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.

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.

A semi-analytical model for coupled flow in stress-sensitive multi-scale shale reservoirs with fractal characteristics

A large number of nanopores and complex fracture structures in shale reservoirs results in multi-scale flow of oil. With the development of shale oil reservoirs, the permeability of multi-scale media undergoes changes due to stress sensitivity, which plays a crucial role in controlling pressure propagation and oil flow. This paper proposes a multi-scale coupled flow mathematical model of matrix nanopores, induced fractures, and hydraulic fractures. In this model, the micro-scale effects of shale oil flow in fractal nanopores, fractal induced fracture network, and stress sensitivity of multi-scale media are considered. We solved the model iteratively using Pedrosa transform, semi-analytic Segmented Bessel function, Laplace transform. The results of this model exhibit good agreement with the numerical solution and field production data, confirming the high accuracy of the model. As well, the influence of stress sensitivity on permeability, pressure and production is analyzed. It is shown that the permeability and production decrease significantly when induced fractures are weakly supported. Closed induced fractures can inhibit interporosity flow in the stimulated reservoir volume (SRV). It has been shown in sensitivity analysis that hydraulic fractures are beneficial to early production, and induced fractures in SRV are beneficial to middle production. The model can characterize multi-scale flow characteristics of shale oil, providing theoretical guidance for rapid productivity evaluation.

Experimental realization of fractal fretwork metasurface for sound anomalous modulation

Natural creatures and ancient cultures are full of potential sources to provide inspiration for applied sciences.Inspired by the fractal geometry in nature and the fretwork frame in ancient culture,here we design the acoustic metasurface to realize sound anomalous modulation,which manifests itself as an incident-dependent propagation behavior:sound wave propagating in the forward direction is allowed to transmit with high efficiency while in the backward direction is obviously suppressed.We quantitatively investigate the dependences of asymmetric transmission on the propagation direction,incident angle and operating frequency by calculating sound transmittance and energy contrast.This compact fractal fretwork metasurface for acoustic anomalous modulation would promote the development of integrated acoustic devices and expand versatile applications in acoustic communication and information encryption.

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.

Experimental investigation on coal pore-fracture variation and fractal characteristics synergistically affected by solvents for improving clean gas extraction

Chemical solvents instead of pure water being as hydraulic fracturing fluid could effectively increase permeability and improve clean methane extraction efficiency.However,pore-fracture variation features of lean coal synergistically affected by solvents have not been fully understood.Ultrasonic testing,nuclear magnetic resonance analysis,liquid phase mass spectrometry was adopted to comprehensively analyze pore-fracture change characteristics of lean coal treated by combined solvent(NMP and CS_(2)).Meanwhile,quantitative characterization of above changing properties was conducted using geometric fractal theory.Relationship model between permeability,fractal dimension and porosity were established.Results indicate that the end face fractures of coal are well developed after CS2and combined solvent treatments,of which,end face box-counting fractal dimensions range from 1.1227 to 1.4767.Maximum decreases in ultrasonic longitudinal wave velocity of coal affected by NMP,CS_(2)and combined solvent are 2.700%,20.521%,22.454%,respectively.Solvent treatments could lead to increasing amount of both mesopores and macropores.Decrease ratio of fractal dimension Dsis 0.259%–2.159%,while permeability increases ratio of NMR ranges from 0.1904 to 6.4486.Meanwhile,combined solvent could dissolve coal polar and non-polar small molecules and expand flow space.Results could provide reference for solvent selection and parameter optimization of permeability-enhancement technology.

A Compact UHF Antenna Based on Hilbert Fractal Elements and a Serpentine Arrangement for Detecting Partial Discharg

Efforts to protect electric power systems from faults have commonly relied on the use of ultra-high frequency(UHF)antennas for detecting partial discharge(PD)as a common precursor to faults.However,the effectiveness of existing UHF antennas suffers from a number of challenges such as limited bandwidth,relatively large physical size,and low detection sensitivity.The present study addresses these issues by proposing a compact microstrip patch antenna with fixed dimensions of 100 mm×100 mm×1.6 mm.The results of computations yield an optimized antenna design consisting of 2nd-order Hilbert fractal units positioned within a four-layer serpentine arrangement with a fractal unit connection distance of 3.0 mm.Specifically,the optimized antenna design achieves a detection bandwidth for which the voltage standing wave ratio is less than 2 that is approximately 97.3%of the UHF frequency range(0.3–3 GHz).Finally,a prototype antenna is fabricated using standard printed circuit board technology,and the results of experiments demonstrate that the proposed antenna is capable of detecting PD signals at a distance of 8 m from the discharge source.

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.

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.

