Search Results (119)

The pore system of coal reservoirs is the fundamental structural basis controlling coalbed methane adsorption, storage, diffusion, migration, and production performance. Its development characteristics are influenced not only by thermal maturity but also by maceral composition. To clarify the differences in mesopore structure and fractal characteristics of coal samples with different maceral compositions, this study focuses on the Xishanyao Formation coals in the Nileke Depression, northeastern Yili Basin. Ten coal samples were collected and analyzed using coal petrology and coal quality tests, low-temperature nitrogen adsorption experiments, and mono-fractal and multifractal characterization. The results show that the samples have R_o,max values ranging from 0.70% to 1.31%, indicating broadly comparable thermal maturity, whereas vitrinite and inertinite contents vary significantly. Accordingly, the samples can be classified into vitrinite-rich Type I and relatively low-vitrinite Type II. Low-temperature nitrogen adsorption results indicate that both types contain mesopores within the 2–100 nm range, but they differ markedly in pore-size distribution and hysteresis-loop characteristics. Type I is mainly controlled by the 10–50 nm pore-size interval and is characterized by a relatively concentrated distribution of medium-scale mesopores, whereas Type II is more strongly influenced by the 50–100 nm interval and shows a more dispersed pore distribution. Fractal analysis further reveals systematic differences between the two types in pore-surface roughness, structural complexity, and scale-dependent heterogeneity. Type I is mainly characterized by enhanced pore-structure complexity governed by medium-scale mesopores, whereas Type II is more strongly characterized by enhanced heterogeneity controlled by variations in larger-scale mesopores. Overall, differences in maceral composition influence the complexity and multiscale heterogeneity of the mesopore system by controlling the dominant pore-size intervals and pore-organization patterns. These findings provide a geological basis for reservoir evaluation of the Xishanyao Formation coals and for understanding the mechanisms of coalbed methane occurrence and migration. Full article

The prominent multi-scale pore heterogeneity widely developed in deep-buried coal reservoirs, which seriously restricts accurate reservoir characterization and precise resource evaluation for deep coalbed methane exploitation. Taking deep-buried coal reservoirs of the Benxi Formation in the Ordos Basin as the research target, this study integrates data from low-temperature CO₂/N₂ adsorption and high-pressure mercury intrusion experiments. Segmented monofractal quantification and multi-fractal singularity analysis are further adopted. Fractal differentiation characteristics, scale effects of pores at different scales, and the synergistic control mechanism of multiple geological factors were indicated. Eight segmented single-fractal dimensions (D1~D8) are defined. The results indicate an obvious scale-dependent zonal distribution of pore heterogeneity in deep-buried coal. Micropores smaller than 1.2 nm possess the largest fractal dimension and the strongest heterogeneity. They act as the primary adsorption and storage space for coalbed methane. Mesopores ranging from 1.1 nm to 8 nm have the lowest fractal dimension with the most homogeneous structure, serving as major gas migration pathways. The heterogeneity of macropores gradually increases with pore size. A continuous full-scale pore size distribution curve is reconstructed. Coal reservoirs exhibit a typical bimodal pore structure dominated by micropores and macropores, with the micropore-dominated storage and macropore-dominated seepage. Key multi-fractal indicators including Δα, Δf and the Hurst index are used for quantitative comparison. Micropores display strong aggregation and weak interpore connectivity. Mesopores own superior connectivity, while their heterogeneity differs greatly between individual samples. Macropores feature moderate aggregation and connectivity. Coalification degree, organic macerals, clay minerals and industrial parameters are associated with the formation and differentiation of pore heterogeneity. Each factor differentially regulates the fractal evolution of pores across various scales. Full article

