Search Results (119)

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

In this paper we analyze complex systems dynamics using a multifractal framework derived from Scale Relativity Theory (SRT). By extending classical differential geometry to accommodate non-differentiable, scale-dependent behaviors, we formulate Schrödinger-type equations that describe multifractal geodesics. These equations reveal deep analogies between quantum mechanics and macroscopic complex dynamics. A key feature of this approach is the identification of hidden symmetries governed by multifractal analogs of classical groups, particularly the SL(2ℝ) group. These symmetries help explain universal dynamic behaviors such as double period dynamics, damped dynamics, modulated dynamics, or chaotic dynamics. The resulting framework offers a unified geometric and algebraic perspective on the emergence of order within complex systems, highlighting the fundamental role of fractality and scale covariance in nature. 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

Supercritical CO₂ (ScCO₂) injection has emerged as a promising method to enhance shale gas recovery while simultaneously achieving CO₂ sequestration. This research investigates how ScCO₂ interacts with water and shale rock, altering the pore structure characteristics of shale reservoirs. The study examines shale samples from three marine shale formations in southern China under varying thermal and pressure regimes simulating burial conditions at 1000 m (45 °C and 10 MPa) and 2000 m (80 °C and 20 MPa). The research employs multiple analytical techniques including XRD for mineral composition analysis, MICP, N₂GA, and CO₂GA for comprehensive pore characterization, FE–SEM for visual observation of mineral and pore changes, and multifractal theory to analyze pore structure heterogeneity and connectivity. Key findings indicate that ScCO₂–water–shale interactions lead to dissolution of minerals such as kaolinite, calcite, dolomite, and chlorite, and as the reaction proceeds, substantial secondary mineral precipitation occurs, with these changes being more pronounced under 2000 m simulation conditions. Mineral dissolution and precipitation cause changes in pore structure parameters of different pore sizes, with macropores showing increased PV and decreased SSA, mesopores showing decreased PV and SSA, and micropores showing insignificant changes. Moreover, mineral precipitation effects are stronger than dissolution effects. These changes in pore structure parameters lead to alterations in multifractal parameters, with mineral precipitation reducing pore connectivity and consequently enhancing pore heterogeneity. Correlation analysis further revealed that H and D₋₁₀–D₁₀ exhibit a significant negative correlation, confirming that reduced connectivity corresponds to stronger heterogeneity, while mineral composition strongly controls the multifractal responses of macropores and mesopores, with micropores mainly undergoing morphological changes. However, these changes in micropores are mainly manifested as modifications of internal space. Siliceous shale samples exhibit stronger structural stability compared to argillaceous shale, which is attributed to the mechanical strength of the quartz framework. By integrating multifractal theory with multi–scale pore characterization, this study achieves a unified quantification of shale pore heterogeneity and connectivity under ScCO₂–water interactions at reservoir–representative pressure–temperature conditions. This novelty not only advances the methodological framework but also provides critical support for understanding CO₂–enhanced shale gas recovery mechanisms and CO₂ geological sequestration in depleted shale gas reservoirs, highlighting the complex coupling between geochemical reactions and pore structure evolution. 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

Accurate carbon market price prediction is crucial for promoting a low-carbon economy and sustainable engineering. Traditional models often face challenges in effectively capturing the multifractality inherent in carbon market prices. Inspired by the self-similarity and scale invariance inherent in fractal structures, this study proposes a novel multifractal-aware model, MF-Transformer-DEC, for carbon market price prediction. The multi-scale convolution (MSC) module employs multi-layer dilated convolutions constrained by shared convolution kernel weights to construct a scale-invariant convolutional network. By projecting and reconstructing time series data within a multi-scale fractal space, MSC enhances the model’s ability to adapt to complex nonlinear fluctuations while significantly suppressing noise interference. The fractal attention (FA) module calculates similarity matrices within a multi-scale feature space through multi-head attention, adaptively integrating multifractal market dynamics and implicit associations. The dynamic error correction (DEC) module models error commonality through variational autoencoder (VAE), and uncertainty-guided dynamic weighting achieves robust error correction. The proposed model achieved an average R² of 0.9777 and 0.9942 for 7-step ahead predictions on the Shanghai and Guangdong carbon price datasets, respectively. This study pioneers the interdisciplinary integration of fractal theory and artificial intelligence methods for complex engineering analysis, enhancing the accuracy of carbon market price prediction. The proposed technical pathway of “multi-scale deconstruction and similarity mining” offers a valuable reference for AI-driven fractal modeling. Full article

