# A Method to Estimate Dynamic Pore Water Pressure Growth of Saturated Sand-Gravel Materials

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## Abstract

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## 1. Introduction

_{L}for most pore water pressure models is not standardized, even though most use the number of cycles required to cause liquefaction N

_{L}to normalize the cyclic loading numbers N. Secondly, compared with sand, there is not sufficient liquefaction test data on sand-gravel materials, and the pore water pressure growth model needs to be further verified. Studying the liquefaction characteristics of sand-gravel materials is, therefore, necessary to provide a basis for the analysis of the liquefaction resistance of dam foundations and dam bodies.

_{L}that considers the relative density ${D}_{\mathrm{r}}$. The pore water pressure growth model of saturated sand-gravel materials was established and its applicability was verified.

_{L}. Section 5 introduces the pore water pressure growth model for saturated sand-gravel materials. Section 6 provides the conclusions.

## 2. Test Design

#### 2.1. Test Apparatus and Sample Preparation

#### 2.2. Test Conditions

_{r}of 0.9 were consolidated under pressures of 50, 100, 200, and 300 kPa. A sine wave with a loading frequency of 0.33 Hz was employed to apply dynamic stress through vibration until failure occurred. Failure was defined as either an axial strain of 5% or excess pore water pressure reaching the confining pressure. This was performed to study the dynamic characteristics of the sand-gravel materials under different initial effective confining pressures ${\sigma}_{3\mathrm{c}}$, consolidation stress ratios ${K}_{\mathrm{c}}$, and stress levels. Table 3 lists static and dynamic stress control conditions of this test. In dynamic triaxial tests, initial dynamic stress values are determined based on empirical experience. Once the samples meet the failure criteria, ${\sigma}_{\mathrm{d}}$ for each condition is recorded in Table 3. In general engineering practice, the minimum principal stress of dam materials is estimated based on the dam height, and the maximum experimental confining pressure is set to exceed this value. This research aims to investigate the dynamic pore pressure of sand-gravel materials under low confining pressures; therefore, confining pressures ${\sigma}_{3\mathrm{c}}$ are set at 50, 100, 200, and 300 kPa. Most natural soils are in an anisotropic consolidation state; thus, soil strength parameters derived from such conditions are more reflective of real-world scenarios. Accordingly, the consolidation stress ratios, ${K}_{\mathrm{c}}$, are set at 1.5 and 2.0.

## 3. Analysis of Dynamic Triaxial Test Results of Saturated Sand-Gravel Material

#### 3.1. Dynamic Pore Water Pressure Curve

#### 3.2. Relationship between the Strain and the Cyclic Numbers

#### 3.3. Effective Stress Path

## 4. The Estimation Model of the Number of Cycles Required to Reach Liquefaction, N_{L}

#### 4.1. The Verification of N_{L}

_{L}is the number of cycles required to reach liquefaction. N

_{L}is mostly obtained from the liquefaction test curve or empirical formula, such as that proposed by Xu [30]:

_{L}and ${\sigma}_{\mathrm{d}}$ can be established. It has been widely used in the field of engineering practice [30,31,32,33]. Figure 9 verifies the accuracy of the empirical formula by using the Chen et al. [34] test data. This empirical formula provides the relationship curves of the different groups of test materials, and the goodness of fit is greater than 0.99. This formula can, however, not consider the influence of other factors, such as relative density, on the cyclic numbers required to reach liquefaction. The number of cycles required to cause liquefaction is often related to various factors such as the soil’s relative density, porosity, consolidation ratio, and initial effective consolidation stress. Therefore, this empirical formula must be corrected to consider these factors.

