# An Assessment Model for the Erosion Occurrence of Gap-Graded Sand-Gravel Soils under Variable Seepage Direction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{cr}is the hydraulic gradient that is critical for erosion to occur. ${G}_{\mathrm{s}}$ is the relative density of the soil, ${G}_{\mathrm{s}}={\gamma}_{\mathrm{s}}/{\gamma}_{\mathrm{w}}$, where ${\gamma}_{\mathrm{s}}$ is the particle weight, ${\gamma}_{\mathrm{w}}$ is the water weight, d

_{1}is the equivalent pore diameter of non-cohesive soils, and d is the particle diameter of mobile particles. In addition, a large number of model test studies of erosion have shown that particle size [15], particle shape [15,16], particle gradation [15,17,18], porosity [19,20], and fines content [21,22] all have an effect on the critical hydraulic conditions of particle initiation.

_{1}pore diameter, spherical particles, and laminar flow in the pore channel.

## 2. Prediction of Critical Seepage Velocity for Erosion Occurrence under Variable Seepage Direction

#### 2.1. Force Analysis of Movable Particles

_{1}, and maximum diameter, d

_{2}, of the skeleton pore channel are calculated using Formulas (2) and (3) [30].

_{h}is the effective particle diameter of skeletal particles, and n

_{a}is the porosity of the hypothetical soil composed of skeletal particles.

_{1}, these particles can be taken out of the soil, as long as the hydraulic conditions are met, and the part of the loose particles will be called movable particles. When the particle diameter is between d

_{1}and d

_{2}, this part of the loose particles only moves within a certain range of the pore channel, and may even block the pore. Therefore, this study assumed that the pore diameter is d

_{1}.

_{D}, of the flow; the underwater gravity, G′; the hydrostatic pressure, F

_{P}; the support force, N, of the surrounding particles; and the frictional force, F

_{f}, between the particles. The drag force can be calculated by the resistance that the particles overcome when moving in the viscous laminar flow [33], its magnitude is related to the flow velocity of the pore, and the direction is along the theoretical bed formed by the surrounding particles, which is determined by Formula (4). The hydrostatic pressure is the water pressure difference acting on the projected area of particles [29]; its magnitude is related to the hydraulic gradient. Combined with Darcy’s law, its direction is the same as the flow velocity of the pore, which is determined by Formula (5). The underwater gravity is determined by Formula (6). When particle A rolls around the contact point with the surrounding B particles, the moment generated by the rolling friction force, F

_{f}, between the particles and the supporting force, N, of the surrounding particles is zero, which is ignored in the calculation.

#### 2.2. Analysis of the Relative Exposure Degree and Relative Hidden Degree

#### 2.3. Critical Seepage Velocity for Particle Initiation

_{1}, according to Poiseuille theory [23], and assuming the number of pores per unit area [30], the flow velocity, v, of the pore system per unit area is:

## 3. Simulation of Erosion Occurrence under Variable Seepage Direction

#### 3.1. Numerical Model

_{B}) is represented by the following formula.

_{i}and I

_{i}are the mass and moment of inertia of particle i, respectively; ${u}_{i}$ and ${\omega}_{i}$ are the linear and angular velocities of particle, respectively; $\mathit{g}$ is the acceleration of gravity. The forces involved include the contact force, ${F}_{ij}^{c}$, and torque, ${M}_{ij}$, acting on particle i by particle j or the wall and the particle–fluid interaction forces, ${F}_{i}^{f}$, acting on particle i; ${n}_{i}^{c}$ is the number of contacts of particle i.

^{2}, and the direction was vertical downward. The sample was prepared using the layered under-compaction method [38], and the sample size was 20 × 20 × 20 mm. The linear model was selected for the contact model, and Table 1 shows the parameters related to DEM particles and contacts.

#### 3.2. Erosion Simulation Test

## 4. Verification of Existing Test Data

_{5}. Similarly, d = d

_{10}was chosen for the Ahlinhan piping test. The parameters of test soil samples are shown in Table 3.

