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A novel analytical model is derived based on fractal geometry theory to characterize the permeability of power-law fluids through fractured porous media with fractal asperities on surfaces. The proposed model is expressed as a function of structure parameters of fractured porous media, fractal dimensions for porous matrix and for fracture as well as the index $n$ of power-law fluid behavior. The influences of these parameters on the permeability are analyzed in detail. It is found that the relationship between permeability and roughness exhibits nonlinear characteristics. A higher power-law fluid behavior index significantly affects the reduction rate of permeability with relative roughness. Moreover, permeability is sensitive to the perturbation due to the rate of aperture to length. The proposed fractal model can clearly reveal more physical mechanisms for power-law fluid in fractured porous media than the existing models. Compared with the experimental data available in the literature, the validity of model predictions is verified.

The rock cores of low permeability reservoirs have special pore structures, which are the essential factors to determine the seepage capacity and oil displacement efficiency and directly affect oil and gas reserves and oil well productivity. This paper studies 16 digital rock core samples. Based on the fractal theory for porous media, we carried out the fractal characterization of the pore structure of the samples by an approach combined with binarized CT images and fractal theory. The computing software based on the Box-counting method was used to measure and calculate the fractal dimension and porosity of the samples, and the calculated results were compared with the binarized CT data from the laboratory. It was found that among the tested samples, the results of 12 rock core samples were in good agreement with the experimental data, indicating that the fractal theory is effective in the measurement of fractal dimension and in the calculation of porosity of digital rock core samples. For the results with larger errors compared to laboratory data, we also analyzed and elaborated the reasons from the relevant binarized images and pore distribution images of samples. It is also found that the minimum pore size has a significant impact on the results when the fractal theory was applied to analyze the digital rock core samples. Finally, a standard or criterion is established whether the pore/particle size distribution in a digit rock core/porous medium is fractal or non-fractal.

This paper analyzed the coupling among the reactions of eyes and brain in response to visual stimuli. Since eye movements and electroencephalography (EEG) signals as the features of eye and brain activities have complex patterns, we utilized fractal theory and sample entropy to decode the correlation between them. In the experiment, subjects looked at a dot that moved on different random paths (dynamic visual stimuli) on the screen of a computer in front of them while we recorded their EEG signals and eye movements simultaneously. The results indicated that the changes in the complexity of eye movements and EEG signals are coupled ($r=-0.8043$ in case of fractal dimension and $r=-0.9259$ in case of sample entropy), which reflects the coupling between the brain and eye activities. This analysis could be extended to evaluate the correlation between the activities of other organs versus the brain.

An important category of studies in vision science is related to the analysis of the influence of environmental changes on human eye movement. In this way, scientists analyze human eye movement in different conditions using different methods. An important category of works is devoted to the decoding of eye reaction to color tonality. In this research for the first time, we examined the application of fractal theory for decoding of eye reaction to variations in color intensity of visual stimuli. Three green visual stimuli with different color intensities have been applied to subjects and accordingly the fractal dimension of their eye movements has been analyzed. We also tested the eye movement in non-stimulation condition (rest). Based on the obtained results, increasing the color intensity of visual stimuli caused a lower complexity in subject’s eye movement. We also observed that eye movement is less complex in case of non-stimulation compared to different stimulation conditions. The application of fractal theory in analysis of eye movement can be extended to analyze the effect of other stimulation conditions on eye movement to investigate about the decoding behavior of human eye, which is very important in vision science.

In this paper, we analyzed the variations in brain activation between different activities. Since Electroencephalogram (EEG) signals as an indicator of brain activation contain information and have complex structures, we employed complexity and information-based analysis. Specifically, we used fractal theory and Shannon entropy for our analysis. Eight subjects performed three different activities (standing, walking, and walking with a brain–computer interface) while their EEG signals were recorded. Based on the results, the complexity and information content of EEG signals have the greatest and smallest values in walking and standing, respectively. Complexity and information-based analysis can be applied to analyze the activations of other organs in different conditions.

Analysis of leg muscle activation and gait variability during locomotion is an important area of research in physiological and sport sciences. In this paper, we analyzed the coupling between the alterations of leg muscle activation and gait variability in single-task and dual-task walking. Since leg muscle activation in the form of electromyogram (EMG) signals and gait variability in the form of stride interval time series have complex structures, fractal theory and approximate entropy were used to evaluate their correlation at various walking conditions. Sixty subjects walked at their preferred speed for 10 min under the single-task condition and for 90s under the cognitive dual-task condition, and we evaluated the variations of the fractal dimension and approximate entropy of EMG signals and stride interval time series. According to the results, dual-task walking caused reductions in the complexity of EMG signals and stride interval time series than single-task walking. This technique can be used to evaluate the correlation between other organs during different locomotion.

