Theoretical Fractal Dimension Research Articles

STUDY ON FRACTAL CHARACTERISTICS OF FISSURE SPACE STRUCTURE AND TIGHT SOLID STRUCTURE OF COAL

The microstructure of a coal is a kind of fissured fractal tight porous media. The formation and action mechanism between its space and solid structure has a strong fundamental significance for engineering seepage problem. Currently, the research and experimental theory of the quantitative characterization of the fissure space structure is relatively complete. However, the physical meaning of the fractal dimension used to characterize structural features remains unclear. At the same time, the tight feature is attributed to the original complexity, and fractal topography dominates the behavioral complexity. Therefore, it is of great significance to clarify the distribution of tight solid structure from the fractal perspective for clarifying the complexity of coal structure. In this paper, the proportion of space structure and solid structure in coal body is accurately measured by scanning electron microscope (SEM) experiment combined with nuclear magnetic resonance (NMR) experiment, and a theoretical model which can be used to measure the fractal dimension of solid structure is proposed based on Menger sponge theory. In the process, the physical meaning of fractal dimension of the fissure space structure measured by the image analysis method and fluid intrusion method is proposed. The research shows that for the fissure space structure, the fractal dimension of fissure space structure calculated based on the SEM image and the box dimension method is between 1 and 2, showing an obvious positive correlation with the proportion of the fissure space structure area. The fractal dimension based on NMR [Formula: see text] spectrum curve fitting mainly characterizes the complexity of the pore size distribution. The fractal dimension increases with the increasing complexity of pore size distribution. For the tight solid structure, the fractal dimension obtained by theoretical calculation is less than 3, the compound fractal scaling law. The two-dimensional fractal dimension obtained through experiments and related algorithms is greater than 2, which do not conform to the fractal scaling law in general, but it can represent the proportion of solid structure in a limited scale as the theoretical fractal dimension.

Capacitance for fractal-like disordered dielectric slab

In this paper, we model a heterogeneous dielectric medium exhibiting fractal geometry or disordered random structures by applying non-integer dimensions to determine its capacitance between two parallel plates. The capacitance depends on the fractional dimensions of the fractal or disordered dielectric slab, which may be obtained from the theoretical fractal dimension or box-counting method. The findings are verified by CST Studio Suite (Electromagnetic field simulation software), experimental measurements, and the equivalent capacitance method. Five common types of fractals (Cantor bars/plates, Sierpinski carpet, Sierpinski triangle, Haferman carpet, and Menger sponge) and random structures are tested with good agreement. There is also an effective gain of capacitance in using less amount of dielectric materials, which may be useful in material-savings of dielectrics. This research shows a useful tool in modeling the capacitance of heterogeneous materials, where fractals and disordered structures may be commonly encountered in organic materials and any dielectrics where precision and fabrication are not perfect.

Fractal characteristics of the microstructure of loaded coal based on NMR and μ-CT

ABSTRACT The change in the fracture microstructure of loaded coal affects the gas seepage characteristics of the coal. Using micron CT scanning (μ-CT) and nuclear magnetic resonance (NMR) experiments, the fractal characteristics of coal fracture structure under triaxial compression are studied from two aspects of theoretical and experimental fractal dimension for the first time. VG Studio MAX software was used to reconstruct the three-dimensional (3D) fracture structure of the coal samples, and the 3D fracture structure was modified by the porosity measured by NMR. Through the analysis of the reconstruction model and the T2 relaxation spectrum, the spatial distribution of internal cracks in coal was obtained. The fractal geometry theory, NMR experimental fractal dimension algorithm and 3D box counting method were used to calculate the theoretical and experimental fractal dimensions and then comprehensively analyze the fractal characteristics of cracks in loaded coal. The results show that an external load promotes the development of pores/cracks in coal. With increasing theoretical fractal dimension, CT porosity increases from 0.0630% to 0.101%, and NMR porosity increases from 0.0655% to 0.105%. The relationship between porosity and theoretical fractal dimension can be expressed by y = ax2+ bx+c (fractal dimension is an independent variable). With increasing porosity, the experimental fractal dimension of NMR decreases linearly from 2.975 to 2.951, and the experimental fractal dimension of μ-CT increases linearly from 2.191 to 2.324. The different changing trends of theoretical and experimental fractal dimensions verify the differences in their representational meanings.

