Fractals

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.

The permeability coefficient and air-entry value of saturated soil are important hydraulic properties, which play an important role in engineering applications. Subsoil supporting foundation is subjected to stress and undergoes deformation; the saturated permeability coefficient of such deformed soil is of practical importance. With the help of fractal theory, based on the different fractal forms of Tao–Kong model, CCG model, Mualem model, and soil–water characteristic curve, this paper derives the saturated permeability coefficient models under four deformation conditions, considering the saturation permeability coefficient is inversely proportional to the square of the air-entry value and directly proportional to the square of the maximum aperture of the soil. Combining the prediction method of air-entry value under deformation conditions, four prediction methods for the permeability coefficient of deformed saturated soil are established and the method proposed in this paper is validated by the measured value of the saturated permeability coefficient of the deformed soil. As observed, the predicted values of the four methods for clay, silt loam, sandy loam and sandy soil under deformed conditions are in acceptable agreement with the measured values, and the prediction results of the second prediction method are the closest to the measured values.

The fractal characteristic of cement paste has been investigated for decades. In this paper, a systematic study referring to analysis, modeling as well as application is presented with respect to the pore size-dependent fractality of the porous structure in cement paste. In particular, the multiscale fractal dimensions other than the traditional single fractal dimension are analyzed for a measure of incomplete fractality. Then, the multiscale solid mass fractal model is introduced and implemented. It is demonstrated that the modeled pore size distribution agrees well with that obtained from the experimental measurement. Moreover, in combination with other analytical (the effective medium theory) and/or numerical (the finite element method) tools, such a pore size-dependent fractality could provide a fast assessment of some basic mechanical and physical properties.

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.

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.

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.

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.

The numerical methods such as the artificial neural networks with greater probability and nonlinear configurations are more suitable for estimation and modeling of the problem parameters. The numerical methods are easy to use in applications as these methods do not require costly and time-consuming tests like the experimental study. In this study, we use the Levenberg–Marquardt-based backpropagation Process (LMP) to create a computing paradigm that makes use of the strength of artificial neural networks (ANN), known as (ANN-LMP). Here we use the ANN-LMP to obtain the solution of the second-grade fluid in a rotating frame in a porous material with the impact of a transverse magnetic field. The 1000 data set points in the interval $[0,1]$ are used for the network training to determine the effect of various physical parameters of the flow problem under consideration. The experiment is executed of six scenarios with different physical paramaters. ANN-LMP is used for evaluating the mean square errors (MSE), training (TR), validation (VL), testing (TT), performance (PF) and fitting (FT) of the data. The problem has been verified by error histograms (EH) and regression (RG) measurements, which show high consistency with observed solutions with accuracy ranging from E-5 to E-8. Characteristics of various concerned parameters on the velocity and temperature profiles are studied.

With the increasing demand for energy, heat and mass transfer through porous media has been widely studied. To achieve accuracy in studying the behavior of heat transfer, a good knowledge of the effective thermal conductivity (ETC) of porous materials is needed. Because pore structure dominates the ETC of porous materials and effective stress leads to a change in pore structure, effective stress is one of the key influencing factors affecting ETC. In this study, considering the structure of surface roughness and pore size, based on fractal theory, a novel analytical solution at the pore scale for ETC of porous materials under stress conditions is proposed. Furthermore, in this model, capillaries in porous materials saturated with multiple phases have sinusoidal periodically constricted boundaries. The derived ETC model is validated against available experimental data. Moreover, the influences of the effective stress, initial effective porosity, roughness structure characterization, and wetting phase saturation on the ETC are analyzed. Compared with previous models, the rough surfaces of porous materials and the coupling of heat conduction and mechanics are taken into consideration to make the model more reasonable. As a result, this ETC model can better reveal the mechanism of heat conduction in porous media under stress conditions.

