# 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

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**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