Fractal Characteristics Research Articles

Fractal characterization of soil particle size distribution under different land use patterns on the north slope of Wula Mountain in China

Fractal characterization of soil particle size distribution under different land use patterns on the north slope of Wula Mountain in China

Pore Structure and Migration Ability of Deep Shale Reservoirs in the Southern Sichuan Basin

The migration phenomenon of deep shale gas is a subject that has yet to be fully comprehended, specifically regarding the migration ability of deep shale gas. This study focuses on the Longmaxi Formation in the southern Sichuan Basin, utilizing it as an example. Various experimental techniques, such as temperature-driven nitrogen and carbon dioxide adsorption, high-pressure mercury intrusion, XRD, and TOC analysis, are employed. The goal is to analyze the pore structure and fractal characteristics of the Longmaxi Formation shale. Additionally, the study aims to calculate its Knudsen number based on parameters like temperature gradient and pressure coefficient. The migration ability of the Longmaxi Formation shale in southern Sichuan Basin is also discussed. The results show the following: (1) The pore volume distribution of the Longmaxi Formation shale in the study area ranges from 0.0131 to 0.0364 cm3/g. Mesopores contribute approximately 56% of the pore volume, followed by micropores with a contribution rate of about 26%, and macropores contributing approximately 18%. Additionally, the Longmaxi Formation shale exhibits strong heterogeneity, with the fractal dimension (D1) of mesopores ranging from 2.452 to 2.8548, with an average of 2.6833, and the fractal dimension (D2) of macropores ranging from 2.9626 to 2.9786, averaging 2.9707. (2) The fractal dimensions of shale were significantly influenced by organic matter, inorganic minerals, and pore structure parameters. D1 and D2 were positively correlated with TOC, clay mineral content, and specific surface area, while exhibiting negative correlation with quartz. However, the correlations with calcite content, pore volume, and average pore size were not significant. (3) The proportion of pores satisfying Darcy flow in the Longmaxi Formation shale was approximately 3.7%–11.8%, with an average of 6.6%. Consequently, the migration capability of shale gas can be calculated using Darcy’s law. Furthermore, the migration capability of shale gas is controlled by D2, specifically the surface area, and the connectivity of macropores.

Geochemical Characteristics of Deep-Sea Sediments in Different Pacific Ocean Regions: Insights from Fractal Modeling

Exploration of mineral resources in the deep sea has become an international trend. However, deep-sea mineral exploration faces challenges such as complex offshore drilling and the weak and mixed signals of ore deposits. Therefore, studying methods for identifying weak and mixed anomalies and extracting composite information in the deep sea is crucial for innovative prediction and evaluation of deep-sea mineral resources. In this study, the Central Pacific Ocean, Northwestern Pacific Ocean, and Eastern Pacific Ocean were selected as research areas. Drawing upon the fractal self-similarity exhibited by rare earth minerals in the deep-sea sediments within the Pacific Ocean, we conducted an analysis and comparison of the fractal geochemical characteristics in various regions of the Pacific Ocean’s deep-sea sediments. Thereafter, we studied the spatial distribution of rare earth elements (REEs) in deep-sea sediments in these regions to explore the mechanisms responsible for rare earth enrichment in the Pacific Ocean. The results revealed that the geochemical fractal characteristics of deep-sea sediments in the Northwestern Pacific Ocean Basin and the Central Pacific Ocean Basin were similar, whereas there were slight differences in the fractal characteristics observed in the Eastern Pacific Ocean Basin. By calculating the singularity index of CaO/P2O5, it was found that the singularity index in the Central and Northwestern Pacific Ocean basins was lower than that in the Eastern Pacific Ocean Basin, suggesting that the phosphorus content in the Eastern Pacific Ocean Basin was lower than that in the Central and Northwestern Pacific Ocean basins. In the Eastern Pacific Ocean, we found that phosphorus content in deep-sea sediments was the primary controlling factor for REE enrichment. Conversely, in the Central and Northwestern Pacific Ocean, both the phosphorus and calcium content in deep-sea sediments played significant roles in REE enrichment.

