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A Multidisciplinary Approach to Triangular Shapes: Philosophy, Art, Mathematical Properties, and Application Purposes for High-Frequency Signal Processing Using Sierpiński Geometry
A Unified Framework for Fractional and Non-Fractional Operators in Some Function Spaces

Journal Description

Fractal and Fractional

Fractal and Fractional is an international, scientific, peer-reviewed, open access journal of fractals and fractional calculus and their applications in different fields of science and engineering published monthly online by MDPI.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, and other databases.
Journal Rank: JCR - Q1 (Mathematics, Interdisciplinary Applications) / CiteScore - Q1 (Analysis)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 19.9 days after submission; acceptance to publication is undertaken in 2.7 days (median values for papers published in this journal in the first half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.

Impact Factor: 3.3 (2024); 5-Year Impact Factor: 3.2 (2024)

Imprint Information Journal Flyer Open Access ISSN: 2504-3110

Latest Articles

15 pages, 3280 KiB

Open AccessArticle

Fractal Scaling of Storage Capacity Fluctuations in Well Logs from Southeastern Mexican Reservoirs

by Sergio Matias-Gutierres, Edgar Israel García-Otamendi, Hugo David Sánchez-Chávez, Leonardo David Cruz-Diosdado and Roberto Cifuentes-Villafuerte

Fractal Fract. 2025, 9(8), 548; https://doi.org/10.3390/fractalfract9080548 - 21 Aug 2025

This study focuses on a hydrocarbon reservoir located in southeastern Mexico. The analysis uses well log data derived from petrophysical evaluations of storage capacity. The structural complexity of the reservoir and observed heterogeneity in Cretaceous units motivate a fractal-based characterization of spatial fluctuations. [...] Read more.

This study focuses on a hydrocarbon reservoir located in southeastern Mexico. The analysis uses well log data derived from petrophysical evaluations of storage capacity. The structural complexity of the reservoir and observed heterogeneity in Cretaceous units motivate a fractal-based characterization of spatial fluctuations. The objective is to assess the fractal scaling of storage capacity fluctuations using the dynamic Family–Vicsek framework. Critical exponents

α

(roughness),

β

(growth), and z (dynamic) are obtained through structure function metrics. Data collapse techniques and local Hurst exponent distributions are used to explore long-range memory and spatial heterogeneity across wells. This study aims to classify storage capacity fluctuation records based on Euclidean or fractal geometries. This analysis allows a novel characterization of storage trends in the reservoir. The analysis reveals persistent scaling behavior, indicating long-range correlations in the storage capacity fluctuations. Multiscale patterns and variations in local Hurst exponents highlight the presence of multifractality and regional heterogeneity. Specifically, the spatial distribution of local Hurst exponents obtained in this study enables the inference of statistical properties in synthetic wells, providing key input for the structural and functional characterization of the reservoir’s geological model. This approach aims to identify preferential subsurface flow pathways for hydrocarbons and gas. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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19 pages, 295 KiB

Open AccessArticle

On Some Inequalities with Higher Fractional Orders

by Lakhdar Ragoub

Fractal Fract. 2025, 9(8), 547; https://doi.org/10.3390/fractalfract9080547 - 19 Aug 2025

The novelty herein pertains to a class of fractional differential equations involving the Hadamard fractional derivative of higher order. Our investigation encompasses the fractional integral operator of a logarithmic function. The mathematical tools utilized in this study are derived from an important function, [...] Read more.

The novelty herein pertains to a class of fractional differential equations involving the Hadamard fractional derivative of higher order. Our investigation encompasses the fractional integral operator of a logarithmic function. The mathematical tools utilized in this study are derived from an important function, wherein its behavior in terms of maximum value facilitates the establishment of bounds necessary for proving the existence of solutions, specifically through Green’s function. Based on this, we endeavor to estimate the bounds of Green’s function as well as analyze its properties within the considered interval. This approach enables us to establish the Hartman–Wintner- and Lyapunov-type inequalities for a class of fractional Hadamard problems. Furthermore, we introduce a novel technique to determine the maximum value of Green’s function. Finally, we illustrate these findings through two applications. Full article

30 pages, 2186 KiB

Open AccessArticle

Dynamic Analysis of a Fractional-Order SINPR Rumor Propagation Model with Emotional Mechanisms

by Yuze Li, Ying Liu and Jianke Zhang

Fractal Fract. 2025, 9(8), 546; https://doi.org/10.3390/fractalfract9080546 - 19 Aug 2025

The inherent randomness and concealment of rumors in social networks exacerbate their spread, leading to significant societal instability. To explore the mechanisms of rumor propagation for more effective control and mitigation of harm, we propose a novel fractional-order Susceptible-Infected-Negative-Positive-Removed (SINPR) rumor propagation model, [...] Read more.

