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

Abstract

The soil-water characteristic curve (SWCC) is an essential tool to determine hydraulic and mechanical properties of unsaturated soils. As an inherent influencing factor, soil texture controls the characteristics of SWCCs. Fractal theory can quantitatively describe the physical characteristics of soil. This study used particle size distribution data and water content data contained in the UNSODA2.0 database to explore the fractal characteristics of 12 soil types with different textures under different matrix suctions. The SWCC fractal model was adopted to characterize the hydraulic properties of soil with various soil textures. The findings revealed that the mass fractal dimensions of particles from these 12 different soil types significantly differed and were closely related to the clay content. Fractal dimension increased with increasing clay content. The fractal dimension established a good relationship between soil structure and hydraulic properties. Fractal analysis can be used to determine the connection between physical properties and soil hydraulic parameters. The estimated results of the SWCC fractal model indicated that it had a good performance regarding the description of SWCCs for the 12 soil textures. The soil structure could be described through fractal dimensions, which can effectively indicate soil hydraulic characteristics. The estimated fractal dimension of this model could be obtained by particle size distribution. Furthermore, using the SWCC fractal model, we found that the SWCC of coarse textured soil changed sharply in the low suction stage and its residual water content was small, and the SWCC of fine textured soil changed gently with a large residual water content. The water retention capacity followed the order clay > silty clay > sandy clay > clay loam > silty clay loam > sandy clay loam > loam > silt loam > sandy loam > silt > loamy sand > sand.

1. Introduction

Unsaturated soils have been extensively studied in environmental science and geotechnical engineering for revealing their hydraulic and mechanical characteristics [1,2,3,4,5,6]. Some of the major properties of unsaturated soil could be represented by the soil-water characteristic curve (SWCC), which describes the complex relationship between matrix suction and water content in the soil [7,8,9,10]. Scientists have carried out numerous studies on the SWCC to better understand water infiltration, soil erosion, shear strength, and slope stability of soils [11]. For example, Zeng et al. [12] measured the SWCC of disintegrated carbonaceous mudstone (DCM) for highway embankments in southwest China to determine stability problems. Wang et al. [13] established the SWCC model of mine wastes to assess the incidents of tailing dams under conditions of rainfall or flood. Moazeni-Noghondar et al. [14] predicted the soil water conditions after rainfall in eastern Iran by transforming the SWCC from the soil water-suction curve to the suction-time curve to study soil erosion. These studies indicated that the strength and hydraulic properties of soil can be described quantitatively by the SWCC, which plays a crucial role in engineering construction and environmental improvement.

The SWCC of unsaturated soils is greatly influenced by different conditions, such as drying and wetting cycles [15] soil texture [16], soil dry density [17,18], initial water content, particle sizes [19], and pore structure [20,21]. Iyer et al. [22] measured the two finely grained soils’ SWCCs (kaolin soil and bentonite soil) under different initial water contents by changing the ratio of water weight to dry soil weight, and found that its SWCCs showed significant differences only when soil suction was below 500 kPa, whereas the SWCCs of soils overlap when soil matrix suction was above 1000 kPa. Li et al. [23], analyzing the SWCCs of 120 soil samples of seven soil types, concluded that the available water content (AW) of finely textured soil was higher, whereas the values for sandy soils were lower. Mahmoodlu et al. [24] employed the van Genuchten model to estimate the mixed sand’s SWCC curves with different particle sizes and found that the shape parameters of the SWCC (α) decreased with decreasing porosity, which indicated that sand with a larger porosity had a smaller air-entry value. Shen et al. [25] explored SWCCs of calcareous silty sand, and they showed some differences in the boundary effect stage (0–5 kPa) and the residual stage (10–750 kPa), whereas the SWCCs were similar in the transition stage. Fine particle content affects SWCCs by mainly controlling the number of soil pores. When fine particle content increases, the pore size formed in the soil decreases, but the porosity may increase [26]. Zhu et al. [27] researched the impacts of soil particle size on SWCCs for the Gaomiaozi (GZM) bentonite pellet-contained material, a Na-bentonite found from Inner Mongolia, China, under different suction ranges. Based on their results, the SWCC was less affected by soil particle distribution under high suction, but was more sensitive to particle size under low suction. These findings showed that the SWCCs of soils with different textures are significantly different, which will further affect the soil water retention capacity. Soil texture was universally divided into 12 types based on the weight ratio of clay fraction (<2 µm), silt fraction (2–50 µm), and sand fraction (>50 µm) [28], and any soil sample can be classified into one of these types. However, there are few studies on the differences of the SWCCs for all soil textures, and identifying the differences of SWCC performance in all soil textures can help in understanding the variations in the water retention capacity of soils and the problems of soil pollution and soil stability.

