Sedimentation motion of sand particles in moving water (I): The resistance on a small sphere moving in non-uniform flow

In hydraulics, when we deal with the problem of sand particles moving relative to the surrounding water, Stokes’ formula of resistance has usually been used to render the velocity of sedimentation of the particles. But such an approach has not been proved rigorously, and its accuracy must be carefully considered. In this paper, we discuss the problem of a sphere moving in a non-uniform flow field, on the basis of the fundamental theory of hydrodynamics. We introduce two assumptions: i) the diameter of the sphere is much smaller than the linear dimension of the flow field, and ii) the velocity of the sphere relative to the surrounding water is very small. Using these two assumptions, we solve the linearized Navier-Stokes equations and equations of continuity by the method of Laplace transform, and finally we obtain a formula for the resistance acting on a sphere moving in a non-uniform flow field.     

Motion of a cloud of particles under gravity and its sedimentation on a flat horizontal surface

On the basis of the nonstationary two-dimensional equations of the mechanics of heterogeneous media a numerical investigation is made of the motion of a cloud of particles under the influence of gravity in an unbounded medium and the interaction of the descending cloud of particles with a flat horizontal surface. Depending on the degree of hydrodynamic interaction between the particles, there are different regimes of motion of the cloud during sedimentation; the change in its spatial configuration determined by the large-scale vortex motion of the carrier medium is determined. The concentration distributions of the particles on the sedimentation surface are obtained. A coefficient of dispersal is introduced for the cloud of particles on the flat horizontal surface, and its dependence on the concentration of the particles, their diameters, and other parameters is investigated.     

Percolation through clay crusts

During drilling of oil and gas wells, drilling muds (a suspension of clay particles in water) are pumped into them. During drilling, the muds perform various functions: They free the borehole from fragments of drilled rock, carry them to the surface, and create pressure on the borehole walls to prevent the entry of oil and gas into the borehole. The solid particles of the mud are deposited on the borehole walls, forming a clay crust, and the liquid phase percolates into the permeable bed. The clay crust represents an inhomogeneous deformable porous medium that is denser near the borehole wall. Knowledge of the properties of clay crusts is very important in practice, since it permits determining certain parameters of the bed; furthermore, the very properties of the clay crust depend on the properties of the drilling muds. In this article we will consider percolation of the drilling mud through the crust. Experimental data are used which show that the permeability and compressibility of the clay crust depend on the stresses acting in the particles of the clay crust. The problem is reduced to a self-similar problem. An ordinary differential equation of the second order is derived for the function in terms of which the stress and other characteristics of the crust are expressed. The distributions of permeability and porosity over the thickness of the crust are found and the quantity of fluid that penetrated into the bed at each instant is also determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 70–75, November–December, 1976.     

Oil Recovery by Low Salinity Water Injection into a Reservoir: A New Study of Tertiary Oil Recovery Mechanism

Low salinity water injections for oil recovery have shown seemingly promising results in the case of clay-bearing sandstones saturated with asphaltic crude oil. Reported data showed that low salinity water injection could provide up to 20% pore volume (PV) of additional oil recovery for core samples and up to 25% PV for reservoirs in near wellbore regions, compared with brine injection at the same Darcy velocity. The question remains as to whether this additional recovery is also attainable in reservoirs. The answer requires a thorough understanding of oil recovery mechanism of low salinity water injections. Numerous hypotheses have been proposed to explain the increased oil recovery using low salinity water, including migration of detached mixed-wet clay particles with absorbed residual oil drops, wettability alteration toward increased water-wetness, and emulsion formation. However, many later reports showed that a higher oil recovery associated with low salinity water injection at the common laboratory flow velocity was neither necessarily accompanied by migration of clay particles, nor necessarily accompanied by emulsion. Moreover, increased water-wetness has been shown to cause the reduction of oil recovery. The present study is based on both experimental and theoretical analyses. Our study reveals that the increased oil recovery is only related to the reduction of water permeability due to physical plugging of the porous network by swelling clay aggregates or migrating clay particles and crystals. At a fixed apparent flow velocity, the value of negative pressure gradient along the flow path increases as the water permeability decreases. Some oil drops and blobs can be mobilized under the increased negative pressure gradient and contribute to the additional oil recovery. Based on the revealed mechanism, we conclude that low salinity water injection cannot be superior to brine injection in any clay-bearing sandstone reservoir at the maximum permitted injection pressure. Through our study of low salinity water injection, the theory of tertiary oil recovery has been notably improved.     

