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This study presents 2D-PECLSM, a novel hyperchaotic map that integrates the 2D Logistic and the Sine map to enhance chaotic behavior. We propose a new image encryption algorithm (IEA) that leverages this map alongside fractal theory and global cross-coupled diffusion techniques. The 2D-PECLSM generates two essential chaotic sequences for secure encryption and based on this map associated with fractal theory processes. Each pixel of the plain image is effectively confused using a fractal matrix and globally diffused, ensuring robust protection of visual data. Experimental results demonstrate that the 2D-PECLSM-based IEA significantly outperforms the existing state-of-the-art IEAs, highlighting its potential as a powerful tool for enhancing image security against digital threats. This work contributes valuable insights to the field of cryptography, emphasizing the need for innovative approaches to safeguard sensitive visual information.

This work proposes and studies the concept of Functional Data Analysis transform, applying it to the performance improving of volumetric Bouligand–Minkowski fractal descriptors. The proposed transform consists essentially in changing the descriptors originally defined in the space of the calculus of fractal dimension into the space of coefficients used in the functional data representation of these descriptors. The transformed descriptors are used here in texture classification problems. The enhancement provided by the FDA transform is measured by comparing the transformed to the original descriptors in terms of the correctness rate in the classification of well known datasets.

A molecular dynamics simulation (MDS) was performed to investigate the characteristics of the interfacial feature between coexisting phases in equilibrium. It is considered that the interface is a fractal surface. The fractal configuration and the dimension of it were presented in the paper.

The industry is the key to economic development. This paper proposed a fractal evolutionary dynamic model of industry based on the propagation dynamics theory from the perspective of fractal theory. The model quantitatively portrays the evolutionary trend of industries and provides reasonable solutions to the problems faced by industrial structure optimization. First, the fractal characteristics of the industrial system are analyzed, and the proposed model is used to study the evolution of industry and enterprise impact values at different granularities. Subsequently, the impact of micro firms on the macro industrial development was further investigated. Finally, the correctness of the model simulation results was verified by real data. The results showed that the evolution of industries under different granularities shows self-similarity and that micro industries with small and medium-sized enterprises (SMEs) were more influential in the development of industries. Finally, the proposed fractal evolutionary dynamics model of the industry has proven to be reasonable and effective through empirical comparison. This study provided new insights for solving industrial development problems with high theoreticality and reliability.

Nowadays many techniques are being used to increase the reliability of human identification systems. Iris is a part of human body that is desirable for biometric identification and has favorable factors. We have focused on the reality that iris is a fractal phenomenon in this paper. During the production of new fractals, some features will be extracted by Chaos Game mechanism. These features are useful and effective in iris identification. There are three steps for iris identification with fractal and Chaos Game Theory. The first step is making a new fractal. The second step includes extracting features during the first step. Finally, the iris identification based on extracted features is the third step. We have named this technique Iris Identification based-Fractal and Chaos Game Theory (Iris-IFCGT). This technique has some fractal properties like stability against zoom, removing part of the iris image, no sensitivity on rotation and so on as well as desirable speed which helps preventing time consuming process of pattern recognition.

The neutral particle motion in a composite field (gravitoelectromagnetic field overlapping a constant external gravitomagnetic field) is analyzed. A detailed nonlinear dynamics description, using temporal series, Poincaré sections, phase space, Lyapunov exponents, bifurcation diagrams and fractal analysis, is performed. New phenomena, e.g. the gravitational gun-type, gravitational chaotic gun-type and gravitational multigun-type effects are discovered, if the amplitude of the gravitoelectromagnetic field is sufficiently large. These effects induce a high acceleration of the neutral particles which lead to sudden jumps between different Larmor-type orbits. It results both in a chaotic behavior and patterns formation in gravitational systems.

To improve the robustness and imperceptibility of the existing coverless image information hiding, a generative coverless image information hiding algorithm based on fractal theory is proposed in this paper. Firstly, four fractal image generation methods are analyzed, and the relationship between the coverless information hiding and these methods is discussed. Secondly, based on the fractal image generation algorithm, secret information is hidden by controlling pixel rendering during the generation process. The robustness, imperceptibility, and capability of resisting steganalysis are balanced by adjusting the rendering distance. As it directly generates stego images, this can resist the detection of most existing steganalysis methods. Meanwhile, different capacities can be achieved by adjusting the size of the generated image. Experimental results and analysis show that the proposed scheme can effectively resist steganalysis and has good robustness against various image attacks. Furthermore, it can achieve large capacity, and it has broad prospects for covert communication.