#### 4.2. The Modification of N_{L}

_{L}. The influence of the relative density on the dynamic strength of soil is clear, in that the dynamic strength of the material increases with an increase in the relative density [35]. The empirical formula proposed by Xu [30] reflects the liquefaction of materials when the dynamic stress is near zero. At extremely small amplitudes, the value of a is associated with the relative density of the material and the number of cycles required to induce liquefaction. The larger the relative density, the larger the a value and the number of cycles required to cause liquefaction. Based on the relationship between a and D

_{r}[19,27,34,36,37,38,39,40,41,42,43,44,45,46,47], the a value is distributed within a curve band. This implies that the mean curve of the relationship between a and D

_{r}can be calculated, as depicted in Figure 10.

_{L}estimation model so that the empirical formula of Xu [30] can be modified by the relationship between the a value and the D

_{r}mean curve. The expressions are given by the following:

_{L}is the number of cycles required to reach liquefaction, a is related to the relative density of the test material, b is the empirical constant, $\overline{\tau}$ is the mean shear stress, ${\sigma}_{0}^{\prime}$ is the initial normal stress on the 45° slope of the sample, and D

_{r}is the relative density.

## 5. Estimation of the Dynamic Pore Water Pressure of the Saturated Sand-Gravel Material

_{u}and b

_{u}are test constants related to the soil properties, ${u}_{\mathrm{d}}$ refers to excess pore water pressure, ${\sigma}_{3\mathrm{c}}$ is the initial effective confining pressure, N is the number of dynamic cycles, and N

_{L}is the number of cycles required to reach liquefaction. In practical engineering, when the axial strain reaches a certain value, the soil can be judged to be liquefied. Therefore, a

_{u}and b

_{u}are identified as independent parameters in this paper.

## 6. Conclusions

- (1)
- Dislocation and slip occur between the grains of the saturated sand-gravel material under cyclic loading. This causes both reversible and irreversible two-part volume strains, causing an increase in the periodic fluctuation of the dynamic pore water pressure. The increase in the dynamic pore water pressure is more obvious and the amplitude is larger due to the large pores between the sand-gravel materials. The dense saturated sand-gravel material with its large consolidation stress ratio shows a small volume shrinkage and a large volume expansion, where dilatancy can inhibit the increase in the pore water pressure in the sand-gravel.
- (2)
- During cyclic loading, volume expansion or shrinkage occurs alternately in the saturated sand-gravel material. The mean effective stress of the sand-gravel material decreases, causing the dynamic shear modulus of the material to decrease and the dynamic strain of the material to increase under the same dynamic stress conditions.
- (3)
- Even if the relative density of the sand-gravel material reaches 90%, there is still the possibility of liquefaction under dynamic loading. Earth-rock dams that are built on sand-gravel overburdens and earth-rock dams filled with sand-gravel materials in earthquake-prone areas should use our proposed analysis and evaluation method to investigate the possibility of sand-gravel liquefaction. We used test data and considered the influence of relative density to modify the existing empirical formula and provide an evaluation method for the number of cycles required to reach liquefaction. We propose a pore water pressure growth model that is suitable for both sand-gravel and sandy materials. Comparisons and validations demonstrated that the proposed model accurately describes the three types of pore water pressure growth processes during liquefaction. The pore pressure growth model proposed in this paper can be used to analyze the liquefaction resistance of the dam foundations and dam structure of high earth-rock dams and high sand-gravel dams located on deep overburdens.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Pore water pressure growth curves of saturated sand-gravel. (

**a**) S2 pore water pressure; (

**b**) S4 pore water pressure; (

**c**) S7 pore water pressure.

**Figure 6.**Upper and lower envelope lines of saturated sand-gravel: (

**a**) saturated sand-gravel upper envelope line; (

**b**) saturated sand-gravel lower envelope line.

**Figure 7.**Relationship between sand-gravel material strain and cyclic numbers. (

**a**) S2 and S12; (

**b**) S7 and S17.

**Figure 8.**Mean effective stress paths of sand-gravel. (

**a**) S1 mean effective stress path; (

**b**) S2 mean effective stress path; (

**c**) S4 mean effective stress path; (

**d**) S7 mean effective stress path.

**Figure 11.**Relationship between $\mathrm{lg}{N}_{\mathrm{L}}$ and $\overline{\tau}/{\sigma}_{0}^{\prime}$.

**Figure 12.**Pore pressure ratio fitting curve of Xinjiang sand-gravel material. (

**a**) Confining pressure is 50 or 100 kPa; (

**b**) confining pressure is 200 or 300 kPa.

Pore Water Pressure Model | Equation | Explanation | Reference |
---|---|---|---|

