Numerical solution and sensitivity analysis of filter cake permeability and resistance to model parameters

The migration and capture of solid particles in porous media occur in fields as diverse as water and wastewater treatment, well drilling, and in various liquid-solid separation processes. Filter cakes are formed when a liquid containing solid particles is forced through a pervious surface which allows the liquid transport while retaining solid particles. Following a literature survey, a governing equation for the cake thickness is obtained by considering the instantaneous mass balance. Later, numerical solutions for the cake thickness, cake permeability, cake resistance, solid particle velocity (cake compression rate) and concentration of suspended particles are obtained and a sensitivity analysis is conducted. The sensitivity analysis shows that the cake permeability and cake resistance are more sensitive to the rate constant of cake erosion than they are to the rate constant of particle capture. However, the concentration of suspended solid particles, and the solid velocity are mostly sensitive to the slurry parameter and the rate constant of particle trapping. Moreover, cake permeability, compressibility, concentration of suspended particles, and the solid velocity are very sensitive to the concentration at the filter septum. Finally, as expected, with a thicker slurry, more particles are captured inside the cake, thus forming a thicker and more resistant cake. Also, as more particles are being filtered at the filter septum, a thinner cake is formed and a smaller effluent concentration is achieved.     

Removal of Particle Bridges From a Porous Material by Ultrasonic Irradiation

Particle bridge formation during the flow of a liquid with particles through a porous material is a fouling mechanism that can block the pores and, hence, decrease the permeability of the material. Ultrasonic irradiation of the material is a cleaning method that can restore the permeability. We make a numerical study of this cleaning method using the lattice-Boltzmann method. We start from a pore blocked by two spherical particles attached to the pore wall by colloidal adhesion forces, thus forming a particle bridge. Next we calculate the hydrodynamic force exerted by a high-frequency acoustic wave on the two particles. By comparing the hydrodynamic force and the adhesion force we investigate, whether the particle bridge will be removed by the ultrasonic irradiation. A sensitivity study is carried out to investigate the influence of some relevant parameters, such as the acoustic wave amplitude, the acoustic frequency, the fluid flow velocity and the ratio of particle diameter and pore diameter. An upscaling procedure is applied to translate the microscopic results for the removal of the particles at the pore level to the permeability improvement of the material at the macroscopic level. A comparison is made between numerical results and experimental data. The agreement is reasonable.     

Permeability Reduction in Qishn Sandstone Specimens due to Particle Suspension Injection

Produced water re-injection is one of the viable methods to manage the large amount of water produced from oil recovery operations. Permeability reduction caused by the particles being retained within porous formations is a major problem affecting the effectiveness of the process. This article presents an experimental investigation of permeability reduction in Qishn sandstone specimens due to particle suspension injection. Test results indicated that the particle deposition and permeability damage varied along the test specimen length depending on the imposed testing conditions. The filtration coefficient was difficult and complex to delineate from the transient injection data. However, it was found that the permeability reduction was uniquely correlated with the amount of particles retained within the specimen independent of the injected influent concentration, injected velocity, injected fluid volume, and particle deposition profile. It was possible to determine the permeability reduction coefficient directly from the laboratory measurement without solving the particle deposition profile from the coupled particle deposition fluid-flow equations.     

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.     

A Two-Dimensional Network Model to Simulate Permeability Decrease Under Hydrodynamic Effect of Particle Release and Capture

A model was developed to simulate permeability decrease induced by hydrodynamic effects when injecting a fluid in a reservoir with respect to particle release and capture mechanisms and the parameters of the fluid–rock system. The kinetics of particle release and capture were integrated after computing the initial permeability of the porous medium with a square lattice of a two–dimensional network model. The rate of particle release is related to the difference between a microscopic velocity of the fluid and a critical velocity. The permeability decrease shows a direct link to the reduction of pore throat radii by three mechanisms of particle capture: straining and particle accumulation through direct interception or diffusion. Comparison between the simulations and the experimental results shows that the model reproduces the physics of the permeability decrease phenomenon, although the values are overestimated. The difference between the two sets of results can be explained by the fact that the simulations are realized at constant pressure whereas the experiments are realized at constant flow rate, and that re–entrainment of the trapped particles was not taken into account in the model.     