The inherent randomness and concealment of rumors in social networks exacerbate their spread, leading to significant societal instability. To explore the mechanisms of rumor propagation for more effective control and mitigation of harm, we propose a novel fractional-order Susceptible-Infected-Negative-Positive-Removed (SINPR) rumor propagation model, which simultaneously incorporates emotional mechanisms by distinguishing between positive and negative emotion spreaders, as well as memory effects through fractional-order derivatives. The proposed model extends traditional frameworks by jointly capturing the bidirectional influence of emotions and the anomalous, history-dependent dynamics often overlooked by integer-order models. First, we calculate the equilibrium points and thresholds of the model, and analyze the stability of the equilibrium, along with the sensitivity and transcritical bifurcation associated with the basic reproduction number. Next, we validate the theoretical results through numerical simulations and analyze the individual effects of fractional-order derivatives and emotional mechanisms. Finally, we predict the rumor propagation process using real datasets. Comparative experiments with other models demonstrate that the fractional-order SINPR model achieves R-squared values of 0.9712 and 0.9801 on two different real datasets, underscoring its effectiveness in predicting trends in rumor propagation. Full article

(This article belongs to the Topic Modeling, Stability, and Control of Dynamic Systems and Their Applications)

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30 pages, 7018 KiB

Open AccessArticle

Real-Time Efficiency Prediction in Nonlinear Fractional-Order Systems via Multimodal Fusion

by Biao Ma and Shimin Dong

Fractal Fract. 2025, 9(8), 545; https://doi.org/10.3390/fractalfract9080545 - 19 Aug 2025

Rod pump systems are complex nonlinear processes, and conventional efficiency prediction methods for such systems typically rely on high-order fractional partial differential equations, which severely constrain real-time inference. Motivated by the increasing availability of measured electrical power data, this paper introduces a series [...] Read more.

Rod pump systems are complex nonlinear processes, and conventional efficiency prediction methods for such systems typically rely on high-order fractional partial differential equations, which severely constrain real-time inference. Motivated by the increasing availability of measured electrical power data, this paper introduces a series of prediction models for nonlinear fractional-order PDE systems efficiency based on multimodal feature fusion. First, three single-model predictions—Asymptotic Cross-Fusion, Adaptive-Weight Late-Fusion, and Two-Stage Progressive Feature Fusion—are presented; next, two ensemble approaches—one based on a Parallel-Cascaded Ensemble strategy and the other on Data Envelopment Analysis—are developed; finally, by balancing base-learner diversity with predictive accuracy, a multi-strategy ensemble prediction model is devised for online rod pump system efficiency estimation. Comprehensive experiments and ablation studies on data from 3938 oil wells demonstrate that the proposed methods deliver high predictive accuracy while meeting real-time performance requirements. Full article

(This article belongs to the Special Issue Artificial Intelligence and Fractional Modelling for Energy Systems)

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20 pages, 434 KiB

Open AccessArticle

Large Deviation Principle for Hilfer Fractional Stochastic McKean–Vlasov Differential Equations

by Juan Chen, Haibo Gu, Yutao Yan and Lishan Liu

Fractal Fract. 2025, 9(8), 544; https://doi.org/10.3390/fractalfract9080544 - 19 Aug 2025

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the [...] Read more.

This paper studies the large deviation principle (LDP) of a class of Hilfer fractional stochastic McKean–Vlasov differential equations with multiplicative noise. Firstly, by making use of the Laplace transform and its inverse transform, the solution of the equation is derived. Secondly, considering the equivalence between the LDP and the Laplace principle (LP), the weak convergence method is employed to prove that the equation satisfies the LDP. Finally, through specific example, it is elaborated how to utilize the LDP to analyze the behavioral characteristics of the system under small noise perturbation. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

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23 pages, 11219 KiB

Open AccessArticle

Texture Feature Analysis of the Microstructure of Cement-Based Materials During Hydration

by Tinghong Pan, Rongxin Guo, Yong Yan, Chaoshu Fu and Runsheng Lin

Fractal Fract. 2025, 9(8), 543; https://doi.org/10.3390/fractalfract9080543 - 19 Aug 2025

This study presents a comprehensive grayscale texture analysis framework for investigating the microstructural evolution of cement-based materials during hydration. High-resolution X-ray computed tomography (X-CT) slice images were analyzed across five hydration ages (12 h, 1 d, 3 d, 7 d, and 31 d) [...] Read more.