Stress model | $\frac{{u}_{\mathrm{d}}}{{\sigma}_{3\mathrm{c}}}}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\mathsf{\pi}}}\mathrm{arcsin}\left[2{\left({\displaystyle \frac{N}{{N}_{\mathrm{L}}}}\right)}^{{\displaystyle \frac{1}{\theta}}}-1\right]$ | ${u}_{\mathrm{d}}$ is the dynamic pore water pressure; ${\sigma}_{3\mathrm{c}}$ is the initial confining pressure; ${N}_{\mathrm{L}}$ is the number of cycles required to cause liquefaction; $\theta $ is the experimental constant related to the type of material. | Seed [6] |

Strain model | $\left\{\begin{array}{l}\Delta u={\overline{E}}_{r}\Delta {\epsilon}_{vd}\\ \Delta {\epsilon}_{vd}={c}_{1}\left(\gamma -{c}_{2}{\epsilon}_{vd}\right)+{\displaystyle \frac{{c}_{3}{\epsilon}_{vd}^{2}}{\gamma +{c}_{4}{\epsilon}_{vd}}}\end{array}\right.$ | ${\overline{E}}_{r}$ is the tangent modulus of the one-dimensional unloading curve at a point corresponding to the initial vertical effective stress; ${c}_{1},{c}_{2},{c}_{3},{c}_{4}$ are test constants. | Martin et al. [11] |

Transient model | $\begin{array}{l}u(t+\Delta t)=u(t)+\Delta u\\ =u(t)+\Delta {u}_{\sigma}+\Delta {u}_{0}+\Delta {u}_{\mathrm{T}}\\ =u(t+\Delta t)*+\Delta {u}_{\mathrm{T}}\end{array}$ | $\Delta {u}_{\sigma},\Delta {u}_{0},\Delta {u}_{\mathrm{T}}$ are increments of the pore water pressure during $\Delta T$; $u{(t+\Delta t)}^{*}$ is the pore water pressure without dissipation at $(t+\Delta t)$; $u(t+\Delta t)$ is the dissipation pore water pressure at $(t+\Delta t)$. | Xie [12] |

Energy model | $\left\{\begin{array}{l}{\displaystyle \frac{u}{{\sigma}_{0}}}=K{W}_{R}^{\beta}\\ {W}_{R}=\left[1-\mathrm{lg}\left({K}_{c}^{3}\right)\right]{W}_{0}\\ {W}_{0}={\displaystyle \frac{{\displaystyle \sum W}}{{\sigma}_{0}}}\end{array}\right.$ | $\sum W$ is the dissipated energy; $K$ and $\beta $ are test constants. | Cao et al. [13] |

Endochronic theory | $\frac{{u}_{d}}{{\sigma}_{v}^{\prime}}}={\displaystyle \frac{A}{B}}\mathrm{ln}(1+Bk)$ | K is the damage parameter; the variable includes the shear strain amplitude and number of cycles. A and B are test parameters. | Finn [14] |

Effective stress path model | $\left\{\begin{array}{l}q/{p}^{\prime}\ge {\left(q/{p}^{\prime}\right)}_{\mathrm{max}},\mathrm{loading}\\ q/{p}^{\prime}{\left(q/{p}^{\prime}\right)}_{\mathrm{max}},\mathrm{unloading}\end{array}\right.$ | Under loading conditions, additional plastic shear deformation occurs in the soil. The variation of pore water pressure is equal to the variation of the mean effective stress. | Ishihara [15] |

