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We analyze the dynamical behavior of an eighth-order Sharma's iterative scheme, which contains a single parameter, with respect to an arbitrary quadratic polynomial using complex analysis. The eighth-order Sharma's iterative scheme is analytically conjugated to a rational operator on the Riemann sphere. We discuss the strange fixed points of the rational operator and present its stable region graph. Additionally, we briefly investigate the superattracting point and the critical point, which have an impact on the Sharma's iterative scheme discussed. Finally, we present the dynamical planes for different parameter values using the complex dynamics tool, which helps us select more effective members of the Sharma's iterative scheme. Numerical experiments are conducted to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Tree-like branching networks are widespread in nature and have found wide applications in engineering, where Murray's law is generally adopted to optimally design tree-like systems, but it may become invalid in some cases. Here we give an energy approach to the analysis of the law and re-find Li–Yu's law for the optimal ratio of the square root of 2 with a suitable constraint. When the cross-section of each branch is considered as a fractal pattern, a modified Murray's law is obtained, which includes the original Murray's law for a Peano-like pore and Li–Yu's law for cylindrical branches, furthermore a useful relationship between the diameter and length of each hierarchy is obtained, which is contrary to the tree-like fractal patterns, and the new hierarchy is named as "fractal Murray tree", which also has many potential applications in science, engineering, social science and economics. This paper is intended to serve as a foundation for further research into the fractal Murray tree and its applications in various fields. [ABSTRACT FROM AUTHOR]

Mandelbrot set, which was provided as a highlight in fractal and chaos, is studied by many researchers. With the extension of Mandelbrot set to generalized M set with different kinds of exponent k (k − M set), properties are hard to understand when k is a complex number. In this paper, fractal property of generalized M set with complex exponent z is studied. First, a relation is constructed between generalized M set with complex and real exponent. Then, distribution of z − M set on complex plane is researched. Meanwhile, symmetry of generalized M set is proved. Finally, graphics, generated by escape time algorithm, are the validated results of this paper. [ABSTRACT FROM AUTHOR]

This paper investigates a type of iterated implicit function systems composed of equations F n (x , y) = c , where F n (x , y) is a continuous function, and c is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation F n (x , y) = c containing the implicit function or being α n -contractive about y. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated. [ABSTRACT FROM AUTHOR]

In this paper, in the first place, a strong coupled fixed point result is established for couplings satisfying Pata-type contractive conditions along with certain corollaries and a supportive illustration. Couplings are combinations of coupled and cyclic maps which have appeared in recent literature and have several applications. The illustrative example shows that the theorem properly extends some of the existing results. In the second part of the paper we construct a special type of iterated function system and, utilizing the coupled fixed point theorem introduced in this work, we show that such systems lead to strong fractal sets. This part of the paper is a part of the Hutchinson–Barnsley theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the equivalence of connectedness for the Sierpinski-like sponge and skeleton networks, and find out the relation between the geodesic distance on the sponge and renormalized shortest path distance on the skeleton networks. Furthermore, under some assumption on the IFS, we obtain the comparability of the Manhattan distance and the geodesic distance on the sponge. [ABSTRACT FROM AUTHOR]

Based on the fractal geometry science and the model of flow friction resistance when fluid flows through rough microcapillaries, the formula of diffusion layer thickness and the spatial charge density distribution of the pipe were derived. The relationship between the amount of space charge and the structural parameters of rough elements, relative roughness, liquid conductivity, pipeline diameter, and other physical quantities was discussed. It was found that the height of rough elements is directly proportional to the thickness of the diffusion layer, and also positively proportional to the amount of current. The predicted results of the model are in good agreement with existing models and various published experimental data, verifying the accuracy and reliability of the model. [ABSTRACT FROM AUTHOR]

Scientists worldwide have always been interested in the study of networks that resemble trees and porous materials. Therefore, based on fractal theory, this paper systematically studies the heat transfer problem of the damaged tree-like networks under different saturations in multi-medium composite materials and derives their dimensionless thermal conductivity (DTC). The research has shown that the dimensionless thermal conductivity (DTC) decreases with an increase in damaged channels. The saturation significantly impacts the heat transfer of the damaged tree networks, and it exhibits linear changes with the ratio of the gas phase, liquid phase, and solid phase. Additionally, it can be observed that the fractal dimension of length distribution and diameter distribution, bifurcation numbers, bifurcation series, porosity, and other parameters in the networks are related to thermal conductivity. The comparison between the data derived from this model and experimental data shows that the proposed model can effectively deepen our understanding of the heat transfer mechanism of the damaged networks in composite materials under different saturation levels. Additionally, the model in this paper does not have an empirical constant, which avoids the influence of potential factors. [ABSTRACT FROM AUTHOR]

Many scholars have studied the interporosity flow function (IFF) and shape factor. However, the existing research does not consider the complex microstructure of dual media and the influence of fluid types on interporosity flow. In this paper, the anisotropic IFF and shape factor model of power-law fluid in smooth and rough fracture dual media are established by using the rough capillary bundle model and considering the influence of the microstructure of dual media and the type of fluid. The relationship between IFF, shape factor, and relative roughness in dual media is studied. It is found that the relative roughness is inversely proportional to the IFF and inversely proportional to the shape factor. In addition, the model is compared with the existing research and experimental data to verify the rationality of the model. [ABSTRACT FROM AUTHOR]

