Multifractal Parameters Research Articles

Multifractal analysis of geometric parameters of nanoforms formed on the surface of ultrathin layers of ZnO and ZnO–CdO synthesized by the sol-gel method

Superthin layers in ZnO and ZnO–CdO systems, which are intended for the use as transparent electrodes of modern optoelectronic devices, were synthesized by modified sol-gel method. The ranges of parameters of the technological process were established to obtain high-quality layers of material on glass substrates. Surface morphology depending on the synthesis conditions was studied by scanning electron microscopy. SEM images of the surface served as a basis for the multifractal analysis of the surface areas and volumes of nanoforms that are formed on the surface of the obtained layers. The dependences of the multifractal parameters of the surface nanoforms characterizing the surface state were determined as a function of the temperature of the final annealing of the layers. The relationships between Rainier numbers and fractal ordering parameters describing the surface geometry of the layers with the temperatures of the final annealing were established. The numerical values of the fractal characteristics of the obtained surface were used to estimate the influence of the fractal surface geometry on the value of the molar surface energy of the layers.

On the multifractal analysis of air quality index time series before and during COVID-19 partial lockdown: A case study of Shanghai, China

Due to the COVID-19 pandemic, human activities are largely restricted in Shanghai, China and it is a valuable experiment to testify the correlation of air quality and human activities. In consideration of the complexity of air pollution, this study aims to compare the multifractal characteristics of air quality index (AQI) time series before and during COVID-19 partial lockdown, and analyze the correlations between multifractal parameters of AQI time series and human activities in Shanghai, China. The hourly AQI series in Shanghai from November 27, 2019 to March 23, 2020 is used for this study. Firstly, using the MF-DFA method, the multifractal characteristics of the AQI series are explored. Secondly, the causes of the multifractality of the AQI series are determined. Finally, the correlations between multifractal parameters of AQI time series and human activities are investigated. The multifractal analysis results reveal that the AQI series during COVID-19 partial lockdown also has multifractal characteristics, and the slightly weaker multifractal characteristics and marginally smaller multifractal degree are obtained in comparison with the pre-lockdown phase. However, the contribution of the effective or intrinsic multifractality before and during COVID-19 partial lockdown are very close. The results via the sliding window procedure indicate that the multifractal parameters (ΔH,Δα,Δf) show the similar fluctuations along with the fluctuations of passenger volume in Shanghai Metro. Furthermore, it is found that ΔH,Δα and Δf and adjusted passenger volume in Shanghai Metro are positively correlated. The possible trend is that the higher adjusted passenger volume is, the larger the value of ΔH,Δα, Δf becomes, which means the stronger multifractal characteristics and larger multifractal degree of air quality system.

Evolution of the multifractal parameters along different steps of a seismic activity. The example of Canterbury 2000–2018 (New Zealand)

The multifractal detrended fluctuation algorithm is applied to a series of distances and elapsed times between consecutive earthquakes recorded along the years 2000–2018 in the Canterbury region (New Zealand). The time evolution of several multifractal parameters (Hurst exponent, Hölder central and maximum exponents, spectral amplitude, asymmetry, and complexity index) is analyzed. Peaks of multifractal parameters, with statistical significance exceeding 95%, are associated with three earthquakes of notable magnitude (equaling or exceeding Mw = 5.7). Additionally, some other peaks are also associated with the swarm of earthquakes of moderate magnitude. Possible shortcomings created by this assignment to mainshocks or swarms can be removed by comparing the results corresponding to elapsed times and interevent distances between consecutive events. Additionally, the Buishand test, which is used to verify the statistical significance of the detected peaks, also discriminates between mainshocks of notable seismic magnitude and swarms of earthquakes with moderate magnitude. The obtained results, based on the multifractal structure of the seismic activity, could also represent some advances in predicting, with sufficient time, forthcoming mainshocks of high magnitude and mitigate their destructive effects.