AN EULERIAN FINITE ELEMENT METHOD FOR TIME-DEPENDENT FREE SURFACE PROBLEMS IN HYDRODYNAMICS

A numerical method based on the finite element method is presented for simulating the two-dimensional transient motion of a viscous liquid with free surfaces. For ease of numerical treatment of the free surface expressed by a multiple-valued function, the marker particle method is employed. Numerous virtual particles are spread over all regions occupied by liquid. They move about on a fixed finite element mesh with the liquid velocity at their positions. These particles contribute nothing to the dynamics of the liquid and only serve as markers of liquid regions. The velocity field within liquid regions is calculated by solving the Navier– Stokes equations and the equation of continuity by the finite element method based on quadrilateral elements. A detailed discussion is given of the methodological problems arising in the implementation of the marker particle method on an unstructured finite element mesh and of the solutions to these problems. The proposed method is demonstrated on three sample problems: the broken dam problem, the impact of a falling liquid drop on a still liquid and the entry of a rigid block into water. Good agreement has been obtained in the comparison of the present numerical results with available experimental data.     

Permeability anisotropy due to consolidation of compressible porous media

The characterisation of flow through porous media is important for all solid–liquid separation and fluid transport realms. The permeability of porous media can be anisotropic and furthermore, the extent of anisotropy can be increased as a result of an applied compressive force. However, the understanding of how anisotropy develops is incomplete. An overview of research on permeability anisotropy is given and an expression for predicting anisotropy as a function of void ratio is offered. The two underlying assumptions of the proposed model are: flow in different directions occurs within the same network of pores and deformation is primarily due to the compression of the particles in the direction of the applied force rather than due to particle rearrangement. The assumption of network connectivity allows permeability anisotropy to be described as a function of flow path tortuosity only. Results are presented for hydraulic anisotropy measured in lignite that has been upgraded by a compression dewatering method known as mechanical thermal expression. The lignite permeability is shown to be up to eight times greater in the direction perpendicular to compression, suggesting that the rate of dewatering could be significantly increased by choosing the drainage to also be perpendicular to the direction of the applied compressive force. It is illustrated that the proposed anisotropy model can be used to accurately predict the experimentally determined permeability anisotropy ratios for lignite, as well as for other materials including sand, clay and kaolin.     

Non-isothermal Permeability Impairment by Fines Migration and Deposition in Porous Media including Dispersive Transport

Particle migration and deposition, and resulting permeability impairment occurring in porous media are described by a practical phenomenological model considering temperature variation and particle transport by advection and dispersion. Variation of the filter coefficient and permeability of porous matrix by temperature and particle deposition, and other essential factors are considered by means of the special correlations of the relevant variables and dimensionless numbers. Comparison of the numerical results, obtained using a finite-difference numerical scheme with and without considering the dispersion mechanism and temperature variation, reveals the significance of such effects on fines migration and deposition, and consequent permeability impairment in porous media. Improved model presented in this article can be instrumental for scientifically guided experimentation, analysis, and optimal design of processes involving in transport of colloidal and fine particles through geological subsurface formations.     

Smoothed Particle Hydrodynamics Modeling of Transverse Flow in Randomly Aligned Fibrous Porous Media

The Lagrangian smoothed particle hydrodynamics (SPH) method is employed to obtain a meso-/micro-scopic pore-scale insight into the transverse flow across the randomly aligned fibrous porous media in a 2D domain. Fluid is driven by an external body force, and a square domain with periodic boundary conditions imposed at both the streamwise and transverse flow direction is assumed. The porous matrix is established by randomly embedding a certain number of fibers in the square domain. Fibers are represented by position-fixed SPH particles, which exert viscous forces upon, and contribute to the density variations of, the nearby fluid particles. An additional repulsive force, similar in form to the 12-6 Lennard-Jones potential between atoms, is introduced to consider the no-penetrating restraint prescribed by the solid pore structure. This force is initiated from the fixed solid material particle and may act on its neighboring moving fluid particles. Fluid flow is visualized by plotting the local velocity vector field; the meandering fluid flow around the porous microstructures always follow the paths of least resistance. The simulated steady-state flow field is further used to calculate the macroscopic permeability. The dimensionless permeability (normalized by the squared characteristic dimension of the fiber cross section) exhibits an exponential dependence on the porosity within the intermediate porosity range, and the derived dimensionless permeability—porosity relation is found to have only minor dependence on either the relative arrangement condition among fibers or the fiber cross section (shape or area).     

