A novel box-counting method for quantitative fractal analysis of threedimensional pore characteristics in sandstone

Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.

Fractal analysis of major faults and fractal dimension of lineaments in the Indo-Gangetic Plain on a regional scale

The Indo-Gangetic Plain(IGP)is one of the most seismically vulnerable areas due to its proximity to the Himalayas.Geographic information system(GIS)-based seismic characterization of the IGP was performed based on the degree of deformation and fractal dimension.The zone between the Main Boundary Thrust(MBT)and the Main Central Thrust(MCT)in the Himalayan Mountain Range(HMR)experienced large variations in earthquake magnitude,which were identified by Number-Size(NS)fractal modeling.The central IGP zone experienced only moderate to low mainshock levels.Fractal analysis of earthquake epicenters reveals a large scattering of earthquake epicenters in the HMR and central IGP zones.Similarly,the fault fractal analysis identifies the HMR,central IGP,and south-western IGP zones as having more faults.Overall,the seismicity of the study region is strong in the central IGP,south-western IGP,and HMR zones,moderate in the western and southern IGP,and low in the northern,eastern,and south-eastern IGP zones.

Computational Quantification of Map Projection Distortion by Fractal Dimension of Coastlines

Maps, essential tools for portraying the Earth’s surface, inherently introduce distortions to geographical features. While various quantification methods exist for assessing these distortions, they often fall short when evaluating actual geographic features. In our study, we took a novel approach by analyzing map projection distortion from a geometric perspective. We computed the fractal dimensions of different stretches of coastline before and after projection using the divide-and-conquer algorithm and image processing. Our findings revealed that map projections, even when preserving basic shapes, inevitably stretch and compress coastlines in diverse directions. This analysis method provides a more realistic and practical way to measure map-induced distortions, with significant implications for cartography, geographic information systems (GIS), and geomorphology. By bridging the gap between theoretical analysis and real-world features, this method greatly enhances accuracy and practicality when evaluating map projections.

The Global Attractor and Its Dimension Estimation of Generalized Kolmogorov-Petrovlkii-Piskunov Equation

In this paper, the initial boundary value problem of a class of nonlinear generalized Kolmogorov-Petrovlkii-Piskunov equations is studied. The existence and uniqueness of the solution and the bounded absorption set are proved by the prior estimation and the Galerkin finite element method, thus the existence of the global attractor is proved and the upper bound estimate of the global attractor is obtained.

Novel Path Counting-Based Method for Fractal Dimension Estimation of the Ultra-Dense Networks

Next-generation networks,including the Internet of Things(IoT),fifth-generation cellular systems(5G),and sixth-generation cellular systems(6G),suf-fer from the dramatic increase of the number of deployed devices.This puts high constraints and challenges on the design of such networks.Structural changing of the network is one of such challenges that affect the network performance,includ-ing the required quality of service(QoS).The fractal dimension(FD)is consid-ered one of the main indicators used to represent the structure of the communication network.To this end,this work analyzes the FD of the network and its use for telecommunication networks investigation and planning.The clus-ter growing method for assessing the FD is introduced and analyzed.The article proposes a novel method for estimating the FD of a communication network,based on assessing the network’s connectivity,by searching for the shortest routes.Unlike the cluster growing method,the proposed method does not require multiple iterations,which reduces the number of calculations,and increases the stability of the results obtained.Thus,the proposed method requires less compu-tational cost than the cluster growing method and achieves higher stability.The method is quite simple to implement and can be used in the tasks of research and planning of modern and promising communication networks.The developed method is evaluated for two different network structures and compared with the cluster growing method.Results validate the developed method.

Fractal Study on the Evolution of Micro-Pores in Concrete Under Freeze-Thaw

After exposure to freeze-thaw cycles, scanning electron microscopy(SEM) and nuclear magnetic resonance(NMR) were used to test the four mixtures. The microstructure is qualitatively analyzed from the 2D SEM image and the 3D pore distribution curve before and after freezing and thawing. The fractal dimension is utilized to characterize the two-dimensional topography image and the three-dimensional pore distribution, quantitatively. The results reveal that the surface porosity and volume porosity increase as the freeze-thaw action increases. Self-similarity characteristics exist in micro-damage inside the concrete. In the fractal dimension, it is possible to characterize pore evolution quantitatively. The fractal dimension correlates with pore damage evolution. The fractal dimension effectively quantitatively characterizes micro-damage features at various scales from the local to the global level.