Study on rock fracturing in smooth blasting under initial stress

In practical smooth blasting, rock fracturing is influenced by in-situ stress, which frequently leads to underbreak/overbreak and excavation damaged zone and consequently poses a serious threat to the safety of the excavated structure. This study systematically investigates the behaviour and mechanism of rock cracking under coupled initial stress and smooth blast utilizing experimental, theoretical, and numerical methods. Model experiments involving three-hole smooth blasting on 200 × 200 × 50 mm granite specimens are conducted under various static stresses. The resulting fracturing patterns are identified using the digital image correlation technique. The model testing examines rock cracking behaviours, fractal characteristics of rock fractures, and underbreak/overbreak. Experimental results indicate that the extents of rock fracturing and the fractal dimension in both the excavation and reserved zones show an overall decreasing trend as the static stress increases. Meanwhile, the area and depth of underbreak/overbreak also present a decreasing trend, suggesting that static stress is beneficial to the performance of smooth blasting. Then, the variations of crack patterns in model tests are theoretically analysed based on the transmission and superimposition of detonation stress waves combined with the static stress distribution near perimeter holes. Theoretical results emphasize the crucial role of the second hole and indicate no evidence of the third hole influencing the generation of connection fractures. The initial stress induces hoop tensile stress concentration on borehole-wall and generates the smallest hoop compression stress distributed vertically to the excavation line, which controls the initiation and extension of fractures in smooth blasting along the connection line of boreholes. Subsequently, the fracture evolution under coupled initial stress and smooth blast is numerically modelled using LS-DYNA. The numerical results vividly illustrate the necessity of a small burden to efficiently fragment the rock within the excavation zone when smooth blasting is applied to rock excavation under high in-situ stress.

Change in the Microstructure and Fractal Characteristics of Intact and Compacted Loess Due to Its Collapsibility

This work aimed to examine the fractal dimension and the difference in the law between intact and compacted loess before and after collapse. Uniaxial compression tests were performed to obtain specimens under various vertical stresses, and mercury intrusion porosimetry (MIP) tests were conducted to determine the pore size distribution (PSD). Three models were selected to determine the fractal dimensions based on PSD. As a result, the pores were classified into ultra-micropores (d < 0.1 μm), micropores (0.1 μm < d < 2 μm), small pores (2 μm < d < 10 μm), and large pores (d > 10 μm). When the fractal dimensions were determined using the capillary pressure model, there were three fractal intervals (Ds1, Ds2, and Ds3), with only Ds1 and Ds2 meeting the definition of fractal dimension. Ds1 increased considerably after the collapse, but Ds2 declined. The thermodynamic law-based model presented the best linear fit, and there was only one fractal interval. The fractal dimension Dn increased dramatically after the specimen underwent wet collapse. In conjunction with fractal theory, it revealed that collapse changed the uniformity of the pore system, making the microscopic pores coarser and more intricate after collapse.

Fractal feature selection model for enhancing high-dimensional biological problems

The integration of biology, computer science, and statistics has given rise to the interdisciplinary field of bioinformatics, which aims to decode biological intricacies. It produces extensive and diverse features, presenting an enormous challenge in classifying bioinformatic problems. Therefore, an intelligent bioinformatics classification system must select the most relevant features to enhance machine learning performance. This paper proposes a feature selection model based on the fractal concept to improve the performance of intelligent systems in classifying high-dimensional biological problems. The proposed fractal feature selection (FFS) model divides features into blocks, measures the similarity between blocks using root mean square error (RMSE), and determines the importance of features based on low RMSE. The proposed FFS is tested and evaluated over ten high-dimensional bioinformatics datasets. The experiment results showed that the model significantly improved machine learning accuracy. The average accuracy rate was 79% with full features in machine learning algorithms, while FFS delivered promising results with an accuracy rate of 94%.