Temporal evolution of multifractality in the Madrid Metro subway network

Abstract Temporal evolution of the (multi-)fractality in the Madrid Metro subway (Spain) is explored by using it as a study case of a public transport network (PTN). By considering this subway as a graph network, a box-covering method for multifractal analysis of complex networks is employed for assessing the evolution of multifractal parameters along time from two perspectives. On the one hand, the development of the topological structure of the subway (unweighted network), and on the other, the influence of the travelling times in its multifractal properties (weighted network). Multifractal results are in agreement with the evolution of the system, which begins with the monofractal structure of a one-dimensional solely line, but increasing in complexity until 1980s-1990s. Nowadays, the system is slightly multifractal due to certain heterogeneity in node density distribution. Scaling behaviour in the current Madrid Metro subway is evaluated by means of the generation of synthetic networks, in which topology is kept but travelling times are reassigned. Additionally, multifractality is also evaluated by studying the effect of the commuter railway system and the deletion of the circular line in the network.

Comparative Evaluation of Pore Structure Heterogeneity in Low-Permeability Tight Sandstones Using Different Fractal Models Based on NMR Technology: A Case Study of Benxi Formation in the Central Ordos Basin

Based on the NMR T2 spectra of all of the samples, the applicability of the single fractal model and the multifractal model for the quantitative investigation of pore diameter distribution heterogeneity is analyzed. Then, the relationships between pore structures, physical properties, and fractal dimensions calculated by each model are discussed. The following results have been achieved. (1) Model 1 can be used to study the heterogeneity of different fluid states (including saturation and bound water) or partial pore size distribution, and the calculated fractal dimension has the best correlation with permeability. This model is mainly suitable for type I samples with macropores developed. (2) Models 2, 3, and 4 can quantitatively analyze the heterogeneity of the overall pore diameter distribution. These calculated fractal dimensions have a good linear relationship with porosity, T2cutoff, T2gm, and other physical parameters, which are mainly suitable for type II, III, and IV samples with micropores developed. (3) There is a negative correlation between multifractal dimensions calculated by model 4 and single fractal parameters calculated by models 2 and 3. Micropores and macropores are the key pore sizes that affect the variation of single fractal parameters, while mesopores are the key size affecting the variation of multifractal parameters. (4) The selection of a reasonable T2 spectrum range has become a key to determine the calculation accuracy of models 2 and 3. Above all, models 1 and 4 should be given priority in characterizing pore distribution heterogeneity of tight sandstone gas reservoirs.

Multifractal analysis of coal pore structure based on NMR experiment: A new method for predicting T2 cutoff value

Nuclear magnetic resonance (NMR) is a commonly used method to quantitatively characterize the pore structure of coal. T2 cutoff value is closely related to the microscopic pore structure of coal reservoirs, which directly affects the calculation of porosity, permeability, movable and irreducible fluid saturation. In this study, the pore structure of four types of coal samples were studied based on NMR experiments and the multifractal characteristics of saturated coal samples were analyzed with multifractal theory. Furthermore, the correlation between T2 cutoff values and multifractal characteristic parameters were discussed. The results show that the pore structure characteristics of CZ, GD and GHS coal samples are basically the same, all of which are mainly micropores, with relatively less content of mesopores, macropores and microcracks, while PDS coal samples have relatively more content of mesopores, macropores and microcracks. Different types of coal samples have similar multifractal characteristics and all coal samples are dominated by micropores. There is an obvious linear relationship between T2 cutoff values and the multifractal characteristic parameters. Five models of T2 cutoff values changing with the multifractal characteristic parameters are obtained by linear fitting. The error range between T2 cutoff values calculated by the five models and the experimental values are within 0.1 ms, among which T2 cutoff values calculated by model 5 are highly consistent with the experimental values, indicating that model 5 can better predict T2 cutoff values. The results can provide theoretical reference for the characterization of coal pore structure and the prediction of T2 cutoff values.