The effective thermal conductivity of unsaturated fibrous media is simulated by Fractal-Monte Carlo simulations in this work. Based on the probability model and thermal-electrical analogy method, the proposed model of the effective thermal conductivity can be expressed as an explicit function of fractal dimensions, D t , D f , intrinsic thermal conductivities, k s , k w , k g , the straight capillary length of a unit cell L 0 , liquid saturation S , pore size λ i as well as porosity ϕ. Each parameter of the proposed model has physical meaning and the function has no empirical constant. The results calculated by Fractal-Monte Carlo simulations have been compared with the experimental data available, and an excellent agreement has been shown for an appropriate range of porosity except for the condition at a lower liquid saturation. It is found that, when k s / k g > 1 and k w / k g > 1 , the dimensionless effective thermal conductivity of unsaturated fibrous media decreases with increment in fractal tortuosity dimension and porosity and decrement in liquid saturation. On the other hand, when k s / k g < 1 and k w / k g < 1 , the correlations are opposite. The present Fractal-Monte Carlo technique may have its own advantages in predicting other transports in unsaturated fibrous media. [ABSTRACT FROM AUTHOR]

The Kozeny–Carman (KC) equation is a well-known semi-empirical formula, which is used to calculate the permeability of porous media in the seepage field. The KC constant is an empirical constant in the KC equation. In this paper, based on the fractal theory, the Fractal-Monte Carlo technique is used to simulate the KC constant of the roughened fibrous porous media (RFPM) with micro-scale effects. There is no empirical constants in the proposed model, and each parameter has its physical meaning. The KC constant model of RFPM can be expressed as a function of structural parameters, including relative roughness (), porosity (ϕ), pore area fractal dimension ( D f ), tortuosity fractal dimension ( D T ), capillary diameter (λ) and Knudsen number (Kn). The result shows that the KC constant increases with increases in ϕ , , D f and D T . On the contrary, with increases in λ and Kn , the KC constant decreases. In addition, the KC constant model constructed in the paper agrees well with the existing experimental data and the model. According to the proposed Fractal-Monte Carlo technique, it is possible to better clarify the transmission physical mechanism in RFPM with micro-scale effects. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Zagreb eccentricity index of the level- n Sierpinski gasket K n . Based on the self-similarity and finite pattern, we compute the Zagreb eccentricity index of K n , which is the integral of square of eccentricity in terms of the self-similar singular measure. [ABSTRACT FROM AUTHOR]

Predicting the electrical conductivity of porous media is important for oil reservoirs, rock physics, and fuel cells. In this work, a cuboid embedded with a damaged tree-like network is employed to denote a portion of porous media. Analytical expressions for the electrical conductivity are then derived with the fractal theory. Various structural parameters have been examined in detail for the influence of electrical conductivity. It is found that the increased number of damaged channels means more difficult ion migration and lower electrical conductivity of porous media. In addition, a decreasing length ratio or an increasing diameter ratio will increase electrical conductivity. Moreover, both the channel distribution fractal dimension and tortuosity fractal dimension result in a decrease in electrical conductivity. A comparison of the results predicted by other models yields a good agreement, validating our proposed model. These results may further interpret the transport properties of porous media. [ABSTRACT FROM AUTHOR]

Although the hydraulic features of the tree-like branching network have been widely investigated, the seepage characteristics of the networks have not been studied sufficiently. In this study, the seepage characteristics of porous media embedded with a tree-like branching network with the effects of roughness are studied based on fractal theory. Then, the Kozeny–Carman (KC) constant of the composite network is derived. The KC constant of porous media embedded with a tree-like branching network with roughened surfaces is in good agreement with the experimental data in the literature. The effects of structural parameters on seepage characteristics are also discussed. Notably, the results show that the KC constant of the composite network increases with an increasing volume porosity, and decreases with an increase in the relative roughness. Besides, the model established in this paper contains no empirical constants to ensure that each parameter has its physical significance. Thus, the proposed model can facilitate a better understanding of the seepage characteristics of fluid transport through a tree-like branching network embedded in porous media. [ABSTRACT FROM AUTHOR]

This paper presents a new fractal seepage model to study the decrease in seepage capacity caused by fluid erosion and pressure changes in porous media. The normalized permeability obtained by our model is in good agreement with the experiments and models in literature and has high accuracy and usability. The results demonstrate that: (1) the larger the instability coefficient, the higher the particle stability, the better the permeability of the porous medium, the later the normalized permeability decreases, and the slower it goes down, and (2) the fractal dimension of the solid skeleton set mainly affects the porosity and permeability of porous media, independent of the stability of solid particles. (3) The fractal dimension of movable particles is one of the main factors affecting the normalized permeability, it does not affect the instability conditions solid-phase particles in the porous media. The proposed model makes full use of the microstructure parameters of porous media, further reveals the mechanism of fluid flow in porous media with particle shedding, and provides a sound theoretical basis for various practical applications in petroleum engineering and other industries. [ABSTRACT FROM AUTHOR]

