# Evaluation of Soil-Water Characteristic Curves for Different Textural Soils Using Fractal Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fractal Theory

_{f}is the fractal parameter, M(L) can represent the mass, volume, or length of a fractal object, and L is the scale.

_{max}are the grain size and the maximum particle size, respectively. Following Turcotte [55], a fractal model of the soil particle number was proposed to characteristic soil structure, which can be shown as follows:

_{1}is the particles’ fractal dimension of soil particles’ number. If the sum number of soil particles of different particle sizes and each particle size are obtained, the fractal dimension of the soil particle size can be solved by using Equation (3). However, it is rather difficult to obtain the number of each soil particle, because it is often subject to large error in the actual operation process.

_{a}and λ

_{a}are both constants related to soil particle shape and size, and D

_{1}is the same as in Equation (3). The formula indicates that area A tends to be constant as the particle size decreases. By introducing it into a three-dimensional domain, we can obtain the expression of soil particle size and particle volume as follows:

_{V}and λ

_{V}describe the particle shape and size. If we assume soil density ρ being a constant, the particle mass with particle size greater than R is expressed as follows:

_{V}can be calculated through the upper limit of the maximum particle scale R

_{L}, and when R is equal to R

_{L}, we could get M (r > R)/M

_{T}= 0, and λ

_{V}equals R

_{L}. Consequently, Equation (8) could also be expressed by:

_{1}in Equation (10) is the mass particle fractal dimension. Through the linear fitting calculation of Equation (10), taking M (r > R)/M

_{T}and R/R

_{L}as the vertical and horizontal coordinates, respectively, the mass fractal dimensions with different soil textures can be obtained.

#### 2.2. Fractal Model of the SWCC

_{f}sub-regions of size r

_{f}

_{(1)}. Correspondingly, the pore set (P) and the solid set (S) are also made up of N

_{p}sub-regions of size r

_{p}

_{(1)}and Ns sub-regions of size r

_{s}

_{(1)}, respectively. Based on the assumption, Bird et al. [50] presented a fractal model to predict SWCC as follows:

_{2}is fractal dimension, h is matrix suction, and h

_{min}is air-entry value of soil. θ and θ

_{s}represent the volumetric water content and saturated volumetric water content, respectively.

_{2}is the fractal dimension of mass or volume, and the other parameters have the same meanings as Equation (11). Compared to other complex SWCC models, such as the Van Genuchten model and the Fredlund & Xing model [58,59], this fractal model is simple in its expression form. The model is fully consistent with the empirical Brooks-Corey model [30] and the Campbell model [60]. The advantage of this model is that it gives clear physical meaning to the fitting parameters and closely links the soil’s basic physical properties with its hydraulic properties.

_{s}), fractal dimension (D

_{2}), and air-entry value (h

_{min}). The UNSODA 2.0 database provided the values of saturated volumetric water content (θ

_{s}) for different types of soil, and the fractal dimension (D

_{2}) could be theoretically computed using Equation (10). The air-entry value (h

_{min}) is also researched by many researchers, and refers to the difference value between the water pressure and air pressure in soil when the maximum pore starts to drain water [61]. It is a threshold on the SWCC and can display soil pore structure features and water infiltration characteristics in soil [62]. The determination of soil air-entry value is of important meaning to explore the permeability of unsaturated soil. Soils with different textures have different air-entry values; for example, the air-entry value of clay is larger than sand [63]. Rawls et al. [64] provided a series of typical air-entry values for soils with different textures (Table 1).

#### 2.3. Data Sources

_{s}), particle size distribution data, and volumetric water content (θ) under different pressure (h) values [65]. Soil particle size distribution (PSD) was measured via dry sieving, and the relationship between volumetric water content (θ) and pressure head (h) was measured using pressure plate extractors. As per the American Soil Texture Classification Standard, the soil types in the database can be divided into sand (0.05–2 mm), silt (0.002–0.05 mm), and clay (< 0.002 mm) [66]. Therefore, these 790 soil samples from the UNSODA 2.0 database can be classified into 12 types of soil textures, namely sandy clay, silty clay, silty clay loam, sandy clay loam, sand, loamy sand, sandy loam, silt, silt loam, clay loam, loam, and clay.

_{1}) of each soil texture was initially obtained by using soil particle distribution data and the logarithmic fitting of Equation (10). Subsequently, the changes in the SWCC with different soil textures were analyzed using the above same data (θ–h) and the SWCC fractal model.

^{2}) to quantify the performance of this SWCC fractal model, and the expression of goodness of fitting (R

^{2}) could be shown as follows:

_{i}is measured soil volumetric water content, ${\theta}_{i}^{\prime}$ is estimated soil water content, $\overline{\theta}$ is average of measured soil water content, and ${\overline{\theta}}^{\prime}$ is the average of estimated soil water content.

