APPLICATION OF THE PRECONDITIONED CONJUGATE GRADIENTS METHOD IN THE SIMULATION OF RELATIVE PERMEABILITY PROPERTIES OF POROUS MEDIA

A model that is after the determination of the relative permeability characteristics of porous media is presented. It is part of a general model that deals with the simulation of capillary phenomena and immiscible fluid flow behaviour in porous media. The relative permeability characteristics in a water wet porous medium are simulated with the use of three-dimensional (3-D) network models of pore structure with pore body and pore throat size distributions. The major assumption involved is that a cubic network of pore bodies connected by pore throats with pore body and pore throat size distribution respectively, is a realistic representation of the pore structure of a porous medium. The physical laws that apply in real media are simulated in the network analysis, and the computed results are compared with experimental findings. A new algorithm was developed that leads to the solution of a large set of linear equations, with a sparse and positive definite coefficient matrix. Results obtained with the application of the Preconditioned Conjugate Gradients method and numerical aspects of the simulation are presented and discussed. Comparison of this method with other available numerical methods is also made. It is concluded that the Preconditioned Conjugate Gradients method is advantageous for large networks with regards to time of solution, convergence, and accuracy. The validity of the algorithm is tested against other methods in the literature.     

ANALYSIS OF PERMEABILITY FOR ELLIS FLUID FLOW IN FRACTAL POROUS MEDIA

This work studies the flow characteristics of Ellis fluid in saturated porous media. A fractal model is developed for the effective permeability of Ellis fluid flow in porous media based on the assumptions that porous media consist of a bundle of tortuous capillaries, whose size distribution and tortuosity follow the fractal scaling laws. The average flow velocity and the effective permeability for Ellis fluid flow in porous media are derived. The proposed fractal model does not contain any empirical constant, and every parameter in the model has clear physical meaning. The model predictions are compared with the measured data, and good agreement between them is obtained.     

A STUDY OF EXPERIMENTAL DESIGN OPTIMALITY FOR WIENER SYSTEMS

A weakly nonlinear theory of wave propagation in two superposed dielectric fluids streaming through porous media in the presence of vertical electric field and in the absence of surface charges at their interface is investigated in three dimensions. The method of multiple scales is used to obtain a dispersion relation for the linear problem and a Ginzburg-Landau equation with complex coefficients for the nonlinear problem, describing the behavior of the system. The stability of the system is discussed both analytically and numerically in both cases, and the corresponding stability conditions are obtained. It is found, in the linear case, that the stability criterion is independent of the medium permeability and that the medium porosity, surface tension, and dimension all have stabilizing effects the fluid viscosities, velocities, and depths have destabilizing effects, and the electric field has a dual role in the stability of the system. In the nonlinear analysis, it is found that the electric field has a stabilizing effect in two-dimensional disturbances and destabilizing effect in three-dimensional disturbances cases. The surface tension, fluid depths, and medium porosity have stabilizing effects in both two- and three-dimensional disturbance cases and the fluid viscosities, velocities, and medium permeability have destabilizing effects in both cases, and this stability or instability occurs faster for three-dimensional disturbance cases. It is found also that the system is unstable in the absence of fluid velocities or for nonporous media. Finally, the dimension was found to have a dual role (stabilizing as well as destabilizing) in the considered system, while it has a destabilizing effect in the case of nonporous media.     

多孔介质/纯流体耦合区域内可压缩气体的流动

采用统一形式的修正N-S方程描述多孔介质/纯流体耦合区域的流动,提出了方程中用于处理多孔介质内流动的源项确定方法,并利用成熟的CFD技术对圆管内具有前后台阶的耦合区域内可压缩气体的流动进行了数值模拟,得到了与实验吻合的计算结果.     