Multiphase flow in porous media is very important in various scientific and engineering fields. It has been shown that relative permeability plays an important role in determination of flow characteristics for multiphase flow. The accurate prediction of multiphase flow in porous media is hence highly important. In this work, a novel predictive model for relative permeability in porous media is developed based on the fractal theory. The predictions of two-phase relative permeability by the current mathematical models have been validated by comparing with available experimental data. The predictions by the proposed model show the same variation trend with the available experimental data and are in good agreement with the existing experiments. Every parameter in the proposed model has clear physical meaning. The proposed relative permeability is expressed as a function of the immobile liquid film thickness, pore structural parameters (pore fractal dimension D_f and tortuosity fractal dimension D_T) and fluid viscosity ratio. The effects of these parameters on relative permeability of porous media are discussed in detail.

Nanopore structure and its multiscale feature significantly affect the shale-gas permeability. This paper employs fractal theory to build a shale-gas permeability model, particularly considering the effects of multiscale flow within a multiscale pore space. Contrary to previous studies which assume a bundle of capillary tubes with equal size, in this research, this model reflects various flow regimes that occur in multiscale pores and takes the measured pore-size distribution into account. The flow regime within different scales is individually determined by the Knudsen number. The gas permeability is an integral value of individual permeabilities contributed from pores of different scales. Through comparing the results of five shale samples, it is confirmed that the gas permeability varies with the pore-size distribution of the samples, even though their intrinsic permeabilities are the same. Due to consideration of multiscale flow, the change of gas permeability with pore pressure becomes more complex. Consequently, it is necessary to cover the effects of multiscale flow while determining shale-gas permeability.

Prediction of fractional flow in fractal porous medium is important for reservoir engineering and chemical engineering as well as hydrology. A physical conceptual fractional flow model of transient two-phase flow is developed in fractal porous medium based on the fractal characteristics of pore-size distribution and on the approximation that porous medium consist of a bundle of tortuous capillaries. The analytical expression for fractional flow for wetting phase is presented, and the proposed expression is the function of structural parameters (such as tortuosity fractal dimension, pore fractal dimension, maximum and minimum diameters of capillaries) and fluid properties (such as contact angle, viscosity and interfacial tension) in fractal porous medium. The sensitive parameters that influence fractional flow and its derivative are formulated, and their impacts on fractional flow are discussed.

X-ray computed tomography (CT) scanning method is the most accurate method to construct digital core, which can reflect the microscopic pore structure of real cores; therefore, it is widely used and researched by experts and scholars all over the world. However, there are few reports about how to select the CT scan core samples at present, and the current practice is to make CT scan samples by visually observing rocks or core columns to select a region that is considered representative or interesting, which can lead to a large difference between the selected sample and the whole rock and a digital core that cannot represent the real rock as a whole. In order to construct the digital cores that can reflect the whole rock structure and reservoir properties, combining with fractal theory, a scientific and reasonable method was proposed to select representative rock samples for digital core modeling. First of all, a core column is scanned by X-ray CT at a certain resolution and CT gray scale images are obtained and stored in the order of scan. Secondly, the fractal dimension (FD) of each image is calculated by box-counting method, and the calculated porosity of each image is achieved by the existing formula. Then, according to the size of the digital core to be constructed, the CT gray scale images are grouped, and the average FD and the average porosity of each combination are calculated by the derived equations. Finally, based on the proposed criteria the best image combination is selected and the preferred sample is determined accordingly. At the same time, a facile experiment was conducted to test the effectiveness of this method. The experimental results show that there are some errors between the subjectively selected cores and the long core in terms of permeability and porosity, and the petrophysical parameters of the core selected by the proposed method are close to those of the long core; as a consequence, the validity of this method was verified and it is feasible and practical to select the representative rock samples for digital core modeling by this method.