## 5. Analysis of the Differences in Particle Initiation Conditions

#### 5.1. Seepage Direction

#### 5.2. The Relative Exposure Degree and the Relative Hidden Degree

## 6. Conclusions

- The model comprehensively considered the effects of the seepage direction and the relative position of particles. Based on the rolling theory of soil particles, the formula for calculating the critical velocity of particles under the variable seepage condition was derived. Further, the hydraulic conditions of seepage erosion in different directions were verified using discrete element numerical simulation and existing test data, which created conditions for subsequent application to two-dimensional seepage.
- In the analysis of the influencing factors, the relative hidden degree was positively correlated with the critical hydraulic conditions of particle initiation, while the relative exposure degree and the seepage direction interact and jointly influence each other. Different from the existing conclusions, the effect of seepage direction on the critical hydraulic conditions was not linear; however, the critical velocity showed a trend of increasing and then decreasing with the increase in seepage direction, due to the influence of relative exposure. Meanwhile, the critical velocity also showed a trend of increasing and then decreasing with the increase in relative exposure in different seepage directions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Foster, M.; Fell, R.; Spannagle, M. The statistics of embankment dam failures and accidents. Can. Geotech. J.
**2000**, 37, 1000–1024. [Google Scholar] [CrossRef] - Zhou, H.-B.; Cai, L.-B.; Gao, W.-j. Statistical analysis of the accidents of foundation pit of the urban mass rail transit station. Hydrogeol. Eng. Geol.
**2009**, 36, 67–71. [Google Scholar] [CrossRef] - Zheng, G.; Zhu, H.-H.; Liu, X.-R.; Yang, G.-H. Control of safety of deep excavations and underground engineering and its impact on surrounding environment. China Civ. Eng. J.
**2016**, 49, 1–24. [Google Scholar] [CrossRef] - Moffat, R.; Herrera, P. Hydromechanical model for internal erosion and its relationship with the stress transmitted by the finer soil fraction. Acta Geotech.
**2015**, 10, 643–650. [Google Scholar] [CrossRef] - Li, J.; Zhang, J.; Yang, X.; Zhang, A.; Yu, M. Monte Carlo simulations of deformation behaviour of unbound granular materials based on a real aggregate library. Int. J. Pavement Eng.
**2023**, 24, 2165650. [Google Scholar] [CrossRef] - Li, J.; Bi, W.; Yao, Y.; Liu, Z. State-of-the-Art Review of Utilization of Microbial-Induced Calcite Precipitation for Improving Moisture-Dependent Properties of Unsaturated Soils. Appl. Sci.
**2023**, 13, 2502. [Google Scholar] [CrossRef] - Kezdi, A. Soil Physics: Selected Topics-Developments in Geotechnical Engineering; Elsevier: Amsterdam, The Netherlands, 1979. [Google Scholar]
- Kenney, T.; Lau, D. Internal stability of granular filters: Reply. Can. Geotech. J.
**1986**, 23, 420–423. [Google Scholar] [CrossRef] - Aberg, B. Washout of grains from filtered sand and gravel materials. J. Geotech. Eng.
**1993**, 119, 36–53. [Google Scholar] [CrossRef] - Liu, Z.-Y.; Miao, T.-D. Assessment for the noncohesive piping-typed soils. Yantu Lixue/Rock Soil Mech.
**2004**, 25, 1072–1076. [Google Scholar] [CrossRef] - Li, H.; Zhu, H.; Wei, X.; Liu, B.; Shao, M. Soil erosion leads to degradation of hydraulic properties in the agricultural region of Northeast China. Agric. Ecosyst. Environ.
**2021**, 314, 107388. [Google Scholar] [CrossRef] - Parhizkar, M.; Shabanpour, M.; Lucas-Borja, M.E.; Zema, D.A.; Li, S.; Tanaka, N.; Cerdà, A. Effects of length and application rate of rice straw mulch on surface runoff and soil loss under laboratory simulated rainfall. Int. J. Sediment Res.
**2021**, 36, 468–478. [Google Scholar] [CrossRef] - Kudai, K.; Sassa, S.; Yang, S.; Takada, K. Influence of soil and hydraulic conditions on the processes of internal erosion, cavity formation and collapse behind coastal structures. Coast. Eng.
**2021**, 170, 104013. [Google Scholar] [CrossRef] - Liu, J. The Stability and Control of Seepage of Soil; China Water&Power Press: Beijing, China, 1992. [Google Scholar]
- Fleshman, M.S.; Rice, J.D. Laboratory Modeling of the Mechanisms of Piping Erosion Initiation. J. Geotech. Geoenvironmental Eng.
