# A Simple Fractal-Based Model for Soil-Water Characteristic Curves Incorporating Effects of Initial Void Ratios

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{a}), while the fractal dimension (D) could be assumed to be constant. As a result, in contrast to the complexity of existing models, a simple and efficient model with only two parameters (i.e., D and ψ

_{a}) was established for predicting the SWCC considering the effects of initial void ratio. The procedure for determining the model parameters with clear physical meaning were then elaborated. The applicability and accuracy of the proposed model were well demonstrated by comparing its predictions with four sets of independent experimental data from the tests conducted in current work, as well as the literature on a wide range of soils, including Wuhan Clay, Hefei and Guangxi expansive soil, Saskatchewan silt, and loess. Good agreements were obtained between the experimental data and the model predictions in all of the cases considered.

## 1. Introduction

## 2. Proposed Model

#### 2.1. Fractal Description of a Soil

_{min}and r

_{max}, in which r

_{min}and r

_{max}are the smallest and largest pores, respectively, the probability density function of pore size r is written as [39]

_{min}is near zero. Then, the total volume V (≤r) of pores having size less than or equal to r can be expressed as

_{v}is a pore volume shape-related constant corresponding to the volume of the pores. Assuming that pores having a size less than or equal to r are fully filled with water, then the gravimetric water content in the pores of soil particles weighing 1 g is

_{w}is the mass density of water.

_{max}are filled with water. Then, substituting r with r

_{max}in Equation (3) yields the following expression

_{s}is gravimetric water content of the soil in saturated condition.

#### 2.2. Fractal-Based Model for Variation of Soil-Water Characteristic Curve (SWCC) with Initial Void Ratio

_{s}is the surface tension and α is the contact angle. In the constant temperature condition, 2T

_{s}cosα can be assumed as a constant. The matrix suction ψ corresponding to maximum pore size can be approximately regarded as the air-entry value ψ

_{a}, which can be captured by substituting r with r

_{max}in Equation (5)

_{a}. If matrix suction ψ is less than ψ

_{a}, the soil sample is assumed to be fully saturated. Then, the fractal model for soil-water characteristic curve is written as

_{s}is the relative density and e refers to the initial void ratio.

#### 2.3. Determination of Model Parameters

_{0}, while the SWCCs at the deformed state refer to the experimental soil samples with the other arbitrary initial void ratio e

_{1}(e

_{1}< e

_{0}). As observed in Equation (12), the prediction of SWCCs at arbitrary initial void ratio is mainly governed by only two parameters: (i) fractal dimension D and (ii) air-entry value ψ

_{a}at the deformed state. The procedure for determining the model parameters (i.e., D and ψ

_{a}) are now elaborated as follows.

#### 2.3.1. Fractal Dimension at Reference State

#### 2.3.2. Fractal Dimension at Deformed State

_{0}, e

_{1}, and e

_{2}(i.e., a-b-c-d, e-b-c-d, and f-c-d) almost overlap at the tail. It should be noted that the SWCC is expressed by gravimetric water content herein. As a result, the fractal dimension D

_{1}for e

_{1}at deformed state is approximatively equal to the fractal dimension D

_{0}for e

_{0}at reference state. That is, the fractal dimension D of saturated soil at deformed state can be assumed as a constant (i.e., D

_{1}= D

_{0}). It should be highlighted that similar ideas have been presented by Bird et al. [34] and Russell and Buzzi [41].

#### 2.3.3. Air-Entry Value at Reference State

_{0}can be calculated by following the procedure described above, and the air-entry value ψ

_{a}

_{0}can be determined by best fitting Equation (12) to the experimental SWCCs. Then, the SWCC at reference state can be expressed as

#### 2.3.4. Air-Entry Value at Deformed State

_{1}/G

_{s}, the horizontal line would have an intersection with the SWCC at reference state. The abscissa of this intersection can be approximately regarded as ψ

_{a}

_{1}corresponding to e

_{1}, as shown in Figure 1. By substituting w = e

_{1}/G

_{s}into the first formula in Equation (14), the following expression is obtained

_{a}

_{1}for e

_{1}.

