# Estimation of Unsaturated Hydraulic Conductivity of Granular Soils from Particle Size Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Estimation of Saturated Hydraulic Conductivity

#### 2.1. Estimation Equations Based on Grain Size Parameters

^{2}) and $\mathsf{\nu}$ is the fluid kinematic viscosity (m

^{2}/s) ($\mathsf{\nu}=0.89\times {10}^{-6}{\mathrm{m}}^{2}/\mathrm{s}$ at $25{}^{\xb0}\mathrm{C}$ for water). Empirically, ${C}_{H}$ is a unitless coefficient about $6.54\times {10}^{-4}$ [13]. Meter can be used as the length unit in this equation to keep the unit consistency (to obtain hydraulic conductivity value in m/s). Furthermore, the effect of particle size uniformity is considered in another equation proposed by Beyer [5], which can be written as:

^{−5}m/s to 2 × 10

^{−3}m/s. From the dimensional analysis, the above equation already counts as part of the porosity effect as porosity has an intrinsic correlation with particle size uniformity ${C}_{u}$. Following Vukovic and Soro [17], the intrinsic porosity of a sand is a function of its ${C}_{u}$ empirically as:

#### 2.2. Applicability and Validity of the Estimation Equation

## 3. Prediction of Unsaturated Relative Permeability

#### 3.1. Van Genuchten’s Closed-Form Equation

#### 3.2. Prediction of Van Genuchten’s Parameters from Particle Size Distribution

#### 3.3. Effect of Porosity Variation on the Air Entry Value

## 4. Verification of the Estimation Model

#### 4.1. The Effect of Porosity on Predictions of Water Retention Curve (WRC) and Relative Permeability

#### 4.2. Verification on a Set of Field Test Data by Instantaneous Profile Method

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Prediction on saturated hydraulic conductivity of different models on soils from the UNSODA database. (

**a**) Hazen model; (

**b**) Beyer model; (

**c**) Kozeny–Carman model; (

**d**) Chapuis model; (

**e**) model of Equation (6); (

**f**) model of Equation (8).

**Figure 2.**Sketch of the fine content effect which may change the pore size distribution to a dual-structure.

**Figure 3.**Prediction of different models on sandy soils with ${d}_{10}$ > 0.02 mm and ${C}_{u}$ < 20 from the UNSODA database. (

**a**) Hazen model; (

**b**) Beyer model; (

**c**) Kozeny–Carman model; (

**d**) Chapuis model; (

**e**) model of Equation (6); (

**f**) model of Equation (8).

**Figure 4.**Effect of initial porosity on air-entry value: relationship between porosity variation and logarithm scale air-entry value difference.

**Figure 5.**Prediction performance of parameter $\alpha $ by the original method and the corrected method. (

**a**) original method (Equation (15)); (

**b**) corrected method (Equation (19)).

**Figure 7.**Prediction performance of coarse and medium Berlin sands. (

**a**) water retention curve of Berlin coarse sand; (

**b**) relative permeability of Berlin coarse sand; (

**c**) water retention curve of Berlin medium sand; (

**d**) relative permeability of Berlin medium sand.

**Figure 8.**Comparison between predicted and measured parameters for Berlin sands. (

**a**) parameter $\alpha $; (

**b**) parameter $n$.

**Figure 9.**Comparison of model performance between using the original estimation of $\alpha $ (Equation (15)) and using the corrected estimation of $\alpha $ (Equation (19)) (experimental measurements are based on the instantaneous profile method).

**Figure 10.**Comparisons between the corrected model predictions and instantaneous profile method measured results of relative permeability of sands in the UNSODA database. (

**a**) sands 1023 and 1024; (

**b**) sands 1241 and 2105; (

**c**) sands 3134 and 3162; (

**d**) sands 3163 and 3164.

**Table 1.**Soil gradation parameters, porosity values and best fitted water retention curve (WRC) parameters of $\alpha $ and $n$ for the 18 sandy soils.

