# Mechanism of Steady and Unsteady Piping in Coastal and Hydraulic Structures with a Sloped Face

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Unsteady Piping Experiments

#### 2.1. Experimental Setup

#### 2.2. Soil Characteristics

#### 2.3. Experimental Procedure

## 3. Force-Balance Equation on a Soil Element

#### 3.1. Force-Balance Equation for a Sloped-Face Soil Parcel Without a Filter Layer

#### 3.2. Force-Balance Equation for a Sloped-Face Soil Parcel Protected by a Filter Layer

## 4. Results

#### 4.1. Comparison of the Steady Piping Criterion with the Available Data

#### 4.2. Experimental Findings on Unsteady Piping

## 5. Discussion and Remarks for Practical Applications

- The grain size ratio of the filter layer to the base soil must comply with the filter criteria given in Equations (1)–(4). This would ensure that the base soil will not be able to move through the grains of the filter layer without clogging the pores but the pore water can be drained easily through the filter. Hence, piping resistance would be enhanced.
- It was seen that the hydraulic conductivity is an important parameter in piping, especially when the hydraulic gradient acts unsteadily. Therefore, the selection of low hydraulic conductivity base soils (especially for coastal structures) must be omitted if possible (see also [2]).
- The thickness of the filter layer must be carefully determined. To this end, the criterion given with Equation (20) can be used to define the minimum filter layer thickness ${B}_{f}$. The thickness of the base soil parcel, $\Delta z$, can be taken as zero to omit the uncertainties and to stay on the safe side. Equations (18) and (22) can be used to determine the active lateral earth pressure coefficient (${K}_{a}$) and the filter pore size (${D}_{p}$), respectively. It should be noted that the criterion for the total failure of the entire soil column given by [24] and defined in Equation (24) comes out to be less critical than the piping criterion (${i}_{cr}<{\left({i}_{cr}\right)}_{col}$).

## 6. Conclusions

- In the case of steady hydraulic loading, the comparison of the force-balance equation (Equation (17)) and the available data from the literature shows that friction forces are effective in resisting against piping. However, there are quite many uncertainties in estimating the friction forces. Therefore, it may be safer to resort to the force balance equation with zero friction (Equation (9)) for design purposes when there is no filter layer. The effect of filter layer on increasing the friction forces is formulated (Equation (20)) in terms of the filter layer thickness.
- The mechanisms of piping under steady and unsteady hydraulic loadings are quite different since the inertia terms become effective in the unsteady loading case. Generally speaking, soil is more prone to piping in the case of unsteady hydraulic loading. This was also shown by Tomlinson and Vaid [3] when they increased the hydraulic gradient in a rapid manner and got piping under a much lower hydraulic gradient compared to the gradually increasing case.
- Experimental findings showed that the period of oscillatory hydraulic loading did not have a significant effect on the results for the tested periods of loading ($T=8\sim 12$ s). Additionally, the sudden hydraulic loading did not generate a distinctly different effect on the soil compared to the oscillatory.
- The hydraulic conductivity of the soil becomes very important for piping vulnerability under unsteady hydraulic gradients, such that low conductivity soil was seen to be much more susceptible to piping. As the hydraulic conductivity gets higher, the soil becomes more resistant to piping under unsteady hydraulic loading. Therefore, the usage of low hydraulic conductivity soils in coastal structures must be omitted, as also suggested by the design guidelines [2].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The unsteady pressure gradient generated by wave action on a slope (only one layer of filter is shown to keep the figure simple): (

**a**) Run-up stage; (

**b**) Run-down stage (vulnerable to piping).

**Figure 3.**Definition sketch of force-balance during piping: (

**a**) Soil parcel without a granular filter; (

**b**) Soil parcel underlying a granular filter with thickness ${B}_{f}$.

**Figure 4.**Variation of the critical hydraulic gradient for piping (${i}_{cr}$) as a function of $\Delta z/D$ for different values of face slopes, cf. Equation (17). The soil parameters are ${\phi}^{\prime}=40\xb0$, $s=2.65$ and $n=0.4$.

**Figure 6.**(

**a**) Pressure time series recorded at Test No. 103, sudden hydraulic loading; (

**b**) Pressure profile with depth at the instant of maximum pressure gradient ($=\frac{\partial \left(p/\gamma \right)}{\partial z}$) measured, $t=26.3$ s.