Selective pore regulation of activated carbon using trace carbonate-assisted catalytic activation: Revealing the effect of cation catalysis on pore topology

Activated carbon is an important adsorbent for the adsorption removal of water or gas pollutants due to its low cost, high efficiency, and large-scale production potential; however, currently commercial activated carbon prepared through physical or chemical activation commonly shows a single microporous characteristic, making it difficult to match different kinds of pollutants. Here, we propose a convenient and cost-effective pore regulation strategy by introducing a trace carbonate catalysis process into the activation reaction, in which we find that the types of cations (Na, K, Ca) in carbonate have a significant impact on the pore topology of coal-based activated carbon, further revealing the role of cation induced catalytic gasification in pore regulation. Therein, K and Ca enhance the formation of micropores and meso-/macropores respectively, while Na promotes the pore development across the entire size range resulting in a hierarchically porous structure with high specific surface area. Kinetic analysis, density functional theory calculations, and three-dimensional pore fractal network characterization reveal that the pore development is collectively determined by the gasification reaction rate and depth. Evaluated as adsorbents, the obtained NaAC sample by Na catalysis demonstrates excellent adsorption capacities for both small-molecule toluene (467 mg g−1) and large-molecule tetracycline (374 mg g−1). This work for the first time elucidates the cation-induced gasification effect and provides a simple and cost-effective method for regulating the pore topology of activated carbon with only trace catalyst dosage.

Synthesizing nuclear power plant fouling with fractal characteristics enables an in-depth study of concerned nuclear safety issues

Fouling deposit on nuclear fuel cladding causes wick boiling and boron hideout, resulting in localized corrosion and power shift with great potential security and economic risks. Herein, a cost-effective time-saving adjustable reproduction method combining sol-gel with ceramic sintering is presented to enable wide coverage of fouling's morphologies and microstructures. Based on fractal analysis, structurally self-similar fouling deposits from different reactors conform to proposed porosity-fractal dimension law under 3% relative error. Wick boiling and boron hideout numerical simulation based on fractal dimension is implemented to treat different morphologies and structures in a unified way. Cladding surface underneath fouling deposit has a maximum 9.243 K temperature increasement due to thermal resistance, and H3BO3 is concentrated 11.274 times by mean of wick boiling, causing Li2B4O7 precipitation under extreme conditions with low porosity and high heat flux. The insights in this study provide a precise approach for quantitative evaluation of localized corrosion and power shift.

A Binary Medium Constitutive Model for Frozen Solidified Saline Soil in Cold Regions and Its Fractal Characteristics Analysis

In addressing the issue of strength degradation in saline soil foundations under the salt-freeze coupling effects, a binary medium constitutive model suitable for un-solidified and solidified frozen saline soil is proposed considering both bonding and friction effects. To verify the validity of the constitutive model, freezing triaxial tests are carried out under different negative temperatures, confining pressures, and water contents. The pore structure and fractal characteristics of saline soil are analyzed using mercury intrusion porosimetry (MIP) and the fractal dimension D qualitatively and quantitatively, which shed light on the strength enhancement mechanism during the solidification of frozen saline soils. The results show that the constitutive model for frozen solidified saline soil based on binary medium theory aptly captures the stress–strain relationship before and after the solidification of frozen saline soil. The stress–strain relationship of frozen saline soil before and after solidification can be delineated into linear elasticity, elastoplasticity, and strain-hardening or -softening phases. Each of these phases can be coherently interpreted through the binary medium constitutive model. The un-solidified and solidified frozen both show pronounced fractal characteristics in fractal analysis. Notably, the fractal dimension D of the solidified saline soil exhibits a significant increase compared to that of un-solidified ones. In Regions I and III, the values of D for solidified saline soil are lower than those for untreated saline soil, which is attributed to the filling effect of hydration products and un-hydrated solidifying agent particles. In Region II, the fractal dimensions DMII and DNII of the solidified saline soil exhibit a “non-physical state”, which is mainly caused by the formation of a significant number of inkpot-type pores due to the binding of soil particles by hydration products.

Classification of drought severity in contiguous USA during the past 21 years using fractal geometry

AbstractDrought is characterized by a moisture deficit that can adversely impact the environment, economy, and society. In North America, like many regions worldwide, predicting the timing of drought events is challenging. However, our novel study in climate research explores whether the Drought Monitor database exhibits fractal characteristics, represented by a single scaling exponent. This database categorizes drought areas by intensity, ranging from D0 (abnormally dry) to D4 (exceptional drought). Through vibration analysis using power spectral densities (PSD), we investigate the presence of power-law scaling in various statistical moments across different scales within the database. Our multi-fractal analysis estimates the multi-fractal spectrum for each category, and the Higuchi algorithm assesses the fractal complexity, revealing that D4 follows a multi-fractal pattern with a wide range of exponents, while D0 to D3 exhibit a mono-fractal nature with a narrower range of exponents.

