Fractals

Current hydraulic fracture conductivity evolution models fail to incorporate the fractures microscopic petrophysical properties, and notable discrepancies consequently exist between predictions and observations. We present a new conductivity model considering the irregular fracture undulation and channel roughness in propped fractures. The propped fracture networks are treated as bundles of tortuous capillaries with a fractal distribution of sizes with the size of a single capillary calculated using a constitutive model representing contacting rock surfaces under normal cyclic loading. The capillaries tortuosity is described by the effective inclination angle, and the fracture closure is calculated by the history of the in situ stress distribution and rock property changes. The fracture surface roughness, number of capillaries per unit width of the hydraulic fracture, and total cross-sectional area are obtained using fractal theory. The proposed model is validated by comparison with experimental data and other analytical solutions. The results indicate that the apparent permeability and conductivity of the fracture significantly decrease to 58.0% and 48.2% within 2 years of production, respectively, and then remain steady for the remainder of the well life. Compared with fixed fracture conductivity, the temporal variability in conductivity leads to a lower formation pressure drop and reduction in final production. Furthermore, the influences of the effective inclination angle, relative roughness of micro-channel, fracture porosity, and microchannel fractal dimension on the conductivity are investigated and the conductivity proves to be largely controlled by the fracture porosity, while the influence of the relative roughness ratio on the conductivity is least significant.

The average geodesic distance is an important index in the study of complex networks. In this paper, we investigate the weighted average distance of Pentadendrite fractal and Pentadendrite networks. To provide the formula, we use the integral of geodesic distance in terms of self-similar measure with respect to the weighted vector.

In order to address the overall properties of the Brazilian agricultural commodity market and the intricate effects of political and economic instabilities, in this work, we provide a comprehensive study of the multifractal properties of the Brazilian commodities using multifractal detrended fluctuation analysis (MFDFA). We focus on the daily price of 12 Brazilian agricultural commodities over the last two decades, and four commodities (sugar, soybean, coffee and cattle) are also studied in terms of time-dependent MFDFA to address the effects of particular political and economic instability events. All commodities exhibit multifractal properties, which are then used to evaluate market efficiency. We find that all commodities except coffee show lower market efficiency for prices in Brazilian Reals (BRL) than in US Dollars (USD) reflecting the presence of constrains in domestic agricultural market, such as minimum price policy. From time-dependent MFDFA, we find that after the 2007/2008 food crisis, the market efficiency increases, as indicated by the changes in price dynamics towards lower persistency and weaker multifractality, with the dominance of small fluctuations.

Harmonic functions possess the mean value property, that is, the value of the function at any point is equal to the average value of the function in a domain that contain this point. It is a very attractive problem to look for analogous results in the fractal context. In this paper, we establish a similar results of the mean value property for the harmonic functions on the higher-dimensional Sierpinski gasket.

In this paper, a 1D chain network with a reverse weighted edge is introduced. We focus on studying the relationships including the convergence rate and the length, the convergence rate and weight of adding reverse edge relationships. Laplacian characteristic determinant is calculated and subsequently, the sum of the reciprocals of all nonzero Laplacian eigenvalues is obtained. Hence, the analytic expression of global mean weighted first-passage time (GMWFPT) can be deduced. The obtained results show that there exists a linearly positive relationship between GMWFPT and the weight $r$.

Microstructures dominate the physical properties of fractal porous media, which means the clarification of complexity types and their assembly are of fundamental importance for static or dynamic purposes. In this work, we identified fractal porous media to dual-complexity systems composed of stationary and scale-invariant complexities as per fractal topography theory, and proposed an open mathematical framework to characterize complexity assembly in microstructures, realized the original complexity, such as random, multi-phase, and multi-type features by the quartet structure generation set (QSGS) algorithm, and unified the behavioral complexity, including the self-similar and self-affine properties by fractal topography model. For demonstration, the control mechanisms on the microstructures from different complexities are discussed, with their physical implications and relations to the physical properties of porous media clarified in principle. The results indicate that our framework is open to arbitrary original and behavioral complexities, and eases the modeling of multi-scale microstructures and the property estimation significantly.

