Mechanisms of Particle Transport Acceleration in Porous Media

Experimental data show that the groundwater transport of radionuclides in porous media is frequently facilitated when accompanied with colloid particles. This is usually explained by the size exclusion mechanism which implies that the particles move through the largest pores where the flow velocity is higher. We call attention to three other mechanisms which influence the colloid particle motion, while determining both the probable transport facilitation and retardation. First of all, it is shown that the transport facilitation may be significantly reduced and even transformed into a retardation due to the growth of the effective suspension viscosity (a friction-limited facilitation). Secondly, we will show that the transport of particles through the largest pores can be retarded due to a reduced connectivity of the large-pore cluster (a percolation-breakup retardation). Thirdly, we highlight the Fermi mechanism of acceleration known in statistical physics which is based on the elastic collisions between particles. All three effects are analyzed in terms of the velocity enhancement factor, by using statistical models of porous media in the form of a capillary bundle and a 3D capillary network. Optimal and critical regimes of velocity enhancement are quantified. Estimations show that for realistic parameters, the maximal facilitation of colloid transport is close to the experimentally observed data.     

Effects of Flow Velocity and Nonionic Surfactant on Colloid Straining in Saturated Porous Media Under Unfavorable Conditions

Knowledge of colloid straining mechanism in porous media is of importance for protecting groundwater from being contaminated by biocolloids (e.g., bacteria and protozoa) and by contaminants whose transport can be facilitated by mobile particles. This study examined effects of flow velocity on colloid straining in porous media under unfavorable chemical conditions. Saturated column experiments were conducted using glass beads as collector and a $3\,\mu \text{ m}$ carboxylate-modified polystyrene latex microsphere as model colloid. To unambiguously examine colloid straining mechanisms, attachment was minimized by extensively cleaning the collectors and adopting deionized water as solution. Results show that increasing flow velocity decreases colloid straining under unfavorable chemical conditions, in agreement with to theoretical finding in literature. This study additionally examined effects of nonionic surfactant (Triton X-100) on colloid straining in porous media under unfavorable chemical conditions. Results show that the addition of Triton X-100 decreases colloid straining and the decrease is enhanced by increasing the concentration of Triton X-100.     

Effect of an unsteady swirled turbulent flow on the motion of a single solid particle

An unsteady swirled turbulent flow between two rotating flat disks is modeled. The flow is directed along the radius toward the rotation axis. A quasi-steady character of the turbulent flow, caused by oscillations of the radial velocity at the entrance to the gap between the disks with a period close to the time of dynamic relaxation of the particle, is studied with the use of the known two-equation Wilcox’s k-ω model of turbulence. The influence of the Stokes number and the frequency and amplitude of oscillations of the carrier medium on the motion of single particles in the field of centrifugal and aerodynamic forces is considered.     

Multiphase Flow Fields in In-situ Air Sparging and its Effect on Remediation

In-situ air sparging (IAS) is used for the clean-up of soil and groundwater that are contaminated with volatile organic compounds in relatively permeable subsurface environments. In this study, we investigated the secondary groundwater and gas flow fields that develop in the vicinity of single and multiple air sparging wells. The purpose is to evaluate their effects on contaminant plume migration and thus, remediation. Governing equations describing multiphase flow and contaminant transport in a three-dimensional domain were formulated and solved using the Galerkin finite element technique. Trichloroethylene was selected as a target contaminant. The increase in air injection contributed to an increase in the size of the IAS cone of influence and the gas saturation levels within the cone. This reduced the groundwater velocity within the cone and increased the zone of detour of groundwater around the air sparging wells. This outcome was quantified and compared under several IAS operations. Different soil permeability characteristics also affected the groundwater and gas flow patterns, and this impacted the remedial performance of the IAS system. Under high ambient groundwater velocity, an air sparging system that uses a single injection well caused the detour of contaminant plumes around injection wells, regardless of air injection rates, and failed to meet the remedial goal specified here. This system was successful for relatively low ambient groundwater velocity environments used here. An IAS system with multiple injection wells was effective in capturing and remediating the detoured contaminant plume, and showed superior performance when compared to a single injection well IAS system. Using IAS simulation, we also analyzed the impact of injection rates on site remediation using single or multiple wells. Design criteria that are based on the results of this study would be useful in enhancing the performance of the IAS systems.     

