Deep Bed Filtration Model Research Articles

Modelling uniform asymptotics of the filtration problem in a porous medium

The solution of the filtration problem allows calculating the injection of the grout into a loose ground to create a solid foundation. The basic deep bed filtration model of a monodisperse suspension in a porous medium is considered. Based on the synthesis of the standard asymptotic solution for a short time and the asymptotics of the filtration problem for a long time, a uniform asymptotic solution is constructed. A numerical calculation shows that the asymptotics is close to the solution for any time.

Asymptotic model of size-exclusion grouting

In the construction of foundations of buildings and structures on fragile ground, various technologies of soil grouting are used. When pouring fine-grained concrete into the porous soil, the concrete grains are filtered in the pores of the soil. The filtration process depends on the ratio of the pore sizes of the soil and the solid particles of the injected concrete mortar.Injection of a carrier fluid with small solid particles in a porous medium forms a dynamic concentrations front of suspended and retained particles, separating the suspended particles and the hollow part of the porous frame. The purpose of the study is to construct and calculate an asymptotic model near the concentrations front for the filtration of monodisperse suspension in a porous medium with size-exclusion mechanism of particles retention.The classical mathematical model for one-dimensional filtration of suspensions and colloids in a porous medium is based on the geometric ratio of the particles and pores sizes: the particles freely pass through the large pores and get stuck in the pore throats with sizes smaller than the particles diameter. The model is determined by a system of two quasilinear first-order partial differential equations with the gap between boundary and initial conditions. To construct an asymptotic expansion in the vicinity of the concentrations front, a special small parameter is used that specifies the distance to the front. This parameter provides direct determination of the asymptotic terms from the recurrent system of ordinary linear differential equations of the first order.Near the concentrations front of the suspended and retained particles, a nonlinear high-order asymptotics is constructed for the filtration problem of solid particles transported by a carrier fluid in a porous medium. The obtained solution is zero before the front and nonzero behind the front. Approbation of the asymptotic expansion is carried out. It is shown that, for a linear blocking filtration coefficient, the asymptotics coincides with the exact solution.The asymptotic model of deep bed filtration makes it possible to obtain exact formulas for the high-order asymptotic expansion near the dynamic concentrations front of suspended and retained particles. The asymptotics improves the ability to fine-tune the filtration model depending on the properties of the porous soil and the grout.

Particle transport in a porous medium with initial deposit

Modelling of the suspended particles transport in a porous medium is used in the analysis of methods for strengthening foundations. To strengthen the porous soil, a low concentration cement-based grout is pumped into it under pressure. The suspension is filtered in a porous medium and fills the cavity of the soil. Grains of the grout are distributed along the network structure of the porous medium and strengthen the soil. The transfer of particles by a flow of a carrier fluid is accompanied by an uneven formation of a deposit on the porous medium frame.The purpose of the paper is to determine the mobile two-phase boundary between water and the particles during the injection of a suspension into a porous medium and to obtain an analytical solution of the nonlinear filtration problem for a general case of variable porosity and permeability.The mathematical model of one-dimensional deep bed filtration with size-exclusion particles retention includes the equations of mass balance and kinetic rate of a deposit and unsteady boundary conditions with unknown dimensionless concentrations of suspended and retained particles. Methods of non-linear asymptotic analysis are used to obtain the analytical solution and to construct an asymptotics near the porous medium inlet. The asymptotics is determined on the basis of a local exact solution of the problem.It is shown that the mobile two-phase boundary moves with variable speed. At the boundary of two phases, the concentration of suspended particles is discontinuous, and the concentration of retained particles is continuous and loses its smoothness. The exact explicit formula for the two-phase boundary and its asymptotics in a form convenient for calculations are obtained. An exact solution is obtained on the boundary of the porous medium and the asymptotics of the filtration problem is constructed near the porous medium inlet. Numerical calculation of the asymptotic solution is performed; graphs of the dependence of concentrations on time and coordinate are presented.In contrast to the numerical solution, analytical methods make it possible to determine the dependence of the solution of the filtration problem on the controlled external parameters. This allows the construction engineers to choose the best size of injected grout grains and the properties of the carrier fluid, optimize the filtration process and form a grouted porous soil of the required strength and density.

