Showing papers on "Fractal dimension published in 2020"
TL;DR: In this paper, the authors consider an advection-dispersion model, where the velocity is considered to be 1 and the kernels are power law, exponential decay law and the generalized Mittag-Leffler kernel.
Abstract: Nonlocal differential and integral operators with fractional order and fractal dimension have been recently introduced and appear to be powerful mathematical tools to model complex real world problems that could not be modeled with classical and nonlocal differential and integral operators with single order. To stress further possible application of such operators, we consider in this work an advection-dispersion model, where the velocity is considered to be 1. We consider three cases of the models, when the kernels are power law, exponential decay law and the generalized Mittag-Leffler kernel. For each case, we present a detailed analysis including, numerical solution, stability analysis and error analysis. We present some numerical simulation.
TL;DR: In this paper, the effect of fractal dimensions of pore-throat structures on petrologic and physical properties of tight sandstone was investigated, and the optical observations, fractal theory, and prediction model were integrated to explore the qualities of various reservoir types in tight sandstones.
Abstract: The pore-throat structures play a dominant role in the evaluation of properties of tight sandstone, but it remains difficult to determine the related parameters and understand their impact on reservoir quality Hence, toward this end, we analyze the experimental data that are indicative of the pore-throat system, then we investigate the effect of fractal dimensions of pore-throat structures on petrologic and physical properties, and finally, the optical observations, fractal theory, and prediction model were integrated to explore the qualities of various reservoir types in tight sandstones The results show that the fractal dimensions of the mercury intrusion curve correspond to three pore-throat types and those of the mercury extrusion curve could correspond to two pore-throat types Five types of reservoirs were identified, the best reservoir type has a high percentage of interparticle and dissolution pores but a low proportion of clay-related pores, and the differences in pore-throat connectivity of various types affect storage capacity significantly The storage ability prediction models of various reservoir types are raised by integrated experimental data This work employed a comprehensive fractal theory based on capillary pressure curves and helps to explore how pore-throat systems influence reservoir quality in tight sandstones
15 Feb 2020-Powder Technology
TL;DR: In this article, the pore structure of different metamorphic coal was analyzed by mercury intrusion porosimetry and N2 adsorption, and a new pore classification scheme was put forward by fractal dimensions.
Abstract: The pore structure of different metamorphic coal was analyzed by mercury intrusion porosimetry and N2 adsorption. Combined with the fractal dimension models of the Frenkel-Halsey-Hill and Menger, the pore size range of the experiment was spliced reasonably. A new pore classification scheme is put forward by fractal dimensions, revealing the pore structure and fractal characteristics of different metamorphic coal. The results suggest that pores exhibit piecewise fractal characteristics. A new pore classification scheme is put forward by piecewise fractal dimensions: I (0–2.0 nm, corresponding to D1); II (2.0–50 nm, corresponding to D2); III (50–2000 nm, corresponding to D3); IV (2000–20,000 nm, corresponding to D4); V (>20,000 nm, corresponding to D5). It is not inconsistent with the international union of pure and applied chemistry for pore classification, which shows that the new pore classification scheme is reliable.
01 Feb 2020-Materials Characterization
TL;DR: In this article, the influence of exposure temperature and resolution of X-ray CT on the determination of microstructure parameters of heat-treated mortar was focused, and it was found the porosity and pore size increased slightly when the exposure temperature varied from 105°C to 200°C and significant pore coarsening and micro-damage occurred once the temperature exceeded 400°C.
Abstract: In this contribution, the microstructure features of cement mortar exposed to various temperatures (105 °C, 200 °C, 400 °C, 600 °C, 800 °C) was investigated by combining Mercury Intrusion Porosimetry (MIP) and multi-scale X-ray computed tomography. The influence of exposure temperature and resolution of X-ray CT on the determination of microstructure parameters of heat-treated mortar was focused. Based on results of MIP test, it was found the porosity and pore size increased slightly when the exposure temperature varied from 105 °C to 200 °C and significant pore coarsening and micro-damage occurred once the temperature exceeded 400 °C. Bimodal pore size distribution (PSD) of the heat-treated mortar specimens was observed when the temperature reached 400 °C. To interpret the results of MIP test, the microstructure of heat-damaged mortar specimens was imaged using X-ray CT with a reconstructed voxel size of ~4.0 μm3 and then local volume inside the specimen was focused and scanned with a reconstructed voxel size of 1.5 μm3. A method was proposed to select proper threshold based on the MIP results for segmenting the void space from the X-ray CT images. The fracture aperture, 2D/3D fractal dimension, connectivity and tortuosity of the heat-damaged mortar specimens were further determined at different scale. By analyzing the fracture aperture determined from X-ray CT images, it was found the bimodal PSD revealed by MIP test can be associated with the creation of thermal micro-fractures. The fractal dimension increased remarkably when exposure temperature was raised from 400 °C to 600 °C while it varied slightly from 600 °C to 800 °C. Linear dependences between the fractal dimension and the volume fraction/tortuosity of micro-scale pores and fractures were found. The scale-dependent fractal properties of the heat-treated mortar were revealed with the capillary pressure data measured by MIP. The fractal dimension of micro-scale pores and fractures measured by MIP exhibited good consistency with that determined based on by X-ray CT images with a reconstructed voxel size of 1.5 μm3.