Effective Solution through the Streamline Technique and HT-Splitting for the 3D Dynamic Analysis of the Compositional Flows in Oil Reservoirs

In this article we suggest a new phenomenological mathematical model for the groundwater transport of colloid particles through porous media which is able to describe some significant effects experimentally observed but not captured within the framework of the classic approach. Our basic idea is to consider both the pure water and the colloid suspension as two thermodynamic phases. Using the network models of porous media, we simulated numerically the transport process at the pore-scale. By averaging the result derived, we have obtained the relative permeabilities for both phases, the percolation threshold for suspension flow, and the effective suspension viscosity. Due to specific laws of colloid particles repartition between various classes of pores, the relative permeability of suspension happens to be a highly nonlinear function of saturation, very far from the diagonal straight line. This determines a difference between the macroscale phase velocities. The suspension velocity is shown to be higher than that of water in major cases, only if the colloid particles are not too large. The suggested model predicts and describes in a closed form the effect of colloid transport facilitation observed experimentally.     

双柱状颗粒在线性剪切流场中的动力学分析

为研究柱状颗粒在线性剪切流场中的运动状态和受力情况,本文以颗粒长径比为2,颗粒之间的初始距离ΔS_Py=4D为例,基于直接力浸入边界法数值模拟了双柱状颗粒在三维线性剪切流场中的运动过程。根据模拟结果分析了柱状颗粒周围流场参数分布,在考虑壁面对颗粒的影响和颗粒之间相互影响的条件下,研究了颗粒的受力和运动的变化,探索了流体曳力导致柱状颗粒迁移和转动的规律。研究结果表明,双柱状颗粒在线性剪切流场中易向速度大的流体区域运动;前后两颗粒运动状态和轨迹不同,颗粒之间距离较近时,曳力会产生较大的波动;只有当颗粒在壁面附近时,滞后颗粒才能追上领先颗粒,两颗粒发生牵引、翻滚和分离过程。     

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.     

Simulation of particles released near the wall in a turbulent boundary layer

A direct numerical simulation was used along with a Lagrangian particle tracking technique to study particle motion in a horizontal, spatially developing turbulent boundary layer along an upper-wall (with terminal velocity directed away from the wall). The objective of the research was to study particle diffusion, dispersion, reflection, and mean velocity in the context of two parametric studies: one investigated the effect of the drift parameter (the ratio of particle terminal velocity to fluid friction velocity) for a fixed and finite particle inertia, and the second varied the drift parameter and particle inertia by the same amount (i.e. for a constant Froude number). A range of drift parameters from 10⁻⁴ to 10⁰ were considered for both cases. The particles were injected into the simulation at a height of four wall units for several evenly distributed points across the span and a perfectly elastic wall collision was specified at one wall unit.Statistics collected along the particle trajectories demonstrated a transition in particle movement from one that is dominated by diffusion to one that is dominated by gravity. For small and intermediate sized particles (i.e. ones with outer Stokes numbers and drift parameters much less than unity) transverse diffusion away from the wall dominated particle motion. However, preferential concentration is seen near the wall for intermediate-sized particles due to inhomogeneous turbulence effects (turbophoresis), consistent with previous channel flow studies. Particle–wall collision statistics indicated that impact velocities tended to increase with increasing terminal velocity for small and moderate inertias, after which initial conditions become important. Finally, high relative velocity fluctuations (compared to terminal velocity) were found as particle inertia increased, and were well described with a quasi-one-dimensional fluctuation model.     

Flow and migration of nanoparticle in a single channel

A numerical simulation based on a combined Euler and Lagrange method is investigated in this work to simulate the flow and migration of nanoparticles in a single channel. The motion of discrete nanoparticles is determined by the Lagrangian trajectory method based on the Newton’s second law that includes the influence of the body force, various hydrodynamic forces, the Brownian motion and the thermophoresis force. The coupling of discrete particles with continuous flow is realized through the modification of the source term of the continuous equation. The results reveal the two-phase flow nature of nanoparticle suspensions and their implications to the convective heat transfer of nanofluids.     