Improved box-counting methods to directly estimate the fractal dimension of a rough surface

Rough surfaces exist widely in nature, and the measurement of these surfaces’ fractal dimension is beneficial for understanding their formation mechanism and effect. The basic principles of previous cubic covering methods are described, and then the stable cubic covering method (SCCM) and stable differential cubic covering method (SDCCM) are proposed based on the box-counting method. Afterward, a real sandstone fracture surface and Takagi surfaces are adopted to compare these methods’ differences. The results show that different measurement methods are needed to measure surfaces with different theoretical fractal dimensions. Meanwhile, the SCCM and SDCCM can obtain more stable measurement results than previous methods. Then, the recommended methods and their corresponding scale factors are summarized. Finally, the potential of these cubic covering methods to measure a thick film is described. The research is of great significance to the fractal dimension measurement of a rough surface or a thick film.

Clustering, Connectivity and Flow Responses of Deterministic Fractal-Fracture Networks

Abstract. It is well known that fracture networks display self-similarity in many cases and the connectivity and flow behavior of such networks are influenced by their respective fractal dimensions. In the past, the concept of lacunarity, a parameter that quantifies spatial clustering, has been implemented by one of the authors in order to demonstrate that a set of seven nested natural fracture maps belonging to a single fractal system, but of different visual appearances, have different clustering attributes. Any scale-dependency in the clustering of fractures will also likely have significant implications for flow processes that depend on fracture connectivity. It is therefore important to address the question as to whether the fractal dimension alone serves as a reasonable proxy for the connectivity of a fractal-fracture network and hence, its flow response or, if it is the lacunarity, a measure of scale-dependent clustering, that may be used instead. The present study attempts to address this issue by exploring possible relationships between the fractal dimension, lacunarity and connectivity of fractal-fracture networks. It also endeavors to study the relationship between lacunarity and fluid flow in such fractal-fracture networks. A set of deterministic fractal-fracture models generated at different iterations and, that have the same theoretical fractal dimension are used for this purpose. The results indicate that such deterministic synthetic fractal-fracture networks with the same theoretical fractal dimension have differences in their connectivity and that the latter is fairly correlated with lacunarity. Additionally, the flow simulation results imply that lacunarity influences flow patterns in fracture networks. Therefore, it may be concluded that at least in synthetic fractal-fracture networks, rather than fractal dimension, it is the lacunarity or scale-dependent clustering attribute that controls the connectivity and hence the flow behavior.

Fractal analysis of recurrence networks constructed from the two-dimensional fractional Brownian motions.

In this study, we focus on the fractal property of recurrence networks constructed from the two-dimensional fractional Brownian motion (2D fBm), i.e., the inter-system recurrence network, the joint recurrence network, the cross-joint recurrence network, and the multidimensional recurrence network, which are the variants of classic recurrence networks extended for multiple time series. Generally, the fractal dimension of these recurrence networks can only be estimated numerically. The numerical analysis identifies the existence of fractality in these constructed recurrence networks. Furthermore, it is found that the numerically estimated fractal dimension of these networks can be connected to the theoretical fractal dimension of the 2D fBm graphs, because both fractal dimensions are piecewisely associated with the Hurst exponent H in a highly similar pattern, i.e., a linear decrease (if H varies from 0 to 0.5) followed by an inversely proportional-like decay (if H changes from 0.5 to 1). Although their fractal dimensions are not exactly identical, their difference can actually be deciphered by one single parameter with the value around 1. Therefore, it can be concluded that these recurrence networks constructed from the 2D fBms must inherit some fractal properties of its associated 2D fBms with respect to the fBm graphs.

Digital evaluation of nanoscale-pore shale fractal dimension with microstructural insights into shale permeability

Permeability significantly affects the production of shale oil and shale gas, and shale microstructures characterized by pore and fracture spaces innately affect the permeability. However, the quantitative characterization of permeability related to pore and facture spaces is not fully understood. This work aims to propose 3D spatial fracture-pore fractal dimensions to predict shale permeability and their effects on fluid flow behaviors. First, triaxial compressive stress and X-ray CT imaging tests are conducted on shale samples to establish fractural models. The digital surface roughness segmentation (DSRS) method is then proposed to obtain the fracture-pore microstructures. Next, spatial fractal dimensions of self-similarity microstructures are proposed to predict the microstructural permeability. Finally, two-phase fluid flows are simulated to study the hydrocarbon flow behaviors in fractural microstructures using the level set method. The results show that the average relative errors between the microstructural spatial dimensions and theoretical fractal dimensions are all less than 3%, highlighting the accuracy of the proposed method. The numerical results for permeability are very close to the analytical solutions, in which fracture permeability is almost 100 times the order of magnitude of the pore structure permeability in the nanoscale pore shale, and the facture and pore structure permeabilities both increase with increasing spatial fractal dimension. The changes of fluid flow behaviors are similar to the permeability variations, and the fluid phase fraction increases with increasing fractal dimension.