Sample ID * | ${\mathit{d}}_{10}$ (mm) | ${\mathit{d}}_{30}$ (mm) | ${\mathit{d}}_{60}$ (mm) | ${\mathit{C}}_{\mathit{u}}$ | ϕ | Fitted Parameters | Goodness of Fitting | |||
---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}\left(\mathbf{kPa}\right)$ | $\mathit{n}$ | SSE | RMSE | R^{2} | ||||||

1011 | 0.00946 | 0.10699 | 0.15511 | 19.743 | 0.43 | 0.43 | 2.75 | 0.016 | 0.047 | 0.989 |

1014 | 0.02078 | 0.11454 | 0.16293 | 9.350 | 0.45 | 0.94 | 2.66 | 0.007 | 0.027 | 0.995 |

1461 | 0.21825 | 0.30949 | 0.43887 | 2.395 | 0.37 | 9.47 | 3.70 | 0.050 | 0.085 | 0.933 |

1462 | 0.12691 | 0.23000 | 0.30867 | 2.818 | 0.43 | 5.70 | 3.43 | 0.037 | 0.068 | 0.955 |

1463 | 0.12733 | 0.23915 | 0.31552 | 2.846 | 0.40 | 6.21 | 3.65 | 0.025 | 0.056 | 0.970 |

1464 | 0.10089 | 0.14356 | 0.20548 | 2.552 | 0.37 | 6.15 | 3.13 | 0.049 | 0.078 | 0.950 |

1465 | 0.02491 | 0.07375 | 0.10463 | 5.000 | 0.38 | 2.08 | 1.88 | 0.005 | 0.025 | 0.996 |

1466 | 0.05631 | 0.07855 | 0.09897 | 2.034 | 0.41 | 5.44 | 4.56 | 0.009 | 0.034 | 0.993 |

1467 | 0.02932 | 0.20852 | 0.31649 | 13.299 | 0.31 | 2.13 | 1.58 | 0.011 | 0.037 | 0.989 |

3330 | 0.04041 | 0.20388 | 0.28925 | 8.526 | 0.42 | 1.62 | 1.65 | 0.024 | 0.069 | 0.971 |

3331 | 0.11858 | 0.22780 | 0.29709 | 2.861 | 0.44 | 4.53 | 2.58 | 0.026 | 0.072 | 0.975 |

3332 | 0.20284 | 0.25656 | 0.32451 | 1.799 | 0.43 | 7.78 | 3.48 | 0.014 | 0.054 | 0.987 |

3340 | 0.12617 | 0.18315 | 0.26612 | 2.549 | 0.46 | 3.95 | 2.26 | 0.086 | 0.055 | 0.973 |

4523 | 0.12133 | 0.16532 | 0.21988 | 2.106 | 0.41 | 8.83 | 7.04 | 0.072 | 0.081 | 0.969 |

4650 | 0.07221 | 0.23130 | 0.31953 | 5.201 | 0.38 | 2.20 | 2.01 | 0.032 | 0.037 | 0.992 |

4651 | 0.08383 | 0.22687 | 0.32525 | 4.646 | 0.38 | 1.95 | 2.01 | 0.029 | 0.036 | 0.992 |

4660 | 0.06469 | 0.21709 | 0.30134 | 5.488 | 0.46 | 0.45 | 1.48 | 0.036 | 0.039 | 0.986 |

4661 | 0.07221 | 0.22944 | 0.31132 | 5.022 | 0.43 | 0.79 | 1.74 | 0.015 | 0.026 | 0.995 |

^{2}: coefficient of determination. *: numbered IDs are from the UNSODA database.

**Table 2.**Soil gradation parameters, porosity values and best fitted WRC parameters ($\alpha $ and $n$) for the three sandy soils which are employed for the model verification.

