Search Results (137)

This study aims to compare different classifiers in the context of distinguishing two classes of signals: nonlinear electrocardiography (ECG) signals and stochastic artifacts occurring in ECG signals. The ECG signals from a single-lead wearable Movesense device were analyzed with a set of eight features: variance (VAR), three fractal dimension measures (Higuchi fractal dimension (HFD), Katz fractal dimension (KFD), and Detrended Fluctuation Analysis (DFA)), and four entropy measures (approximate entropy (ApEn), sample entropy (SampEn), and multiscale entropy (MSE) for scales 1 and 2). The minimum-redundancy maximum-relevance algorithm was applied for evaluation of feature importance. A broad spectrum of machine learning models was considered for classification. The proposed approach allowed for comparison of classifier features, as well as providing a broader insight into the characteristics of the signals themselves. The most important features for classification were VAR, DFA, ApEn, and HFD. The best performance among 34 classifiers was obtained using an optimized RUSBoosted Trees ensemble classifier (sensitivity, specificity, and positive and negative predictive values were 99.8, 73.7%, 99.8, and 74.3, respectively). The accuracy of the Movesense device was very high (99.6%). Moreover, the multifractality of ECG during sleep was observed in the relationship between SampEn (or ApEn) and MSE. Full article

Various structures that exist worldwide are three-dimensional. Consequently, evaluating only two-dimensional cross-sectional structures is insufficient for analysing all worldwide structures. In this study, we interpreted the generalised fractal-dimensional formula of two-dimensional multifractal analysis and proposed three computational extension methods that consider the structure of three-dimensional slices. The proposed methods were verified using Monte Carlo and Gray–Scott simulations; the pixel-existence probability (PEP)-averaging method, which averages the pixel-existence probability in the slice direction, was confirmed to be the most suitable for analysing three-dimensional structures in two dimensions. This method enables a stable quantitative evaluation, regardless of the direction from which the three-dimensional structure is observed. Full article

This paper presents a systematic methodology for identifying optimal scaling regions in segment-based box-counting fractal dimension calculations through a three-phase algorithmic framework combining grid offset optimization, boundary artifact detection, and sliding window optimization. Unlike traditional pixelated approaches that suffer from rasterization artifacts, the method used directly analyzes geometric line segments, providing superior accuracy for mathematical fractals and other computational applications. The three-phase optimization algorithm automatically determines optimal scaling regions and minimizes discretization bias without manual parameter tuning, achieving significant error reduction compared to traditional methods. Validation across the Koch curve, Sierpinski triangle, Minkowski sausage, Hilbert curve, and Dragon curve demonstrates substantial improvements: excellent accuracy for the Koch curve (0.11% error) and significant error reduction for the Hilbert curve. All optimized results achieve $R^2\ge 0.9988$. Iteration analysis establishes minimum requirements for reliable measurement, with convergence by level 6+ for the Koch curve and level 3+ for the Sierpinski triangle. Each fractal type exhibits optimal iteration ranges where authentic scaling behavior emerges before discretization artifacts dominate, challenging the assumption that higher iteration levels imply more accurate results. Application to a Rayleigh–Taylor instability interface (D = 1.835 ± 0.0037) demonstrates effectiveness for physical fractal systems where theoretical dimensions are unknown. This work provides objective, automated fractal dimension measurement with comprehensive validation establishing practical guidelines for mathematical and real-world fractal analysis. The sliding window approach eliminates subjective scaling region selection through systematic evaluation of all possible linear regression windows, enabling measurements suitable for automated analysis workflows. Full article

