Asymptotics for filtration of polydisperse suspension with small impurities

A model for deep bed filtration of a polydisperse suspension with small impurities in a porous medium is considered.Different suspended particles move with the same velocity as the carrier water and get blocked in the pore throats due to the size-exclusion mechanism of particle retention.A solution of the model in the form of a traveling wave is obtained.The global exact solution for a multiparticle filtration with one high concentration and several low concentrations of suspended particles is obtained in an explicit form.The analytic solutions for a bidisperse suspension with large and small particles are constructed.The profiles of the retained small particles change monotony with time.The global asymptotics for the filtration of a polydisperse suspension with small kinetic rates is constructed in the whole filtration zone.     

A model of two-velocity particles transport in a porous medium

A one-dimensional flow of suspension with two types of solid particles moving with different velocities in a porous medium is considered. A mathematical model of deep bed filtration which generalizes the known equations of mass balance and particle capture kinetics for a flow of fluid with identical particles is developed. The exact solution is evaluated at the filter inlet and on the concentration front of fast suspended and retained particles, asymptotic solutions are provided in certain vicinities of these lines. A global asymptotic solution to the problem with a small limit deposit is constructed. The asymptotics rapidly converges to the numerical solution.     

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.     

Tomographic Study of Internal Erosion of Particle Flows in Porous Media

In particle-laden flows through porous media, porosity and permeability are significantly affected by the deposition and erosion of particles. Experiments show that the permeability evolution of a porous medium with respect to a particle suspension is not smooth, but rather exhibits significant jumps followed by longer periods of continuous permeability decrease. Their origin seems to be related to internal flow path reorganization by avalanches of deposited material due to erosion inside the porous medium. We apply neutron tomography to resolve the spatiotemporal evolution of the pore space during clogging and unclogging to prove the hypothesis of flow path reorganization behind the permeability jumps. This mechanistic understanding of clogging phenomena is relevant for a number of applications from oil production to filters or suffosion as the mechanisms behind sinkhole formation.     

Numerical modeling of deposition-release mechanisms in long-term filtration: validation from experimental data

A numerical phenomenological filtration model based on the combination of existing modeling approaches for simulating the transport of suspended particles in saturated porous medium is presented. The model accounts for the decreased physical straining with the distance from the inlet and the amount of deposited particles in the deposition kinetics. The particle release flux is a function of the local shear stress exerted by the flow on the pore surfaces. The proposed model is validated by interpreting a series of experimental data, realized in a laboratory sand column. The results show that the present model allows simulating the presence of a plateau in the breakthrough curves in the light of the shear stress conditions, and the spatial profile of deposited particles in the porous medium in the light of the straining profile.     

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.     

Investigation of the porous drag and permeability at the porous-fluid interface: Influence of the filtering parameters on Darcy closure

The porous-fluid interface encompasses a region bridging the flow inside a porous medium and a free-flowing fluid. In the context of volume-averaged simulations, it can be described by a set of gradually changing parameters defining the porous medium, mainly porosity and permeability. In this paper, both the permeability and the porous-induced drag force are evaluated a-priori, by explicitly filtering a set of Particle-Resolved Simulations (PRS) of the flow in the channel partially occupied by the porous medium. Different porous matrices are considered and the influence of the geometry and filtering parameters on the macroscopic quantities is studied. Especially, the focus is placed on the requirements for the kernel type and size to perform filtering accurately, and their impact on the distribution of permeability at the interface. The performance of the typically used models for the permeability is compared to the explicitly filtered results. Lastly, a new model for permeability and the drag force is introduced, taking into account the information about the filtering size and non-uniformity of the velocity field. The model greatly improves the prediction of velocity at the porous-fluid interface and serves as a proof of concept that a successful porous drag model should strive to include information about both parameters.     

Changes of a porous structure in flow of a monodisperse medium

The motion of fluids with suspended particles in porous media is considered. A mathematical model for the interaction of a monodisperse suspension with a porous structure is proposed. Changes in the parameters of the medium and the flow are studied for equilibrium regimes. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 113–121, March–April, 2000.     

Fluid flow simulation in a random elliptical porous medium by using the lattice Boltzmann method

A fluid flow through an isotropic porous medium with randomly arranged elliptical particles is simulated by the lattice Boltzmann method. The dimensionless pressure drop and the dimensionless permeability are evaluated as functions of the Reynolds number. The effect of the aspect ratio of the major to minor semi-axis of the ellipse on the dimensionless permeability is considered for different values of porosity. The pressure drop is thoroughly investigated as a function of fluid viscosity for different values of the aspect ratio and porosity. The influence of various parameters of the problem on the mean tortuosity of the medium is considered.     