Study on fractal and energy dissipation characteristics of rock impact fragmentation in Junggar basin

In order to study the rock energy absorption and crushing fractal characteristics under impact load, the dynamic impact tests of rock in J1b formation in Junggar Basin were carried out by split Hopkinson bar. The strain rate effect on the crushing energy consumption and failure mode were analyzed. Combined with fractal dimension calculation and energy density analysis, the energy density and fractal characteristics of rocks during the impact process are studied. The results showed that the strain rate and energy density of the specimens are in line with the power relationship, the energy density increases with the strain rate, and the fractal dimension increases significantly with the increase of the strain rate, the energy density and fractal dimension of the specimen under dynamic load conform to a power relationship. The more energy absorbed by the rock, the more obvious the fractal characteristics are. The energy density combined with the fractal dimension calculation of rock fragmentation can quantitatively study the energy absorption law of rock per unit time.

A 3D FRACTAL MODEL COUPLED WITH TRANSPORT AND ACTION MECHANISMS TO PREDICT THE APPARENT PERMEABILITY OF SHALE MATRIX

The permeability of shale controls gas transport in shale gas reservoirs. The shale has a complex pore structure at the nanoscale and its permeability is affected by multiple transport and action mechanisms. In this study, a 3D fractal model for predicting the apparent gas permeability of shale matrix is presented, accounting for the effects of the transport mechanisms (bulk gas transport and adsorption gas diffusion) and action mechanisms (gas adsorption, real gas properties, water film, stress dependence, and total organic carbon (TOC) content). The proposed model is validated with the published experimental data. A series of sensitivity analyses are performed to investigate the influence of fractal characteristics and action mechanisms on the apparent permeability caused by each transport mechanism. The results show that the real gas properties, water film, and stress dependence cause different effects on the total apparent permeability of shale under different fractal characteristics. The maximum pore diameter is inversely proportional to the effects of these action mechanisms, and the porosity is positively proportional to the effects of real gas properties and water film but inversely proportional to the effects of stress dependence. An increase in TOC content can cause an improvement in the total apparent permeability. Furthermore, the effects of action mechanisms on the apparent permeability caused by different transport mechanisms are differently affected by the fractal characteristics. Changes in fractal characteristics mainly affect the apparent permeability caused by slip flow in the real gas effect, slip flow and Knudsen diffusion in the water film effect, and all transport mechanisms in the stress dependence effect.

What is the variant of fractal dimension under addition of functions with same dimension and related discussions

<abstract><p>This paper attempts to explore possible box dimension of two added fractal continuous functions with the same dimension. Two interesting and meaningful results are obtained. Let $ g(x) $ and $ h(x) $ have the same box dimension $ t\; (1 < t\leq 2) $, the box dimension of $ g(x)+h(x) $ may or may not exist. If it exists, it can take an arbitrary real number $ \gamma $ satisfying $ 1 < \gamma\leq t $. If it does not exist, its lower and upper box dimensions can reach arbitrary different real numbers $ t_1\; {\rm and}\; t_2 $ that satisfy $ 1 < t_1 < t_2 < t\leq 2 $. These unexpected conclusions drive us to probe into the characteristics of collection of all fractal continuous functions with the same box dimension under ordinary linear operations (scalar multiplication and addition). Following the known fractal features of some typical fractal functions such as the Weierstrass function $ W_t(x) $, we classify the fractal functions into three types: consistent fractal functions, non-consistent fractal functions, and simple fractal functions. By utilizing these classifications and fractal feature descriptions, the causality of the box dimension of two added fractal functions can be partially revealed. We hope that these initial superficial discussions will lead deeper consideration on the essence of variants of fractal dimension under linear combinations of fractal functions. Moreover, these fractal features may be applied further in other fields of fractals.</p></abstract>

Fractal Geometry Meets Computational Intelligence: Future Perspectives.