一种基于Fractal-DenseNet的电磁血管断层图像重建算法

针对电磁血管断层图像重建中存在的欠定性、病态性,提出一种基于Fractal-DenseNet的电磁血管断层图像重建算法.基于血流磁电耦合效应的血管断层图像重建原理,将FractalNet的分形思想与DenseNet的密集紧密连接思想相结合,构建了一种适用于反演血液流速分布的Fractal-DenseNet网络模型,用于血管断层图像重建.通过人体前臂尺动脉血流磁电耦合测量模型,建立血管断层剖面流速和血流磁电效应引起的电压信号的对应数据对,分别作为网络模型输入和输出;通过监督学习,实现基于血管断层流速分布的血管断层图像重建.结果表明:Fractal-DenseNet重建结果的均方根误差和相关系数分别为0.0078,99.28%;本文算法具有良好的抗噪性,可在复杂边界条件下实现血管断层图像重建.

Multi-scale pore fractal characteristics of differently ranked coal and its impact on gas adsorption

Well-developed pores and cracks in coal reservoirs are the main venues for gas storage and migration.To investigate the multi-scale pore fractal characteristics,six coal samples of different rankings were studied using high-pressure mercury injection(HPMI),low-pressure nitrogen adsorption(LPGA-N2),and scanning electron microscopy(SEM)test methods.Based on the Frankel,Halsey and Hill(FHH)fractal theory,the Menger sponge model,Pores and Cracks Analysis System(PCAS),pore volume complexity(D_(v)),coal surface irregularity(Ds)and pore distribution heterogeneity(D_(p))were studied and evaluated,respectively.The effect of three fractal dimensions on the gas adsorption ability was also analyzed with high-pressure isothermal gas adsorption experiments.Results show that pore structures within these coal samples have obvious fractal characteristics.A noticeable segmentation effect appears in the Dv1and Dv2fitting process,with the boundary size ranging from 36.00 to 182.95 nm,which helps differentiate diffusion pores and seepage fractures.The D values show an asymmetric U-shaped trend as the coal metamorphism increases,demonstrating that coalification greatly affects the pore fractal dimensions.The three fractal dimensions can characterize the difference in coal microstructure and reflect their influence on gas adsorption ability.Langmuir volume(V_(L))has an evident and positive correlation with Dsvalues,whereas Langmuir pressure(P_(L))is mainly affected by the combined action of Dvand Dp.This study will provide valuable knowledge for the appraisal of coal seam gas reservoirs of differently ranked coals.

Effect of shear-induced contact area and aperture variations on nonlinear flow behaviors in fractal rock fractures

This study experimentally analyzes the nonlinear flow characteristics and channelization of fluid through rough-walled fractures during the shear process using a shear-flow-visualization apparatus.A series of fluid flow and visualization tests is performed on four transparent fracture specimens with various shear displacements of 1 mm,3 mm,5 mm,7 mm and 10 mm under a normal stress of 0.5 MPa.Four granite fractures with different roughnesses are selected and quantified using variogram fractal dimensions.The obtained results show that the critical Reynolds number tends to increase with increasing shear displacement but decrease with increasing roughness of fracture surface.The flow paths are more tortuous at the beginning of shear because of the wide distribution of small contact spots.As the shear displacement continues to increase,preferential flow paths are more distinctly observed due to the decrease in the number of contact spots caused by shear dilation;yet the area of single contacts in-creases.Based on the experimental results,an empirical mathematical equation is proposed to quantify the critical Reynolds number using the contact area ratio and fractal dimension.

Gapless Spin Liquid and Nonlocal Corner Excitation in the Spin-1/2 Heisenberg Antiferromagnet on Fractal

Motivated by the mathematical beauty and the recent experimental realizations of fractal systems,we study the spin-1/2 antiferromagnetic Heisenberg model on a Sierpiński gasket.The fractal porous feature generates new kinds of frustration to exhibit exotic quantum states.Using advanced tensor network techniques,we identify a quantum gapless-spin-liquid ground state in fractional spatial dimension.This fractal spin system also demonstrates nontrivial nonlocal properties.While the extremely short-range correlation causes a highly degenerate spin form factor,the entanglement in this fractal system suggests a long-range scaling behavior.We also study the dynamic structure factor and clearly identify the gapless excitation with a stable corner excitation emerged from the ground-state entanglement.Our results unambiguously point out multiple essential properties of this fractal spin system,and open a new route to explore spin liquid and frustrated magnetism.

Cushioning Performance of Hilbert Fractal Sandwich Packaging Structures under Quasi-Static Compressions

The sandwich structure of cushioning packaging has an important influence on the cushioning performance.Mathematical fractal theory is an important graphic expression.Based on Hilbert fractal theory,a new sandwich structure was designed.The generation mechanism and recurrence formula of theHilbert fractal were expressed by Lin’s language,and the second-orderHilbert sandwich structure was constructed fromthermoplastic polyurethane.The constitutive model of the hyperelastic body was established by using the finite element method.With the unit mass energy absorption as the optimization goal,the fractal sandwich structure was optimized,and the best result was obtained when the order was 2.5 and the unit layer thickness was 0.75 mm.TheHilbert sandwich structure was compared with the rice-shaped sandwich structure commonly used in industry,and the Hilbert fractal structure had better energy absorption.This has practical significance for the development and application of newcushioning packaging structures.