Three-dimensional fractal characteristics of soil pore structure and their relationships with hydraulic parameters in biochar-amended saline soil

Due to poor soil quality, coastal saline soil limits soil productivity. Therefore, it is of great importance to find appropriate and efficient methods to improve the quality of coastal saline soil. Biochar has been widely used as a soil amendment to improve soil properties, but the effects of biochar amendment on soil pore structure and soil hydraulic properties are unclear. In recent years, with the development of fractal theory and computer technology, fractal geometry based on image analysis has been employed for quantitatively characterizing the structure and heterogeneity of soil pores. In this study, field experiments were established with three proportions of straw biochar (0, 2 and 5 %, w/w) to study the biochar amendment effects on soil structure and soil hydraulic properties. Three replicates were set for each treatment. The results showed that the addition of biochar to saline soil improved soil pore structures and soil hydraulic characteristics. The biochar amendment led to the strong pore spatial variation, complex pore structure and irregular pore shape. The biochar-amended saline soil had a high considerable water retention capacity (θS, AWC). Biochar had a long-term effect on saline soil improvement. For multifractal analysis, it is better to use the ΔD and Δα values to analyze the multifractal characteristics of soil. The heterogeneity of the soil pore distribution became stronger after biochar amendment. Multifractal parameters had significant changes with the time of biochar amendment. The changes of soil pore structure resulted in the improvement of soil hydraulic characteristics. The Ks value improved as the pore connectivity increased. The strong pore spatial variation, complex pore structure and irregular pore shape aided in increasing the θs value. The available water content value was improved as the number of soil pores, especially the micropores, increased. In response, this study proposes a new approach to analyzing soil pore structure to explain the association between the soil pore structure and soil hydraulic characteristics in biochar-amended saline soil.

Do reconstructed landforms have multifractal characteristics? A case study at an opencast coalmine dump in the loess area of China

Opencast coal mining causes large changes in landform. The geomorphological features of the reconstructed landform have not been fully understood in the loess area. To quantify the characteristics of the reconstructed landform, multifractal analysis and Principal Component Analysis (PCA) were applied to classify three soil dumps, located at the Antaibao opencast coalmine in China. Most parts of the reconstructed soil dumps had multifractal characteristics. Multifractal parameters reflected heterogeneity of the reconstructed landform from surface relief, regularity and variance in elevation, and the results of PCA further provided a good explanation of the differences and similarities among the three soil dumps. Sub-areas of the South Dump (S1, S2, S3, and S4) were clustered together and this grouping was explained by relatively high D(0), ɑmax and D(q)min, whereas sub-areas of the remaining dumps were sparsely distributed. Long-term restoration and soil erosion may cause the homogenization of the reconstructed landform. Moreover, multifractal parameters, D(q)min, D(0), D(1), D(2), ɑmin, and f(ɑmin) were significantly correlated with Normalized Difference Vegetation Index (NDVI), suggesting susceptibility of the restored vegetation to landform development. Therefore, to improve revegetation practice, micro-topographic measures and silviculture technology are both needed in post-mining areas. The micro-geomorphological patterns could not only improve the efficiency of water use, but also provide a basis for the evolution of geomorphological research in the loess area of China.

Multi-scaling Properties of 2D Reservoir Micro-pore Heterogeneity Based on Digital Casting Thin-Section Images

In recent years, in order to study the micro-pore structures of reservoirs, to define the types of reservoir space and to classify and evaluate reservoir capabilities, the technology of numerical image analysis has been concerned widely with the study of pore heterogeneity with fractals and multifractals. This is of great challenge due to developments in image resolutions and quantitative characterization methods. In this paper, the pore structure of digital casting thin-section images of six samples of volcanic and volcano-sedimentary rocks from the Dagang area (Huanghua depression, China) was studied, and then the pore spaces of the reservoir samples were characterized automatically using multifractal parameters advantageous to describe the irregularity and disorder of geometric objects. The method of moments was used to derive the parameters of multifractality and heterogeneity of pore structures. The results show well that, together with cluster analysis, 12 fractal and multifractal parameters can be applied to describe effectively the irregular degrees of pore spaces. The six rock samples have been classified into three groups with various reservoir capabilities. The work presented here can provide new avenues for analysis of the spatial arrangement of mineral particles, aggregates and pores in spaces, and that the multi-scaling nature of casting thin-section images of rocks suggests using these algorithms as a basis for reservoir capability assessment.