01 May 2020-Powder Technology
TL;DR: In this paper, the pore structure and fractal characteristics of acid-treated coal were analyzed using the low-temperature nitrogen adsorption method, scanning electron microscopy and the fractal theory.
Abstract: To study the pore structure and fractal characteristics of acid-treated coal, in this paper, the low-temperature nitrogen adsorption method, scanning electron microscopy and the fractal theory were used to analyze the coal samples. The results show that the acid treatment enlarged the wedge-shaped hole into a cylindrical hole. Under the influence of HNO3, the ash, volatile and sulfur contents are negatively correlated with the fractal dimension of the pore surface (D1) and positively correlated with the fractal dimension of the pore structure (D2). And the specific surface area and total pore volume of the coal samples are positively correlated with D1 and negatively correlated with D2 after acid treatment. Influenced by the specific surface area and total pore volume, the relationship between the fractal dimension and adsorption is consistent with the relationship between the fractal dimension and specific surface area/total pore volume.
TL;DR: Wang et al. as mentioned in this paper used a rough capillary bundle to characterize the coal structure and combined with fractal theory, a model including the tortuosity fractal dimension and the specific surface area of the pores was established based on the traditional Kozeny-Carman equation.
Abstract: It is of great significance to quantitatively describe the change in the permeability of the water-injected coal to improve the effect of the coal seam water injection technology. However, the current permeability model often assumes the pores of the porous medium are smooth, which is a large difference from the coarse coal matrix-pore interface. Therefore, a rough capillary bundle is used as the physical model to characterize the coal structure in this paper. Combined with fractal theory, a permeability model including the tortuosity fractal dimension and the specific surface area of the pores is established based on the traditional Kozeny-Carman equation, and the degree of influence of each factor on the permeability was obtained. Then, liquid permeability and structural parameters of the coal samples from the Daliuta Coal Mine and the Qincheng Coal Mine in China were obtained by nuclear magnetic resonance experiments, which verified the accuracy of the model. The research show that the tortuosity fractal dimension has the greatest influence on the theoretical permeability, and the theoretical permeability decreases rapidly when the tortuosity fractal dimension is between 1.05 and 1.20. Increasing the specific surface area of the pores will lead to an increase in the tortuosity fractal dimension and a decrease in the theoretical permeability. Under the different nuclear magnetic resonance experimental conditions, the theoretical permeability of the coal samples is consistent with the change in the liquid permeability and is closer to the measured value compared with the permeability models of Xu and Liu.
TL;DR: Wang et al. as discussed by the authors proposed a set of empirical equations based on the fractal dimension and different pore throat radius, in which the equation based on maximum permeability contribution radius has the highest correlation.
Abstract: Permeability estimation and quantitative evaluation of pore microstructure have always been hotspots and difficulties in tight sandstone reservoirs research. Taking the fractal dimension as the breakthrough, 10 core samples of Chang 7 section in the Upper Triassic Yanchang Formation were collected from different wells in Longdong area of Ordos Basin. The fractal dimension was calculated by mercury intrusion capillary pressure curve which can be obtained from high pressure mercury injection experiment, and samples were divided into four groups as group I, II, III and IV according to the value of fractal dimension estimated. The observation of casting thin sections and field emission scanning electron microscope show that group I mainly has dissolved-intergranular pores and original intergranular pores, group II is mainly dissolved-intergranular pores, group III mainly has dissolved-intragranular pores, and group IV is mainly micropores in clay matrix. Meanwhile, there is a highly correlation between the fractal dimension and the reservoir physical properties. With the fractal dimension of tight sandstone samples approaching 3, the compaction and cementation show a gradual increase, while the pore sorting, physical parameters and percolation capacity display a gradual decrease. Through multiple regression analysis, a set of empirical equations based on the fractal dimension and different pore throat radius were constructed, in which the equation based on maximum permeability contribution radius has the highest correlation. It shows that micropore (0.1–0.2 μ m) contributes most to permeability in tight sandstone reservoir. 10 blind samples were included in regression analysis to improve the prediction accuracy of the new model. A total of 28 samples, including 8 referenced data, were used to verify the predictive accuracy of the new model. The new empirical equation can predict the permeability of tight sandstone relatively accurate. Thus, the fractal dimension can effectively predict the permeability and evaluate the quality of tight sandstone reservoirs.
20 Jul 2020-Light-Science & Applications
TL;DR: Fractal lattices structured as Sierpinski gasket composed of an array of evanescently coupled helical waveguides, found theoretically that photonic topological insulators can also exist in fractal lattice, comprising only edges—with no bulk at all.
