# Hydraulic Modeling and Evaluation Equations for the Incipient Motion of Sandbags for Levee Breach Closure Operations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equations for Sandbag Incipient Motion

_{cr}= depth averaged velocity at the location of the particle at which the particle starts moving, W = weight of the particle, and C

_{1}= an empirical constant.

_{2}= an empirical constant. Novak and Nalluri [52] proposed an equation for the critical motion of single particles on smooth and rough beds in the form

_{3}and C

_{4}are empirical constants. This relation is stated to be valid for D/R < 0.3. On the other hand, for prism sandbags, Gulcu [42] used force balance on single solid particles and the proposed relation between critical velocity and particle dimensions as follow:

_{8}is a constant to be calibrated from experimental data.

## 3. Experimental Setup

^{3}/s (2500 gallons per minutes) axial pumps were used for flow supply from an underground sump to an overhead tank that supplies the flume through a 0.305 m supply pipe. To ensure uniformly distributed flow at the breach location, perforated screens and a 60 mm long honeycomb were placed at the flume inlet.

^{3}. Rectangular prisms were divided into two groups according to their orientation to the flow direction. Geometrical properties and dimensional details of the sandbags used in this investigation are shown in Table 1. It should be noted that a, b, and c are the width, height and length of an imaginary rectangular prism enfolding sandbag, and ∀ and S are the volume and surface area, respectively. All sandbags were placed such that the shortest dimension was always the height of the sandbag.

## 4. Experimental Procedure

_{b}were considered (0.25 m and 0.40 m), and for each the downstream channel gate was altered to change flow through the breach and thus the breach flow ratio, Q

_{r}$={Q}_{b}/{Q}_{u}$, was changed from 1 to 0.80, where Q

_{b}is the breach discharge and Q

_{u}is the main channel discharge upstream of the breach. Fr

_{u}is the main channel Froude number upstream of the breach, and Fr

_{d}is the main channel Froude number downstream of the breach. Table 2 presents the hydraulic parameters which were recorded in the experimental study.

## 5. Results and Discussions

#### 5.1. Water Surface Mapping

_{b}= 40 cm, and Q

_{r}= 1, water depth in the main channel upstream of the breach was 38 cm, which increased longitudinally across the main channel to 42 cm downstream of the breach, and decreased rapidly in the transverse direction at the breach location to 29 cm at the breach. Figure 2 shows the mapping of water depth for L

_{b}= 40 cm.

_{r}= 1 m

^{3}/s, the recirculation effects and the separation zone were highest when compared to those in the other case of Q

_{r}< 1. Unlike for open channel dividing flow, the breach flow separation occurred at the near side of the breach in the main channel, causing a flow contraction. This is obvious from the velocity vectors across the breach. However, for the same Q

_{r}, the larger the breach opening, the larger the angle of entry of flow, and the extent of flow separation in the main channel just before the flow exits from the breach. Also, the larger the breach opening, the smaller the contraction zone. Michelazzo et al. [7,53] discussed more details on the influence of breach width on channel flow. These characteristics are similar to those of the dividing open channel flows [20]. Figure 3 shows the hydraulics of an open channel levee breach for the two considered cases, showing clearly the separation and stagnation zones.

#### 5.2. Velocity Fields

_{r}= 1 m

^{3}/s. The x-velocity decreases at the mid–section of the breach and reaches zero by the end of the breach due to the complete curvature of the vectors exiting the breach. Experimental results showed that the vectors at the tip of the stagnation zone at the breach end took a steep curvilinear path to exit the breach and were forced by the flume side to exit in the negative x-direction.

_{r}= 1 (m

^{3}/s). These velocities gave a better view of the separation and contraction zones. In the separation zone that started to form in the main channel before the breach flow exit, the exit velocities in the y-direction were the lowest along the breach section.

_{r}= 1 m

^{3}/s and L

_{b}= 0.25 m, Figure 5 shows the velocity field for y- and z-velocity components at two locations: x = 0.10 m and x = 0.20 m. Both sections were after the region of the flow separation at the breach, and they showed clearly the impact of the third vertical component of velocity (z-velocity) in the flow pattern near the breach section. This secondary downward current is very strong and obvious at section x = 0.20 m; however, near the end of the breach, its intensity at the other sections in the contraction zone is low. This secondary current caused the vector to drop rapidly, leading to a rapid decrease in the water level in the vicinity of the breach, until the mid-width of the channel, where it started to diminish along the channel width. The z-velocity component decreased with a decrease in z distance at the contraction zone, at x = 0.10 m and 0.20 m.