To alleviate environmental pressures, manufactured sand (MS) are increasingly being used in the production of ultra-high-performance concrete (UHPC) due to their consistent supply and environmental benefits. However, manufactured sand properties are critically influenced by processing and production techniques, resulting in substantial variations in fundamental characteristics that directly impact UHPC matrix pore structure and ultimately compromise performance. Traditional testing methods inadequately characterize UHPC’s pore structure, necessitating multifractal theory implementation to enhance pore structural interpretation capabilities. In this study, UHPC specimens were fabricated with five types of MS exhibiting distinct properties and at varying water to binder (w/b) ratios. The flowability, mechanical strength, and pore structure of the specimens were systematically evaluated. Additionally, multifractal analysis was conducted on each specimen group using mercury intrusion porosimetry (MIP) data to characterize pore complexity. SM-type sands have a more uniform distribution of pores of different scales, better pore structure and matrix homogeneity due to their finer particles, moderate stone powder content, and better cleanliness. Both excessively high and low stone powder content, as well as low cleanliness, will lead to pore aggregation and poor closure, degrading the pore structure. 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 effect of the pore–fracture structure on the porosity and permeability affects the production process of tight sandstone gas. In this paper, 12 groups of tight sandstone samples are selected as the object, and the pore–fracture volume of a tight reservoir is quantitatively characterized by a high-pressure mercury injection test. The multifractal and single fractal characteristics of different types of samples are calculated by fractal theory. On this basis, the pore volume variation under stress is discussed through the overlying pressure pore permeability test, and the pore–fracture compressibility is calculated. Finally, the main factors affecting the stress sensitivity of tight sandstone are summarized from the two aspects of the pore structure and mineral composition. The results are as follows. (1) The samples could be divided into types A and B by using the mercury-in and mercury-out curves. There is a significant hysteresis loop in the mercury inlet and outlet curves of type A, and the efficiency of the mercury inlet and outlet in the pores is relatively higher. The mercury removal curve of type B is almost parallel, and its mercury removal efficiency is relatively lower. (2) The applicability of singlet fractals in characterizing the heterogeneity of micropores is higher than that of multifractals. This is because the single fractal characteristics of the two types of samples have significant differences, while the differences in the multifractals are relatively weak. (3) A pore diameter of 100–1000 nm provides the main compression space for the type A samples. A pore distribution heterogeneity of 100–1000 nm affects the compression effect and stress sensitivity of this type B sample. Full article

This study presents a comprehensive multifractal characterization of full-scale pore structures in middle- to high-rank coal reservoirs from the Western Guizhou–Eastern Yunnan Basin and establishes a permeability prediction model integrating fractal heterogeneity and pore throat parameters. Eight coal samples were analyzed using mercury intrusion porosimetry (MIP), low-pressure gas adsorption (N₂/CO₂), and multifractal theory to quantify multiscale pore heterogeneity and its implications for fluid transport. Results reveal weak correlations (R² < 0.39) between conventional petrophysical parameters (ash yield, volatile matter, porosity) and permeability, underscoring the inadequacy of bulk properties in predicting flow behavior. Full-scale pore characterization identified distinct pore architecture regimes: Laochang block coals exhibit microporous dominance (0.45–0.55 nm) with CO₂ adsorption capacities 78% higher than Tucheng samples, while Tucheng coals display enhanced seepage pore development (100–5000 nm), yielding 2.5× greater stage pore volumes. Multifractal analysis demonstrated significant heterogeneity (Δα = 0.98–1.82), with Laochang samples showing superior pore uniformity (D₁ = 0.86 vs. 0.82) but inferior connectivity (D₂ = 0.69 vs. 0.71). A novel permeability model was developed through multivariate regression, integrating the heterogeneity index (Δα) and effective pore throat diameter (D₁₀), achieving exceptional predictive accuracy. The strong negative correlation between Δα and permeability (R = −0.93) highlights how pore complexity governs flow resistance, while D₁₀’s positive influence (R = 0.72) emphasizes throat size control on fluid migration. This work provides a paradigm shift in coal reservoir evaluation, demonstrating that multiscale fractal heterogeneity, rather than conventional bulk properties, dictates permeability in anisotropic coal systems. The model offers critical insights for optimizing hydraulic fracturing and enhanced coalbed methane recovery in structurally heterogeneous basins. Full article