Multifractal Analysis of Pore Structure and Evaluation of Deep-Buried Cambrian Dolomite Reservoir with Image Processing: A Case from Tarim Basin, NW China

Reservoir pore space assessment is of great significance for petroleum exploration and production. However, it is difficult to describe the pore characteristics of deep-buried dolomite reservoirs with the traditional linear method because these rocks have undergone strong modification by tectonic activity and diagenesis and show significant pore space heterogeneity. In this study, 38 dolostone samples from 4 Cambrian formations of Tarim Basin in NW China were collected and 135 thin section images were analyzed. Multifractal theory was used for evaluation of pore space heterogeneity in deep-buried dolostone based on thin section image analysis. The physical parameters, pore structure parameters, and multifractal characteristic parameters were obtained from the digital images. Then, the relationships between lithology and these parameters were discussed. In addition, the pore structure was classified into four categories using K-means clustering analysis based on multifractal parameters. The results show that the multifractal phenomenon generally exists in the pore space of deep-buried dolomite and that multifractal analysis can be used to characterize the heterogeneity of pore space in deep-buried dolomite. For these samples, multifractal parameters, such as αmin, αmax, ΔαL, ΔαR, Δf, and AI, correlate strongly with porosity but only slightly with permeability. However, the parameter Δα, which is usually used to reveal heterogeneity, does not show an obvious link with petrophysical properties. Of dolomites with different fabrics, fine crystalline dolomite and medium crystalline dolomite show the best petrophysical properties and show significant differences in multifractal parameters compared to other dolomites. More accurate porosity estimations were obtained with the multifractal generalized fractal dimension, which provides a new method for porosity prediction. The various categories derived from the K-means clustering analysis of multifractal parameters show distinct differences in petrophysical properties. This proves that reservoir evaluation and pore structure classification can be accurately performed with the K-means clustering analysis method based on multifractal parameters of pore space in deep-buried dolomite reservoirs.

Linking deforestation patterns to soil types: A multifractal approach

AbstractIn the three last decades, continental Ecuador biosphere reserves (CEBRs) have undergone important land cover transformations because of human‐managed systems such as agriculture, livestock, forestry, and urbanization. Thus, the native landscape structure has been altered, exhibiting a mosaic of patches with varying sizes and shapes. The resulting landscape could shape some patterns over time, reflecting the complex interactions between human activities and biotic‐abiotic factors, such as soil, climate, and water availability. Quantifying the landscape patterns and their dynamics is essential for the monitoring and assessment of the ecological consequences of land‐cover change in time. Based on the soil maps and raster land‐cover maps from 1990 to 2016 of two CEBRs, the spatial deforestation patterns were quantified through lacunarity and multifractal analysis. To conduct these analyses, the deforested areas were identified and overlapped to predominant soil types. The results showed that the deforestation process pattern and soil types interact through complex relations due to specific site characteristics. Thereby, both soil and land cover interacted in fractal structures. The patch area distribution of deforestation patterns linked to soil types reveals a power‐law relationship in which, as the area of the deforested patches increased, their frequency decreased. Also, this relation presents self‐similarity and multifractal behaviour over a wide range of scales from 46–2,674 pixels, with a pixel size of 30 m × 30 m. However, each CEBR showed varying degrees of heterogeneity and complexity, which were reflected through lacunarity and multifractal parameters. The dynamics of deforestation patterns linked to soil types can be spatially dependent on the initial patchiness state, the distance to the adjacent patches, and the suitability of the soil for agrarian activities. Altogether, these were the spatial underlying factors of the deforestation evolution in the two CEBRs.Highlights Deforestation patterns show spatial self‐similarity Deforestation patterns are linked to soil types Landscape ecology, lacunarity, and multifractal analyses are useful to understand deforestation patterns Initial patch location, distance to the nearest neighbour and soil suitability boost the deforestation expansion