Abstract: We present Floquet fractal topological insulators: photonic topological insulators in a fractal-dimensional lattice consisting of helical waveguides. The helical modulation induces an artificial gauge field and leads to a trivial-to-topological phase transition. The quasi-energy spectrum shows the existence of topological edge states corresponding to real-space Chern number 1. We study the propagation of light along the outer edges of the fractal lattice and find that wavepackets move along the edges without penetrating into the bulk or backscattering even in the presence of disorder. In a similar vein, we find that the inner edges of the fractal lattice also exhibit robust transport when the fractal is of sufficiently high generation. Finally, we find topological edge states that span the circumference of a hybrid half-fractal, half-honeycomb lattice, passing from the edge of the honeycomb lattice to the edge of the fractal structure virtually without scattering, despite the transition from two dimensions to a fractal dimension. Our system offers a realizable experimental platform to study topological fractals and provides new directions for exploring topological physics. Photonic topological insulators are currently a subject of great interest because they support edge states that can propagate without being affected by defects and disorder. All topological insulators discovered thus far have a bulk surrounded by edges. Now, Zhaoju Yang and coworkers from Technion in Israel, found theoretically that photonic topological insulators can also exist in fractal lattices, comprising only edges—with no bulk at all. They studied fractal lattices structured as Sierpinski gasket composed of an array of evanescently coupled helical waveguides. Despite the lack periodicity in such structures, tight-binding simulations and quasienergy analysis predict the existence of topological edge states, residing either on outer or on inner edges, exhibiting a Chern number of 1 and displaying scattering-free propagation. The fractal symmetries of such lattices are found to be crucial for the existence of the topological properties. Such fractal lattices could be fabricated by femtosecond laser writing technology.
01 Feb 2020-Marine and Petroleum Geology
TL;DR: Fractal geometry provides an effective method for characterization of the complex and irregular pore structure of Eocene Shahejie low permeability sandstones in the Raoyang Sag, the Bohai Bay Basin, China as mentioned in this paper.
Abstract: Fractal geometry provides an effective method for characterization of the complex and irregular pore structure of Eocene Shahejie low permeability sandstones in the Raoyang Sag, the Bohai Bay Basin, China. Laboratory measurements including porosity, permeability, scanning electron microscope (SEM), thin sections, nuclear magnetic resonance measurements (NMR) and X-ray computed tomography (CT) technology are used to provide insights into the fractal characteristics of pore structure in sandstones of Eocene Shahejie Formation in the Bohai Bay Basin, China. Quantitative CT analysis reveals the pore radius is not always linear to T2 (transverse relaxation time) value obtained from the NMR tests, but instead power function of T2. Fractal analysis was performed on the T2 distribution using various fractal models, and the related fractal dimensions are calculated. The fractal dimensions calculated using various fractal models are correlated with NMR parameters and permeability. The fractal curves break into two segments at the T2cutoff (T2 separating the immovable and movable fluids) value or smaller when using fractal model Ⅰ and fractal model Ⅱ, and only the large-scale pore networks can be described by the fractal geometry. Mostly the entire pore size distributions (micro-pores to large-scale pore networks) can be described by the fractal model Ⅲ, and the calculated fractal dimensions are in accordance with the CT scanning and thin section data, and are strongly correlated with the T2gm (geometric mean of T2), permeability and BVI (bulk volume of immovable fluids). The fractal behaviors of pore size distributions from NMR analysis have implications for pore structure evaluation in low permeability sandstones with similar geological settings.
01 Jan 2020
TL;DR: It is derived that fractal analysis is important in understanding the growth of structures during different manufacturing processes (and parameters) and developing fractal-like structures for enhanced performance in various engineering applications.
Abstract: A review of the applications of fractal theory in modern manufacturing is presented in this chapter. A brief conceptual foundation, typical examples, and methods of computing fractals are summarized. The most common methods of computing fractal dimensions include box-counting, area-based measurements, and fractional Brownian motion (fBm) methods and have been briefly discussed along with the proposed improvements in the literature. It is noted that there are concerted efforts to improve on the known methods to enhance their accuracy and application in various fields. Finally, applications of fractals in thin films, laser processing, machining, and friction stir processes/welding are illustrated based on the published data. It is derived that fractal analysis is important in (1) understanding the growth of structures during different manufacturing processes (and parameters) and (2) developing fractal-like structures for enhanced performance in various engineering applications. Directions for future research and applications of fractal theory in manufacturing and their potentials are described in respective sections of the chapter. The chapter is a useful resource for academic and industry in studying, developing, and manufacturing of fractal-like engineering components and the interrelationships among the manufacturing process, parameters, and fractal characteristics of the engineering product.
01 Oct 2020-Chaos Solitons & Fractals
TL;DR: A reconstruction of the epidemic curves from the fractal interpolation point of view is proposed to retrieve missing pieces of information due to insufficient testing and predict the evolution of the disease.
Abstract: The present paper proposes a reconstruction of the epidemic curves from the fractal interpolation point of view. Looking at the epidemic curves as fractal structures might be an efficient way to retrieve missing pieces of information due to insufficient testing and predict the evolution of the disease. A fractal approach of the epidemic curve can contribute to the assessment and modeling of other epidemics. On the other hand, we have considered the spread of the epidemic in countries like Romania, Italy, Spain, and Germany and analyzed the spread of the disease in those countries based on their fractal dimension.