Each financial market carries a unique “fractal signature” with its own distinct risk and return pattern. This study comparatively deciphers these fractal signatures of the leading stock market indices of the Emerging Seven (E7) countries (Turkey, India, Brazil, Mexico, Russia, China, Indonesia), using Multifractal Detrended Fluctuation Analysis (MFDFA) with data covering the 2021–2025 period. The findings reveal that all examined markets deviate from the classical random walk model and exhibit distinct multifractal characteristics. However, significant differences were observed among these signatures: in contrast to Russia’s chaotic structure, which showed extreme fragility to geopolitical shocks, the Chinese and Mexican markets presented a more stable and homogeneous risk profile. In all indices, it was found that small-scale fluctuations carry a strong long-memory effect (stable trends), while large-scale fluctuations assume a more random character (sudden shocks). This asymmetric behavior confirms the heterogeneous nature of investor expectations. For example, the generalized Hurst exponents H(q) ranged from 0.22 (RTS, Russia) to 0.73 (BIST100, Turkey), and the spectrum width Δα varied between 0.10 (Mexico) and 0.45 (Russia), confirming significant heterogeneity in market complexity. Turkey’s BIST100 index, with its structure encompassing both predictable and sudden-shock-prone dynamics, occupies a balanced position within this spectrum. Consequently, the study confirms that understanding these unique fractal signatures of emerging markets is a fundamental prerequisite for formulating effective risk management strategies and achieving global portfolio diversification. Full article

In recent years, it has become inevitable to dig underneath existing highways when excavating tunnels. The soil settlement induced by ground excavation may adversely affect existing highways. In this study, a settlement monitoring system is used to obtain the settlement sequence of multiple measurement points on the pavement. Multifractal detrended fluctuation analysis (MF-DFA) is used to focus on analyzing the multiple fractal features of the pavement settlement rate. The results show that the settlement rates of the highway caused by the tunnel excavation and construction process all show multiple fractal characteristics. The fluctuations in the measurement points above and near the entrance of the tunnel are more complex and intense. Based on the moving-average method (MA), convolutional neural network (CNN), and Extreme Learning Machine (ELM), MA-CNN and MA-ELM prediction models are constructed to predict the settlement value sequences of the fluctuating points. The results indicate that the MA-ELM prediction model demonstrates superior predictive performance (with R² values of 0.956, 0.950, and 0.979 on the test set). Further, with the help of the Dung Beetle Optimizer (DBO), a meta-heuristic algorithm for parameter optimization, the hybrid model DBO-MA-ELM greatly improves the prediction performance (R² of 0.975, 0.997, 0.998 for the testing set). Full article

Glaucoma is a leading cause of irreversible vision loss, and its early diagnosis remains critically important yet challenging. Traditional assessment based on the cup-to-disc ratio is often insufficient at early stages, whereas the retinal vascular network can provide additional quantitative biomarkers. This study develops and validates a binary classification method for distinguishing healthy from glaucomatous fundus images by combining deep-learning-based vessel segmentation, fractal and multifractal analysis, and textural features. The public ORIGA dataset is utilized. Images are converted to grayscale using three alternative approaches, followed by Gray-Level Co-occurrence Matrix texture analysis and fractal analysis based on the differential box-counting method. Vessel segmentation is implemented via a U-Net neural network trained on a combination of public datasets, after which multifractal analysis is performed on the resulting binary masks. The extracted features are used to train and compare several machine learning models with hyperparameter optimization. The best-performing model among ONH-based features (Random Forest) achieves 75.00%; however, a logistic regression model using multifractal parameters and CDR reaches 86.17%, substantially outperforming the CDR-only baseline (66.15%). Notably, while classical fractal dimension shows only marginal differences (1–2% relative change) between groups, multifractal parameters reveal distinct changes: the multifractal spectrum width $\Delta \alpha$ increases markedly and the minimum singularity exponent ${\alpha }_{\min }$ decreases in glaucomatous eyes, indicating increased heterogeneity of the vascular network. These findings suggest that multifractal characteristics of the vascular network can serve as reliable and sensitive biomarkers for automated glaucoma screening, offering clear advantages over classical fractal analysis. Full article