## 3. Model Validation

#### 3.1. Wuhan Clay

^{3}. The specific test procedure is as follows: (1) The sample with different dry densities, together with the HAE ceramic disc, was saturated. (2) The specimen, together with the stainless steel cutting rings, was placed on the HAE ceramic disc in the pressure cell. (3) The applied air pressure was imposed on the specimen when the pressure cell was sealed. (4) The water drained from the specimens was recorded during the whole process of the test. It was assumed to reach the equilibrium state at the current suction level when the water drainage of specimen was constant, then the next suction level would be imposed. (5) At the end of each suction level step, the drainage valve was closed and then the applied air pressure was released. Meanwhile, the weights of specimens needed to be measured. (6) The above-mentioned procedure was repeated until the whole test was completed.

_{0}= 1.115) was regarded as the reference state. Following the procedure presented previously, the fractal dimension for the Wuhan Clay at reference state was determined as D

_{0}= 2.869, with a fitting correlation coefficient of up to 0.99, while the air-entry value at reference state was estimated to be ψ

_{a}

_{0}= 1.66 kPa by fitting Equation (12) to the experimental SWCCs, as shown in Figure 3 and Figure 4, respectively.

_{a}of Wuhan Clay at deformed state were determined using Equation (16), as shown in Table 2.

_{0}= 1.037, 0.964, 0.897, 0.833, 0.719, and 0.613, respectively).

#### 3.2. Hefei and Guangxi Expansive Soils

^{3}, 1.48 g/cm

^{3}, 1.54 g/cm

^{3}, respectively. The physical properties of Hefei and Guangxi expansive soils are summarized in Table 3 and Table 4, respectively. The void ratios of the specimens at the loosest state (e

_{0}= 0.915 for Hefei expansive soil, e

_{0}= 0.901 for Guangxi expansive soil) were regarded as the initial void ratios at reference state.

_{0}= 2.514, ψ

_{a}

_{0}= 50.56 kPa) were obtained as shown in Figure 6 and Figure 7, respectively, while the corresponding air-entry values ψ

_{a}at deformed state were determined using Equation (16), as shown in Table 2. As can be seen in Figure 6, the fitting correlation coefficients for fractal dimension are up to 0.98, indicating that the SWCCs of Hefei specimens have obvious fractal features.

_{0}= 2.589, ψ

_{a}

_{0}= 26.78 kPa) were obtained as shown in Figure 9 and Figure 10, respectively, while the corresponding air-entry values ψ

_{a}at deformed state were determined using Equation (16), as shown in Table 2. As can be seen in Figure 9, the fitting correlation coefficients for fractal dimension are up to 0.97, indicating that the SWCCs of Guangxi specimens have obvious fractal features.

#### 3.3. Saskatchewan Silt

_{0}= 0.692 for T1 was regarded as the initial void ratio at reference state. The calibrated SWCC parameters of T1 at reference state for the first tests (D

_{0}= 2.640, ψ

_{a0}= 7.78 kPa) were obtained as shown in Figure 12 and Figure 13, respectively. The corresponding air-entry values at deformed state are presented in Table 2.

_{0}= 0.525 for T2 was regarded as the initial void ratio at reference state. The calibrated SWCC parameters of Saskatchewan silt of T2 at reference state for the second tests (D

_{0}= 2.604, ψ

_{a0}= 17.71 kPa) were obtained as shown in Figure 15 and Figure 16, respectively. The corresponding air-entry values at deformed state are presented in Table 2.

