# A New Practical Method to Simulate Flood-Induced Bridge Pier Scour—A Case Study of Mingchu Bridge Piers on the Cho-Shui River

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Field Instrument for Scour Depth Measurement

#### 1.2. Hydrodynamic and Sediment Transport Modeling

_{s}) is simply an arithmetic summation of the general scour (G

_{s}) and local scour (L

_{s}). The former is estimated by using a proposed 2D finite-volume hydraulic model coupled with a general scour-computation equation. The latter is in the form of a simple local-scour computation algorithm. The accuracy of the proposed method is verified using the measured field data. The proposed method is then applied by simulating nine different return period discharges to investigate the relationship between the total scour depth and discharge at the bridge pier.

## 2. Field Measurements, Procedures and Results

#### 2.1. Site Description

_{50}= 35 mm, with a geometric standard deviation of sediment size (${\mathsf{\sigma}}_{g}=\sqrt{\left({d}_{84}/{d}_{16}\right)}$) 15.33, where d

_{16}and d

_{84}= particle size for which 16% and 84% are finer by weight, respectively. However, the survey results in 2014 indicated a median sediment size of d

_{50}= 1.52 mm, with a geometric standard deviation of sediment size σ

_{g}= 7.45. The data show that the sediment particle distribution at the bridge site clearly has become more uniformly distributed. Moreover, the median grain size has changed from that of coarse gravel to coarse sand. The change in the bed material composition at the Mingchu Bridge over the past decade is significant as it reflects the effect of the Chi-Chi Weir in reducing sediment supply.

#### 2.2. Procedures of Scour Measurements

#### 2.3. Results of Scour Measurements

^{3}/s, whereas the Monsoon yielded a peak flow discharge of 1446 m

^{3}/s, which is nearly one-third that of Typhoon Matmo. The scour depth, including the general and total scour depths, was related to the magnitude of the peak flow discharge and duration of the flood hydrograph. The duration of the hydrograph associated with Typhoon Matmo is evidently longer than that with the Monsoon.

^{2}/s to 33.1 m

^{2}/s. The median sediment particle size ranged from 1.52 mm to 136 mm. For most of the measured data, the median particle size of the bed materials exceeds 2 mm; therefore, the bed may be classified as a gravel bed. However, at the field site at the Silo Bridge, which is located in the lower Cho-Shui River with a channel bed slope of approximately 0.001, the bed is a sandy bed. Furthermore, at the field site of the current study (Mingchu Bridge), the bed sediment varies from gravel to gravel with large-grained sand because the bed has been incised over the recent 10 years (Figure 3, Figure 4 and Figure 5).

#### 2.4. Proposed General-Scour Computation Equation

_{s}induced by a flood may be expressed in a functional form as follows:

_{50}is the median particle diameter; σ

_{g}is the geometric standard deviation of the particle size distribution. Using Buckingham’s π theorem, G

_{s}in Equation (1) may be expressed in the following dimensionless form:

_{s}is the density of sediment particles and ρ is the density of fluid. Analyses of the field data indicate that Equation (3) may be expressed in the following form:

_{0}, a

_{1}, a

_{2}, and a

_{3}are coefficients. The field data shown in Table 1 were used in the regression analysis to obtain the coefficients in Equation (4). The equation thus obtained is

## 3. Method for Bridge Scour Simulation

#### 3.1. 2D Finite-Volume Hydraulic Model

**Q**is the conserved physical vector;

**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; h is the water depth; u and v are the depth-averaged velocity components in the x- and y-directions, respectively; ρ is the density of water; T

_{xx}, T

_{xy}and T

_{yy}are the depth-averaged turbulent stresses; g is the gravitational acceleration; s

_{0x}and s

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

_{fx}and s

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

^{m}is the length of the m side for the cell;

**T**(θ)

^{−1}is the inverse of 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;

**Q**

^{n}is the vector of conserved variables for a cell at time index n.

**Q′**is the vector of conserved variables at the predictor step; $\mathbf{F}(\overline{\mathbf{Q}})={\mathbf{F}}_{I}^{}(\overline{\mathbf{Q}})-{\mathbf{F}}_{V}(\overline{\mathbf{Q}})$ is the numerical flux; ${\mathbf{F}}_{I}^{}(\overline{\mathbf{Q}})$ represents the inviscid numerical flux; ${\mathbf{F}}_{V}(\overline{\mathbf{Q}})$ denotes the viscous numerical flux; ${\mathbf{S}}^{\prime}({\mathbf{Q}}^{\prime})$ is the well-balanced source term:

_{Mann}is the Manning roughness coefficient.