**2014**, 140, 04014017. [Google Scholar] [CrossRef] - Ojha, C.S.P.; Singh, V.P.; Adrian, D.D. Determination of Critical Head in Soil Piping. J. Hydraul. Eng.
**2003**, 129, 511–518. [Google Scholar] [CrossRef] - Okeke, A.C.-U.; Wang, F. Critical hydraulic gradients for seepage-induced failure of landslide dams. Geoenvironmental Disasters
**2016**, 3, 9. [Google Scholar] [CrossRef][Green Version] - Skempton, A.W.; Brogan, J.M. Experiments on piping in sandy gravels. Géotechnique
**1994**, 44, 449–460. [Google Scholar] [CrossRef] - Wan, C.F.; Fell, R. Assessing the Potential of Internal Instability and Suffusion in Embankment Dams and Their Foundations. J. Geotech. Geoenvironmental Eng.
**2008**, 134, 401–407. [Google Scholar] [CrossRef] - Ahlinhan, M.F.; Achmus, M. Experimental Investigation of Critical Hydraulic Gradients for Unstable Soils. In Scour and Erosion; ASCE: San Francisco, CA, USA, 2010; pp. 599–608. [Google Scholar]
- Prasomsri, J.; Shire, T.; Takahashi, A. Effect of fines content on onset of internal instability and suffusion of sand mixtures. Geotech. Lett.
**2021**, 11, 209–214. [Google Scholar] [CrossRef] - Wang, P.; Ge, Y.; Wang, T.; Liu, Q.-w.; Song, S.-x. CFD-DEM modelling of suffusion in multi-layer soils with different fines contents and impermeable zones. J. Zhejiang Univ. Sci. A
**2023**, 24, 6–19. [Google Scholar] [CrossRef] - Indraratna, B.; Radampola, S. Analysis of Critical Hydraulic Gradient for Particle Movement in Filtration. J. Geotech. Geoenvironmental Eng.
**2002**, 128, 347–350. [Google Scholar] [CrossRef] - Xu, B.-q.; Chen, J.-s.; Liang, Y. Damage Test of Fine Sand Piping and Seepage Deformation Analysis. Water Resour. Power
**2012**, 30, 66–69. [Google Scholar] [CrossRef] - Zhang, C.-H.; Ji, E.-Y.; Wang, B.-T. Research on a Critical Hydraulic Gradient of Piping in Noncohesive Soils. Adv. Civ. Eng.
**2021**, 2021, 6217101. [Google Scholar] [CrossRef] - Huang, Z.; Bai, Y.; Xu, H.; Cao, Y.; Hu, X. A Theoretical Model to Predict the Critical Hydraulic Gradient for Soil Particle Movement under Two-Dimensional Seepage Flow. Water
**2017**, 9, 828. [Google Scholar] [CrossRef][Green Version] - Richards, K.S.; Reddy, K.R. True Triaxial Piping Test Apparatus for Evaluation of Piping Potential in Earth Structures. Geotech. Test. J.
**2009**, 33, 83–95. [Google Scholar] [CrossRef] - Liang, Y.; Yeh, T.-C.J.; Ma, C.; Zhang, Q.; Yang, D.; Hao, Y. Experimental investigation of internal erosion behaviours in inclined seepage flow. Hydrol. Process.
**2020**, 34, 5315–5326. [Google Scholar] [CrossRef] - Indraratna, B.; Vafai, F. Analytical Model for Particle Migration within Base Soil-Filter System. J. Geotech. Geoenvironmental Eng.
**1997**, 123, 100–109. [Google Scholar] [CrossRef] - Kovács, G. Seepage Hydraulics; Elsevier Scientific Publishing Company: Amsterdam, The Netherlands, 1981. [Google Scholar]
- Choi, S.-U.; Kwak, S. Theoretical and probabilistic analyses of incipient motion of sediment particles. KSCE J. Civ. Eng.
**2001**, 5, 59–65. [Google Scholar] [CrossRef] - Wang, M.-N.; Jiang, Y.-T.; Yu, L.; Dong, Y.-C.; Duan, R.-Y. Analytical solution of startup critical hydraulic gradient of fine particles migration in sandy soil. Yantu Lixue/Rock Soil Mech.
**2020**, 41, 2515–2524. [Google Scholar] [CrossRef] - Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1973. [Google Scholar]
- Yang, F.-G.; Liu, X.-N.; Huang, E.; Yang, K.-J.; Cao, S.-Y. Study on the incipient velocity of Tangjiashan Barrier lake downstream area sediment. Sichuan Daxue Xuebao (Gongcheng Kexue Ban)/J. Sichuan Univ. (Eng. Sci. Ed.)
**2009**, 41, 84–89. [Google Scholar] - Zhou, S.; Zhang, G.-g.; Liang, Z.-x.; Xing, R. Incipient velocity of non-cohesive uniform sediment on slopes. J. Sediment Res.
**2015**, 4, 7–13. [Google Scholar] [CrossRef] - Felice, R.D. The voidage function for fluid-particle interaction systems. Int. J. Multiph. Flow
**1994**, 20, 153–159. [Google Scholar] [CrossRef] - Itasca. Particle Flow Code, Version 5.0 User Manuals; Itasca Consulting Group Inc.: Minneapolis, MN, USA, 2018. [Google Scholar]
- Jiang, M.J.; Konrad, J.M.; Leroueil, S. An efficient technique for generating homogeneous specimens for DEM studies. Comput. Geotech.
**2003**, 30, 579–597. [Google Scholar] [CrossRef] - Wu, M.-X.; Gao, G.-Y.; Yang, J.-X.; Zhan, Z.-G. A method of predicting critical gradient for piping of sand and gravel soils. Yantu Lixue/Rock Soil Mech.
**2019**, 40, 861–870. [Google Scholar] [CrossRef] - Zhou, X.-j.; Jie, Y.-x.; Li, G.-x. Numerical simulation of the developing course of piping. Comput. Geotech.
**2012**, 44, 104–108. [Google Scholar] [CrossRef]

