# A Theoretical Model to Predict the Critical Hydraulic Gradient for Soil Particle Movement under Two-Dimensional Seepage Flow

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## Abstract

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## 1. Introduction

_{c}. This parameter is usually defined as the critical condition at which the effective stress of the soil becomes negligible [7]. Apparently, a large number of theoretical and experimental approaches have been used to obtain critical hydraulic gradients in water-retaining structures [8,9,10,11,12]. The critical gradient is generally calculated as the ratio of the buoyant unit weight of soil to the unit weight of water, as established by Terzaghi [8] according to the following Equation (1):

_{s}is the specific weight of soil, γ

_{w}is the specific weight of water, and n is the porosity. Terzaghi suggested that the critical gradient depends only on the void ratio and on the specific gravity of the solids. However, both in the field and in the laboratory, it has been observed that seepage erosion can initiate at gradients much lower than those determined with Terzaghi’s classical approach [6,9,10]. For example, Skempton and Brogan [9] carried out tests with upward flow for unstable soils. They found gradient values from one-third to one-fifth smaller than the theoretical values established by Terzaghi. Critical hydraulic gradients lower than 0.1 have also been observed in the field, for example in the failures of the Herbert Hoover Dike and of the A. V. Watkins Dam [10]. Other parameters have been proposed, for instance i

_{f}, defined by Samani [11] as the critical hydraulic gradient of seepage failure. In addition, Wan and Fell [12] introduced i

_{start}and i

_{boil}to represent the critical hydraulic gradients for the starting of internal erosion and soil boiling, respectively. However, the critical hydraulic gradients described above are generally used for the conventional one-dimensional upward seepage flow; moreover, the resulting estimates are usually not accurate [6,9,10].

_{ini}and the failure i

_{f}of landslide dams under different geometrical and hydraulic conditions, were determined by Okeke and Wang [7]. The influences of several factors on the gradients were discussed in their research. The critical hydraulic gradients were studied by Adel et al. [19] through tests with stable and unstable soils (based on the Kenney and Lau's criterion [20]). Wan and Fell found that smaller critical hydraulic gradients apply tendentially to soils in a loose state, rather than to soils in a dense state [21]. Similar conclusions were obtained by Ahlinhan et al. in their experimental investigations [22]. Five soil specimens were examined in their experiments to determine the gradient for vertical and horizontal seepages. The internal stability of the soil was proved to be a decisive factor for the critical gradient. However, only few criteria for the soil particle movement in 2-D seepage have been studied so far; in addition, the influence of various parameters on these criteria also requires further study.

## 2. Methods

#### 2.1. Particle Movement Process

#### 2.2. Forces on the Movable Particles

_{L}, contact force of motionless particle (B) F

_{n}, friction F

_{τ}, and gravity G. The forces G and F

_{L}are described in Equations (3) and (4).

_{W}and ρ

_{S}are the densities of the water and the soil particles, respectively, g is the gravity acceleration, i is the hydraulic gradient, e is the void ratio of the soil, d

_{θk}is the equivalent diameter (defined by 1/Σ(p

_{i}/d

_{i}), where p

_{i}is the percent of grains with the diameter of d

_{i}), C

_{L}is the lift coefficient, and V

_{b}is the seepage velocity in the void. The forces F

_{n}and F

_{τ}have no computational method, and they are obtained from the mechanical equilibrium of a movable particle.

#### 2.3. Equilibrium Equations

_{n}and F

_{τ}are equal to zero.

_{L}is the horizontal distance from the center of A to the locus of F

_{L}, L

_{F}is the vertical distance from the center of A to the locus of F; β is the angle referred to the position of the movable particle, and R is the radius of A. In this study, the forces F

_{L}and F were assumed to be located at the particle center.

#### 2.4. Analysis of the Relative Exposure Degree

#### 2.5. Equation for the Critical Hydraulic Gradient

_{L}is much lower than that of the seepage force F and of the gravity force G. Thus, the lift force F

_{L}will be ignored in the following sections. The critical hydraulic gradient i

_{c}can be obtained from Equation (5) as follows:

_{S}− ρ

_{W}is the particle density in water. In Equation (8), the critical hydraulic gradient is influenced by the soil particle density, the movable particle size, the seepage direction, the soil void ratio, the soil gradation, and the particle relative positions. When the soil parameters are constant, the critical hydraulic gradient varies only with Δ′ and θ. Equation (8) can be expressed in a simpler form, as Equation (9)

## 3. Results

#### 3.1. Qualitative Analysis

^{−4}m/s, which is consistent with the empirical value in Table 1. The seepage field of a vertical slope model (1 m × 0.5 m), with water levels of 0.5 m upstream and 0 m downstream, respectively, is shown in Figure 3. The seepage direction θ can be obtained from the seepage field.