Soil Type * | Sample ID ** | ${\mathit{d}}_{10}$ (mm) | ${\mathit{d}}_{30}$ (mm) | ${\mathit{d}}_{60}$ (mm) | ${\mathit{C}}_{\mathit{u}}$ | ϕ | Fitted Parameters | Goodness of Fitting | |||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}\left(\mathbf{kPa}\right)$ | $\mathit{n}$ | SSE | RMSE | R^{2} | |||||||

Wagram sand | 1140 | 0.051 | 0.147 | 0.25 | 4.9 | 0.428 | 3.752 | 3.657 | 0.011 | 0.031 | 0.995 |

1141 | 0.336 | 4.318 | 3.340 | 0.024 | 0.044 | 0.989 | |||||

1142 | 0.272 | 4.889 | 2.881 | 0.023 | 0.043 | 0.989 | |||||

Berlin coarse sand | 1460 | 0.217 | 0.308 | 0.522 | 2.4 | 0.297 | 5.510 | 8.236 | 3.950 | 0.703 | 0.444 |

1461 | 0.373 | 3.123 | 3.702 | 0.050 | 0.085 | 0.933 | |||||

Berlin medium sand | 1462 | 0.127 | 0.235 | 0.360 | 2.8 | 0.43 | 3.233 | 3.424 | 0.037 | 0.068 | 0.955 |

1463 | 0.399 | 3.514 | 3.654 | 0.025 | 0.056 | 0.970 |

^{2}: coefficient of determination. *: Soil gradation parameters are average values for each soil. **: numbered IDs are from the UNSODA database.

Sample ID | ${\mathit{d}}_{10}$ (mm) | ${\mathit{d}}_{30}$ (mm) | ${\mathit{d}}_{50}$ (mm) | ${\mathit{d}}_{60}$ (mm) | ${\mathit{d}}_{90}$ (mm) | ${\mathit{C}}_{\mathit{u}}$ |
---|---|---|---|---|---|---|

1014 | 0.021 | 0.115 | 0.163 | 0.194 | 0.469 | 9.35 |

1023 | 0.125 | 0.555 | 0.713 | 0.808 | 1.473 | 6.48 |

1024 | 0.115 | 0.515 | 0.665 | 0.755 | 1.347 | 6.59 |

1241 | 0.237 | 0.415 | 0.598 | 0.689 | 1.252 | 2.90 |

2105 | 0.022 | 0.106 | 0.161 | 0.198 | 0.430 | 8.83 |

3134 | 0.107 | 0.163 | 0.250 | 0.289 | 0.444 | 2.70 |

3162 | 0.031 | 0.085 | 0.132 | 0.160 | 0.358 | 5.11 |

3163 | 0.051 | 0.087 | 0.129 | 0.154 | 0.289 | 2.99 |

3164 | 0.052 | 0.091 | 0.135 | 0.161 | 0.338 | 3.09 |

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**MDPI and ACS Style**

Wang, J.-P.; Zhuang, P.-Z.; Luan, J.-Y.; Liu, T.-H.; Tan, Y.-R.; Zhang, J. Estimation of Unsaturated Hydraulic Conductivity of Granular Soils from Particle Size Parameters. *Water* **2019**, *11*, 1826.
https://doi.org/10.3390/w11091826

**AMA Style**

Wang J-P, Zhuang P-Z, Luan J-Y, Liu T-H, Tan Y-R, Zhang J. Estimation of Unsaturated Hydraulic Conductivity of Granular Soils from Particle Size Parameters. *Water*. 2019; 11(9):1826.
https://doi.org/10.3390/w11091826

**Chicago/Turabian Style**

Wang, Ji-Peng, Pei-Zhi Zhuang, Ji-Yuan Luan, Tai-Heng Liu, Yi-Ran Tan, and Jiong Zhang. 2019. "Estimation of Unsaturated Hydraulic Conductivity of Granular Soils from Particle Size Parameters" *Water* 11, no. 9: 1826.
https://doi.org/10.3390/w11091826