**Figure 7.**Hydraulic gradient time series recorded at Test No. 103. The maximum pressure gradient is attained at $t=26.3$ s.

**Figure 8.**(

**a**) Pressure time series recorded at Test No. 1003, oscillatory hydraulic loading; (

**b**) Pressure profile with depth at the instant of maximum and minimum pressure gradient ($i=\frac{\partial \left(p/\gamma \right)}{\partial z}$) measured, $t=33.0$ s and $130.4$ s, respectively.

**Figure 9.**Hydraulic gradient time series recorded at Test No. 1003. The maximum pressure gradient is attained at $t=33.0$ s.

**Figure 11.**Driving pressure gradient ($\Delta H/L$) vs. experienced maximum pressure gradient ${i}_{max}=\frac{\partial \left(p/\gamma \right)}{\partial z}$. Data points corresponding to piping are shown with red markers. (

**a**) Tests with medium sand $\alpha =0\xb0$; (

**b**) Tests with silty fine sand $\alpha =0\xb0$.

**Figure 12.**Driving pressure gradient ($\Delta H/L$) vs. experienced maximum pressure gradient ${i}_{max}=\frac{\partial \left(p/\gamma \right)}{\partial z}$. Data points corresponding to piping are shown with red markers. Tests with silty fine sand (

**a**) $\alpha =18.5\xb0$; (

**b**) $\alpha =34\xb0$.

**Figure 13.**Driving pressure gradient ($\Delta H/L$) vs. experienced maximum pressure gradient ${i}_{max}=\frac{\partial \left(p/\gamma \right)}{\partial z}$. Tests with gravel (

**a**) $\alpha =18.5\xb0$; (

**b**) $\alpha =34\xb0$.

Soil Type | $\mathbf{Specific}\text{}\mathbf{Gravity},\text{}\mathit{s}$ | $\mathbf{Median}\text{}\mathbf{Grain}\text{}\mathbf{Size},\text{}{\mathit{d}}_{50}\text{}\left(\mathbf{mm}\right)$ | $\mathbf{Standard}\text{}\mathbf{Geometric}\text{}\mathbf{Deviation},\text{}{\mathit{\sigma}}_{\mathit{g}}=\sqrt{\frac{{\mathit{d}}_{84}}{{\mathit{d}}_{16}}}$ | $\mathbf{Hydraulic}\text{}\mathbf{Conductivity},\text{}\mathit{k}$ (m/s) | $\mathbf{Porosity},\text{}\mathit{n}$ | $\mathbf{Angle}\text{}\mathbf{of}\text{}\mathbf{Friction},\text{}\mathit{\phi}\prime $$(\xb0)$ |
---|---|---|---|---|---|---|

Silty Fine Sand | 3.00 | 0.2 | 3.1 | $1.2\times {10}^{-5}$ | 0.37 | 40 |

Medium Sand | 2.65 | 0.7 | 2.5 | $2.8\times {10}^{-4}$ | 0.39 | 38 |

Gravel | 3.40 | 12 | 1.8 | $6.6\times {10}^{-2}$ | 0.40 | 44 |

No. | Soil Type | Type of Hydraulic Loading | Surface Angle, $\mathit{\alpha}$ $(\xb0)$ | $\mathbf{Unsteady}\text{}\mathbf{Hydraulic}\text{}\mathbf{Head},\text{}\Delta \mathit{H}\text{}\left(\mathbf{m}\right)$ | $\mathbf{Period}\text{}\mathbf{of}\text{}\mathbf{Motion},\text{}\mathit{T}$ (s) | Piping? |
---|---|---|---|---|---|---|

