# Predicting River Embankment Failure Caused by Toe Scour Considering 1D and 2D Hydraulic Models: A Case Study of Da-An River, Taiwan

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Bend Scour Simulation Model

#### 2.1. WASH1D Model

_{S}is the human-induced source; S

_{R}is the source caused by rainfall; S

_{E}is the sink caused by evapotranspiration; S

_{F}is the source caused by exfiltration from the subsurface media; S

_{1}and S

_{2}are the source terms contributing to overland flow; V is the river velocity; g is gravity; Z

_{0}is the bottom elevation; h is the water depth; ρ is the water density; Δρ = ρ − ρ

_{0}is the density deviation from the reference density (ρ

_{0}), which is a function of temperature and salinity as well as other chemical concentrations; c is the shape factor of the cross-sectional area; F

_{x}is the momentum flux caused by eddy viscosity; M

_{S}is the external momentum–impulse from artificial sources or sinks; M

_{R}is the momentum–impulse from rainfall; M

_{E}is the momentum–impulse lost to evapotranspiration; M

_{F}is the momentum–impulse from the subsurface by exfiltration; M

_{1}and M

_{2}are the momenta–impulses from the overland flow; B is the top width of the cross-section; τ

^{S}is the surface shear stress; P is the wet perimeter; τ

^{b}is the bottom shear stress, which can be assumed proportional to the flow rate as τ

^{b}/ρ = κV

^{2}where κ = g(n

_{M})

^{2}/(R

_{H})

^{1/3}, where R

_{H}is the hydraulic radius and n

_{M}is the Manning roughness.

_{S}and S

_{S}are considered as lateral flows.

#### 2.2. SFM2D Model

**Q**is the vector of conserved variables;

**F**

_{I}and

**G**

_{I}are the inviscid flux vectors in the x and y directions, respectively;

**F**

_{V}and

**G**

_{V}are the viscous flux vectors in the x and y directions, respectively;

**S**is the source term; u and v are the depth-averaged velocity components in the x and y directions, respectively; T

_{xx}, T

_{xy}, and T

_{yy}are the depth-averaged turbulent stresses; s

_{0x}and s

_{0y}are the bed slopes in the x and y directions, respectively; and s

_{fx}and s

_{fy}are the friction slopes in the x and y directions, respectively.

_{cell}is the area of the cell; dW

_{cell}is the area element; is the control volume; m is the index representing the side of the cell; M is the total number of sides for the cell;

**T**(θ)

^{−1}is the inverse of the rotation matrix corresponding to the m side; θ is the angle between the outward unit vector

**n**and the x axis;

**n**is the outward unit vector normal to the boundary of the control volume; L

^{m}is the length of the m side of the cell; $\mathbf{F}(\overline{\mathbf{Q}})={\mathbf{F}}_{I}(\overline{\mathbf{Q}})-{\mathbf{F}}_{V}(\overline{\mathbf{Q}})$ is the numerical flux; ${\mathbf{F}}_{I}(\overline{\mathbf{Q}})$ is the inviscid numerical flux; ${\mathbf{F}}_{V}(\overline{\mathbf{Q}})$ is the viscous numerical flux; $\overline{\mathbf{Q}}={\left[h,h{u}_{u},h{v}_{t}\right]}^{\mathrm{T}}$ is the vector of conserved variables transformed from the original vector of conserved variables

**Q**, in which ${u}_{n}=u\mathrm{cos}\theta +v\mathrm{sin}\theta $ and ${v}_{t}=v\mathrm{cos}\theta -u\mathrm{sin}\theta $ are respectively the flow velocity components in normal and tangential directions.

**Q**

^{n}is the vector of conserved variables for a cell at time index n. Equation (9) is then solved with the Euler method for a homogeneous part and the hydrostatic reconstruction method for source terms. The upstream flux-splitting finite-volume (UFF) scheme proposed by Lai et al. [29] has been verified and is especially suitable for resolving shallow-water free-surface flow involving discontinuities as well as hybrid flow regimes. Therefore, the UFF scheme was utilized to compute numerical inviscid fluxes of mass and momentum. Further detailed descriptions of the SFM2D model have been reported [29,30].