Motion of a slender body near the free surface of a compressible fluid

An analytic solution to the problem of motion of a slender rigid body in a semi-infinite domain of a compressible fluid is obtained for the case when the body moves in parallel to the free surface at a constant velocity. This problem is similar to the problem of motion of a hydrofoil ship whose wing-like device allows it to lift its hull above the water surface and to decrease the friction and drag forces limiting the speed of usual ships. During its motion in water, a hydrofoil produces a lift force. The obtained analytic solution allows one to derive explicit expressions for the drag force and for the lift force in the limiting cases of relatively small and large depths. When depth is small, the drag force is greater than that in an infinite medium, since the wave drag is additionally evolved. When the velocity increases and approaches the sound velocity, the forces exerted on the body increase without limit, which is typical for a linear formulation of the problem.     

Modelling Low-Salinity Oil Recovery Mechanisms Using an Ion Dissociation Model

Experimental evidence shows that injecting low-salinity water during the oil recovery process can lead to an increase in the amount of oil recovered. Numerous mechanisms have been proposed to explain this effect, and, in recent years, two which have gained notable support are multicomponent ionic exchange (MIE) and pH increase. Both mechanisms involve ion exchange reactions within the thin film of water separating the oil in a reservoir from the clay minerals on the surface of the reservoir rock. Since the reactions occur on the molecular scale, an upscaled model is required in order to accurately determine the dominant mechanism using centimetre-scale experiments. In this paper, we develop the first stages of this upscaling process by modelling the pore-scale motion of an oil slug through a clay pore throat. We use a law-of-mass-action approach to model the exchange reactions occurring on the oil–water and clay–water interfaces in order to derive expressions for the surface charges as functions of the salinity. By balancing the disjoining pressure in the water film with the capillary pressure across the oil–water interface, we derive an expression for the salinity-dependent film thickness. We compare the two mechanisms by modifying an existing model for the velocity of an oil slug through a pore throat. Numerical results show that the velocity increases as the salinity decreases. The percentage increase is larger for the MIE mechanism, suggesting that MIE may be the dominant causal mechanism; however, this will vary depending on the particular clay and oil being studied.

     

Statistical model of collisions of bidisperse particles in an anisotropic turbulent flow

A statistical model for describing the motion and collisions of a bidisperse mixture of particles in anisotropic turbulent flows is presented. The model is based on a kinetic equation for the particle velocity probability density function (PDF). The results are compared with the data of a direct numerical simulation of the sedimentation of a bidisperse mixture of particles under the action of the gravity force.     

Experiments on transport of hydrophobic particles and gas bubbles in porous media

Adhesion of hydrophobic colloids (clay minerals) on the surface of bubbles of air and the transport of the composite units formed by bubbles and mineral particles were observed in a glass micro model.When a clay mineral suspension flowed in a porous medium that contained bubbles of air trapped in small pores, particles accumulated preferentially on the upstream portion of the bubbles, and quasi-stable bubble-mineral particle units were formed. With an increase in the flow velocity, the particles moved along the interface between the bubble and the liquid and accumulated on the downstream portion of the bubbles. A large stress could mobilize the units which, occasionally, accumulated in larger voids.The mechanism suggested is adhesion of the particles on the surface of the bubble due to compression of their diffuse electrical double layer. The adsorbed particles can be moved by shear stresses which act in the region of water molecules between the well-organized layers of water on the surfaces of the bubble and the clay particles. A large enough shear stress causes the bubbles to become more streamlined, allowing them to move in the channel system. If in contact, the common lamina of the bubbles can withdraw and rupture.Bubbles transport from 20 to 50 times more particles than can be transported by average suspension.     

Effect of Elevated Velocity of Particles in Groundwater Flow and Its Role in Colloid-facilitated Transport of Radionuclides in Underground Medium