Onset velocity of circulating fluidization and particle residence time distribution:A CFD-DEM study

Until now,the onset velocity of circulating fluidization in liquid-solid fluidized beds has been defined by the turning point of the time required to empty a bed of particles as a function of the superficial liquid velocity,and is reported to be only dependent on the liquid and particle properties.This study presents a new approach to calculate the onset velocity using CFD-DEM simulation of the particle residence time distribution(RTD).The onset velocity is identified from the intersection of the fitted lines of the particle mean residence time as a function of superficial liquid velocity.Our results are in reasonable agreement with experimental data.The simulation indicates that the onset velocity is influenced by the density and size of particles and weakly affected by riser height and diameter.A power-law function is proposed to correlate the mean particle residence time with the superficial liquid velocity.The collisional parameters have a minor effect on the mean residence time of particles and the onset velocity,but influence the particle RTD,showing some humps and trailing.The particle RTD is found to be related to the particle trajectories,which may indicate the complex flow structure and underlying mechanisms of the particle RTD.     

Experimental investigation of microsuspension filtration in a highly-permeable porous medium

Filtration of a microsuspension of aluminosilicate particles in a highly-permeable porous medium formed by a glass-sphere filling is investigated. The particle-to-filling granule diameter ratio varied from 0.046 to 0.109 and the volumic particle concentration ratio from 0.001 to 0.02. The data on the specimen permeability dynamics and its dependence on the initial concentration of the suspension microparticles and the ratio of their diameter to that of the porous medium granules are obtained. The permeability is shown to considerably vary along the specimen. Two stages of the porous medium damage by the trapped microparticles are established. The critical parameter of the formation of an impermeable cake in the initial region of the specimen are determined.     

Local gas and liquid phase velocity measurement in a miniature stirred vessel using PIV combined with a new image processing algorithm

A method which combines standard two-dimensional particle image velocimetry (PIV) with a new image processing algorithm has been developed to measure the average local gas bubble velocities, as well as the local velocities of the liquid phase, within small stirred vessel reactors. The technique was applied to measurements in a gas–liquid high throughput experimentation (HTE) vessel of 45 mm diameter, but it is equally suited to measurements in larger scale reactors. For the measurement of liquid velocities, 3 μm latex seeding particles were used. For gas velocity measurements, a separate experiment was conducted which involved doping the liquid phase with fluorescent Rhodamine dye to allow the gas–liquid interfaces to be identified. The analysis of raw PIV images enabled the detection of bubbles within the laser plane, their differentiation from obscuring bubbles in front of the laser plane, and their use in lieu of tracer particles for gas velocity analysis using cross-correlation methods. The accuracy of the technique was verified by measuring the velocity of a bubble rising in a vertical glass column. The new method enabled detailed velocity fields of both phases to be obtained in an air–water system. The overall flow patterns obtained showed a good qualitative agreement with previous work in large scale vessels. The downward liquid velocities above the impeller were greatly reduced by the addition of the gas, and significant differences between the flow patterns of the two-phases were observed.     

Multi-fluid model of suspension filtration in a porous medium

In the framework of a three-fluid approach, a new model of suspension filtration in a porous medium is constructed with account for the formation of a dense packing of trapped particles with finite permeability and porosity. The following three continua are considered: the carrier fluid, the suspended particles, and the deposited particles. For a one-dimensional transient flow of suspension, a system of equations for the concentrations of the suspended and deposited particles, the suspension velocity, and the pressure is constructed. Two cases of the flow in a porous medium are considered: plane and radial. Numerical solution is found using a finite-difference method. Numerical calculations are shown to be in agreement with an analytical solution for the simplest case of filtration with a constant velocity and constant porosity and permeability. A comparison is performed with the classic filtration models for five sets of experimental data on the contamination of a porous sample. It is shown that near the inlet boundary, where an intense deposition of particles takes place, the new model describes the concentration profile of the deposited particles more accurately than the classical model.     