Multifractal characterisation of particulate matter ([formula omitted]) time series in the Caribbean basin using visibility graphs

A good knowledge of pollutant time series behavior is fundamental to elaborate strategies and construct tools to protect human health. In Caribbean area, air quality is frequently deteriorated by the transport of African dust. In the literature, it is well known that exposure to particulate matter with an aerodynamic diameter of 10 μm or less (PM10) have many adverse health effects as respiratory and cardiovascular diseases. To our knowledge, no study has yet performed an analysis of PM10 time series using complex network framework. In this study, the so-called Visibility Graph (VG) method is used to describe PM10 dynamics in Guadeloupe archipelago with a database of 11 years. Firstly, the fractal nature of PM10 time series is highlighted using degree distribution for all data, low dust season (October to April) and high dust season (May to September). Thereafter, a profound description of PM10 time series dynamics is made using multifractal analysis through two approaches, i.e. Rényi and singularity spectra. Achieved results are consistent with PM10 behavior in the Caribbean basin. Both methods showed a higher multifractality degree during the low dust season. In addition, multifractal parameters exhibited that the low dust season has the higher recurrence and the lower uniformity degrees. Lastly, centrality measures (degree, closeness and betweenness) highlighted PM10 dynamics through the year with a decay of centrality values during the high dust season. To conclude, all these results clearly showed that VG is a robust tool to describe times series properties.

Dynamics of meteorological time series on the base of ground measurements and retrospective data from MERRA‐2 for Poland

AbstractA comparison study has been performed to assess the dynamics of meteorological processes in Poland on the basis of meteorological time series of air pressure, air temperature and wind speed coming from 35 synoptic stations belonging to the Institute of Meteorology and Water Management—National Research Institute (IMGW‐PIB) and from the nearest grid points of the Modern‐Era Retrospective Analysis for Research and Applications, Version 2 (MERRA‐2) produced at NASA's Global Modelling and Assimilation Office from January 1, 2007 to October 31, 2016. Apart from comparative statistics, the differences in the multifractal properties of the time series were evaluated with the use of MultiFractal Detrended Fluctuation Analysis (MFDFA), both for hourly and daily data, showing a high degree of similarity between the MERRA‐2 and IMGW‐PIB series. For the air pressure and air temperature, not only were high determination coefficients (close to .99) between the time series coming from the two sources noticed, but there were also similarities with the MFDFA parameters. Lower correlations between the time series of the wind speed obtained from the two studied databases were observed, which was related to differences in the data structure and methodology of the measurements for specific IMGW‐PIB stations. Additionally, to verify data similarities coming from the IMGW‐PIB and MERRA‐2 databases, the correlations between specific multifractal parameters and the orography were estimated and compared. For the air pressure and temperature, a remarkably high correlation was found between the multifractal parameter α0 and the height above sea level of the measurement site. An analysis of the source of multifractality was performed, indicating that, for all studied meteorological elements and both data sources, the long‐range correlations prevail.

Tool condition monitoring in milling process using multifractal detrended fluctuation analysis and support vector machine

Tool wear will lead to the reduction of surface quality and machining accuracy. Therefore, tool condition monitoring is vital to the improvement of industrial production efficiency and quality. In this paper, first of all, the internal mechanism of the milling process and the performance characteristics of the milling signals were analyzed. It is found that there are different trends between the large and small fluctuations of milling signals in the process of tool wear increasing. This property can be characterized by various parameters of multifractal spectrum to establish the relationship between tool wear and multifractal parameters. By analyzing the changes of multifractal spectrum parameters, the tool wear monitoring can be realized. Then, the multifractal detrended fluctuation analysis (MFDFA) method is used to calculate the mean square error, generalized Hurst exponent, and multifractal spectrum parameters, which are the eigenvectors, and establish its relationship with tool wear. Finally, the tool condition diagnosis is conducted by a support vector machine (SVM). The results show that the tool condition monitoring method of MFDFA combined with SVM is proved to be effective and the multifractal parameters of MFDFA are very sensitive to tool wear.