Damage Characteristics Analysis and Fractal Study of Shale With Prefabricated Fractures under Thermal-mechanical Coupling

To study the damage and failure of shale with different fracture inclination angles under uniaxial compression loading,in this work,RFPA2D-Thermal,a two-dimensional real failure process analysis software,was used for numerical simulation.Numerical simulation results show that quartz in shale mainly affects the tensile and compressive strength of shale by increasing rock brittleness.The coupling of temperature and pressure will cause lateral and volume destruction of shale,which enables the shale body to be more easily broken.Fracture inclination is the key factor affecting shale damage patterns.The failure mode of shale with low-and high-angle fractures is mainly shear failure,and the compressive strength does not vary with crack inclination.The damage mode of obliquely intersecting fractured shale is slip damage along the fracture face,the compressive strength decreases and then increases with the fracture inclination,and a minimum value exists.The acoustic emission simulation results of the damage process effectively reflect the accumulated internal damage and macroscopic crack appearance until fracture instability when the prefabricated fractured shale is subjected to uniaxial compressive loading.The crack inclinations of 0°and 120℃ corresponds to the most complex"N"shape damage mode.The crack inclinations of 30°and 60°,and the damage mode is an inverted"λ"shape.

Microstructure Characterization of Bubbles in Gassy Soil Based on the Fractal Theory

The microscopic characterization of isolated bubbles in gassy soil plays an important role in the macroscopic physical properties of sediments and is a key factor in the study of geological hazards in gas-bearing strata.Based on the box-counting method and the pore fractal features in porous media,a fractal model of bubble microstructure parameters in gassy soil under different gas con-tents and vertical load conditions is established by using an industrial X-ray CT scanning system.The results show that the fractal di-mension of bubbles in the sample is correlated with the volume fraction of bubbles,and it is also restricted by the vertical load.The three-dimensional fractal dimension of the sample is about 1 larger than the average two-dimensional fractal dimension of all the slices from the same sample.The uniform porous media fractal model is used to test the equivalent diameter,and the results show that the variation of the measured pore diameter ratio is jointly restricted by the volume fraction and the vertical load.In addition,the measured self-similarity interval of the bubble area distribution is tested by the porous media fractal capillary bundle model,and the fitting curve of measured pore area ratio in a small loading range is obtained in this paper.

Fractal Analysis on Methane Hydrate Formation in Porous Media

The study of the hydrate formation process in porous media is of great significance for hydrate application.In this work,the formation process of methane hydrate in porous media was monitored in situ by a low-field magnetic resonance imaging(MRI)system.The formation characteristics of methane hydrate in porous media and the change of fractal dimension of pore space were studied through the change of residual water saturation and T2 relaxation time distribution.The experimental results show that the hydrate formation process is divided into two stages:fast and slow.During the formation process,the water in the pores is continuously consumed and transformed into hydrate,and the overall T2 distribution gradually shifts to the left.In the formation process of hydrate,the pore space becomes more complex,the change of fractal dimension from top to bottom of the reactor gradually increases,and the hydrate formation rate also gradually increases.

A Technique for Estimation of Box Dimension about Fractional Integral

This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals.

GIS-based landslide susceptibility mapping using hybrid integration approaches of fractal dimension with index of entropy and support vector machine

The loess area in the northern part of Baoji City, Shaanxi Province, China is a region with frequently landslide occurrences. The main aim of this study is to quantitatively predict the extent of landslides using the index of entropy model(IOE), the support vector machine model(SVM) and two hybrid models namely the F-IOE model and the F-SVM model constructed by fractal dimension. First, a total of 179 landslides were identified and landslide inventory map was produced, with 70%(125) of the landslides which was optimized by 10-fold crossvalidation being used for training purpose and the remaining 30%(54) of landslides being used for validation purpose. Subsequently, slope angle, slope aspect, altitude, rainfall, plan curvature, distance to rivers, land use, distance to roads, distance to faults, normalized difference vegetation index(NDVI), lithology, and profile curvature were considered as landslide conditioning factors and all factor layers were resampled to a uniform resolution. Then the information gain ratio of each conditioning factors was evaluated. Next, the fractal dimension for each conditioning factors was calculated and the training dataset was used to build four landslide susceptibility models. In the end, the receiver operating characteristic(ROC) curves and three statistical indexes involving positive predictive rate(PPR), negative predictive rate(NPR) and accuracy(ACC) were applied to validate and compare the performance of these four models. The results showed that the F-SVM model had the highest PPR, NPR, ACC and AUC values for training and validation datasets, respectively, followed by the F-IOE model.Finally, it is concluded that the F-SVM model performed best in all models, the hybrid model built by fractal dimension has advantages than original model, and can provide reference for local landslide prevention and decision making.