_{c}

_{,15}/d

_{f}

_{,85}was used to determine the stability of the soils, with d

_{c}

_{,15}= grain diameter with 15% w/w of the grains of the coarse soil being finer, and d

_{f}

_{,85}= grain diameter with 85% w/w of the grains of the fine soil being finer. The finer particles can be removed by the flow easily. As suggested by Kezdi [33], d

_{15}is assumed to be the largest movable particle in Equation (8), and, thus, D = d

_{15}. Figure 4 shows the distribution of the critical hydraulic gradients that were predicted under different relative exposure degrees (Δ′ = 0.05, 0.1, 0.2) and different porosities (e = 0.43, 0.67, 1).

#### 3.2. Experimental Validation

_{15}was set at 0.1 because a few movements of particles can lead to seepage failure. The value of the soil particle density was 2.65 g/cm

^{3}. According to the analyses above, Equation (8) can be reduced to Equation (12).

^{2}= 0.98).

^{2}= 0.53). The data-points above the solid line indicate that the predicted values were greater than the measured values. This is because the gradients of the soil specimens gradually increased during the test. In smaller gradients, fine particles were removed by the flow, which made the soil much looser. In a looser soil, the measured gradients are smaller than the real gradients. The points below the line indicate that the predicted i were underestimated. This may result from the obstruction caused by the finer particles, which affects the other particles of the group when the hydraulic gradient is small.

^{2}= 0.91), though there was some discrepancies between them. The results from these experiments are presented in Table 2. For the experiment of Ahlinhan et al., the predicted results were more accurate than for the other two experiments. In the Fleshman and Rice’s test, the critical hydraulic gradient was overestimated.

## 4. Discussion

#### 4.1. Effect of the Soil Internal Instability

_{θk}is much larger than d

_{15}, which leads to a smaller Π. Therefore, the calculated critical hydraulic gradient was significantly smaller than that of the stable soil. Our theoretical model also agrees with experimental data found in the literature. In the horizontal seepage, Adel [19] demonstrated that the critical hydraulic gradient for unstable soils should be around 0.2, while in Ahlinhan’s test [22], the gradients were between 0.14 and 0.29. In our study, the gradients calculated for unstable soils were between 0.16 and 0.28. For stable soils, Adel [19] showed that the critical hydraulic gradient was about 0.7, while in Ahlinhan’s test, it was shown to be between 0.52 and 0.66. Our calculated gradients were between 0.42 and 0.64, and, thus, they were similar to those determined experimentally, with only some deviations. In the vertical seepage, the critical hydraulic gradients were between 0.2 and 0.34 for the unstable soil [9]; in Ahlinhan’s test [22], they were between 0.19 and 0.25. According to our calculations, the gradients were between 0.21 and 0.48, and, thus, slightly different from the experimental results, but consistent on the whole. For stable soils, Skempton’s [9] experimental critical hydraulic gradient was 1.0, while the values determined by Ahlinhan’s [22] were between 0.7 and 0.97, and those found by Fleshman’s [36,37] were between 1.3 and 1.73. In our study, the gradients varied from 0.51 to 1.36, thus showing that the range of values for the critical hydraulic gradients is relatively large in this case for different soils. In any case, the gradients’ values determined in our study were more accurate than Terzaghi’s, which were reported to be higher than the real ones [6]. In our analysis, we could also determine the influence of soil instability on the critical hydraulic gradients. We concluded that the critical hydraulic gradients of unstable soil were lower than those of stable soil.

#### 4.2. Effects of Other Parameters

_{θk}on the critical gradients are shown in Figure 9. In this figure, the experimental data were obtained from Ahlinhan’s vertical seepage test [22].

_{θk}are inversely correlated to those of the critical gradients. Larger d

_{θk}values indicate that the soil particles are relatively coarse, and the soil is more inhomogeneous and more unstable. This effect is consistent with Skempton and Ahlinhan’s results [9,22].

