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An analytical expression for the fractal dimension for tortuous streamtubes in porous media is derived and found to be a function of porosity and microstructures of porous media. There is no empirical constant in the proposed fractal dimension. The present model for the fractal dimension is verified by a comparison with the available analogy model.

The tortuosity is a very important parameter for description of fluid flow in porous media, and it has been shown that porous media in nature have the fractal characteristics. The Sierpinski carpet is an exactly self-similar fractal model, which is often used to simulate fractal porous media. In this work, the tortuosity of different generations of Sierpinski carpet is calculated and analyzed by the finite volume method. A simple linear relation between the generations and tortuosity in pore fractal model of porous media is obtained. The results are compared with the available conclusions and show a more realistic tortuosity predication for fluid flow in the two-dimensional pore fractal models of porous media.

In this work, we have given an analogical method for estimating the fractal dimension for three-dimensional fracture tortuosity (3D-FT). The comparison and error analysis of analogical and rigorous methods on fractal dimension for 3D-FT were carried out in this work. The fractal dimension $D_{T-R}$ for 3D-FT from the proposed analogical method is the function of 3D fracture average tortuosity (${\tau }_{\mathrm{\text{av}}})$ and average fracture length ($L_{\mathrm{\text{av}}})$. The analogical method for estimating fractal dimension ($D_{T-A})$ with high accuracy indicates good consistency with the rigorous method ($D_{T-R})$. The fractal dimension ($D_{T-R})$ from the rigorous method is the embodiment of the physical meaning of $D_{T-R}$. The fractal dimension ($D_{T-A})$ from the analogical method is relatively convenient for calculating the premise of ensuring accuracy.

Hydraulic tortuosity is one of the key parameters for evaluating effective transport properties of natural and artificial porous media. A pore-scale model is developed for fluid flow through porous media based on fractal geometry, and a novel analytical tortuosity–porosity correlation is presented. Numerical simulations are also performed on two-dimensional Sierpinski carpet model. The proposed fractal model is validated by comparison with numerical results and available experimental data. Results show that hydraulic tortuosity depends on both statistical and morphological characteristics of porous media. The exponents for the scaling law between tortuosity and porosity depend on pore size distribution and tortuous fractal dimension. It has been found that hydraulic tortuosity indicates evident anisotropy for asymmetrical particle arrangements under the same statistical characteristics of porous media. The present work may be helpful to understand the transport mechanisms of porous materials and provide guidelines for the development of oil and gas reservoir, water resource and chemical engineering, etc.

This paper proposed a method for the fractal characterization of the three-dimensional (3D) fracture tortuosity ($D_{T3})$ in coal based on CT scanning experiment. The methodology was deduced in detail, and the values of $D_{T3}$ of four coal samples were calculated by the rigorous derivation equation established by Feng and Yu. The values of $D_{T3}$ by the proposed method fit the relation of $D_{T3}$ versus the fractal dimension for 3D fracture number $D_{f3}$, and the relation of $D_{T3}$ versus the 3D fracture porosity, indicating the rationality and accuracy of the proposed method on estimation of the $D_{T3}$. The results show that the proposed $D_{T3}$ can comprehensively character the fractal characteristics of fractures tortuosity in 3D space. It is worth to further study for establishing an analytical fractal equation for fluid mass transfer in 3D fractures of porous media based on the $D_{T3}$.

Large-scale hydraulic fracturing is the critical technology for effective shale oil production. However, the imbibition flow mechanisms of fracturing fluid in shale micropores and the influence of shale microstructure and physical properties are still indistinct, which makes the optimization goal of fracturing flowback unclear and restricts the enhancement of shale oil recovery. Therefore, based on SEM and XRD experiments, it is analyzed that shale has the characteristics of multiple pores, which are divided into organic pores, brittle mineral pores, and clay pores. Nonetheless, how the tube cross-section controls the interface displacement is not well discussed in the available literature, especially in irregular triangles, rectangles and other non-circular shapes. This paper studies the influence of cross-section shapes on the capillary force by considering the corner flow of the wetting phase, and it analyzes the imbibition dynamics of different types of pores. Using the shale multi-pores physical model and fractal theory, the shale semi-analytical solution models of SI and FI are established. Theoretical analysis of the water imbibition mechanisms shows that the key factors controlling SI and FI volume include imbibition time, fluid properties, pore cross-section shapes, tortuosity, and forced pressure.

