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Fractal Fract
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Fractal Analysis and Its Applications in Geophysical Science
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Fractal Analysis and Its Applications in Geophysical Science

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (15 January 2024) | Viewed by 18296

Special Issue Editors

Dr. Mei Yin

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Guest Editor
College of Engineering, Design and Physical Sciences, Brunel University London, Uxbridge, UK
Interests: numerical analysis; soil mechanics; geothermal
Prof. Dr. Mengxi Zhang

E-Mail Website
Guest Editor
School of Mechanics and Engineering Science, Shanghai University, Shanghai, China
Interests: tunnel and underground structure; reinforced soil calculation theory
Prof. Dr. Yi Rui

E-Mail Website
Guest Editor
School of Civil Engineering, Tongji University, Shanghai 200092, China
Interests: energy underground engineering; intelligent perception of underground infrastructure; geotechnical computational mechanics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The fractal theory provides a general framework for the study of irregular sets and shapes, where the geometry of matter can be described in a set of real noninteger numbers known as the fractal dimension, differing from the more familiar Euclidean or topological dimensions. In recent years, fractal analysis and fractal dimension have been widely adopted in geophysical science to characterize the complicated structures of geophysical materials, such as soil, rock, concrete and other aggregates. The fractality of the geometrical features of these materials, such as microstructures, granular aggregates and cracking behavior, can be quantified in terms of their fractal dimensions. Several techniques, including digital image analysis and optical and scanning electron microscopy, have been used to establish the relevant fractal dimension, and then used as the key parameter in some conventional approaches of geophysical science, such as analytical analysis, continuum mechanics, discrete element analysis and racking mode analysis.

In this Special Issue, “Fractal Analysis and Its Applications in Geophysical Science", we would like to solicit your innovative ideas and work regarding the investigation and application of fractal dimensions in geophysical science in the form of original articles. In addition, your study could focus on any aspect of geophysical science, such as geophysical material properties, numerical analysis, experimental and theoretical verifications, etc. The purpose of this Special Issue is to promote the deeper and wider investigation and application of the fractal theory in fields of geophysical science. The submitted manuscripts will be peer reviewed, and those accepted will be published in the open access journal Fractal and Fractional. The topics to be considered in this Special Issue include, but are not limited to, the following:

  • Earth science;
  • Geotechnical engineering;
  • Engineering geology;
  • Microstructures of rock and soil;
  • Granular aggregate properties;
  • Modelling of cracking behavior;
  • Continuum mechanics and numerical analysis;
  • Experimental and theoretical study.

Dr. Mei Yin
Prof. Dr. Mengxi Zhang
Prof. Dr. Yi Rui
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (17 papers)

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Research

21 pages, 8368 KiB  
Article
Analysis of the Fractal Dimension, b-value, Slip Ratio, and Decay Rate of Aftershock Seismicity Following the 6 February 2023 (Mw 7.8 and 7.5) Türkiye Earthquakes
by Sherif M. Ali and Kamal Abdelrahman
Fractal Fract. 2024, 8(5), 252; https://doi.org/10.3390/fractalfract8050252 - 25 Apr 2024
Viewed by 419
Abstract
On 6 February 2023, Türkiye experienced a pair of consecutive earthquakes with magnitudes of Mw 7.8 and 7.5, and accompanied by an intense aftershock sequence. These seismic events were particularly impactful on the segments of the East Anatolian Fault Zone (EAFZ), causing extensive [...] Read more.