Linear displacement of a wetting fluid by an immiscible non-wetting fluid in a porous medium: A predictive algorithm

The linear displacement of a wetting fluid by an immiscible non-wetting fluid in a two-dimensional porous medium composed of a network of sites multi-connected by bonds has been simulated mathematically. The algorithm involves Monte Carlo decision making, random walks and principles of the percolation theory. The algorithm described in the present work successfully predicts the three distinct behaviours of immiscible displacement in porous media. This algorithm is tested against experiments available in the literature for two-dimensional porous media. The agreement between the numerical results and the experiments is very good.     

Numerical Simulation for Particle Penetration Depth Distribution in Deep Bed Filtration

A two‐dimensional model has been developed to simulate particle penetration through porous media. The particle penetration depends on many parameters including the Reynolds number, particle drag coefficient, the ratio of the diameter of injected to filtered particles, fluid velocity, and pore size, etc. The numerical model for separation efficiency in periodic porous media was studied. Previous work has described the effects of injected particle size, Reynolds number and particle drag coefficient. In this study, the porous media flow is modeled (solution of the Navier‐Stokes equations) by using the finite element method, and the analysis is restricted to the case of two‐dimensional periodic porous media. The effects of these factors and particle depth distribution in porous media are investigated. It is noted that the results for the three Reynolds numbers 1, 16.56, and 100, are qualitatively similar, and about 40 % of particles are trapped in the top part of the filter.     

SINGLE FLUID FLOW IN POROUS MEDIA

A comprehensive study on single fluid flow in porous media is carried out. The volume averaging technique is applied to derive the governing flow equations. Additional terms appear in the averaged governed equations related to porosity ε, tortuosity τ, shear factor F and hydraulic dispersivity D _h. These four parameters are uniquely contained in the volume averaged Navier-Stokes equation and not all of them are independent. The tortuosity can be related to porosity through the Brudgemann equation, for example, for unconsolidated porous media.

The shear factor models are reviewed and some new results are obtained concerning high porosity cases and for turbulent flows. It is known that there are four regions of flow in porous media: pre-Darcy's flow, Darcy's flow, Forchheimer flow and turbulent flow. The transitions between these regions arc smooth. The first region, the pre-Darcy's flow region represents the surface-interactive flows and hence is strongly dependent on the porous media and the flowing fluid. The other flow regions are governed by the flow strength of inertia. For Darcy's flow, the pressure gradient is found to be proportional to the flow rate. The Forchheimer flow, however, is identified by a strong inertia! effects and the pressure gradient is a parabolic function of flow rate. Turbulent flow is unstable and unsteady flow characterized by chaotic flow patterns. The pressure drop is slightly lower than that predicted using the laminar flow equation.

The hydraulic dispersivity is a property of the porous media. It may be considered as the connectivity of the pores in a porous medium. It characterizes the dispersion of mementum, heat and mass transfer. In this paper, only the dispersion of momentum is studied.

Single fluid flow through cylindrical beds of fibrous mats and spherical particles has been used to show how to solve the single fluid flow problems in porous media utilizing the knowledge developed in this communication. Both the pressure drop and axial flow velocity profiles are computed using the developed shear factor and hydraulic dispersion models. Both the predicted velocity profile and pressure drop compare fairly well with the published experimental data.     

ROUTE TO CHAOS IN CHEMICALLY ENHANCED THERMAL CONVECTION IN POROUS MEDIA

Exothermic chemical reactions can influence natural convection effects in a porous medium. Such phenomena may occur in tubular reactors, oxidation of solid materials in large containers, chemical vapor deposition systems, liquid explosives, and others. Experimental evidence indicates that the influence of natural convection in many chemically reacting systems cannot be neglected.