It is found that the pore microstructures of Juncus effusus (JE), a yearly renewable and high-yield cellulose fiber with unique three-dimensional network structure and interconnected channels, exhibit the fractal characters. The cross-sectional images are measured by SEM and the fractal dimensions are calculated by the box-counting method. In order to reveal more details of JE, the multi-fractal spectrum method is employed to analyze its local features, which further shows the fractal structures. The results indicate that the fractal characteristics are obvious although JE is complex and disordered in structure. The present fractal study may have the potential and significance on fractal analysis of mechanical properties and transport properties of these natural fibers. [ABSTRACT FROM AUTHOR]

The flow law of carbon dioxide in the pores of the novel porous LiOH adsorbent is studied, and the intrinsic relationship between the pore structure of the absorbent and the fractal dimension is analyzed. In this paper, the fractal structure — Sierpinski–Menger sponge structure — is used to establish a three-dimensional fractal pore model to achieve different porosities and pore structures. The flow process of the working fluid is numerically simulated, and the flow law under different flow velocities (1, 0.1, and 0.01 mm/s) is analyzed. The simulation results show that the flow of gas in the Sierpinski–Menger sponge structure has fractal characteristics. The streamlines and velocity clouds of different pore structures and porosities reflect the distribution characteristics of the flow velocity in the porous media. The velocity field and pressure field distribution of sponge models of different series are consistent, which has obvious self-similarity symmetry. It provides a numerical reference for the study of the pore adsorption law of real LiOH absorbent. [ABSTRACT FROM AUTHOR]

Deliverability evaluation plays an important role in the reservoir exploitation. In this study, a new seven-region semi-analytical mathematical model considering the influences of fractal, imbibition and non-Darcy flow is proposed to evaluate the deliverability of multiply-fractured horizontal wells in tight oil reservoirs. The Laplace transformation, perturbation method and Stehfest numerical inversion are employed to solve the model. The reliability and accuracy of the analytical solution are verified by the field example. The sensitivity analysis of the major influencing factors on the deliverability is specifically analyzed. The numerical results indicate that the seven-region semi-analytical model can better explain the heterogeneity of fracture network, and its solution can provide an effective algorithm for the deliverability evaluation. It is found that the fractal plays a predominant influence on the productivity of tight oil reservoirs. The larger the fractal dimension and the smaller the fractal index, the higher the accumulative production rate. The imbibition also has an important effect on the deliverability of tight oil reservoirs. As the rising of wetting angle, both daily and accumulative production rates can obviously decrease. The imbibition has a positive impact on the production rate in the water-wet formations, while it has a negative impact on the production rate in the oil-wet formations. Compared with the fractal and imbibition, the threshold pressure gradient has less influence on the production of tight oil reservoirs. There exists a negative correlation between the threshold pressure gradient and the production performance. This work provides a new approach to understand the fractal tight oil reservoirs, which is of great significance for the deliverability evaluation. [ABSTRACT FROM AUTHOR]

The pores and fissures in loaded rock masses are the main channels for underground flow, and may cause serious accidents during the development of groundwater resources. This work presents an efficient method for analyzing the microstructure of the loaded rock mass using fractal theory and computed tomography (CT) scanning. A relation between the microstructure features of the sandstone porosity, fractal dimension, and loading stress is developed using an image identification technique. The results demonstrate that the distribution trends of sandstone samples' slice porosities in the xz- and yz-directions are nearly identical, and the distribution in the xy-direction differs significantly from those in xz- and yz-directions. The total and connected porosities increase with the increase of stress, and the change can be fitted to straight lines. The fractal dimensions of the pores change significantly with stress or loading stress in the xy-direction. [ABSTRACT FROM AUTHOR]

Gas permeability is an important parameter for gas transport in microporous and nanoporous media. A probability model of gas permeability of fractal porous media with rough surfaces is proposed and numerically simulated by the Monte Carlo technique. This model consists of two gas flow mechanisms: the Poiseuille flow and the Knudsen flow, and can be expressed by structural parameters, such as the pore fractal dimension, the tortuosity fractal dimension, the relative roughness and porosity. The validity of the proposed model is investigated through the available experimental data, and a good agreement is obtained. The predicted results indicate that gas permeability of microporous and nanoporous media with rough surfaces decreases with the increase of the relative roughness and the tortuosity fractal dimension, and increases with the increase of porosity and the pore fractal dimension. Our gas permeability model could reveal the physical mechanisms of gas transport in porous media with rough surfaces. [ABSTRACT FROM AUTHOR]