On 6 February 2023, Türkiye experienced a pair of consecutive earthquakes with magnitudes of Mw 7.8 and 7.5, and accompanied by an intense aftershock sequence. These seismic events were particularly impactful on the segments of the East Anatolian Fault Zone (EAFZ), causing extensive damage to both human life and urban centers in Türkiye and Syria. This study explores the analysis of a dataset spanning almost one year following the Turkiye mainshocks, including 471 events with a magnitude of completeness (Mc) ≥ 4.4. We employed the maximum likelihood approach to estimate the b-value and Omori-Utsu parameters (K, c, and p-values). The estimated b-value is 1.21 ± 0.1, indicating that the mainshocks occurred in a region characterized by elevated stress levels, leading to a sequence of aftershocks of larger magnitudes due to notable irregularities in the rupture zone. The aftershock decay rate (p-value = 1.1 ± 0.04) indicates a rapid decrease in stress levels following the main shocks. However, the c-value of 0.204 ± 0.058 would indicate a relatively moderate or low initial productivity of aftershocks. Furthermore, the k-value of 76.75 ± 8.84 suggests that the decay of aftershock activity commenced within a range of approximately 68 to 86 days following the mainshocks. The fractal dimension (Dc) was assessed using the correlation integral method, yielding a value of 0.99 ± 0.03. This implies a tendency toward clustering in the aftershock seismicity and a linear configuration of the epicenters. The slip ratio during the aftershock activity was determined to be 0.75, signifying that 75% of the total slip occurred in the primary rupture, with the remaining fraction distributed among secondary faults. The methodologies and insights acquired in this research can be extended to assist in forecasting aftershock occurrences for future earthquakes, thus offering crucial data for future risk assessment. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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19 pages, 5158 KiB  
Article
Unravelling the Fractal Complexity of Temperature Datasets across Indian Mainland
by Adarsh Sankaran, Thomas Plocoste, Arathy Nair Geetha Raveendran Nair and Meera Geetha Mohan
Fractal Fract. 2024, 8(4), 241; https://doi.org/10.3390/fractalfract8040241 - 20 Apr 2024
Viewed by 387
Abstract
Studying atmospheric temperature characteristics is crucial under climate change, as it helps us to understand the changing patterns in temperature that have significant implications for the environment, ecosystems, and human well-being. This study presents the comprehensive analysis of the spatiotemporal variability of scaling [...] Read more.
Studying atmospheric temperature characteristics is crucial under climate change, as it helps us to understand the changing patterns in temperature that have significant implications for the environment, ecosystems, and human well-being. This study presents the comprehensive analysis of the spatiotemporal variability of scaling behavior of daily temperature series across the whole Indian mainland, using a Multifractal Detrended Fluctuation Analysis (MFDFA). The analysis considered 1° × 1° datasets of maximum temperature (Tmax), minimum temperature (Tmin), mean temperature (Tmean), and diurnal temperature range (DTR) (TDTR = Tmax − Tmin) from 1951 to 2016 to compare their scaling behavior for the first time. Our results indicate that the Tmin series exhibits the highest persistence (with the Hurst exponent ranging from 0.849 to unity, and a mean of 0.971), and all four-temperature series display long-term persistence and multifractal characteristics. The variability of the multifractal characteristics is less significant in North–Central India, while it is highest along the western coast of India. Moreover, the assessment of multifractal characteristics of different temperature series during the pre- and post-1976–1977 period of the Pacific climate shift reveals a notable decrease in multifractal strength and persistence in the post-1976–1977 series across all regions. Moreover, for the detection of climate change and its dominant driver, we propose a new rolling window multifractal (RWM) framework by evaluating the temporal evolution of the spectral exponents and the Hurst exponent. This study successfully captured the regime shifts during the periods of 1976–1977 and 1997–1998. Interestingly, the earlier climatic shift primarily mitigated the persistence of the Tmax series, whereas the latter shift significantly influenced the persistence of the Tmean series in the majority of temperature-homogeneous regions in India. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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32 pages, 8093 KiB  
Article
Urban Meteorology, Pollutants, Geomorphology, Fractality, and Anomalous Diffusion
by Patricio Pacheco, Eduardo Mera, Gustavo Navarro and Carolina Parodi
Fractal Fract. 2024, 8(4), 204; https://doi.org/10.3390/fractalfract8040204 - 30 Mar 2024
Viewed by 624
Abstract
The measurements, recorded as time series (TS), of urban meteorology, including temperature (T), relative humidity (RH), wind speed (WS), and pollutants (PM10, PM2.5, and CO), in three different geographical morphologies (basin, mountain range, and coast) are analyzed through chaos [...] Read more.
The measurements, recorded as time series (TS), of urban meteorology, including temperature (T), relative humidity (RH), wind speed (WS), and pollutants (PM10, PM2.5, and CO), in three different geographical morphologies (basin, mountain range, and coast) are analyzed through chaos theory. The parameters calculated at TS, including the Lyapunov exponent (λ > 0), the correlation dimension (DC < 5), Kolmogorov entropy (SK > 0), the Hurst exponent (0.5 < H < 1), Lempel–Ziv complexity (LZ > 0), the loss of information (<ΔI> < 0), and the fractal dimension (D), show that they are chaotic. For the different locations of data recording, CK is constructed, which is a proportion between the sum of the Kolmogorov entropies of urban meteorology and the sum of the Kolmogorov entropies of the pollutants. It is shown that, for the three morphologies studied, the numerical value of the CK quotient is compatible with the values of the exponent α of time t in the expression of anomalous diffusion applied to the diffusive behavior of atmospheric pollutants in basins, mountain ranges, and coasts. Through the Fréchet heavy tail study, it is possible to define, in each morphology, whether urban meteorology or pollutants exert the greatest influence on the diffusion processes. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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15 pages, 14513 KiB  
Article
Fractal Analysis of Polarizability in Graphite Deposits: Methodological Integration for Geological Prediction and Exploration Efficiency
by Yuqi Liang, Qinglin Xia, Kenan Jiang and Ercheng Pang
Fractal Fract. 2024, 8(4), 198; https://doi.org/10.3390/fractalfract8040198 - 29 Mar 2024
Viewed by 500
Abstract
Most geophysical and geochemical data are commonly acknowledged to exhibit fractal and multifractal properties, but the fractal characteristics of polarizability have received limited attention from the literature. The present study demonstrates that the polarizability data of the graphite deposits have fractal characteristics and [...] Read more.
