Fractal Fluid Research Articles

Variational approaches to Dirichlet gradient-type systems on the Sierpiński gasket

In this paper, we study the multiple solutions of parametric quasi-linear systems of the gradient-type on the Sierpiński gasket arising in physical problems leading to nonlinear models involving reaction-diffusion equations, in problems on elastic fractal media or fluid flow through fractal regions. By using some critical point theorems, we give some new results to establish one, two and three weak solutions for our system. Finally, we give four examples to illustrate the main results.

Effects of Pore Geometry and Saturation on the Behavior of Multiscale Waves in Tight Sandstone Layers

AbstractGeometric heterogeneities in tight reservoir rocks saturated with a fluid mixture may exhibit different scale distribution characteristics. Conventional models of rock physics based on poroelasticity, which usually consider single‐scale pore structure and fluid patches, are inadequate for describing elastic wave responses. A major challenge is to establish the relationship between the wave response at different spatial scales and frequencies. To address this problem, three sets of observational data over a wide frequency range were obtained from a tight oil reservoir in the Ordos Basin, China. Ultrasonic measurements were made on eight sandstone samples at partial oil‐water saturation at 0.55 MHz. Data from six borehole measurements and seismic profiles were acquired and analyzed at about 10 kHz and 30 Hz, respectively. Analysis of the cast thin sections shows that dissolution pores and microcracks generally develop, with fractal dimensions of the pores ranging from 2.45 to 2.67 for the samples with porosities between 5.1% and 10.2%. Compressional wave velocity and attenuation were estimated from the observed data. The results show that the velocity dispersion from seismic to ultrasonic frequencies is 10.02%, mostly occurring between sonic and ultrasonic frequencies. The attenuation is stronger at higher oil saturation. The relationships between velocity, attenuation, and wavelength were established and can be used for further forward modeling and seismic interpretation studies. A partial saturation model has been derived based on effective differential medium theory and a double double‐porosity model, assuming that the medium contains fractal cracks and fluid patches. The effects of scale and saturation on wave responses are prevalent. Modeling results consistent with observed data show that the radii of cracks and fluid patches range from 0.1 μm to 2.8 mm, affecting ultrasonic, acoustic, and seismic attenuation. The multiscale data and proposed model quantify the relationship between fracture and fluid distributions and attenuation and could be useful for upscaling to the reservoir scale. The study helps improve the understanding of seismic wave propagation in partially saturated rocks, which has potential applications in seismic exploration, hydrocarbon production in reservoirs, and CO2 sequestration in aquifers.

A Generalization of Poiseuille’s Law for the Flow of a Self-Similar (Fractal) Fluid through a Tube Having a Fractal Rough Surface

In this paper, a generalization of Poiseuille’s law for a self-similar fluid flow through a tube having a rough surface is proposed. The originality of this work is to consider, simultaneously, the self-similarity structure of the fluid and the roughness of the tube surface. This study can have a wide range of applications, for example, for fractal fluid dynamics in hydrology. The roughness of the tube surface presents a fractal structure that can be described by the surface fractal noninteger dimensions. Complex fluids that are invariant to changes in scale (self-similar) are modeled as a continuous medium in noninteger dimensional spaces. In this work, the analytical solution of the Navier–Stokes equations for the case of a self-similar fluid flow through a rough “fractal” tube is presented. New expressions of the velocity profiles, the fluid discharge, and the friction factor are determined analytically and plotted numerically. These expressions contain fractal dimensions describing the effects of the fractal aspect of the fluid and of that of the tube surface. This approach reveals some very important results. For the velocity profile to represent a physical solution, the fractal dimension of the fluid ranges between 0.5 and 1. This study also qualitatively demonstrates that self-similar fluids have shear thickening-like behavior. The fractal (self-similarity) nature of the fluid and the roughness of the surface both have a huge impact on the dynamics of the flow. The fractal dimension of the fluid affects the amplitude and the shape of the velocity profile, which loses its parabolic shape for some values of the fluid fractal dimension. By contrast, the roughness of the surface affects only the amplitude of the velocity profile. Nevertheless, both the fluid’s fractal dimension and the surface roughness have a major influence on the behavior of the fluid, and should not be neglected.