#### 2.3. Boundary Connection of 1D and 2D Models

- When the inflow at the connection boundary is subcritical, the unknown unit width discharge ${(h{u}_{n})}_{R}^{2\mathrm{D}}$ has to be given. The three unknown variables (${h}_{R}^{2\mathrm{D}}$, ${{u}_{n}}_{R}^{2\mathrm{D}}$, and ${{v}_{t}}_{R}^{2\mathrm{D}}$) can be solved simultaneously from the following three equations [36,37]:$${(h{u}_{n})}_{R}^{2\mathrm{D}}={(h{u}_{n})}_{R}^{1\mathrm{D}},\text{}{({u}_{n})}_{L}^{2\mathrm{D}}+2\sqrt{g{h}_{L}^{2\mathrm{D}}}={({u}_{n})}_{R}^{2\mathrm{D}}+2\sqrt{g{h}_{R}^{2\mathrm{D}}},\text{}{({v}_{t})}_{R}^{2\mathrm{D}}={({v}_{t})}_{L}^{2\mathrm{D}}$$$${(h{u}_{n})}_{R}^{1\mathrm{D}}={(h{u}_{c,ol,or})}_{R}^{1\mathrm{D}}=\frac{{Q}_{c,ol,or}}{{A}_{c,ol,or}}$$$${Q}_{c,ol,or}=Q\frac{{K}_{c,ol,or}}{{K}_{c}+{K}_{ol}+{K}_{or}},\text{}{K}_{c,ol,or}={A}_{c,ol,or}\frac{{({R}_{H})}_{c,ol,or}^{2/3}}{{({n}_{M})}_{c,ol,or}}$$
- When the outflow at the connection boundary is subcritical, the unknown water depth ${h}_{R}^{2\mathrm{D}}$ should be given. Thus, other two unknown variables (${{u}_{n}}_{R}^{2D}$ and ${{v}_{t}}_{R}^{2D}$) can be obtained from the following two relationships:$${({u}_{n})}_{R}^{2\mathrm{D}}={({u}_{n})}_{L}^{2\mathrm{D}}+2\sqrt{g}\left(\sqrt{{h}_{L}^{2\mathrm{D}}}-\sqrt{{h}_{R}^{2\mathrm{D}}}\right),\text{}{({v}_{t})}_{R}^{2\mathrm{D}}={({v}_{t})}_{L}^{2\mathrm{D}}$$
- When the inflow at the connection boundary is supercritical, all three unknown variables (i.e., ${h}_{R}^{2\mathrm{D}}$, ${{u}_{n}}_{R}^{2D}$ and ${{v}_{t}}_{R}^{2D}$) should be given from measured data. However, as the field data are not available, one can use the relation (${h}_{R}^{2D}={h}_{R}^{1D}$) and employ Equations (11) and (12) to achieve variables ${{u}_{n}}_{R}^{2D}$ and ${{v}_{t}}_{R}^{2D}$.
- When outflow at the connection boundary is supercritical, the transmissive boundary conditions are specified as follows:$${h}_{R}^{2\mathrm{D}}={h}_{L}^{2\mathrm{D}},\text{}{({u}_{n})}_{R}^{2\mathrm{D}}={({u}_{n})}_{L}^{2\mathrm{D}},\text{}{({v}_{t})}_{R}^{2\mathrm{D}}={({v}_{t})}_{L}^{2\mathrm{D}}$$

#### 2.4. Empirical Equation for Bend Scour Prediction

_{bs}is the bend scour depth; S

_{0}is the channel slope; σ

_{g}is the geometric standard deviation of the particle size distribution; R

_{c}(= R

_{o}− 0.5 W) is the centerline radius of the bend, where R

_{o}is the outer radius of the bend and W is the water surface width.

_{c}) to peak flow discharge (Q

_{p}) in the rising limb of a flood event. For practical application, Q

_{c}can be expressed as the width-full flow discharge. From the field study by Su and Lu [43], the higher duration value tends to cause embankment damage during flood-induced scouring. Thus, the effective peak–flow duration is one of the significant factors in affecting embankment toe failure.