Flow instability in a curved porous channel formed by two concentric cylindrical surfaces

Stability of laminar flow in a curved channel formed by two concentric cylindrical surfaces is investigated. The channel is occupied by a fluid saturated porous medium; the flow in the channel is driven by a constant azimuthal pressure gradient. The momentum equation takes into account two drag terms: the Darcy term that describes friction between the fluid and the porous matrix, and the Brinkman term, which allows imposing the no-slip boundary condition at the channel walls. An analytical solution for the basic flow velocity is obtained. Numerical analysis is carried out using the collocation method to investigate the onset of instability leading to the development of a secondary motion in the form of toroidal vortices. The dependence of the critical Dean number on porosity and the channel width is analyzed.     

Tomography-Based Characterization and Optimization Of Fluid Flow Through Porous Media

Numerical simulations to characterize fluid flow through porous media have been carried out using tomography-derived real geometry data that has been manipulated using digital image processing techniques to obtain a wide range of porosities. Two kinds of porous media have been analyzed: (a) a reticulated porous ceramic (RPC) foam and (b) a packed bed of CaCO₃ particles. The porosity of the media is varied via morphological operations between 0.727 and 0.913 in case of the RPC and between 0.329 and 0.824 in case of the packed bed. A mesh generator based on the pore space indicator function is then used to generate unstructured tetrahedral grids from the processed tomography data. Fluid flow simulations are carried out for Reynolds numbers ranging from 0.1 to 200 and the results are used to determine the permeability and the Dupuit?CForchheimer coefficient in each case. The results are then compared with existing analytical models and the applicability of the models is examined. In the RPC case, the Happel?CBrenner (parallel-flow) model predicts the permeability with a normalized root mean square error (NRMSE) of 11.8 % across the porosity range and Modified Ergun (Macdonald et. al) model predicts the Dupuit?CForchheimer coefficient within a NRMSE of 13.5 %. In the packed-bed case, the Brinkman drag model predicts the permeability within a NRMSE of 8.26 % across the porosity range and the Modified Ergun model predicts the Dupuit?CForchheimer coefficient within an NRMSE of 5.94 %. For each material, an adjusted Kozeny constant is determined. For the RPC, the Kozeny constant is evaluated at 7.73 and for the CaCO₃ packed bed, it is found to be 6.10, leading to predictions of the permeability with an NRMSE of 4.16 and 3.37 %, respectively.     

Filtration model of longitudinal flow in a finned cylindrical channel

The problem of laminar flow of a viscous incompressible fluid in a finned circular tube is considered. A solution is obtained in the form of series in eigenfunctions of the Laplace operator; the coefficients in the series are found numerically. For the same problem, a simpler filtration approximation is proposed in which the system of fins is modeled by a radially inhomogeneous porous layer, and fluid flow in it is described by the Brinkman equation. A formula for the effective permeability of the porous medium is obtained by varying the number and height of fins. The formula provides an accurate evaluation of the mean flow velocity and viscous drag coefficient in finned channels.     

Particle suspension in a simple shear flow

A hydrodynamic model describing the particle distribution over the cross-section of a finely dispersed flow is proposed. The model is constructed on the basis of notions concerning the diffusion of particles induced by their random displacements in the process of relative motion of neighboring layers at constant shear velocity. It is shown that the suspension capacity of the flow is large for small particles due to thermal fluctuations and for relatively large particles due to shear-induced particle pulsations. There are critical particle sizes for which the particles are suspended and transported by the flow less effectively than larger or smaller particles.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–121, January–February, 1995.     

On the Permeability of Fractal Tube Bundles

The permeability of a porous medium is strongly affected by its local geometry and connectivity, the size distribution of the solid inclusions, and the pores available for flow. Since direct measurements of the permeability are time consuming and require experiments that are not always possible, the reliable theoretical assessment of the permeability based on the medium structural characteristics alone is of importance. When the porosity approaches unity, the permeability?Cporosity relationships represented by the Kozeny?CCarman equations and Archie??s law predict that permeability tends to infinity and thus they yield unrealistic results if specific area of the porous media does not tend to zero. The aim of this article is the evaluation of the relationships between porosity and permeability for a set of fractal models with porosity approaching unity and a finite permeability. It is shown that the tube bundles generated by finite iterations of the corresponding geometric fractals can be used to model porous media where the permeability?Cporosity relationships are derived analytically. Several examples of the tube bundles are constructed, and the relevance of the derived permeability?Cporosity relationships is discussed in connection with the permeability measurements of highly porous metal foams reported in the literature.     

A numerical study of heat transfer of a porous block with the random porosity model in a channel flow

 In this paper, heat transfer of a hot plate with a porous block in a channel flow is numerically investigated. A porous block is simulated as a fin type heat sink. The random/artificial porosity models are used to generate the distribution of porosity. In fact, the distribution of porosity in porous medium is irregular, thus the random porosity model is more realistic than the constant or variable porosity model to describe the phenomena happening in porous medium. Therefore, the distribution of porosity of porous block obeys the random porosity model, and the factors of mean porosity and standard deviation are taken into consideration. The variations of the porosity and the velocity in porous block are no longer smooth. For obtaining more heat transfer rate, the artificial porosity model is proposed. The heat transfer rates of the several cases derived by the artificial porosity model are better than those of the random porosity model. The thermal performance of porous block is larger than that of solid block as the mean porosity is larger than 0.5. Received on 5 March 2001 / Published online: 29 November 2001     