Experimental Study on the Fractal Pattern of a Coal Body Pore Structure Around a Water Injection Bore

Abstract In order to enhance the disaster prevention effect of coal seam water injection technology, in this paper, the structural characteristics of the coal sample under the true mechanical environment of coal seam water injection are measured via nuclear magnetic resonance technology, and the quantitative relation between the theoretical and the experimental pore volume fractal dimension is analyzed based on fractal geometrical theory. The results show that there is a large difference between the porosity of seepage pores and absorption pores, 1.345–2.818% and 6.840–7.940%, respectively, indicating obvious inhomogeneity of the internal structure development. However, their porosities’ overall change with pore water pressure and confining pressure is consistent, that is, increasing confining pressure decreases porosity, while for increasing pore water pressure it is the opposite, and confining pressure and pore water pressure have a greater impact on the seepage pores’ porosity; meanwhile, based on the pore size distribution curves, it can be found that pore water pressure can enlarge pore volume, and confining pressure can reduce pore volume. In addition, seepage pores’ experimental and theoretical fractal dimension values are between 2.920–2.968 and 2.0737–2.2327, respectively, and adsorption pores’ experimental and theoretical fractal dimensions are between 2.296–2.343 and 2.4146–2.4471 respectively. The quantitative relation between theoretical and experimental fractal dimensions is established to achieve a common characterization of the pore structure of a coal body under load via both the theoretical and experimental fractal dimensions.

Differential box counting methods for estimating fractal dimension of gray-scale images: A survey

Abstract Fractal dimension is extensively in use as features in computer vision applications to characterize roughness and self-similarity of objects in an image for many years. These features have been adopted successfully mainly in texture segmentation and classification. Differential box counting method is one of the widely accepted approaches, those exist in literature to estimate fractal dimension of an image. In this work, we comprehensively reviewed the available differential box counting methods. First, the differential box counting method is discussed in detail along with its computer vision applications and drawbacks. Second, various variants of differential box counting method are thoroughly studied and grouped using different parameters of differential box counting method. Third, the synthetic and real-world databases, considered for demonstrating experimental results by the state-of-the-art methods have been presented. Fourth, some of the state-of-the-art methods have been implemented and corresponding results obtained in this study are reported. Fifth, three evaluation metrics have also been reviewed. However, these metrics work only for synthetic fractal Brownian motion images because the theoretical fractal dimension values for these images are known and have been used as a set of ground truths. Finally, we concluded the status of differential box counting methods and explored the possible future directions.

A NEW METHOD FOR CALCULATING THE FRACTAL DIMENSION OF SURFACE TOPOGRAPHY

A new method termed as three-dimensional root-mean-square (3D-RMS) method, is proposed to calculate the fractal dimension (FD) of machined surfaces. The measure of this method is the root-mean-square value of surface data, and the scale is the side length of square in the projection plane. In order to evaluate the calculation accuracy of the proposed method, the isotropic surfaces with deterministic FD are generated based on the fractional Brownian function and Weierstrass–Mandelbrot (WM) fractal function, and two kinds of anisotropic surfaces are generated by stretching or rotating a WM fractal curve. Their FDs are estimated by the proposed method, as well as differential boxing-counting (DBC) method, triangular prism surface area (TPSA) method and variation method (VM). The results show that the 3D-RMS method performs better than the other methods with a lower relative error for both isotropic and anisotropic surfaces, especially for the surfaces with dimensions higher than 2.5, since the relative error between the estimated value and its theoretical value decreases with theoretical FD. Finally, the electrodeposited surface, end-turning surface and grinding surface are chosen as examples to illustrate the application of 3D-RMS method on the real machined surfaces. This method gives a new way to accurately calculate the FD from the surface topographic data.