## 3. Results and Discussion

#### 3.1. Fractal Dimension of Different Textural Soil

_{T}) and log(R/R

_{L}), and we can know that the value goodness of fitting (R

^{2}) varied from 0.6503 for silt to 0.9120 for sandy clay loam (Table 2). Figure 1 shows that the slopes of the fitting lines (k) for soil with different soil textures were different. The mass fractal dimensions (D

_{1}) of soils with different textures were calculated by D

_{1}= 3-k, and the mass fractal dimension (D

_{1}) decreased as the slope of this fitting line increased.

_{1}, whereas the sand has the smallest D

_{1}. Overall, the values of D

_{1}for the soils with different textures ranged between 2.4024 and 2.8928 (Table 2) and followed the order (from largest to the smallest) clay, silty clay, sandy clay, clay loam, silty clay loam, sandy clay loam, loam, silt loam, sandy loam, silt, loamy sand, and sand. Huang and Zhang [41] obtained some similar results by investigating the fractal characteristics with 12 textural soils.

_{1}and three particle size fractions (clay content, silt content, and sand content) of 12 different soil textures. As shown in Table 3, the mass fractal dimension (D

_{1}) had different degrees of correlation with three particle contents. Specifically, the value of D

_{1}was correlated with clay content and silt content positively (p < 0.01), while it was correlated with sand content negatively (p < 0.01). Particularly, fractal dimension was strongly dependent on clay content. The finding of this research is similar to the results of Millán et al. [67] and Zhao et al. [68]. Millán et al. [67] even explored a linear relationship between the fractal dimensions of soils with different textures and the clay content.

#### 3.2. Fractal Modeling of the SWCC

^{2}) ranged from 0.8584 to 0.9419, showing that the SWCC fractal model exhibited a good prediction performance. The estimated fractal dimension (D

_{2}) of soils is identical with that of the mass fractal dimension (D

_{1}) calculated. However, we found that the mass fractal dimension (D

_{1}) is not equivalent to the estimated fractal dimension (D

_{2}), and the D

_{1}value calculated by using PSD data was slightly lower than the fitted D

_{2}value (Table 4).

_{1}) and estimated fractal dimension (D

_{2}), suggested that the D

_{2}of the SWCC fractal model can be replaced by D

_{1}. In the subsequent research on the differences in hydraulic properties in soils with different textures, the SWCC fractal model expressed in Equation (13) was adopted, and the values of D

_{1}calculated in Section 3.1 were used as parameters in this model. Considering the applicability of the fractal model in this study, the air-entry values (h

_{min}) in Equation (13) were selected from the empirical values recommended by Rawls et al. [64], as shown in Table 1.

_{s}, D

_{1}, and h

_{min}) of each sample into Equation (13) to obtain the expression of the SWCC and generated the corresponding graph using Origin 8.0. Figure 2 shows the SWCCs for these 12 representative soil samples.

## 4. Conclusions

- The fractal characteristic of soil with different textures was significantly different, and fractal dimension was strongly dependent on the clay content in the soil. The average fractal dimensions of the 12 different textures ranged from 2.4024 to 2.8928. Clay had the largest fractal dimension, whereas sand had the lowest one. The particle size composition of soil will significantly change the fractal dimension of soil. The fractal dimension of finely textured soil was larger than that of medium and coarse textured soil. Correlation analysis also suggests that the fractal dimension of soil particles is intensively related to the contents of clay, silt, and sand (p < 0.01). A higher ratio of clay content in the soil can produce a greater mass fractal dimension of soil particles. Fractal theory can quantitatively describe the features of soil particle composition.
- The SWCC of unsaturated soil was strongly dependent on soil texture. The relationship of soil structure with hydraulic properties can be established using fractal analysis. The fractal model representing SWCC has good fitting results for soils of different textures, and the estimated fractal dimension (D
_{2}) in this fractal model can be obtained by particle size distribution. The fitting results of the SWCC fractal model showed that soil with different textures had different changes in SWCC. The soil water retention capacity gradually increased with increasing fractal dimension. In the low suction stage, the changes in the SWCC of coarse textured soil were steeper than those of fine textured soil. Fine textured soil had a larger residual moisture content, while coarse textured soil had a smaller one.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Fitting relationships between log (M(r < R)/M

_{T}) and log(R/R

_{L}) for all soil textures in UNSODA 2.0.

**Figure 2.**Soil-water characteristic curves (SWCCs) of 12 soil textures predicted using the SWCC fractal model.

Soil Texture | Clay | Silty Clay | Sandy Clay | Clay Loam | Silty Clay Loam | Sandy Clay Loam | Loam | Silt Loam | Sandy Loam | Silt | Loamy Sand | Sand |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Value | 37.30 | 34.19 | 29.17 | 25.89 | 32.56 | 28.08 | 11.15 | 20.76 | 14.66 | 20.00 | 8.69 | 7.26 |

Soil Textures | Number | Slope | Fractal Dimension | Clay Content | R^{2} |
---|---|---|---|---|---|