#### 5.3. Critical Velocity for Sandbags Incipient Motion

^{2}of 0.90:

^{2}of 0.95 as shown in Figure 7, where ${V}_{cr}=0.381{W}^{1/6}$ using metric units, and W is the sandbag weight in Newton.

^{2}of 0.83 as follows:

#### Uncertainty Analysis

_{i}is the prediction error, P

_{i}is the predicted value of parameter, and T

_{i}is the measured value of the parameter. Data were then used to calculate main indicators defined as mean prediction error $\overline{e}={\displaystyle \sum}_{i=1}^{n}{e}_{i}$, the width of uncertainty band, ${B}_{ub}=\pm 1.96{S}_{e}$ and the confidence band around the predicted value:

_{e}is the standard deviation of prediction errors and P

_{i}is the predicted value and is taken as unity. Table 5 summarizes the results of the uncertainty analysis performed on both calibrated and developed models for prediction of critical velocity for sandbag motion in flowing water. All models were tested on experimental data in this study and Zhu et al. [43] data.

#### 5.4. 17th Street Canal Levee Breach Closure

^{3}/s. Using these data, the critical velocity is 2.5 m/s (as measured experimentally by Sattar et al. [29] and calculated numerically by Jia et al. [66]) and ${V}_{cr}$ is 0.78 m/s for 1359 kg (3000 lb) and 0.70 m/s for 2718 kg (6000 lb) sandbags.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

b | sandbag height (cm) |

${B}_{ub}$ | the width of the uncertainty band |

c | sandbag length parallel to the flow direction (cm) |

C_{1–8} | empirical constants |

D | the diameter of the spherical particle (mm) |

e_{i} | the prediction error |

$\overline{e}$ | mean prediction error |

H | water depth above sandbag (m) |

L_{b} | breach width (m) |

P_{i} | the predicted value of the parameter |

Q_{b} | the breach discharge (m^{3}/s) |

Q_{u} | main channel discharge (m^{3}/s) |

Q_{r} | breach flow ratio = ${Q}_{b}/{Q}_{u}$ |

R | channel hydraulic radius (m) |

R^{2} | the coefficient of determination |

S_{e} | the standard deviation of prediction errors |

T_{i} | the measured value of the parameter |

${V}_{cr}^{\ast}$ | ${V}_{cr}/\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gb}$; |

${\rho}_{s}$ | sandbag density (g/cm^{3}) |

${\rho}_{w}$ | water density (g/cm^{3}) |

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**Figure 1.**Schematic description of the experimental setup. (

**a**) The experimental levee breach model, (

**b**) the grid used for velocity measurements.

**Figure 3.**Schematics of the hydraulics of channel levee breach (L

_{B}and W are breach width and channel width respectively).

**Figure 4.**x- and y-velocity field at the breach section in the main flume. Q

_{u}= 0.16 m

^{3}/s, Q

_{r}= 1 m

^{3}/s, z = 0.15 m, for L

_{b}= 0.40 m.

**Figure 5.**y-z velocity vector profile along the breach, Q

_{u}= 0.16 m

^{3}/s, Q

_{r}= 1 m

^{3}/s, for L

_{b}= 0.25 m, at x = 0.2 m.

**Figure 6.**Relationship between the non-dimensional critical velocity (${V}_{cr}^{\ast})$ at the initiation of sandbag motion and D/R for circular sandbags.

**Figure 9.**${V}_{cr}/\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gb}$ versus the height to length ratio (b/c) for prism sandbags.

**Figure 10.**Black hawk dumping sandbags to close the 17th Street Canal levee breach [29].

Sandbag Shape | Statistical Parameter | a (cm) | b (cm) | c (cm) | ∀ (cm ^{3}) | S (cm ^{2}) |
---|---|---|---|---|---|---|

Prism (29 cases) | Minimum | 3 | 2.5 | 4 | 40 | 72 |

Maximum | 18 | 7 | 18 | 378 | 384 | |

Average. | 6.9 | 3.82 | 9.1 | 214 | 236 | |

Standard deviation | 4.2 | 1.15 | 4.3 | 103 | 93 | |

Sphere (13 cases) | Minimum | 5 | 58 | 72 | ||

Maximum | 22 | 5575 | 1521 | |||

Average. | 13 | 1739 | 618 | |||

Standard deviation | 6 | 1788 | 475 |

Breach Width | Q_{u} (m^{3}/s) | Q_{b} (m^{3}/s) | Q_{r} | Fr_{u} | Fr_{d} |
---|---|---|---|---|---|

0.25 m | 0.16 | 0.16 | 1 | 0.62 | - |

0.16 | 0.128 | 0.8 | 0.62 | 0.11 | |

0.4 m | 0.16 | 0.16 | 1 | 0.62 | - |

0.16 | 0.128 | 0.8 | 0.62 | 0.11 |

**Table 3.**Sandbag geometric and dimensional properties versus the critical velocity ${V}_{cr}/\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gb}$.