The Shape of Moving Boundary of Fluid Flow in Sandstone: Video Microscopic Investigation and Stochastic Modeling Approach

Many engineering problems such as exploitation of petroleum and gas, deposition of nuclear waste, and groundwater contamination by organic liquids are closely related to the movement of fluid in rocks. In this paper, a video microscope is employed to investigate the shape of moving front boundary of fluid flow in sandstone. The experimental results show that the fronts of the moving boundary display a fractal behavior. Based on the experimental results, a stochastic differential equation is proposed to describe the moving boundary. By decomposing the velocity of a given point into a drift term and a fluctuation term, the effect of the mesoscope structure of porous media on fluid flow is taken into account. The stochastic approach is in agreement with the experimental results. The analysis shows that the front of the moving boundary of fluid flow in rocks is a comprehensive result caused by the average tendency of fluid flow, which can be described by the classical Darcys Law, and the fluctuation tendency of fluid flow, which is closely related to the mesoscope structure of rocks.     

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).     

Dynamics of dual-particles settling under gravity

As a first step towards understanding particle–particle interaction in fluid flows, the motion of two spherical particles settling in close proximity under gravity in Newtonian fluids was investigated experimentally for particle Reynolds numbers ranging from 0.01 to 2000. It was observed that particles repel each other for Re>0.1 and that the separation distance of settling particles is Reynolds number dependent. At lower Reynolds numbers, i.e. for Re<0.1, particles settling under gravity do not separate.The orientation preference of two spherical particles was found to be Reynolds number dependent. At higher Reynolds numbers, the line connecting the centres of the two particles is always horizontal, regardless of the way the two particles are launched. At lower Reynolds numbers, however, the particle centreline tends to tilt to an arbitrary angle, even of the two particles are launched in the horizontal plane. Because of the tilt, a side migration of the two particles was found to exist. A linear theory was developed to estimate the side migration velocity. It was found that the maximum side migration velocity is approximately 6% of the vertical settling velocity, in good agreement with the experimental results.Counter-rotating spinning of the two particles was observed and measured in the range of Re=0–10. Using the linear model, it is possible to estimate the influence of the tilt angle on the rate of rotation at low Reynolds numbers. Dual particles settle faster than a single particle at small Reynolds numbers but not at higher Reynolds numbers, because of particle separation. The variation of particle settling velocity with Reynolds number is presented. An equation which can be used to estimate the influence of tilt angle on particle settling velocity at low Reynolds number is also derived.     

Inertial migration of sedimenting particles in a suspension flow through a Hele-Shaw cell

Within the framework of the model of two interpenetrating continua, a horizontal laminar dilute-suspension flow in a vertical Hele-Shaw cell is investigated. Using the method of matched asymptotic expansions, an asymptotic model of the transverse migration of sedimenting particles is constructed. The particle migration in the horizontal section of the cell is caused by an inertial lateral force induced by the particle sedimentation and the shear flow of the carrier phase. A characteristic longitudinal length scale is determined, on which the particles migrate across the slot through a distance of the order of the slot half-width. The evolution of the particle number concentration and velocity fields along the channel is studied using the full Lagrangian method. Depending on the particle inertia parameter, different particle migration regimes (with and without crossing of the channel central plane by the particles) are detected. A critical value of the particle inertia parameter corresponding to the change in migration regime is found analytically. The possibility of intersection of the particle trajectories and the formation of singularities in the particle number concentration is demonstrated.     