The effective thermal conductivity of unsaturated porous media is of interest in a number of applications of heat transfer. In this paper, a novel fractal solution for effective thermal conductivity is derived based on the fractal distribution of surface roughness and pore size in unsaturated porous media with roughened surfaces. The proposed fractal model explicitly relates the effective thermal conductivity to the microstructural parameters (relative roughness, porosity and fractal dimensions). The proposed fractal model is verified by a satisfying agreement of the effective thermal conductivity predicted by our model and that reported as existing experimental data in the literature. A parametric study is also elaborated to investigate the influences of the microstructural parameters on the effective thermal conductivity. The results demonstrate that our proposed fractal model improves our understanding of the physical mechanisms of heat transport through unsaturated porous media with roughened surfaces. One advantage of our fractal analytical model is that it contains no empirical constant, while it is usually required in previous models.

This paper presents a modified complete normal contact stiffness model of a fractal surface considering contact friction. We use this model to study the influence of fractal dimensions and fractal roughness on normal contact stiffness. The fractal micro-contact model of an asperity and the complete length scale contact model of fractal surface (both contrasting classical mechanics) are revised. The influence of frictional resistance at micro-contact interfaces on normal contact stiffness is also considered. Predictions of the new model are found to be in greater agreement with the results of the experiments than the predictions of the original model. The study analyzes the influence of fractal dimensions and fractal roughness on the normal contact stiffness. With the increase of these two fractal parameters, their influences on the normal contact stiffness are opposite and are different under high pressure and low pressure.

The present paper characterizes the resistive and inductive loads of an electric distribution system by Julia fractal sets, in order to discover other observations enabling the elevation of new theoretical approaches. The result shows that indeed the electrical load reflects a clear graphic pattern in the fractal space of the Julia sets. This result, then, is a new contribution that extends the universal knowledge about fractal geometry.

For alleviating energy shortage and environmental problems, it is of great importance to improve the energy charging and discharging efficiency of thermal energy storage systems. In this context, an innovative phase change heat exchanger (PCHE) with fractal tree-shaped fins is presented in this paper. A numerical investigation of the solidification behaviors in the PCHE with fractal tree-shaped fins is conducted. The dynamic temperature response and the solidification front evolution in the PCHE are analyzed and discussed. Furthermore, two evaluation criteria, including total solidification time and energy charging efficiency, are introduced to quantitatively study the effect of fin material on the solidification heat transfer characteristics. The results indicate that the fractal tree-shaped fin leads to a uniform temperature distribution of phase change material (PCM). The temperature response of fin is faster than that of PCM due to its high thermal conductivity. Moreover, the fractal tree-shaped fin breaks the restriction of gradually forward fashion of solidification front in the traditional PCHE and dramatically improves the energy discharging performance. The material of fractal tree-shaped fins is an essential factor affecting the solidification performance of the PCHE. The energy discharging performance of PCHE with pure copper fins is the best, whereas that with cupronickel fins is the worst. However, from the perspective of practical application, aluminum is the best potential alternative because of its relatively high thermal conductivity, lighter weight, and low cost.

This paper focus on the compact Ahlfors–David regular subsets of fractals. For this, we introduce the Ahlfors–David regular dimension and study its basic properties. Based on these properties and some known results in dimension theory of fractals, we compute Ahlfors–David Regular dimension for Moran sets and McMullen sets.

Since the directional permeability of fractured rock masses is significantly dependent on the geometric properties of fractures, in this work, a numerical study was performed to analyze the relationships between them, in which fracture length follows a fractal distribution. A method to estimate the representative elementary volume (REV) size and directional permeability ($K_f)$ by extracting regular polygon sub-models with different orientation angles (${𝜃}_r)$ and side lengths ($L_n)$ from an original discrete fracture network (DFN) model was developed. The results show that the fracture number has a power law relationship with the fracture length and the evolution of the exponent agrees well with that reported in previous studies, which confirms the reliability of the proposed fractal length distribution and stochastically generated DFN models. The $K_f$ varies significantly due to the influence of random numbers utilized to generate fracture location, orientation and length when $L_n$ is small. When $L_n$ exceeds some certain values, $K_f$ holds a constant value despite of ${𝜃}_r$, in which the model scale is regarded as the REV size and the corresponding area of DFN model is represented by $S_{\mathrm{\text{REV}}}$ (in 2D). The directional permeability contours for DFN models plotted in the polar coordinate system approximate to circles when the model size is greater than the REV size. The $S_{\mathrm{\text{REV}}}$ decreases with the increment of fractal dimension of fracture length distribution ($D_f)$. However, the decreasing rate of $S_{\mathrm{\text{REV}}}$ (79.5%) when $D_f$ increases from 1.4 to 1.5 changes more significantly than that (34.8%) when $D_f$ increases from 1.5 to 1.6 for regular hexagon sub-models. This indicates that the small non-persistent fractures dominate the preferential flow paths; thereafter, the flow rate distribution becomes more homogeneous when $D_f$ exceeds a certain value (i.e. 1.5). A larger $D_f$ results in a denser fracture network and a stronger conductivity.