In the present work, transient effects of a two-dimensional convection generated and sustained by an exothermic chemical reaction and a constant boundary wall temperature are studied. The Darcy-Boussinesq equations are used to describe fluid flow through porous media with a zero-order chemical reaction. A recently developed method is used to compute the singular points (such as limit and symmetry breaking bifurcation points) precisely as a function of Rayleigh (Ra) and Frank-Kamenetskii (Fk) numbers. Fold curves are drawn as a function of Ra, and Fk. Flow behavior is governed by two natural parameters, namely, Ra and Fk. Multiple stationary solutions arise over an intermediate range of Ra. The solution structure, i.e., the interconnections between the various branches appear quite complicated. Determination of linear stability on the these solution branches reveal that all two-dimensional, stationary solutions develop some form of instability at higher values of Ra. Transient simulations reveal the emergence of time periodic solutions at higher Ra. The nature and frequency of these periodic solutions depend on the route followed in the parameter space Fk and Ra. The various routes to chaos are identified in this parameter space.     

Supercritical fluid flow in porous media: modeling and simulation

The present work aims the modeling and simulation of supercritical fluid flow through porous media. This type of flow appears in several situations of interest in applied science and engineering, as the supercritical flow in porous materials employed in chromatography, supercritical extraction and petroleum reservoirs. The fluid is constituted of one pure substance, the flow is monophasic, highly compressible and isothermal. The porous media is isotropic, possibly heterogeneous, with rectangular format and the flow is two-dimensional. The heterogeneities of porous media are modeled by a simple power law, which describes the relationship between permeability and porosity. The modeling of the hydrodynamic phenomena incorporates the Darcy's law and the equation of mass conservation. Appropriated correlations are used to model, in a realistic form, the density and the viscosity of the fluid. A conservative finite-difference scheme is used in the discretization of the differential equations. The nonlinearity is treated by Newton method, together with the conjugate gradient method. The results of the simulation for pressure and mobility of supercritical and liquid propane flowing through porous media are presented, analyzed and graphically depicted.     

LIQUID-LIQUID RELATIVE PERMEABILITY: NETWORK MODELS AND EXPERIMENTS

Two different but related two-dimensional network models are used to elucidate the concept of relative permeability in simultaneous liquid-liquid flow in porous media. The first model is designed for monosized sphere packs while the other, consisting of spherical pore chambers interconnected by capillaries of variable length and radius, is intended for more general porous media. The effects of various operating conditions and physical factors—pressure gradient, wettability, surface tension, throat and pore size distributions, imbibition or drainage—on relative permeability are studied. Experimental data are also presented to confirm various aspects of the theory.     

Study on the effect of micro geometric structure on heat conduction in porous media subjected to pulse laser

Based on fractal theory, different two-dimensional fractal structures were constructed to simulate the practical porous media. Effective thermal conductivity for porous media was calculated by means of the finite volume method. Theoretical analysis of thermal response in the porous media under various heating conditions was performed with a multi-layer hyperbolic heat conduction model with volumetric heat generation. The results obtained in this paper indicate that pore size and micro distribution have a far-reaching impact on the heat conduction in porous media. If we assumed that both the thermal conductivity and the heat capacity of the solid phase is larger than those of liquid phase, decreasing the pore size and porosity is helpful to enhance the heat transfer in porous media and the peak of temperature increases with pore size and porosity. With the same pore size and porosity, the effect of the pore micro-geometric distribution on heat conduction in porous media is obvious. The method presented in this paper may suggest a valuable approach to theoretically evaluate the effect of pore micro-geometric structure on heat conduction in porous media.     

基于改进BISQ模型双相介质波场数值模拟

Biot理论和喷射流动理论是表述动态孔隙弹性性质的基础理论。弹性波在多孔隙双相介质中传播时固-流相互作用的这两种力学机制是同时存在的,Biot流动力学机制描述的是宏观现象,喷射流动机制反映的是局部特征,Dvorkin和Nur将其两者有机结合,提出了统一的BISQ(Biot-Squirt)模型。Mamadou Sanou和Erwin Appel又提出了不含特征喷射流动长度的改进BISQ模型,更便于描述且计算简单,正演数值模拟表明:改进BISQ模型与BISQ模型的结果保持一致,具有较好的可行性。     