This study presents a comprehensive grayscale texture analysis framework for investigating the microstructural evolution of cement-based materials during hydration. High-resolution X-ray computed tomography (X-CT) slice images were analyzed across five hydration ages (12 h, 1 d, 3 d, 7 d, and 31 d) using three complementary methods: grayscale histogram statistics, fractal dimension calculation via differential box-counting, and texture feature extraction based on the gray-level co-occurrence matrix (GLCM). The average value of the mean grayscale value of slice (Mean_{G_AVE}) shows a trend of increasing and then decreasing. Average fractal dimension values (D_{B_AVE}) decreased logarithmically from 2.48 (12 h) to 2.41 (31 d), quantifying progressive microstructural homogenization. The trend reflects pore refinement and gel network consolidation. GLCM texture parameters—including energy, entropy, contrast, and correlation—captured the directional statistical patterns and phase transitions during hydration. Energy increased with hydration time, reflecting greater spatial homogeneity and phase continuity, while entropy and contrast declined, signaling reduced structural complexity and interfacial sharpness. A quantitative evaluation of parameter performance based on intra-sample stability, inter-sample discrimination, and signal-to-noise ratio (SNR) revealed energy, entropy, and contrast as the most effective descriptors for tracking hydration-induced microstructural evolution. This work demonstrates a novel, integrative, and segmentation-free methodology for texture quantification, offering robust insights into the microstructural mechanisms of cement hydration. The findings provide a scalable basis for performance prediction, material optimization, and intelligent cementitious design. Full article

(This article belongs to the Special Issue Fractal Analysis and Its Applications in Materials Science)

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21 pages, 2993 KiB

Open AccessArticle

ABAQUS Subroutine-Based Implementation of a Fractional Consolidation Model for Saturated Soft Soils

by Tao Zeng, Tao Feng and Yansong Wang

Fractal Fract. 2025, 9(8), 542; https://doi.org/10.3390/fractalfract9080542 - 17 Aug 2025

This paper presents a finite element implementation of a fractional rheological consolidation model in ABQUS, in which the fractional Merchant model governs the mechanical behavior of the soil skeleton, and the water flow is controlled by the fractional Darcy’s law. The implementation generally [...] Read more.

This paper presents a finite element implementation of a fractional rheological consolidation model in ABQUS, in which the fractional Merchant model governs the mechanical behavior of the soil skeleton, and the water flow is controlled by the fractional Darcy’s law. The implementation generally involves two main parts: subroutine-based fractional constitutive models’ development and their coupling. Considering the formal similarity between the energy equation and the mass equation, the fractional Darcy’s law was implemented using the UMATHT subroutine. The fractional Merchant model was then realized through the UMAT subroutine. Both subroutines were individually verified and then successfully coupled. The coupling was achieved by modifying the stress update scheme based on Biot’s poroelastic theory and the effective stress principle in UMAT, enabling a finite element analysis of the fractional consolidation model. Finally, the model was applied to simulate the consolidation behavior of a multi-layered foundation. The proposed approach may serve as a reference for the finite element implementation of consolidation models incorporating a fractional seepage model in ABAQUS. Full article

(This article belongs to the Special Issue Fractional Derivatives in Mathematical Modeling and Applications)

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28 pages, 4465 KiB

Open AccessArticle

Neural Networks-Based Analytical Solver for Exact Solutions of Fractional Partial Differential Equations

by Shanhao Yuan, Yanqin Liu, Limei Yan, Runfa Zhang and Shunjun Wu

Fractal Fract. 2025, 9(8), 541; https://doi.org/10.3390/fractalfract9080541 - 16 Aug 2025

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in [...] Read more.

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 3rd Edition)

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29 pages, 3058 KiB

Open AccessArticle

Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation

by Sahar Abbas, Abdul Ahad Abro, Syed Muhammad Daniyal, Hanaa A. Abdallah, Sadique Ahmad, Abdelhamied Ashraf Ateya and Noman Bin Zahid

Fractal Fract. 2025, 9(8), 540; https://doi.org/10.3390/fractalfract9080540 - 16 Aug 2025

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of [...] Read more.