The key to shale gas exploration is the characterization of gas migration under the combination of multiple factors. To address the long-standing energy challenge of rapidly and accurately quantifying the behavior of natural fractures and matrix pores in shale at an engineering scale in interaction with gas migration. This study proposes an interdisciplinary model for shale gas extraction by adopting fractal theory. Five innovative microstructural parameters are developed to characterize the size and scale of natural matrix pores/fractures in shale, so as to investigate the contributions of fractal distributed pores and fractal power-law distributed fractures to shale gas extraction. The present results of the proposed model are consistent with the exploitation state of the UK Bowland Shale #114 well. The evolution of the shale microstructure will lead to changes in gas migration behavior throughout the reservoir and in turn affect shale stress, temperature and gas adsorption–desorption effect, and finally have a significant impact on permeability. It is found that in the present analysis of the entire Bowland Shale, the overall permeability changes by 10.8% with the evolution of fractal distributed pores and by 41.3% with the evolution of fractal power-law fractures. This work provides a new approach for rapidly exploring the behavior of shale fractures and matrix pores at engineering scales. This work also offers a new and practical baseline for shale gas extraction assessment and fossil energy management. [ABSTRACT FROM AUTHOR]

In this work, we have given an analogical method for estimating the fractal dimension for three-dimensional fracture tortuosity (3D-FT). The comparison and error analysis of analogical and rigorous methods on fractal dimension for 3D-FT were carried out in this work. The fractal dimension D T − R for 3D-FT from the proposed analogical method is the function of 3D fracture average tortuosity ( τ av ) and average fracture length ( L av ). The analogical method for estimating fractal dimension ( D T − A) with high accuracy indicates good consistency with the rigorous method ( D T − R). The fractal dimension ( D T − R) from the rigorous method is the embodiment of the physical meaning of D T − R . The fractal dimension ( D T − A) from the analogical method is relatively convenient for calculating the premise of ensuring accuracy. [ABSTRACT FROM AUTHOR]

Much information on the structural properties and some relevant dynamical aspects of a graph can be provided by its normalized Laplacian spectrum, especially for those related to random walks. In this paper, we aim to present a study on the normalized Laplacian spectra and their applications of weighted level- 4 Sierpiński graphs. By using the spectral decimation technique and a theoretical matrix analysis that is supported by symbolic and numerical computations, we obtain a relationship between the normalized Laplacian spectra for two successive generations. Applying the obtained recursive relation, we then derive closed-form expressions of Kemeny's constant and the number of spanning trees for the weighted level- 4 Sierpiński graph. [ABSTRACT FROM AUTHOR]

The hydro-mechanical coupling behavior of the fractured rock is constitutive in accurately modeling the seepage properties such as permeability and porosity. In this work, the stress-dependent models for permeability and porosity of the fractured rock are proposed by employing the fractal geometry theory and the two-part Hooke's model (TPHM). The proposed models for the permeability and porosity are related to the effective stress and the microstructures of fractures, such as the fractal dimension ( D 0 , f and D σ , f) , the maximum fracture length l σ , max , and fracture orientations ( and φ) as well as porosities ( ϕ 0 and ϕ σ). The validity of the proposed permeability model coupled with stress is validated by comparison with the available experimental data. The model predictions agree well with the experimental data. The effects of microstructures of fracture networks in fractured rock on the dimensionless stress-dependent permeability and porosity as well as the fractal dimension for fracture areas are also discussed in detail. The proposed models may shed much light on the fundamental behavior of coupled hydro-mechanical properties of fractured rock. [ABSTRACT FROM AUTHOR]

Stress field is a key external factor affecting the thermodynamic properties of rock in many scientific and engineering fields ranging from geoscience to practical problems in geothermal resources, oil and gas exploitation, mineral exploitation, etc. In this work, the stress-dependent models for the effective thermal conductivity of fractured rock are proposed by combining the fractal geometry theory and the two-part Hooke's model. The proposed model for the effective thermal conductivity is a function of the effective stress and structure-parameters and physical parameters of fractured rock, such as porosity ( ϕ σ) , fracture orientations ( and φ) as well as k s and k f . The results show that thermal conductivity increases nonlinearly with increasing stress at low stress range due to the closure of fractures, then, it exhibits an almost linear behavior of pressure dependence at high stress range for most fractures closing completely. The proposed thermal conductivity model coupled with stress is evaluated by comparison with the available experimental data. The model predictions show a good agreement with the available experimental data. A more detailed investigation of the essential influence of structures parameters of a set of fractures on the dimensionless stress-dependent thermal conductivity is performed. The proposed model may provide a good quantitative understanding of the stress effect on thermal conductivity of fractured rock. [ABSTRACT FROM AUTHOR]