Determination of effective macropore diffusion coefficients by digital image processing

In this contribution, a new method is proposed for determining effective macropore diffusion coefficients in porous media by digitized microphotographs of porous support materials and random walk simulations. The method introduced allows calculation of the effective diffusion coefficient as a function of the mean free path length over a wide variety of values. A versatile method for the preparation of porous substances for light microscopy is described. The frequently applied model of an active shell, which is used to model particle/solid collisions, was found to give incorrect results in conjunction with the application of the first passage time algorithm, which was applied to save computing time in the simulation of gas diffusion. It was possible to show that more realistic results are obtained if a Knudsen layer is used to model particle/solid collisions. Furthermore, in investigating diffusion in two- and three-dimensional representations of capillary tubes, it was found that results of simulations performed to calculate transport properties of fluids in porous media, based on two-dimensional model systems, cannot in any case be transferred to the corresponding three-dimensional systems.     

Numerical investigation on the flow characteristics and permeability of three-dimensional reticulated foam materials

In this paper, a three-dimensional (3D) model was proposed for predicting the flow characteristics and permeability of three-dimensional reticulated foam materials, which were prepared by replication process. Parameters, such as permeability, inertia coefficient, and friction factor, were obtained in order to describe the fluid flow characteristics of porous media. The influence of foam structure on the fluid flow characteristics was elucidated. Three flow regimes in porous media, including Darcy's regime, Forchheimer's regime and Froude's regime, were visualized and discussed. The flow transition from linear (Darcy's regime) to nonlinear (Forchheimer's regime) behavior, which is typical of experiments, was founded in the simulation. The data presented revealed the fact that the numerical results are in agreement with the experimental ones published previously.     

HEAT AND MASS TRANSFER IN STAGNATION-POINT FLOW OF A POLAR FLUID TOWARDS A STRETCHING SURFACE IN POROUS MEDIA IN THE PRESENCE OF SORET,DUFOUR AND CHEMICAL REACTION EFFECTS

This work is focused on the numerical solution of steady boundary-layer stagnation-point flow of a polar fluid towards a stretching surface embedded in porous media in the presence of the effects of Soret and Dufour numbers and first-order homogeneous chemical reaction. The governing boundary-layer equations of the problem are formulated and transformed into a self-similar form. The obtained equations are solved numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. Both assisting and opposing flow conditions are considered. Comparisons of the present numerical results with previously published work under limiting cases are performed and found to be in excellent agreement. Representative results for the fluid velocity, angular velocity, temperature, and solute concentration profiles as well as the local heat and mass transfer rates for various values of the physical parameters are displayed in both graphical and tabular forms.     

Simulation of saturated fluid flow in packed particle beds—The lattice-Boltzmann method for the calculation of permeability from XMT images

Cone beam X-ray microtomography (XMT) instrumentation is a state-of-the-art non-invasive technology now used for several years to describe important characteristics of packed particle beds in three-dimensional (3D) detail. Many process engineering operations involve the transport of fluid in porous media. It is well known that the flow in porous media depends on the geometric properties of the pore network structure and in this regard X-ray microtomographic imaging captures the porous network structure of opaque packed particle beds which is later used for fluid flow analysis. The coupling of XMT 3D imaging with a novel fluid flow simulation method, known as the lattice-Boltzmann model (LBM), allows for direct local flow determination and micro-permeability calculations for complex porous structures. In this paper the methodology is briefly explained, implementations for some practical problems are addressed, the application of the technique from results for packed particle beds of interest are presented, and a comparison with experimental data is made.     