Most geophysical and geochemical data are commonly acknowledged to exhibit fractal and multifractal properties, but the fractal characteristics of polarizability have received limited attention from the literature. The present study demonstrates that the polarizability data of the graphite deposits have fractal characteristics and introduces the fractal method for its quantitative analysis to indicate and predict the properties of graphite deposits. The results show that the concentration-area (C-A) method is superior to classical interpolation in anomaly extraction but inferior to the spectrum-area (S-A) method in the coverage region. Because the type of graphite ore is sedimentary-metamorphic in this area, the graphite ore-bodies can be regarded as a special stratum, which is different from most metal deposits, and the anomaly of graphite ore are shown in the background mode of the S-A method. The high values of the background mode effectively indicate the potential areas where the graphite-bearing strata occur, while observing a decrease in the power-law exponent (β) of the background mode as the width of ore-bodies increases. The validity of this conclusion was confirmed based on the vertical profiles of the predicted area, and the uncharted ore vein was thereby identified. Furthermore, it was found that the anomaly mode can serve as a grade indicator of graphite ore rather than delineating the fault. By integrating the background and anomaly modes of the S-A method, we can quantitatively predict and effectively identify high-grade targets from sedimentary deposits containing minerals in future exploration. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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19 pages, 9522 KiB  
Article
Analysis of Pore Characterization and Energy Evolution of Granite by Microwave Radiation
by Keping Zhou, Yifan Zhang, Chun Yang, Niange Yang and Zheng Pan
Fractal Fract. 2024, 8(3), 161; https://doi.org/10.3390/fractalfract8030161 - 12 Mar 2024
Viewed by 834
Abstract
To study the dynamic response of granite to different levels of microwave power, an intelligent microwave rock-breaking instrument is used to irradiate different power from three directions. The servo universal testing machine is used to carry out a uniaxial compression test on the [...] Read more.
To study the dynamic response of granite to different levels of microwave power, an intelligent microwave rock-breaking instrument is used to irradiate different power from three directions. The servo universal testing machine is used to carry out a uniaxial compression test on the granite after microwave damage to analyze the strength damage characteristics and the degree of pore damage. Pore fractal characteristics are analyzed based on nuclear magnetic resonance to establish the microwave damage degradation model. In parallel, the energy evolution process of granite under the influence of various power levels is analyzed using the theory of energy dissipation. Simultaneously, based on the energy dissipation theory, we analyze the energy evolution process of granite under the action of different powers. The results show that with higher microwave power, the peak strength and modulus of elasticity show a linear decreasing law. The degree of fragmentation is more obvious, showing the damage characteristics with two big ends and little in the middle. The higher the power, the greater the porosity and the more sensitive the micropore becomes to microwaves. Additionally, the damage degradation model established to evaluate the microwave damage of the rock showed that it was feasible. The higher the power, the lower the total energy, elastic energy, and dissipation energy, and the granite is gradually transformed from elastic deformation to plastic deformation. The elastic energy ratio decreases, the dissipation energy ratio increases, and the degree of damage becomes more and more serious. This study provides theoretical support for exploring the mechanical behavior and mechanism of microwave-assisted rock breaking and is of great practical significance. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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20 pages, 4189 KiB  
Article
Statistical Study of the Bias and Precision for Six Estimation Methods for the Fractal Dimension of Randomly Rough Surfaces
by Jorge Luis Flores Alarcón, Carlos Gabriel Figueroa, Víctor Hugo Jacobo, Fernando Velázquez Villegas and Rafael Schouwenaars
Fractal Fract. 2024, 8(3), 152; https://doi.org/10.3390/fractalfract8030152 - 7 Mar 2024
Viewed by 1061
Abstract
The simulation and characterisation of randomly rough surfaces is an important topic in surface science, tribology, geo- and planetary sciences, image analysis and optics. Extensions to general random processes with two continuous variables are straightforward. Several surface generation algorithms are available, and preference [...] Read more.