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractional calculus. Employing Banach’s and Krasnoselskii’s fixed-point theorems, we establish the existence and uniqueness of solutions. Stability is analyzed in the Ulam–Hyers (UHS), generalized Ulam–Hyers (GUHS), and Ulam–Hyers–Rassias (UHRS) senses. A modified Adomian decomposition method (MADM) is introduced to derive explicit solutions without linearization, preserving the problem’s original structure. The first numerical example validates the theoretical findings on existence, uniqueness, and stability, supplemented by graphical results obtained via the MADM. Further examples illustrate fuzzy solutions by varying the uncertainty level (r), the variable (x), and both parameters simultaneously. The numerical results align with the theoretical analysis, demonstrating the efficacy and applicability of the proposed method. Full article

(This article belongs to the Special Issue Recent Developments in Multidimensional Fractional Differential Equations and Fractional Difference Equations)

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29 pages, 5533 KiB

Open AccessArticle

Automated First-Arrival Picking and Source Localization of Microseismic Events Using OVMD-WTD and Fractal Box Dimension Analysis

by Guanqun Zhou, Shiling Luo, Yafei Wang, Yongxin Gao, Xiaowei Hou, Weixin Zhang and Chuan Ren

Fractal Fract. 2025, 9(8), 539; https://doi.org/10.3390/fractalfract9080539 - 16 Aug 2025

Microseismic monitoring has become a critical technology for hydraulic fracturing in unconventional oil and gas reservoirs, owing to its high temporal and spatial resolution. It plays a pivotal role in tracking fracture propagation and evaluating stimulation effectiveness. However, the automatic picking of first-arrival [...] Read more.

Microseismic monitoring has become a critical technology for hydraulic fracturing in unconventional oil and gas reservoirs, owing to its high temporal and spatial resolution. It plays a pivotal role in tracking fracture propagation and evaluating stimulation effectiveness. However, the automatic picking of first-arrival times and accurate source localization remain challenging under complex noise conditions, which constrain the reliability of fracture parameter inversion and reservoir assessment. To address these limitations, we propose a hybrid approach that combines optimized variational mode decomposition (OVMD), wavelet thresholding denoising (WTD), and an adaptive fractal box-counting dimension algorithm for enhanced first-arrival picking and source localization. Specifically, OVMD is first employed to adaptively decompose seismic signals and isolate noise-dominated components. Subsequently, WTD is applied in the multi-scale frequency domain to suppress residual noise. An adaptive fractal dimension strategy is then utilized to detect change points and accurately determine the first-arrival time. These results are used as inputs to a particle swarm optimization (PSO) algorithm for source localization. Both numerical simulations and laboratory experiments demonstrate that the proposed method exhibits high robustness and localization accuracy under severe noise conditions. It significantly outperforms conventional approaches such as short-time Fourier transform (STFT) and continuous wavelet transform (CWT). The proposed framework offers reliable technical support for dynamic fracture monitoring, detailed reservoir characterization, and risk mitigation in the development of unconventional reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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4 pages, 163 KiB

Open AccessEditorial

Fractal and Fractional Theories in Advancing Geotechnical Engineering Practices

by Shao-Heng He, Zhi Ding and Panpan Guo

Fractal Fract. 2025, 9(8), 537; https://doi.org/10.3390/fractalfract9080537 - 16 Aug 2025

Fractal and fractional theories have developed over several decades and have gradually grown in popularity, with significant applications in geotechnical engineering [...] Full article

(This article belongs to the Special Issue Fractal and Fractional in Geotechnical Engineering)

23 pages, 5400 KiB

Open AccessArticle

Quantitative Analysis of Multi-Angle Correlation Between Fractal Dimension of Anthracite Surface and Its Coal Quality Indicators in Different Regions

by Shoule Zhao and Dun Wu

Fractal Fract. 2025, 9(8), 538; https://doi.org/10.3390/fractalfract9080538 - 15 Aug 2025

The nanoporous structure of coal is crucial for the occurrence and development of coalbed methane (CBM). This study, leveraging the combined characterization of atomic force microscopy (AFM) and Gwyddion software (v2.62), investigated six anthracite samples with varying degrees of metamorphism (R_o = [...] Read more.

The nanoporous structure of coal is crucial for the occurrence and development of coalbed methane (CBM). This study, leveraging the combined characterization of atomic force microscopy (AFM) and Gwyddion software (v2.62), investigated six anthracite samples with varying degrees of metamorphism (R_o = 2.11–3.36%). It revealed the intrinsic relationships between their nanoporous structures, surface morphologies, fractal characteristics, and coalification processes. The research found that as R_o increases, the surface relief of coal decreases significantly, with pore structures evolving from being macropore-dominated to micropore-enriched, and the surface tending towards smoothness. Surface roughness parameters (R_a, R_q) exhibit a negative correlation with R_o. Quantitative data indicate that area porosity, pore count, and shape factor positively correlate with metamorphic grade, while mean pore diameter negatively correlates with it. The fractal dimensions calculated using the variance partition method, cube-counting method, triangular prism measurement method, and power spectrum method all show nonlinear correlations with R_o, moisture (M_ad), ash content (A_ad), and volatile matter (V_daf). Among these, the fractal dimension obtained by the triangular prism measurement method has the highest correlation with R_o, A_ad, and V_daf, while the variance partition method shows the highest correlation with M_ad. This study clarifies the regulatory mechanisms of coalification on the evolution of nanoporous structures and surface properties, providing a crucial theoretical foundation for the precise evaluation and efficient exploitation strategies of CBM reservoirs. Full article