Relevance between abutment pressure and fractal dimension of crack network induced by mining

Based on the geological conditions of coal mining face No.15-14120 at No.8 mine of Pingdingshan coal mining group,the real-time evolution of coal-roof crack network with working face advancing was collected with the help of intrinsically safe borehole video instrument.And according to the geology of this working face,a discrete element model was calculated by UDEC.Combining in situ experimental data with numerical results,the relationship between the fractal dimension of boreholes'wall and the distribution of advanced abutment pressure was studied under the condition of mining advance.The results show that the variation tendency of fractal dimension and the abutment pressure has the same characteristic value.The distance between working face and the peak value of the abutment pressure has a slight increasing trend with the advancing of mining-face.When the working face is set as the original point,the trend of fractal dimension from the far place to the origin can be divided into three phases:constant,steady increasing and constant.And the turning points of these phases are the max-influencing distance(50 m)and peak value(15 m)of abutment pressure.

Volume fractal dimension of soil particles and relationships with soil physical-chemical properties and plant species diversity in an alpine grassland under different disturbance degrees

Fractal geometry is an important method in soil science,and many studies have used fractal theory to examine soil properties and the relationships with other eco-environmental factors.However,there have been few studies examining soil particle volume fractal dimension in alpine grasslands.To study the volume fractal dimension of soil particles （D） and its relationships with soil salt,soil nutrient and plant species diversity,we conducted an experiment on an alpine grassland under different disturbance degrees：non-disturbance （N0）,light disturbance （L）,moderate disturbance （M） and heavy disturbance （H）.The results showed that （1） Ds varied from 2.573 to 2.635 among the different disturbance degrees and increased with increasing degrees of disturbance.（2） Shannon-Wiener diversity index,Pielou＇s evenness index and Margalef richness index reached their highest values at the M degree,indicating that moderate disturbance is beneficial to the increase of plant species diversity.（3） In the L and M degrees,there was a significant positive correlation between D and clay content and a significant negative correlation between D and soil organic matter （SOM）.In the H degree,D was significantly and positively correlated with total salt （TS）.The results suggested that to a certain extent,D can be used to characterize the uniformity of soil texture in addition to soil fertility characteristics.（4） For the L degree,there was a significant negative correlation between D and the Shannon-Wiener diversity index; while for the M degree,there was a significant negative correlation between D and Pielou＇s evenness index.

Fractal dimension of coal particles and their CH_4 adsorption

We describe the fractal analysis of three differently sized coal samples(0.350-0.833 mm,0.245-0.350 mm,and 0.198-0.245 mm).The influence of fractal dimension on CH 4 adsorption capacity is investigated.The physical parameters of the samples were determined via the Brunauer-Emmett-Teller(BET) theory.A CH 4 adsorption study over the pressures range from 0 to 5 MPa was carried out with a new volumetric measurement system.The CH 4 adsorption was measured using the differently sized coal.Two fractal dimensions,D 1 and D 2 were determined over the pressure ranges from 0 to 0.5 MPa and from 0.5 to 1 MPa,using the Frenkel-Halsey-Hill(FHH) method.We conclude that the two fractal dimensions correlate with the CH 4 adsorption capacity of the coal:increasing CH 4 adsorption capacity occurs with a corresponding increase in fractal dimension.Furthermore,D 1 and D 2 are positively correlated with surface area,pore volume,and samples size.The size distribution of the samples has fractal characteristics.

CONNECTION BETWEEN THE ORDER OF FRACTIONAL CALCULUS AND FRACTIONAL DIMENSIONS OF A TYPE OF FRACTAL FUNCTIONS

The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.

