Slow Motion of a Porous Eccentric Spherical Particle-in-Cell Models

A combined analytical?Cnumerical method is presented for the quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous eccentric spherical particle-in-cell models. The flow inside the porous particle is governed by the Brinkman model and the flow in the fictitious envelope region is governed by Stokes equations. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based on both the porous particle and fictitious spherical envelope. Boundary conditions on the particle??s surface and fictitious spherical envelope that correspond to the Happel, Kuwabara, Kvashnin, and Cunningham/Mehta-Morse models are satisfied by a collocation technique. The drag of these eccentric porous particles relative to the drag experienced by a centered porous particle are investigated as functions of the effective distance between the center of the porous particle and the fictitious envelope, the volume ratio of the porous particle over the surrounding sphere and a coefficient that is proportional to the inverse of the permeability. In the limits of the motions of the porous particle in the concentric position with cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature.     

The Penetration Processes of Red Mud Filtrate in a Porous Medium by Seepage

This study investigated the effect of flow velocity, the concentration of red mud particles, and the concentration of \(\hbox {OH}^{-}\) ions on the penetration processes of red mud filtrate with fine particles in a porous medium by seepage. The results show that the peak concentrations of the breakthrough curves (BTCs) of red mud particles with high alkalinity are much higher than that with low alkalinity, indicating that the existence of \(\hbox {OH}^{-}\) ions enhances the repulsive interaction between red mud particles and between red mud particles and the matrix and promotes the migration of red mud particles. The red mud particles are more easily absorb onto the surface of porous medium or embedded in the matrix due to the greater adsorption between red mud particles and porous dielectric matrix than silicon powders. The penetration velocity of these red mud particles is often slower than water velocity due to the capture effect by straining and the detours path effect, especially in the case of high injection concentration and low alkalinity. Both the recovery rate and modal size of recovered particles increase with the increase in flow velocity, and the recovery rate of particles with high alkalinity is higher than that of particles with low alkalinity, which can be attributed to the stronger repulsive interaction between particles and between particles and the matrix. An analytical solution for the migration of particles in a porous medium in which the contaminant intensity varies with time has been developed from the elementary solution, and the predicted BTCs for a repeated three-pulse injection are in good accordance with the experimental results.     

Nonlinear Effects on Electrophoresis of a Soft Particle and Sustained Solute Release

A numerical study is made on the electrophoresis of a core-shell soft particle based on the first principle of electrophoresis. The soft particle consists of a charged rigid core coated with a polymer shell. Numerical computations for the electrophoretic velocity are obtained and compared with the existing analytical solution. The analytical solutions, based on the Boltzmann distribution of ions and the Debye–Huckel approximation, are valid for lower range of charge density, weak applied electric field and thin double layer. Discrepancy from the existing analytical solution is found when the Debye layer extends beyond the porous shell. This discrepancy becomes larger for higher values of the rigid core surface potential, fixed charge density of the soft shell and stronger imposed electric field. The double-layer polarization is found to have a strong impact when the shell thickness is lower than the Debye length. The electrophoretic velocity is found to vary nonlinearly with the imposed electric field when the imposed field strength is large enough to create a potential drop across the particle bigger than the thermal potential. We have also analyzed the mechanism of sustained solute release from the soft particle. Our results show that the rate of solute release is large compared to a pure diffusion dominated process.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.     

Slow Motion of a Porous Sphere Translating Along the Axis of a Circular Cylindrical Pore Subject to a Stress Jump Condition

The coupled flow problem of an incompressible axisymmetrical quasisteady motion of a porous sphere translating in a viscous fluid along the axis of a circular cylindrical pore is discussed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stress along with continuity of normal stress and velocity components are employed. The flow through the porous particle is governed by the Brinkman model and the flow in the outside porous region is governed by Stokes equations. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The boundary conditions are satisfied first at the cylindrical pore wall by the Fourier transforms and then on the surface of the porous particle by a collocation method. The collocation solutions for the normalized hydrodynamic drag force exerted by the clear fluid on the porous particle is calculated with good convergence for various values of the ratio of radii of the porous sphere and pore, the stress jump coefficient, and a coefficient that is proportional to the permeability. The shape effect of the cylindrical pore on the axial translation of the porous sphere is compared with that of the particle in a spherical cavity; it found that the porous particle in a circular cylindrical pore in general attains a lower hydrodynamic drag than in a spherical envelope.     