The investigation of gas transport in microfractures of tight/shale reservoirs can provide potential applications in predicting shale gas production rates. In this paper, analytical expressions for flow rate and apparent permeability are derived based on the fractal theory and the superposition of convection and molecular diffusion transfer. The proposed model relates the flow rate and apparent permeability to the microstructural parameters of tight/shale reservoirs, gas properties, the ambient pressure as well as temperature. The model predictions from the present model are compared with existing experimental data sets and are found to be consistent with existing experimental measurements. The effects of microstructural parameters of tight/shale reservoirs on apparent permeability are also investigated. The results show that apparent permeability increases with temperature, the pore area fractal dimension, the porosity as well as the maximum microfracture width and decreases with the tortuosity fractal dimension and the mean pressure.

A fractal permeability model coupling non-flowing boundary-layer effect for tight oil reservoirs was proposed. Firstly, pore structures of tight formations were characterized with fractal theory. Then, with the empirical equation of boundary-layer thickness, Hagen–Poiseuille equation and fractal theory, a fractal torturous capillary tube model coupled with boundary-layer effect was developed, and verified with experimental data. Finally, the parameters influencing effective liquid permeability were quantitatively investigated. The research results show that effective liquid permeability of tight formations is not only decided by pore structures, but also affected by boundary-layer distributions, and effective liquid permeability is the function of fluid type, fluid viscosity, pressure gradient, fractal dimension, tortuosity fractal dimension, minimum pore radius and maximum pore radius. For the tight formations dominated with nanoscale pores, boundary-layer effect can significantly reduce effective liquid permeability, especially under low pressure gradient.

Based on the basic principle of the porosity method in image segmentation, considering the relationship between the porosity of the rocks and the fractal characteristics of the pore structures, a new improved image segmentation method was proposed, which uses the calculated porosity of the core images as a constraint to obtain the best threshold. The results of comparative analysis show that the porosity method can best segment images theoretically, but the actual segmentation effect is deviated from the real situation. Due to the existence of heterogeneity and isolated pores of cores, the porosity method that takes the experimental porosity of the whole core as the criterion cannot achieve the desired segmentation effect. On the contrary, the new improved method overcomes the shortcomings of the porosity method, and makes a more reasonable binary segmentation for the core grayscale images, which segments images based on the actual porosity of each image by calculated. Moreover, the image segmentation method based on the calculated porosity rather than the measured porosity also greatly saves manpower and material resources, especially for tight rocks.

For the purpose of transmitting data in smart grids, broadband communication over power lines (PLC) is considered to be one of the feasible technologies. Due to the multi-path effect of the signal propagation in complicate electrical networks, PLC signals present some dynamic features in time and space domains. The mono-fractal and multi-fractal theories are introduced to understand such dynamic features in PLC signals. Four common methods, namely, re-scaled range analysis, variance–time plot method, periodic diagram analysis and wavelet-based method are used to study the nonlinear properties and self-similarity. Fractal analysis at different frequencies and times is also performed to verify further. The paper also tests multi-fractal properties of PLC signals by the means of multi-fractal detrended fluctuation analysis (MFDFA). The multi-fractal spectrum of power low exponents is estimated from the measured PLC signals. We also proposed a new algorithm to improve the performance of the traditional MFDFA, where wavelet theory is integrated. By simulations, the better performance of the proposed method is verified.

Duo to different transport mechanisms and gas storage in organic and inorganic systems, a new triple-continuum model coupling Discrete Fracture Model (DFM) was established to investigate gas flow in shale gas reservoir. Considering the multi-scale and heterogeneity of shale matrix, fractal theory was used to calculate the apparent permeability of organic and inorganic systems while multiple gas transport mechanisms such as viscous flow, Knudsen diffusion, surface diffusion, gas absorption/desorption effect and real gas effect were incorporated. This coupled mathematical model was solved by Finite Element Method (FEM) and the presented fractal apparent permeability model was validated with the experimental data. The results show that fractal characteristics of shale matrix have great impact on gas reservoir performance. The model without considering the influence of fractal characteristics could lead to underestimate gas production by approximately 17%. Viscous flow is the dominate transport mechanisms of shale gas and Knudsen diffusion has an impact on gas flow when the pressure declines. Surface diffusion should be only considered in organic systems and can be ignored. Then the results of sensitivity analysis show that the characteristic parameters of inorganic matter have a greater impact than those of organic matter and establishing a triple-continuum model with considering comprehensive effect of organic and inorganic matter is necessary. In addition, gas production would decrease as the pore fractal dimension and tortuosity fractal dimension increase, which results from the increasing number of small pores and more tortuous path for gas flow.