Considering the importance of SWCCs in geotechnical engineering, the SWCC has been estimated using experimental methods and mathematical models [29,30,31,32]. For example, Matlan et al. [33] evaluated the performance of four empirical models (Van Genuchten model, Kosugi model, Brooks and Corey model, and Modified Gardner model) to forecast the SWCCs of sand, clay, and silt. Soltani et al. [34] developed a new method to determine two parameters describing SWCC changes using soil of different textures, including air-entry value and residual matrix suction. Tao et al. [35] optimized two prediction methods of the SWCC for sandy loam with different dry densities considering soil water evaporation, reducing the workload of measuring SWCCs. However, these above empirical models contain parameters whose physical meanings are unclear. In 1967, Mandelbrot [36] proposed the fractal theory to specifically describe complex and irregular objects. The application of this theory solves the problem of the unclear physical meanings of parameters in the empirical models. In this theory, soil is considered a complex natural fractal medium with self-similarity, and its particles and pores both have fractal characteristics [37,38,39,40]. Fractal dimension, a parameter characterizing the fractal property of an object, can describe soil structure quantitatively, such as pore and particle of soil [41,42,43,44,45,46]. Furthermore, fractal dimension can also establish a link between soil structure and soil-water characteristic curves (SWCC) [47,48], thus providing a convenient method to settle the problem of time-consuming and laborious SWCC measurement. For example, Tyler and Wheatcraft [49] deduced a fractal model of SWCC to represent the volumetric distribution of soil pores through the Sierpinski carpet. Bird et al. [50] established the relationship of the SWCC with the fractal characteristics of soil particle mass distribution. Huang et al. [51] analyzed the fractal dimensions of the soil structure with different textures based on fractal theory and established an equation for the connection between the fractal dimensions of soil particle mass and the clay content. Using the SWCC fractal model, Tao et al. [52] conducted a study on the impacts of clay content and initial porosity on hydraulic properties of soil and provided a theoretical explanation for the effects of the two elements on the SWCCs. Zhou et al. [53] provided a straightforward physical model using fractal theory to evaluate the effect of initial porosity on the soil-water characteristic curve and the hydraulic conductivity of unsaturated soil. Zhang et al. [54] applied this theory to loess and reported that the fractal dimension of soil particles closely depended on soil water distribution, which was an effective tool to explain soil hydraulic characteristics in the loess region. The application of fractal theory in soil property studies simplifies the quantitative expression of soil complex structures (particles and pores) and provides a reliable parameter for accurately predicting soil water holding capacity and for revealing soil-water transport law by establishing the relationship between soil’s physical properties and hydraulic properties. In general, fractal theory is an effective analytical method in soil property studies and should be worthy of extensive application.

The water retention capacity and hydraulic parameters of soil are different for soil with different textures. Traditionally, the hydraulic properties can be expressed directly by the SWCC, the determination of which, however, is time-consuming. The SWCC can also indirectly reflect the property of soil structure distribution (pore and particle). Considering this, the fractal theory, which is a simple tool to combine soil structure with hydraulic properties, was used in this research to study the SWCCs for various types of soil, hoping to finding an easier way to quantify the hydraulic properties of soil. In this sense, the purposes of this article are (1) to analyze fractal properties of soil particles to study the structural differences of soil with various textures, and (2) to establish an SWCC fractal model to study the difference of hydraulic characteristics for soil with various textures. The research could provide a basis for accurate prediction of SWCCs and contribute to solving some engineering problems and the management of the ecological environment.

