Relationship between Water Infiltration Indices and Fractal Dimension of Soil Particle Size Distribution in Semi-Arid Areas of Zanjan Province

Background and Objectives
Water infiltration into the soil is regarded as one of the key processes in the hydrological cycle and water resources management. This process not only plays a crucial role in the efficient utilization of water resources, but is also fundamental to runoff modeling, groundwater recharge, and soil conservation. Among the factors influencing soil infiltration behavior, soil physical structure, particularly particle size distribution (PSD) has a direct effect on total porosity, pore size distribution and continuity, and hydraulic conductivity. However, these properties result from the complex interaction between PSD and other soil attributes such as aggregate stability, organic matter content, and bulk density. Assessment of soil structure solely through classical geometric approaches is insufficient to describe inherent natural complexity of the soil. In this context, the concept of fractal dimension has been introduced as a tool for characterizing irregular, complex, and self-similar structures of the soils. The application of fractal theory, particularly in relation to soil hydraulic properties such as water infiltration indices still requires further investigation. Given the complex and multifactorial nature of the infiltration process, analyzing the fractal dimension of primary particle size distribution may help to clarify the role of this characteristic as a complementary and usefull variable in advanced and multivariate infiltration models. This issue is of particular importance in soils of semi-arid regions, which often exhibit weak structural development. Therefore, the objective of this study was to investigate the possible relationship between soil water infiltration indices and the fractal dimension of particle size distribution.
 
Methodology
In this study, to investigate the relationship between soil water infiltration indices and the fractal dimension of particle size distribution (PSD), a wide range of soil particle size distributions in a semi-arid region was employed. Accordingly, 68 sampling sites were selected across Zanjan province, Iran, a representative semi-arid region with a mean annual precipitation of approximately 324 mm. Soil water infiltration at each site was measured using a double-ring infiltrometer. At each location, 14 consecutive readings of infiltration depth were recorded at specified time intervals (0, 0.5, 1, 2, 3, 5, 10, 15, 20, 30, 45, 60, 80, and 90 min), and measurements continued until the infiltration rate reached to steady-state conditions, allowing the calculation of soil water infiltration indices. To evaluate soil infiltration behavior, the following indices were examined: (1) cumulative infiltration at 90 min (CI), (2) initial infiltration at 30 s (II), (3) final infiltration rate (FIR), and (4) mean infiltration rate (MIR). For the determination of soil properties, disturbed soil samples were collected using a hand shovel from the 0–60 cm soil depth, considered the active hydrological layer of the soil. Particle size distribution was determined according to the USDA classification using the hydrometer method. Initial soil moisture content prior to infiltration experiments was determined by collecting soil samples from the 0–20 cm depth using metal cylinders (5 cm × 5 cm) and oven-drying them at 105 °C. Bulk density was measured using undisturbed soil samples collected with metal cylinders of known volume, following the method described by Culley (1993).The fractal dimension of soil particle size distribution was calculated using the models proposed by Bird et al. (2000) and Tyler and Wheatcraft (1992). As these model may describe different aspects of soil structure, both were applied and evaluated in this study. To assess the effects of the fractal dimension of PSD on soil water infiltration characteristics: (1) mean comparisons were performed to examine statistically significant differences among fractal dimension classes and infiltration indices using Tukey’s honestly significant difference (HSD) test; and (2) Pearson correlation analysis was employed to evaluate correlations among  variables with different distributions.
 
Results
The results of Pearson correlation analysis and analysis of variance (ANOVA) indicated that no statistically significant correlations existed between the fractal dimension of particle size distribution and the various soil water infiltration indices. Moreover, differences in infiltration indices among the fractal dimension classes (< 2.7, 2.7–2.8, and > 2.8) were not statistically significant. Nevertheless, examination of the mean values of infiltration indices across different fractal dimension ranges revealed relatively consistent increasing or decreasing trends for some infiltration indices. Comparison of the corresponding fractal dimension values further showed that the model proposed by Bird et al. (2000) generally estimated lower fractal dimension values of the primary particle size distribution than the model of Tyler and Wheatcraft (1992), particularly at lower fractal dimension ranges.
 
Conclusions
Overall, findings of the present study disclosed lack of statistically significant correlations  between soil water infiltration indices and the fractal dimension of primary particle size distribution which can be attributed to the complex, multifactorial, and nonlinear nature of the soil water infiltration process. Soil water infiltration results from the simultaneous interaction of soil physical, chemical, and biological properties. Therefore, the fractal dimension of particle size distribution, as a single index, is insufficient to fully explain variations in water infiltration or differences among soils. Under such conditions, this parameter may be more appropriately employed as a complementary explanatory variable within multivariate frameworks or advanced modeling approaches.
Author Contributions
Conceptualization, A.V. and M.Y.; methodology, A.V. and M.Y.; software, A.V. and M.Y.; validation, A.V. and M.Y.; formal analysis, A.V. and M.Y.; investigation, A.V. and M.A.; resources, A.V. and M.Y.; data curation, A.V. and M.Y.; writing-original draft preparation, A.V. and M.Y.; writing–review and editing, A.V.; visualization, A.V.; supervision, A.V..; project administration, A.V.; funding acquisition, A.V. and M.A. All authors have read and agreed to the published version of the manuscript.
Acknowledgements
This paper is published as a part of a PhD thesis supported by the Soil Science and Engineering Department of the University of Zanjan, Iran. The authors are thankful to the University of Zanjan for financial supports.
Data Availability Statement
Data is available on reasonable request from the authors.
Conflict of interest
The authors declare no conflict of interest.
Ethical considerations
The authors avoided data fabrication, falsification, plagiarism, and misconduct.