank beach. Although the concave bank also experiences siltation, the quantity is relatively minor.
- During the drawdown period of the reservoir, the original sedimentation is scoured, with scouring concentrated over a small width. The flow gradually erodes a deep channel in the river-bed, forming compound channels with a beach and multiple channels, which reshapes the channel of the low flow period. The deposition of bed-load from upstream to downstream and the slope of the longitudinal section of the river bed in the fluctuating backwater area are generally gradually reduced.
- There is an element of randomness in the location and morphology of sedimentation due to the effect of the downstream water level and fluctuating backwater. Under type 3, the location and bed morphology of the end of the backwater vary under the same inlet flow conditions and different downstream water levels. The location and direction of upstream flow are changed under differences in location and morphology, resulting in large differences in sedimentation under different flow conditions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic diagram of element; (

**b**) Schematic diagram of element interface sediment transport rate.

**Figure 2.**The location and survey area of the current study. (

**a**) A shows map of China, whereas B is Sichuan Province and C is a map, showing the Longxi River (Zipingpu reservoir tail); (

**b**) Satellite map of Longxi river (Zipingpu reservoir tail) in 2018 in Google Earth (scale = 1:10,000); (

**c**) Onsite view of the downstream reach of Longxi River (Zipingpu reservoir tail) in 2017; (

**d**) Onsite view of the upstream reach of Longxi River (Zipingpu reservoir tail) in 2017.

**Figure 3.**(

**a**) Flood processes of 10- and 50-year frequencies; (

**b**) Grain size distribution of numerical simulation.

**Figure 5.**(

**a**) Physical model developed by the State Key Laboratory of Hydraulics and Mountain River Engineering (SKLH) of Sichuan University; (

**b**) Layout of measuring sections; (

**c**) The position of the initial paved sediments of the physical model; (

**d**) The local schematic of the initial paved sediments of the physical model; (

**e**) Test to validate the physical model.

**Figure 6.**The water level of the river talweg in the longitudinal section and the horizontal distance, represents the distance from the reference section CS39. (

**a**) The upstream discharge = 707.1 m

^{3}s

^{−1}, the downstream water level = 890 m; (

**b**) The upstream discharge = 707.1 m

^{3}s

^{−1}, the downstream water level = 885 m.

**Figure 7.**The x-coordinate represents the distance from left to right of the varied river cross section and the y-coordinate represents the elevation of the river bed. (

**a**) The compared elevation of CS29 when the upstream discharge = 707.1 m

^{3}s

^{−1}and the downstream water level = 890 m; (

**b**) The compared elevation of CS22 when the upstream discharge = 707.1 m

^{3}s

^{−1}and the downstream water level = 890 m; (

**c**) The compared elevation of CS24 when the upstream discharge = 707.1 m

^{3}s

^{−1}and the downstream water level = 885 m; (

**d**) The compared elevation of CS20 when the upstream discharge = 707.1 m

^{3}s

^{−1}and the downstream water level = 885 m.

**Figure 8.**Variation in the contours of bed elevation under the same upstream discharge and different downstream water levels of reservoir storage; (

**a**) Time of simulation is 12 h and downstream water level is 870 m; (

**b**) Time of simulation is 36 h and downstream water level is 875 m; (

**c**) Time of simulation is 60 h and downstream water level is 880 m; (

**d**) Time of simulation is 84 h and downstream water level is 885 m; (

**e**) Time of simulation is 108 h and downstream water level is 890 m.

**Figure 9.**The river bed elevation of river talweg in the longitudinal section during reservoir storage, and the horizontal distance is the distance from the reference section CS39.

**Figure 10.**The variation in the contours of bed elevation under the same upstream discharge and different downstream water levels during reservoir drawdown; (

**a**) Time of simulation is 48 h and downstream water level is 885 m; (

**b**) Time of simulation is 72 h and downstream water level is 880 m; (

**c**) Time of simulation is 96 h and downstream water level is 875 m; (

**d**) Time of simulation is 120 h and downstream water level is 870 m.

**Figure 11.**The river bed elevation of river talweg in the longitudinal section during reservoir drawdown, and the horizontal distance is the distance from the reference section CS39.

**Figure 12.**The variation in the contours of bed elevation of Case A under a typical continuous flood process with the flood discharge of a 50-year frequency set at approximately 669 m

^{3}s

^{−1}and the downstream water level as 875 m in the fluctuating backwater area. (

**a**) Time of simulation is 12 h; (

**b**) Time of simulation is 22 h; (

**c**) Time of simulation is 28 h; (

**d**) Time of simulation is 48 h.

**Figure 13.**The variation in the contours of bed elevation of Case B under a typical continuous flood process and the flood discharge of a 50-year frequency set to approximately 669 m

^{3}s

^{−1}and the downstream water level as 880 m in the fluctuating backwater area. (

**a**) Time of simulation is 12 h; (

**b**) Time of simulation is 22 h; (

**c**) Time of simulation is 28 h; (

**d**) Time of simulation is 48 h.

**Figure 14.**The river bed elevation of river talweg in the longitudinal section under a typical constant flood process, and the horizontal distance is the distance from the reference section CS39. (

**a**) Case A: the flood discharge of a 50-year frequency set to approximately 669 m

^{3}s

^{−1}and the downstream water level as 875 m; (

**b**) Case B: the flood discharge of a 50-year frequency set to approximately 669 m

^{3}s

^{−1}and the downstream water level as 875 m.

Name | Zhang RJ | Tang CB | Dou GR | Sha YQ | Sharmov | B.H. | Shields |
---|---|---|---|---|---|---|---|

K | 1.53 | 1.789 | 1.314~1.343 | 1.277 | 1.33 | 1.58 | 1.272 |

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**MDPI and ACS Style**

Luo, M.; Yu, H.; Huang, E.; Ding, R.; Lu, X. Two-Dimensional Numerical Simulation Study on Bed-Load Transport in the Fluctuating Backwater Area: A Case-Study Reservoir in China. *Water* **2018**, *10*, 1425.
https://doi.org/10.3390/w10101425

**AMA Style**

Luo M, Yu H, Huang E, Ding R, Lu X. Two-Dimensional Numerical Simulation Study on Bed-Load Transport in the Fluctuating Backwater Area: A Case-Study Reservoir in China. *Water*. 2018; 10(10):1425.
https://doi.org/10.3390/w10101425

**Chicago/Turabian Style**

Luo, Ming, Heli Yu, Er Huang, Rui Ding, and Xin Lu. 2018. "Two-Dimensional Numerical Simulation Study on Bed-Load Transport in the Fluctuating Backwater Area: A Case-Study Reservoir in China" *Water* 10, no. 10: 1425.
https://doi.org/10.3390/w10101425