Electron conductivity of a thermally ionized gas in an electric field

It is well known that the current carriers in a thermally ionized gas vary in composition, but that electrons [1] make the fundamental contribution to the conductivity of the gas, since their mobility is incomparably larger than that of other current-carrying particles. We shall thus be concerned only with electron conductivity. If the gas is under a high pressure in a weak electric field, then in estimating its electrical conductivity by classical means the same concepts are usually employed as those which Drude applied in the theory of metallic conduction. The Drude-Lorentz formula for electrical conductivity was subsequently perfected by Cowling and Chapman who introduced a coefficient to take into account the rate at which the particle interaction forces decrease with distance [2], For electron Coulomb interaction this coefficient takes the value 0.532 instead of 0.500 as compared with the Drude-Lorentz formula.For high pressures and low electric field strengths the electron drift velocity in the field is vanishingly small compared with the mean velocity of random motion, and so it is logical to suppose that the electron free time is independent of the drift velocity, and this supposition leads in the end to the conclusion that Ohm's law is applicable to gases at high pressure in very weak fields.However, we must not overlook the fact that even under the conditions mentioned the conclusion concerning the validity of Ohm's law is only an approximation which becomes less accurate, the lower the gas pressure and the greater the field strength.In what follows the conductivity of the gas is also determined by Drude's method, but with the refinement that in determining the electron free time the drift velocity of these particles in the field is considered.     

Influence of large eddies on the suspension of solid particles

The phenomena of solid particles suspensions, in a turbulent flow, can more conveniently be described by stochastic models than by diffusion models, particularly in the case of relatively coarse particles.

The fundamental difficulties of using such models are principally due to the difficulty of performing direct measurements of probabilities, because the number of observations (or tests) necessary to obtain physically representative values is important (theoretically infinite).

We have used such a model to describe the movement of spheres in an inclinable pipe.

To do so, we have identified the movement through a Markov process which permits us to show that we can characterize it by the limit distribution for passage probabilities in a cross section. We have used a special system of close-circuit television to measure it, doing a sufficiently large number of observations for the measurements to be significant.

In the case of a vertical pipe, the phenomena is one-dimensional. By using the model stochastic displacement, we obtain a differential equation which it is possible to integrate by assuming an obviously constant radial dispersion. The interpretation of limit distributions for passage probabilities and visual observations of particles movement in the pipe have caused us to conclude that the mean displacment is due, on one hand, to a radial acceleration bounded to a stochastic rotation of the flow and, on the other hand, to the effect of the mean velocity gradient. The experimental results show that the radial dispersion is a function of the relative dimension of particles with respect to the macroscale of the turbulence.

In the case of an inclined pipe, a two-dimensional stochastic model of the displacement is possible, but the integration of the equation is quite complicated and may be done numerically. We have prefered a two-dimensional simulation model. The results of the simulations permit us to obtain a limit repartition of passage probabilities, the moments of which we have compared with those that we have measured. These comparisons show that the model obviously represents the phenomena when the pipe is horizontal or very slightly inclined but differs in the near vertical case. This is due to the simplicity of the model in which we neglect the radial acceleration we have considered previously and the effect of which is negligible in comparison with gravity when the pipe is inclined.

The interpretation of the measurements by comparison of moments with the two-dimensional model shows that the angular dispersion of solid particles is essentially due to big eddies and that the particle diameters are not essential parameters in this case.

By associating this conclusion with that obtained previously concerning the radial dispersion, it seems that the eddies bigger than the macroscale of turbulence may be of capital importance in the dispersion of solid particles and that it will be of practical interest to characterize them as a function of a mean parameter of the flow.

The study of the movement of sufficiently large particles seems to be a method which is able to give this result.     

Effects of demixing on suspension rheometry

It is well known that particles in initially well-mixed suspensions subjected to inhomogeneous shear flows can migrate and establish particle concentration gradients and non-Newtonian velocity profiles. In this study we introduce a modified version of the shear-induced migration model to predict transient torque reductions in torsional startup flows and transient pressure drop reductions in capillary developing flows. Special attention is devoted to the relationship between the evolution of the driving forces and various mechanisms that contribute to the shear-induced migration of the suspended particles. This analysis reveals that suspension rheometry can complement other techniques, such as nuclear magnetic resonance (NMR) imaging, in qualitatively and quantitatively evaluating the model parameters. Received: 20 May 1997 Accepted: 21 January 1998     

Rheology and structure of a suspension of particles subjected to Quincke rotation

We report some experimental results that show that, if the particles of a suspension rotate faster than the surrounding liquid, the apparent viscosity is lessened. The increase of the particles’ spin rate is induced by Quincke rotation, which is the spontaneous rotation of an insulating particle immersed in a slightly conducting liquid when submitted to a high enough DC electric field. We study the flow of such a suspension with internal spin rate in various geometries, and we observe, depending on the values of the shear rate and of the electric field, that the suspension can form a layered structure or present a decomposition into two phases. To predict the appearance of the layered structure, we propose to introduce a modified Mason number that takes into account the rotation of the particles. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29 2006, Crete, Greece.     