#### 3.4. Loess

^{3}, 1.4 g/cm

^{3}, 1.5 g/cm

^{3}, and 1.6 g/cm

^{3}, respectively, were prepared and tested by the high-speed centrifuge method. The void ratio of a specimen at the loosest state (e

_{0}= 1.23) was regarded as the initial void ratio at reference state. The experimental temperatures were controlled at 5 °C, 15 °C, 25 °C, and 35 °C, respectively. For model validation, experimental data at 5 °C was employed. The parameters of the loess at reference state (D

_{0}= 2.825, ψ

_{a0}= 0.55 kPa) were obtained as shown in Figure 18 and Figure 19, respectively, while the corresponding air-entry values at deformed state with various initial void ratios (i.e., e

_{0}= 0.88, 0.75, and 0.72, respectively) are shown in Table 2. It is shown in Figure 18 that the fitting correlation coefficient is up to 1.00, which highlights that the SWCCs of loess specimens have significant fractal features.

## 4. Discussion

_{r}–ψ curve). In the current study, the theoretical principle of the proposed model is illustrated in Figure 1, where the SWCC presents a type of “broom shape” distribution in terms of the gravimetric water content. If it is necessary to obtain the SWCCs expressed by volumetric water content or degree of saturation, the gravimetric water content can be converted to the volumetric water content or degree of saturation. Respectively, the transformation formulas are expressed as

## 5. Conclusions

_{a}) were employed in the current model. Determination of the model parameters with clear physical meaning were elaborated. The application of the model to a wide range of experimental data from the tests conducted in the current work, as well as the literatures, was examined. Good agreement was obtained between the experimental data and the model predictions in all of the cases considered.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic sketch of soil-water characteristic curves (SWCCs) in terms of gravimetric water content of unsaturated soils with different initial void ratios.

**Figure 3.**Determination of the values of fractal dimension at reference state through plotting experimental data of lnw against (−lnψ) for Wuhan Clay with e

_{0}= 1.115.

**Figure 4.**Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for Wuhan Clay with e

_{0}= 1.115.

**Figure 5.**Measured and predicted SWCCs for Wuhan clay at deformed state at (

**a**) e = 1.037; (

**b**) e = 0.964; (

**c**) e = 0.897; (

**d**) e = 0.833; (

**e**) e = 0.719; (

**f**) e = 0.613.

**Figure 6.**Determination of the values of fractal dimension at reference state through plotting experimental data of lnw against (–lnψ) for Hefei expansive soil with e

_{0}= 0.915.

**Figure 7.**Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for Hefei expansive soil with e

_{0}= 0.915.

**Figure 8.**Measured and predicted SWCCs for Hefei expansive soils at deformed state at (

**a**) e = 0.838; (

**b**) e = 0.766 (data after Miao et al. [42]).

**Figure 9.**Determination of the values of fractal dimension at reference state through plotting experimental data of lnw against (−lnψ) for Guangxi expansive soil with e

_{0}= 0.901.

**Figure 10.**Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for Guangxi expansive soil with e

_{0}= 0.901.

**Figure 11.**Measured and predicted SWCCs for Guangxi expansive soils at deformed state at (

**a**) e = 0.824; (

**b**) e = 0.753 (data after Miao et al. [42]).

**Figure 12.**Determination of the values of fractal dimension at reference state through plotting experimental data of lnw against (−lnψ) for Saskatchewan silt (T1) with e

_{0}= 0.692.

**Figure 13.**Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for Saskatchewan silt (T1) with e

_{0}= 0.692.

**Figure 14.**Measured and predicted SWCCs for Saskatchewan silt specimens (T1) at deformed state at (

**a**) e = 0.540; (

**b**) e = 0.528; (

**c**) e = 0.501; (

**d**) e = 0.483; (

**e**) e = 0.466 (data after Huang [43]).

**Figure 15.**Determination of the values of fractal dimension at reference state through plotting experimental data of lnw against (−lnψ) for Saskatchewan silt (T2) with e

_{0}= 0.525.

**Figure 16.**Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for Saskatchewan silt (T2) with e

_{0}= 0.525.

**Figure 17.**Measured and predicted SWCCs for Saskatchewan silt specimens (T2) at deformed state at (

**a**) e = 0.513; (

**b**) e = 0.490; (

**c**) e = 0.474; (

**d**) e = 0.454; (

**e**) e = 0.426 (data after Huang [43]).

**Figure 18.**Determination of the values of fractal dimension at reference state through plotting experimental data of lnw against (−lnψ) for loess with e

_{0}= 1.230.