Flow of a Self-Similar Non-Newtonian Fluid Using Fractal Dimensions

In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid is depicted as a continuum in a noninteger dimensional space. Involving vector calculus for the instance of a noninteger dimensional space, we determine an analytical solution of the Cauchy equation for the instance of a non-Newtonian self-similar fluid flow in a cylindrical pipe. The plot of the velocity profile obtained shows that the rheological behavior of a non-Newtonian power-law fluid is essentially impacted by its self-similar structure. A self-similar shear thinning fluid and a self-similar Newtonian fluid take on a shear-thickening way of behaving, and a self-similar shear-thickening fluid becomes more shear thickening. This approach has many useful applications in industry, for the investigation of blood flow and fractal fluid hydrology.

Evolution of Fractal Pore Structure in Sedimentary Rocks

AbstractGeological processes alter pore spaces over time, and their analysis can shed light on the dynamic fractal structure and fluid flow of rocks over time. This study presents experimental evidence to illustrate that the pore fractal structure evolves with sedimentation, carbonate cementation, clay growth, and dissolution. It examines, describes and characterizes a suite of core samples from the Gaotaizi oil layer of the second and third members of the Qingshankou Formation, Songliao Basin, China. The effects of mechanical compaction and other diagenesis effects on fractal pore structure on sedimentary rocks are discussed. A schematic diagram is proposed that describes the impacts of these diagenetic processes on fractal pore structure at the microscopic scale in sedimentary rocks. This work links the state of diagenetic alteration and fractal pore structure, which can guide practical applications such as predicting the permeability of sedimentary rocks.

NEW RESULTS OF FRACTAL FRACTIONAL MODEL OF DRILLING NANOLIQUIDS WITH CLAY NANOPARTICLES

In recent years, a new idea of the fractal fractional derivative has been introduced. However, it is not used for the free convection flow drilling nanoliquid with clay nanoparticles. In this paper, we have deliberated this new approach of the fractal fractional derivative with a power-law kernel for heat transfer in a drilling nanofluid with clay nanoparticle in a vertical channel. Water is taken as the base fluid. The flow of the fluid is between two fixed vertical parallel plates such that one of them is constantly heated. The resulting problem is modeled with a fractal fractional derivative operator with a power-law kernel. As the exact solution to this problem is not possible, therefore, the Crank–Nicolson Finite Difference Scheme (CNFDS) is used for numerical solutions. This idea for the fractal fractional fluid problem is used here for the first time in the literature. Results are portrayed in the various graphs using Maple-15 software. The importance of both fractal and fractional parameters is discussed in detail. Results showed that the fractal fractional parameter gives a more general solution of the velocity and temperature for the fixed fractional operators. Therefore, a mutual approach of fractal fractional clarifies the memory of the function better than fractional only.

On nonlocal fractal laminar steady and unsteady flows

In this study, we join the concept of fractality introduced by Li and Ostoja-Starzewski with the concept of nonlocality to produce a new set of nonlocal fractal fluid equations of motion. Both the unsteady and steady laminar flows are discussed. It is revealed that a damped wave equation emerges from the nonlocal fractal Navier–Stokes equation, a result which could lead to a better understanding of fluids turbulence.

A Fractal-Fractional Model for the MHD Flow of Casson Fluid in a Channel

An emerging definition of the fractal-fractional operator has been used in this study for the modeling of Casson fluid flow. The magnetohydrodynamics flow of Casson fluid has cogent in a channel where the motion of the upper plate generates the flow while the lower plate is at a static position. The proposed model is non-dimensionalized using the Pi-Buckingham theorem to reduce the complexity in solving the model and computation time. The non-dimensional fractal-fractional model with the power-law kernel has been solved through the Laplace transform technique. The Mathcad software has been used for illustration of the influence of various parameters, i.e., Hartman number, fractal, fractional, and Casson fluid parameters on the velocity of fluid flow. Through graphs and tables, the results have been implemented and it is shown that the boundary conditions are fully satisfied. The results reveal that the flow velocity is decreasing with the increasing values of the Hartman number and is increasing with the increasing values of the Casson fluid parameter. The findings of the fractal-fractional model have elucidated that the memory effect of the flow model has higher quality than the simple fractional and classical models. Furthermore, to show the validity of the obtained closed-form solutions, special cases have been obtained which are in agreement with the already published solutions.

A FRACTAL TWO-PHASE FLOW MODEL FOR THE FIBER MOTION IN A POLYMER FILLING PROCESS

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.

