Method For Calculation Of Fractal Dimension Research Articles

STUDY ON QUANTIFICATION OF ROCK FRACTURE NETWORK TO PROMOTE SHALE GAS DEVELOPMENT

Shale gas is produced by fracturing the reservoir to increase production. Quantifying the complex structure of the rock fracture network has a very important role in shale gas development. In order to quantify the complex structure of rock fracture network, in this paper, combined with the hierarchical structure of fractal figure, long tail distribution and fractal self-similarity, the current calculation method of fractal dimension is improved, and the basic fractal dimension calculation method is proposed based on the principle. By removing the limit calculation method which causes the deviation of numerical results, a reasonable and convenient calculation model for engineering application is proposed. A set of two-dimensional fracture pictures are used to test the model. In the 10 pictures, the more complex the figure is, the larger the new fractal dimension value is, which is in line with the characteristics of fractal dimension describing the complexity of graphics. Furthermore, lattice Boltzmann method (LBM) is used to simulate the velocity field, and the permeability of the fracture network is calculated according to Darcy’s law. The permeability and fractal dimension of the fracture network are compared and analyzed. The conclusion that the permeability is positively related to the fractal dimension of the fracture network is obtained, which provides theoretical support for the further study of shale mining problems such as hydraulic fracturing.

Fractal Dimension Calculation Methods and Their Use in the Textile Industry

From the methods of calculating the fractal dimension, the prism method, the cube method, the method of counting cells, instructions on the methods of calculating the fractal dimension using set theory and the block diagrams of algorithms were developed. A software tool was created based on the Hausdorff method of calculating the fractal dimension. Fractal measurement determined by this software can be used in medicine for changes in the vascular system, in the accurate diagnosis of tumors in the brain, in the textile industry, in the textural analysis of the permeability and roughness of fabrics.

Wavelet Analysis-Based Texture Analysis of Ceramic Surface Images

This paper is conducted to explore a new characterization method as a supplement to the traditional roughness characterization. The main research includes the extraction and evaluation of damage features of ceramic surface morphology by applying wavelet methods, the extraction of damage features in surface contours by using wavelet analysis, and the quantitative evaluation of damage degree by using damage rate and damage mean spacing. By comparing various fractal dimension calculation methods, a fractal dimension method suitable for calculating the ceramic surface was selected, and the fractal method was used to describe the ceramic surface topography as a whole. By comparing different methods of calculating the fractal dimension and further verifying them with the measured three-dimensional morphology, it is found that the vibrational method is more suitable for calculating the fractal dimension of ceramic surface, and its calculation accuracy is investigated, and the results show that the method is a reliable one. Based on the fractal theory, a mathematical model of surface wear and surface sealing was established. Further study of the model shows that the surface with a large fractal dimension has a good sealing effect; the surface corresponding to the best fractal dimension is the most resistant to wear. The fractal method can characterize the complexity of the surface profile as a whole. The wavelet method can describe the ceramic surface profile from a local perspective, and the combination of the two methods can characterize the ceramic surface well. Finally, the experimental device of the ceramic surface defect detection system is constructed, and the joint debugging of hardware and software is completed. Under different light source intensities, ceramic image samples are collected, and the accuracy detection experiments of sample defective edges are conducted, and the results show that the light source has a small impact on the accuracy of ceramic defective edge detection. The results show that the light source has more influence on the accuracy of scratch detection. The results show that the system constructed in this thesis has good applicability for different ceramic sample detection.

Study on Mesoscopic Damage Evolution Characteristics of Single Joint Sandstone Based on Micro-CT Image and Fractal Theory

The different directions of joints in rock will lead to great differences in damage evolution characteristics. This study utilizes DIP (digital image processing) technology for characterizing the mesostructure of sandstone and combines DIP technology with RFPA2D. The mesoscale fracture mechanics behavior of 7 groups of jointed sandstones with various dip angles was numerically studied, and its reliability was verified through theoretical analysis. According to digital image storage principle and box dimension theory, the box dimension algorithm of rock mesoscale fracture is written in MATLAB, the calculation method of fractal dimension of mesoscale fracture was proposed, and the corresponding relationship between mesoscale fractal dimension and fracture damage degree was established. Studies have shown that compressive strength as well as elastic modulus of sandstone leads to a U-shaped change when joint dip increases. There are a total of six final failure modes of joint samples with different inclination angles. Failure mode and damage degree can be quantified by D (fractal dimension) and ω (mesoscale fracture damage degree), respectively. The larger the ω, the more serious the damage, and the greater the D, the more complex the failure mode. Accumulative AE energy increases exponentially with the increase of loading step, and the growth process can be divided into gentle period, acceleration period, and surge period. The mesoscale fracture damage calculation based on the fractal dimension can be utilized for quantitatively evaluating the spatial distribution characteristics of mesoscale fracture, which provides a new way to study the law of rock damage evolution.