_{b})

_{LR}= max[(Z

_{b})

_{L}, (Z

_{b})

_{R}] is the bed elevation at the cell interface LR.

- The model is suitable for modeling flow hydraulics involving irregular bed topography;
- Both steady or unsteady flows can be simulated;
- The solution is accurate and the numerical algorithm is efficient;
- An unstructured arbitrarily shaped mesh is used; and
- All flow regimes (i.e., subcritical, transcritical, and supercritical flows) can be resolved.

#### 3.2. Local-Scour Computation Algorithm

_{S}is the local pier-scour depth; D

_{p}is the pier diameter; ${F}_{d}(=U/\sqrt{{g}^{\prime}{d}_{50}})$ is the densimetric particle Froude number; g′[=(ρ

_{s}/ρ − 1)g] is the reduced gravitational acceleration; σ

_{g}is the geometric standard deviation of the distribution of sediment particles; and t is the time.

- For the first flow discharge Q
_{1}with duration t_{1}, the evolution of the scour depth follows the OA curve under a steady flow condition. The cumulative scour depth is denoted as L_{s1}. If the duration of flow discharge Q_{1}is sufficiently long, the local scour hole may reach the equilibrium condition, and the corresponding scour depth is L_{se1}, where L_{se1}> L_{s1.} - When the flow discharge increases from Q
_{1}to Q_{2}, the evolution of the scour depth follows the AB curve under the steady flow condition. Point C represents the virtual origin for the scouring process associated with the discharge Q_{2}. Because the scouring process can “memorize” the previous scour depth, and because Q_{2}> Q_{1}, the time (t_{*,1}) required for the scour depth to reach L_{s1}is less than t_{1}. The AB curve represents the corresponding evolution of the scour depth from t_{1}to t_{2}. - Similar to the calculation procedure (2), when the flow rate increases from Q
_{2}to Q_{3}(> Q_{2}), the evolution of the scour depth follows the BD curve under the steady flow condition. Likewise, Point E represents the virtual origin for the scouring process associated with the discharge Q_{3}. Because Q_{3}> Q_{2}, the time (t_{*,2}) required for the scour depth to reach L_{s2}is less than t_{*,1}+ (t_{2}− t_{1}). The BD curve indicates the corresponding evolution of the scour depth from t_{2}to t_{3}. - Repeat the preceding procedure until all of the subdivisions are completed.
- Obtain the temporal variation of scour depth under unsteady flow conditions.

#### 3.3. Method for Simulating Total Scour-Depth Evolution

- Using the proposed 2D finite-volume hydraulic model to simulate the 2D flow field near the bridge piers, one obtains the hydraulic properties including the water levels, velocities, and water depths upstream of the bridge piers.
- The evolution of the general scour is obtained when these hydraulic properties are inputted to the proposed general-scour computation equation (i.e., Equation (5)).
- On the basis of the evolution of the general scour depth and the revised approach flow conditions, the local-scour computation algorithm, i.e., Equations (13)~(15), can be used to estimate the evolution of the local scour depth.
- The evolution of the total scour depth can be obtained by summing the general depth and local scour depth.

## 4. Hydraulic and Bridge Scour Simulations

#### 4.1. Verification of Finite-Volume Hydraulic Model

_{p}and ET

_{p}are used herein. The peak water level error Eη

_{p}is defined as follows:

_{p}is defined as:

_{p}value, the proposed model presents good solutions in capturing the time to peak water level. The peak water level errors, Eη

_{p}indicate that the proposed model performs good solutions near the peak flow. Figure 14a,b present the simulated results under peak-flood condition, showing the velocity contours for the Monsoon and Typhoon Matmo, respectively. Comparing to the flood event of the Monsoon, the value of the simulated velocity during Typhoon Matmo is higher. The results also show that the velocity in the braided channel can be reasonable simulated. In addition, the proposed model is capable of resolving the flow hydraulics in irregular bed topography.