TL;DR: In this article, the pore size distribution and fractal dimension on adsorption and desorption values were systematically analyzed by scanning electron microscopy (SEM) and low pressure adaption N2, and CO2 methods for better evaluation of coalbed Methane and its pores geometrical mechanism.
Abstract: The significance of pore structure and fractal characteristics is important for understanding untapped Coalbed Methane (CBM) exploration. In this study, these features of low rank coal in the lower Indus Basin, SE Pakistan, were evaluated using drill core samples from several locations. The pore size distribution and fractal dimension on adsorption (AdDs) and desorption (DeDs) values were systematically analyzed by scanning electron microscopy (SEM), low pressure adsorption N2, and CO2 methods for better evaluation of CBM and its pores geometrical mechanism. Coal petrographic properties, were investigated for macerals composition revealed a huminite domination. SEM analysis of various pore structures, showed first-developed pores present in the larger pores and thermogenic gas pores. CO2 density functional theory revealed micropore particle size peaks in three adsorption phases on incremental pore volume (0.47–0.64 nm,0.64–0.83 nm, 0.83–11.11 nm), where gas molecules distort bonding with increasing pore size and define the micropore surface complexities. The mesopore size distribution analysis of differential and incremental pore areas indicated a transition zone at 22.8 nm, suggesting major micro-mesopores within smaller size particles. Nonhomogeneous particle sizes caused a dynamic range of peak trends due to different kinds of porespheres are interconnected with macrosphere, which can trap accessible and inaccessible gas molecules in the different pores structures. Fractal dimension analysis demonstrated that Ds values linear correlation showed a good fit for the AdDs value was 0.98–0.99, and 0.96 to 0.99 for the DeDs value. This finding showed that average obtained surface complexity of AdDs was 2.35, and the DeDs was 2.44 with notably more complicated pore roughness and high semifusinite content, possibly resulting in the higher DeDs value because larger pore spheres consist of microspheres.
08 Jul 2020
TL;DR: Variable-order space-time fractional diffusion equations, in which the variation of the fractional orders determined by the fractal dimension of the media via the Hurst index characterizes the media, are solved for LaSalle's inequality.
Abstract: Variable-order space-time fractional diffusion equations, in which the variation of the fractional orders determined by the fractal dimension of the media via the Hurst index characterizes the stru...
TL;DR: In this article, a 3D-Laser profiler was used to characterize the coal cleat surface and the fractal dimension of the surface was quantified by the Weierstrass-Mandelbrot function.
Abstract: Unconventional energy such as coal seam gas is trapped in a low porosity/permeability environment and poses challenges for production. Of particular interest is the mode of microscale gas transport through so-called coal cleats which form a network of micro-channels/fractures and allow for gas transfer through impermeable coal seams. The problem has attracted interest from many fields such as microstructural characterization, permeability-, porosity-, fluid flow-, diffusion-, adsorption/desorption- and adsorption-induced deformation studies. The main mode of gas flow is supported by the cleat network pattern for which the surface roughness is a very significant property of gas transport. Roughness plays an important role because adsorption is very sensitive to the change in the area. Direct studies on the effects of surface roughness for methane migration in coal by experimental methods are difficult to perform because the micro-scale heterogeneous pore structure and methane sorption properties are difficult to observe and therefore robust numerical simulations provide the most comprehensive method of investigation. In this paper, we develop a numerical model of coal seam gas diffusion and adsorption considering surface roughness based on fractal statistics and the lattice Boltzmann method. The rough surface is characterized by a 3D-Laser profiler and the fractal dimension of the cleat surface is quantified. Generic variants of the measured rough profiles are generated by the Weierstrass-Mandelbrot function through altering the fractal dimension D and length-scale parameter G. The fractal-based lattice Boltzmann method is introduced to solve the governing equations for the gas flow and diffusion processes and scaling laws may be a standardized analysis method and are applied for the non-dimensionalization of gas adsorption. By considering D ranging from 1.5 to 1.8 and G ranging from 1 to 20 it is found that the fractal dimension D has a small effect on the flow but a significant effect on the gas diffusion. In addition, the length-scale parameter G significantly affects gas transport through varying characteristic cleat aperture, and gas diffusion and adsorption by changing the contact area. The data rescaled by scaling laws provide a better analysis of gas adsorption through simplifying the assessment of the effects of multiple variables. Results of this work allow direct quantification of the effect of surface roughness on gas diffusion and adsorption in coal cleats and can provide an improved assessment of the effect of microscopic mechanisms of gas transfer/adsorption in coal on the overall large recovery of gas in the reservoir.
TL;DR: In this paper, the size and spatial distributions of mining-induced coal microfractures and the anisotropic tortuosity characteristics of coal fracture networks have been quantitatively determined based on fractal theory.