Fractal Characteristics and Fractal Dimension Measurement on Broken Surfaces of Aluminum Electric Porcelain

The characteristics of broken surfaces were r esearched by a scanning electron microscope (SEM) and a reflection microscope, a nd the fractal dimensions of broken surfaces were measured by the Slit Island me thod. The experimental results indicate that the broken surface of aluminum elec tric porcelain is a fractal body in statistics, and the fractal dimensions of br oken surfaces are different with the different amplification multiple value.In a ll of measured fractal dimensions,both of values measured in 100× under reflect ion microscope and in 500× under SEM are maximum, whereas the values measur ed in 63× under reflection microscope and in 2000× under SEM are obviously min imum. The fractal dimensions of broken surfaces are also affected by the degrees of gray comparison and the kinds of measuring methods. The relationships betwee n the fractal dimensions of broken surfaces and porcelain bend strengths are tha t they are in positive correlation on the low multiples and in negative correlat ion on the high multiples.

Characterizing the Spheroidization Grade and Strength of 15CrMo Steel through Determining Fractal Dimension

The fractal dimension（FD） of surfaces has been widely used to characterize the properties of materials.However,most of the previous researches were concentrated on the correlation between the FD of surfaces and mechanical properties of materials,such as impact energy and fracture toughness,etc.The aim of this paper is to characterize the spheroidization grade and strength of 15CrMo steel through determination of FD of cementite phase on the basis of two-dimension microstructural image.Two methods,namely slit-island method（SIM） and box-counting method（BCM）,are used to determine the value of FD.It is found that the FD value evaluated by using BCM is generally higher than that evaluated by SIM.This phenomenon may be due to the difference in the principles used in different methods.Whether SIM or BCM is used,the spheroidization grade in 15CrMo steel linearly increases with decreasing the value of FD.The relationship between the FD value,D,and spheroidization grade,Sg,can be approximately expressed as D≌-0.11Sg＋A,where A is a constant value which is depended on the evaluation method.Both the ultimate strength and the yielding strength of 15CrMo steel increase with increasing FD of cementite phase.There may be a common relationship between the FD of cementite phase and strength of 15CrMo steel.When the FD of cementite phase in 15CrMo steel is determined,the strength of this steel can be evaluated.The present paper can provide a novel method to evaluate the strength and spheroidization grade of carbon steel through determination of fractal dimension（FD） of cementite phase.

A Fractal Dimension Based Framework for Night Vision Fusion

In this paper, a novel fusion framework is proposed for night-vision applications such as pedestrian recognition,vehicle navigation and surveillance. The underlying concept is to combine low-light visible and infrared imagery into a single output to enhance visual perception. The proposed framework is computationally simple since it is only realized in the spatial domain. The core idea is to obtain an initial fused image by averaging all the source images. The initial fused image is then enhanced by selecting the most salient features guided from the root mean square error(RMSE) and fractal dimension of the visual and infrared images to obtain the final fused image.Extensive experiments on different scene imaginary demonstrate that it is consistently superior to the conventional image fusion methods in terms of visual and quantitative evaluations.

A Research on Fractal Dimension and Nonlinear Dynamic Model of Xintan Landslide，China

The dynamic research of landslide is one of the key Points in landslidology. In the paper,from the view point of nonlinear dynamic theory. some features of Xintan landslide, such as the distribution regularities of spatial and temporal fractal dimensions and their corresponding relationships to landslide occurring are researched. The accumulative principles of fractal dimension reduction is exploratively pointed out. The nonlinear dynamic equation of the landslide is built by analyzing the relationship between the correlation dimension and the phase space. Finally, the forecasted results and error analysis are listed. The research results are satisfactory.

Fractal Dimension of Voice-Signal Waveforms

The fractal dimension is one important parameter that characterizes waveforms. In this paper, we derive a new method to calculate fractal dimension of digital voice-signal waveforms. We show that fractal dimension is an efficient tool for speaker recognition or speech recognition. It can be used to identify different speakers or distinguish speech. We apply our results to Chinese speaker recognition and numerical experiment shows that fractal dimension is an efficient parameter to characterize individual Chinese speakers. We have developed a semiautomatic voiceprint analysis system based on the theory of this paper and former researches.