#### 4.2. Simulations of Total Scour-Depth Evolution

_{s}is the sum of the general scour G

_{s}and local pier-scour depths L

_{s}). To compare the local pier-scour depth with the estimated depths by using different formulas, the measured local pier-scour depth was first calculated by subtracting the measured total scour depth at Pier 4 from the general scour depth. Following the proposed method shown in Figure 10, the general scour depth was calculated by using Equation (5), whereas the local pier-scour depth was evaluated using Equations (13)~(15). Numerous equations have already been derived for estimation of the maximum pier scour depth (see Gaudio et al. [27] and Gaudio et al. [28]). In the present study, four formulas (Laursen [29], Shen et al. [30], Jain and Fischer [31] and Hong et al. [15]) were selected for comparison.

#### 4.3. Scoured Bed Level-Discharge Relationship

_{b}) is uniquely related to discharge (Q) and can be fitted to z

_{b}= 170.72Q

^{−0.021}(R

^{2}= 0.983), showing that z

_{b}decreases with an increase in the discharge. Using this equation, one can easily deduce the scour depth for a given flow discharge. To provide a reference of the scoured bed level at the Mingchu Bridge, three meaningful levels, namely the top and bottom levels of the pile cap (142.75 m and 139.75 m), and the level of the pile extension (129.75 m), also are superimposed in Figure 17. With these three reference levels, four bridge warning stages, namely safe, low risk, moderate risk and high risk, are identified. For instance, when discharge = 10,000 m

^{3}/s, the scoured bed level, z

_{b}= 170.72Q

^{−0.021}, yielding z

_{b}= 140.69 m. The computed scoured bed level lies between the top and bottom levels of the pile cap, which constitutes the “low risk” condition. It is to be noted that for a complex foundation shape as shown in Figure 6b a more accurate prediction of scour depth may be obtained through the methods presented in Ferraro et al. [32]. In summary, the proposed bridge safety curve (Figure 17) enables DGH to make a rational decision on bridge closure during typhoon seasons based solely on the value of discharge because a determinate relationship between discharge and scour has been established.

## 5. Conclusions

- The “numbered-brick” method is a useful tool for measuring the total scour and general scour depth, especially for ephemeral/intermittent rivers even though it only can provide the maximum scour depth after a flood event. However, it should be noted that the stability of the pier may be threatened by digging around the pier foundation to place the bricks. To this end, extreme care must be taken when digging around the pier foundation. A dimensionless formula was established for calculating the general scour depth using the measured field data and based on the unit width peak flow discharge, sediment particle size and standard deviation of the particle size distribution. The dimensionless general scour formula gives a reasonably good prediction compared with the field data. The formula may be used for the assessment of general scour under a design flood with certain return period, and this information may be further used for the design of protection for bridge piers or the toe of levees.
- For practical engineering practices, the total scour depth at a bridge pier may be the sum of the general scour, contraction scour and local-pier scour depth. When the contraction scour is ignored, the total scour is then the sum of general scour and local scour. The percentage of flood induced short-term general scour is found to increase (percentage of local scour decreases) with the flow discharge. It may therefore be inferred that for high flow conditions, the contribution of general scour to the total scour is more important, thus rendering a careful monitoring of the evolution of general scour to be essential.
- For simulating the total scour evolution, this study proposed a straight-forward and accurate estimation method. The accuracy of the proposed method is verified from both short-term general scour and total pier-scour depth collected from field measurements. The satisfactory simulated results show that the method can be successfully used to evaluate the development of the total pier-scour depth. In future studies, this method can be coupled with hydro-meteorological modeling for bridge scour forecasting, which can be applied to a real-time bridge scour warning system. This development will provide bridge owners with a superior method to estimate the scoured bed level towards a rational approach for bridge closure decision with a higher level of confidence.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 4.**The near field of Mingchu Bridge (

**a**) looking downstream; (

**b**) looking upstream, photos taken in May 2014.

**Figure 6.**Schematic site plan of the Mingchu Bridge for the installation of numbered-brick columns (

**a**) Plan view; (

**b**) Site view.

**Figure 9.**Schematic diagram of superposition for computing local pier-scour evolution under a stepwise hydrograph: (

**a**) stepwise hydrograph; (

**b**) evolution of scour depth.

**Figure 12.**Comparison of the measured water level and the simulated result at Mingchu gauging station during Monsoon (May, 2014).

**Figure 13.**Comparison of the measured water level and the simulated result at Mingchu gauging station during Typhoon Matmo (July, 2014).