Heat transport in tree-like bifurcation networks has been widely studied in various fields. In this work, we investigate heat conduction in the dual porous media embedded with asymmetric tree-like bifurcation networks. In addition, considering the effects of nonuniform tube shape, we assume that the bifurcated tube shows sinusoidal fluctuations. Based on the fractal distribution of pore size and bifurcation structure, we established a dimensionless effective thermal conductivity (ETC) model of the dual porous media. The dimensionless ETC ( λ +) obtained is related to the porosity (ϕ) , the fluid–solid thermal conductivity ratio ( λ f / λ s) , the pore area fractal dimension D f and the structural parameters of the bifurcation network (bifurcation level i , length ratio γ , radius ratio η , fluctuation amplitude factor ξ , bifurcation angle ). To verify the validity of this model, a comparison of the present dimensionless ETC model with available experimental data was carried out and the results were in good agreement. We have discussed the effects of each parameter on the dimensionless thermal conductivity in detail and constructed parametric planes to evaluate the structural parameters more directly. The model has positive implications for revealing the heat transport mechanism in asymmetric tree-like bifurcation dual porous media. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the fractal version of Zagreb eccentricity index which is a singular integral of the eccentricity squared in terms of self-similar measure. Based on the geometric characteristics of the farthest point, we calculate the fractal version of Zagreb eccentricity index on the (3,1) flower fractal to be 1 8 2 9 3 6 0 by using the technique of finite pattern. [ABSTRACT FROM AUTHOR]

Permeability hysteresis under cyclic pressure loading and unloading has received a lot of attention in both science and engineering. But most of the existing model is only for a one-time pressure cycle. Therefore, a permeability hysteresis model is established based on the theory of elastic-structural deformation of capillary cross section and the fractal theory of porous media. Both the triangular and quadrilateral structures are considered. The stress sensitivity of structural deformation decreases with the increase in the cycle. The porosity hysteresis can also be predicted by the proposed model. Compared with experimental data with different permeability hysteresis, the prediction of the proposed model is consistent with the experimental results. Compared with other models, the proposed model has a smaller average error and a better agreement with experimental data. The proposed model can predict the permeability hysteresis under not only a one-time pressure cycle like the existing model but also multiple pressure cycles. The influence of parameters shows that the decrease in Young's modulus and Poisson's ratio of the solid cluster increases the permeability stress sensitivity but does not influence the permeability hysteresis. The increase in the proportion of quadrilateral structure and stress sensitivity of structural deformation increases the permeability hysteresis and stress sensitivity at the same time, while the capillary fractal dimension, tortuosity fractal dimension, and the decay rate of stress sensitivity of structural deformation during the cycles show the opposite. The proposed model has a significant meaning in underground resource mining and the study of permeability hysteresis mechanism. [ABSTRACT FROM AUTHOR]

Fermat point of a triangle is the point with the minimal total distance from the three vertices in a triangle. In this paper, we discuss the average Fermat distance for a class of hierarchical networks. First, the unweighted hierarchical scale-free network is established in an iterative way. Applying the recursive method, we deduce the analytical expression of average Fermat distance and average geodesic distance. Then we reveal the linear relation of the leading terms for average Fermat distance and average geodesic distance. Finally, we obtain the small-world property of the hierarchical scale-free network, which indicates that average Fermat distance can be a valuable index of small-word property. [ABSTRACT FROM AUTHOR]

Spontaneous imbibition has attracted considerable attention due to its extensive existence in nature. In this study, we theoretically explored the spontaneous imbibition dynamics in a damaged V-shaped tree-like branching network by comparing with a parallel net with fixed constraints. Additionally, the imbibition capacity is characterized by two dimensionless quantities: imbibition potential and dimensionless imbibition time. The fractal theory is then used to generate the analytical expressions of these two dimensionless quantities. After that, the influence of structural parameters on the imbibition process is systematically investigated. It is found that a larger number of damaged channels will correspond to the lower imbibition potential and dimensionless imbibition time. Notably, the branching number N has an evident enhancement effect on the imbibition potential. A parameter plane is introduced to visualize parameter combinations, enabling the direct evaluation of the imbibition process in a specific network system. The physical mechanisms revealed by the proposed model provide effective guidance for imbibition process analysis in the damaged tree-like networks. [ABSTRACT FROM AUTHOR]

The pull-in instability is the inherent property of a micro-electromechanical system (MEMS) when the voltage is larger than its threshold value. Recently, a fractal MEMS system was proposed to overcome the pull-in instability with great success, and it has opened a total new path for the so-called pull-in stability. This paper suggests a pull-in plateau, a novel concept for qualifying the pull-in stability. The plateau's basic properties are elucidated, and the effect of the fractal dimensions on the plateau width is elucidated, and the paper concludes that there exists a critical condition for an ever pull-in stability when both the acceleration and the speed of the system equal zero. [ABSTRACT FROM AUTHOR]

In the petroleum and soil science literature, fractal models of porous systems abound. Consequently, any process, be it physicochemical or geomechanical, that imparts pore structure will equally impact the fractal nature of the system. Several studies have reported the effect of physicochemical processes, such as the water–rock interactions, on the fractal dimension of the porous system using different approaches involving sophisticated instrumentation. Others have also reported the effect of geomechanical processes such as compaction on fractal dimension using similar approaches. To the best of our knowledge, none or little has been reported regarding the use of multiphase flow hydraulic models for revealing the effect of water–rock interactions on the fractal dimension and pore size distribution of the porous system. Consequently, to fill the knowledge gap in this regard, data on two-phase flow of acid gas and brine in porous media have been used to show that the fractal dimension and pore size distribution of the porous system change in response to water–rock interactions. In view of the injection of acid gas, such as anthropogenic carbon dioxide, into saline aquifers being a feasible means for carbon immobilization by mineral carbonation and biogeochemical means, the effect of change in the fractal properties of the porous system on the microbial transport and distribution in porous system has also been reviewed in relationship to the principal findings of this paper. Most importantly, the results of this study show that data on two-phase flow in porous media involving acid gas and brine can be used to prove changes in the fractal nature of the porous system, and this change can also impact subsurface distribution of methanogens that have the potential to trap carbon by methanogenesis. [ABSTRACT FROM AUTHOR]