To investigate the flow characteristics of movable water in coal under the influence of micro-nano pore fractures with multiple fractal structures, this study employed nuclear magnetic resonance (NMR) and multifractal theory to analyze gas–water seepage under different injection pressures. Then, the scale threshold for mobile water entering coal pores and fractures was determined by clarifying the relationship among “injection pressure-T₂ dynamic multiple fractal parameter seepage resistance-critical pore scale”. The results indicate that coal samples from Yiwu (YW) and Wuxiang (WX) enter the nanoscale pore size range at an injection pressure of 8 MPa, while the coal sample from Malan (ML) enters the nanoscale pore size range at an injection pressure of 9 MPa. During the water injection process, there is a significant linear relationship between the multiple fractal parameters log X(q, ε) and log(ε) of the sample. The generalized fractal dimension D(q) decreases monotonically with increasing q in an inverse S-shape. This decrease occurs in two distinct stages: D(q) decreases rapidly in the low probability interval q < 0; D(q) decreases slowly in the high probability interval q > 0. The multiple fractal singularity spectrum function f(α) has an asymmetric upward parabolic convex function relationship with α, which is divided into a rapidly increasing left branch curve and a slowly decreasing right branch curve with α₀ as the boundary. Supporting evidence indicates the feasibility of a methodology for identifying the variation in multiple fractal parameters of gas–water NMR seepage and the critical scale transition conditions. This investigation establishes a methodological foundation for analyzing gas–water transport pathways within porous media materials. Full article

Shale reservoirs offer significant potential for CO₂ geological sequestration due to their extensive nanopore networks and heterogeneous pore systems. This study comparatively assessed the CO₂ storage potential of the Lower Silurian Longmaxi and Lower Cambrian Qiongzhusi shales through an integrated approach involving organic geochemical analysis, mineralogical characterization through X-ray diffraction (XRD), mercury intrusion capillary pressure (MICP), low-pressure nitrogen and carbon dioxide physisorption, field-emission scanning electron microscopy (FE-SEM), stochastic 3D microstructure reconstruction, multifractal analysis, and three-dimensional succolarity computation. The results demonstrate that mineral assemblages and diagenetic history govern pore preservation: Longmaxi shales, with moderate maturity and shallower burial, retain abundant organic-hosted mesopores, whereas overmature and deeply buried Qiongzhusi shales are strongly compacted and mineralized, reducing pore availability. Multifractal spectra and 3D reconstructions reveal that Longmaxi develops broader singularity spectra and higher succolarity values, reflecting more isotropic meso-/macropore connectivity at the SEM scale, while Qiongzhusi exhibits narrower spectra and lower succolarity, indicating micropore-dominated and anisotropic networks. Longmaxi has nanometer-scale throats (D₅₀ ≈ 10–25 nm) with high CO₂ breakthrough pressures (P₁₀ ≈ 0.57 MPa) and ultra-low RGPZ permeability (mean ≈ 1.5 × 10⁻² nD); Qiongzhusi has micrometer-scale throats (D₅₀ ≈ 1–3 μm), very low breakthrough pressures (P₁₀ ≈ 0.018 MPa), and much higher permeability (mean ≈ 4.63 × 10³ nD). Storage partitioning further differs: Longmaxi’s median total capacity is ≈15.6 kg m⁻³ with adsorption ≈ 93%, whereas Qiongzhusi’s median is ≈12.8 kg m⁻³ with adsorption ≈ 70%. We infer Longmaxi favors secure adsorption-dominated retention but suffers from injectivity limits; Qiongzhusi favors injectivity but requires reliable seals. Full article