Analytical model for deep bed filtration with multiple mechanisms of particle capture

A model for deep bed filtration of a monodisperse suspension in a porous medium with multiple geometric particle capture mechanisms is considered. It is assumed that identical suspended particles can block pores of different sizes. The pores smaller than the particle size are clogged by single particles; if the pore size exceeds the diameter of the particles, it can be blocked by bridging — several particles forming various stable structures.An exact solution is obtained for constant filtration coefficients. Exact solutions for non-constant filtration functions are obtained on the concentrations front of the suspended and retained particles and at the porous medium inlet. Asymptotic solutions are constructed near these lines. For small and close to constant filtration functions, global asymptotic solutions are obtained.A basic model with two mechanisms of particle capture is studied in detail. Asymptotic solutions are compared to the results of numerical simulation. The applicability of various types of asymptotics is analyzed.

Filtration and backwashing behaviors of the deep bed filtration using long length poly-propylene fiber filter media

Abstract The filtration and backwashing performances of the deep bed filtration system using unique filter media made of long length poly-propylene fibers weaved into ring shape were investigated. The dynamic behavior of concentration of particles in filter bed during filtration, i.e. breakthrough curve, was observed. The filtration behavior was evaluated with the deep bed filtration model as filtration coefficient λ0, which is an index of particle capture efficiency, and filtration capacity σs. The effects of feed concentration and flow rate on the filtration performance were analyzed with the deep bed filtration model. The results showed that the filtration performances depended on the amount of filtered particles regardless of feed concentration and flow rate. The backwashing of filtration column was conducted with upward two-phase flow, which consisted of water and air. The change in turbidity of effluent showed two stage exponential changes, suggesting that, the particles can be classified into two groups which are particles easy to remove and particles difficult to remove. A mathematical model evaluating the backwashing behavior was developed for optimizing of backwashing condition. The effects of water flow rate and air flow rates were analyzed with this model and the optimal backwashing condition was determined.

Model of suspension displacement in a porous medium

The displacement of monodisperse suspension by clean water in a porous medium, accompanied by the formation of deposit is considered. A flow of water is supplied at the inlet of the porous medium filled with suspension. The suspension is displaced by water moving at a constant velocity. A mathematical model of deep bed filtration of suspension in a porous medium based on size-exclusion mechanism of particle retention and pore clogging is formulated. It is proved that in the suspension area the solution depends only on time, and in the clean water area – only on the distance to the porous medium inlet. For constant initial conditions an exact solution of the problem is constructed. In the case of linear filtration coefficient, the analytic solution is given in explicit form. The properties of the obtained solutions are analyzed.

Calculation of the filtration at the porous medium outlet

The study of fluid filtration with solid impurities in a porous medium is necessary for the construction of tunnels, hydraulic structures and underground storage of radioactive waste. The model of deep bed filtration of a monodisperse suspension in a homogeneous porous medium with variable porosity and permeability is considered. An asymptotic solution is constructed at the porous medium outlet. Calculations show the proximity of the high order asymptotics to the numerical solution.

A model of two-velocity particles filtration with variable injection

The filtration problems are actual in the design of underground storage facilities for hazardous waste. When the grout is injected into the porous soil, the fluid penetrates deep into the rock and, when solidified, blocks the pores. A model of suspension flow with 2-size particles moving with different velocities in a porous medium is considered. The proposed model of deep bed filtration generalizes the known equations of mass balance and particle capture kinetics for a fluid flow with various particles. The injection of a suspension with periodically changing concentration is considered. Exact and asymptotic solutions are obtained. The asymptotics rapidly converges to the numerical solution.

Modeling of Rock Permeability Damage and Repairing Dynamics Due to Invasion and Removal of Particulate from Drilling Fluids

The investigation discussed in this paper was motivated by the need for model which is able to simulate both permeability reduction of hydrocarbon formations due to the mud-component invasion during over-balance drilling, casing/cementing, workover operations, and dynamics of permeability repairing during well cleanup. The paper focuses on development and validation of model to describe internal mud cake (IMC) dynamics, placing special emphasis on dynamics of the IMC removing during well cleanup procedure. Set of laboratory experiments with clay slurry injection and subsequent brine water backflow in samples of Bentheimer sandstone is discussed. The specific of these experiments is that backflow was carried out with alternating rates (“multirate” backflow). It is shown that the conventional deep-bed filtration model is not able to reproduce the dynamics of multirate backflow. The stochastic model we suggest takes into account pore size distribution and describes the mobilization of trapped particles within individual groups of pores within a “pore ensemble.” We provide simulation results to show that the suggested model reasonably reproduces permeability dynamics during both clay slurry injection and brine water backflow stages.