**Figure 3.**Numerical model of particle flow: (

**a**) gradation sample, (

**b**) fluid elements, and measurement regions.

**Figure 4.**Initiation state of fine particles under the seepage. (

**a**) v = 0.00 cm/s, (

**b**) v = 0.40 cm/s, (

**c**) v = 0.50 cm/s, (

**d**) v = 0.65 cm/s, (

**e**) v = 0.70 cm/s, (

**f**) v = 0.80 cm/s.

**Figure 7.**Comparison of predicted and test values of critical velocity: (

**a**) horizontal seepage and (

**b**) vertical seepage.

Model | Parameter Type | Value |
---|---|---|

Particle model | Fine particle diameter/mm | 0.6 |

Coarse particle diameter/mm | 2.0~5.0 | |

Particle density/(kg/m^{3}) | 2650 | |

Friction coefficient | 0.5 | |

Effective modulus/Pa | 1 × 10^{9} | |

Normal-to-shear stiffness ratio | 1.25 | |

Initial porosity | 0.3 | |

Fluid model | Fluid density/(kg/m^{3}) | 1000 |

Dynamic viscosity/($\mathrm{Pa}\cdot \mathrm{s}$) | 0.001 |

**Table 2.**The critical velocity, calculated by formula and numerical simulation under different conditions.

Sample | Direction of Seepage/° | Calculation Results/(cm/s) | Simulation Results/(cm/s) |
---|---|---|---|

Sample 1 | 90 | 0.557 | 0.55 |

120 | 0.622 | 0.60 | |

135 | 0.678 | 0.65 | |

150 | 0.815 | 0.75 | |

180 | 0.706 | 0.65 | |

Sample 2 | 90 | 0.664 | 0.65 |

120 | 0.742 | 0.75 | |

135 | 0.808 | 0.80 | |

150 | 0.971 | 0.95 | |

180 | 0.841 | 0.85 | |

Sample 3 | 90 | 0.864 | 0.85 |

120 | 0.965 | 0.95 | |

135 | 1.050 | 1.05 | |

150 | 1.263 | 1.20 | |

180 | 1.094 | 1.05 |

Sample | Initial Relative Density | Natural Void Ratio e | Limiting Particle Diameter x_{a} [9]/mm | Effective Particle Diameter of Skeletal Particles D_{h}/mm | Porosity of the Hypothetical Soil n_{a} | Minimum Pore Diameter d_{1}/mm | Particle Diameter d |
---|---|---|---|---|---|---|---|

A | - | 0.52 | 1.000 | 3.235 | 0.44 | 0.568 | ${d}_{5}$ |

B | - | 0.59 | 0.431 | 2.488 | 0.43 | 0.417 | ${d}_{5}$ |

E1 | 0.5 | 0.62 | 0.537 | 1.279 | 0.47 | 0.500 | ${d}_{10}$ |

0.6 | 0.60 | 0.46 | 0.486 | ${d}_{10}$ | |||

0.85 | 0.55 | 0.44 | 0.452 | ${d}_{10}$ | |||

E2 | 0.41 | 0.55 | 0.439 | 1.191 | 0.46 | 0.443 | ${d}_{10}$ |

0.5 | 0.52 | 0.45 | 0.427 | ${d}_{10}$ | |||

0.85 | 0.41 | 0.41 | 0.361 | ${d}_{10}$ | |||

E3 | 0.75 | 0.52 | 0.96 | 2.166 | 0.44 | 0.386 | ${d}_{10}$ |

0.95 | 0.46 | 0.42 | 0.355 | ${d}_{10}$ |

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**MDPI and ACS Style**

Li, D.; Zhao, Y.; Liu, N.; Gao, X.
An Assessment Model for the Erosion Occurrence of Gap-Graded Sand-Gravel Soils under Variable Seepage Direction. *Water* **2023**, *15*, 1487.
https://doi.org/10.3390/w15081487

**AMA Style**

Li D, Zhao Y, Liu N, Gao X.
An Assessment Model for the Erosion Occurrence of Gap-Graded Sand-Gravel Soils under Variable Seepage Direction. *Water*. 2023; 15(8):1487.
https://doi.org/10.3390/w15081487

**Chicago/Turabian Style**

Li, Da, Yaowei Zhao, Ningyi Liu, and Xiaojuan Gao.
2023. "An Assessment Model for the Erosion Occurrence of Gap-Graded Sand-Gravel Soils under Variable Seepage Direction" *Water* 15, no. 8: 1487.
https://doi.org/10.3390/w15081487