Foam fluid has found wide applications in oilfield development, such as profile control, water plugging, gas channeling control, fracturing, and so on. As a non-Newtonian fluid, the successful application of foam is significantly influenced by its structure. The foam texture, however, is complex and irregular, and becomes even more complicated in porous media by the boundary effects. Therefore, the description of dynamic foam structure is crucial and a quantitative description method for foam fluid is worth exploring. In this paper, the fractal characteristics of foam in porous media are verified and combined with foam microdisplacement experiment, and the fractal rule of foam is found. The relationship between fractal dimension and pressure is also discussed. The results show that foam has dynamic fractal characteristics during transport in porous media and the box-counting fractal dimension ranges from 1 to 2. Furthermore, the dynamic change of foam fractal dimension during transport in porous media could be divided into three stages. In the first stage when no foam forms, the fractal dimension is about 2; in the second unsteady foam stage, the fractal dimension is reduced from 1.9 to 1.6; the last one is the steady stage and the fractal dimension is almost constant (about 1.6). Besides, the fractal dimension of foam fluid is closely related to displacement pressure. Low pressure corresponds to higher fractal dimension, and high pressure corresponds to lower fractal dimension. Pressure is negatively linearly correlated with fractal dimension. These results are expected to enrich the understanding of the foam dynamic characteristics in their advanced applications.

The fracture has great impact on the flow behavior in fractured reservoirs. Fracture traces are usually self-similar and scale-independent, which makes the fractal theory become a powerful tool to characterize fracture. To obtain three-dimensional (3D) digital rocks reflecting the properties of fractured reservoirs, we first generate discrete fracture networks by stochastic modeling based on the fractal theory. These fracture networks are then added to the existing digital rocks of rock matrixes. We combine two low-permeable cores as rock matrixes with a group of discrete fracture networks with fractal characteristics. Various types of fractured digital rocks are obtained by adjusting different fracture parameters. Pore network models are extracted from the 3D fractured digital rock. Then the permeability is predicted by Darcy law to investigate the impacts of fracture properties to the absolute permeability. The permeability of fractured rock is subject to exponential increases with fracture aperture. The relationship between the permeability and the fractal dimension of fracture centers is exponential, as well as the relationship between permeability and the fractal dimension of fracture lengths.

Rubner 1880 surface law reveals that the basal metabolic rate scales with body mass raised to the power of 2/3, which is geometrically correct and biologically relevant. However, Kleiber 1932 scaling law experimentally found that the scaling index was 3/4 instead of 2/3. There is no theory that can explain the Kleiber's data, explanations in Science in 1997 and later in Nature in 2002 for 3/4 scaling law for all life were apparently wrong. Here we show that Rubner's surface law was approximately correct, and it requires modification due to the fact that a cell is porous. Using fractal theory, the scaling index is about 0.7, 0.73, and 0.83, respectively, for inactive, active and motion statuses, and Kleiber's exponent can be fully explained by Rubner's law.

The prediction of permeability in rough fracture under stress condition presents ever more of a challenge in various scientific and engineering fields. However, up to now, the essential controls on stress-dependent permeability of rough fracture are not determined. In order to find a relationship between the microstructure and the permeability of rough fracture, an analytical method for the permeability of roughened fracture under stress condition is proposed based on the fractal model. The validity of the proposed model is obtained by the good agreement between the simulated results and the experimental data. Compared with the previous models, our model takes into account more factors, including the influence of the microstructural parameters of rough fracture and rock lithology. This paper presents that (1) the rock with soft lithology can yield smaller normalized permeability, (2) normalized permeability decreases with the increases of percent of smaller rough elements. The fractal permeability model can reveal more mechanisms that affect the coupled flow deformation behavior in the fractured porous media.