2. Materials and Methods

2.1. Fractal Theory

Fractal theory can quantitatively characterize soil structural characteristics, such as soil particles and pore structures, by the fractal dimension (D), according to Mandelbrot [36]. There are two methods to figure out the fractal dimension of soil particles, including the particle mass fractal dimension and the particle volume fractal dimension. Generally, soil particle mass can be obtained easily and is therefore generally used for calculation. Fractal objects have two main features. One is its magnitude M(L), in which the measured scale L obeys the following scaling relationship:

$$M(L)\sim L^{D_f}$$

(1)

In the above equation, D_f is the fractal parameter, M(L) can represent the mass, volume, or length of a fractal object, and L is the scale.

The second feature is that the cumulative number N can be expressed by its size distribution, using the following relation:

$$N(L\ge \lambda )={(\frac{{\lambda }_{\max }}{\lambda })}^{D_f}$$

(2)

For soil particles, λ and λ_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:

$$N(>R)=C{R}^{-D_1}$$

(3)

In Equation (3), N (> R) indicates the number of particles with grain size greater than R, C is a constant dependent upon the soil properties, and D₁ 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.

Generally, the particle size distribution and its cumulative percentage curve can be used to represent a soil’s particle size distribution. Tyler and Wheatcraft [56] believed that the distribution of soil particle mass could account for fractal dimension in the model, which is easy to obtain, to derive a fractal model of particle mass distribution.

Mandelbrot [36] gave the fractal formula for particle surface area, in which the particle size is larger than a certain scale R:

$$A(r>R)=C_a\left[1-{\left(\frac{R}{{\lambda }_a}\right)}^{2-D_1}\right]$$

(4)

In this formula, A (r > R) is the particle surface area with a particle size greater than R, R is particle size, C_a and λ_a are both constants related to soil particle shape and size, and D₁ 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:

$$V(r>R)=C_V\left[1-{\left(\frac{R}{{\lambda }_V}\right)}^{3-D_1}\right]$$

(5)

where, C_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:

$$M(r>R)=\rho C_V\left[1-{\left(\frac{R}{{\lambda }_V}\right)}^{3-D_1}\right]$$

(6)

Therefore, the total soil particle mass is:

$$M_T=M(r>0)=\rho C_V\left[1-{\left(\frac{0}{{\lambda }_V}\right)}^{3-D_1}\right]=\rho C_V$$

(7)

The combination of Equations (5) and (6) can yield:

$$\frac{M(r>R)}{M_T}=1-{\left(\frac{R}{{\lambda }_V}\right)}^{3-D_1}$$

(8)

Here, λ_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:

$$\frac{M(r>R)}{M_T}=1-{\left(\frac{R}{R_L}\right)}^{3-D_1}$$

(9)

$$\frac{M(r<R)}{M_T}={\left(\frac{R}{R_L}\right)}^{3-D_1}$$

(10)

Equation (10) is the fractal model based on particle mass distribution established by Tyler and Wheatcraft [56], and the D₁ 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

Many fractal models for understanding SWCCs were established. A widely used SWCC fractal model was derived by Bird [50] using the soil particle mass distribution and combined the pore-solid fractal (PSF) model with the Young–Laplace equation. The Pore-Solid Fractal (PSF) model was established by Perrier et al. [57], which was a generalized fractal model describing soil particle mass distribution. This model describes soil complexity and divides the soil into three parts, i.e., pore set (P), solid set (S), and fractal set (F). The proportions of these three parts satisfy the relation of p + s + f = 1. Assuming that there is a fractal object with a characteristic length of L, the fractal set (F) consists of N_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:

$$\theta =\Phi -\frac{p}{p+s}\left\{1-{\left(\frac{h}{h_{\min }}\right)}^{D_2-d}\right\}$$

(11)

where, Φ is soil total porosity, D₂ 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.

$$\Phi =\frac{p}{p+s}$$

(12)

Consequently, Equation (11) suits the three dimensions:

$$\theta ={\theta }_s{\left(\frac{h}{h_{\min }}\right)}^{D_2-3}$$

(13)

Equation (13) is a generalized fractal model of SWCCs according to the PSF model. D₂ 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.

Three main parameters are important in Equation (13), namely saturated volumetric water content (θ_s), fractal dimension (D₂), 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 (D₂) 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. Empirical values of the air-entry values for all soil textures.

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

).