The root causes forcing the differential pore-throat performances and crude oil recoverability in heterogeneous shale lithofacies of saline-lacustrine fine-grained mixed sedimentary sequences are still debated. Especially application cases of fractal theory in characterizing pore-throat heterogeneity are still lacking and the significance of differential multifractal distribution patterns on reservoir assessment remains controversial. This present study focuses on the shale-oil reservoirs in saline-lacustrine fine-grained mixed depositional sequences of the Middle Permian Lucaogou Formation (southern Junggar Basin, NW China), and presents a set of new results from petrographical investigation, field-emission scanning electron microscopy (FE-SEM) imaging, fluid injection experiments (low-pressure N₂ adsorption and high-pressure mercury intrusion porosimetry (HMIP)), nuclear magnetic resonance (NMR) spectroscopy and T₁-T₂ mapping, directional spontaneous imbibition, as well as contact angle measurements. Our results demonstrated that the investigated lithofacies are mainly divided into a total of five lithofacies categories: felsic siltstones, sandy dolomitic sandstones, dolarenites, micritic dolomites, and dolomitic mudstones, respectively. More importantly, the felsic siltstone and sandy dolomitic siltstones can be identified as the most advantageous lithofacies categories exhibiting the strongest movable oil-bearing capacity owing to an acceptable complexity and heterogeneity of mesopore-throat structures, as evidenced by the corresponding moderate fractal dimension of mesopores (D₂) from HMIP and apparently lower fractal dimension of movable fluids’ pores (D₂) from NMR results. Particularly noteworthy is the relatively poor shale-oil movability recognized in the dolarenites, micritic dolomites, and dolomitic mudstones due to heterogeneous and unfavorable pore-throat systems, even though an acceptable micro-connectivity and a more oleophilic interfacial wettability prevails in crucial dolomitic components. Finally, a comprehensive and conceptual model is established for an effective and characteristic parameter system for assessing differential reservoir petrophysical properties, interfacial wettability, and shale-oil movability concerning heterogeneous lithofacies categories. Our achievements can serve as an analog for investigating saline-lacustrine mixed shale-oil reservoirs to gain a more comprehensive understanding of differential recoverability of dessert reservoir intervals, and to guide the assessment of “sweet spots” distribution and optimization of engineering technique schemes for commercial exploitation. Full article

In the present work, we consider three branching random walk $S_n\mathsf{Z}\left(\mathsf{t}\right),$$\mathsf{Z}\in \{\mathsf{X},\mathsf{Y},\Phi \}$ on a supercritical random Galton–Watson tree $\partial \mathsf{T}$. We compute the Hausdorff and packing dimensions of the level set ${\mathrm{E}}_{\chi }(\alpha ,\beta )=\left\{\mathsf{t}\in \partial \mathsf{T}:{\lim }_{n\to \infty }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}=\alpha \mathrm{and}{\lim }_{n\to \infty }{\displaystyle \frac{S_n\mathsf{Y}\left(\mathsf{t}\right)}{n}}=\beta \right\},$ where $\partial \mathsf{T}$ is endowed with random metric using $S_n\Phi \left(\mathsf{t}\right)$. This is achieved by constructing a suitable Mandelbrot measure supported on $\mathrm{E}(\alpha ,\beta )$. In the case where $\Phi =1$, we develop a formalism that parallels Olsen’s framework (for measures) and Peyrière’s framework (for the vectorial case) within our setting. Full article

Reservoirs and caprocks overlap with each other in heterogeneous carbonate rocks. The sealing capacity of caprocks and their controlling factors are not clear, which restricts the prediction, exploration, and development of carbonate hydrocarbon reservoirs. We selected core samples from the Ordovician reservoirs and caprocks in the Tarim Basin, China, for scanning electron microscopy, thin section, breakthrough pressure (BP), high-pressure mercury intrusion porosimetry (HMIP), and nitrogen adsorption method (N₂GA). The experimental results show that the reservoir and caprock can be distinguished by BP. The BP of the reservoir is less than 3.0 MPa, and the BP of the caprock is less than 3.0 Mpa. We analyzed the heterogeneity characteristics and differences in reservoirs and caprocks with different lithologies from the perspectives of monofractal and multifractal. The results indicate that the differences in pore structure of grainstone, dolomite, and micrite/argillaceous limestone result in significant heterogeneity differences between samples. The correlation analysis between the fractal parameters and BP indicates that the characteristics of reservoir microporous structures have a decisive impact on BP (correlation coefficient > 0.7). The pore structure of the carbonate reservoir–caprock system exhibits self-similarity. The heterogeneity of the caprock has no significant control effect on BP (correlation coefficient < 0.3), while the higher the heterogeneity of the reservoir, the greater the BP. The sealing capacity of the caprock depends on the heterogeneity differences in pore types and pore structures between the reservoirs and caprocks. When both the reservoir and the caprock are grainstone, the micropores in the reservoirs and caprocks are dispersed but evenly distributed, and little heterogeneous differences can achieve sealing. When the lithology of reservoirs and caprocks is different, the enhancement of heterogeneity differences in micropores will improve the sealing capacity of the caprock. In summary, fractal dimension is an effective method for studying the heterogeneous structure and sealing capacity of pore–throat in carbonate caprocks. This study proposes a new perspective that the difference between the heterogeneity of micropore structures of reservoirs and caprocks affects the sealing capacity of carbonate rocks, and provides a new explanation and model for the sealing mode of carbonate rock caprocks. Full article

