# Numerical Simulation and Experiment Study on the Characteristics of Non-Darcian Flow and Rheological Consolidation of Saturated Clay

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Study on the Permeability Characteristics of Saturated Clay

#### 2.1. Constant-Head Permeability Test Device

#### 2.2. Permeability Test and Data Processing

_{0}= i

_{1}(m − 1)/m, c = k/(mi

_{1}

^{m}

^{−1}); c and k are the coefficient of permeability of the exponential flow at low gradients and the linear flow at high gradients, respectively; m is the exponent of exponential flow at low gradients; i

_{0}is the initial the hydraulic gradient of the linear percolation section; and i

_{1}is the threshold hydraulic gradient for the linear relationship. When i

_{0}= 0 or m = l, Equation (1) degenerates into Darcy’s flow law.

^{m}to obtain the values of c and m. (2) Equation $v=cm{i}_{1}^{m-1}\left[i-{i}_{1}\left(m-1\right)/m\right]$ was obtained by substituting c and m into formula $v=k\left(i-{i}_{0}\right)$, then the g to n experimental points were non-linearly fitted through equation $v=cm{i}_{1}^{m-1}\left[i-{i}_{1}\left(m-1\right)/m\right]$ by Origin software, and the value of i

_{1}was obtained. (3) By substituting the values of c, m, and i

_{1}into the Equation (1), the calculation points of Hansbo’s flow mathematical model were obtained, and the correlation between experimental and the calculation values was compared to obtain maximum R

^{2}. Otherwise, the above methods were repeated at 0 to g + 1 and g + 1 to n experimental points. where e

_{0}is the initial void ratio, k is the permeability coefficient of soil samples, R

^{2}is the correlation coefficient.

^{2}above 0.95. Hansbo’s flow parameter m was also within the ranges suggested by Hansbo [3], Dubin et al. [8], and Sun et al. [11] (between 1 and 2). However, due to the lack of relevant experimental data, there is no unified statement on the range of i

_{0}in current studies, Hansbo [3] stated that i

_{0}ranged from 1 to 4; Dubin et al. [8] reported it was between 3 and 25, and Sun et al. [11] reported it to be between 0.5 and 14, and the results obtained in this paper were basically within the above range, which proves the applicability of Hansbo’s flow model to clay samples.

_{0}is the initial permeability coefficient corresponding to a initial void ratio e

_{0}and C

_{k}is the permeability index.

_{k}= 0.193 was also in the reasonable range of permeation index given by Berry [39] (C

_{k}value between 0.02 and 5.00). Therefore, it was confirmed that Equation (2) is applicable to the clay samples studied in this paper, i.e., logk and the void ratio e were linearly related.

## 3. Experimental Study on Rheological Consolidation Characteristics of Saturated Clay

#### 3.1. One-Dimensional Rheological Consolidation Test

_{0}is the initial void ratio corresponding to the initial thickness H

_{0}of the soil sample.

#### 3.2. One-Dimensional Rheological Consolidation Test Data Processing

_{c}and C

_{s}, and it was found that the average value of C

_{c}for the two samples was 0.42. The values of C

_{c}/C

_{s}for samples LB-1 and LB-2 were 4.32–4.79 and 5.20–7.65, respectively. The corresponding C

_{c}/C

_{s}value of saturated clay was between 4.32 and 7.65, which was also obtained by Li [41] based on the GDS consolidation instrument who found C

_{c}/C

_{s}value of 3.65 to 7.88 for clay samples collected from the Xiaoshan in Zhejiang province. Based on the above analysis, the effectiveness of the improved oedometer proposed in this paper was verified.

## 4. Derivation of Governing Equations

#### 4.1. UH Model Considering Time Effect

_{0}is reciprocal of the over-consolidation ratio (OCR). ${\epsilon}_{\mathrm{v}}^{\mathrm{p}}$ is the plastic strain produced by stress and time, ${{p}^{\prime}}_{\mathrm{c}}$ is the vertical stress at the intersection of rebound and instantaneous compression lines, similar to pre-consolidation pressure.