_{0}= 0.486%–1.11%, R

_{c}= 500–985 m, σ

_{g}= 3.28–19.96, tp = 1.5–44 h, q = 8.90–23.88 m

^{2}/s, and W = 60–450 m, respectively. For modelling bend scour, physical variables expressed in Equation (17) are used to predict bend scour depth. The values of W and q can be obtained from WASH1D and SFM2D models. Others (e.g., σ

_{g}, S

_{0}, R

_{c}, tp) are achieved from the field data.

## 3. Bend Scour Simulation Results and Discussions

#### 3.1. Study Site, Field Data, and Model Setup

^{2}and is located in central Taiwan. As displayed in Figure 3, the riverbed elevation ranges from 0 m to 3877 m. The average bed slope in the main channel reach of Da-An River is approximately 0.011. During typhoons, bend scours may form rapidly because of river flooding, causing severe damage to the river embankment toe. The survey data from the WRA [21] indicated that over three embankment failure incidents have been reported at the Shuiwei Embankment. Therefore, the Shuiwei Embankment was chosen as the study site for bend scour modeling.

^{®}Core™ i5-8500 CPU and 8.0 GB RAM from Taipei, Taiwan.

#### 3.2. Performance Verification of WASH1D Model

_{p}is the peak water depth error; ET

_{p}is the error of time to peak water level; R

^{2}denotes the coefficient of determination; ${h}_{p}^{sim}$ and ${h}_{p}^{mea}$ denote the simulated and measured peak water depths, respectively; ${T}_{p}^{sim}$ and ${T}_{p}^{mea}$ are the simulated and measured time to the peak water depth, respectively; ${h}_{i}^{sim}$ and ${h}_{i}^{mea}$ are the simulated and measured water depths at each time; and ${\overline{h}}^{sim}$ and ${\overline{h}}^{mea}$ denote the average simulated and measured water depths.

_{p}(1.64%). The overall performance for Eh

_{p}was less than 18.24%. Furthermore, the overall performance regarding the R

^{2}coefficient for two flood events was larger than 0.75. The quantitative results for the three criteria indicated that the WASH1D model achieved satisfactory overall numerical accuracy in modeling 1D river flow.

#### 3.3. Performance Verification of SFM2D Model

^{2}coefficient was larger than 0.8. The overall performance of the Eh

_{p}may smaller than 19%.

_{p}of 57.65% and 34.13% at Yili and Shuiwei scour stations, respectively.

#### 3.4. Comparisons of WASH1D and SFM2D Models for Scour Prediction

- WASH1D-CS approach, which is constructed using cross-sectional bed-elevation data.
- WASH1D-HR approach, formed using high-resolution bed-elevation data. The high-resolution DEM was used to recreate the cross-sectional topography for the WASH1D model.
- SFM2D model based on high-resolution bed-elevation data.

_{p}, ET

_{p}, and R

^{2}, whereas the WASH1D-CS approach obtained the most erroneous solution. Figure 12 displays the simulated velocity contours under the peak flood condition of Typhoon Nesat obtained using the SFM2D model. The SFM2D model achieves a reasonable velocity component in the main channel, which is faster than those in the floodplain. These results indicate that incorporating high-resolution data into simulations could improve numerical performance.

_{g}, S

_{0}, R

_{c}, tp) into Equation (17) to obtain the bend scour depth at each simulation time step.

## 4. Model Application for River Embankment Failure Assessment

#### 4.1. New Assessment Methodology: Approach 1

^{2}, which is smaller than the critical value of 588.6 N/m

^{2}[47]. Thus, the embankment toe protections were stable.