Effects of pore size distribution and coordination number on filtration coefficients for straining-dominant deep bed filtration from percolation theory with 3D networks

A power law relation between the filtration coefficients of straining and flux through small pores has been reported. The effluent concentrations of the colloidal particles and the exponents derived from the experiment are inconsistent with those obtained from 2D network models simulation in their researches. In this study, a straining-dominant deep bed filtration (DBF) model is proposed based on 3D network simulation. The effects of simulation parameters such as lattice type, lattice coordination number z, pore size distribution (PSD), and particle capture scheme on DBF were investigated. Consistent with the power law formula and the normalized effluent concentrations (Ce/C0), simulation results indicate that the exponents increase with increasing coordination number z. The change in the PSD parameters alters the flux and weight of path type linked to node, thereby influencing the numerical simulation of staining-dominate DBF on 3D networks. The effects of capture scheme and coordination number on the simulated normalized effluent concentration could be due to variations in the total capture probability for different lattices with the same PSD. The results were obtained from simulation on hexagonal close packing (HCP) and 2D triangular lattices under the optimal conditions. The simulated Ce/C0 and exponents of the HCP lattice are similar to those of 2D triangular lattice and consistent with the experimental data.

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.

CALCULATION OF THE FILTRATION PROBLEM AT THE FILTER INLET

Filtration of the suspension in a porous medium with a geometric particle capture mechanism is con-sidered. The porous medium has an initial deposit unevenly distributed across the filter. The nonlinear model of deep bed filtration suggests that the porosity and permeability of the porous medium depend on the deposit. The asymptotics of the movable boundary of the two phases is determined. The asymptotic solution of the problem is constructed and calculated near the filter inlet.

Mathematical model of deep-bed filtration of a two-component suspension through a porous medium

A model of deep-bed filtration of a two-component suspension through a porous medium with formation of two types of the deposit which have different structures and properties is constructed. The influence of the parameters of fluid and particle flux densities which determine mass transfer between different components of the suspension and deposits on the filtration characteristics and properties of the resulting deposits is estimated on the basis of numerical experiments for the suspensions with contrast particle fractions.

Calculation of filtration of polydisperse suspension in a porous medium

The problem of filtering the suspension in a porous medium is considered. The proposed model for the transport of particles of different sizes is a generalization of deep bed filtration model for a monodisperse suspension with size-exclusion particle capture mechanism. Exact and asymptotic solutions are constructed at the filter inlet and on the concentration front of the suspended and retained particles. Numerical calculation for a suspension with 2-size particles shows that the distribution of deposit in the filter depends on the particle size.

Percolation-based model for straining-dominant deep bed filtration

A straining-dominant filtration model based on directed bond percolation is proposed. The infinite cluster structure of this model is relevant to filtration properties. Thus, a direct electrifying algorithm is modified for infinite cluster backbone identification, and the geometrical characteristic of the infinite cluster is analyzed. In addition, the relationship between the normalized effluent concentration and infinite cluster is established. The model is validated by comparing the particle concentrations estimated from analyzing the property of filtration cluster of network model with the normalized effluent concentration obtained from filtration process simulation and the experimental data. The results show that normalized effluent concentrations estimated from cluster analysis are consistent with the experimental data and filtration simulated result. Finally, the sensitivity of pore size distribution (PSD) parameters to normalized effluent concentrations estimation is analyzed. Assuming that the PSD is lognormal, the normalized effluent concentrations increase as the location parameter μ increases and scale parameter σ decreases.