#### 4.2. One-Dimensional Rheological Consolidation Equation of UH Model

#### 4.3. Discretization of the Governing Equation

_{j}to t

_{j+1}), Equation (15) was integrated in the control volume of bth:

## 5. Verification of the Suitability of UH Model

#### 5.1. Verification of the Experimental Results in this Paper

_{1}= 0. The compression index C

_{c}, swelling index C

_{s}, and secondary consolidation coefficient C

_{α}are taken according to the above test results. Sorensen et al. [43] summarized some clay experimental data on clay and provided the relationship between the internal friction angle φ and plasticity index I

_{P}, as shown in Equation (27). Therefore, φ = 25° can be estimated based on Equation (27) and Table 1 (I

_{P}= 17.1), and then M = 0.984 can be obtained by substituting φ into Equation (8). Moreover, the initial permeability coefficient k

_{0}, permeation index C

_{k}, and initial over-consolidation parameter R

_{0}were selected by trial calculation, as shown in Table 6 and Table 7. Numerical simulation results are shown in Figure 9, which shows that the predicted results from the UH model, considering the time effect, are in good agreement with the experimental results. However, the predicted results of Terzaghi’s theory are quite different from the experimental results in the later stage of loading. The above results fully prove the necessity of considering the soil’s rheological characteristics in predicting foundation settlement in saturated clay areas, and also proved the UH model’s applicability and the necessity of considering variable permeability coefficient.

#### 5.2. Verification of Experimental Results obtained by Li's

_{0}= 0.82, C

_{C}= 0.356, C

_{k}= 0.528, C

_{S}= 0.073, C

_{α}= 0.011, k

_{0}= 2.0 × 10

^{−11}m/s, M = 0.984, and R

_{0}= 0.575. The comparison of theoretical and experimental values based on the proposed model is shown with Figure 11. It can be seen that the results obtained from the UH model considering the variable permeability coefficient are closer to experimental values compared to those from the component model, which fully confirms superiority of the UH model and the necessity of considering the permeability coefficient in similar studies.

## 6. Conclusions

- (1)
- Compared to the traditional falling-head permeability experiment, the improved constant-head experimental device for insignificant amounts of water flow saturated clays requires shorter time and has less experimental error.
- (2)
- The water flow of saturated clay pores deviated from Darcy flow and Hansbo’s flow equations had good applicability to clay samples used in this experiment.
- (3)
- Saturated clay had significant rheological properties. The predicted results of Terzaghi’s theory were quite different from the experimental results, and the UH constitutive model agreed well with experimental results and had good applicability in predicting the rheological deformation of saturated clay samples.

## 7. Patents

- Zhongyu Liu; Yangyang Xia; Jiachao Zhang; Xinmu Zhu; Jiandong Wei. A device for preparing saturated remolded clay ring knife sample. China Patent, CN201,721,820,336.8, 2018-08-07.
- Jiandong Wei; Yangyang Xia; Zhongyu Liu; Jiachao Zhang; Xinmu Zhu. A triple saturated clay constant-head permeability test device. China Patent, CN201,821,335,366.4, 2019-04-12.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**Comparison of theoretical and experimental settlement values for samples LB-1and LB-2. CUHM: unified hardening model with constant permeability coefficient, VUHM: unified hardening model with variable permeability coefficient.

**Figure 10.**Schematic diagram of rheological model (

**a**): three-element rheological model, (

**b**) Four-element rheological model. where E

_{0}and E

_{1}are the moduli of the independent spring and the spring of the Kelvin–Voigt body, respectively; η

_{0}and η

_{1}are the viscosity coefficient of the independent dashpot and the dashpot of the Kelvin–Voigt body, respectively.

**Figure 11.**Comparison of prediction values and experimental results of the settlement theory under different models.

Specific Gravity, G_{s} | Liquid Limit, w_{L} (%) | Plastic Limit, w_{P} (%) | Plastic Index, I_{P} (%) |
---|---|---|---|

2.71 | 39.4 | 22.3 | 17.1 |

Soil Sample Number | e_{0} | k (cm/s) | i_{0} | m | R^{2} |
---|---|---|---|---|---|

SL-1 | 1.26 | 6.73 × 10^{−8} | 22.20 | 1.45 | 0.985 |

SL-2 | 1.29 | 1.28 × 10^{−7} | 23.54 | 1.58 | 0.978 |

SL-3 | 1.36 | 2.31 × 10^{−7} | 27.81 | 1.63 | 0.964 |

SL-4 | 1.44 | 6.85 × 10^{−7} | 31.25 | 1.81 | 0.972 |

SL-5 | 1.47 | 8.95 × 10^{−7} | 24.37 | 1.64 | 0.967 |

SL-6 | 1.50 | 1.29 × 10^{−6} | 30.41 | 1.83 | 0.956 |

_{0}is the initial void ratio , k is the permeability coefficient of soil samples, R

^{2}is the correlation coefficient.