The simulation and characterisation of randomly rough surfaces is an important topic in surface science, tribology, geo- and planetary sciences, image analysis and optics. Extensions to general random processes with two continuous variables are straightforward. Several surface generation algorithms are available, and preference for one or another method often depends on the specific scientific field. The same holds for the methods to estimate the fractal dimension D. This work analyses six algorithms for the determination of D as a function of the size of the domain, variance, and the input value for D, using surfaces generated by Fourier filtering techniques and the random midpoint displacement algorithm. Several of the methods to determine fractal dimension are needlessly complex and severely biased, whereas simple and computationally efficient methods produce better results. A fine-tuned analysis of the power spectral density is very precise and shows how the different surface generation algorithms deviate from ideal fractal behaviour. For large datasets defined on equidistant two-dimensional grids, it is clearly the most sensitive and precise method to determine fractal dimension. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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17 pages, 6208 KiB  
Article
Changes in Pore Structure and Fractal Characteristics of Solvents Pretreated High-Rank Coal under Supercritical CO2
by Yong Li, Xiaodong Zhang, Yijuan Sun, Zhen Wang, Shuo Zhang and Binghui Li
Fractal Fract. 2024, 8(3), 141; https://doi.org/10.3390/fractalfract8030141 - 28 Feb 2024
Viewed by 874
Abstract
CO2 injection in coal seams, which is a significant initiative to mitigate environmental problems caused by greenhouse gases, often leads a sequence of changes in the physical properties of coal reservoirs. To look into how the pore structure changes in the process [...] Read more.
CO2 injection in coal seams, which is a significant initiative to mitigate environmental problems caused by greenhouse gases, often leads a sequence of changes in the physical properties of coal reservoirs. To look into how the pore structure changes in the process of CO2 sequestration, we selected fresh coal from Huoerxinhe coal mine in China as the object. Then, acid treatment and SC-CO2 extraction were used to dissolve Organic and inorganic components in coal. Thus, by using SEM, LTGA-N2 apparatus and XRD, the characteristics of pore parameter and fractal dimension variation were discussed. The research results show that, the APS of samples THF, HCL-HF and Y-C increase, while the total PV decreases and the pore connectivity deteriorates. The pore connectivity of Samples THF and HCL-HF is improved (THF-C, HCL-HF-C), but the total pore volume continuously reduces. In addition, solvents treatment and SC-CO2 extraction mainly act on the microporous fraction. After solvents pretreatment, the changes in the pore size distribution curves are mainly manifested in the reduction of number of micropores, especially in the micropores around 3–4 nm. There is a small increase in micropores for samples Y-C and HCL-HF-C, with the pore size mainly concentrated around 4 nm, while the pores of the sample THF-C mainly show an increase within the scope of 3–16 nm. Generally, solvent pretreatment and SC-CO2 extraction help to simplify pore structure. However, the sample HCL-HF-C shows opposite change characteristics. In a short period of time, the larger pore fractal dimension, the less beneficial it is to the flow of CO2, while pore fractal dimension becomes progressively less useful in assessing pore connectivity with increasing time. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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15 pages, 5089 KiB  
Article
An Improved Rock Resistivity Model Based on Multi-Fractal Characterization Method for Sandstone Micro-Pore Structure Using Capillary Pressure
by Weibiao Xie, Qiuli Yin, Jingbo Zeng, Fan Yang, Pan Zhang and Binpeng Yan
Fractal Fract. 2024, 8(2), 118; https://doi.org/10.3390/fractalfract8020118 - 16 Feb 2024
Viewed by 891
Abstract
Micro-pore structures are an essential factor for the electrical properties of porous rock. Theoretical electrical conductivity models considering pore structure can highly improve the accuracy of reservoir estimation. In this study, a pore structure characterization method based on a multi-fractal theory using capillary [...] Read more.