Study of high Deborah number flow of polyisobutylene in square die

A large capacity RAM extruder was designed which provides the opportunity to study high Deborah number (D) flows, with D < 1,000. A modified version of particle image velocimetry was developed to enable the measurement of the velocity field in dies of arbitrary cross section. During the measurement process, tracer particles were simultaneously illuminated by both a focused laser beam locally and a lamp globally. Only those particles that passed through the laser beam were taken into account. The beam was scanned to achieve full field measurements. This method of measurement allowed us to find the location of a particle in the direction of the optical axis of the lens, i.e. that which makes the particle image on the CCD detector of the video camera. A device employing this method was designed and used to measure velocity profiles. The results of these measurements in two cross sections of the square die, at three values of pressure, are presented. The velocity was defined as the ratio of displacement to the elapsed time during which this displacement occurred. Errors in measurements of the coordinates and the observation time of particles were estimated as ±20 μm and less than 0.1%, respectively. A large plateau in the axial velocity profile was found at relatively small Deborah numbers, e.g. D ≈ 28. In flows with higher Deborah numbers, e.g. D ≈ 766, an almost flat velocity profile was detected. Two components of velocity, one longitudinal and one transversal, were measured simultaneously. However, the transversal component appeared to be less than the error of measurements and less than 1% of the axial velocity. Received: 4 August 1998 Accepted: 5 April 1999     

Two-phase redistribution in horizontal subchannel flow—turbulent mixing and gravity separation

The redistribution of two-phase flow in two horizontal interconnected subchannels caused by gravity separation and turbulent mixing has been investigated experimentally using an air—water loop. The measured redistribution data included the axial distribution of void fraction and liquid and gas flow rates in the two subchannels. The redistribution data exhibited an asymptotic behaviour, approaching certain flow distributions independent of the inlet distribution. The observed equilibrium distributions were explained as a balance between gravity forces (which tend to cause flow stratification) and turbulent diffusion (which tends to homogenize the two-phase mixture). A constitutive equation for transverse vapour drift, to account for both gravity separation and turbulent diffusion, was presented and a turbulent mixing coefficient was identified. The experimental data were used to obtain the best estimate of the empirical constants.     

Comparison of Kinetic and Hybrid-Equilibrium Models Simulating Colloid-Facilitated Contaminant Transport in Porous Media

The presence of colloidal particles in groundwater can enhance contaminant transport by reducing retardation effects and carrying them to distances further than predicted by a conventional advective/dispersive equation with normal retardation values. When colloids exist in porous media and affect contaminant migration, the system can best be simulated as a three-phase medium. Mechanisms of mass transfer from one phase to another by colloids and contaminants can be kinetic or equilibrium-based, depending on the sorption–desorption reaction rate between the aqueous and solid phases. When the rate of sorption between the water phase and the solid phase(s) is not much greater than the rate of change in contaminant concentration in the water phase, kinetic sorption models may better describe the phenomenon. In some cases of modeling one or more mass transfer processes, a useful simplification may be to introduce the local equilibrium assumption. In this study, the local equilibrium assumption for sorption processes on colloidal surfaces (hybrid equilibrium model) was compared with kinetic-based models. Sensitivity analyses were conducted to deduce the effect of major parameters on contaminant transport. The results obtained from the hybrid equilibrium model in predicting the transport of colloid-facilitated groundwater contaminant are very similar to those of the kinetic model, when the point of interest is not at contaminant and colloid source vicinities and the time of interest is sufficiently long for imposed sources.