THE MEASUREMENT OF THE FLOW AROUND A SPHERE SETTLING IN A RECTANGULAR BOX USING 3-DIMENSIONAL PARTICLE IMAGE VELOCIMETRY

A novel three-dimensional particle image velocimetry technique is used to measure the planar three-dimensional flow field about the centreline of a sphere sedimenting in a rectangular shaped box. Measurements are made in the center of the container and also one diameter from a plane wall. Results are presented for a sphere falling in both a constant viscosity elastic (Boger) fluid and a shear-thinning elastic liquid. In the center of the box, the flow field is essentially two-dimensional as expected. Near the wall, there is substantial out-of-plane motion in the shear-thinning solution due to the presence of the wall. Surprisingly, there is little out-of-plane motion for a sphere sedimenting near the wall in the Boger fluid. There are significant qualitative differences in the flow field for the sphere sedimenting in the shear-thinning and constant viscosity elastic liquids. The results are compared with previously published work for a sphere settling in a non-Newtonian fluid and also with results obtained in an identical geometry for a Newtonian fluid. Reasons for the differences in the velocity maps are discussed. The drag coefficient for each geometry and fluid is calculated.     

THE MEASUREMENT OF THE FLOW AROUND A SPHERE SETTLING IN A RECTANGULAR BOX USING 3-DIMENSIONAL PARTICLE IMAGE VELOCIMETRY

A novel three-dimensional particle image velocimetry technique is used to measure the planar three-dimensional flow field about the centreline of a sphere sedimenting in a rectangular shaped box. Measurements are made in the center of the container and also one diameter from a plane wall. Results are presented for a sphere falling in both a constant viscosity elastic (Boger) fluid and a shear-thinning elastic liquid. In the center of the box, the flow field is essentially two-dimensional as expected. Near the wall, there is substantial out-of-plane motion in the shear-thinning solution due to the presence of the wall. Surprisingly, there is little out-of-plane motion for a sphere sedimenting near the wall in the Boger fluid. There are significant qualitative differences in the flow field for the sphere sedimenting in the shear-thinning and constant viscosity elastic liquids. The results are compared with previously published work for a sphere settling in a non-Newtonian fluid and also with results obtained in an identical geometry for a Newtonian fluid. Reasons for the differences in the velocity maps are discussed. The drag coefficient for each geometry and fluid is calculated.     

高温金属熔液在旋转多孔介质内的渗流传热过程

针对转动坐标系中铝熔液在SiC多孔介质内的流动传热现象 ,考虑离心力对渗流传热过程的影响 ,采用局部非热平衡假设建立多孔介质渗流传热数理模型 ,研究不同工况下流体的流速、压力损失及铝熔液和多孔介质的温度变化 .计算结果表明 :在渗透区域 ,液固两相存在温差 ,且液固温差随渗透界面的移动而减小 ;在非渗透区域 ,固体的温度曲线基本不变 .离心转速或孔隙率的增加都使渗透前沿区域液固两相温差增大 .孔隙率对流场和压力损失有较大影响.     

MOMENTUM INTEGRAL METHOD FOR FORCED CONVECTION IN THERMAL NONEQUILIBRIUM POWER-LAW FLUID-SATURATED POROUS MEDIA

Forced convection heat transfer for power-law fluid flow in porous media was studied analytically. The analytical solutions were obtained based on the Brinkman-extended Darcy model for fluid flow and the two-equation model for forced convection heat transfer. As a closed-form exact velocity profile is unobtainable for the general power-law index, an approximate velocity profile based on the parabolic model is proposed by subscribing to the momentum boundary layer integral method. Heat transfer analysis is based on the two-equation model by considering local thermal nonequilibrium between fluid and solid phases and constant heat flux boundary conditions. The velocity and temperature distributions obtained based on the parabolic model were verified to be reasonably accurate and improvement is justified compared to the linear model. The expression for the overall Nusselt number was derived based on the proposed parabolic model. The effects of the governing parameters of engineering importance such as Darcy number, power-law index, nondimensional interfacial heat transfer coefficient, and effective thermal conductivity ratio on the convective heat transfer characteristics of non-Newtonian fluids in porous media are analyzed and discussed.