Micro-pore structures are an essential factor for the electrical properties of porous rock. Theoretical electrical conductivity models considering pore structure can highly improve the accuracy of reservoir estimation. In this study, a pore structure characterization method based on a multi-fractal theory using capillary pressure is developed. Next, a theoretical electrical conductivity equation is derived based on the new pore structure characterization method. Furthermore, a distinct interrelationship between fractal dimensions of capillary pressure curves (Dv) and of resistivity index curves (Dt and Dr) is obtained. The experimental data of 7 sandstone samples verify that the fitting result by the new pore structure characterization method is highly identical to the experimental capillary pressure curves, and the accuracy of the improved rock resistivity model is higher than the Archie model. In addition, capillary pressure curves can be directly converted to resistivity index curves according to the relationship model between fractal dimensions of capillary pressure curves (Dv) and resistivity index curves (Dt and Dr). This study provides new ideas to improve the accuracy of pore structure characterization and oil saturation calculation; it has good application prospects and guiding significance in reservoir evaluation and rock physical characteristics research. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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19 pages, 7491 KiB  
Article
Pore Structure Quantification and Fractal Characterization of MSA Mortar Based on 1H Low-Field NMR
by Zhen Jiang, Huan He, Guanglin Tian, Weizuo Guo, Yingzhen Li and Zheng Pan
Fractal Fract. 2024, 8(1), 42; https://doi.org/10.3390/fractalfract8010042 - 9 Jan 2024
Cited by 1 | Viewed by 1133
Abstract
With the gradual depletion of natural sand due to over-exploitation, alternative building materials, such as manufactured sand aggregate (MSA), have attracted much attention. In order to interpret the evolution of pore structure and fractal characteristics in MSA mortar over long-term water saturation, the [...] Read more.
With the gradual depletion of natural sand due to over-exploitation, alternative building materials, such as manufactured sand aggregate (MSA), have attracted much attention. In order to interpret the evolution of pore structure and fractal characteristics in MSA mortar over long-term water saturation, the 1H low-field nuclear magnetic resonance (LF-NMR) relaxation method was used to investigate the temporal evolution of the pore structure in five single-graded MSA mortars and synthetic-graded mortars with small amplitudes in particle size. MSA presents a fresh rock interface characterized by a scarcity of pores, which significantly reduces the porosity of the mortar. The surface-to-volume ratio (SVR) is employed for characterizing the MSA gradation. Through an analysis of parameters, such as total porosity, pore gradation, pore connectivity, and pore fractal dimension of mortar, a correlation model between pore structure parameters and aggregate SVR is constructed. The fractal characteristics of pores and their variations are discussed under three kinds of pore gradations, and the correlation model between fractal dimension and porosity is established. These results demonstrate the high impermeability and outstanding corrosion resistance of synthetic-graded mortar. The fractal model of the pore structure evolution of MSA mortar has a guiding effect on the pore distribution evolution and engineering permeability evaluation of MSA mortar in engineering. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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10 pages, 2685 KiB  
Article
Experimental Study on the Cross-Scale Relationship of Cemented Backfill under the Action of an Air-Entraining Agent
by Xiaosheng Liu, Dongjie Yang and Weijun Wang
Fractal Fract. 2023, 7(11), 821; https://doi.org/10.3390/fractalfract7110821 - 15 Nov 2023
Viewed by 845
Abstract
Air-entraining agents have the function of optimizing pores and improving the performance of backfill. In this study, we used tailings and cement as the main raw materials and added different amounts of air-entraining agents to make backfill samples. By testing the uniaxial compressive [...] Read more.
Air-entraining agents have the function of optimizing pores and improving the performance of backfill. In this study, we used tailings and cement as the main raw materials and added different amounts of air-entraining agents to make backfill samples. By testing the uniaxial compressive strength (UCS) and microstructure, macro- and micro characteristics were studied. Nuclear magnetic resonance technology was used to explore pore characteristics, and fractal theory was used to quantitatively discuss the complexity of pore structure. Finally, a cross-scale relationship model between UCS and pores was established. The main conclusions are as follows: (1) Adding the appropriate amount of air-entraining agents can optimize pore structure and increase the UCS of backfill materials, which is beneficial to backfill materials. (2) The pores of backfill materials have fractal characteristics, the fractal effects of pores with different pore size ranges are different, and the air-entraining agent has a certain influence on the fractal characteristics of the pores. (3) There are inverse relationships between UCS and different pore size ranges. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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18 pages, 20234 KiB  
Article
Fractal Characterization of Multiscale Fracture Network Distribution in Dolomites: Outcrop Analogue of Subsurface Reservoirs
by Ivica Pavičić, Željko Duić, Anja Vrbaški and Ivan Dragičević
Fractal Fract. 2023, 7(9), 676; https://doi.org/10.3390/fractalfract7090676 - 7 Sep 2023
Cited by 1 | Viewed by 1101
Abstract
Fractured aquifers, especially dolomites, are important hydrocarbon reservoirs and sources of thermal and groundwater in many parts of the world, especially in the Alpine-Dinaric-Carpathian region. The most dominant porosity type is fracture porosity, which acts as the preferential fluid pathway in the subsurface, [...] Read more.