This study examines the particle size distribution (PSD) of calcareous soils under cultivated and natural conditions in Chenggong District of Kunming, Yunnan Province, China, using single-fractal and multifractal analyses. Soil samples were collected from the profiles of both land use types, and the PSD parameters, organic matter, and total nitrogen were determined. Single-fractal analysis showed that the single-fractal dimension (D) was mainly influenced by the clay content, with higher clay fractions corresponding to larger D values. The generalized dimension spectrum revealed clear differences between natural and cultivated soils: natural soils exhibited greater sensitivity to probability density weight index(q) changes and a more compact particle distribution, whereas cultivation led to broader PSD ranges and higher heterogeneity. The ratio D₁/D₀ was negatively correlated with the clay content, and multifractal spectrum asymmetry (Δf) indicated that fine particles dominate the variability in deeper layers. Compared with natural soils, cultivated soils had higher organic matter and total nitrogen, reflecting the influence of fertilization and tillage on the soil aggregation and PSD. These findings demonstrate that fractal theory provides a sensitive tool for characterizing soil structural complexity and land use impacts, offering a theoretical basis for soil quality assessment and the sustainable management of calcareous soils. Full article

The dynamic changes in shale pore structure due to tectonic uplift are crucial for understanding the processes of shale gas enrichment and accumulation, particularly in complex tectonic regions. To explore the heterogeneous changes in pore structure and their influencing factors during the last tectonic uplift of Longmaxi shale, triaxial creep experiments were performed under varying confining pressure conditions. In addition, FE-SEM, MIP, LN2GA, and LCO2GA experiments were employed to both qualitatively and quantitatively characterize the pore structure of three distinct groups of Longmaxi shale samples. To further investigate pore heterogeneity, the multifractal dimension was applied to examine the evolution of the shale pore structure under the influence of the last tectonic uplift. The results revealed that the primary pore types in Longmaxi shale include organic matter pores, microfractures, intergranular pores, and intragranular pores. The shale’s mechanical properties and mineral content have a significant impact on the heterogeneity of these pores. Notably, the shale pores exhibit distinct multifractal characteristics, highlighting the complex nature of pore heterogeneity. The singular index (α₀) and ten other multifractal dimension parameters provide valuable insights into the heterogeneity characteristics of shale pores from various perspectives. Additionally, dynamic changes in pore heterogeneity are primarily controlled by the mineral composition. Under identical creep pressure variation conditions, significant differences are observed in the pore rebound behavior of Longmaxi shale with different mineral compositions. Under high-pressure conditions, the content of TOC and quartz plays a dominant role in controlling pore heterogeneity, with their influence initially decreasing and then increasing as pressure decreases. The reduction in creep pressure emphasizes the controlling effect of TOC, quartz, and feldspar content on pore connectivity. This study introduces high-pressure triaxial creep experiments to simulate the stress response behavior of pore structures during tectonic uplift, offering a more comprehensive reflection of pore evolution in organic-rich shale under realistic geological conditions. Full article