The unsaturated permeability coefficient is of importance to study the behavior of the water seepage and contaminant transport in unsaturated soils. The direct measurement of the unsaturated permeability coefficient is time consuming and laborious. Therefore, indirect approaches are usually applied to obtain the unsaturated permeability coefficient. However, indirect methods still have some drawbacks. An indirect method is usually fitted through the soil–water characteristic curve of a special type of soil, which also requires a large workload and its application is limited. To overcome these drawbacks, a fractal model of the relative permeability coefficient of unsaturated soils has been deduced in this study based on the relation between clay contents and fractal dimensions in terms of mass. A new fractal simple prediction method of unsaturated soil absolute permeability coefficient associated with the new fractal model has also been proposed by combining the simple relation between saturated permeability coefficient and air-entry value. The method is validated with the experimental data of existing literature, and the result shows that the proposed model has a higher prediction accuracy.

Unconventional gas is a momentous energy source due to its considerable reserves and eco-friendly properties, where relative permeability is a key evaluative parameter of unconventional gas extraction. However, the geo-complexity, multiscale and multieffect of the unconventional gas reservoir challenge the relative permeability evaluation and production enhancement. Here, we establish a gas–water flow model by integrating multifractal theory, covering from nanoscale to macroscale and regarding the effects of slip, gas desorption–diffusion and water film separation, to reliably evaluate the relative permeability evolution during unconventional gas development. Based on our model, we describe the permeability of the unconventional reservoir with an 88% less evaluation error compared to the single fractal Darcy with the literature benchmark. Moreover, we characterize the gas–water relative permeability with a no more than 10% evaluation error based on the experimental data. The slip effect plays the most crucial role in the evaluation precision of relative permeability. We reveal that the permeability of the unconventional gas reservoir is decreased by the increase of generalized fractal dimension which enhances the heterogeneity and tortuosity of pores. We uncover that the slip effect facilitates the relative permeability of gas and water. Besides, the gas desorption–diffusion boosts gas relative permeability while limiting water relative permeability, whereas water film separation enhances water relative permeability but hinders gas relative permeability. This work brings insights into the precise description of multiscale and multieffect gas–water porous flow in unconventional gas development.

Due to a great majority of lung cancer patients dying within one year after being diagnosed with apparent symptoms, developing a diagnostic/monitoring technique for early-stage lung cancer is in critical demand. Conventionally, lung cancer diagnostic approaches are costly, and they increase the health risks caused by invasiveness and radiation hazards. In this work, a new diagnostic technique using aerosol fingerprints in the breath test is explored based on computational fluid dynamics (CFD) modeling and fractal analysis. At first, the three-dimensional symmetrical human lung model is constructed by Solidworks. Then, the large eddy simulation-discrete-phase model (LES-DPM) approach is used in CFD modeling to model the airflow pattern and aerosol behaviors in the human lung model. After that, the box-counting method is employed in fractal analysis to calculate the fractal dimension of exhaled aerosol patterns in the human lung model. Finally, the fractal distributions of exhaled aerosols in the breath test at different respiratory intensities and aerosol-releasing positions are specifically investigated. The results show that the coupled CFD modeling and fractal analysis is a reliable method for deciphering the complexity of exhaled fingerprints which is shown to quantify and differentiate the exhaled aerosol patterns adequately. Meanwhile, the exhaled aerosol fingerprints are found to be relevant to the respiratory intensity, and the distribution of exhaled aerosols exhibits a unique pattern at different respiratory intensities. Therefore, the abnormal respiration of patients, which hints at the severity of lung cancer, can be judged by aerosol fingerprints and fractal dimensions. In addition, the exhaled aerosol fingerprints are associated with the aerosol-releasing positions. Different aerosol-releasing positions lead to different aerosol distribution patterns. It is feasible to locate the site of lung cancer by judging the aerosol fingerprints and fractal dimension. This study is helpful in determining the respiratory abnormalities caused by lung cancer and diagnosing the location where the lung cancer occurs.

Insufficient consideration of the complex morphology of hydraulic fractures (HF) and heterogeneous physical properties of fractured reservoirs in seepage models can result in unreliable well testing analyses. The fractal porosity and permeability (FPP) model provides an effective method for characterizing reservoir heterogeneity in the near-wellbore zone. However, its application to scenarios involving irregularly-shaped hydraulic fracture networks and multiple fracture clusters is challenging due to the lack of spatial symmetry. To address this issue, this paper proposes a combined approach of FPP and conformal mapping (FPP-CM) to transform the region of fractured formation into the exterior of the unit disk domain using numerical conformal mapping. The transient seepage flow model of the vertical fracture well (VFW) is then established by coupling it with the FPP model. The typical curve of pressure transient behavior with the division of flow stages was plotted, and the model verification and sensitivity analysis of parameters were conducted. The results indicate that the fractal dimension primarily affects the formation linear flow stage and its subsequent flow stages; with a decrease in fractal dimension resulting in an increase in the position of the typical curve. For VFW with multiple HF wings, a decrease in the included angle of fracture wings causes an increase in the heterogeneity distribution of microfracture physical properties, resulting in an increase in the position of the pseudo-pressure derivative curve during the late flow stage.