The venation-finned porous network has been demonstrated as a promising method to maximize the thermal transport access. To improve the heat storage efficiency, an innovative venation-finned porous network is employed here to enhance the melting performance of phase change materials (PCMs). The venation-finned porous network is quantitatively described by Murray's law and Voronoi method, and a modified thermal lattice Boltzmann model of PCM melting processes in a square cavity is developed and numerically analyzed to optimize the venation-finned porous network. The melting performance of composite PCMs with three configurations (venation fin, porous network and venation-finned porous network) are compared and analyzed. Moreover, the effects of branching angle and Murray coefficient on melting performance are comprehensively studied. It indicates that venation-finned porous network is favorable to melting performance improvement due to venation's inherent efficient heat transfer paths. Compared to venation fins and porous networks, the melting duration time of venation-finned porous networks is reduced by 78.4% and 21.4%, respectively. Furthermore, the branching angle of 45 ∘ and Murray coefficient of 3 are suggested for maximizing the melting efficiency. Importantly, the melting mechanism is conduction and convective conjugated heat transfer in composite PCMs with venation fins, however, it is dominated by heat conduction for those with porous networks or venation-finned porous networks. [ABSTRACT FROM AUTHOR]

In this paper, a fractal model is proposed for gas diffusion in dry and wet fibrous media with tortuous converging–diverging capillary bundle on the basis of the fractal theory. The proposed theoretical model for the normalized gas diffusivity (NGD) can be expressed as an explicit functional relation of porosity, φ , fluid saturation, s , fractal dimensions, D t and D f , the minimum average radius, R ¯ min , the maximum average radius, R ¯ max , the straight capillary length of a unit cell L 0 as well as fluctuation amplitude a. The predictions of the proposed model have been compared with the existing experimental data and the available model predictions, and a good agreement can be observed. The effect of various parameters on the NGD is studied alone. It is observed that the NGD decreases with an increase in the fluctuation amplitude. Also, it is seen that the NGD decreases with an increase in the tortuosity fractal dimension. Moreover, it is found that the NGD in wet fibrous media decreases with an increase in the fluid saturation. The present model has no empirical constant and each parameter contains clear physical meaning. These may better reveal the physical mechanisms of gas diffusion in fibrous media. [ABSTRACT FROM AUTHOR]

The low-medium maturity shale oil in China has large resource potential. The occurrence state of such resource is mainly retained hydrocarbon and unconverted organic substance, making the in-situ conversion process a requisite technology. Meanwhile, the naturally fractured shale reservoir has multiple types of pore space, making the hydrothermal coupling process more complex. In this paper, a coupled thermal-reactive compositional flow model and numerical solution method is proposed. Specifically, the apparent permeability of fractured shale oil reservoir is derived based on fractal theory. Besides, the multistage parallel chemical kinetic reactions during in-situ conversion process and evolution of reservoir fluid composition and shale porosity are incorporated. Then the accuracy of the proposed model is tested through numerical cases. Finally, the impact of key operation parameters and fractal parameters on shale oil exploitation is analyzed, such as heating temperature, heating spacing, pore size fractal dimension, fracture length fractal dimension and tortuosity fractal dimension. The proposed model is capable of accurate and efficient prediction of the fractured low-medium maturity shale oil reservoir production. [ABSTRACT FROM AUTHOR]

Permeability analysis is an essential process in reservoir characterization. As one of the most development potential kind of reservoirs, the coexistence of fractures and vugs brings great challenges to permeability analysis of fractured-vuggy carbonate reservoirs due to its highly heterogeneity on multi scales. In this work, a novel permeability analysis model is developed for fractured-vuggy carbonate reservoirs based on fractal theory, which can predict the permeability of fractured-vuggy porous media within a reliable range. Besides, a Monte Carlo simulation has also been implemented to obtain feasible values of permeability based on experimental sample parameters. The validity of the proposed fractal model is verified by both the experimental data and numerical simulation results. Each parameter in the model has clear physical meaning. The results show that the permeability of a fractured-vuggy carbonate reservoir is related to the fracture fractal dimension, fracture fractal tortuosity dimension, the maximum vug diameter and the maximum fracture aperture. [ABSTRACT FROM AUTHOR]