The Upper Lianggaoshan (LGS) Formation in the northeastern Sichuan Basin, composed of shale with interbedded siltstone, is a promising target layer for shale oil. Accurate evaluation of pore structures is essential for effective exploration of shale oil. This study investigated pore structures of siltstone and shale samples from the Upper LGS Formation using low-pressure CO₂ adsorption (LTCA), low-temperature N₂ adsorption (LTNA), high-pressure mercury intrusion (HPMI), and nuclear magnetic resonance (NMR) methods. The single-exponent and multifractal dimensions of samples were determined, and the relationships between fractal dimensions and pore structures were explored. Results show that the pore size distribution (PSD) of siltstone and shale samples exhibits multi-peak characteristics, with mesopores (2–50 nm) being dominant in the total pore volumes. The multi-scaled pores in shale and siltstone samples exhibit fractal characteristics. The average values of single-fractal dimensions (D₁, D₂) obtained by LTNA data are 2.39 and 2.62 for shale samples, and 2.24 and 2.59 for siltstone samples, respectively. Compared to siltstones, the pore structures of shale samples exhibit greater complexity, indicated by larger fractal dimensions. The samples from subsections Liang 2 and Liang 3 exhibit greater heterogeneity compared to subsection Liang 1. The single-fractal dimensions of micropores and mesopores show positive correlations with specific surface area (SSA) and pore volume (PV), while the fractal dimension of macropores shows a negative correlation with average pore diameter and median radius. The average values of single-fractal dimension D₃ obtained from HPMI data are 2.9644 and 2.9471 for shale and siltstone samples, respectively, indicating more complex structures of macropores in shale samples compared to siltstone samples. The average value of $\Delta D^{NMR}$ and singularity strength range Δα obtained by a multifractal model for core samples from subsection Liang 1 are 1.868 and 2.155, respectively, which are the smallest among all of the three subsections, indicating that the heterogeneity of pore structures of subsection Liang 1 is the weakest. This research provides valuable guidance for shale oil development in the northeastern Sichuan Basin, China. Full article

Deep tight sandstone reservoirs are characterized by low porosity and permeability, complex pore structure, and strong heterogeneity. Conducting research on the heterogeneity characteristics of reservoirs could lay a foundation for evaluating their effectiveness and accurately identifying advantageous reservoirs, which is of great significance for searching for “sweet spot” oil and gas reservoirs in tight reservoirs. In this study, the deep tight sandstone reservoir in the Dibei area, northern Kuqa depression, Tarim Basin, China, is taken as the research object. Firstly, statistical methods are used to calculate the coefficient of variation (CV) and coefficient of heterogeneity (TK) of core permeability, and the heterogeneity within the reservoir is evaluated by analyzing the variations in the reservoir permeability. Then, based on fractal theory, the fractal and multifractal parameters of the GR and acoustic logs are calculated using the box dimension, correlation dimension, and the wavelet leader methods. The results show that the heterogeneity revealed by the box dimension, correlation dimension, and multifractal singular spectrum calculated based on well logs is consistent and in good agreement with the parameters calculated based on core permeability. The heterogeneity of gas layers is comparatively weaker, while that of dry layers is stronger. In addition, the fractal parameters of GR and the acoustic logs of three wells with the oil test in the study area were analyzed, and the relationship between reservoir heterogeneity and production was determined: As reservoir heterogeneity decreases, production increases. Therefore, reservoir heterogeneity can be used as an indicator of production; specifically, reservoirs with weak heterogeneity have high production. Full article

The fractal dimension quantitatively describes the complexity of the shale pore structure and serves as a powerful tool for characterizing the evolution of shale reservoirs. Thermal simulation experiments were conducted on the low-maturity transitional shale from the Shanxi Formation in the Ordos Basin. The initial samples consisted mainly of quartz (39.9%) and clay minerals (49.9%) with moderate-to-good hydrocarbon generation potential. Samples from different thermal maturation stages were analyzed through geochemical, mineralogical, and pore structure experiments to reveal the evolution of mineral compositions and pore structure parameters. The fractal dimensions of the pore structure were calculated using both the FHH and capillary bundle models. Correlation coefficients and principal component analysis (PCA) were employed to explore the factors influencing the fractal dimension and its evolutionary patterns during reservoir development. The results indicate that (1) with increasing thermal maturity, the quartz content gradually increases while the contents of clay minerals, carbonate minerals, pyrite, and feldspar decrease. (2) The evolution of porosity follows five stages: a slow decrease (0.78 < Ro < 1.0%), a rapid increase (1.0% < Ro < 2.0%), a relatively stable phase (2.0% < Ro < 2.7%), a rapid rise (2.7% < Ro < 3.2%), and a slow decline (Ro > 3.2%). The evolution of the pore volume (PV) and specific surface area (SSA) indicates that the proportion of pores in the 5–20 nm and 20–60 nm ranges gradually increases while the proportion of pores smaller than 5 nm decreases. (3) The fractal dimension of shale pores (D₁, average value 2.61) derived from the FHH model is higher than D₂ (average value 2.56). This suggests that the roughness of pore surfaces is greater than the complexity of the internal pore structure at various maturities. The D_M distribution range calculated from the capillary bundle model was broad (between 2.47 and 2.94), with an average value of 2.84, higher than D₁ and D₂. This likely indicates that larger pores have more complex structures. (4) D₁ shows a strong correlation with porosity, PV, and SSA and can be used to reflect pore development. D₂ correlates well with geochemical parameters (TOC, HI, etc.) and minerals prone to diagenetic alteration (carbonates, feldspar, and pyrite), making it useful for characterizing the changes in components consumed during pore structure evolution. (5) Based on the thermal maturation process of organic matter, mineral composition, diagenesis, and pore structure evolution, an evolutionary model of the fractal dimension for transitional shale was established. Full article