Modeling Deep Bed Filtration Considering Limited Particle Retention

The classic deep bed filtration model (CDBFM) has been widely used for predicting transport and retention of particulate suspensions in porous media. In the CDBFM, the filtration coefficient function is assumed to be dependent of the retained particle concentration, and the boundary conditions are fixed at the porous media inlet cross section. However, if particle retention is limited, no more retention occurs (filtration coefficient equals zero) in regions where maximum retention (jamming limit) is reached. In this article, a model for deep bed filtration considering limited retention is proposed and analytical solutions are obtained. In cases where jamming limit is reached, the well-known CDBFM and the proposed model significantly differ from each other. Contrary to the CDBFM solutions, the obtained solutions show that concentrations along porous media are highly influenced by maximum retention front, which propagates with velocity independent of filtration coefficient.

Asymptotic model for deep bed filtration

Asymptotic solution for non-linear stage of colloidal suspension flow in porous media is developed. The expansion is performed behind the concentration front; the zero order approximation coincides with the short-time solution of the linearised system. Using the first order approximation allows significantly enhancing the validity time period for the analytical model if compared with the linearised solution, allowing using the long-term experimental breakthrough concentration history for the model adjustment. Laboratory injection tests for three size colloids are performed. The asymptotic solution is used to tune the model parameters from the breakthrough histories of two size particles; good quality of matching is observed. The breakthrough concentration history for the third size particles is compared with the prediction by the adjusted model; good quality prediction is observed. The above serves for validation of the asymptotic method for the model tuning and prediction of non-linear colloidal suspension flow in porous media.

Mathematical Modeling of Colloidal Particles Transport in the Medium Treated by Nanofluids: Deep Bed Filtration Approach

A deep bed filtration model has been developed to quantify the effect of nanoparticles (NPs) on mitigating fines migration in porous media. The filtration coefficients representing the total kinetics of particles capture were obtained by fitting the model to the laboratory data. Based on the optimum filtration coefficients, the model was utilized to history match the particle concentration breakthrough profiles observed in twelve core flood tests. In the flooding experiments, the effect of five types of metal oxide NPs, \(\upgamma \hbox {-Al}_{2}\hbox {O}_{3}\), CuO, MgO, \(\hbox {SiO}_{2}\), and ZnO, on migrating fines were investigated. In each test, a stable suspension was injected into the already NP-treated core and effluents’ fines concentration was measured based on turbidity analysis. In addition, zeta potential analysis was done to obtain the surface charge (SC) of the NP-treated medium. It was found that the presence of NPs on the medium surface results in SC modification of the bed and as a result, enhances the filter performance. Furthermore, the ionic strength of the nanofluid was recognized as an important parameter which governs the capability of NPs to modify the SC of the bed. The remedial effect of NPs on migrating fines is quantitatively explained by the matched filtration coefficients. The SC of the medium soaked by \(\upgamma \hbox {-Al}_{2}\hbox {O}_{3}\) nanofluid is critically increased; therefore, the matched filtration coefficient is of remarkably high value and as a result, the treated medium tends to adsorb more than 70 % of suspended particles. The predicted particle concentration breakthrough curves well matched with the experimental data.

Modelling the capture of gasoline engine exhaust particulate matter in three-way catalytic converters

The aim of this work was to numerically model the effect of a three-way catalytic converter on nano-scale particulate matter from a gasoline direct injection engine. The work used a deep bed filtration model to simulate and validate experimental findings. A 1.6L, gasoline direct injection, spark ignition, turbocharged and intercooled Euro IV engine was used for experimentation. The capture efficiency was experimentally determined by measuring pre and post-catalyst particulate matter numbers using a DMS-500 differential mobility spectrometer at different dilution ratio settings. Despite variable experimental results at lower engine speeds and loads, the model was capable of predicting catalytic converter particle capture efficiency in the majority of cases. This indicates that deep bed filtration theory can indeed be used to explain nano-scale particle capture within three-way catalytic converters. The results also suggest that there is no significant agglomeration of particles within the catalytic converter. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Improved population balance model for straining-dominant deep bed filtration using network calculations

Colloidal-suspension flow in porous media is modelled simultaneously by the large scale population balance equations and by the microscale network model. The phenomenological parameter of the correlation length in the population balance model is determined from the network modelling. It is found out that the correlation length in the population balance model depends on the particle size. This dependency calculated by two-dimensional network has the same tendency as that obtained from the laboratory tests in engineered porous media.