Since the brain controls heart activations, there should be a correlation between their activities in different conditions. This study investigates the correlation between heart and brain responses to olfactory stimulation. We employed fractal theory and sample entropy to evaluate the complexity of EEG signals and Heart Rate Variability (HRV) in the form of R–R time series. We applied four different pleasant odors with different molecular complexities to 13 participants and analyzed their EEG and ECG signals. The results demonstrated that the complexities of HRV and EEG signals are strongly correlated; a bigger alteration in the complexity of olfactory stimuli is mapped to a bigger alteration in the complexity of HRV and EEG signals. This investigation can be similarly done to examine the correlation between various organs and the brain by quantifying the complexity of their signals versus brain signals.

One of the important areas of heart research is investigating how heart activity changes during aging. In this research, we employed complexity-based techniques to analyze how heart activity varies based on the age of subjects. For this purpose, the heart rate variability (HRV) of 54 healthy subjects (30 M, 24 F, 28.5–76 years old) in three different age groups was analyzed using fractal theory, sample entropy, and approximate entropy. We showed that the fractal dimension, sample entropy, and approximate entropy of the RR interval time series (as HRV) are related to the age of the subjects. In other words, as subjects get older, the complexity of their RR interval time series decreases. Therefore, we decoded the variations in HRV during aging. The method of analysis that was employed in this research can be used to analyze the variations of other physiological signals (e.g. Electroencephalogram (EEG) signals) during aging.

Analysis of the brain activity to external stimulation is an important area of research in biomedical engineering. In this paper, for the first time, we analyzed the brain reaction to visual stimuli with different frequencies using three complexity methods. For this purpose, we utilized fractal theory, sample entropy, and approximate entropy to study the variations of the complexity EEG signals while subjects received visual stimuli at 7, 9, 11, and 13 Hz. The results showed that, in general, by moving from 9 Hz to 13 Hz stimuli, the complexity of EEG signals increases, except in the case of 11 Hz stimulus. The statistical analysis also supported the results of the analysis. The conducted analysis in this research can be performed in the case of other types of external stimuli to study how the brain reacts in different conditions.

Due to the increasing growth of economy in the last few years, environmental problems have become a prominent difficulty in Chinese economic growth and social development. Therefore, in order to respond to the environmental protection policy of Anhui Province, all prefecture-level cities need to be evaluated for their environmental efficiency. A study of environmental efficiency in Anhui Province is presented in this paper. The time period is from 2015 to 2020. By exploiting MaxDEA software, the input–output system in different years is taken as the decision-making unit, and slacks-based model (SBM-DEA) which is based on an unexpected output is used to construct the index system by selecting two output indexes and four input indexes, so as to calculate the environmental efficiency of Anhui Province, including 16 prefecture-level cities, under the condition of constant non-radial and non-angle scale benefits. In addition, based on the persistence and anti-persistence theory of fractal theory, in this paper, we have analyzed the future trend of environmental efficiency of prefecture-level cities in Anhui Province by rescaled range analysis (R/S). The results show that Anhui province has 16 prefectures that have not achieved an optimal level of environmental performance and fluctuates downward from 2015 to 2020. The average value of comprehensive efficiency is 0.846. As for technical efficiency, the mean value is 0.954, and scale efficiency has an average value of 0.887. The technical progress of environmental efficiency development in Anhui Province is small and the room for improvement is large. Finally, according to the results of the study, some suggestions on environmental technology are put forward.

One of the important areas of research in neuroscience is to investigate how brain activity changes during aging. In this research, we employ complexity techniques to analyze how brain activity changes based on the age of subjects during sleep. For this purpose, we analyze the Electroencephalogram (EEG) signals of 22 subjects induced by sleep medication using fractal theory and sample entropy. The analysis showed that the fractal dimension and sample entropy of EEG signals decrease due to aging. Therefore, we concluded that aging causes lower complexity in EEG signals during sleep. The employed method of analysis could be applied to analyze the effect of aging on the variations of the activity of other organs (e.g. heart, muscle) during aging by studying their related physiological signals (e.g. ECG, EMG).