Heterogeneity of pore and fracture structures has become an important factor affecting the migration of methane and water in coal reservoirs. However, controlling factors of pore and fracture structure collected from coal seam 10 in Taliqike Formation, Kubai Coalfield, Xinjiang need to be studied. In this paper, carbon dioxide adsorption, cryogenic liquid nitrogen, and high-pressure mercury intrusion, as well as coal microscopic components, were used to study pore volumes and characterize pore diameter distribution heterogeneity. By the theory of single weight and multiple fractal formations, the heterogeneity of the pore fracture structure of coal reservoir is expressed, and the influencing factors of the heterogeneity of the pore fracture structure and the pore volume are also discussed. The results are as follows. (1) Micro-pore distribution presents a distinct bidirectional state, with the main peak at approximately 0.6 nm and 0.85 nm. R_o,max has an obvious influence on micro-pore volume. The single-fractal dimension of micro-pore is not affected by a micro-pore volume but is influenced by other factors such as R_o,max and microscopic composition. The heterogeneity of the low-value area controls the heterogeneity of micro-pore diameter distribution. (2) For lower R_o,max samples, mesopores of these samples are ink bottle-shaped pores, and the pore connectivity is poor. In contrast, meso-pore of higher thermal evolution coal samples are mostly simple pores, such as parallel plates. The main mesopores are 10–100 nm pores, accounting for 75% of the total meso-pore volume. For the single fractal dimension, D₁ is greater than D₂, which also shows that the heterogeneity of a pore structure greater than 4 nm is much stronger than that of a pore structure less than 4 nm in these samples. (3) For lower R_o,max samples, double S-shaped curves with distinct hysteresis loop are obtained, while samples of higher R_o,max samples show parallel curves, suggesting that macro-pore of this type of sample develops parallel plate-like pore. There is a positive relationship between D₋₁₀–D₀ and D₋₁₀–D₁₀, while D₀–D₁₀ and D₋₁₀–D₀ have a weak correlation. With the increase of 2–10 nm pore volume, pore distribution heterogeneity of lower value area (D₋₁₀–D₀) weakens. This indicates that pore volume is an important factor affecting the multifractal variation. Full article

Fractal dimension has proven to be a valuable tool in the analysis of geological data. For instance, it can be used for assessing the distribution and connectivity of fractures in rocks, which is important for evaluating hydrocarbon storage potential. However, while calculating a single fractal dimension for an entire geological profile provides a general overview, it can obscure local variations. These localized fluctuations, if analyzed, can offer a more detailed and nuanced understanding of the profile’s characteristics. Hence, this study proposes a fractal characterization procedure using a new strategy based on moving windows applied to the analysis domain, enabling the evaluation of data multifractality through the Dynamical Approach Method. Validations for the proposed methodology were performed using controlled artificial data generated from Weierstrass–Mandelbrot functions. Then, the methodology was applied to real geological profile data measuring permeability and porosity in oil wells, revealing the fractal dimensions of these data along the depth of each analyzed case. The results demonstrate that the proposed methodology effectively captures a wide range of fractal dimensions, from high to low, in artificially generated data. Moreover, when applied to geological datasets, it successfully identifies regions exhibiting distinct fractal characteristics, which may contribute to a deeper understanding of reservoir properties and fluid flow dynamics. Full article