Fractured aquifers, especially dolomites, are important hydrocarbon reservoirs and sources of thermal and groundwater in many parts of the world, especially in the Alpine-Dinaric-Carpathian region. The most dominant porosity type is fracture porosity, which acts as the preferential fluid pathway in the subsurface, thus strongly controlling fluid flow. Outcrops provide valuable information for the characterization of fracture networks. Dolomite rock properties and structural and diagenetic processes result in fractured systems that can be considered fractals. The fracture network was analyzed on 14 vertical outcrops in 35 digitized photographs. The values of the fractal dimensions varied slightly by the software and method used, but the trends were consistent, which confirms that all methods are valid. Small values of fractal dimension indicate the dominance of a few small or large fractures, and high values of fractal dimension result from a combination of large numbers of small fractures accompanied by a few large fractures. The mean value of the fractal dimension for analyzed fracture networks was 1.648. The results indicate that the fracture network of the Upper Triassic dolomites can be approximated by fractal distribution and can be considered a natural fractal, and values can be extrapolated to higher and lower scales (1D and 3D). Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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24 pages, 9467 KiB  
Article
The Impact of Fractal Dimension, Stress Tensors, and Earthquake Probabilities on Seismotectonic Characterisation in the Red Sea
by Sherif M. Ali and Kamal Abdelrahman
Fractal Fract. 2023, 7(9), 658; https://doi.org/10.3390/fractalfract7090658 - 31 Aug 2023
Cited by 3 | Viewed by 1163
Abstract
The frequency–magnitude statistics of 6527 earthquakes with 1.0 ≤ ml ≤ 5.7 and focal depths between 0 and 49 km in the Red Sea region between 1980 and 2021 show that the threshold magnitude, above which most of the Red Sea earthquakes are [...] Read more.
The frequency–magnitude statistics of 6527 earthquakes with 1.0 ≤ ml ≤ 5.7 and focal depths between 0 and 49 km in the Red Sea region between 1980 and 2021 show that the threshold magnitude, above which most of the Red Sea earthquakes are precisely located, is 1.5. The b-value, which identifies regional stress situations and associated energy release modalities, has a value of 0.75, less than in historical data, and averages between 0.4 and 0.85 as it varies over time, indicating modest stress accumulation. We utilised these instrumental data to examine dynamic stress patterns in the Red Sea region, shedding light on the region’s geodynamics and providing a foundation for estimating the region’s seismic hazard. The computed fractal dimension (Dc) has a relatively high value of 2.3, which is significant for the Red Sea’s geological complexity and structural diversity. This result indicates the regular distribution of Red Sea earthquakes, which occur in clusters or along fault lines. The low b-value and comparatively high Dc were most likely due to major earthquakes in the past and the greater stress they caused. The focal mechanisms of the big earthquakes, predominantly normal solutions, are consistent with the movement and extensional regime. The pressure and tension (P-T) axes show a compression axis trending NW-SE and a tension axis trending NE-SW. According to the stress inversion results, the maximum principal stress (σ1) is oriented vertically, the minimum stress axis (σ3) is subhorizontal and strikes in the NE-SW direction, and the intermediate principal stress (σ2) is trending in the NE-SW direction. The variance in the region that characterises the homogeneity of stress directions within the range is 0.19. The stress ratio (R), which identifies the faulting type, is 0.76, suggesting a normal faulting pattern for the region. The hazard parameters are expressed by the probability of exceedance for 1-, 10-, 50-, and 100-year return periods. The highest probability that an earthquake will occur within a 50-year period is thought to be around 6.0. The largest observed catalogue and instrumental magnitudes in the area, 5.7 and 6.7, respectively, show average recurrence intervals of 36 and 142 years. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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18 pages, 3853 KiB  
Article
Determination of Integrity Index Kv in CHN-BQ Method by BP Neural Network Based on Fractal Dimension D
by Qi Zhang, Yixin Shen, Yuechao Pei, Xiaojun Wang, Maohui Wang and Jingqi Lai
Fractal Fract. 2023, 7(7), 546; https://doi.org/10.3390/fractalfract7070546 - 15 Jul 2023
Cited by 3 | Viewed by 984
Abstract
The integrity index Kv is the quantitative index in the CHN-BQ method, which can be determined by the acoustic wave test, volume joint number Jv, or empirical judgment. However, these methods are not convenient and require the practitioner to have [...] Read more.