Multifractal characteristics of shale and tight sandstone pore structures with nitrogen adsorption and nuclear magnetic resonance

Based on the experiments of nitrogen gas adsorption (N2GA) and nuclear magnetic resonance (NMR), the multifractal characteristics of pore structures in shale and tight sandstone from the Chang 7 member of Triassic Yanchang Formation in Ordos Basin, NW China, are investigated. The multifractal spectra obtained from N2GA and NMR are analyzed with pore throat structure parameters. The results show that the pore size distributions obtained from N2GA and NMR are different, and the obtained multifractal characteristics vary from each other. The specific surface and total pore volume obtained by N2GA experiment have correlations with multifractal characteristics. For the core samples with the similar specific surface, the value of the deviation of multifractal spectra Rd increases with the increase in the proportion of large pores. When the proportion of macropores is small, the Rd value will increase with the increase in specific surface. The multifractal characteristics of pore structures are influenced by specific surface area, average pore size and adsorption volume measured from N2GA experiment. The multifractal characteristic parameters of tight sandstone measured from NMR spectra are larger than those of shale, which may be caused by the differences in pore size distribution and porosity of shale and tight sandstone.

Relationship between granitic soil particle-size distribution and shrinkage properties based on multifractal method

The mechanical properties of granitic residual soils vary with depth due to changes in soil type and heterogeneity caused by weathering. The purpose of this study was to relate the spatial variation of particle-size distribution (PSD) of granitic soils with soil shrinkage parameters using multifractal theory. The heterogeneity of PSD and pedogenic processes were depicted in detail by multifractal dimensions. The PSD generally increased with the increase of profile depth in accordance with the variation of single fractal dimension (D) ranging from 2.45 to 2.65. The shrinkage limit was greatly influenced by the multifractal dimension parameters, including information dimension (D1) and capacity dimension (D0) (Adjusted R2 = 0.998, P < 0.01), and the maximum linear extensibility (κν) was determined by spectral width (Δα) and bulk density, with the latter explaining 89% of the total variance of κν (P < 0.01). Soil shrinkage characteristic curve was fitted by the modified logistic model (R2 > 0.97, root sum of squares < 0.1), and the water variation corresponding to the maximum change rate of linear extensibility was determined by the silt content (R2 = 0.81, P < 0.01). Overall, the shrinkage of granitic soils was primarily influenced by PSD and soil compactness.

Investigation of the Pore Structure of Tight Sandstone Based on Multifractal Analysis from NMR Measurement: A Case from the Lower Permian Taiyuan Formation in the Southern North China Basin

Understanding the pore structure can help us acquire a deep insight into the fluid transport properties and storage capacity of tight sandstone reservoirs. In this work, a series of methods, including X-ray diffraction (XRD) analysis, casting thin sections, scanning electron microscope (SEM), nuclear magnetic resonance (NMR) experiment and multifractal theory were employed to investigate the pore structure and multifractal characteristics of tight sandstones from the Taiyuan Formation in the southern North China Basin. The relationships between petrophysical properties, pore structure, mineral compositions and NMR multifractal parameters were also discussed. Results show that the tight sandstones are characterized by complex and heterogenous pore structure, with apparent multifractal features. The main pore types include clay-dominated micropores and inter- and intragranular dissolution pores. Multifractal parameters of sandstone samples were acquired by NMR and applied to quantitatively describe the pore heterogeneity in higher and lower probability density regions (with respect to small and large pore-scale pore system, respectively). The multifractal parameter (D−10) of lower probability density areas has better correlation with the petrophysical parameters, which is more suitable for evaluating the reservoir properties of tight sandstone. However, the multifractal parameter (D10) of higher probability density areas is more conducive to characterize the pore structure of tight sandstone. Additionally, the mineral compositions of sandstone have a complex effect on multifractal characteristics of pores in different probability density areas. The D10 increases with the decrease of quartz content and increase in clay mineral content, whereas D−10 decreases with the increase in clay minerals and decrease of authigenic quartz content and feldspar content.

Morphology and multifractal characteristics of soil pores and their functional implication

As an important inherentcondition for an eroded landform and fragile ecosystem, the erodible loess in China’s Loess Plateau is a clastic and predominantly silt-sized sediment. Understanding the pore complexity of the loess may be crucial for exploring soil functional properties and improving knowledge on hydrological dynamic processes occurring in the soil plant atmosphere continuum. Twelve undisturbed columns of four typical soil types, namely, aeolian sandy, light dark loessial, loessial, and Lou soils, were collected for the multifractal analysis of X-ray computed tomography images. The heterogeneity of soil pore distribution and the measure of porosity dispersion relative to the pore size dispersion in the loessial and Lou soils were significantly lower than those of the aeolian sandy and light dark loessial soils. The loessial soil exhibited the highest symmetry of singularity spectra and presented the lowest variability for pore arrangement. Most correlations among parameters of soil properties, multifractal characteristics, as well as soil macropore morphology and connectivity characteristics, were significant (p < 0.05 or 0.01). The multiple regression models of the theoretical saturated hydraulic conductivity and total porosity incorporated all types of the parameters, and the multifractal parameter ΔD was the dominant variable. Macropore connectivity parameters were commonly superior to the morphology ones when the links between macropore characteristics and multifractal parameters were considered. Meantime, the number of macropores acted as a crucial parameter influencing the complexity of soil pore structures. Overall, it is of great significance to integrate the soil pore complexity into the comprehensive understanding of pore function. This study also provides an effective reference for optimizing soil structures, improving the soil water holding capacity, and balancing the relationship between water and ecosystems in the Loess Plateau.

Multifractal Analysis in Characterizing Adsorption Pore Heterogeneity of Middle- and High-Rank Coal Reservoirs.

Nanopore heterogeneity has a significant effect on adsorption, desorption, and diffusion processes of coalbed methane. The adsorption pore size distribution heterogeneity was calculated by combining N2 with CO2 adsorption data, and factors affecting multifractal and single-fractal dimensions were studied. The results indicate that pore size distribution of micropores (with pore diameters smaller than 2 nm) and meso–macro-pores (with pore diameters between 2 and 100 nm) in coal samples exhibit typical multifractal behavior. The overall heterogeneity of micropores in high-rank coal samples is higher than that in the middle-rank coal samples. The low-probability measure areas control the overall heterogeneity of pores with diameters of 0.40–1.50 nm. The high-probability measure area heterogeneity and spectral width ratio have a higher linear correlation with coal rank and pore structure parameters than those of low-probability measure areas. Heterogeneity of high-probability measure areas and overall pore size distribution are controlled by pores with diameters of 0.72–0.94 nm. Multifractal parameters of meso–macro-pores have no clear relationship with coal rank. The pore volume of 2–10 nm diameter shows a good linear correlation with heterogeneity of low-probability measure areas, and pores of this diameter range are the key interval that affected pore size distribution heterogeneity. The single-fractal dimension obtained using the Frenkel–Halsey–Hill (FHH) model shows a positive linear correlation with heterogeneity of the low-probability measure areas. It indicates that this parameter can effectively characterize the pore size distribution heterogeneity of low-probability measure areas in meso–macro-pores.

EXPERIMENTAL INVESTIGATION ON MULTI-FRACTAL CHARACTERISTICS OF ACOUSTIC EMISSION OF COAL SAMPLES SUBJECTED TO TRUE TRIAXIAL LOADING–UNLOADING

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 [Formula: see text] and [Formula: see text] 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 [Formula: see text] and [Formula: see text] can reflect the stress and damage degree of coal samples. The dynamic change process of [Formula: see text] well accords with the damage evolution process of coal samples. A gradual decrease of [Formula: see text] 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 [Formula: see text] mutation point. The change of [Formula: see text] 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.