In this paper, we study the existence of numerical solution and stability of a chemostat model under fractal-fractional order derivative. First, we investigate the positivity and roundedness of the solution of the considered system. Second, we find the existence of a solution of the considered system by employing the Banach and Schauder fixed-point theorems. Furthermore, we obtain a sufficient condition that allows the existence of the stabling of solutions by using the numerical-functional analysis. We find that the proposed system exists as a unique positive solution that obeys the criteria of Ulam–Hyers (U-H) and generalized U-H stability. We also establish a numerical analysis for the proposed system by using a numerical scheme based on the Lagrange interpolation procedure. Finally, we provide two numerical examples to verify the correctness of the theoretical results. We remark that the structure described by the considered model is also sometimes called side capacity or cross-flow model. The structure considered here can be also seen as a limiting case of the pattern chemostats in parallel with diffusion connection. Moreover, the said model forms in natural and engineered systems and can significantly affect the hydrodynamics in porous media. Fractal calculus is an excellent tool to discuss fractal characteristics of porous media and the characteristic method of the porous media.

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.

This study focuses on the analytical and numerical solutions of the convexity analysis for fractional differences with exponential and Mittag-Leffler kernels involving negative and nonnegative lower bounds. In the analytical part of the paper, we will give a new formula for ${\nabla }^2$ of the discrete fractional differences, which can be useful to obtain the convexity results. The correlation between the nonnegativity and negativity of both of the discrete fractional differences,

The in-situ foam technology has been extensively applied in the complex reservoir reconstruction since it improves the sweep efficiency by diverting the flow of injected fluids into areas with lower permeability and as a result enhances the oil recovery. The in-situ foam structure inside the pores can significantly affect the sweep efficiency, however, quantitative characterizations on foam structure are inadequate. Here, we propose a quantitative method based on fractal theory and the two-dimensional (2D) micro physical simulation experiment for the study of fractal characteristic, evolution law and sensitivity analysis. The findings demonstrate that foam confined within porous media exhibits fractal characteristics, as evidenced by the measured box-counting fractal dimensions ranging between 1.05 and 1.752 based on acquired structural images. Notably, a higher fractal dimension corresponds to a more irregular in-situ foam structure. Besides, in-situ foam in the porous media presents the “quasi check sign” evolution law, which can be divided into three time-dependent stages. Moreover, the evolution laws of in-situ foam within porous media remains consistent across varying temperatures and concentrations of foaming agents, and increasing temperature and decreasing concentration can shorten the time to reach the inflection point.

Tree-like branching structures occur in both natural and artificial transport systems, which have fascinated multidisciplinary researchers to study and apply the transport mechanisms of tree-like branching structures for decades. In this paper, the flow field of a proton exchange membrane fuel cell (PEMFC) with a Y-shaped tree-like fractal structure was studied utilizing Murray’s law in fractal theory. The polarization curve, gas concentration distribution, water concentration distribution, pressure drop distribution, and current density distribution of the PEMFC are numerically simulated in this research, and the transmission law is analyzed with different channel branching angles $(70^{\circ },75^{\circ },80^{\circ })$. The results demonstrate that the optimal branching angle of the designed tree-like fractal flow field is $75^{\circ }$. When compared to a parallel flow field under identical conditions, the maximum output power density of the fractal flow field with a branching angle of $75^{\circ }$ is 26.7% higher. The optimal angle of $75^{\circ }$ for symmetric branching flow derived from Murray’s law was shown to be applicable to the flow field design of the fuel cell, improving the transport characteristics of the reaction gases and the overall performance of the PEMFC. This research may provide further references for the design of flow fields in fuel cells.

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.