The integrity index Kv is the quantitative index in the CHN-BQ method, which can be determined by the acoustic wave test, volume joint number Jv, or empirical judgment. However, these methods are not convenient and require the practitioner to have extensive experience. In this study, a new quantitative evaluation of Kv is proposed to determine Kv accurately and conveniently. A method for determining the fractal dimension D based on the structural plane network simulation is proposed. A quantitative relationship between fractal dimension D and integrity index Kv is established based on the geological information from 80 sampling windows in Mingtang Tunnel. To further consider the effect of structural plane conditions on Kv, a BP neural network is constructed with the fractal dimension D and structural plane condition index R3 as input and Kv as output. The BP neural network is trained by 260 groups of tunnel data and validated by 39 groups of test data. The results show that the correlation coefficient R2 between the predicted Kvp and measured Kvm is 0.93, and the average relative error is 7.51%. In addition, the predicted Kvp from the 39 groups of data is compared with the Kvd determined directly by fractal dimension D. It can be found that the Kvd has a larger error compared with the Kvp, especially in the case of a Kv less than 0.5. Finally, the BP neural network for predicting Kv is applied to the Jiulaopo Tunnel. The maximum relative error between the measured Kvm and the predicted Kvp is 5.13%, and the average relative error is 2.71%. The BP neural network is well trained and can accurately predict Kv based on the fractal dimension D and the structural plane condition index R3. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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18 pages, 7893 KiB  
Article
A New Perspective on Predicting Roughness of Discontinuity from Fractal Dimension D of Outcrops
by Qi Zhang, Yuechao Pei, Yixin Shen, Xiaojun Wang, Jingqi Lai and Maohui Wang
Fractal Fract. 2023, 7(7), 496; https://doi.org/10.3390/fractalfract7070496 - 22 Jun 2023
Cited by 1 | Viewed by 936
Abstract
In tunnel construction, predicting the roughness of discontinuity is significant for preventing the collapse of the excavation face. However, currently, we are unable to use a parameter with invariant properties to quantify and predict the roughness of discontinuity. Fractal dimension D is one [...] Read more.
In tunnel construction, predicting the roughness of discontinuity is significant for preventing the collapse of the excavation face. However, currently, we are unable to use a parameter with invariant properties to quantify and predict the roughness of discontinuity. Fractal dimension D is one such parameter that be used to characterize the roughness of discontinuity. The study proposes a new method to predict the roughness of discontinuity from the fractal dimension D of outcrops. The measurement method of the coordinates of outcrops is firstly summarized, and the most suitable method of calculating fractal dimension D is then provided. For characterizing the spatial variability of fractal dimension D, the random field of fractal dimension D is discretized, and the prediction model is then established based on Bayesian theory. The proposed method is applied to one tunnel for predicting the roughness of discontinuity, and the results indicate that the relative errors of prediction are less than 1.5%. The sensitivities of correlation function and discontinuity size are analyzed. It is found that the different correlation functions have no obvious effect on the prediction results, and the proposed method is well applied to relatively large sizes of discontinuity. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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17 pages, 6812 KiB  
Article
Analysis of Damage Characteristics for Skarn Subjected to Freeze-Thaw Cycles Based on Fractal Theory
by Jielin Li, Shuaijie Tan, Chun Yang, Hui Chen and Yun Lin
Fractal Fract. 2023, 7(5), 354; https://doi.org/10.3390/fractalfract7050354 - 27 Apr 2023
Viewed by 1119
Abstract
A large number of rock works in cold areas suffer from long-term freeze-thaw damage, and it seriously affects the stability of mine slopes. In this paper, the XRD component measurement, P-wave velocity, freeze-thaw cycling test at different times, uniaxial compression test, and scanning [...] Read more.