(This article belongs to the Special Issue Applications of Fractal Dimensions in Rock Mechanics and Geomechanics)

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19 pages, 2887 KiB

Open AccessArticle

Multifractal Characterization of Heterogeneous Pore Water Redistribution and Its Influence on Permeability During Depletion: Insights from Centrifugal NMR Analysis

by Fangkai Quan, Wei Lu, Yu Song, Wenbo Sheng, Zhengyuan Qin and Huogen Luo

Fractal Fract. 2025, 9(8), 536; https://doi.org/10.3390/fractalfract9080536 - 15 Aug 2025

The dynamic process of water depletion plays a critical role in both surface coalbed methane (CBM) development and underground gas extraction, reshaping water–rock interactions and inducing complex permeability responses. Addressing the limited understanding of the coupling mechanism between heterogeneous pore water evolution and [...] Read more.

The dynamic process of water depletion plays a critical role in both surface coalbed methane (CBM) development and underground gas extraction, reshaping water–rock interactions and inducing complex permeability responses. Addressing the limited understanding of the coupling mechanism between heterogeneous pore water evolution and permeability during dynamic processes, this study simulates reservoir transitions across four zones (prospective planning, production preparation, active production, and mining-affected zones) via centrifugal experiments. The results reveal a pronounced scale dependence in pore water distribution. During low-pressure stages (0–0.54 MPa), rapid drainage from fractures and seepage pores leads to a ~12% reduction in total water content. In contrast, high-pressure stages (0.54–3.83 MPa) promote water retention in adsorption pores, with their relative contribution rising to 95.8%, forming a dual-structure of macropore drainage and micropore retention. Multifractal analysis indicates a dual-mode evolution of movable pore space. Under low centrifugal pressure, D₋₁₀ and Δα decrease by approximately 34% and 36%, respectively, reflecting improved connectivity within large-pore networks. At high centrifugal pressure, an ~8% increase in D₀−D₂ suggests that pore-scale heterogeneity in adsorption pores inhibits further seepage. A quantitative coupling model establishes a quadratic relationship between fractal parameters and permeability, illustrating that permeability enhancement results from the combined effects of pore volume expansion and structural homogenization. As water saturation decreases from 1.0 to 0.64, permeability increases by more than 3.5 times. These findings offer theoretical insights into optimizing seepage pathways and improving gas recovery efficiency in dynamically evolving reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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30 pages, 8331 KiB

Open AccessArticle

Fracture Complexity and Mineral Damage in Shale Hydraulic Fracturing Based on Microscale Fractal Analysis

by Xin Liu, Jiaqi Zhang, Tianjiao Li, Zhengzhao Liang, Siwei Meng, Licai Zheng and Na Wu

Fractal Fract. 2025, 9(8), 535; https://doi.org/10.3390/fractalfract9080535 - 15 Aug 2025

The geological structural complexity and microscale heterogeneity of shale reservoirs, characterized by the brittleness index and natural fracture density, exert a decisive effect on hydraulic fracturing’s effectiveness. However, the mechanisms underlying the true microscale heterogeneity of shale structures, which is neglected in conventional [...] Read more.

The geological structural complexity and microscale heterogeneity of shale reservoirs, characterized by the brittleness index and natural fracture density, exert a decisive effect on hydraulic fracturing’s effectiveness. However, the mechanisms underlying the true microscale heterogeneity of shale structures, which is neglected in conventional models and influences fracture evolution, remain unclear. Here, high-resolution scanning electron microscopy (SEM) was employed to obtain realistic distributions of mineral components and natural fractures, and hydraulic–mechanical coupled simulation models were developed within the Realistic Failure Process Analysis (RFPA) simulator using digital rock techniques. The analysis examined how the brittleness index and natural fracture density affect the fracture morphology’s complexity, mineral failure behavior, and flow conductivity. Numerical simulations show that the main fractures preferentially propagate toward areas with high local brittleness and dense natural fractures. Both the fracture’s fractal dimension and the stimulated reservoir volume increased with the brittleness index. A moderate natural fracture density promotes the fracture network’s complexity, whereas excessive densities may suppress the main fracture’s propagation. Microscopically, organic matter and silicate minerals are more prone to damage, predominantly tensile failures under external loading. These findings highlight the dominant role of microscale heterogeneity in shale fracturing and provide theoretical support for fracture control and stimulation optimization in complex reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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17 pages, 4064 KiB