Arteries often demonstrate geometric variations such as elliptic and eccentric cross sections, stenosis, and tapering along the longitudinal axis. Effects of these variations on the mechanical stability of the arterial wall have not been investigated. The objective of this study was to determine the buckling behavior of arteries with elliptic, eccentric, stenotic, and tapered cross sections. The arterial wall was modeled as a homogeneous anisotropic nonlinear material. Finite element analysis was used to simulate the buckling process of these arteries under lumen pressure and axial stretch. Our results demonstrated that arteries with an oval cross section buckled in the short axis direction at lower critical pressures as compared to circular arteries. Eccentric cross sections, stenosis, and tapering also decreased the critical pressure. Stenosis led to dramatic pressure variations along the vessel and reduced the buckling pressure. In addition, tapering shifted the buckling deformation profile of the artery towards the distal end. We conclude that geometric variations reduce the critical pressure of arteries and thus make the arteries more prone to mechanical instability than circular cylindrical arteries. These results improve our understanding of the mechanical behavior of arteries.

Several results of lattice-gas and lattice-Boltzmann simulations of single-fluid flow in 2D and 3D porous media are discussed. Simulation results for the tortuosity, effective porosity and permeability of a 2D random porous medium are reported. A modified Kozeny–Carman law is suggested, which includes the concept of effective porosity. This law is found to fit well the simulated 2D permeabilities. The results for fluid flow through large 3D random fibre webs are also presented. The simulated permeabilities of these webs are found to be in good agreement with experimental data. The simulations also confirm that, for this kind of materials, permeability depends exponentially on porosity over a large porosity range.

Spontaneous imbibition (SI) is a capillary-driven flow process, in which a wetting fluid moves into a porous medium displacing an existing non-wetting fluid. This process likely contributes to the loss of fracking fluids during hydraulic fracturing operations. It has also been proposed as a method for an enhanced recovery of hydrocarbons from fractured unconventional reservoirs. Numerous analytical and numerical approaches have been employed to model SI. Invariably, these idealize a fracture as the gap formed between parallel flat surfaces. In reality, rock fracture surfaces are rough over multiple scales, and this roughness will influence the contact angle and rate of fluid uptake. We derived an analytical model for the early-time SI behavior within a fracture bounded by parallel impermeable surfaces with fractal roughness assuming laminar flow. The model was tested by fitting it to experimental data for the SI of deionized water into air-filled rock fractures. Twenty cores from two rock types were investigated: a tight sandstone (Crossville) and a gas shale (Mancos). A simple Mode I longitudinal fracture was produced in each core by compressive loading between parallel flat plates using the Brazilian method. Half of the Mancos cores were fractured perpendicular to bedding, while the other half were fractured parallel to bedding. The two main parameters in the SI model are the mean separation distance between the fracture surfaces, $\bar{x}$, and the fracture surface fractal dimension $2\le D<3$. The $\bar{x}$ was estimated for each core by measuring the geometric mean fracture aperture width through image analysis of the top and bottom faces, while $D$ was estimated inversely by fitting the SI model to measurements of water uptake obtained using dynamic neutron radiography. The $\bar{x}$ values ranged from 45$\mu$m to 190$\mu$m, with a median of 93$\mu$m. The SI model fitted the height of uptake versus time data very well for all of the rock cores investigated; medians of the resulting root mean squared errors and coefficients of determination were 0.99mm and 0.963, respectively. Estimates of $D$ ranged from 2.04 to 2.45, with a median of 2.24. Statistically, all of the $D$ values were significantly greater than two, confirming the fractal nature of the fracture surfaces. Future research should focus on forward prediction through independent measurements of $D$ and extension of the existing SI model to late times (through the inclusion of gravity) and fractures with permeable surfaces.

Permeability is one of the most important parameters for accurately predicting water flow in reservoirs and quantifying underground water inrush into coal mines. This study developed a predictive permeability model by considering the microstructural parameters and tortuosity effects of low-permeability sandstone. The model incorporates the fractal geometry theory, Darcy’s law, and Poiseuille equation into a multistep inversion framework for systematic interpretation of sandstone scanning electron microscopy (SEM) images. A threshold segmentation algorithm is applied to transform SEM images into binary images. Then, we used an improved statistical algorithm with binary image data to estimate the geometric parameters of each pore, such as the perimeter and area. The fractal parameters of pore microstructure were determined by fitting the data of pore perimeters and areas. Finally, the effects of tortuosity on microscopic percolation were considered, and a conventional model was modified for quantifying the relationship between microscopic pore structures parameters and macroscopic permeability. Eight groups of sandstone samples from the Xingdong coal mine in North China were collected for estimating permeability by the developed inversion framework. A direct permeability measurement was also conducted on each sample with an AP-608 automatic measuring instrument. The measured permeability values were compared with results from theoretical models, and we found that the accuracy of the newly developed predictive model is better than that of a conventional permeability model. The predictive model developed in this study provides a useful tool for estimating permeability in low-permeable sandstone reservoirs.