Abstract: Coal permeability is a key issue in CO2 injection and enhanced coalbed methane (CBM) recovery, and it is determined by the fracture network, which is strongly influenced by mining-induced stress evolutions. For a comprehensive understanding of the size and spatial distribution characteristics of coal fracture networks under different mining-induced stress conditions, a series of laboratory experiments, computed tomography (CT) scans and image analyses focusing on 3D coal fracture systems have been conducted considering the stress conditions induced by three typical mining layouts, i.e., top-coal caving mining (TCM), non-pillar mining (NM) and protective coal-seam mining (PCM). The size and spatial distributions of mining-induced coal microfractures and the anisotropic tortuosity characteristics of coal fracture networks have been quantitatively determined based on fractal theory. The results show that the size distributions of microfracture geometries, the fractal dimensions of fracture size distribution and the tortuosity of the mining-induced coal fracture networks vary according to the mining-induced stress conditions, resulting in differences in the seepage capacity of coal masses. The coal samples subjected to PCM conditions have the highest percentage of large microfractures, and those exposed to NM conditions have the lowest percentage. The fractal dimensions (Da and Dd) of the microfracture size distributions of the typical coal specimens decrease in the order of PCM, NM and TCM conditions, and the Da and Dd of coal specimens without pre-mining unloading-expansion simulation (PUES) are much lower than those of specimens with PUES. The tortuosity fractal dimensions (δ) of the mining-induced fracture network are spatially anisotropic. The ranges of δ for the coal masses exposed to PCM, NM and TCM conditions are 1.0–1.2, 1.2–1.3 and 1.25–1.4, respectively. Using fractal theory and the measured fracture network data, the anisotropic spatial distribution of coal permeability can be theoretically estimated.
16 Apr 2020-Physical Review Letters
TL;DR: In this article, the scaling properties of geometric operators such as lengths, areas, and volumes in models of higher derivative quantum gravity were determined by renormalizing appropriate composite operators. And they used these results to deduce the fractal dimensions of such hypersurfaces embedded in a quantum spacetime at very small distances.
Abstract: We determine the scaling properties of geometric operators such as lengths, areas, and volumes in models of higher derivative quantum gravity by renormalizing appropriate composite operators. We use these results to deduce the fractal dimensions of such hypersurfaces embedded in a quantum spacetime at very small distances.
TL;DR: In this paper, a new model for 2D complex tortuous fractured porous media based on fractal theory, and two important characteristics of fractures, i.e. branching and tortuosity are considered in the developed permeability model.
Abstract: The complex fracture networks are of significance to petroleum production in unconventional reservoirs, but effectively predicting its permeability is challenging due to the complex distribution and geometry of fracture networks. This paper proposes a new permeability model for 2D complex tortuous fractured porous media based on fractal theory, and two important characteristics of fractures, i.e. branching and tortuosity are considered in the developed permeability model. In this paper, the fractured porous media is assumed to be made up of a bundle of tortuous fractal-like tree fracture networks. First, the modified cubic law for describing flow rate in a fracture with fractal distribution of tortuosity is derived. Secondly, the fractal-like tree fracture network is used to represent the distribution of complex fracture networks, and then flow rate in a tortuous fractal-like tree fracture network is derived. After that, with the assumption of the fractal scaling law between fracture number and fracture apertures, the total flow rate in a bundle of tortuous fractal-like tree fracture networks is given and then the analytical expression for permeability of 2D complex tortuous fractured porous media is derived. Finally, the model reliability is verified with a case study and the key parameters influencing permeability of 2D fracture porous media are investigated with sensitivity study. The research results show that the fractal dimension of aperture distribution D f , tortuosity fractal dimension D T , branch level m and fracture length ratio γ have significant influence on fracture network permeability; the effects of branch angle θ and the straight length of the zero-level fracture l 0 on fracture network permeability are insignificant.
TL;DR: In this article, the authors investigate the influence of the artificial crack density on the shear fracturing and fractal behavior of rock bridges in jointed rock slopes, and find strong correlations that display an exponential decay function.
Abstract: Shear-box experiments, Rock Failure Process Analysis (RFPA) simulations, and box-counting fractal analysis on rock-like models are conducted to investigate the influence of intermittent artificial crack density on the shear fracturing and fractal behavior of rock bridges in jointed rock slopes. The artificial crack geometry of the conceptual rock bridge model is a combination of two edge-notched artificial cracks and imbedded artificial cracks with different intermittent artificial crack densities. By numerical shear-box tests, deep insight into the mesoscopic mechanism of crack evolution is gained, and the simulated failure patterns are in accordance with experimental results. Three types of failure patterns are identified: shear mode, mixed shear/tensile mode, and tensile mode. The RFPA simulations demonstrate that macroscale shear cracks form as damage belts consisting of many tensile/shear mesocracks, as typically observed in microscopic experimental work. The failure pattern is mostly influenced by the intermittent artificial crack density, whereas the peak shear strength is related to the failure pattern. As the intermittent artificial crack density increases, the failure pattern changes from shear mode to mixed shear/tensile mode and then to tensile mode, resulting in a decrease in the peak shear strength. The regression analysis shows that the relationship between the peak shear strength and intermittent artificial crack density can be expressed by an exponential decay model. Furthermore, digital image processing and box-counting fractal analyses are performed on the shear fracture surfaces of the physical and numerical models to describe the fractal behavior. The relationships between the fractal dimension and peak shear strength are analyzed, and strong correlations that display an exponential decay function are found.