A large number of rock works in cold areas suffer from long-term freeze-thaw damage, and it seriously affects the stability of mine slopes. In this paper, the XRD component measurement, P-wave velocity, freeze-thaw cycling test at different times, uniaxial compression test, and scanning electron microscope (SEM) test were carried out to obtain the mechanical properties and microstructure evolution of skarn under the effect of freeze-thaw cycles. The results of the study indicate that with an increase in the number of freeze-thaw cycles, the mass of the rock gradually increases and the P-wave velocity, uniaxial compressive strength, elastic modulus, and Poisson’s ratio all decrease. Based on the SEM image of the rock after crushing, fine pores and fissures gradually developed, expanded, and penetrated each other under the action of freezing and thawing; the inter-particle bonding force decreased; and the cement gradually loosened. The fractal dimension of the specimens under different numbers of freeze-thaw cycles was obtained using the box dimension method, and the degradation of the fine structure of the rock was quantitatively elaborated. By establishing the relationship between the compressive strength of rocks and the fractal dimension, the mechanism of damage to skarn under freeze-thaw action was further investigated. It provides some theoretical basis for the characterization of freeze-thaw damage of rocks in cold regions. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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24 pages, 8294 KiB  
Article
The Influence of the Fractal Dimension on the Mechanical Behaviors of the Soil–Rock Mixture: A Case Study from Southwest China
by Zhenping Zhang, Xiaodong Fu, Wei Yuan, Qian Sheng, Shaobo Chai and Yuxiang Du
Fractal Fract. 2023, 7(2), 106; https://doi.org/10.3390/fractalfract7020106 - 18 Jan 2023
Cited by 3 | Viewed by 1217
Abstract
As the typical multi-phase geotechnical material, the particle size distribution of the natural soil–rock mixture (S–RM) has a significant impact on the structural and mechanical properties. The coarse grain content used in the laboratory and simulation tests falls short of accurately describing the [...] Read more.
As the typical multi-phase geotechnical material, the particle size distribution of the natural soil–rock mixture (S–RM) has a significant impact on the structural and mechanical properties. The coarse grain content used in the laboratory and simulation tests falls short of accurately describing the particle size distribution feature of the entire material. The main subject of this article is the influence of the fractal dimension on mechanical behaviors based on the fractal theory. The double fractal characteristics were principally discussed along with the typical particle size distribution characteristics of the S–RM in the Three Gorges Reservoir and southwest China. The influence of the various fractal dimensions on the mechanical behaviors of S–RM was then investigated using three groups of large–scale triaxial tests, and the responses of the linear and nonlinear strength indexes were analyzed. The results show that the stress–strain curves of S–RM in the hyperbolic shape are visible under various confining pressure, and the nonlinear strength characteristics can be observed. The coarse grain content exhibits a negative correlation to the average fraction dimension. The difference between the coarse and fine grain fraction dimensions becomes considerably more obvious as the coarse grain content increases, which also increases the error when using the average fractal dimension. The voids between the coarse grains cannot be filled with the fine grains as the grain coarseness grows, resulting in a loose structure and a contact frictional effect, which lowers cohesion and raises the peak friction angle. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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13 pages, 4453 KiB  
Article
Correlation Analysis between Rail Track Geometry and Car-Body Vibration Based on Fractal Theory
by Xiao-Zhou Liu, Zai-Wei Li, Jun Wu, Cheng-Jie Song and Jun-Hua Xiao
Fractal Fract. 2022, 6(12), 727; https://doi.org/10.3390/fractalfract6120727 - 9 Dec 2022
Cited by 2 | Viewed by 2058
Abstract
The effect of track geometry on vehicle vibration is a major concern in high-speed rail (HSR) operation from the perspectives of ride comfort and safety. However, how to quantitatively characterize the relation between them remains a problem to be solved in track quality [...] Read more.
The effect of track geometry on vehicle vibration is a major concern in high-speed rail (HSR) operation from the perspectives of ride comfort and safety. However, how to quantitatively characterize the relation between them remains a problem to be solved in track quality assessment. By using fractal analysis, this paper studies the detailed correlation between track surface and alignment irregularities and car body vertical and lateral acceleration in various wavelength ranges. The time-frequency features of the track irregularity and car-body acceleration are first analyzed based on empirical mode decomposition (EMD). Then, the fractal features of the inspection data are determined by calculating the Hurst exponent of their intrinsic mode functions (IMFs). Finally, the fractal dimensions of the track irregularity and car-body acceleration are obtained, and the correlation between their fractal dimensions with respect to different IMFs is revealed using regression analysis. The results show that the fractal dimension is only related to the roughness of the IMF waveforms of the track irregularity and car-body vibration and is irrelevant to the amplitude of the time series of the data; the correlation coefficient of the fractal dimension of the track irregularity and car-body acceleration is greater than 0.7 for wavelengths greater than 30 m, indicating that the relationship between track irregularity and car-body vibration acceleration is more obvious for long wavelengths. The findings of this research could be used for optimizing HSR track maintenance work from the viewpoint of the ride quality of high-speed trains. Full article
(This article belongs to the Special Issue Fractal Analysis and Its Applications in Geophysical Science)
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