**Figure 14.**Simulated velocity contours in the study reach under peak-flood condition for (

**a**) Monsoon and (

**b**) Typhoon Matmo.

River | Site | Flood Event | Q_{p} (m^{3}/s) | B (m) | q_{p} (m^{2}/s) | S | d_{50} (mm) | G_{s} (m) | Bed Features | Remark |
---|---|---|---|---|---|---|---|---|---|---|

Cho-Shui River | Mingchu Bridge | 1. Typhoon Dujuan (September 2003) | 2146 | 247 | 8.69 | 0.01 | 35 | 2.1 | Gravel bed | Lu et al. [3] |

2. Typhoon Mindulle (July 2004) | 7250 | 275 | 21.54 | 0.01 | 35 | 6 | Lu et al. [3] | |||

3. Typhoon Soulik (July 2013) | 7285 | 300 | 24.28 | 0.007 | 37.73 | 3.80 | Gravel/sand bed | Current study | ||

4. Monsoon (May 2014) | 1446 | 148.5 | 9.74 | 0.00518 | 1.52 | 1.82 | Current study | |||

5. Typhoon Matmo (July 2014) | 4980 | 150.4 | 33.1 | 0.00518 | 1.52 | 3.25 | Current study | |||

Silo Bridge | 6. Typhoon Dujuan (September 2003) | 2268 | 506 | 4.48 | 0.001 | 2 | 1.2 | Sand bed | Lu et al. [3] | |

7. Typhoon Mindulle (July 2004) | 8050 | 758 | 10.62 | 0.001 | 2 | 1.65 | Lu et al. [3] | |||

Da-Chia River | Houfeng Bridge | 8. Typhoon Sinlaku (September 2008) | 5410 | 230 | 23.52 | 0.011 | 136 | 1.56 | Gravel bed | Su and Lu [12] |

Dachia Highway Bridge | 9. Typhoon Morakot (August 2009) | 4225 | 400 | 10.56 | 0.011 | 96 | 4.5 | Su and Lu [12] |

**Table 2.**Comparisons of simulated water levels with measured results by using 2D finite-volume hydraulic model.

Events | Two Criteria | |
---|---|---|

Eη_{p} (%) | ET_{p} (h) | |

Monsoon (May, 2014) | 0.0057 | 0 |

Typhoon Matmo (July, 2014) | 0.0521 | 1 |

Variables | Monsoon (May, 2014) | Typhoon Matmo (July, 2014) | ||||
---|---|---|---|---|---|---|

Q_{p} (m^{3}/s) | 1446 | 4980 | ||||

Measured | Simulated | Error (%) | Measured | Simulated | Error (%) | |

G_{s} (m) | 1.815 | 1.881 | 3.64 | 3.245 | 3.137 | −3.33 |

L_{S} (m) | 0.770 | 0.707 | −8.18 | 0.880 | 1.362 | 54.77 |

T_{s} (m) | 2.585 | 2.588 | 0.12 | 4.125 | 4.499 | 9.07 |

G_{s}/T_{s} | 0.70 | 0.73 | 4.29 | 0.79 | 0.70 | −11.39 |

L_{s}/T_{s} | 0.30 | 0.27 | 10.00 | 0.21 | 0.30 | 42.86 |

_{s}: General scour upstream of Mingchu Bridge; L

_{S}: Local pier-scour at the pier foundation of Mingchu Bridge; T

_{s}: Total scour at the pier foundation of Mingchu Bridge.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hong, J.-H.; Guo, W.-D.; Chiew, Y.-M.; Chen, C.-H.
A New Practical Method to Simulate Flood-Induced Bridge Pier Scour—A Case Study of Mingchu Bridge Piers on the Cho-Shui River. *Water* **2016**, *8*, 238.
https://doi.org/10.3390/w8060238

**AMA Style**

Hong J-H, Guo W-D, Chiew Y-M, Chen C-H.
A New Practical Method to Simulate Flood-Induced Bridge Pier Scour—A Case Study of Mingchu Bridge Piers on the Cho-Shui River. *Water*. 2016; 8(6):238.
https://doi.org/10.3390/w8060238

**Chicago/Turabian Style**

Hong, Jian-Hao, Wen-Dar Guo, Yee-Meng Chiew, and Cheng-Hsin Chen.
2016. "A New Practical Method to Simulate Flood-Induced Bridge Pier Scour—A Case Study of Mingchu Bridge Piers on the Cho-Shui River" *Water* 8, no. 6: 238.
https://doi.org/10.3390/w8060238