The fractal dimension of random walker (FDRW) is an important parameter for description of electrical conductivity in porous media. However, it is somewhat empirical in nature to calculate FDRW. In this paper, a simple relation between FDRW and tortuosity fractal dimension (TFD) of current streamlines is derived, and a novel method of computing TFD for different generations of two-dimensional Sierpinski carpet and three-dimensional Sierpinski sponge models is presented through the finite element method, then the FDRW can be accordingly predicted; the proposed relation clearly shows that there exists a linear relation between pore fractal dimension (PFD) and TFD, which may have great potential in analysis of transport properties in fractal porous media.

Tortuosity (τ) of two-dimensional fractal model of porous media is investigated to study their relationship with porosity. Square full-walk technique is applied to obtain τ in a two-dimensional fractal model of porous substance constructed by Randomized Sierspinski Carpets. The numerical result is in good agreement with previous results and empirical relation between tortuosity and porosity given by τ ~ p(1 - ϕ) + 1 that was found by other using Lattice Gas Automata method for solving flow equation on two-dimensional porous substance constructed by randomly placed rectangles of equal size and with unrestricted overlap. Average tortuosity of the flow path decreases linearly as fractal dimension of pore increases at each fractal iteration. Both fractal dimension and iteration give almost the same linearly tortuosity–porosity relation. The type of random algorithm for constructing Randomized Sierspinski Carpets has no significant influence on the tortuosity–porosity relation.

Accurate characterization of pore-scale structures of porous media is necessary for studying their transport mechanisms and properties. An analytical model for pore and capillary structures of porous media is developed based on fractal theory in this study. The pore and tortuosity fractal dimensions are introduced to characterize the pore size distribution and tortuous flow paths. A power law scaling between fractal probability function and pore diameter is proposed, which can be applied to determine the pore fractal dimension. The explicit expression for tortuosity fractal dimension is derived based on exactly self-similar fractal set and fractal capillary bundle model. The present fractal model has been validated by comparison with that of experiments and numerical simulations as well as theoretical models. The results show that the tortuosity fractal dimension decreases as porosity and pore fractal dimension increase, it increases with the increment of tortuosity. Both the particle shape and pore size range take important effect on the tortuosity fractal dimension under certain porosity. The proposed pore-scale model can present a conceptual tool to study the transport mechanisms of porous media and may provide useful guideline for oil and gas exploitation, hydraulic resource development, geotechnical engineering and chemical engineering.

In gas-bearing coal seam mining projects, the pivotal considerations encompass the assessment of gas migration, emission trends, and coal seam stability, which are crucial for ensuring both the safety and efficiency of the project. The accurate evaluation of the nonlinear evolution of the fracture network, acting as the primary conduit for gas migration and influenced by mining disturbances, coal seam stress, overlying strata pressure, and gas pressure, emerges as a key determinant in gauging coal seam stress and safety. To address the industry challenge of quantitatively assessing the complex behaviors of fracture networks during gas-bearing coal seam extraction, this study introduces a novel, interdisciplinary fractal analysis model. Drawing upon fractal theory for classical porous media, four fractal parameters capable of quantitatively characterizing the microscopic behaviors of fractures are proposed and defined as functions of permeability. Subsequently, the gas pressure in gas-bearing coal seams, coal seam deformation and stress, in-situ stress, overlying strata pressure, and adsorption–desorption effects are comprehensively coupled and applied to the classic gas-bearing coal seam at the Jianxin Coal Mine’s 4301 working face in Shaanxi, China. Upon the robust validation of the proposed model, the present computational results reveal: (1) the proposed micro-parameters adeptly characterize the number, roughness, tortuosity, and length of fractures in gas-bearing coal seams; (2) a larger fractal dimension of fractures leads to increased coal seam stress and strain, while the fractal dimensions of fracture tortuosity and roughness are inversely proportional to coal seam stress and strain; (3) these fractal parameters directly induce evolutionary changes in gas seepage behavior, leading to varying degrees of mechanical property evolution in the coal seam. When $D_S$ and $D_T$ increased from 1.2 to 1.8, the maximum change in coal seam deformation was 16.9% and 13.8%, respectively, and when $𝜀$ increases from 0.03 to 0.12, the coal seam deformation changes by 15.1%. This represents a quantitative characterization unattainable by previously published coal seam analysis models, including mainstream fractal computation models.