#### 4.3. Exposure Degree and Particle Initiation Probability

_{15}, the initiation probability of the particles is large, and the variation is drastic. This suggests that the movable particles are mostly smaller than d

_{15}, which, reasonably, represents the largest possible diameter for movable particles. In addition, the initiation probability of the particles is an important factor in the sediment transport studies regarding river dynamics. In the seepage erosion, also the effects of sediment transport need to be determined. Hence, the theoretical model needs to be further developed. The initiation probability determined in this study will allow to develop the theoretical model.

## 5. Conclusions

_{15}. The initiation probability is a factor contributing to the development of sediment transport under seepage flow.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Notations

β | Angle referred to the position of the movable particle |

L_{L} | Arm of lift force |

L_{F} | Arm of seepage force |

F_{n} | Contact force |

i_{c} | Critical hydraulic gradient |

ρ_{S} | Densities of the soil particles |

ρ_{W} | Densities of the water |

d_{θk} | Equivalent diameter |

Δ | Exposure degree |

F_{τ} | Friction force |

d_{15} | Grain diameter for which 15% of the grains by weight are finer |

d_{c,15} | Grain diameter for which 15% of the grains by weight of the coarse soil are finer |

d_{f,85} | Grain diameter for which 85% of the grains by weight of the fine soil are finer |

G | Gravity |

g | Gravity acceleration |

K | Hydraulic conductivity |

i | Hydraulic gradient |

i_{f} | Hydraulic gradient at slopes or dams failure |

i_{boil} | Hydraulic gradient at soil boil |

i_{start} | Hydraulic gradient at soil starting movement |

i_{ini} | Hydraulic gradient at initiation of landslide dams |

P | Initiation probability of particles |

C_{L} | Lift coefficient |

F_{L} | Lift force |

D | Movable particle diameter |

Θ | Parameter refered to the seepage field and the relative exposure degree |

Π | Parameter refered to the soil parameters |

ρ′ | Particle density in water |

n | Porosity of the soil |

f (Δ′) | Probability density function of Δ′ |

R | Radius of the movable particle |

Δ′ | Relative exposure degree |

θ | Seepage direction |

F | Seepage force |

V_{b} | Seepage velocity |

γ_{s} | Specific weight of soil |

γ_{w} | Specific weight of the water |

e | Void ratio of the soil |

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**Figure 4.**The distribution of the critical hydraulic gradients. In I, II, and III Δ′ are 0.05, 0.1. and 0.2, respectively; in (i), (ii), and (iii) e are 0.3, 0.4, and 0.5, respectively.

Soils | K (m/s) | Soils | K (m/s) |
---|---|---|---|

Coarse gravel | 1~2 × 10^{−3} | Clay sand | 2 × 10^{−5}~10^{−6} |

Sandy gravel | 10^{−3}~10^{−4} | Sandy loess | 10^{−5}~10^{−6} |

Grit sand | 5 × 10^{−4}~10^{−4} | Muddy loess | 10^{−7}~10^{−8} |

Silver sand | 5 × 10^{−5}~10^{−5} | Clay | 10^{−8}~10^{−10} |

Predicted i | Measured i | Author |
---|---|---|

0.39 | 0.20 | Skempton and Brogan |

0.48 | 0.34 | |

0.50 | 1.00 | |

0.68 | 1.00 | |

0.60 | 0.70 | Ahlinhan et al. |

0.63 | 0.80 | |

0.71 | 0.90 | |

0.73 | 0.95 | |

0.70 | 0.90 | |

0.72 | 0.96 | |

0.75 | 0.97 | |

0.42 | 0.30 | |

0.43 | 0.36 | |

0.52 | 0.60 | |

0.25 | 0.19 | |

0.24 | 0.21 | |

0.31 | 0.25 | |

0.21 | 0.19 | |

0.23 | 0.21 | |

1.10 | 1.30 | Fleshman and Rice |

0.93 | 1.47 | |

0.91 | 1.40 | |

0.81 | 1.38 | |

1.36 | 1.73 |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/w9110828

**AMA Style**

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(11):828.
https://doi.org/10.3390/w9110828

**Chicago/Turabian Style**

Huang, Zhe, Yuchuan Bai, Haijue Xu, Yufen Cao, and Xiao Hu.
2017. "A Theoretical Model to Predict the Critical Hydraulic Gradient for Soil Particle Movement under Two-Dimensional Seepage Flow" *Water* 9, no. 11: 828.
https://doi.org/10.3390/w9110828