Landslides constitute a globally pervasive and highly destructive natural hazard. Although artificial intelligence (AI)-driven landslide susceptibility mapping has emerged as an effective tool for delineating high-risk zones, its predictive performance is frequently constrained by inherent data noise and insufficient characterization of landslide triggering factors, restricting the credibility of the mapping results. In this study, to remedy this limitation, we adopt fractal analysis to extract latent inherent information from topographic features. Specifically, the box-counting method and multifractal analysis are applied to excavate the intrinsic nonlinear characteristics embedded in eight topographic factors, and an improved K-means algorithm is utilized to perform feature selection and construct a dedicated fractal feature dataset, which is fed to advanced AI models. Our results indicate that the information dimension ($D_1$) of the slope gradient, the correlation dimension ($D_2$) of aspect, land relief, the $D_2$ of roughness, the $D_2$ of plan curvature, the multifractal spectrum width ($∆\mathsf{\alpha }$) of profile curvature, the $D_2$ of elevation, and the surface cutting depth were the most effective features, demonstrating superior performance in capturing landslide targets. Comparative performance evaluations reveal that AI models trained on fractal features demonstrate substantially superior predictive capabilities compared to AI models trained on raw features. This superiority is consistently evidenced across key evaluation metrics, including overall accuracy, kappa coefficient, F1-score, and predictive efficiency, demonstrating that the integration of fractal characteristics significantly augments model robustness and predictive efficacy. To mitigate the ‘black-box’ problem of AI modeling, Shapley additive explanations were employed to quantify individual feature contributions and elucidate the underlying predictive mechanisms. Our findings indicate that the integration of fractal analysis yields highly discriminative and robust feature representations, thereby expanding the representational capacity of the models and improving predictive accuracy. Furthermore, a joint assessment of spatial uncertainty and susceptibility maps demonstrates that these models exhibit low predictive variance and high spatial stability when delineating high-susceptibility zones. Notably, models utilizing fractal-derived features achieve superior spatial capture efficiency. The resultant topographic features characterized by fractal representation and selected via the improved K-means algorithm can significantly improve the predictive performance of trained AI models in landslide susceptibility mapping tasks, offering a scientific and viable technical approach for future landslide prediction and prevention. Full article

The pore-throat structure is a key factor in the exploration and development of tight sandstone reservoirs. In the present study, 14 tight sandstone samples from the Chang 8 member of the Ordos Basin were analyzed using high-pressure mercury intrusion, cast thin section analysis, scanning electron microscopy and cathodoluminescence imaging techniques. Fractal dimensions, obtained from the slopes of log(S_W) versus log(P_c) double-logarithmic plots, were applied to quantitatively characterize pore-throat structures and classify reservoirs through multifractal analysis, and discuss the diagenetic controlling factors affecting the pore-throat structure of different reservoir types. The results showed that the Chang 14 tight sandstones are characterized as two segments fractal features, which indicated that these samples have complex pore-throat structure and consist of two types of spaces: mesopore-throat spaces and micropore-throat spaces. The mesopore-throat system shows a higher fractal dimension (D1: 2.74–2.99), indicating greater heterogeneity and irregularity, while the micropore-throat system exhibits a lower dimension (D2: 2.28–2.61). D1 exhibits a negative correlation with the porosity and permeability of mesopores, while D2 shows a weak positive correlation with the properties of micropores. The total fractal dimension (D) is weakly correlated with overall reservoir properties, confirming that reservoir storage and flow capacity are primarily governed by the mesopore system rather than the micropore system. By analyzing the contribution of pore throats to sample physical properties, the results indicate that the 14 samples can be classified into two types based on 35% porosity contribution and 60% permeability contribution thresholds. Type 1, reservoirs dominated by microporous throat space (D values ranging from 2.603 to 2.644); Type 2, reservoirs dominated by mesoporous throat space (D values ranging from 2.544 to 2.598). Type 1 is characterized by primary intergranular pores, residual intergranular pores and intergranular dissolution pores, which enhance connectivity and reduce network complexity, thereby improving fluid permeability. In contrast, Type 2 consists mainly of intragranular dissolution pores, intergranular gap pores and micro-dissolution pores in clay minerals, which significantly inhibit fluid mobility. Diagenesis, including compaction, dissolution and cementation, exerts a significant control on the fractal characteristics and pore-throat structure evolution. The fractal characteristics exhibited in the pore-throat structure could provide a desirable analytical method, distinguishing from classification based on scale or size, for the evaluation and classification of tight sandstone reservoirs. Full article