Steady rotation of a composite sphere in a concentric spherical cavity

The problem of steady rotation of a compositesphere located at the centre of a spherical container has beeninvestigated.A composite particle referred to in this paperis a spherical solid core covered with a permeable sphericalshell.The Brinkman’s model for the flow inside the composite sphere and the Stokes equation for the flow in the spherical container were used to study the motion.The torque experienced by the porous spherical particle in the presence ofcavity is obtained.The wall correction factor is calculated.In the limiting cases,the analytical solution describing thetorque for a porous sphere and for a solid sphere in an unbounded medium are obtained from the present analysis.     

Analysis of Multiphase Non-Darcy Flow in Porous Media

Recent laboratory studies and analyses (Lai et al. Presented at the 2009 Rocky Mountain Petroleum Technology Conference, 14–16 April, Denver, CO, 2009) have shown that the Barree and Conway model is able to describe the entire range of relationships between flow rate and potential gradient from low- to high-flow rates through porous media. A Buckley and Leverett type analytical solution is derived for non-Darcy displacement of immiscible fluids in porous media, in which non-Darcy flow is described using the Barree and Conway model. The comparison between Forchheimer and Barree and Conway non-Darcy models is discussed. We also present a general mathematical and numerical model for incorporating the Barree and Conway model in a general reservoir simulator to simulate multiphase non-Darcy flow in porous media. As an application example, we use the analytical solution to verify the numerical solution for and to obtain some insight into one-dimensional non-Darcy displacement of two immiscible fluids with the Barree and Conway model. The results show how non-Darcy displacement is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, and injection rates. Overall, this study provides an analysis approach for modeling multiphase non-Darcy flow in reservoirs according to the Barree and Conway model.     

A stochastic model for filtration of particulate suspensions with incomplete pore plugging

A population balance model for particulate suspension transport with capture of particles by porous medium accounting for complete and incomplete plugging of pores by retained particles is derived. The model accounts for pore space accessibility, due to restriction on finite size particle movement through the overall pore space, and for particle flux reduction, due to transport of particles by the fraction of the overall flux. The novel feature of the model is the residual pore conductivity after the particle retention in the pore and the possibility of one pore to capture several particles. A closed system of governing stochastic equations determines the evolution of size distributions for suspended particles and pores. Its averaging results in the closed system of hydrodynamic equations accounting for permeability and porosity reduction due to plugging. The problem of deep bed filtration of a single particle size suspension through a single pore size medium where a pore can be completely plugged by two particles allows for an exact analytical solution. The phenomenological deep bed filtration model follows from the analytical solution.     

Homotopy analysis method for thermophoretic particle deposition effect on magnetohydrodynamic mixed convective heat and mass transfer past a porous wedge in the presence of suction

Homotopy analysis method is used to analyze the effect of thermophoretic particle deposition on magnetohydrodynamic mixed convection flow with heat and mass transfer over a porous wedge. An explicit analytical solution is obtained which is valid throughout the solution domain and is consistent with numerical results.     

A solution to the hydraulic fracture problem in the traveling wave form

Solutions to the system of equations describing the propagation of hydraulic fracture cracks in a porous medium are obtained in the traveling wave form. The only sought solution is the separatrix of integral curves on the “penetration depth-crack width” plane. Some necessary dependencies that should be given at the crack inlet are found for the fluid flow rate and the fluid pressure. The crack width and the fluid penetration depth are related by power laws in the limiting cases when the crack propagation processes or the fluid penetration processes are dominant.     