The effect of the coal seam water infusion to reduce dust is directly related to the development degree of the coal structure under the action of the effective stress. Therefore, this paper first summarizes several basic effective-stress models and proposes the effective-stress model considering the capillary force. Thereafter, the nuclear magnetic resonance (NMR) experiment system high-voltage seepage module is used to achieve the saturation coal sample T${}_{2c}$ and water infusion coal sample T₂ test under different mechanical environments. And the fractal dimension was calculated based on the results of porosity and pore size test. Finally, the parameters were sequentially substituted into effective-stress models. The calculation results were compared and analyzed. The results show that the saturation levels of the irreducible fluid of Sample DLT and Sample XLZ are 92.93% and 93.66%, respectively, indicating that these samples are dense porous media structures. Meanwhile, the porosity test shows the overall porosity with the water infusion pressure and confining pressure changes being consistent with the conventional law; the partial porosity caused a slight fluctuation due to different sensitivities of different pores to stress. The effective stresses of the capillary force and the calculated effective stress of the body are closest to the fractal effective stress, so the difference between the two values is negligible due to the small capillary force, which indicates that during the high-pressure water infusion, the capillary force in the meso-structure hardly affects the mechanical environment. Based on the theoretical research results, this paper can provide a theoretical reference for the determination of water infusion parameters under different on-site mechanical environment during the process of coal seam water infusion, which is conducive to the popularization and application of this technology, thus, providing the scientific basis for safe mining of the deep coal seam and ensuring the safe and efficient production of mines.

In this paper, the evolving networks are created from a series of Sierpinski-type polygon by applying the encoding method in fractal and symbolic dynamical system. Based on the self-similar structures of our networks, we study the cumulative degree distribution, the clustering coefficient and the standardized average path length. The power-law exponent of the cumulative degree distribution is deduced to be ${log}_{2}n$ and the average clustering coefficients have a uniform lower bound $\frac{2n-3}{3n}$. Moreover, we find the asymptotic formula of the average path length of our proposed networks. These results show the scale-free and the small-world effects of these networks.

In this paper, we analyze the extreme events of non-stationary time series in the framework of horizontal visibility graph (HVG). We give a new definition of extreme events, which incorporates the temporal structure of the series and the degree of the nodes in the HVG. An advantage of the new concept is that it does not require ad hoc treatment even when the non-stationarity arises in time series. We also use the information-theoretic methods to analyze the degree of nodes in the HVG. In the numerical analysis, we study the statistical characterizations of the extreme events of synthetic time series, including the random noises, periodic time series, random walk processes, and the long-range auto-correlated time series. Then, we study 9 time series in stock markets to identify the extreme events evolving in these non-stationary systems. Interestingly, we find that the daily closing price series perform rather close to the random walk processes, while the daily trading volume series behave quite similar to the random noises.

This paper studied the level-3 Sierpinski gasket. We solved Dirichlet problem of Poisson equations and proved variational principle on the level-3 Sierpinski gasket by expressing Green’s function explicitly.

Let the self-similar measure ${\mu }_{M,D}$ be generated by an expanding real matrix $M={\rho }^{-1}I\in M_2(ℝ)$ and a digit set $D=\{{(0,0)}^t,{(1,0)}^t,{(0,1)}^t,{(-1,-1)}^t\}$ in space ${ℝ}^2$. In this paper, we only consider $\rho >0$ and the case $\rho <0$ is similar. We show that there exists an infinite orthogonal set of exponential functions in $L^2({\mu }_{M,D})$ if and only if $\rho ={(q/(2p))}^{\frac{1}{r}}$ for some $p,q,r\in \mathbb{N}$ with $gcd(2p,q)=1$. Furthermore, for the cases that $L^2({\mu }_{M,D})$ does not admit any infinite orthogonal set of exponential functions, the exact cardinality of orthogonal exponential functions in $L^2({\mu }_{M,D})$ is given.