The relationship between pore structure heterogeneity in tight sandstone reservoirs and their fractal characteristics is well documented. However, the impact of differential diagenesis across lithofacies on pore-throat structure and fractal properties remains unclear. In this study, we investigate the Carboniferous Benxi Formation in the Ordos Basin using a suite of experiments to characterize pore-throat structure and multifractal behavior, and to assess the influence of diagenesis. The results reveal significant differences among lithofacies in mineral composition, pore types, pore throat structure, fractal dimensions, and petrophysical properties, primarily attributed to variations in sedimentary environments and diagenesis. Fractal characteristics were quantified by converting the T₂ spectra into pore-throat size distributions. Macropores exhibit the highest fractal dimensions, indicating the greatest structural complexity and heterogeneity, followed by mesopores, whereas micropores show the lowest heterogeneity (D₃ > D₂ > D₁). Quartz content mainly controls the fractal properties of macropores by enhancing structural stability, whereas clay minerals govern the fractal behavior of micropores and mesopores by increasing pore-throat complexity. High-energy depositional conditions promote sediment transportation and sorting, leading to quartzarenite lithofacies (QL) and sublitharenite lithofacies (SL) with lower fractal dimensions, more uniform pore structures, and better connectivity. In contrast, feldspathic litharenite lithofacies (FL) and litharenite lithofacies (LL) exhibit higher fractal dimensions due to stronger compaction, reduced primary porosity, and higher clay content, resulting in poorer reservoir quality. This study improves understanding of pore structure heterogeneity in tight sandstones and provides useful insights for predicting high-quality reservoirs in similar geological settings. Full article

The deep marine shale of the Wufeng–Longmaxi (WF–LMX) Formation in the Sichuan Basin is characterized by laterally continuous thickness, high porosity, and significant gas content, making it a representative shale reservoir with considerable resource potential. This study investigates the heterogeneity of pore structures and their controlling factors using shale samples from three representative wells, based on low-temperature nitrogen adsorption and mercury intrusion data. The reservoir can be classified into three main lithofacies: mixed siliceous shale (MSS), clay-rich siliceous shale (CSS), and siliceous clay mixed shale (SMS). The results show that siliceous shales (MSS and CSS) exhibit higher total organic carbon and quartz contents, with more developed pore systems. Among them, the CSS exhibits the highest specific surface area and the largest mesopore and macropore volumes, indicating a greater development of larger pores and superior reservoir quality. All three shale facies exhibit clear single and multifractal characteristics. The average D₁ and D₂ values (fractal dimensions from nitrogen adsorption at P/P₀ < 0.45 and >0.45, respectively) are higher than D_Hg, (fractal dimension from mercury intrusion), indicating greater pore-surface roughness than internal pore structure complexity and stronger heterogeneity in larger pores. The D(q)–q spectrum shows a left-wide/right-narrow pattern, whereas the α–f(α) spectrum exhibits the opposite trend. The branch-width ratios S_kd and S_ka (indices of pore-size distribution complexity and heterogeneity) are both <0.1, suggesting that heterogeneity is more pronounced in low-probability regions. Fractal and multifractal analyses reveal significant pore structure heterogeneity across different lithofacies, with CSS showing relatively more homogeneous pore structures, whereas MSS exhibits stronger heterogeneity and poorer connectivity. The heterogeneity of shale reservoirs is primarily controlled by pore development, especially micropores and mesopores, and is strongly influenced by total organic carbon and quartz content. Full article