Clay | 15 | 0.1072 | 2.8928 | 51.01 | 0.8509 |

Silty clay | 12 | 0.1327 | 2.8673 | 40.33 | 0.7425 |

Sandy clay | 3 | 0.1470 | 2.8530 | 39.13 | 0.9116 |

Clay loam | 14 | 0.1483 | 2.8517 | 37.32 | 0.8993 |

Silty clay loam | 20 | 0.1595 | 2.8405 | 36.65 | 0.7565 |

Sandy clay loam | 24 | 0.2154 | 2.7846 | 24.60 | 0.9120 |

Loam | 50 | 0.2482 | 2.7519 | 21.49 | 0.8655 |

Silt loam | 78 | 0.2832 | 2.7168 | 18.59 | 0.7190 |

Sandy loam | 50 | 0.3439 | 2.6561 | 13.16 | 0.8654 |

Silt | 3 | 0.3589 | 2.6411 | 9.10 | 0.6503 |

Loamy sand | 34 | 0.4767 | 2.5233 | 6.72 | 0.8489 |

Sand | 53 | 0.5976 | 2.4024 | 3.02 | 0.7875 |

Index | D_{1} | Clay Content | Silt Content | Sand Content |
---|---|---|---|---|

D_{1} | 1.000 | |||

Clay content | 0.943 ** | 1.000 | ||

Silt content | 0.449 ** | 0.474 ** | 1.000 | |

Sand content | −0.678 ** | −0.726 ** | −0.917 ** | 1.000 |

**Table 4.**Average estimated fractal dimension (D

_{2}) and mass fractal dimension (D

_{1}) values for 12 soil textures.

Soil Textures | Number | D_{1} | D_{2} | Relative Error (%) | R^{2} |
---|---|---|---|---|---|

Clay | 15 | 2.8928 | 2.9497 | 1.92 | 0.8911 |

Silty clay | 12 | 2.8673 | 2.9152 | 1.64 | 0.9365 |

Sandy clay | 3 | 2.8530 | 2.9113 | 2.00 | 0.8539 |

Clay loam | 14 | 2.8517 | 2.9105 | 2.02 | 0.9419 |

Silty clay loam | 20 | 2.8405 | 2.8865 | 1.59 | 0.9222 |

Sandy clay loam | 24 | 2.7846 | 2.8854 | 3.49 | 0.9083 |

Loam | 50 | 2.7519 | 2.8836 | 4.56 | 0.9208 |

Silt loam | 78 | 2.7168 | 2.8798 | 5.66 | 0.9008 |

Sandy loam | 50 | 2.6561 | 2.8650 | 7.29 | 0.8786 |

Silt | 3 | 2.6411 | 2.8504 | 7.34 | 0.8584 |

Loamy sand | 34 | 2.5233 | 2.6751 | 5.03 | 0.9304 |

Sand | 53 | 2.4024 | 2.5712 | 6.56 | 0.8625 |

Soil Textures | Code | Particle Density | Porosity | D_{1} | Clay Content |
---|---|---|---|---|---|

Clay | 2362 | 2.65 | 0.56 | 2.9249 | 63.0% |

Silty clay | 3030 | 2.66 | 0.50 | 2.8916 | 42.0% |

Sandy clay | 1135 | — | — | 2.8568 | 41.0% |

Clay loam | 2701 | 2.61 | 0.34 | 2.8325 | 31.2% |

Silty clay loam | 3110 | 2.54 | 0.47 | 2.8195 | 31.0% |

Sandy clay loam | 2630 | 2.72 | 0.53 | 2.8028 | 25.8% |

Loam | 2591 | 2.65 | 0.43 | 2.7809 | 21.8% |

Silt loam | 2671 | 2.78 | 0.49 | 2.7513 | 17.3% |

Sandy loam | 2560 | 2.61 | 0.48 | 2.6567 | 9.4% |

Silt | 4670 | 2.65 | 0.46 | 2.4931 | 9.0% |

Loamy sand | 2763 | 2.67 | 0.43 | 2.4852 | 2.8% |

Sand | 4660 | 2.56 | 0.46 | 2.3291 | 2.0% |

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## Share and Cite

**MDPI and ACS Style**

Yang, C.; Wu, J.; Li, P.; Wang, Y.; Yang, N.
Evaluation of Soil-Water Characteristic Curves for Different Textural Soils Using Fractal Analysis. *Water* **2023**, *15*, 772.
https://doi.org/10.3390/w15040772

**AMA Style**

Yang C, Wu J, Li P, Wang Y, Yang N.
Evaluation of Soil-Water Characteristic Curves for Different Textural Soils Using Fractal Analysis. *Water*. 2023; 15(4):772.
https://doi.org/10.3390/w15040772

**Chicago/Turabian Style**

Yang, Chunliu, Jianhua Wu, Peiyue Li, Yuanhang Wang, and Ningning Yang.
2023. "Evaluation of Soil-Water Characteristic Curves for Different Textural Soils Using Fractal Analysis" *Water* 15, no. 4: 772.
https://doi.org/10.3390/w15040772