Particle Arrangement Design for Predicting the Percolation Threshold of Silver/Epoxy Composite for Electrically Conductive Adhesive Application

The early use of electrically conductive adhesives (ECAs) provided an alternative to solder because of their several advantages, such as good electrical conductivity, low cost, extendability to fine pitch of interconnecting material and environmental friendliness. According to previous works, an optimal particle volume fraction became a major objective of many researchers in order to obtain highly conductive ECAs, realizing that the need for transitions from an insulator to a conductor is controlled by the geometric arrangement of particles. In the current study, particle arrangement models of ECAs are developed by establishing the effects of van der Waals’ attraction energy and particle motion, which act as a kind of particle interaction to generate a conducting structure. The methodology is divided into three major parts: the formulation of a particle arrangement technique, and numerical and experimental studies. The formulation of particle arrangement is developed in an epoxy colloidal system. During verification, the particle arrangement model is validated by the theoretical fractal dimension and guided by a morphological study of the experimental assessments. The model was simulated through representative volume elements with the volume fraction factor, which was set in the range of 2–8 vol.%, while electrical conductivity was an observed parameter. The numerical results showed good agreement with the experiments in which the percolation threshold occurred between 4 and 6% of the volume of filler loading.

The application of fractal box dimensions in predicting the emission characteristics of colliding sawdust particles for sustainable sawmilling

The prediction of emission characteristics of sawdust particles immediately after the cutting operation from the interaction of band saw?s blade and plank is a growing research area. Still, a wide gap exists with respect to understanding the behaviour of sawdust particles as they collide with one another. Previous efforts have focused on non-collision states of sawdust particles. However, in real life, collision of particles must occur, with several particles colliding after the cutting operation. This paper establishes a new perspective of the fractal properties of sawdust particles in motion as a motivation to understanding how to control its toxicity of effects on sawmill workers and maintain sustainable sawmilling activities. In particular, the possibility of predicting the fractal dimension of the randomly moving sawdust particles in sawmills that is generated as fractal curves using the combination of probabilities and theoretical fractal dimensions is investigated for the first time. Cases were established on the possible representations of the theory and practice. As an example, four cases were designed around varied number of fractal pattern combinations drawn out of five and fifty different probabilities combinations, ten different random number generating seed values and maximum of four fractal curves generation iterations as driven parameters. Preliminary study of the differences between theoretical fractal box dimension recorded a maximum absolute percentage error of 7.24% for fractal curve associated with fractal pattern five (i.e. Koch 5). In all the cases studied, average absolute percentage error decreases between 3.52 � 1.18 and 1.51 � 1.14 while the correlation coefficient (R2) decreases between 0.9315 and 0.7365 from case 1 to case 4, respectively. It is concluded that the model is a good predictor of sawdust particle emission at colliding states from cutting operation. This is reflected in the fact that the higher the number of fractal patterns (generators) in a study case, the smaller the correlation coefficient between average estimated fractal box dimension and predicted fractal dimension of the sawdust particles in motion in the sawmill.

A novel method to identify the scaling region for chaotic time series correlation dimension calculation

To obtain more accurate correlation dimension estimations for chaotic time series, a novel scaling region identification method is developed. First, points that obviously do not belong to the scaling region associated with the whole double logarithm correlation integral curve are removed using the K-means algorithm. Second, a point-slope-error algorithm is developed to recognize a possible scaling region. Third, the K-means algorithm is used again to further remove a small interval of interfering points in the possible scaling region to obtain a more precise scaling region. The correlation dimension of four typical chaotic attractors and five curves generated by the Weierstrass-Mandelbrot fractal function were calculated using the proposed method. These calculated values were very close to the respective theoretical fractal dimensions. Moreover, the effectiveness of our method in identifying the scaling region was compared with existing methods. Results show that our method can distinguish the scaling region objectively, accurately, automatically and quickly, making estimations of the correlation dimension more precise and affording significant improvements in nonlinear analysis.

Macroscopic and microscopic characterizations of a cellulosic ultrafiltration (UF) membrane fouled by a humic acid cake deposit: First step for intensification of reverse osmosis (RO) pre-treatments