Open AccessArticle

Study on Multi-Scale Damage Evolution of Sandstone Under Freeze–Thaw Cycles: A Computational Perspective Based on Pore Structure and Fractal Dimension

by Jianhui Qiu, Keping Zhou, Guanglin Tian and Taoying Liu

Fractal Fract. 2025, 9(8), 534; https://doi.org/10.3390/fractalfract9080534 - 15 Aug 2025

Understanding the intrinsic relationship between microscopic structures and macroscopic mechanical properties of rock under freeze–thaw (F-T) conditions is essential for ensuring the safety and stability of geotechnical engineering in cold regions. In this study, a series of F-T cycle tests, nuclear magnetic resonance [...] Read more.

Understanding the intrinsic relationship between microscopic structures and macroscopic mechanical properties of rock under freeze–thaw (F-T) conditions is essential for ensuring the safety and stability of geotechnical engineering in cold regions. In this study, a series of F-T cycle tests, nuclear magnetic resonance (NMR) measurements, and uniaxial compression tests were conducted on sandstone samples. The mechanisms by which F-T cycles influence pore structure and mechanical behavior were analyzed, revealing their internal correlation. A degradation model for peak strength was developed using mesopore porosity as the key influencing parameter. The results showed that with increasing F-T cycles, the total porosity and mesopore and macropore porosities all exhibited increasing trends, whereas the micropore and different fractal dimensions decreased. The compaction stage in the stress–strain curves became increasingly prominent with more F-T cycles. Meanwhile, the peak strength and secant modulus decreased, while the peak strain increased. When the frost heave pressure induced by water–ice phase transitions exceeded the ultimate bearing capacity of pore walls, smaller pores progressively evolved into larger ones, leading to an increase in the mesopores and macropores. Notably, mesopores and macropores demonstrated significant fractal characteristics. The transformation in pore size disrupted the power-law distribution of pore radii and reduced fractal dimensions. A strong correlation was observed between peak strength and both the mesopore and mesopore fractal dimensions. The increase in mesopores and macropores enhanced the compaction stage of the stress–strain curve. Moreover, the expansion and interconnection of mesopores under loading conditions degraded the deformation resistance and load-bearing capacity, thereby reducing both the secant modulus and peak strength. The degradation model for peak strength, developed based on changes in mesopore ratio, proved effective for evaluating the mechanical strength when subjected to different numbers of F-T cycles. Full article

(This article belongs to the Special Issue Applications of Fractal Dimensions in Rock Mechanics and Geomechanics)

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22 pages, 4009 KiB

Open AccessArticle

A Multi-Dimensional Feature Extraction Model Fusing Fractional-Order Fourier Transform and Convolutional Information

by Haijing Sun, Wen Zhou, Jiapeng Yang, Yichuan Shao, Le Zhang and Zhiqiang Mao

Fractal Fract. 2025, 9(8), 533; https://doi.org/10.3390/fractalfract9080533 - 14 Aug 2025

In the field of deep learning, the traditional Vision Transformer (ViT) model has some limitations when dealing with local details and long-range dependencies; especially in the absence of sufficient training data, it is prone to overfitting. Structures such as retinal blood vessels and [...] Read more.

In the field of deep learning, the traditional Vision Transformer (ViT) model has some limitations when dealing with local details and long-range dependencies; especially in the absence of sufficient training data, it is prone to overfitting. Structures such as retinal blood vessels and lesion boundaries have distinct fractal properties in medical images. The Fractional Convolution Vision Transformer (FCViT) model is proposed in this paper, which effectively compensates for the deficiency of ViT in local feature capture by fusing convolutional information. The ability to classify medical images is enhanced by analyzing frequency domain features using fractional-order Fourier transform and capturing global information through a self-attention mechanism. The three-branch architecture enables the model to fully understand the data from multiple perspectives, capturing both local details and global context, which in turn improves classification performance and generalization. The experimental results showed that the FCViT model achieved 93.52% accuracy, 93.32% precision, 92.79% recall, and a 93.04% F1-score on the standardized fundus glaucoma dataset. The accuracy on the Harvard Dataverse-V1 dataset reached 94.21%, with a precision of 93.73%, recall of 93.67%, and F1-score of 93.68%. The FCViT model achieves significant performance gains on a variety of neural network architectures and tasks with different source datasets, demonstrating its effectiveness and utility in the field of deep learning. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Fourier Transforms and Applications, 2nd Edition)