Experimental investigation on the anisotropic fractal characteristics of the rock fracture surface and its application on the fluid flow description

Based on the anisotropic fractal characteristics of natural fracture surfaces, a fractal fluid flow model was developed. Three-dimensional printed fracture models were built using the morphological information of an actual tensile and shear fracture surface of a rock. The impacts of the fractal dimension on the fluid flow characteristics were investigated. The results demonstrated that the concentration distribution of the undulating height of tensile fracture was larger than that of shear fracture. The fractal dimension of fracture lines with different directional sections ranges from 1.01 to 1.03, which results in the fluid flow anisotropy of fracture surfaces. As predicted by the theoretical model, the fluid flows along the tensile fracture surface and the shear fracture surface in different directions are 0.7–1.2 times and 0.9–1.1 times the average flow rates, respectively. With increasing pressure head, the Reynolds number linearly increased, and the Euler number first rapidly decreased and then slowly decreased.

Liver Backscatter and the Hepatic Vasculature’s Autocorrelation Function

Ultrasound imaging of the liver is an everyday, worldwide clinical tool. The echoes are produced by inhomogeneities within the interrogated tissue, but what are the mathematical properties of these scatterers? In theory, the spatial correlation function and the backscatter coefficient are linked by a Fourier transform relationship, however direct measures of these are relatively rare. Under the hypothesis that the fractal branching vasculature and fluid channels are the predominant source of scattering in normal tissues, we compare theory and experimental measures of the autocorrelation function, the frequency dependence of scattering, and fractal dimension estimates from high contrast 3D micro-CT data sets of rat livers. The results demonstrate a fractal dimension of approximately 2.2 with corresponding power law estimates of autocorrelation and ultrasound scattering. These results support a general framework for the analysis of ultrasound scattering from soft tissues.

On the separation of particle flow during pulse laser deposition of heterogeneous materials - A multi-fractal approach

The problem of plasma self-structuring during the pulse laser deposition of multicomponent materials is discussed from both theoretical and experimental perspectives. A non-differential mathematical model was used to build the set of equation to characterize a complex fluid, assimilated with the laser produced plasmas in the fractal paradigm. Numerical simulations regarding the flow of a fractal fluid were performed reflecting real multi-component plasmas. These revealed that the structuring of a fluid is strongly related to its fractality and thus to individual properties of its components. The theoretical results were confronted with experimental data regarding the flow of multi-component plasma in a fluid representation. The laser produced plasma was investigated by means of ICCD fast camera imaging and optical emission spectroscopy. The experimental results showcase a splitting process on the two main directions coupled with a separation of the ejected particles with respect to their excitation energy. Both theoretical and experimental data are in good agreement.

Residual Multiparticle Entropy for a Fractal Fluid of Hard Spheres.

The residual multiparticle entropy (RMPE) of a fluid is defined as the difference, , between the excess entropy per particle (relative to an ideal gas with the same temperature and density), , and the pair-correlation contribution, . Thus, the RMPE represents the net contribution to due to spatial correlations involving three, four, or more particles. A heuristic “ordering” criterion identifies the vanishing of the RMPE as an underlying signature of an impending structural or thermodynamic transition of the system from a less ordered to a more spatially organized condition (freezing is a typical example). Regardless of this, the knowledge of the RMPE is important to assess the impact of non-pair multiparticle correlations on the entropy of the fluid. Recently, an accurate and simple proposal for the thermodynamic and structural properties of a hard-sphere fluid in fractional dimension has been proposed (Santos, A.; López de Haro, M. Phys. Rev. E 2016, 93, 062126). The aim of this work is to use this approach to evaluate the RMPE as a function of both d and the packing fraction . It is observed that, for any given dimensionality d, the RMPE takes negative values for small densities, reaches a negative minimum at a packing fraction , and then rapidly increases, becoming positive beyond a certain packing fraction . Interestingly, while both and monotonically decrease as dimensionality increases, the value of exhibits a nonmonotonic behavior, reaching an absolute minimum at a fractional dimensionality . A plot of the scaled RMPE shows a quasiuniversal behavior in the region .

Semi-numerical solution to a fractal telegraphic dual-porosity fluid flow model

We present a semi-numerical solution to a fractal telegraphic dual-porosity fluid flow model. It combines Laplace transform and finite difference schemes. The Laplace transform handles the time variable whereas the finite difference method deals with the spatial coordinate. This semi-numerical scheme is not restricted by space discretization and allows the computation of a solution at any time without compromising numerical stability or the mass conservation principle. Our formulation results in a non-analytically-solvable second-order differential equation whose spatial discretization yields a tridiagonal linear algebraic system. Moreover, we describe comparisons between semi-numerical and semi-analytical solutions for particular cases. Results agree well with those from semi-analytic solutions. Furthermore, we expose a parametric analysis from the coupled model in order to show the effects of relevant parameters on pressure profiles and flow rates for the case where neither analytic nor semi-analytic solutions are available.