Fractal nature of human gastrointestinal system:Exploring a new era

The morphological complexity of cells and tissues,whether normal or pathological,is characterized by two primary attributes:Irregularity and self-similarity across different scales.When an object exhibits self-similarity,its shape remains unchanged as the scales of measurement vary because any part of it resembles the whole.On the other hand,the size and geometric characteristics of an irregular object vary as the resolution increases,revealing more intricate details.Despite numerous attempts,a reliable and accurate method for quantifying the morphological features of gastrointestinal organs,tissues,cells,their dynamic changes,and pathological disorders has not yet been established.However,fractal geometry,which studies shapes and patterns that exhibit self-similarity,holds promise in providing a quantitative measure of the irregularly shaped morphologies and their underlying self-similar temporal behaviors.In this context,we explore the fractal nature of the gastrointestinal system and the potential of fractal geometry as a robust descriptor of its complex forms and functions.Additionally,we examine the practical applications of fractal geometry in clinical gastroenterology and hepatology practice.

Activation Functions Effect on Fractal Coding Using Neural Networks

Activation functions play an essential role in converting the output of the artificial neural network into nonlinear results,since without this nonlinearity,the results of the network will be less accurate.Nonlinearity is the mission of all nonlinear functions,except for polynomials.The activation function must be dif-ferentiable for backpropagation learning.This study’s objective is to determine the best activation functions for the approximation of each fractal image.Different results have been attained using Matlab and Visual Basic programs,which indi-cate that the bounded function is more helpful than other functions.The non-lin-earity of the activation function is important when using neural networks for coding fractal images because the coefficients of the Iterated Function System are different according to the different types of fractals.The most commonly cho-sen activation function is the sigmoidal function,which produces a positive value.Other functions,such as tansh or arctan,whose values can be positive or negative depending on the network input,tend to train neural networks faster.The coding speed of the fractal image is different depending on the appropriate activation function chosen for each fractal shape.In this paper,we have provided the appro-priate activation functions for each type of system of iterated functions that help the network to identify the transactions of the system.

Influence of Acid Treatment on Pore Structure and Fractal Characterization of a Tight Sandstone:A Case Study from Wudun Sag,Dunhuang Basin

In this study,X-ray diffraction,N_(2)adsorption(N_(2)A),and mercury intrusion(MI)experiments were used to investigate the influence of acid treatment on pore structure and fractal characterization of tight sandstones.The results showed that acid treatment generated a certain number of ink-bottle pores in fine sandstone,aggravated the ink-bottle effect in the sandy mudstone,and transformed some smaller pores into larger ones.After the acid treatment,both the pore volume in the range of 2–11 nm and 0.271–8μm for the fine sandstone and the entire pore size range for the sandy mudstone significantly increased.The dissolution of sandstone cement causes the fine sandstone particles to fall off and fill the pores;the porosity increased at first but then decreased with acid treatment time.The fractal dimension obtained using the Frenkel-Halsey-Hill model was positively correlated with acid treatment time.However,the total fractal dimensions obtained by MI tests showed different changes with acid treatment time in fine sandstone and sandy mudstone.These results provide good guiding significance for reservoir acidification stimulation.

Research on Pore Structure and Fractal Characteristics of Tight Sandstone Reservoir Based on High-Pressure Mercury Injection Method

Pore structure is the key element of tight sandstone reservoir, which restricts the accumulation and flow of oil and gas in the reservoir. At present, reservoir pore structure is the focus and difficulty of unconventional oil and gas exploration and development research. The tight sandstone reservoir in the Chang 4 + 5 member of the Upper Triassic Yanchang Formation is the main reservoir for oil and gas exploration in G area. At present, there is little research on its pore structure and fractal characteristics, which to some extent affects the progress of exploration and development. This paper selects the tight core samples of the Chang 4 + 5 member in the southern edge of the Ordos Basin, and based on the high-pressure mercury intrusion experiment, uses fractal theory to study the pore structure and fractal characteristics of the reservoir in the study area, thus providing theoretical basis for the evaluation and exploration and development of the Chang 4 + 5 tight reservoir in the G area. The research results show that the lithology of the Chang 4 + 5 tight sandstone reservoir in the southern edge of the Ordos Basin is mainly feldspathic sandstone, with the highest feldspar content, followed by quartz, and the clay mineral is mainly chlorite. The reservoir has poor physical properties and strong heterogeneity. There are three main fractal characteristics in Chang 4 + 5 reservoir in G area: the fractal curve of Type I reservoir sample is in two segments, the relatively large pore has certain fractal characteristics, the pore structure is relatively regular, and the heterogeneity is weak;Relatively small pores have no fractal characteristics and pore structure is irregular. The fractal curve of Type II reservoir samples shows a three-segment pattern, and each pore size range has certain fractal characteristics, and it gradually gets better with the increase of pore size. The fractal curve of Type III reservoir samples presents a similar one-segment pattern, and the fractal dimension exceeds the upper limit of 3. It is considered that the full pore size of this type of reservoir does not have fractal characteristics, the pore throat is completely irregular or the surface is rough, and the heterogeneity is very strong.