Colloid-facilitated transport of radionuclides by groundwater can increase the level of ecological hazard from radioactive contaminant migration in geological medium. The reason for this is that the migration velocity of a radioactive colloid can be higher than that of radionuclides carried by the groundwater as a solute. On the basis of their field and laboratory studies, a few researchers have concluded that the velocity of the colloid can even exceed the interstitial velocity of the groundwater by a few times. A theoretical analysis of this effect is carried out in this article. The analysis is based on the assumption that the only mechanism responsible for the effect is caused by a redistribution of the colloid in a cross section of the groundwater flow in a representative volume to such domains of the cross section where the local velocity of the flow is higher than the average velocity over the whole cross section. This redistribution can be caused by drift forces arising as a result of the Magnus effect. The influence of these drift forces on a particle’s movement is considered in two extreme cases; that of relatively large and relatively small colloid particles. Particles are considered relatively small if the thermal motion of water molecules exerts a decisive influence on the particles’ movement. Otherwise the particles are considered as being relatively large. It is shown that in the case of relatively large particles this redistribution can be caused by the instability of their movement. The redistribution in the case of relatively small particles can be caused by an influence of the drift forces on characteristics of Brownian motion. It follows from the results of the theoretical analysis that an influence of the drift forces in both cases does not lead to an increase in the particles’ migration velocity at near-horizontal direction of the groundwater flow. Data from experimental studies of the elevated velocity of colloids in porous medium are analyzed. It is shown that some findings of colloidal migration velocity exceeding the interstitial velocity of the groundwater are a result of misinterpretation of experimental results.     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.     

The supersonic motion of a rigid body in an elastic medium with friction

The plane problem of supersonic steady motion of a body in an elastic medium is solved. Two possible cases of body motion are considered depending on its velocity. In the first case, the body moves at a velocity greater than the velocity of transverse waves but smaller than the velocity of longitudinal waves. In the second case, the body moves at a velocity greater than the velocity of longitudinal waves. An analytic solution of the problem under study is obtained and analyzed. It is shown that friction substantially influences the penetration process.     

The theory of similarity and the profile of vertical distribution of concentration of sand particles

In this article, we treat the problem of two-dimensional uniform steady channel flow of turbid water with theory of similarity. Under the condition of similarity of turbulent fluctuation velocity and fluctuation of concentration of sand particles, we obtain the profile of the vertical distribution of concentration of sand particles. This profile of vertical distribution of concentration of sand particles is slightly different from that obtained by diffusion theory, and departs from that obtained by gravitational theory.     

Simultaneous measurement of particle and fluid velocities in a plane mixing layer with dispersion

 An optical method is presented to measure simultaneously and separately the velocity field of both phases in particle-laden flows. The fluid is seeded with flow markers which are fluorescent at a specific wavelength and thus can be distinguished from the sediment particles by applying an optical filter. The motion of each phase is recorded by two CCD cameras, which are triggered such that a high correlation between subsequent images is guaranteed. The velocity fields are determined by means of least-square matching of a group of particles. The whole set-up was applied to study the sedimentation of particles through a rapidly evolving mixing layer. Received: 14 June 1999/Accepted: 15 January 2000     

Integrating thermal-concentration smoothed profile with lattice Boltzmann methods for simulating sedimentation of nonisothermal circular particles

A thermal-concentration smoothed profile-lattice Boltzmann method is proposed to study the effect of the concentration field on the dynamic behavior of nonisothermal cylindrical particles during the sedimentation process. The velocity, temperature, and concentration equations are solved using the lattice Boltzmann method. Moreover, the smoothed profile method is employed to enforce the nonslip boundary condition as well as constant temperature and constant concentration boundary conditions at the particles surfaces. Moreover, the Boussinesq approximation is used to couple the velocities, temperatures, and concentrations fields. The proposed combined method is validated by comparing the present numerical results with those found in the literature, showing good consistency. Then, the effect of the concentration buoyancy on the behavior of nonisothermal particles is discussed. In addition, the effect of Prandtl, Schmidt, and thermal Grashof numbers on the settling process is investigated. The results show that, by adding the effect of concentration, the maximum settling velocity of hot particles is reduced more relative to the cold ones; accordingly, the cold particles are settled faster than the hot ones. Finally, the sedimentation of two particles in a container at high thermal Grashof is investigated. It is shown that, at high thermal Grashof, there is an intense competition between the buoyancy force and gravity for the hot particles. The buoyancy flow generated leads to the reversal of the drafting-kissing-tumbling motion of the hot particles, making the particles move upward.     

Equations of hydromechanics and compression shock in two-velocity and two-temperature continuum with phase transformations