01 Jun 2020-Acta Geotechnica
TL;DR: In this article, the microstructure of bentonite from Cerny vrch deposit in the Czech Republic is clarified by quantification of fractal dimension characteristics of pores of different sizes.
Abstract: In this paper, we aim to clarify microstructure of bentonite from Cerny vrch deposit in the Czech Republic. We adopt results of ESEM and MIP experiments performed at various suctions along wetting and drying paths on bentonite samples compacted from powder to two different initial dry densities. The data were used for quantification of fractal dimension characteristics of pores of different sizes. Two different methods of calculating fractal dimension were used for MIP data, and one method was used for evaluation of ESEM images. Fractal dimensions obtained from MIP data, combined with the measured pore size density functions, allowed us to identify two different pore families: micropores and macropores. Macropores can be further subdivided into fine macropores and coarse macropores based on fractal analysis. The pore systems were further distinguished by different responses to suction changes and to compaction effort. In general, we observed slight increase in fractal dimension with increasing suction and with increasing dry density.
15 Oct 2020-NeuroImage
TL;DR: Results show that psychedelic drugs increase the fractal dimension of activity in the brain and this is seen as an indicator that the changes in consciousness triggered by psychedelics are associated with evolution towards a critical zone.
Abstract: Psychedelic drugs, such as psilocybin and LSD, represent unique tools for researchers investigating the neural origins of consciousness. Currently, the most compelling theories of how psychedelics exert their effects is by increasing the complexity of brain activity and moving the system towards a critical point between order and disorder, creating more dynamic and complex patterns of neural activity. While the concept of criticality is of central importance to this theory, few of the published studies on psychedelics investigate it directly, testing instead related measures such as algorithmic complexity or Shannon entropy. We propose using the fractal dimension of functional activity in the brain as a measure of complexity since findings from physics suggest that as a system organizes towards criticality, it tends to take on a fractal structure. We tested two different measures of fractal dimension, one spatial and one temporal, using fMRI data from volunteers under the influence of both LSD and psilocybin. The first was the fractal dimension of cortical functional connectivity networks and the second was the fractal dimension of BOLD time-series. In addition to the fractal measures, we used a well-established, non-fractal measure of signal complexity and show that they behave similarly. We were able to show that both psychedelic drugs significantly increased the fractal dimension of functional connectivity networks, and that LSD significantly increased the fractal dimension of BOLD signals, with psilocybin showing a non-significant trend in the same direction. With both LSD and psilocybin, we were able to localize changes in the fractal dimension of BOLD signals to brain areas assigned to the dorsal-attenion network. These results show that psychedelic drugs increase the fractal dimension of activity in the brain and we see this as an indicator that the changes in consciousness triggered by psychedelics are associated with evolution towards a critical zone.
01 Jan 2020-Journal of Aerosol Science
TL;DR: In this paper, the authors investigated different approaches for the determination of morphological parameters of fractal-like particle aggregates from TEM images of soot and compared the results to derive an optimized evaluation strategy for TEM nanoparticle characterization.
Abstract: For a comprehensive understanding of nanoparticle formation in gas phase processes, such as soot formation, morphological parameters of fractal-like particle aggregates, like the radius of gyration, the fractal dimension and the primary particle size, have to be determined. Often transmission electron microscopy (TEM) is employed for the investigation of particle characteristics as it not only allows to investigate ensemble averages but single particle aggregates and thus also to determine statistical properties, such as the size distribution within a sample. Many different evaluation methods can be found in the literature. We investigated different approaches for the determination of morphological parameters from TEM images of soot and compared the results to derive an optimized evaluation strategy for TEM nanoparticle characterization. We compared four methods for the determination of the radius of gyration – three length- and one pixel-based methods – showing good agreement within 9% deviation for the median of the recovered lognormal size distribution. Furthermore, the fractal dimension was determined via a sample-based and various box counting methods with different limiting box sizes. Here, we could show that the upper and lower bounds (aggregate size in terms of radius of gyration and primary particle size in terms of its radius, respectively) of self-similarity of fractal-like aggregates should be accounted for by choosing corresponding upper and lower box sizes. Using box counting, we could show that for small aggregates the fractal dimension as well as its span are increased, yet with increasing aggregate size the fractal dimension converges towards 1.6. Furthermore, we could show the potential of semi-automatic aggregate detection through Trainable Weka Segmentation. However, image noise resulting in erroneous aggregate splitting often leads to smaller aggregate sizes by automatic detection compared to manual segmentation. Generalized Hough Transformation for the semi-automatic determination of primary particle sizes performs well for large soot particle aggregates as those often show spherical primary particles. For leaner combustion conditions, the primary particles of the formed clumpy soot aggregates cannot be detected well via semi-automatic detection. TEM images were taken on soot samples from premixed laminar flat flames (burner type: McKenna) under various conditions to provide comprehensive reference data.