Dispersion of a filtration flow

In this paper we shall consider the transport of a dynamically neutral impurity in a porous medium containing random inhomogeneities. The original versions of the equations for the mean impurity concentration [1, 2] were based on the hyphothesis that the random motions obeyed the Markov principle, use being made of the diffusion equations of A. N. Kolmogorov. Later [3, 4] the method of perturbations was used to study the complete system of equations for the impurity concentration and random filtration velocity in the case of a constant, nonrandom porosity; after an averaging process this yields a generalized equation for the average concentration. In the limiting cases of small- and large-scale inhomogeneities in the permeability of the medium, the basic integrodifferential equation may be, respectively, reduced to parabolic and hyperbolic equations of the second order. In the present analysis we shall use the perturbation method to study the transport of an impurity by a flow when the filtration velocity of the latter fluctuates around inhomogeneities in the permeability field, the porosity of the medium in which the flow is taking place also constituting a random field, correlating with the field of permeability. We shall derive equations for the average concentration and should formulate the corresponding boundary-value problems for these equations; we shall also calculate the components of the dispersion tensor and shall consider the equilibrium sorption of an impurity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–69, July–August, 1976.The author is grateful to A. I. Shnirel'man for useful discussions.     

Modified Particle Detachment Model for Colloidal Transport in Porous Media

Particle detachment from the rock during suspension transport in porous media was widely observed in laboratory corefloods and for flows in natural reservoirs. A new mathematical model for detachment of particles is based on mechanical equilibrium of a particle positioned on the internal cake or matrix surface in the pore space. The torque balance of drag, electrostatic, lifting and gravity forces, acting on the particle from the matrix and the moving fluid, is considered. The torque balance determines maximum retention concentration during the particle capture. The particle torque equilibrium is determined by the dimensionless ratio between the drag and normal forces acting on the particle. The maximum retention function of the dimensionless ratio (dislodging number) closes system of governing equations for colloid transport with particle release. One-dimensional problem of coreflooding by suspension accounting for limited particle retention, controlled by the torque sum, allows for exact solution under the assumptions of constant filtration coefficient and porosity. The explicit formulae permit the calculation of the model parameters (maximum retention concentration, filtration and formation damage coefficients) from the history of the pressure drop across the core during suspension injection. The values for maximum retention concentration, as obtained from two coreflood tests, have been matched with those calculated by the torque balance on the micro scale.     

Filtration Behaviour of Cement-Based Grout in Porous Media

Filtration behaviour of cement particles, especially under the high grouting pressure with a rapid grout flow velocity, has a significant effect on the grout injection. However, there have been few studies on this field where the governing equation of this behaviour remains unclear. In the present study, a novel experimental procedure for grout injection was adopted to acquire the spatial and temporal variations in porosity and viscosity of high-speed grout flow in coarse sand. Experimental observations showed that there were dramatic variations in viscosity and porosity during the grout penetration within the first 50 s, suggesting that the high velocity had a significant influence on the distribution of the filtration coefficient. A model based on the Stokes–Brinkman (S–B) equation and advection–filtration equations was established to describe the filtration of grout flow in porous media. Meanwhile, numerical solutions from both the proposed model and traditional Darcy’s law were compared with experimental results. The comparative results showed that the proposed approach can match the laboratory tests well; the analysis indicated that Darcy’s law was unable to properly describe high-speed grout flow in porous media due to the lack of a shear force and the inertial term. Nonuniform filtration behaviour of cement grout flowing in porous media was revealed. Due to the nonuniform distribution of the pore velocity isoline caused by Poiseuille flow, it led to a heterogenous distribution of porosity as well. Parametric studies on the applicability of Darcy’s law and S–B equation for grout flow were discussed, in which an error of less than 10% was calculated when the Reynolds number was less than 2.5.     

Mathematical model of convective heat transfer between flows of finely dispersed media in adjacent channels with permeable walls

The discussion is concerned with a mathematical model for convective heat transfer between the flows of finely dispersed media moving in adjacent channels separated by a permeable wall where portions of the fluid phases are exchanged many times between the flows. Numerical solutions are given for a countercurrent flow of a suspension and a liquid. Equations are derived and curves constructed to show the distribution of the flow velocity and the suspension porosity along the length of the channels as well as the dependence on time of the temperatures of the flows.     

Exact solutions to the problem of deep-bed filtration with retardation of a jump in concentration within the framework of the nonlinear two-velocity model

Exact solutions with plane and cylindricalwaves are obtained for one-dimensional problems of injection of a suspension into a porous reservoir when lagging of the suspended particles behind the carrier fluid is taken into account in the case of large change in the porosity. It is shown that taking lagging of the particles behind the fluid into account can lead to slowing-down the motion of jump in concentration. This is in agreement with the results of a series of experiments. It is also noted that, in principle, models in which the particles pass in average ahead of the carrier fluid are possible in the problems of deep bed filtration.