Experiment | b/c | $\frac{{\mathit{V}}_{\mathit{c}\mathit{r}}}{\sqrt{\frac{\left({\mathit{\rho}}_{\mathit{s}}-{\mathit{\rho}}_{\mathit{w}}\right)}{{\mathit{\rho}}_{\mathit{w}}}\mathit{g}\mathit{b}}}$ | Exp. | b/c | $\frac{{\mathit{V}}_{\mathit{c}\mathit{r}}}{\sqrt{\frac{\left({\mathit{\rho}}_{\mathit{s}}-{\mathit{\rho}}_{\mathit{w}}\right)}{{\mathit{\rho}}_{\mathit{w}}}\mathit{g}\mathit{b}}}$ |
---|---|---|---|---|---|

1 | 0.63 | 0.76 | 27 | 1 | 0.49 |

2 | 0.60 | 0.69 | 28 | 1 | 0.51 |

3 | 0.50 | 0.69 | 29 | 1 | 0.52 |

4 | 0.58 | 0.79 | 30 | 1 | 0.61 |

5 | 0.50 | 0.79 | 31 | 1 | 0.52 |

6 | 0.42 | 0.93 | 32 | 1 | 0.61 |

7 | 0.35 | 0.81 | 33 | 1 | 0.57 |

8 | 0.44 | 0.81 | 34 | 1 | 0.53 |

9 | 0.44 | 0.81 | 35 | 1 | 0.50 |

10 | 0.56 | 0.68 | 36 | 1 | 0.55 |

11 | 0.50 | 0.68 | 37 | 1 | 0.53 |

12 | 0.45 | 0.72 | 38 | 1 | 0.58 |

13 | 0.32 | 1.15 | 39 | 1 | 0.56 |

14 | 0.30 | 1.25 | 40 | 1 | 0.55 |

15 | 0.23 | 1.25 | 41 | 1 | 0.52 |

16 | 0.19 | 1.35 | 42 | 1 | 0.53 |

17 | 0.18 | 1.41 | |||

18 | 0.21 | 1.35 | |||

19 | 0.22 | 1.35 | |||

20 | 0.18 | 1.46 | |||

21 | 0.70 | 0.65 | |||

22 | 0.64 | 0.70 | |||

23 | 0.50 | 0.70 | |||

24 | 0.58 | 0.67 | |||

25 | 0.44 | 0.67 | |||

26 | 1.00 | 0.49 |

Study | Model Equation | |
---|---|---|

Novak and Nalluri [52] | Original | ${V}_{cr}={C}_{3}{\left(\frac{D}{R}\right)}^{{C}_{4}}\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gD}$ |

Calibrated | ${V}_{cr}=0.569{\left(\frac{D}{R}\right)}^{-0.34}\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gD}$ | |

Brahms | Original | ${V}_{cr}={C}_{1}{W}^{1/6}$ |

Calibrated | ${V}_{cr}=0.381{W}^{1/6}$ | |

Izbash [44] | Original | ${V}_{cr}=3.76\sqrt{D\left(\frac{{\rho}_{s}-{\rho}_{w}}{{\rho}_{w}}\right)}$ |

Calibrated | ${V}_{cr}=1.77\sqrt{D\left(\frac{{\rho}_{s}-{\rho}_{w}}{{\rho}_{w}}\right)}$ |

Sandbags Shape | Study | Model | The Width of Uncertainty Band | Prediction Interval Around Hypothetical Predicted ${\mathit{V}}_{\mathit{c}\mathit{r}}^{\ast}$ of Unity |
---|---|---|---|---|

Sphere | Novak and Nalluri [52] (Calibrated) | $\frac{{V}_{cr}}{\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gD}}=0.569{\left(\frac{D}{R}\right)}^{-0.34}$ | ±0.14 | (0.78–1.51) |