TL;DR: In this article, the fractal dimension of the seismic spatio and temporal distribution in the Eurasian seismic belt is quantitatively evaluated based on the single fractal model, and it is shown that over time the time interval of seismic activity shortened, and the seismic activity on the seismic belt has a nonlinear structure and self-similar characteristics.
Abstract: The Eurasian seismic belt is the second largest seismic zone in the world. It has numerous seismic activities which have enormous impact on human’s life. It is of importance to study the spatio–temporal characteristics of the Eurasian seismic belt. As we can learn from previous studies, fractal and fractal dimension theories can be used to study seismic activities. In general, the temporal sequence and the spatio sequence of earthquakes both exhibit the fractal structures. When huge earthquakes occur, the fractal dimension of the temporal sequence is very low. As the days went by, the value of fractal dimension fluctuates. The seismic data from 1973 to 2014 in The Eurasian seismic belt are selected as the object of this study. Based on the single fractal model, the complex structure of the seismic spatio and temporal distribution in the seismic belt is quantitatively evaluated. Results show that over time the time interval of the seismic activity shortened, and the seismic activity on the Eurasian seismic belt has a nonlinear structure and self-similar characteristics. From the perspective of space, the fractal dimension of the Eurasian seismic belt tends to grow with time, and it also has a nonlinear structure and self-similar characteristics. When the temporal unit is set as 1 year, the accumulation and release of energy are probably periodic: the minor period might be about 8.5 years, and the major period might be about 13 years.
TL;DR: In this article, the authors investigated the relationship between the fractal characteristics of the acoustic emission (AE) correlation and the mechanical damage of different ratios of tantalum-niobium tailings CPB materials.
Abstract: The motivation for this work was to investigate the relationship between the fractal characteristics of the acoustic emission (AE) correlation and the mechanical damage of different ratios of tantalum–niobium tailings CPB materials. The AE parameters of the tailings CPB materials under uniaxial compression were obtained by an AE test. On this basis, the fractal dimensions of the AE parameters were derived, and the change rule of the fractal parameters during the entire specimen damage was obtained by analysis. The results showed that the minimum value of the fractal dimension was generally within a stress-level range of 70% to 90%. While approaching the peak value of the stress, the amplitude fractal dimension displayed an increasing trend. The AE amplitude, event rate, energy rate, and spatial distribution exhibited fractal characteristics. In the investigation of the AE fractal, the fractal dimension of the amplitude was more concentrated, which exhibited that it was more accurate to determine the failure of a cemented paste backfill (CPB). Therefore, the study focused on using the amplitude fractal dimension value to monitor the failure of the CPB. This feature provides a reference for the control of the stability of the surrounding rock by backfilling the goaf with CPB.
TL;DR: It is suggested that cortical functional connectivity networks display fractal character and that this is associated with level of consciousness in a clinically relevant population, with higher fractal dimensions (i.e. more complex) networks being associated with higher levels of consciousness.
Abstract: Recent evidence suggests that the quantity and quality of conscious experience may be a function of the complexity of activity in the brain and that consciousness emerges in a critical zone between low and high-entropy states. We propose fractal shapes as a measure of proximity to this critical point, as fractal dimension encodes information about complexity beyond simple entropy or randomness, and fractal structures are known to emerge in systems nearing a critical point. To validate this, we tested several measures of fractal dimension on the brain activity from healthy volunteers and patients with disorders of consciousness of varying severity. We used a Compact Box Burning algorithm to compute the fractal dimension of cortical functional connectivity networks as well as computing the fractal dimension of the associated adjacency matrices using a 2D box-counting algorithm. To test whether brain activity is fractal in time as well as space, we used the Higuchi temporal fractal dimension on BOLD time-series. We found significant decreases in the fractal dimension between healthy volunteers (n = 15), patients in a minimally conscious state (n = 10), and patients in a vegetative state (n = 8), regardless of the mechanism of injury. We also found significant decreases in adjacency matrix fractal dimension and Higuchi temporal fractal dimension, which correlated with decreasing level of consciousness. These results suggest that cortical functional connectivity networks display fractal character and that this is associated with level of consciousness in a clinically relevant population, with higher fractal dimensions (i.e. more complex) networks being associated with higher levels of consciousness. This supports the hypothesis that level of consciousness and system complexity are positively associated, and is consistent with previous EEG, MEG, and fMRI studies.
01 Jan 2020-Journal of Earth Science
TL;DR: In this article, a series of experiments were conducted on outcrop samples from the Lower Cambrian Niutitang Formation on Yangtze Platform, including X-ray diffraction (XRD), field-emission scanning electron microscopy (FE-SEM), and low-temperature nitrogen adsorption.