Abstract One of the critical issues for the application of low-pressure membrane processes (microfiltration, MF or ultrafiltration, UF) as pre-treatment processes for freshwater preparation is membrane fouling due to natural organic matter (NOM). The aim of this preliminary study is to contribute to a better understanding of the fouling phenomena occurring on a regenerated cellulose UF membrane fouled with a humic acid cake deposit. The originality of this work is based on a double approach on surface analysis at both macroscopic and microscopic scales. It is presently reported that humic acid fouling is mainly governed by cake formation, which plays a major role in flux decline via the well-known model of resistances in series. We obtained that the adsorbed resistance is 2% of the total resistance while the cake resistance is 52% of the total resistance, which is higher than that of the virgin membrane. From field emission gun scanning electron microscopy (FESEM) it was found for the first time that the humic acid cake is well organized, and particularly in fractal forms. The fractal dimension (FD) of the cake is determined as 2.52, which is in good agreement with the theoretical fractal dimension of particle–cluster aggregation underlying diffusion-limited aggregation (FD = 2.51). This new microscopic fouling index decreases with the presence of cake and can be correlated with the decrease of the hydraulic permeability. The classical silt density index (SDI) and the new modified fouling index (denoted MFI-UF) were obtained and also proved the presence of the cake. To complete this approach transmembrane streaming potential (denoted SP) measurements were conducted with a new homemade apparatus developed in our lab and presented for the first time in the present article, helped us to observe also a penetration of low molecular fractions of humic acid inside the membrane. Indeed the displacement of the isoelectric point (iep) of the membrane from 2.3 to 1.5 for the virgin and fouled membranes, respectively, permitted to illustrate this penetration. This newly designed SP apparatus is a semi-automatic tool assisted by a software denoted as proFluid 1.2. Furthermore, preliminary experiments with seawater were realized in order to estimate the influence of seawater filtration on the hydraulic permeability and SP parameters for the RC 100-kDa membrane.

A three-dimensional fractal analysis method for quantifying white matter structure in human brain

Fractal dimension (FD) is increasingly used to quantify complexity of brain structures. Previous research that analyzed FD of human brain mainly focused on two-dimensional measurements. In this study, we developed a three-dimensional (3D) box-counting method to measure FD of human brain white matter (WM) interior structure, WM surface and WM general structure simultaneously. This method, which firstly incorporates a shape descriptor (3D skeleton) representing interior structure and combines the three features, provides a more comprehensive characterization of WM structure. WM FD of different brain segments was computed to test robustness of the method. FDs of fractal phantoms were computed to test the accuracy of the method. The consistency of the computed and theoretical FD values suggests that our method is accurate in measuring FDs of fractals. Statistical analysis was performed to examine sensitivity of the method in detecting WM structure differences in a number of young and old subjects. FD values of the WM skeleton and surface were significantly greater in young than old individuals, indicating more complex WM structures in young people. These results suggest that our method is accurate in quantifying three-dimensional brain WM structures and sensitive in detecting age-related degeneration of the structures.

Two-dimensional percolation and cluster structure of the random packing of binary disks.

In this paper we study the short-range correlated percolation and the cluster structure of two-dimensional (2D) random packing of binary disks with size ratio lambda in the range of 1-5. A Monte Carlo simulation model is used to generate the configuration of random packing first. Then a from-neighbor-to-neighbor propagation method is used to identify the number and sizes of the clusters. Results show that for lambda=1 the percolation threshold p(c) lies between the square and triangular site percolation thresholds. As lambda increases the percolation threshold p(c) (the area fraction of small disks) decreases. To characterize the cluster structure at the percolation threshold, we scale the cluster size s(c) with the cluster radius R as s(c) proportional, variant R(D). The fractal dimension D obtained lies between 1.86 and 1.88 and is independent of the size ratio lambda. This value is in good agreement with the 2D theoretical fractal dimension which is equal to 91/48.

The use of fractal dimension estimators for enhancing airborne magnetic data

Airborne magnetic data is routinely enhanced by amplitude based filters such as horizontal and vertical derivatives. Texture is defined as the spatial distribution of amplitudes over a region. Textural analysis provides a possible alternative method of data enhancement. This paper investigates the potential of using fractal dimension for quantifying texture and highlighting textural contrasts in airborne magnetic data.Profiles have been created by combining theoretical data with fractal dimensions (FD) of 1.1, 1.3 and 1.5. Estimates of FD using the semi-variogram and variation methods clearly distinguish between sections of the profiles with different theoretical FD. Fractal dimension estimates made on a real airborne magnetic profile, using the variation method, clearly define two regions of visible textural contrast. A series of other variations in the estimated FD suggests that the method is able to resolve subtle contrasts that are not easily detected visually. The semi-variogram method of FD estimation is not able to resolve the obvious textural contrasts in real data, a result that is perhaps due to the stochastic nature of this methodology.The variation method has been used to estimate FD on a series of airborne magnetic profiles. This profile data was then gridded to generate an image of FD that moderately improves structural resolution. Whilst more work needs to be carried out, it is obvious that estimates of FD do detect textural contrasts in both theoretical and real data, and that this information can be used to enhance aeromagnetic data.