Brahms (Calibrated) | ${V}_{cr}=0.381{W}^{1/6}$ | ±0.05 | (0.91–1.14) | |

Present study—Sphere | ${V}_{cr}=1.633{D}^{0.47}$ | ±0.04 | (0.89–1.13) | |

Samov (Original) | ${V}_{cr}=2.5{D}^{0.44}$ | ±0.05 | (1.45–1.85) | |

Izbash [44] (Original) | ${V}_{cr}=3.76\sqrt{D\left(\frac{{\rho}_{s}-{\rho}_{w}}{{\rho}_{w}}\right)}$ | ±0.05 | (1.87–2.39) | |

Izbash [44] (Calibrated) | ${V}_{cr}=1.77\sqrt{D\left(\frac{{\rho}_{s}-{\rho}_{w}}{{\rho}_{w}}\right)}$ | ±0.05 | (0.88–1.12) | |

Prism | Present study—Prism1 | $\frac{{V}_{cr}}{\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gb}}=0.295{\left(\frac{b}{c}\right)}^{-1}$ | ±0.19 | (0.52–1.25) |

Present study—Prism2 | $\frac{{V}_{cr}}{\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gb}}=0.522{\left(\frac{b}{c}\right)}^{-0.67}$ | ±0.08 | (0.91–1.33) | |

Zhu et al. [43] (Original) | $\frac{{V}_{cr}}{\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gb}}=0.44{\left(\frac{H}{b}\right)}^{1/6}{\left(\frac{b}{c}\right)}^{-0.5}$ | ±0.08 | (0.90–1.30) |

**Table 6.**Critical velocity (${V}_{cr}$) for spherical sandbag motion for breach closure in Base Scenario studied by Sattar et al. [29].

Study | Model | V_{cr} 95% Confidence Interval (m/s) | 10,000 lbs Sandbag | 15,000 lbs Sandbag |
---|---|---|---|---|

Novak and Nalluri [52] (Calibrated) | ${V}_{cr}=0.569{\left(\frac{D}{R}\right)}^{-0.34}\sqrt{\frac{\left({\rho}_{s}-{\rho}_{w}\right)}{{\rho}_{w}}gD}$ | Predicted | 0.96 | 0.98 |

Range | (0.73–1.45) | (0.75–1.48) | ||

Brahms (Calibrated) | ${V}_{cr}=0.381{W}^{1/6}$ | Predicted | 2.27 | 2.43 |

Range | (2.01–2.50) | (2.21–2.77) | ||

Present study—Sphere | ${V}_{cr}=1.633{D}^{0.47}$ | Predicted | 2.00 | 2.16 |

Range | (1.78–2.26) | (1.92–2.44) | ||

Samov (Original) | ${V}_{cr}=2.5{D}^{0.44}$ | Predicted | 3.00 | 3.24 |

Range | (4.35–5.55) | (4.70–6.00) | ||

Izbash [44] (Original) | ${V}_{cr}=3.76\sqrt{D\left(\frac{{\rho}_{s}-{\rho}_{w}}{{\rho}_{w}}\right)}$ | Predicted | 5.48 | 5.77 |

Range | (10.25–13.10) | (10.79–13.79) | ||

Izbash [44] (Calibrated) | ${V}_{cr}=1.77\sqrt{D\left(\frac{{\rho}_{s}-{\rho}_{w}}{{\rho}_{w}}\right)}$ | Predicted | 2.20 | 2.37 |

Range | (1.94–2.46) | (2.08–2.66) |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Sattar, A.M.A.; Bonakdari, H.; Gharabaghi, B.; Radecki-Pawlik, A.
Hydraulic Modeling and Evaluation Equations for the Incipient Motion of Sandbags for Levee Breach Closure Operations. *Water* **2019**, *11*, 279.
https://doi.org/10.3390/w11020279

**AMA Style**

Sattar AMA, Bonakdari H, Gharabaghi B, Radecki-Pawlik A.
Hydraulic Modeling and Evaluation Equations for the Incipient Motion of Sandbags for Levee Breach Closure Operations. *Water*. 2019; 11(2):279.
https://doi.org/10.3390/w11020279

**Chicago/Turabian Style**

Sattar, Ahmed M. A., Hossein Bonakdari, Bahram Gharabaghi, and Artur Radecki-Pawlik.
2019. "Hydraulic Modeling and Evaluation Equations for the Incipient Motion of Sandbags for Levee Breach Closure Operations" *Water* 11, no. 2: 279.
https://doi.org/10.3390/w11020279