Understanding the multifractal dynamics of carbon and energy markets is essential for capturing complex cross-market interactions and policy-induced volatility. This study investigates China’s carbon and energy markets from 16 July 2021 to 30 January 2026, integrating macro policy interventions with nonlinear market evolution. We first employ a Generalized Autoregressive Conditional Heteroskedasticity-Dynamic Conditional Correlation (GARCH-DCC) model with exogenous policy variables to quantify volatility spillovers and dynamic correlations under policy shocks. Then, a rolling-window multifractal detrended cross-correlation analysis (MF-DCCA) is applied to reveal multiscale dependencies, characteristic periods, and complex fractal structures in cross-market linkages. The results indicate: (1) pronounced spillover effects exist among carbon and energy markets, with policy interventions amplifying short-term contagion; (2) policy shocks exert a “green-squeezing” effect, particularly in the coal market, while endogenous volatility structures exhibit long-term resilience; (3) cross-market linkages display multifractal characteristics, with turning points between the carbon market and electricity, new energy, and coal markets at approximately 6.28, 5.58, and 6.96 months, respectively. These findings provide insights for policymakers in designing differentiated energy regulations and for investors in multiscale risk management and asset allocation. Full article

The assembly interface topography of precision machinery exhibits complex multi-scale geometric features, including roughness, waviness, and form error, which critically influence assembly accuracy and tolerance analysis. To address the lack of adaptivity in existing separation criteria, this paper proposes a multi-scale digital modeling approach oriented toward tolerance analysis of precision assembly interfaces, based on a fractal-wavelet framework. Firstly, multiple Weierstrass–Mandelbrot functions with independent fractal dimensions are superposed to construct a multi-fractal topography model with controllable multi-scale characteristics, grounded in the power spectral density energy additivity property. Subsequently, wavelet functions are employed to hierarchically decompose the topography height field information. The effects of the compact support length and vanishing moments of the wavelet functions on the decomposition performance are analyzed to establish a clear basis for their selection. Finally, an adaptive multi-scale separation criterion based on wavelet energy K-means clustering is then proposed, with the optimal number of scale classes determined by maximizing the silhouette coefficient, eliminating reliance on empirical thresholds. Case study results show that the fused waviness-and-form-error model retains 94.8% of the original energy while reducing convex peak count by over 90%, significantly simplifying the interface microstructure for downstream tolerance computation. The proposed method provides a high-fidelity, adaptive digital foundation for assembly accuracy prediction of precision interfaces. Full article

Different methodologies have been developed for the analysis and study of dynamical systems, including both theoretical models and natural systems. Examples span a wide range of applications, such as astronomy, financial and economic time series, biophysical systems, physiological phenomena, and Earth sciences, including seismicity and climatic processes. The study of these complex systems is commonly based on the analysis of the signals they generate, using mathematical tools to extract relevant information. A broad spectrum of mathematical disciplines converges in this context, including stochastic, probability and statistical theory, entropic and informational measures, fractal and multifractal analysis, natural time analysis, modeling of non-linearity and recurrence methods, generalized entropies, non-extensive systems, machine learning, and high-dimensional and multivariate complexity. Research in this area is largely focused on the characterization of complex systems, providing indicators of determinism or stochasticity, distinguishing between regularity, chaos, and noise, and identifying topological as well as disorder-regularity features. In addition, short- and long-term forecasting, together with the identification of short- and long-range correlations, play a central role in such characterization. To address these objectives, numerous mathematical tools have been developed for the analysis of time series and point processes, each designed to capture specific signal properties. In this work, many of the most important tools used in time series analysis are compiled and reviewed, highlighting their main characteristics and the different types of complex systems to which they have been applied. Full article

Dew constitutes a component of the near-surface water balance, but its large-scale fractal dynamical properties remain poorly documented across Africa. This study estimates dew amounts and investigates their fractal and multifractal behavior under African climatic conditions using gridded ERA5 datasets from 1993 to 2022. The Rescaled-Range (R/S) method, Multifractal Detrended Fluctuation Analysis (MFDFA), and the Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN) algorithm are used. Hurst exponent (Hu) and the multifractal spectrum width (ω) are evaluated at daily and monthly scales over the full period and two sub-periods (1993–2007 and 2008–2022). The results reveal pronounced spatial heterogeneity in dew distribution. Daily mean amounts range between 0 and 0.18 mm, corresponding to annual accumulations reaching up to ~85 mm·yr⁻¹ in humid coastal, equatorial, and sub-equatorial regions, while remaining below 0.5 mm·yr⁻¹ in hyper-arid deserts. The continental mean annual amount is ~35.5 mm·yr⁻¹. The Hurst exponent exhibits values between zero and one, indicating region-dependent persistent and anti-persistent behaviors. This suggests that prediction schemes based on preceding values may be suitable for dew time series prediction in African regions exhibiting persistent characteristics. The multifractal spectrum width (ω), reaching values of up to 10, highlights strong scaling heterogeneity, particularly at the monthly timescale. These findings indicate that African dew dynamics exhibit significant long-range dependence and multifractal variability, providing new insights into the intrinsic temporal structure of dew and into appropriate approaches for its forecasting. Full article