^{3}/s were employed to conduct scenario simulations. As displayed in Figure 17, the simulated results of shear stress and the river discharge for four scenarios were combined with the field data under two events, revealing the relationships between river discharge, shear stress, and bend scour depth. In Figure 17, two field data of bend scour depth for Typhoons Nesat and Maria are considered. It is noted that the field data for shear stress is not available, thus the shear stress for Typhoons Nesat and Maria and four scenario cases are obtained from the simulated results under peak-flood condition. On the basis of this relationship curve, the critical flow discharge at the embankment toe protection works was estimated to be 2800 m

^{3}/s for a critical shear stress of 588.6 N/m

^{2}. Furthermore, the warning value of river discharge for riverbank failure was selected based on the failure event during Typhoon Soulik, which caused serious damage to the Shuiwei Embankment under peak flow with a discharge of 3120 m

^{3}/s.

_{c}

_{1}of 2800 m

^{3}/s, the embankment toe protection works may fail. Therefore, without protection, the river embankment is at risk. Furthermore, sustained river discharge higher than a Q

_{c}

_{2}value of 3150 m

^{3}/s may lead to failure of the riverbank toe foundation.

#### 4.2. New Assessment Methodology: Approach 2

_{u}in the upstream straight reach is a suitable information as input that can be obtained from approach flow cross section, as shown in Figure 12.

_{u}, R

_{c}, and shear stress, a predictive regression equation to determine shear stress based on water depth is proposed as follows.

^{2}value of 0.95), as illustrated in Figure 18.

_{u}, R

_{c}, W, and bend scour depth, the local scour depth, created by river flood flow in a bend reach, can be proposed as follows:

^{2}value of 0.87.

- Combine Equations (21) and (22) with water depth values to estimate the shear stress and bend scour depth.
- The estimated shear stress can then be compared with the critical value.
- If the estimated shear stress is larger than the critical value of 588.6 N/m
^{2}, the riverbank toe protection works may fail. - The estimated bend scour depth can be compared with the designed value. If the estimated bend scour depth is larger than the designed value, the riverbank may fail.

## 5. Conclusions

- The integrations of WASH1D, SFM2D, and bend scouring computation equations create a bend scour simulation model. The applications of the integrated model indicated that the model was able to successfully resolve the temporal process of bend scour caused by typhoon-induce floods.
- Based on high-resolution bed-elevation data, three approaches (namely WASH1D-CS, WASH1D-HR, and SFM2D) were presented and compared. The results demonstrate that the SFM2D model produces reasonable scour predictions with acceptable accuracy for water level and bend scour depth.
- A new methodology for assessing embankment failure is proposed. The novel aspects of the proposed methodology include the consideration of toe protections and two new predictive equations. The former is implemented to enhance the reliability of riverbank failure assessments. The latter provides more specific deterministic results.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**Comparisons of the measured and simulated water levels at (

**a**) Xiangbi, (

**b**) Shuangqi, (

**c**) Yili, and (

**d**) Dadaian gauging stations for Typhoon Saola (July 2012).

**Figure 8.**Comparisons of the measured and simulated water levels at (

**a**) Xiangbi, (

**b**) Shuangqi, (

**c**) Yili, and (

**d**) Dadaian gauging stations for Typhoon Soulik (July 2013).

**Figure 9.**Comparisons of the measured and simulated water levels at (

**a**) Shuiwei scour and (

**b**) Yili gauging stations for Typhoon Nesat (July 2017).

**Figure 10.**Contour plots showing the water depths based on (

**a**) coarse and (

**b**) fine computational meshes for Typhoon Nesat.

**Figure 11.**Comparisons of the measured and simulated water levels obtained using three approaches at the Shuiwei gauging station for (

**a**) Typhoon Nesat and (

**b**) Typhoon Maria.

**Figure 12.**Contour plot illustrating the flow velocity under peak-flood condition for Typhoon Nesat.

**Figure 13.**The simulated unit width discharges based on three approaches at Shuiwei water stage gauging station for (

**a**) Typhoon Nesat and (

**b**) Typhoon Maria.

**Figure 14.**Comparisons of the measured and simulated bend scour depths obtained using three approaches at Shuiwei scour gauging station for (