2.3. Data Sources

The U.S. Salinity Laboratory has developed a database of UNSODA 2.0 on unsaturated soil from many regions of the world (http://www.ussl.ars.usda.gov/, accessed on 10 November 2022). The database has 790 soil samples and provides details about the basic properties and hydraulic properties of each soil sample, including saturated hydraulic conductivity (K_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.

In this research, 357 soil samples covering 12 soil textures from the UNSODA 2.0 database were randomly selected to study the fractal characteristics of soils of different textures. Moreover, the SWCC fractal model was established by using the water retention data (θ–h) of these 357 samples contained in the database. To do this, the mass fractal dimension (D₁) 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.

For the SWCCs for different soil textures, we used statistical parameter (R²) to quantify the performance of this SWCC fractal model, and the expression of goodness of fitting (R²) could be shown as follows:

$$R^2={\left[\frac{{\displaystyle \sum _{i=1}^n\left({\theta }_i-\overline{\theta }\right)\left({\theta }_i^{\prime }-{\overline{\theta }}^{\prime }\right)}}{\sqrt{{\displaystyle \sum _{i=1}^n{\left({\theta }_i-\overline{\theta }\right)}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{\left({\theta }_i^{\prime }-{\overline{\theta }}^{\prime }\right)}^2}}}\right]}^2$$

(14)

where, n is the sample number, θ_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

The results of the fractal dimensions for soils with different textures, obtained using the particle mass distribution data of 357 soil samples, were shown in Figure 1 and

Table 2. Fractal dimension (D₁) values of soils with different textures.

Soil Textures	Number	Slope	Fractal Dimension	Clay Content	R2
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

. There was a good linear correlation between log (M (r < R)/M_T) and log(R/R_L), and we can know that the value goodness of fitting (R²) 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 (D₁) of soils with different textures were calculated by D₁ = 3-k, and the mass fractal dimension (D₁) decreased as the slope of this fitting line increased.

As shown in Figure 1, clay has the steepest fitting line with the largest D₁, whereas the sand has the smallest D₁. Overall, the values of D₁ 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.

Correlation analysis was applied to study the relationship between D₁ and three particle size fractions (clay content, silt content, and sand content) of 12 different soil textures. As shown in

Table 3. Correlation between fractal dimension of soils and soil mechanical composition.

Index	D1	Clay Content	Silt Content	Sand Content
D1	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

** indicates significant correlation at p < 0.01; n = 357.

, the mass fractal dimension (D₁) had different degrees of correlation with three particle contents. Specifically, the value of D₁ 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.

A soil’s fractal dimension reflects its capacity to fill pore spaces with soil particles. Smaller pore size is usually observed in soil associated with a higher ratio of finer particles (i.e., clay content), and the soil structure with a higher ratio of finer particles would be more compacted, thus showing a larger fractal dimension [69]. Conversely, a looser soil structure leads to a smaller fractal dimension. Therefore, it can be concluded that the fractal dimension is a representative index for soil particle composition. Some scholars have used the fractal dimension to describe the spatial variability of soil textures and to monitor the degree of soil erosion and desertification [70,71].

3.2. Fractal Modeling of the SWCC

The SWCC fractal model presented in Equation (13) was proposed by Bird [50] to estimate soil water retention data with different textures.

Table 4. Average estimated fractal dimension (D₂) and mass fractal dimension (D₁) values for 12 soil textures.

Soil Textures	Number	D1	D2	Relative Error (%)	R2
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

showed the results of estimated SWCCs with different soil textures. The average goodness of fitting (R²) ranged from 0.8584 to 0.9419, showing that the SWCC fractal model exhibited a good prediction performance. The estimated fractal dimension (D₂) of soils is identical with that of the mass fractal dimension (D₁) calculated. However, we found that the mass fractal dimension (D₁) is not equivalent to the estimated fractal dimension (D₂), and the D₁ value calculated by using PSD data was slightly lower than the fitted D₂ value (Table 4).

The relative error range of these two fractal dimensions was 1.59%–7.34%, and the mean relative error of 4.09% was less than 5%. Similarly, Huang and Zhan [72], investigating the relationship between mass fractal dimension (D₁) and estimated fractal dimension (D₂), suggested that the D₂ of the SWCC fractal model can be replaced by D₁. 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 D₁ 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.