Power system blackouts represent typical manifestations of instability in complex systems, whose evolution often exhibits non-stationarity, long-range correlations, and nonlinear scaling behavior. Most reliability assessment methods widely used in engineering practice are built on the core assumptions of event independence and light-tailed distribution, which will inevitably lead to systematic underestimation of extreme tail risks when blackouts actually present long-range memory and power-law heavy-tailed characteristics. Based on long-cycle historical blackout records of China’s power grid spanning 1981–2025, this paper develops an integrated framework combining Self-Organized Criticality (SOC) theory, Hurst exponent analysis, symbolic time-series methods, and Multifractal Detrended Fluctuation Analysis (MFDFA). This study systematically characterizes the evolution law and inherent dependence structure of blackout events from four dimensions: statistical scaling, temporal correlation, nonlinear structure, and multi-scale fractal spectrum. The results show that both the load-loss magnitudes and inter-event intervals of blackouts follow strict power-law distributions, with the system exhibiting scaling behavior consistent with SOC theory. The blackout event sequence presents significant long-range positive correlation and self-similarity, confirming a persistent long-term memory effect in the system evolution. Symbolic analysis further reveals the nonlinear fluctuation patterns and burst clustering behavior of the blackout process, reflecting the intermittency and complexity of blackout risks. MFDFA results verify that the blackout sequence has a broad-spectrum multifractal structure across different temporal scales, and Monte Carlo shuffle tests demonstrate that this multifractality mainly arises from intrinsic long-range temporal correlations, rather than being driven solely by heavy-tailed distribution. This study confirms that blackouts in China’s power grid are not random independent events, but present fractal statistical characteristics consistent with the self-organized critical mechanism. The findings provide a novel fractal perspective and quantitative framework for the statistical characterization, operational security assessment, and multi-scale early-warning modeling of blackout risks in China’s large-scale power systems. Full article

The nature of nonlinear dynamics of financial markets results in fractal geometry and chaotic behavior that can be viewed on a variety of scales in time. This paper conducts research on the fractal characteristics of the stock network and its contribution to the price of assets based on the Fractal Market Hypothesis (FMH). A multiscale network centrality measure is built based on high-frequency return dependencies to measure the self-similar, scale-invariant nature of inter-stock dependencies. The network factor and portfolio returns are then broken down with the wavelet packet decomposition (WPD) to obtain frequency-domain profiles, which characterize the variability of risk transmission in relation to investment horizons. The profiles are consistent with scaling properties of fractal, but the decomposition does not identify causal pathways on its own. Estimation of fractal dimension by use of the box-counting technique aided by the Hurst exponent analysis reveals that the A-share of China market exhibited long-range dependence and multifractal scaling. Network factor has the largest explanatory power in mid-frequency between the D5 and D6 bands of 32 to 128 days. This intermediary frequency concentration is consistent with the hypothesis of heterogeneous markets, in which the groups of investors with varying time horizons generate scale-related price dynamics. The addition of the network factor to a 6-factor specification lowers the GRS under the 5-factor specification by 31.45 to 17.82 on the same test-asset universe, indicating better cross-sectional coverage in the sample. The estimates of the Lyapunov exponents (0.039) as well as the correlation dimension ($D_2=4.7$) confirm the presence of low-dimensional chaotic processes of the network factor series, but these values are specific to the Chinese A-share market over the 2005–2023 sample period. These results provide a frequency-disaggregated use of network-based factor modeling and suggest that it can be applicable in multiscale portfolio risk management where the investor horizon is not uniform. Full article