Coal–rock dynamic disasters seriously threaten safe production in coal mines, and an effective early warning is especially important to reduce the losses caused by these disasters. The occurrence of coal–rock dynamic disasters is determined by mining-induced stress loading and unloading. Therefore, it is of great significance to analyze the precursory information of coal deformation and failure during true triaxial stress loading and unloading. In this study, the deformation and failure of coal samples subjected to true triaxial loading and unloading, including fixed axial stress and unloading confining stress (FASUCS), are experimentally investigated. Meanwhile, acoustic emission (AE) during the deformation of coal samples is monitored, and the multi-fractal characteristics of AE are analyzed. Furthermore, combined with the deformation and failure of coal samples, the precursory information of coal deformation and rupture during true triaxial stress loading and unloading is obtained. Finally, the relationship between multi-fractal characteristics and damage evolution of coal samples under FASUCS is discussed. The results show that the multi-fractal spectral widths of AE time series under the conditions of FASUCS with different initial confining stresses or unloading rates are quite different, but the dynamic changes of multi-fractal parameters $\mathrm{\Delta }\alpha$ and $\mathrm{\Delta }f$ are similar. This indicates that the microscopic complexity of AE events of coal samples under different conditions of FASUCS differs, but the macroscopic generation mechanism of AE events has inherent uniformity. The dynamic changes of $\mathrm{\Delta }\alpha$ and $\mathrm{\Delta }f$ can reflect the stress and damage degree of coal samples. The dynamic change process of $\mathrm{\Delta }\alpha$ well accords with the damage evolution process of coal samples. A gradual decrease of $\mathrm{\Delta }\alpha$ corresponds to a slow increase of damage, while a sharp increase of it corresponds to a rapid growth of damage. At the same time, the mutation point of damage curve at distinct stress difference levels shares the same variation trend with the $\mathrm{\Delta }\alpha$ mutation point. The change of $\mathrm{\Delta }\alpha$ can reflect the damage process of coal samples, which can be used as precursor information for predicting coal–rock rupture. The finding is of great significance for the early warning of coal–rock dynamic disasters.

This paper suggests a fractal two-phase fluid model for the polymer melt filling process to deal effectively with the unsmooth front interface. An infinitesimal fluid element model in a fractal space is proposed to establish the governing equations according to the conservation laws in fluid mechanics, the fractal divergence and fractal Laplace operator are defined. The unsmooth interface is solved numerically, and fibers’ motion properties on the interface are also elucidated. Moreover, the distribution of fibers on the interface at different stages shows the fractal property of the fibers’ motion. However, the motion of fibers is affected by the flow of macroscopic polymer melt, and the fiber orientation in the interface shows a certain statistical regularity. Based on the characters of fiber orientation, the fractal interface can be used for the optimal design of the polymer melt filling process.

COVID-19 is a pandemic disease, which massively affected human lives in more than 200 countries. Caused by the coronavirus SARS-CoV-2, this acute respiratory illness affects the human lungs and can easily spread from person to person. Since the disease heavily affects human lungs, analyzing the X-ray images of the lungs may prove to be a powerful tool for disease investigation. In this research, we use the information contained within the complex structures of X-ray images between the cases of COVID-19 and other respiratory diseases, whereas the case of healthy lungs is taken as the reference point. To analyze X-ray images, we benefit from the concept of Shannon’s entropy and fractal theory. Shannon’s entropy is directly related to the amount of information contained within the X-ray images in question, whereas fractal theory is used to analyze the complexity of these images. The results, obtained in this study, show that the method of fractal analysis can detect the level of infection among different respiratory diseases and that COVID-19 has the worst effect on the human lungs. In other words, the complexity of X-ray images is proportional to the severity of the respiratory disease. The method of analysis, employed in this study, can be used even further to analyze how COVID-19 progresses in affected patients.