We introduce and study mixed triangular labyrinthic fractals, which can be seen as an extension of (generalized) Sierpiński gaskets. This is a new class of fractal dendrites that generalize the self-similar triangular labyrinth fractals studied recently by the authors. A mixed triangular labyrinthic fractal is defined by a triangular labyrinthic pattern and an infinite sequence of triangular labyrinth patterns systems, whereas one triangular labyrinth patterns system is sufficient for generating a self-similar triangular labyrinth fractal. Moreover, a labyrinthic pattern is more general than a labyrinth pattern, and the idea is that the use of many different patterns provides objects with a "richer" structure. After proving that these non-self-similar fractals are dendrites, we study the growth of path lengths in graphs associated with iterations. We prove geometric properties of the fractals, some of which have rather "local" character and on the other hand occur at sites distributed "all over" the fractals. [ABSTRACT FROM AUTHOR]

Mining the hierarchical structure of Internet review topics and realizing a fine classification of review texts can help alleviate users' information overload. However, existing hierarchical topic classification methods primarily rely on external corpora and human intervention. This study proposes a Modified Continuous Renormalization (MCR) procedure that acts on the keyword co-occurrence network with fractal characteristics to achieve the topic hierarchy mining. First, the fractal characteristics in the keyword co-occurrence network of Internet review text are identified using a box-covering algorithm for the first time. Then, the MCR algorithm established on the edge adjacency entropy and the box distance is proposed to obtain the topic hierarchy in the keyword co-occurrence network. Verification data from the Dangdang.com book reviews shows that the MCR constructs topic hierarchies with greater coherence and independence than the HLDA and the Louvain algorithms. Finally, reliable review text classification is achieved using the MCR extended bottom-level topic categories. The accuracy rate (P) , recall rate (R) and F 1 value of Internet review text classification obtained from the MCR-based topic hierarchy are significantly improved compared to four target text classification algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we have employed Fractal formalism to calculate the Fracture Functions of the Leading neutron produced in e p collisions. The Fractal concept describes the self-similar behavior of the proton structure at Leading neutron production of semi-inclusive Deep Inelastic Scattering at the low values of the fractional momentum variable β. The Fracture Functions (FFs) parameterized the non-perturbative part of the fragmentation process at the initial scale of Q 0 2 . In this analysis, we benefit from the Leading neutron (Ln) experimental data published by H1 Collaboration and in order to estimate the uncertainty of neutron FFs and corresponding observables, we used the Hessian method. As a consequence of this Next-to-Leading order QCD analysis, we achieve a nice agreement between the prediction of our model for neutron FFs and experimental data. It seems that the Fractal approach based on the self-similar behavior of these conditional parton distribution functions can nicely describe the experimental data at low x. [ABSTRACT FROM AUTHOR]

We study Bohr's formula for fractal Schrödinger operators on the half line. These fractal Schrödinger operators are defined by fractal measures with overlaps and a locally bounded potential that tends to infinity. We first derive an analog of Bohr's formula for these Schrödinger operators under some suitable conditions. Then we demonstrate how this result can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, m-fold convolution of Cantor-type measures, and a family of graph-directed self-similar measures that are essentially of finite type. [ABSTRACT FROM AUTHOR]

Fractals play an important role in nonlinear science. The most important parameter when modeling a fractal is the fractal dimension. Existing information dimension can calculate the dimension of probability distribution. However, calculating the fractal dimension given a mass function, which is the generalization of probability, is still an open problem of immense interest. The main contribution of this work is to propose an information fractal dimension of mass function. Numerical examples are given to show the effectiveness of our proposed dimension. We discover an important property in that the dimension of mass function with the maximum Deng entropy is ln 3 ln 2 ≈ 1. 5 8 5 , which is the well-known fractal dimension of Sierpiski triangle. The application in complexity analysis of time series illustrates the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, a new computing technique is investigated for the cause of Mycobacterium ulcerans (MU) with the help of differential equations through the SIR model. This study is carried out to find the cause between the ulcer and the environment for the spread of Buruli ulcer (BU). Fractional-order SIR model is applied to discuss in detail the BU disease. The construction and development of this given numerical scheme are performed for the SIR epidemiological model concerning the transmission dynamics of BUs. Mathematical modeling of the presented scheme with the help of stability techniques is accomplished. Further, the behavior of the scheme is accessed by the evaluation of the different parameters of interest. The attained numerical results are observed for the illustration of the benefit of introducing a fractional model. Finally, we profound that the presented numerical scheme maintains all required control measuring features of the corresponding dynamical system and decreases the ancillary cost when studied over a long time. With the help of a given novel and unique approach, a significant observation is noted and recommended for other epidemiological fractional-order models. In future, this novel fractal technique will be appertained in chemical and biomedical engineering applications. [ABSTRACT FROM AUTHOR]