The equations of motion of multiphase mixtures have been considered in [1–10] and several other studies. In [1] it is proposed that the mixture motion be considered as an interpenetrating motion of several continua when velocity, pressure, mean density, concentration, etc., fields for each phase are introduced in the flowfield. The equations of motion are written separately for each phase, and the force effect of the other components is considered by introducing the interaction forces, which for the entire system are internal. The assumption of component barotropy is used to close the system.The energy equations are used in [2, 3] in place of the component barotropy assumption. Moreover, mixtures without phase transformations are considered. In [4] an analysis is made of the equations of turbulent motion with account for viscous forces for a two-velocity, but single-temperature medium in which equilibrium phase transformations are assumed, i. e., a two-phase medium is considered in which the phase temperatures are the same, the composition is equilibrium, but the phase velocities are different. In [5] the equations are written on the interface in a multicomponent medium consisting of barotropic fluids. A discontinuity classification is also presented here. In the aforementioned work [3] the equations on the shock are written for a continuum with particles without the use of the property of barotropy of the carrier fluid. Various different aspects of the motion of multiphase mixtures are considered in [6–11], for example, the effect of particle collisions with one another, the effect of the volume occupied by the particles on the parameters stream, shock waves, etc. In [7] a study is made of the force effect of an agitated medium on a particle on the basis of the Basset-Boussinesq-Oseen equation.In the following we derive the equations of motion of a two-velocity and two-temperature continuum with drops or particles with nonequilibrium phase transformations, i. e., a medium in which the phase velocities and temperatures are different and the composition may be nonequilibrium. In addition, we study the effect of the presence of particles or drops on the gas parameters behind a shock. Further, the equations obtained here are used to study compression waves, and in particular shock waves.The author wishes to thank Kh. A. Rakhmatulin, S. S. Grigoryan, and Yu. A. Buevich for helpful discussions and valuable comments.     

Dynamic simulation of sedimentation of solid particles in an Oldroyd-B fluid

In this paper we present a two-dimensional numerical study of the viscoelastic effects on the sedimentation of particles in the presence of solid walls or another particle. The Navier-Stokes equations coupled with an Oldroyd-B model are solved using a finite-element method with the EVSS formalism, and the particles are moved according to their equations of motion. In a vertical channel filled with a viscoelastic fluid, a particle settling very close to one side wall experiences a repulsion from the wall; a particle farther away from the wall is attracted toward it. Thus a settling particle will approach an eccentric equilibrium position, which depends on the Reynolds and Deborah numbers. Two particles settling one on top of the other attract and form a doublet if their initial separation is not too large. Two particles settling side by side approach each other and the doublet also rotates till the line of centers is aligned with the direction of sedimentation. The particle-particle interactions are in qualitative agreement with experimental observations, while the wall repulsion has not been documented in experiments. The driving force for lateral migrations is shown to correlate with the pressure distribution on the particle's surface. As a rule, viscoelasticity affects the motion of particles by modifying the pressure distribution on their surface. The direct contribution of viscoelastic normal stresses to the force and torque is not important.     

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.     

MULTI-SCALE AGGREGATION OF PARTICLES IN GAS-SOLIDS FLUIDIZED BEDS

The multi-scale characteristics of clusters in a fast fluidized bed and of agglomerates in a fluidized bed of cohesive particles are discussed on the basis of large amounts of experiments.The cluster size and concentration are dominated by the local voidage of the bed.A cluster consists of many sub-clusters with different sizes and discrete par-ticles,and the sub-cluster size probability density distribution appears as a negative exponential function.The agglom-erates in a fluidized bed of cohesive particles also possess the multi-scale nature.The large agglomerates form a fixed bed at the bottom,the medium agglomerates are fluidized in the middle,and the small agglomerates and discrete parti-cles become the dilute-phase region in the upper part of the bed.The agglomerate size is mainly affected by cohesive forces and gas velocity.The present models for prediction the size of clusters and agglomerates can not tackle the in-trinsic mechanism of the multi-scale aggregation,and a challenging problem for establishing mechanistic model is put forward.     

Size segregation and particle velocity fluctuations in settling concentrated suspensions

We investigate the sedimentation of concentrated suspensions at low Reynolds numbers to study collective particle effects on local particle velocity fluctuations and size segregation effects. Experiments are carried out with polymethylmetacrylate (PMMA) spheres of two different mean diameters (190 and 25 μm) suspended in a hydrophobic index-matched fluid. Spatial repartitions of both small and large spheres and velocity fluctuations of particles are measured using fluorescently labelled PMMA spheres and a particle image velocimetry method. We also report measurements of the interstitial fluid pressure during settling. Experiments show that size segregation effects can occur during the sedimentation of concentrated suspensions of either quasi-monodisperse or bidisperse spheres. Size segregation is correlated to the organisation of the sedimentation velocity field into vortex-like structures of finite size. A loss of size segregation together with a significant decrease of the fluid pressure gradient in the bulk suspension is observed when the size of vortex-like structures gets on the order of the container size. However, the emergence of channels through the settling zone prevents a complete loss of size segregation in very concentrated suspensions.