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18 pages, 6449 KiB

Open AccessArticle

Analysis of the Microscopic Pore Structure Characteristics of Sandstone Based on Nuclear Magnetic Resonance Experiments and Nuclear Magnetic Resonance Logging Technology

by Shiqin Li, Chuanqi Tao, Haiyang Fu, Huan Miao and Jiutong Qiu

Fractal Fract. 2025, 9(8), 532; https://doi.org/10.3390/fractalfract9080532 - 14 Aug 2025

This study focuses on the complex pore structure and pronounced heterogeneity of tight sandstone reservoirs in the Linxing area of the Ordos Basin and develops a multi-scale quantitative characterization approach to investigate the coupling mechanism between pore structure and reservoir properties. Six core [...] Read more.

This study focuses on the complex pore structure and pronounced heterogeneity of tight sandstone reservoirs in the Linxing area of the Ordos Basin and develops a multi-scale quantitative characterization approach to investigate the coupling mechanism between pore structure and reservoir properties. Six core samples were selected from the Shiqianfeng Formation (depth interval: 1326–1421 m) for detailed analysis. Cast thin sections and scanning electron microscopy (SEM) experiments were employed to characterize pore types and structural features. Nuclear magnetic resonance (NMR) experiments were conducted to obtain T₂ spectra, which were used to classify bound and movable pores, and their corresponding fractal dimensions were calculated separately. In addition, NMR logging data from the corresponding well intervals were integrated to assess the applicability and consistency of the fractal characteristics at the logging scale. The results reveal that the fractal dimension of bound pores shows a positive correlation with porosity, whereas that of movable pores is negatively correlated with permeability, indicating that different scales of pore structural complexity exert distinct influences on reservoir performance. Mineral composition affects the evolution of pore structures through mechanisms such as framework support, dissolution, and pore-filling, thereby further enhancing reservoir heterogeneity. The consistency between logging responses and experimental observations verifies the regional applicability of fractal analysis. Bound pores dominate within the studied interval, and the vertical variation of the PMF/BVI ratio aligns closely with both the NMR T₂ spectra and fractal results. This study demonstrates that fractal dimension is an effective descriptor of structural characteristics across different pore types and provides a theoretical foundation and methodological support for the evaluation of pore complexity and heterogeneity in tight sandstone reservoirs. Full article

(This article belongs to the Special Issue Multiscale Fractal Analysis in Unconventional Reservoirs)

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20 pages, 6947 KiB

Open AccessArticle

Fractal Evolution Characteristics of Weakly Cemented Overlying Rock Fractures in Extra-Thick Coal Seams Mining in Western Mining Areas

by Cun Zhang, Zhaopeng Ren, Jun He and Xiangyu Zhao

Fractal Fract. 2025, 9(8), 531; https://doi.org/10.3390/fractalfract9080531 - 14 Aug 2025

Coal mining disturbance induces progressive damage and fracturing in overlying rock (OLR), forming a complex fracture network. This process triggers groundwater depletion, ecological degradation, and severely compromises mine safety. Based on field drilling sampling and mechanical experiments, this paper reveals the occurrence properties [...] Read more.

Coal mining disturbance induces progressive damage and fracturing in overlying rock (OLR), forming a complex fracture network. This process triggers groundwater depletion, ecological degradation, and severely compromises mine safety. Based on field drilling sampling and mechanical experiments, this paper reveals the occurrence properties and characteristics of weakly cemented overlying rock (WCOLR). At the same time, similar simulation experiments, DIC speckle analysis system, and fractal theory are used to explain the development and evolution mechanism of mining-induced fractures under this special geological condition. The OLR fracture is determined based on the grid fractal dimension (D) distribution. A stress arch-bed separation (BS) co-evolution model is established based on dynamic cyclic BS development and stress arch characteristics, enabling identification of BS horizons. The results show that the overlying weak and extremely weak rock accounts for more than 90%. During the process of longwall face (LF) advancing, the D undergoes oscillatory evolution through five distinct stages: rapid initial growth, constrained slow growth under thick, soft strata (TSS), dimension reduction induced by fracturing and compaction of TSS, secondary growth from newly generated fractures, and stabilization upon reaching full extraction. Grid-based D analysis further categorizes fracture zones, indicating a water conducting fracture zone (WCFZ) height of 160~180 m. Mining-induced fractures predominantly concentrate at dip angles of 0–10°, 40–50°, and 170–180°. Horizontally BS fractures account for 70.2% of the total fracture population, vertically penetrating fractures constitute 13.1% and transitional fractures make up the remaining 16.7%. The stress arch height is 314.4 m, and the stable BS horizon is 260 m away from the coal seam. Finally, an elastic foundation theory-based model was used to predict BS development under top-coal caving operations. This research provides scientific foundations for damage-reduced mining in ecologically vulnerable Western China coalfields. Full article