In this paper, a highly efficient tree-shaped fractal antenna with multi-broadband resonance characteristics is proposed. The proposed antenna exhibits broad operating bandwidth, high gain, and high efficiency characteristics due to the suggested modifications in the antenna geometry. The suggested circular patch is modified by introducing slots in the form of a tree-shaped fractal structure, and the partial ground plane is modified by incorporating a narrow rectangular slot. The proposed antenna is designed on an FR4 substrate of 4 0 × 2 5. 0 5 × 1. 6 mm3. The prototype of the proposed antenna has been fabricated and tested to justify the simulation results. The measurement results are in good agreement with the simulation that validates the multi-broadband design approach of the proposed fractal antenna. As per the measurement results, the proposed antenna operates at four distinct bands with − 10 dB impedance bandwidths of 600 MHz (2.2–2.8) GHz, 1070 MHz (3.3–4.37) GHz, 2550 MHz (4.75–7.3) GHz, and 2200 MHz (9.7–11.9) GHz. Furthermore, a high peak gain of 10.23 dB and a peak radiation efficiency of 96.65% are recorded for the suggested fractal antenna. The suggested compact bandwidth enhanced multi-band antenna can be useful for several wireless communication systems such as wireless fidelity (Wi-Fi), wireless local area network (WLAN), Bluetooth, long-term evolution (LTE), C, S and X bands. [ABSTRACT FROM AUTHOR]

In this paper, we study the iterated function systems (IFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. We define a weak* separation condition (W*SC), which is strictly weaker than the weak separation condition of Lau and Ngai. By assuming the bounded distortion property, we show that the W*SC is equivalent to the identity limit criterion of Zerner for such IFSs. In particular, in the one-dimensional case, we show that the W*SC is also equivalent to Ahlfors–David regularity, the Assouad dimension is strictly less than 1 and positivity of the s -dimensional Hausdorff measure, where s is the zero of the topological pressure. [ABSTRACT FROM AUTHOR]

The statistical simulation of atmospheric carbon dioxide based on time series analysis plays an important role in the development of refined carbon emission assessment and traceability. A new statistical method based on the random Weierstrass–Mandelbrot function is proposed to simulate the high-Reynolds-number and high-Péclet-number scalar turbulence in the atmospheric flow. In this paper, we focus on the carbon dioxide, which is the most important scalar for the global climate changes. The new statistical method is used to simulate the time series of carbon dioxide concentrations in the high-frequency inertial subrange. Results show that the simulated power spectral densities are almost the same as the observed ones in the inertial subrange. Compared with the Gaussian distribution, the simulated probability density functions are more similar to the observed ones with the skewness characteristic. More importantly, the fractal characteristics of the time series that cannot be simulated by traditional methods in the meteorological engineering can also be well simulated by the new method. [ABSTRACT FROM AUTHOR]

Modeling fluid flow and transport in fracture network is of considerable importance in a wide range of science and engineering. In this work, a probability model of the effective permeability for fracture network is developed based on fractal theory and Monte Carlo method. The proposed numerical model is expressed as a function of the fractal dimension, the minimum/maximum trace length, fracture orientation as well as random number, and the validity is validated by comparison with the analytical fractal model. It is found that the effective permeability increases with the increase of the fractal dimension and fracture density of fracture network, as well as decreases with increase of the dip angle of fractures. The proposed Fractal Monte Carlo method can significantly reduce the computational load compared with conventional numerical techniques and also permits to relate the other flow and transport properties in fracture networks. [ABSTRACT FROM AUTHOR]

The point within a triangle that meets the minimum sum of the distances to the three vertices is called the Fermat point. The aim of this paper is to obtain the Fermat average distance on a self-similar fractal tree in terms of the self-similar measure and the technique of finite pattern. [ABSTRACT FROM AUTHOR]

In this paper, we relax the definition of Rényi information dimension. The power law of the Entropy-Layer in the Galton board is discovered and we calculate its information fractal dimension. When the Galton board is extended to bias or three-dimensional space, we get the same fractal features. In addition, according to the connection between Pascal's triangle and the Poisson distribution, we find constrained Poisson distribution groups with the same information dimension. This is the first time the information entropy is utilized to explore the fractal features of the Galton board and Pascal's triangle. [ABSTRACT FROM AUTHOR]

We study the heat equation on a bounded open set U ⊂ ℝ d supporting a Borel measure. We obtain asymptotic bounds for the solution and prove the weak parabolic maximum principle. We mainly consider self-similar measures defined by iterated function systems with overlaps. The structures of these measures are in general complicated and intractable. However, for a class of such measures that we call essentially of finite type, important information about the structure of the measures can be obtained. We make use of this information to set up a framework to study the associated heat equations in one dimension. We show that the heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study the propagation speed problem. [ABSTRACT FROM AUTHOR]

The study characterizes power-law fluid through charged fibrous porous media with spatial fractional-derivative and fractal geometry. Seepage flow of power-law fluid across fractal fibrous porous media in the presence of electric double layers (EDLs) is investigated based on the capillary bundle model. The acquired velocity distribution equation in a narrow capillary is then transformed into the form of series with appropriate Taylor approximation. After that, an analytical formula for dimensionless permeability is derived based on the generalized Darcy's law. The effects of diverse parameters, including the fractal dimension of pore area, porosity, fractional order and Zeta potential on dimensionless permeability, are discussed. It can be seen from the results that lower fractional order has an amplification effect on dimensionless permeability with the change in Zeta potential. The results provide some theoretical guidance for revealing the seepage mechanism of a power-law fluid in charged porous media. [ABSTRACT FROM AUTHOR]