Techniques used to calculate shale fractal dimensions involve uncertainties and imprecisions that require more careful consideration

Surface roughness of shales has a key influence on the petroleum resources they are able to store and the fraction of them that can be recovered. The fractal dimension quantifies the degree of roughness and is influenced primarily by the pore surfaces within the shale that typically include micro-, meso-and macro-pores. Isotherms generated by gas adsorption experiments are the common data source used to derive estimates of fractal dimension. The Frenkel-Halsey-Hill fractal technique is the most widely applied fractal dimension estimation method. Other methods can derive fractal dimension from isotherm data but typically the values they generate are different from the Frenkel-Halsey-Hill derived fractal dimension values. Moreover, those differences can vary significantly depending on the type of shales involved. Those shales displaying more complex pore-scale distributions including extensive micro-porosity components tend to be associated with the greatest discrepancies. A comparison of three fractal dimension calculation methods applied to shales reveals aspects of their calculation and interpretation methods that explain the differences in the fractal dimension values they generate. This study identifies the uncertainties that should be taken into account when applying the methods and the appropriate curve fitting optimization configurations that should be evaluated. Taking these factors into account leads to more realistic selections of appropriate fractal dimension values from gas adsorption isotherms of organic-rich shales. Cited as : Wood, D.A. Techniques used to calculate shale fractal dimensions involve uncertainties and imprecisions that require more careful consideration. Advances in Geo-Energy Research, 2021, 5(2): 153-165, doi: 10.46690/ager.2021.02.05

Review about the Application of Fractal Theory in the Research of Packaging Materials.

The work is intended to summarize the recent progress in the work of fractal theory in packaging material to provide important insights into applied research on fractal in packaging materials. The fractal analysis methods employed for inorganic materials such as metal alloys and ceramics, polymers, and their composites are reviewed from the aspects of fractal feature extraction and fractal dimension calculation methods. Through the fractal dimension of packaging materials and the fractal in their preparation process, the relationship between the fractal characteristic parameters and the properties of packaging materials is discussed. The fractal analysis method can qualitatively and quantitatively characterize the fractal characteristics, microstructure, and properties of a large number of various types of packaging materials. The method of using fractal theory to probe the preparation and properties of packaging materials is universal; the relationship between the properties of packaging materials and fractal dimension will be a critical trend of fractal theory in the research on properties of packaging materials.

Study on Mesoscale Damage Evolution Characteristics of Irregular Sandstone Particles Based on Digital Images and Fractal Theory

To study the mesoscale damage evolution law of irregular sandstone particles, based on RFPA2D and digital image processing technology, a real mesostructure numerical model of irregular sandstone particles is established to simulate the breakage process of particles, the effects of loading conditions and mesoscale heterogeneity on irregular sandstone particle damage are studied, and the calculation method of fractal dimension of irregular rock particles mesoscale fracture is proposed. The results show that the fracture damage degree (ω) and fractal dimension (D) maximum values of the constrained particles are 0.733 and 1.466, respectively, and the unconstrained particles are 0.577 and 1.153, respectively. The final failure mode of constrained particles is more complicated than unconstrained particles, the damage is more serious, and the fracture is more complete. Thus, the larger values of D yield a more complicated final failure mode of the particles. Consequently, with the larger ω, the final damage is more serious, and the breakage effect is comparatively better. The study is of great significance for exploring the laws of rock particle breakage and energy consumption, rock breakage mechanism, and searching for efficient and energy‐saving rock‐breaking methods.

土壤表面粗糙度数据采集及分形特性研究

It is of great significance to acquire the soil surface roughness accurately for the study of the interaction between tractors and soil. Based on the laser sensor, this paper proposed the non-contact measuring instrument of the soil surface roughness with the data acquiring system by using Lab-View software. By using W-M theory, three commonly used fractal dimension calculation methods are compared and analyzed.. The result showed that the Root-mean-square method has the highest accuracy and clear physical meaning, which is ideal method to calculate the soil surface roughness characteristics. When the fractal dimension is between 1.4 and 1.6, the acquired data is analysed by the Root-mean-square method to obtain the fractal features of the soil surface roughness. The experiment results indicated that the fractal dimension of the ploughed surface is 1.39, that of disc harrow surface is 1.550, and that of rolled surface is 1.46-1.54. Obviously, the fractal dimension can accurately distinguish the soil surface roughness with the different treatments. However, the fractal dimension selected from different scales showed an obvious instability during calculations. The surface roughness index combined with the two parameters can effectively represent the soil surface roughness, and the larger the surface roughness index is, the greater the surface roughness is.