Solid‐particle erosion in the tube end of the tube sheet of a shell‐and‐tube heat exchanger

Erosion is one of the major problems in many industrial processes, and in particular, in heat exchangers. The effects of flow velocity and sand particle size on the rate of erosion in a typical shell‐and‐tube heat exchanger were investigated numerically using the Lagrangian particle‐tracking method. Erosion and penetration rates were obtained for sand particles of diameters ranging from 10 to 500 µm and for inlet flow velocities ranging from 0.197 to 2.95 m/s. A flow visualization experiment was conducted with the objective of verifying the accuracy of the continuous phase calculation procedure. Comparison with available experimental data of penetration rates was also conducted. These comparisons resulted in a good agreement. The results show that the location and number of eroded tubes depend mainly on the particle size and velocity magnitude at the header inlet. The rate of erosion depends exponentially on the velocity. The particle size shows negligible effect on the erosion rate at high velocity values and the large‐size particles show less erosion rates compared to the small‐size particles at low values of inlet flow velocities. The results indicated that the erosion and penetration rates are insignificant at the lower end of the velocity range. However, these rates were found to increase continuously with the increase of the inlet flow velocity for all particle sizes. The particle size creating the highest erosion rate was found to depend on the flow velocity range. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical analysis of fluid flow and mass transfer in a channel with a porous bottom wall

The flow of a solution between parallel plates is considered. The bottom plate is porous, while the top one is an impermeable solid. A computer program based on the control volume approach was developed to analyse the flow and concentration fields. The effects of the slip at the porous wall on the velocity and particle concentration distributions were investigated. It was observed that as the slip increases, the concentration on the porous wall decreases and the maximum velocity moves towards the porous wall. The concentration on the porous wall increases in the flow direction. This increase in the particle concentration along the porous wall may cause a reduction of the porosity and hence a variation in the suction rate along the porous wall. In order to take this effect into account, a linearly varying transverse velocity along the porous wall was considered. The results were compared with the data available in the literature.     

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.     

The stokes problem for a porous particle with radially nonuniform porosity

The flow past a nonuniform porous spherical particle immersed in a uniform steady-state stream is studied in the Stokes approximation. For a power-law radial dependence of the particle permeability coefficient, an analytical solution for the velocity and pressure fields outside and inside the particle is obtained. Volgograd, Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–184, January–February, 2000.     

Particle imaging techniques for microfabricated fluidic systems

This paper presents the design and implementation of velocimetry techniques applicable to the analysis of microfluidic systems. The application of both micron-resolution particle image velocimetry (micro-PIV) and particle tracking velocimetry (PTV) to the measurement of velocity fields within micromachined fluidic channels is presented. The particle tracking system uses epifluorescent microscopy, CCD imaging, and specialized image interrogation algorithms to provide microscale velocity measurement resolution. The flow field in a straight channel section is measured using cross-correlation micro-PIV and compared to the analytical solution for a measured mass flow rate. Velocity field measurements of the flow at the intersection of a cross-channel are also presented and compared with simulations from a commercially available flow solver, CFD-ACE+. Discussions regarding flow seeding, imaging optics, and the flow setup for measuring flows in microfabricated fluidic devices are presented. A simple process for estimating measurement uncertainty of the in-plane velocity measurements caused by three-dimensional Brownian motion is described. A definition for the measurement depth for PTV measurements is proposed. The agreement between measured and predicted values lends further support to the argument that liquid microflows with characteristic dimensions of order 50-μm dimension channels follow macroscale flow theory.     

Effects of particle migration on suspension flow in a hydraulic fracture

An asymptotic model of a hydraulic-fracture flow of a sedimenting concentrated suspension is formulated on the basis of the two-fluid approach with account of transverse particle migration. In the thin-layer approximation, a two-dimensional system of equations averaged across the fracture is constructed with account for a nonuniform distribution of the particle concentration. As compared to the similar model without particle migration, the averaged two-dimensional equations contain modified coefficients which explicitly depend on the width of the flow core occupied by the particles. Using the model constructed, a numerical simulation is performed, which shows that the particle migration towards the fracture center results in the increase in the depth of particle penetration into the fracture and the suppression of gravitational convection in the vicinity of the leading front. The calculations are compared with available experimental data and an analytical formula for the height of the dense packed sediment. A good agreement between the analytical theory, the experiments, and the two-dimensional calculations is attained.     