(This article belongs to the Special Issue Applications of Fractal Analysis in Underground Engineering, Second Edition)

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26 pages, 3065 KiB

Open AccessArticle

A Kangaroo Escape Optimizer-Enabled Fractional-Order PID Controller for Enhancing Dynamic Stability in Multi-Area Power Systems

by Sulaiman Z. Almutairi and Abdullah M. Shaheen

Fractal Fract. 2025, 9(8), 530; https://doi.org/10.3390/fractalfract9080530 - 14 Aug 2025

In this study, we propose a novel metaheuristic algorithm named Kangaroo Escape optimization Technique (KET), inspired by the survival-driven escape strategies of kangaroos in unpredictable environments. The algorithm integrates a chaotic logistic energy adaptation strategy to balance a two-phase exploration process—zigzag motion and [...] Read more.

In this study, we propose a novel metaheuristic algorithm named Kangaroo Escape optimization Technique (KET), inspired by the survival-driven escape strategies of kangaroos in unpredictable environments. The algorithm integrates a chaotic logistic energy adaptation strategy to balance a two-phase exploration process—zigzag motion and long-jump escape—and an adaptive exploitation phase with local search guided by either nearby elite solutions or random peers. A unique decoy drop mechanism is introduced to prevent premature convergence and ensure dynamic diversity. KET is applied to optimize the parameters of a fractional-order Proportional Integral Derivative (PID) controller for Load Frequency Control (LFC) in interconnected power systems. The designed fractional-order PID controller-based KET optimization extends the conventional PID by introducing fractional calculus into the integral and derivative terms, allowing for more flexible and precise control dynamics. This added flexibility enables enhanced robustness and tuning capability, particularly useful in complex and uncertain systems such as modern power systems. Comparative results with existing state-of-the-art algorithms demonstrate the superior robustness, convergence speed, and control accuracy of the proposed approach under dynamic scenarios. The proposed KET-fractional order PID controller offers 29.6% greater robustness under worst-case conditions and 36% higher consistency across multiple runs compared to existing techniques. It achieves optimal performance faster than the Neural Network Algorithm (NNA), achieving its best Integral of Time Absolute Error (ITAE) value within the first 20 iterations, demonstrating its superior learning rate and early-stage search efficiency. In addition to LFC, the robustness and generality of the proposed KET were validated on a standard speed reducer design problem, demonstrating superior optimization performance and consistent convergence when compared to several recent metaheuristics. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Control with Special Focus to Power Systems)

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14 pages, 356 KiB

Open AccessArticle

Pointwise Error Analysis of the Corrected L1 Scheme for the Multi-Term Subdiffusion Equation

by Qingzhao Li and Chaobao Huang

Fractal Fract. 2025, 9(8), 529; https://doi.org/10.3390/fractalfract9080529 - 14 Aug 2025

This paper considers the multi-term subdiffusion equation with weakly singular solutions. In order to use sparser meshes near the initial time, a novel scheme (which we call the corrected L1 scheme) on graded meshes is constructed to estimate the multi-term Caputo fractional derivative [...] Read more.

This paper considers the multi-term subdiffusion equation with weakly singular solutions. In order to use sparser meshes near the initial time, a novel scheme (which we call the corrected L1 scheme) on graded meshes is constructed to estimate the multi-term Caputo fractional derivative by investigating a corrected term for the nonuniform L1 scheme. Combining this nonuniform corrected L1 scheme in the temporal direction and the finite element method (FEM) in the spatial direction, a fully discrete scheme for solving the multi-term subdiffusion equation is developed. The stability result of the developed scheme is given. Furthermore, the optimal pointwise-in-time error estimate of the developed scheme is derived. Finally, several numerical experiments are conducted to verify our theoretical findings. Full article

(This article belongs to the Special Issue Exploration and Analysis of Higher-Order Numerical Methods for Fractional Differential Equations)

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