Sample Number | Specific Gravity, G_{s} | Bulk Density, ρ (g/cm^{3}) | Water Content, w (%) | Void Ratio, e_{0} |
---|---|---|---|---|

LB-1 | 2.71 | 1.738 | 47.77 | 1.30 |

LB-2 | 2.71 | 1.864 | 34.14 | 0.95 |

Order | Loading (kPa) |
---|---|

First loading | 0-25-50-100-200 |

First Unloading | 200-100-50 |

Second Loading | 50-100-200-400 |

Second Unloading | 400-200-100-50 |

Third Loading | 50-100-200-400-800 |

Third Unloading | 800-400-200-100-50 |

Fourth Loading | 50-100-200-400-800-1600 |

**Table 5.**Fitting results of secondary consolidation coefficient C

_{α}under primary loading conditions.

Soil Sample Number | Loading (kPa) | ||||||
---|---|---|---|---|---|---|---|

25 | 50 | 100 | 200 | 400 | 800 | 1600 | |

LB-1 | 0.0040 | 0.0050 | 0.0120 | 0.0110 | 0.0091 | 0.0085 | 0.0080 |

LB-2 | 0.0009 | 0.0015 | 0.0030 | 0.0092 | 0.0135 | 0.0208 | 0.0134 |

Loading Interval/(kPa) | H/mm | e_{0} | C_{C} | C_{k} | C_{S} | C_{α} | R_{0} | k_{0}/(m/min) |
---|---|---|---|---|---|---|---|---|

50–100 | 18.99 | 1.184 | 0.42 | 0.66 | 0.100 | 0.0120 | 0.65 | 1.2 × 10^{−8} |

100–200 | 18.01 | 1.070 | 0.0110 | 0.92 | 0.70 × 10^{−8} | |||

200–400 | 16.63 | 0.913 | 0.0091 | 0.83 | 0.40 × 10^{−8} | |||

400–800 | 15.44 | 0.776 | 0.0085 | 0.79 | 0.25 × 10^{−8} | |||

800–1600 | 14.29 | 0.644 | 0.0080 | 0.76 | 0.12 × 10^{−8} |

Loading Interval/(kPa) | H/mm | e_{0} | C_{C} | C_{k} | C_{S} | C_{α} | R_{0} | k_{0}/(m/min) |
---|---|---|---|---|---|---|---|---|

50–100 | 1.988 | 0.940 | 0.42 | 0.66 | 0.065 | 0.0030 | 0.58 | 1.0 × 10^{−8} |

100–200 | 1.959 | 0.911 | 0.0092 | 0.53 | 0.85 × 10^{−8} | |||

200–400 | 1.899 | 0.852 | 0.0135 | 0.65 | 0.39 × 10^{−8} | |||

400–800 | 1.789 | 0.745 | 0.0190 | 0.60 | 0.19 × 10^{−8} | |||

800–1600 | 1.512 | 0.622 | 0.0135 | 0.66 | 0.06 × 10^{−8} |

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**MDPI and ACS Style**

Liu, Z.; Xia, Y.; Shi, M.; Zhang, J.; Zhu, X.
Numerical Simulation and Experiment Study on the Characteristics of Non-Darcian Flow and Rheological Consolidation of Saturated Clay. *Water* **2019**, *11*, 1385.
https://doi.org/10.3390/w11071385

**AMA Style**

Liu Z, Xia Y, Shi M, Zhang J, Zhu X.
Numerical Simulation and Experiment Study on the Characteristics of Non-Darcian Flow and Rheological Consolidation of Saturated Clay. *Water*. 2019; 11(7):1385.
https://doi.org/10.3390/w11071385

**Chicago/Turabian Style**

Liu, Zhongyu, Yangyang Xia, Mingsheng Shi, Jiachao Zhang, and Xinmu Zhu.
2019. "Numerical Simulation and Experiment Study on the Characteristics of Non-Darcian Flow and Rheological Consolidation of Saturated Clay" *Water* 11, no. 7: 1385.
https://doi.org/10.3390/w11071385