Oscillatory flow due to eccentrically rotating porous disk and a fluid at infinity embedded in a porous medium

An oscillatory flow due to non-coaxial rotations of an oscillating porous disk and a fluid at infinity rotating about an axis parallel to the axis of rotation of the disk through a fixed point has been investigated. An analytical solution of the unsteady Navier-Stokes equations is obtained for three cases when the frequency parameter is less than, equal to or greater than the rotation parameter. The influences of the physical parameters acting on the flow are explained with the help of the figures. It is found that the depth of the penetration or the wave length of the layers decreases with an increase in porosity parameter.     

A New Non-Darcy Flow Model for Low-Velocity Multiphase Flow in Tight Reservoirs

The pore and pore-throat sizes of shale and tight rock formations are on the order of tens of nanometers. The fluid flow in such small pores is significantly affected by walls of pores and pore-throats. This boundary layer effect on fluid flow in tight rocks has been investigated through laboratory work on capillary tubes. It is observed that low permeability is associated with large boundary layer effect on fluid flow. The experimental results from a single capillary tube are extended to a bundle of tubes and finally to porous media of tight formations. A physics-based, non-Darcy low-velocity flow equation is derived to account for the boundary layer effect of tight reservoirs by adding a non-Darcy coefficient term. This non-Darcy equation describes the fluid flow more accurately for tight oil reservoir with low production rate and low pressure gradient. Both analytical and numerical solutions are obtained for the new non-Darcy flow model. First, a Buckley–Leverett-type analytical solution is derived with this non-Darcy flow equation. Then, a numerical model has been developed for implementing this non-Darcy flow model for accurate simulation of multidimensional porous and fractured tight oil reservoirs. Finally, the numerical studies on an actual field example in China demonstrate the non-negligible effect of boundary layer on fluid flow in tight formations.     

Measurement and simulation of the two-phase velocity correlation in sudden-expansion gas-particle flows

In this paper the present authors measured the gas-particle two-phase velocity correlation in sudden expansion gas-particle flows with a phase Doppler particle anemometer(PDPA) and simulated the system behavior by using both a Reynolds-averaged Navier-Stokes(RANS) model and a large-eddy simulation(LES).The results of the measurements yield the axial and radial time-averaged velocities as well as the fluctuation velocities of gas and three particle-size groups(30 μ m,50 μ m,and 95 μ m) and the gas-particle velocity correlation for 30 μ m and 50 μ m particles.From the measurements,theoretical analysis,and simulation,it is found that the two-phase velocity correlation of sudden-expansion flows,like that of jet flows,is less than the gas and particle Reynolds stresses.What distinguishes the two-phase velocity correlations of sudden-expansion flow from those of jet and channel flows is the absence of a clear relationship between the two-phase velocity correlation and particle size in sudden-expansion flows.The measurements,theoretical analysis,and numerical simulation all lead to the above-stated conclusions.Quantitatively,the results of the LES are better than those of the RANS model.     

Experimental Investigation and Modelling of Colloidal Release in Porous Media

Gas production from underground storage reservoirs is sometimes associated with solid particles eroded from the rock matrix. This phenomenon often called sand production can cause damage to the storage equipments, leading the operator to choke the wells and prevent them from producing at full capacity. Colloid release is often associated as a precursor of larger solid production. Indeed, in sandstone storage sites, clay release induced by the presence of condensed water associated with the gas production in the near-wellbore region can be a forecast of intergranular cement erosion. The objective of this work is twofold: firstly to experimentally investigate colloidal particle detachment through ionic strength reduction (absence of salinity of the condensed water) in porous media and secondly to determine its evolution with time and to model it. Laboratory experiments with model systems are developed to reproduce the particle generation and their transport in porous media. The model porous medium is a packed column of two powders: silicon carbide particles of 50 μm and silica particles of 0.5 μm (3% by weight) initially mixed together. Brine flows at different concentrations are imposed through the porous sample and, at very low salt concentration, colloid silica particles are massively released from the medium. Experimental evolutions of the particle concentration with time are compared to solutions of the advection–dispersion equation including first-order source terms for colloid release. The dispersion coefficients of the porous medium have been determined with tracer tests. The experimental results exhibit a different behaviour at short- and long-time intervals and a model has been built to predict the colloid production evolution with the introduction of two different time scales for the eroded rate. The model can be used in a core test to evaluate the amount of detachable fines and the rate of erosion.