Steady laminar flow of fractal fluids

We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived.

Poiseuille equation for steady flow of fractal fluid

Fractal fluid is considered in the framework of continuous models with noninteger dimensional spaces (NIDS). A recently proposed vector calculus in NIDS is used to get a description of fractal fluid flow in pipes with circular cross-sections. The Navier–Stokes equations of fractal incompressible viscous fluids are used to derive a generalization of the Poiseuille equation of steady flow of fractal media in pipe.

Fractal characterization of pore–fracture in low-rank coals using a low-field NMR relaxation method

To describe a low-field nuclear magnetic resonance (NMR) method for quantifying pore–fracture fractal dimensions and their influence on effective porosity and permeability, we performed modeling comparisons between fractal analysis and pore–fracture physical properties in low-rank coals. The adsorption space fractal (DNMRA), seepage space fractal (DNMRS) and moveable fluid space fractal (DNMRM) were calculated to be 1.62–1.91, 2.77–2.98 and 1.56–2.75, respectively. The DNMRA generally increased with increasing Langmuir volume (VL, 9.54–31.06m3/t), Langmuir pressure (PL, 0.58–8.13MPa), the Brunauer–Emmett–Teller (BET) surface area and its fractal dimension. Higher DNMRA indicated the significant coalbed methane (CBM) adsorption capability. Both the DNMRS and DNMRM decreased with increasing areas of T2>2.5ms distribution (ST and SCT) and sorting coefficient. These phenomena showed that the NMR fractal method could reflect the coal pore–fracture heterogeneity and had significant influence on seepage space content. The correlations of moveable fluid porosity and permeability with DNMRM can be found by performing the models of y=ax+b (a<0), so coals with high DNMRM occur to have low flow capability. Furthermore, the pore–fracture porosity and permeability have positive correlations with ST and SCT, which result from the connection between pores and fractures. These results also show that fractal analysis calculated with T2 can be developed to appraise the physical properties of low-rank coals and supply some reference for a relatively full identification of porous media. We advise that low-field NMR can be employed as a lossless analytic method to quantify moveable fluid space fractal theory.

Erratum to: A Model of the Fractal Fluid Loss Coefficient and its Effect on Fracture Length and Well Productivity

Erratum to: A Model of the Fractal Fluid Loss Coefficient and its Effect on Fracture Length and Well Productivity

A Model of the Fractal Fluid Loss Coefficient and its Effect on Fracture Length and Well Productivity

Loss of the fracturing fluid in hydraulic fracturing is a complex flow process that impacts the effectiveness of the technology, affecting such factors as fracture length and gas-well productivity. Many computational models have been constructed for the fluid loss coefficient, but the models do not account for the fractal character and tortuosity of the porous medium. This study considers these two parameters and proposes a fractal model to calculate the fluid loss coefficient. Fractal permeability is calculated based on the capillary-pressure curves of nine core samples. The permeability values based on fractal theory agree well with the measured values. The fluid loss coefficient determined by the fractal model is higher than the coefficient obtained by standard calculations because of high tortuosity of the porous medium. Predictions of fracture length and gas-well productivity, which depend on the fluid loss coefficient, are discussed and compared, and it is suggested that accurate determination of this coefficient is essential for optimizing hydraulic fracturing.

A fractal model for spherical seepage in porous media

The characters of spherical/radial seepage in porous media have attracted steadily attention in many sciences and technologies such as in oil/water exploration, nuclear waste disposal, and heat and mass transfer in aerospace materials, biological tissue and organs. Considering the effects of capillary pressure, in this work we study the effective radial permeability and relative permeability for spherical seepage in porous media by applying the fractal theory and technique for porous media. The proposed models are related to the structural parameters of porous media, such as fractal dimensions, porosity, tortuosity and fluid properties. The validity of the proposed model is verified by comparing the model predictions with the available model and the existing experimental data. The present results show that the effective radial permeability and radial porosity decrease with the increase of radial distance. The parametrical effects on the radial permeability are also studied. It is found that the relative permeability for spherical seepage of wetting phase increases with the increase of wetting phase saturation, and the relative permeability of the non-wetting phase decreases with the increase of the wetting phase saturation. The contributions for permeability and relative permeability for spherical/radial seepage from capillary pressure can be negligible when pc,av/pm<0.01, otherwise, the effect of capillary pressure on the seepage in porous media should be taken into account.