The dynamics of complex systems often exhibit multifractal properties, where interactions across different scales influence their evolution. In this study, we apply the Multifractal Theory of Motion within the framework of scale relativity theory to explore synchronization phenomena in complex systems. We demonstrate that the motion of such systems can be described by multifractal Schrödinger-type equations, offering a new perspective on the interplay between deterministic and stochastic behaviors. Our analysis reveals that synchronization in complex systems emerges from the balance of multifractal acceleration, convection, and dissipation, leading to structured yet highly adaptive behavior across scales. The results highlight the potential of multifractal analysis in predicting and controlling synchronized dynamics in real-world applications. Several applications are also discussed. Full article

Experimental and theoretical design on thermal and structural properties of Cu-Zn-Al alloys are established. As such, from an experimental point of view, differential thermal analysis has been performed with the help of a DSC Netzsch STA 449 F1 Jupiter calorimeter with high levels of sensitivity, and the structural analysis has been accomplished through X-ray diffraction and SEM analysis. An unusual specific property for a metallic material has been discovered, which is known as “rubber-type behavior”, a characteristic determined by micro-structural changes. From the theoretical point of view, the thermal transfer in Cu-Zn-Al is presented by assimilating this alloy, both structurally and functionally, with a multifractal, situation in which the order–disorder transitions assimilated with thermal “dynamics” of Cu-Zn-Al, are mimed through transitions from non-multifractal to multifractal curves. In such a context, the thermal expansion velocity contains both the propagation speed of the phase transformation (be it a direct one: austenitic–martensitic transformation, or an indirect one: martensitic–austenitic transformation) and the thermal diffusion speed. Then, through self-modulations of the thermal field, the Cu-Zn-Al alloy will self-structure in channel-type or cellular-type thermal patterns, which can be linked to obtained experimental data. Consequently, since the thermal conductivity becomes a function of the observation scale, and heat transfer is modified to reflect the multifractal, non-differentiable paths in the material, it leads to anomalous diffusion and complex thermal behaviors. Full article

The heterogeneity of shale pore systems, which is controlled by thermal maturation, fundamentally governs hydrocarbon storage and migration. Artificial sequence maturity samples of Xiamaling shale were obtained through a temperature–pressure simulation experiment (350–680 °C, 15–41 MPa). In combination with low-pressure CO₂/N₂ adsorption experiments, mercury intrusion porosimetry experiments and fractal theory, the heterogeneity of the pore size distribution of micropores, mesopores and macropores in shale of different maturities was quantitatively characterized. The results reveal that the total porosity follows a four-stage evolution with thermal maturity (R_o = 0.62–3.62%), peaking at 600 °C (R_o = 3.12%). Multifractal parameters indicate that areas with a low probability density are dominant in terms of pore size heterogeneity, while monofractal parameters reflect enhanced uniform development in ultra-over maturity (R_o > 3.2%). A novel Fractal Quality Index (FQI) was proposed to integrate porosity, heterogeneity, and connectivity, effectively classifying reservoirs into low-quality, medium-quality, and high-quality sweet-spot types. The findings contribute to the mechanistic understanding of pore evolution and offer a fractal-based framework for shale gas reservoir evaluation, with significant implications for hydrocarbon exploration in unconventional resources. Full article

The relationships between the pore structure and a single fractal or specific region have been widely reported. However, the intrinsic relationship between multifractal parameters and physical properties have remained uncertain. In this study, micro-computed tomography scanning technology and high-pressure mercury injection technologies were applied to determine the pore structures of tight sandstone at different scales. Subsequently, the multifractal theory was applied to quantitatively evaluate the multiscale pore structure heterogeneity. An evident linear relationship exists between $logX\left(q,\epsilon \right)$ and $\mathit{log}\left(\epsilon \right)$, indicating the pore structure of tight sandstones exhibits significant multifractal characteristics. Multifractal parameters, including $∆\left(\alpha \right), ∆D, D_{min}-D_0,\mathrm{a}\mathrm{n}\mathrm{d}{ D}_0-D_{max}$, can serve as sensitive indicators to assess the multiscale pore structure heterogeneity. In particular, the relative development degree of large-scale pores (>10 μm) can be reflected by $D_{min}-D_{0 }$, which has strong heterogeneity. The heterogeneity of the multiscale structure is closely linked to the mineral components of tight sandstone reservoirs, and the heterogeneity of small-scale pores (1–10 μm) is stronger by clay mineral enrichment. Furthermore, the part of the pore structure corresponding to the combination of pore size range of 10 to 20 μm and throat size range of 20 to 40 μm in a low probability measure area may dominate the permeability of tight sandstone. The findings enhance the understanding of pore structure heterogeneity and broaden the application of multifractal theory. 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