101 | Medium Sand | Sudden | 0 | 0.15 | N/A | No |

102 | Sudden | 0 | 0.30 | N/A | No | |

103 | Sudden | 0 | 0.50 | N/A | Yes | |

104 | Sudden | 0 | 0.75 | N/A | Yes | |

105 | Sudden | 0 | 0.40 | N/A | Yes | |

201 | Oscillatory | 0 | 0.10 | 8 | No | |

202 | Oscillatory | 0 | 0.15 | 8 | No | |

203 | Oscillatory | 0 | 0.20 | 8 | No | |

204 | Oscillatory | 0 | 0.25 | 8 | No | |

205 | Oscillatory | 0 | 0.30 | 8 | No | |

206 | Oscillatory | 0 | 0.35 | 8 | No | |

207 | Oscillatory | 0 | 0.40 | 8 | No | |

208 | Oscillatory | 0 | 0.40 | 12 | Yes | |

209 | Oscillatory | 0 | 0.50 | 12 | Yes | |

901 | Silty Fine Sand | Sudden | 0 | 0.20 | N/A | No |

902 | Sudden | 0 | 0.30 | N/A | No | |

903 | Sudden | 0 | 0.40 | N/A | Yes | |

904 | Sudden | 0 | 0.50 | N/A | Yes | |

905 | Sudden | 0 | 0.75 | N/A | Yes | |

1001 | Oscillatory | 0 | 0.15 | 8 | No | |

1002 | Oscillatory | 0 | 0.20 | 8 | No | |

1003 | Oscillatory | 0 | 0.30 | 10 | Yes | |

1004 | Oscillatory | 0 | 0.40 | 10 | Yes | |

1005 | Oscillatory | 0 | 0.50 | 10 | Yes | |

1006 | Oscillatory | 0 | 0.65 | 12 | Yes | |

1101 | Sudden | 18.5 | 0.20 | N/A | No | |

1102 | Sudden | 18.5 | 0.30 | N/A | Yes | |

1103 | Sudden | 18.5 | 0.40 | N/A | Yes | |

1104 | Sudden | 18.5 | 0.50 | N/A | Yes | |

1201 | Oscillatory | 18.5 | 0.15 | 8 | No | |

1202 | Oscillatory | 18.5 | 0.20 | 8 | Yes | |

1203 | Oscillatory | 18.5 | 0.30 | 8 | Yes | |

1204 | Oscillatory | 18.5 | 0.40 | 8 | Yes | |

1205 | Oscillatory | 18.5 | 0.65 | 12 | Yes | |

1301 | Sudden | 34.0 | 0.20 | N/A | Yes | |

1302 | Sudden | 34.0 | 0.30 | N/A | Yes | |

1303 | Sudden | 34.0 | 0.40 | N/A | Yes | |

1304 | Sudden | 34.0 | 0.50 | N/A | Yes | |

1305 | Sudden | 34.0 | 0.75 | N/A | Yes | |

1306 | Sudden | 34.0 | 1.00 | N/A | Yes | |

1401 | Oscillatory | 34.0 | 0.15 | 8 | Yes | |

1402 | Oscillatory | 34.0 | 0.20 | 8 | Yes | |

1403 | Oscillatory | 34.0 | 0.30 | 8 | Yes | |

1404 | Oscillatory | 34.0 | 0.40 | 8 | Yes | |

1405 | Oscillatory | 34.0 | 0.65 | 12 | Yes | |

301 | Gravel | Sudden | 0 | 0.20 | N/A | No |

302 | Sudden | 0 | 0.30 | N/A | No | |

303 | Sudden | 0 | 0.40 | N/A | No | |

304 | Sudden | 0 | 0.50 | N/A | No | |

305 | Sudden | 0 | 0.75 | N/A | No | |

306 | Sudden | 0 | 1.00 | N/A | No | |

307 | Sudden | 0 | 2.00 | N/A | No | |

401 | Oscillatory | 0 | 0.25 | 8 | No | |

402 | Oscillatory | 0 | 0.50 | 10 | No | |

403 | Oscillatory | 0 | 0.65 | 12 | No | |

501 | Sudden | 18.5 | 0.20 | N/A | No | |

502 | Sudden | 18.5 | 0.30 | N/A | No | |

503 | Sudden | 18.5 | 0.40 | N/A | No | |

504 | Sudden | 18.5 | 0.75 | N/A | No | |

505 | Sudden | 18.5 | 1.00 | N/A | No | |

506 | Sudden | 18.5 | 2.00 | N/A | No | |

601 | Oscillatory | 18.5 | 0.65 | 12 | No | |

701 | Sudden | 34.0 | 2.00 | N/A | No | |

801 | Oscillatory | 34.0 | 0.65 | 12 | No |

Ref. | Data No. | Soil Type | $\mathbf{Specific}\text{}\mathbf{Gravity},\text{}\mathit{s}$ | $\mathbf{Angle}\text{}\mathbf{of}\text{}\mathbf{Friction},\text{}\mathit{\phi}\prime $$(\xb0)$ | $\mathbf{Porosity},\text{}\mathit{n}$ | $\Delta \mathit{z}/\mathit{D}$ | $\mathbf{Critical}\text{}\mathbf{Hydraulic}\text{}\mathbf{Gradient}\text{}\mathbf{for}\text{}\mathbf{Piping},\text{}{\mathit{i}}_{\mathit{c}\mathit{r}}$ |
---|---|---|---|---|---|---|---|