Sand-Gravel Materials in Xinjiang | Mean Grain Sized_{50}/mm | Specific Gravity G_{s} | Minimum Dry Densityρ_{min}/g·cm^{−3} | Maximum Dry Densityρ_{max}/g·cm^{−3} | Relative DensityD_{r} | Dry Densityρ_{d}/g·cm^{−3} |

14 | 2.75 | 1.85 | 2.34 | 0.9 | 2.28 |

Test Number | K_{c} | σ_{3c} (kPa) | σ_{d} (kPa) | $\mathbf{C}\mathbf{S}\mathbf{R}$ |
---|---|---|---|---|

S1 | 1.5 | 50 | 70.50 | 0.56 |

S2 | 100 | 142.90 | 0.57 | |

S3 | 121.25 | 0.49 | ||

S4 | 200 | 291.45 | 0.58 | |

S5 | 245.25 | 0.49 | ||

S6 | 227.65 | 0.46 | ||

S7 | 300 | 438.45 | 0.58 | |

S8 | 393.15 | 0.52 | ||

S9 | 346.30 | 0.46 | ||

S10 | 2.0 | 50 | 95.10 | 0.63 |

S11 | 100 | 190.95 | 0.64 | |

S12 | 175.75 | 0.59 | ||

S13 | 200 | 387.65 | 0.65 | |

S14 | 342.25 | 0.57 | ||

S15 | 296.75 | 0.49 | ||

S16 | 300 | 582.80 | 0.65 | |

S17 | 519.35 | 0.58 | ||

S18 | 447.15 | 0.50 |

**Table 4.**The results for the ${N}_{\mathrm{L}}$ and ${u}_{\mathrm{d}}$ tests on saturated sand-gravel materials.

Test Number | K_{c} | σ_{3c} (kPa) | u_{d} (kPa) | N_{L} |
---|---|---|---|---|

S1 | 1.5 | 50 | 49.93 | 174.4 |

S2 | 100 | 99.95 | 65.4 | |

S3 | 100.01 | 147.5 | ||

S4 | 200 | 197.1 | 42.35 | |

S5 | 198.54 | 74.3 | ||

S6 | 199.79 | 153.3 | ||

S7 | 300 | 295.87 | 30.37 | |

S8 | 298.07 | 43.35 | ||

S9 | 299.94 | 84.3 | ||

S10 | 2.0 | 50 | 49.24 | 145.3 |

S11 | 100 | 95.3 | 44.3 | |

S12 | 95.06 | 105.25 | ||

S13 | 200 | 187.77 | 23.3 | |

S14 | 193.42 | 35.25 | ||

S15 | 195.93 | 78.25 | ||

S16 | 300 | 280.62 | 18.25 | |

S17 | 282.71 | 32.2 | ||

S18 | 281.03 | 73.2 |

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**MDPI and ACS Style**

Chen, J.; Fu, Z.; Chen, S.; Shi, B.
A Method to Estimate Dynamic Pore Water Pressure Growth of Saturated Sand-Gravel Materials. *Appl. Sci.* **2024**, *14*, 7909.
https://doi.org/10.3390/app14177909

**AMA Style**

Chen J, Fu Z, Chen S, Shi B.
A Method to Estimate Dynamic Pore Water Pressure Growth of Saturated Sand-Gravel Materials. *Applied Sciences*. 2024; 14(17):7909.
https://doi.org/10.3390/app14177909

**Chicago/Turabian Style**

Chen, Jinyi, Zhongzhi Fu, Shengshui Chen, and Beixiao Shi.
2024. "A Method to Estimate Dynamic Pore Water Pressure Growth of Saturated Sand-Gravel Materials" *Applied Sciences* 14, no. 17: 7909.
https://doi.org/10.3390/app14177909