**a**) Typhoon Nesat and (

**b**) Typhoon Maria.

**Figure 15.**Procedure of the proposed assessment methodology for predicting reinforced concrete embankment failure.

**Figure 16.**Contour plot illustrating shear stress under peak-flood condition for the Typhoon Maria event.

**Figure 17.**Relationships between shear stress, bend scour depth, and river discharge under peak-flood condition.

Flood Events | t_{p} (h) | S_{0}(%) | σ_{g} | R_{c} (m) |
---|---|---|---|---|

Typhoon Nesat | 1.5 | 1.09 | 3.28 | 489 |

Typhoon Maria | 3.0 | 1.09 | 3.28 | 434 |

Events | Stations | Three Criteria | ||
---|---|---|---|---|

Eh_{p} (%) | ET_{p} (%) | R^{2} | ||

Typhoon Saola (July 2012) | Xiangbi | 1.64 | 2.94 | 0.97 |

Shuangqi | 15.07 | 2.94 | 0.76 | |

Yili | 13.82 | 12.1 | 0.80 | |

Dadaian | 3.07 | 12.1 | 0.93 | |

Typhoon Soulik (July 2013) | Xiangbi | 18.24 | 0 | 0.97 |

Shuangqi | 16.19 | 14.3 | 0.78 | |

Yili | 14.59 | 0 | 0.75 | |

Dadaian | 17.91 | 23.8 | 0.93 |

Stations | Three Criteria | ||
---|---|---|---|

Eh_{p} (%) | ET_{p} (%) | R^{2} | |

Yili | 18.18 | 0 | 0.92 |

Shuiwei scour | 2.41 | 0 | 0.82 |

Computational Mesh | CPU Time (h) | Eh_{p} (%) | |
---|---|---|---|

Yili | Shuiwei Scour | ||

Coarse mesh (1230 elements and 1364 nodes) | 0.46 | 57.65 | 34.13 |

Fine mesh (7465 elements and 7910 nodes) | 1.85 | 18.18 | 2.41 |

**Table 5.**Simulated results at the Shuiwei water gauging station obtained using the three presented approaches.

Events | Approaches | Three Criteria | ||
---|---|---|---|---|

Eh_{p} (%) | ET_{p} (%) | R^{2} | ||

Typhoon Nesat (July 2017) | WASH1D-CS | 108.42 | 0 | 0.88 |

WASH1D-HR | 57.67 | 0 | 0.85 | |

SFM2D | 20.81 | 0 | 0.91 | |

Typhoon Maria (July 2018) | WASH1D-CS | 56.81 | 8.33 | 0.89 |

WASH1D-HR | 26.67 | 8.33 | 0.88 | |

SFM2D | 12.73 | 8.33 | 0.96 |

Events | Ed_{bs} by Different Approaches | ||
---|---|---|---|

WASH1D-CS | WASH1D-HR | SFM2D | |

Typhoon Nesat (July 2017) | 0.142 | 0.104 | 0.080 |

Typhoon Maria (July 2018) | 2.022 | 2.096 | 1.097 |

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## Share and Cite

**MDPI and ACS Style**

Chang, C.-H.; Chen, H.; Guo, W.-D.; Yeh, S.-H.; Chen, W.-B.; Liu, C.-H.; Lee, S.-C.
Predicting River Embankment Failure Caused by Toe Scour Considering 1D and 2D Hydraulic Models: A Case Study of Da-An River, Taiwan. *Water* **2020**, *12*, 1026.
https://doi.org/10.3390/w12041026

**AMA Style**

Chang C-H, Chen H, Guo W-D, Yeh S-H, Chen W-B, Liu C-H, Lee S-C.
Predicting River Embankment Failure Caused by Toe Scour Considering 1D and 2D Hydraulic Models: A Case Study of Da-An River, Taiwan. *Water*. 2020; 12(4):1026.
https://doi.org/10.3390/w12041026

**Chicago/Turabian Style**

Chang, Chih-Hsin, Hongey Chen, Wen-Dar Guo, Sen-Hai Yeh, Wei-Bo Chen, Che-Hsin Liu, and Shih-Chiang Lee.
2020. "Predicting River Embankment Failure Caused by Toe Scour Considering 1D and 2D Hydraulic Models: A Case Study of Da-An River, Taiwan" *Water* 12, no. 4: 1026.
https://doi.org/10.3390/w12041026