No‐slip consistent immersed boundary particle tracking to simulate impaction filtration in porous media

In this paper, we present a new method for simulating the motion of a disperse particle phase in a carrier gas through porous media. We assume a sufficiently dilute particle‐laden flow and compute, independently of the disperse phase, the steady laminar fluid velocity using the immersed boundary method. Given the velocity of the carrier gas, the equations of motion for the particles experiencing the Stokes drag force are solved to determine their trajectories. The ‘no‐slip consistent’ particle tracking algorithm avoids possible numerical filtration of very small particles due to the nonzero velocity field at the solid–fluid interface introduced by the immersed boundary method. This physically consistent tracking allows a reliable estimation of the filtration efficiency of porous filters due to inertial impaction. We illustrate and test our new approach for model porous media consisting of a structured array of aligned rectangular fibers, arranged in line and staggered. In the staggered geometry, the effect of the residual velocity at the solid–fluid interface is significant for particles with low inertia. Without adopting the developed no‐slip consistent numerical method, an artificial numerical filtration is observed, which becomes dominant for small enough particles. For both the in line and the staggered geometries, the filtration rate depends quite strongly and non monotonically on the particle inertia. This is expressed most clearly in the staggered arrangement in which a very strong increase in the filtration efficiency is observed at a well‐defined critical droplet size, corresponding to a qualitative change in the dominant particle paths in the porous medium. Copyright © 2013 John Wiley & Sons, Ltd.     

Onset velocity of circulating fluidization and particle residence time distribution: A CFD–DEM study

Until now, the onset velocity of circulating fluidization in liquid–solid fluidized beds has been defined by the turning point of the time required to empty a bed of particles as a function of the superficial liquid velocity, and is reported to be only dependent on the liquid and particle properties. This study presents a new approach to calculate the onset velocity using CFD–DEM simulation of the particle residence time distribution (RTD). The onset velocity is identified from the intersection of the fitted lines of the particle mean residence time as a function of superficial liquid velocity. Our results are in reasonable agreement with experimental data. The simulation indicates that the onset velocity is influenced by the density and size of particles and weakly affected by riser height and diameter. A power-law function is proposed to correlate the mean particle residence time with the superficial liquid velocity. The collisional parameters have a minor effect on the mean residence time of particles and the onset velocity, but influence the particle RTD, showing some humps and trailing. The particle RTD is found to be related to the particle trajectories, which may indicate the complex flow structure and underlying mechanisms of the particle RTD.     

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.     

Boundary conditions at the interface between fluid layer and fibrous medium

The flow through a channel partially filled with fibrous porous medium was analyzed to investigate the interfacial boundary conditions. The fibrous medium was modeled as a periodic array of circular cylinders, in a hexagonal arrangement, using the boundary element method. The area and volume average methods were applied to relate the pore scale to the representative elementary volume scale. The permeability of the modeled fibrous medium was calculated from the Darcy's law with the volume‐averaged Darcy velocity. The slip coefficient, interfacial velocity, effective viscosity and shear jump coefficients at the interface were obtained with the averaged velocities at various permeabilities or Darcy numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

Probabilistic approach for transport of contaminants through porous media

The channels formed between individual particles in porous media have variable dimensions and orientations. The porosity, permeability and its anisotropy exhibit random spatial distributions. The probabilistic approach can effectively describe the transport of contaminants through porous media and is analysed in this paper. Numerical results are obtained by considering (I) random dispersion coefficients without and with spatial structure, (II) random time distribution of concentration at the inlet boundary, (III) random velocity distribution in the flow field without and (IV) with variable dispersion coefficient, (V) non-linearity of the governing equation and (VI) anisotropy of the dispersion coefficient. Two methods are used for probabilistic predictions: (1) Gaussian field approach in conjunction with Monte Carlo method and (2) random walk method. The input random parameters are assumed to have normal and log-normal distributions according to available experimental data. The probability distribution functions of the contaminant concentration at different locations within the flow domain are calculated and compared with the input distributions as a function of the mean and fluctuation Peclet numbers. The one-dimensional case is analysed in detail and the illustrative numerical predictions are compared with analytical and experimental results. The extension to a two-dimensional domain is discussed in the last part of this paper.