Overall, we selected 12 representative soil samples to evaluate the differences in SWCCs (

Table 5. Basic physical parameters of representative soil samples.

Soil Textures	Code	Particle Density	Porosity	D1	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%

). These 12 representative samples’ basic physical properties were similar, with particle densities between 2.54 and 2.78 and porosity between 0.34 and 0.56. None of these physical properties of sandstone were provided in the database. We introduced the three parameters (θ_s, D₁, 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.

As seen in Figure 2, the volumetric water content decreased gradually as matrix suction increased. In the low matrix suction section (h < 2500 cm), the SWCC changed sharply. It indicates that a smaller increase of matrix suction will lead to a rapid decline in volumetric water content. However, the curve changed gently in the high suction stage (h > 2500 cm), and the volumetric water content changed slightly with the increase of matrix suction. The change trend of coarse textured soil (sand, loamy sand, sandy loam) was the strongest in all 12 types of soils, followed by medium textured soil (silt, loam, silt loam, sandy clay loam, silty clay loam) and fine textured soil (sandy clay, clay loam, silty clay, clay). Generally, soil water can be retained by both soil adsorption force and capillary force. However, the capillary force usually attracts water molecules in the low suction stage, while water molecules were retained by soil particles mainly by the adsorption force in the high suction stage [73]. Therefore, soil with different textures showed different trends in SWCCs.

There was an apparent inflection point in SWCCs of all soil samples, which is in connection with residual water content [74]. The corresponding volumetric water content and matrix suction at the inflection points of soils with different textures significantly differed. The order of the volumetric water content at the inflection point was consistent with the order of the fractal dimension of different textures. In addition, under the same water content, soil with a larger fractal dimension had a larger matrix suction. Similarly, the soil volumetric water content was higher for a larger fractal dimension of soil under the same suction condition. This suggests that the fractal dimension may control the shape of the SWCCS with fractal characteristics [75]. The SWCC reflects the water retention capacity of soil. In this study, clay had the strongest water retention capacity while sand had the weakest, and the order of other textures was consistent with the fractal dimension order. Rusell [76] put forward that the water retention capacity of soil was mostly achieved through the adsorption of soil particles and that smaller particles have a larger surface area and a better capacity to attract water molecules. Consequently, soil with higher clay content has better water retention ability. Correspondingly, the clay contents in silt, sandy loam, loamy sand, and sand are less than 10%, and the SWCC, therefore, reflects the poor soil water retention capacity. The water retention capacities for the 12 soil textures were consistent with soil’s fractal characteristics. It could draw a conclusion that the fractal dimension is sensitive to soil texture and hydraulic properties [77]. The shape and parameter changes of SWCCs with different textures could be determined by the value of fractal dimension. Specifically, soil with larger fractal dimension indicated a higher soil clay content, which corresponded to higher residual water content. This is actually caused by the different compositions and structures of soil particles. The specific surface area of soil particles gradually increased with the thinning of soil particles, but the number of macropores in soil decreased accordingly. Thus, water molecules were retained in small pores and were not easy to seepage. Therefore, fine textured soil usually has higher water retention capacity, which can be indicated using fractal dimension. Furthermore, the SWCC changed gently in the high matrix suction stage. In short, the SWCC changes are obviously due to the difference of soil textures, resulting in different water retention capacities. This finding can be particularly helpful for understanding the changes in hydraulic properties in loess undergoing tremendous human disturbance, such as land creation projects currently underway in China [78], facilitating the sustainable management of water resources and soil erosions in loess areas [79]. In addition, SWCC is also a key tool to understand the infiltration of rainfall and irrigation water, which helps in establishing various hydraulic models for understanding the laws and mechanisms of soil water seepage [80].

4. Conclusions

In the present study, the fractal characteristics of soil particles and the SWCCs of soils with different soil textures were analyzed with the help of the fractal theory. Many soil samples from UNSODA2.0 database containing particle size distribution data and SWCC data were adopted to check the performance of the fractal model. The following conclusions can be achieved:

• 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 (D₂) 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.

This study will provide a theoretical and practical basis for scholars to explore soil-related issues such as soil water infiltration, pollution transport, and soil stability, and will be helpful for water and soil resource management and protection. It will also deepen the application of fractal theory in soil science, benefiting the development of soil science.