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.

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.

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.

Oil–water relative permeability curve is an important parameter for analyzing the characters of oil and water seepages in low-permeability reservoirs. The fluid flow in low-permeability reservoirs exhibits distinct nonlinear seepage characteristics with starting pressure gradient. However, the existing theoretical model of oil–water relative permeability only considered few nonlinear seepage characteristics such as capillary pressure and fluid properties. Studying the influences of reservoir pore structures, capillary pressure, driving pressure and boundary layer effect on the morphology of relative permeability curves is of great significance for understanding the seepage properties of low-permeability reservoirs. Based on the fractal theory for porous media, an analytically comprehensive model for the relative permeabilities of oil and water in a low-permeability reservoir is established in this work. The analytical model for oil–water relative permeabilities obtained in this paper is found to be a function of water saturation, fractal dimension for pores, fractal dimension for tortuosity of capillaries, driving pressure gradient and capillary pressure between oil and water phases as well as boundary layer thickness. The present results show that the relative permeabilities of oil and water decrease with the increase of the fractal dimension for tortuosity, whereas the relative permeabilities of oil and water increase with the increase of pore fractal dimension. The nonlinear properties of low-permeability reservoirs have the prominent significances on the relative permeability of the oil phase. With the increase of the seepage resistance coefficient, the relative permeability of oil phase decreases. The proposed theoretical model has been verified by experimental data on oil–water relative permeability and compared with other conventional oil–water relative permeability models. The present results verify the reliability of the oil–water relative permeability model established in this paper.

In this paper, a novel fractal model for the invasion depth of fluid through the tortuous capillary bundle with roughened surfaces in porous media is proposed. The capillary pressure effect is considered in the proposed model. The proposed model is expressed as a function of structure parameters of porous media, including the relative roughness, the fractal dimension for pore size distribution, the contact angle, the density, the gravitational acceleration, the tortuosity and porosity. The invasion depth can be quantitatively characterized by the proposed model. By using the fractal theory, the effect of relative roughness on the invasion depth is discussed. It is observed that the invasion depth decreases with increasing the relative roughness of the tortuous capillary bundle with roughened surfaces. In addition, it is found that the invasion depth increases with the increase of the invasion time. The proposed model predictions are in good agreement with the available experimental data. Each parameter in our model has clear a distinct physical meaning, which may contribute to comprehend the better understanding of seepage mechanisms.

The analysis of extraocular muscles’ activation is crucial for understanding eye movement patterns, providing insights into oculomotor control, and contributing to advancements in fields such as vision research, neurology, and biomedical engineering. Ten subjects went through the experiments, including normal watching, blinking, upward and downward movements of eyes, and eye movements to the left and right while their electromyogram (EMG) signals were recorded. We analyzed the complexity of recorded EMG signals using fractal theory, sample entropy, and approximate entropy (ApEn). The results showed that the techniques are able to decode the changes in the complexity of EMG signals between different eye movements. In other words, we can use these methods to study extraocular muscle activations in different conditions.

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.

An important research area in physiological and sport sciences is the analysis of the variations of the muscle reaction due to changes in walking speed. In this paper, we investigated the effect of walking speed variations on leg muscle reaction by the analysis of Electromyogram (EMG) signals at different walking inclines. For this purpose, we benefited from fractal theory and sample entropy to analyze how the complexity of EMG signals changes at different walking speeds. According to the results, although fractal theory could not show a clear trend between the variations of the complexity of EMG signals and the variations of the walking speed, however, based on the results, increasing the speed of walking in the case of different inclines is mapped on to the decrement of the sample entropy of EMG signals. Therefore, sample entropy could decode the effect of walking speed on the reaction of leg muscle. This analysis method could be applied to analyze the variations of other physiological signals of humans durin walking.

Analysis of the brain activity in different mental tasks is an important area of research. We used complexity-based analysis to study the changes in brain activity in four mental tasks: relaxation, Stroop color-word, mirror image recognition, and arithmetic tasks. We used fractal theory, sample entropy, and approximate entropy to analyze the changes in electroencephalogram (EEG) signals between different tasks. Our analysis showed that by moving from relaxation to the Stroop color-word, arithmetic, and mirror image recognition tasks, the complexity of EEG signals increases, respectively, reflecting rising brain activity between these conditions. Furthermore, only the fractal theory could decode the significant changes in brain activity between different conditions. Similar analyses can be done to decode the brain activity in case of other conditions.