FRACTAL DIMENSION OF FRACTIONAL BROWNIAN MOTION BASED ON RANDOM SETS

The fractal dimension of fractional Brownian motion can effectively describe random sets, reflecting the regularity implicit in complex random sets. Data mining algorithms based on fractal theory usually follow the calculation of the fractal dimension of fractional Brownian motion. However, the existing fractal dimension calculation methods of fractal Brownian motion have high time complexity and space complexity, which greatly reduces the efficiency of the algorithm and makes it difficult for the algorithm to adapt to high-speed and massive data flow environments. Therefore, several existing fractal dimension calculation methods of fractional Brownian motion are summarized and analyzed, and a random method is proposed, which uses a fixed memory space to quickly estimate the associated dimension of the data stream. Finally, a comparison experiment with existing algorithms proves the effectiveness of this random algorithm. Second, in the sense of two different measures, based on the principle of stochastic comparison, the stability of the stochastic fuzzy differential equations is derived using the stability of the comparison equations, and the practical stability criterion of two measures according to probability is obtained. Then, the stochastic fuzzy differential equations are discussed. The definition of stochastic exponential stability is given and the stochastic exponential stability criterion is proved.

A comparative study of fractal dimension calculation methods for rough surface profiles

Fractal dimension is the most important parameter for surface characterization. In this paper, four methods used to estimate the fractal dimensions of surface profiles and their applications in machined surfaces are studied. These methods are first evaluated using surface profiles created by Weierstrass–Mandelbrot function from the three aspects of fitting accuracy, calculation accuracy and calculation stability, and then applied to the machined rough surfaces. By comparing the results of the four methods, it is found that none of the methods is particularly prominent in all of the three aspects. However, the three point sinuosity method is found to be relatively the most suitable and reliable method among the four tested methods for extracting fractal dimensions of both generated and measured rough surface profiles.

Application of Fractal Theory in Architectural Design

Firstly, this paper introduces the research background of fractal theory. Secondly, the article gives the definition of fractal dimension, and introduces the calculation method of fractal dimension through examples. At last, the paper discusses the fractal structure of several kinds of typical buildings. And then it is proved that the fractal geometry has a deep influence on the development of architecture in the aspects of epistemology and methodology.

BİST100 Endeksi Fiyat ve İşlem Hacminin Fraktallık Analizi

In this study, we examined the fractal structure of the BIST100 index returns and volume during the period of 04.01.2000-19.03.2014. In the fractality tests we used long memory analysis and the fractal dimension calculation methods. Long memory analysis was conducted via Rescaled Range (R/S) analysis, Detrended Fluctuaiton Analysis (DFA) and Smith’s (2005) modified GPH analysis; fractal dimension calculations were performed with Box-Counting, Semi-Periodogram and Variogram methods. Results showed that all findings of the different methods consistent with each other, and there is no fractality in the BIST100 index returns and volume.

Comparison of Fractal Dimension Calculation Methods for Channel Bed Profiles

Research on channel bed profiles is one of the basic topics of river dynamics and fluvial process studies, since it was closely associated with the problems such as turbulent structure, flow resistance and sediment transportation in alluvial rivers. As a result of motion water flow interacting with sediment, channel bed profiles have irregular plane structures and self-similar microstructures, which were better analyzed by fractal method to reflect the intrinsic rough characteristics. In this study, several existing fractal dimension calculation methods were compared quantitatively, along with the discussion of their non-scale zones, correlation coefficients and computational accuracies. The following methods are included in this paper: Yard Stick Method, Box Counting Method, Variation Method, Structure Function Method, Root Mean Square Method and R/S Analysis Method. Finally, it is suggested that the Structure Function Method is suitable for calculating fractal dimension of rough channel bed profiles, which can provide references for future micro-fractal studies on fluvial process, flow resistance and waterway regulations.

A Model of void distribution in collapsed zone based on fractal theory

In order to calculate the grout volume in the collapsed zone in coal mine after mining, the fractal theory is used to study the feature of the void distribution in collapsed zone, and a fractal model forecasting the voidage in collapsed zone is put forward. The fractal dimension of block particles, pile size of rock block, pore, porosity in the model and relation among them are studied. The influence of lithology of overlying rock, mining width and mining height on the void distribution in collapsed zone is revealed. Combined with the characteristics of overburden failure, a simple and applicable calculation method of fractal dimension connected with the parameters of mining technology is proposed. The validity of the model has been verified in the engineering practice of grouting filling in collapsed zone in a coal mine in China. The model provides a theoretical basis for the design of grouting filling capacity in collapsed zone.

New techniques in rock mass classification: application to welded tuffs at the Nevada Yucca Mountain

Many rock mass classification systems exist to assist the engineer in assessing the rock support requirements for underground design. On-going research in this area is directed at attempting to utilize the fractal dimension and the acoustic emission response of the tuffs at the Nevada Yucca Mountain to further aid in rock mass classification. Acoustic emission response is shown to be correlated with the porosity of the sample. Engineering behaviour of the rock varies dramatically with porosity; events and peak amplitude offer a means to distinguish between fracture porosity and pore porosity and consequently the engineering behaviour of the rock. Fractal dimension is used to characterize the roughness of fracture surfaces. Two fractal dimension calculation methods, one based on the semi-variogram for the surface and the other based on the use of dividers, are applied for this purpose. The divider method is shown to resolve deviation from a straight line; the semi-variogram method is shown to identify statistical similarity to various types of noise.