Characterizations in terms of fractals are typically employed for systems with complex and multiscale descriptions. A prominent example of such systems is provided by the human brain, which can be idealized as a complex dynamical system made of many interacting subunits. The human brain can be modeled in terms of observable variables together with their spatio-temporal-functional relations. Computational intelligence is a research field bridging many nature-inspired computational methods, such as artificial neural networks, fuzzy systems, and evolutionary and swarm intelligence optimization techniques. Typical problems faced by means of computational intelligence methods include those of recognition, such as classification and prediction. Although historically conceived to operate in some vector space, such methods have been recently extended to the so-called nongeometric spaces, considering labeled graphs as the most general example of such patterns. Here, we suggest that fractal analysis and computational intelligence methods can be exploited together in neuroscience research. Fractal characterizations can be used to (i) assess scale-invariant properties and (ii) offer numeric, feature-based representations to complement the usually more complex pattern structures encountered in neurosciences. Computational intelligence methods could be used to exploit such fractal characterizations, considering also the possibility to perform data-driven analysis of nongeometric input spaces, therby overcoming the intrinsic limits related to Euclidean geometry.

Energy Dissipation and Fractal Properties of Coal under Cyclic Impact Loading

To investigate the periodic disturbance and damage characteristics of coal under different impact pressures, the stress‒strain failure and damage trends of weak, impact‐prone coal samples were analysed under different impact pressures and different numbers of impacts. Split‐Hopkinson pressure bar (SHPB) tests were conducted to study the stress‒strain curve, dynamic compressive strength, elastic modulus, failure strain, and absorbed energy responses with different numbers of impacts and different strain rates. Fractal theory and methods were applied to analyse the influence of different impact pressures and different numbers of impacts on coal sample failure to reveal the relationship between the number of impacts and the fractal size characteristics of coal fragmentation. The energy modes and failure modes of coal samples under different impact effects were analysed under dynamic loading. Regarding the energy absorption corresponding to different numbers of impacts, the energy absorption per unit volume (EAPUV) of three impacts was calculated to be 0.46 J/cm3. The number of impacts was directly correlated with the energy absorbed by the coal sample. Furthermore, the number of impacts was directly correlated with the severity of the deformation and damage of the coal sample and was inversely correlated with the fragmentation of the coal sample. The degree of coal sample fragmentation was directly proportional to the number of impacts, and the fractal dimension increased with the number of impacts. The study of the failure and fractal characteristics of coal samples under different impact pressures and different numbers of impacts provides important information for the prevention and control of coal mine rock bursts.

A CLUSTERED FRACTAL DISCRETE FRACTURE NETWORK MODEL FOR FRACTURED COAL

The fracture network in fractured coal is the main channel of coal seam gas flow. Not only the geometric topology properties (such as fractal characteristics) of a single fracture but also the connection topology properties (interconnection characteristics between fractures) of the fracture network have an important impact on the fluid flow in fracture networks. In this study, the connection topology properties of the fracture network in the fractured coal are explored based on the complex network theory for the first time. The property parameters such as the fracture node degree, the clustering coefficient, and the average path length are analyzed. It shows that the average clustering coefficient of the fracture network in fractured coal is larger, and the average path length is smaller. The connection property of the fracture network in the fractured coal presents a typical “small-world” clustering model. Further, by considering the fractal characteristics of the single fracture and the clustering characteristics of the fracture network, an improved clustered fractal discrete fracture network (DFN) model is developed. Then, based on the lattice Boltzmann method, the permeability properties of the generated clustered fractal DFNs are analyzed. The results show that the permeability of DFNs is positively correlated with the average clustering coefficient of fracture network, and negatively correlated with the fractal dimension of fracture. Therefore, the topological clustering characteristics of fracture networks and the fractal characteristics of fractures cannot be ignored in describing the fluid flow in the fracture network, and our clustered fractal DFN model provides a new idea for guiding the optimization design in DFN engineering.

Fractal Neurodynamics.