Abstract: Shales from the Lower Cambrian Niutitang Formation of Yangtze Platform have been widely investigated due to its shale gas potential. To better illustrate the pore structure and fractal characteristics of shale, a series of experiments were conducted on outcrop samples from the Lower Cambrian Niutitang Formation on Yangtze Platform, including X-ray diffraction (XRD), field-emission scanning electron microscopy (FE-SEM) and low-temperature nitrogen adsorption. Frenkel-Halsey-Hill (FHH) model was adopted to calculate the fractal dimensions. Furthermore, the relationships between fractal dimensions and pore structure parameters and mineral composition are discussed. FE-SEM observation results show that interparticle pores are most developed in shale, followed by intraparticle pores. This study identified the fractal dimensions D1 (ranging from 2.558 0 to 2.710 2) and D2 (ranging from 2.541 5 to 2.765 2). The pore structure of the Niutitang Formation shale is primarily controlled by quartz and clay content. Fractal dimensions are able to characterize the pore structure complexity of Niutitang Formation shale because D1 and D2 correlate well with average pore diameter and quartz content.
TL;DR: The experimental results illustrate that the proposed FD estimation method is effective and efficient and outperforms the three state-of-the-art methods by observing the values of two proposed metrics, namely average error and average computed FD.
Abstract: Fractal dimension (FD) is a useful metric for the analysis of natural images that exhibit a high degree of complexity, randomness and irregularity in color and texture. Several approaches exist in the literature to measure FD of gray-scale images. The aim of this study is to introduce a FD estimation method for color images with color proximity in Lab space. The proposed method uses a xy-plane partitioning–shifting mechanism, where the divisors of image size are used as grid sizes. The proposed method simulates on synthesized color fractal Brownian motion (FBM) images, publicly available Brodatz database, Google color fractal images and noisy Brodatz database. The random midpoint displacement algorithm for the formation of gray-scale images is extended in this work to synthesize color FBM images. Noisy Brodatz database is obtained by adding salt-and-pepper noise with different noise densities to understand the behavior of FD. The experimental results illustrate that the proposed method is effective and efficient and outperforms the three state-of-the-art methods by observing the values of two proposed metrics, namely average error and average computed FD. A new mathematical expression for estimating FD of a color image is demonstrated, which relies on the number of edge pixels of individual color channel using multiple linear regression.
TL;DR: In this article, a 6-year continuous anemometric data was used to analyze the fractal dimension of wind speed time series recorded under various terrain conditions based on box-counting method.
Abstract: Understanding the persistence in time series is of crucial importance relating to the reliable forecast of wind speed. It has been widely acknowledged that fractal analysis is a useful tool to evaluate the persistence in wind speed time series using the fractal dimension (D) as a quantitative indicator. This paper aims to unveil the persistent characteristics of wind speed time series recorded under various terrain conditions based on 6-year continuous anemometric data. Fractal dimension analysis is carried out using box-counting method. The results indicate that the 10-min wind speed time series analysed in this study exhibit clear fractal behaviour, characterizing a daily fractal dimension between 1.32 and 1.47. Larger D occurs mostly at urban conditions, while the minimum is obtained at offshore condition. The monthly pattern of fractal dimension is strongly correlated with the turbulence intensity, in which the fractal dimension either remains relatively consistent or exhibits marked monthly maxima during hotter months. Furthermore, the fractal dimension is closely tied with the length of data, in which D typically increases with increasing window-width, and decreases as the measurement time interval increases.
01 Mar 2020-Indagationes Mathematicae
TL;DR: In this article, the Hausdorff dimension and box dimension of the graph of a continuous function defined on a rectangular region in R 2, which is of bounded variation according to some of these approaches are studied.
Abstract: In contrast to the univariate case, several definitions are available for the notion of bounded variation for a bivariate function. This article is an attempt to study the Hausdorff dimension and box dimension of the graph of a continuous function defined on a rectangular region in R 2 , which is of bounded variation according to some of these approaches. We show also that the Riemann–Liouville fractional integral of a function of bounded variation in the sense of Arzela is of bounded variation in the same sense. Further, we deduce the Hausdorff dimension and box dimension of the graph of the fractional integral of a bivariate continuous function of bounded variation in the sense of Arzela.
TL;DR: The paper explores an alternative strategy to the standard time analysis, by joining the multidimensional scaling (MDS) computational tool and the concepts of distance, entropy, fractal dimension, and fractional calculus, which highlights the results obtained by the other tools.
Abstract: Financial time series have a fractal nature that poses challenges for their dynamical characterization. The Dow Jones Industrial Average (DJIA) is one of the most influential financial indices, and due to its importance, it is adopted as a test bed for this study. The paper explores an alternative strategy to the standard time analysis, by joining the multidimensional scaling (MDS) computational tool and the concepts of distance, entropy, fractal dimension, and fractional calculus. First, several distances are considered to measure the similarities between objects under study and to yield proper input information to the MDS. Then, the MDS constructs a representation based on the similarity of the objects, where time can be viewed as a parametric variable. The resulting plots show a complex structure that is further analyzed with the Shannon entropy and fractal dimension. In a final step, a deeper and more detailed assessment is achieved by associating the concepts of fractional calculus and entropy. Indeed, the fractional-order entropy highlights the results obtained by the other tools, namely that the DJIA fractal nature is visible at different time scales with a fractional order memory that permeates the time series.