**Figure 19.**Determination of air-entry values at reference state through fitting Equation (12) to the experimental SWCCs for loess with e

_{0}= 1.230.

**Figure 20.**Measured and predicted SWCCs for Xi’an loess specimens at deformed state at (

**a**) e = 0.88; (

**b**) e = 0.75; (

**c**) e = 0.72 (data after Wang et al. [44]).

Natural Density (g/cm^{3}) | Relative Density | Natural Water Content (%) | Liquid Limit (%) | Plastic Limit (%) |
---|---|---|---|---|

2.03 | 2.75 | 21.9 | 38.9 | 20.4 |

Soil Type | Initial Void Ratio | Fractal Dimension | Air-Entry Value/kPa | Soil Type | Initial Void Ratio | Fractal Dimension | Air-Entry Value/kPa |
---|---|---|---|---|---|---|---|

Wuhan clay | 1.037 0.964 0.897 0.833 0.719 0.613 | 2.869 | 2.89 5.04 8.74 15.37 47.28 159.74 | Saskatchewan silt/T1 | 0.54 0.528 0.501 0.483 0.466 | 2.640 | 15.50 16.50 19.08 21.12 23.33 |

Hefei expansive soil | 0.838 0.766 | 2.514 | 60.58 72.89 | Saskatchewan silt/T2 | 0.513 0.490 0.474 0.454 0.426 | 2.604 | 18.77 21.08 22.92 25.56 30.02 |

Guangxi expansive soil | 0.824 0.753 | 2.589 | 33.28 41.44 | Xian Loess (5 °C) | 0.88 0.75 0.72 | 2.825 | 3.73 9.29 11.73 |

**Table 3.**Physical properties of Hefei expansive soils (data after Miao et al. [42]).

Relative Density | Liquid Limit (%) | Plastic Limit (%) | Plasticity Index (%) |
---|---|---|---|

2.72 | 58.6 | 26.4 | 32.2 |

**Table 4.**Physical properties of Guangxi expansive soils (data after Miao et al. [42]).

Relative Density | Liquid Limit (%) | Plastic Limit (%) | Plasticity Index (%) |
---|---|---|---|

2.70 | 61.4 | 30.3 | 31.1 |

**Table 5.**Physical properties of Saskatchewan silt (data after Huang [43]).

Relative Density | Natural Water Content (%) | Liquid Limit (%) | Plastic Limit (%) | Plastic Index (%) |
---|---|---|---|---|

2.68 | 0.86 | 22.2 | 16.6 | 5.6 |

**Table 6.**Physical properties of Xi’an loess (data after Wang et al. [44]).

Depth | Liquid Limit (%) | Plastic Limit (%) | Plastic Index (%) |
---|---|---|---|

2.68 | 30.7 | 18.4 | 12.3 |

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**MDPI and ACS Style**

Tao, G.; Chen, Y.; Kong, L.; Xiao, H.; Chen, Q.; Xia, Y.
A Simple Fractal-Based Model for Soil-Water Characteristic Curves Incorporating Effects of Initial Void Ratios. *Energies* **2018**, *11*, 1419.
https://doi.org/10.3390/en11061419

**AMA Style**

Tao G, Chen Y, Kong L, Xiao H, Chen Q, Xia Y.
A Simple Fractal-Based Model for Soil-Water Characteristic Curves Incorporating Effects of Initial Void Ratios. *Energies*. 2018; 11(6):1419.
https://doi.org/10.3390/en11061419

**Chicago/Turabian Style**

Tao, Gaoliang, Yin Chen, Lingwei Kong, Henglin Xiao, Qingsheng Chen, and Yuxuan Xia.
2018. "A Simple Fractal-Based Model for Soil-Water Characteristic Curves Incorporating Effects of Initial Void Ratios" *Energies* 11, no. 6: 1419.
https://doi.org/10.3390/en11061419