The neuronal ongoing electrical activity in the brain network, the neurodynamics, reflects the structure and functionality of generating neuronal pools. The activity of neurons due to their excitatory and inhibitory projections is associated with specific brain functions. Here, the purpose was to investigate if the local ongoing electrical activity exhibits its characteristic spectral and fractal features in wakefulness and sleep across and within subjects. Moreover, we aimed to show that measures typical of complex systems catch physiological features missed by linear spectral analyses. For this study, we concentrated on the evaluation of the power spectral density (PSD) and Higuchi fractal dimension (HFD) measures. Relevant clinical impact of the specific features of neurodynamics identification stands primarily in the potential of classifying cortical parcels according to their neurodynamics as well as enhancing the effectiveness of neuromodulation interventions to cure symptoms secondary to neuronal activity unbalances.

Influence of manufacturing surface topography on torque and load bearing capacity of hydro-viscous drive clutch under mixed lubrication stage

PurposeHydro-viscous drive (HVD) clutches are widely used in equipment requiring soft start, such as fans and pumps, to transmit torque and adjust speed by changing the gap distance between friction pairs. This paper aims to propose a novel two-parameter evaluation method for HVD during the mixed lubrication stage. The objective is to develop an effective model that establishes the relationship between these parameters and the actual surface topography.Design/methodology/approachIn the presented methods, the fractal features of the real manufacturing surface are calculated based on the power spectrum function by the ultra-depth three-dimensional microscope. After that, the hybrid friction model of the friction plate is established based on mixed elasto-hydrodynamic lubrication theory, boundary friction model and fractal theory. Then the torque and load bearing characteristics of the clutch are obtained, and the influences of the surface fractal features are investigated and discussed. Finally, the Weierstrass–Mandelbrot function is adopted for the surface topography characterization and evaluation.FindingsThe results indicate that the proposed method exhibits good accuracy, while the speed difference between the friction pair exceeds 2,500 rpm. It is concluded that this paper proposed a way to evaluate the torque and loading capacity of HVD considering the real manufacturing surface topography and is helpful for surface optimization.Originality/valueThe originality and value of this study lie in its development of a novel torque and load bearing capacity evaluation method for HVD in mixed lubrication stage, considering manufacturing surface topography and describing the real manufacturing surface.

Nomes, lugares e malocas de dança: unidades sociais entre os Tukano orientais

Abstract Since Irving Goldman’s pioneering work among the Cubeo (Goldman, 1968 [1963]), defining Eastern Tukanoan social units has been a recurring issue in regional ethnology. Many late twentieth-century ethnographies produced solid knowledge on Tukano social structure by showing distinctive and exclusive features within the Amazonian scenario (such as language exogamy, patrilineality, specialized hierarchical clans or sibs, etc.). However, they also indicated some inconsistencies and paradoxes related to a discrepancy between the ideal model of hierarchy and patrilineality and a dynamic reality more attuned to equality and cognatic arrangements. Recent attempts to understand these paradoxes have focused on what a social unit actually means from an indigenous perspective, leading to new ideas about the fractal (multi-scale) character of such units and the way they work. These interpretations take the maloca (communal house) as a model, which evolves as it changes names, places, rituals, and specific narratives that serve to create permanent differences, which in turn produce disputes and alter kinship relations.

EXACT TRAVELING WAVE SOLUTION OF GENERALIZED (4+1)-DIMENSIONAL LOCAL FRACTIONAL FOKAS EQUATION

In this paper, within the scope of the local fractional derivative theory, the (4+1)-dimensional local fractional Fokas equation is researched. The study of exact solutions of high-dimensional nonlinear partial differential equations plays an important role in understanding complex physical phenomena in reality. In this paper, the exact traveling wave solution of generalized functions is analyzed defined on Cantor sets in high-dimensional integrable systems. The results of non-differentiable solutions in different cases are numerically simulated when the fractal dimension is equal to [Formula: see text] = ln 2/ln 3. The results show that the exact solution of the local fractional Fokas equation represents the fractal waves on the shallow water surface. Through numerical simulation, we find that the exact solution of the local fractional Fokas equation can describe the fractal waves and waves characteristics of shallow water surface. It also shows that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics.