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The irregular geometry at all scales of measurement for thermal analysis refers to the thermal fractal models which are suitable to resolve the empirical problems related to heat transport properties. Using the fractal mathematical model of Caputo differential operator, an innovative free convection Newtonian fluid model is proposed and investigated numerically by means of fractal as well as fractional approach. The governing equation of velocity field is established via fractal and fractional domains and then tackled for solution by means of fractal Laplace transforms. The profile of velocity field for heat transfer characteristics is studied in separate fractal and fractional domains and combined fractal as well as fractional domains. The heat transfer characteristics have been emphasized under statistical analysis and mathematical expectations. The obtained numerical and statistical results demonstrate the positive impact of applying the fractal domain to enhance the heat transference process.

Soil–water characteristic curve (SWCC) and suction stress characteristic curve (SSCC), as the core of unsaturated soil mechanics, are important constitutive relationship curves for unsaturated soils, which can accurately describe the stress state in the soil. This paper examines the pore size distribution curves (PSD) of loess collected from Yan’an, China, by performing nuclear magnetic resonance (NMR) tests. In this paper, the Young–Laplace theory and the fractal model are combined with the PSD to establish the relationship between matric suction and volumetric water content. Finally, it establishes a mathematical model for predicting SWCC and SSCC. In addition, the evolution of SWCC and SSCC under different dry densities of compacted loess was studied based on the predictions. The study found that pore volumes with pore diameters smaller than the critical pore diameter could be used to predict the residual water content. Additionally, the water-holding capacity can be expressed in terms of the fractal dimension, which is controlled by the PSD. Based on the rapid characteristics of the NMR test, this paper makes a rapid prediction of SWCC and SSCC. It gives a new idea for the subsequent rapid testing of SWCC and SSCC.

Pore volume/surface area and size distribution heterogeneity are two important parameters of pore structures, which restrict the gas-water-oil migration process in sandstone reservoirs. The fractal theory has been proved to be one of the most effective methods to quantify pore distribution heterogeneity. However, the dynamic variation of porosity and permeability due to fractal characteristics has been rarely studied. In this paper, physical properties, mineral composition, and pore distribution of 18 groups of sandstone samples were analyzed using scanning electron microscope (SEM) and high-pressure mercury injection tests. Then, Sierpinski model, Menger model, thermodynamic model, and multi-fractal model were used to calculate the fractal dimension of the pore volume. Thus, the relationship between fractal dimension and porosity/permeability variation rate, and pore compressibility were studied. The results are as follows. 1) All samples can be divided into three types based on pore volume (0.9 cm 3∙g −1) and mercury removal efficiency (35%), i.e., Type A (< 0.9 cm 3∙g −1and < 35%); Type B (> 0.9 cm 3∙g −1 and <35%); Type C ( > 0.9 cm 3∙g −1 and > 35%). 2) Four fractal models had poor applicability in characterizing fractal characteristics of different sample types. The fractal dimension by the Sierpinski model had a good linear correlation with that of other models. Pores with smaller volumes dominated the overall pore distribution heterogeneity by multi-fractal dimension. The pore diameter between 200−1000 nm and larger than 1000 nm was the key pore size interval that determined the fractal characteristics. 3) With the increase of confining pressures, porosity and permeability decreased in the form of a power function. The compressibility coefficient of typical samples was 0.002−0.2 MPa −1. The compressibility of Types A and B was significantly higher than that of Type C, indicating that the total pore volume was not the key factor affecting the pore compressibility. The correlation of compressibility coefficient/porosity variation rate with pore volume (total and different size pore volume), fractal value and mineral component were not significant. This indicates that these three factors comprehensively restricted pore compression.

This paper proposes a fractal viscoelastic element via He?s fractal derivative, its properties are analyzed in details by the two-scale transform for the first time. The element is used to establish a fractal Maxwell-rheological model, which unifies the fractal creep equation and relaxation equation, and includes the classic elastic model and the classical Maxwell-rheological model as two special cases. This paper sheds a bright light on viscoelasticity, and the model can find wide applications in rock mechanics, plastic mechanics, and non-continuum mechanics.

Research on processing geochemical data and identifying geochemical anomalies has made important progress in recent decades. Fractal/multi-fractal models, compositional data analysis, and machine learning (ML) are three widely used techniques in the field of geochemical data processing. In recent years, ML has been applied to model the complex and unknown multivariate geochemical distribution and extract meaningful elemental associations related to mineralization or environmental pollution. It is expected that ML will have a more significant role in geochemical mapping with the development of big data science and artificial intelligence in the near future. In this study, state-of-the-art applications of ML in identifying geochemical anomalies were reviewed, and the advantages and disadvantages of ML for geochemical prospecting were investigated. More applications are needed to demonstrate the advantage of ML in solving complex problems in the geosciences.

It has been widely recognized that there are influences of static friction characteristics in a bolted joint on pre-tightening and service reliability. This study proposed a static friction coefficient model of joint surfaces based on the modified three-dimensional fractal model. A method of measuring the static friction coefficient of joint surfaces was investigated to verify the accuracy of model. The results showed that calculated value of the model and measured value of experiment are highly matched. Furthermore, the influential mechanisms of material, pressure, and surface topography were investigated. The relative error between experimental results and model calculations are all less than 15%. This research provides a theoretical basis for subsequent research on the friction characteristics of joint surfaces and connection reliability of bolted joints.

In the present paper, we formulate a new mathematical model for the dynamics of COVID-19 with quarantine and isolation. Initially, we provide a brief discussion on the model formulation and provide relevant mathematical results. Then, we consider the fractal-fractional derivative in Atangana–Baleanu sense, and we also generalize the model. The generalized model is used to obtain its stability results. We show that the model is locally asymptotically stable if R0<1. Further, we consider the real cases reported in China since January 11 till April 9, 2020. The reported cases have been used for obtaining the real parameters and the basic reproduction number for the given period, R0≈6.6361. The data of reported cases versus model for classical and fractal-factional order are presented. We show that the fractal-fractional order model provides the best fitting to the reported cases. The fractional mathematical model is solved by a novel numerical technique based on Newton approach, which is useful and reliable. A brief discussion on the graphical results using the novel numerical procedures are shown. Some key parameters that show significance in the disease elimination from the society are explored.

Langmuir?s linear rate equation has limited applications in the adsorption kinetics at solid/solution interface. Considering the fractal properties of adsorption surfaces, a fractal derivative model is proposed, its initial slope agrees well with Azizian-Fallah?s modified rate equation. nema

The impact-induced rock mass fragmentation in a rockfall is analyzed by comparing the in situ block size distribution (IBSD) of the rock mass detached from the cliff face and the resultant rockfall block size distribution (RBSD) of the rockfall fragments on the slope. The analysis of several inventoried rockfall events suggests that the volumes of the rockfall fragments can be characterized by a power law distribution. We propose the application of a three-parameter rockfall fractal fragmentation model (RFFM) for the transformation of the IBSD into the RBSD. A discrete fracture network model is used to simulate the discontinuity pattern of the detached rock mass and to generate the IBSD. Each block of the IBSD of the detached rock mass is an initiator. A survival rate is included to express the proportion of the unbroken blocks after the impact on the ground surface. The model was calibrated using the volume distribution of a rockfall event in Vilanova de Banat in the Cadí Sierra, Eastern Pyrenees, Spain. The RBSD was obtained directly in the field, by measuring the rock block fragments deposited on the slope. The IBSD and the RBSD were fitted by exponential and power law functions, respectively. The results show that the proposed fractal model can successfully generate the RBSD from the IBSD and indicate the model parameter values for the case study.

As China’s third-largest lead–zinc ore field, the Xicheng Ore Field has significant potential for discovering concealed deposits. In this study, a tectono-geochemical survey was conducted, and 1329 composite samples (comprising 5614 subsamples) were collected from the central part of the field. The dataset was analyzed using staged factor analysis (SFA) and concentration–area (C–A) fractal model. Four geochemical factors were extracted from centered log-ratio (CLR)-transformed data: F2-1 (Ag–Pb–Sb–Hg), F2-2 (Mo–Sb–(Zn)), F2-3 (Au–Bi), and F2-4 (W–Sn). Known Pb–Zn deposits coincide with positive F2-1 and negative F2-2 anomalies, as identified by the C–A fractal model, suggesting these factors are reliable indicators of Pb–Zn mineralization. Five Pb–Zn exploration targets were delineated. Statistical analysis and anomaly maps for F2-3 and F2-4 also indicate the potential for Au and W mineralization. Notably, some anomalies from different factors spatially overlap, indicating the possibility of epithermal Pb–Zn mineralization at shallow depths and mesothermal to hyperthermal Au and W mineralization at great depths. Overall, the integration of tectono-geochemistry, targeted and composite sampling, SFA, and C–A fractal modeling proves to be an effective and economical approach for identifying and enhancing ore-related geochemical anomalies.