[13] | 1 | Ottawa 20–30 sand | 2.64 | 35 | 0.35 | 2.49 | 1.95 |

[13] | 2 | Ottawa graded sand | 2.64 | 35 | 0.35 | 2.49 | 2.12 |

[13] | 3 | Angular 20–30 sand | 2.64 | 37 | 0.43 | 2.49 | 2.72 |

[13] | 4 | Angular graded sand | 2.64 | 38 | 0.42 | 2.49 | 2.99 |

[13] | 5 | No. 100 garnet sand | 3.87 | 39 | 0.47 | 2.49 | 2.89 |

[14] | 1 | N/A | 2.65 | 26.57 | 0.36 | 2.47 | 1.67 |

[14] | 2 | N/A | 2.65 | 26.57 | 0.38 | 2.47 | 1.39 |

[14] | 3 | N/A | 2.65 | 26.57 | 0.38 | 2.47 | 1.36 |

[14] | 4 | N/A | 2.65 | 26.57 | 0.40 | 2.47 | 1.29 |

[14] | 5 | N/A | 2.65 | 26.57 | 0.40 | 2.47 | 1.29 |

[14] | 6 | N/A | 2.65 | 26.57 | 0.39 | 2.47 | 1.32 |

[14] | 7 | N/A | 2.65 | 26.57 | 0.38 | 0.59 | 1.26 |

[14] | 8 | N/A | 2.65 | 26.57 | 0.38 | 1.11 | 1.28 |

[14] | 9 | N/A | 2.65 | 26.57 | 0.38 | 1.55 | 1.29 |

[14] | 10 | N/A | 2.65 | 26.57 | 0.37 | 2.53 | 1.45 |

[14] | 11 | N/A | 2.65 | 26.57 | 0.37 | 3.52 | 1.68 |

[14] | 12 | N/A | 2.65 | 0.00 | 0.35 | 2.40 | 1.23 |

[14] | 13 | N/A | 2.65 | 14.04 | 0.37 | 2.51 | 1.38 |

[14] | 14 | N/A | 2.65 | 26.57 | 0.37 | 2.55 | 1.45 |

[14] | 15 | N/A | 2.65 | 36.87 | 0.38 | 2.58 | 1.53 |

[14] | 16 | N/A | 2.65 | 0.00 | 0.36 | 2.47 | 1.32 |

[14] | 17 | N/A | 2.65 | 14.04 | 0.37 | 2.51 | 1.42 |

[14] | 18 | N/A | 2.65 | 26.57 | 0.37 | 2.53 | 1.45 |

[14] | 19 | N/A | 2.65 | 26.57 | 0.37 | 2.55 | 1.45 |

[14] | 20 | N/A | 2.65 | 26.57 | 0.37 | 2.53 | 0.16 |

[14] | 21 | N/A | 2.65 | 26.57 | 0.38 | 2.56 | 1.51 |

[14] | 22 | N/A | 2.65 | 26.57 | 0.39 | 2.58 | 1.38 |

[14] | 23 | N/A | 2.65 | 26.57 | 0.39 | 2.62 | 1.32 |

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**MDPI and ACS Style**

Kirca, V.S.O.; Kilci, R.E.
Mechanism of Steady and Unsteady Piping in Coastal and Hydraulic Structures with a Sloped Face. *Water* **2018**, *10*, 1757.
https://doi.org/10.3390/w10121757

**AMA Style**

Kirca VSO, Kilci RE.
Mechanism of Steady and Unsteady Piping in Coastal and Hydraulic Structures with a Sloped Face. *Water*. 2018; 10(12):1757.
https://doi.org/10.3390/w10121757

**Chicago/Turabian Style**

Kirca, V. S. Ozgur, and R. Evren Kilci.
2018. "Mechanism of Steady and Unsteady Piping in Coastal and Hydraulic Structures with a Sloped Face" *Water* 10, no. 12: 1757.
https://doi.org/10.3390/w10121757