PARTICLE TRANSPORT WITH FINITE FILTRATION TIME

Filtration problems are actual for the design of underground structures and foundations, strengthening of loose soil and construction of watertight walls in the porous rock. A liquid grout pumped under pressure penetrates deep into the porous rock. Solid particles of the suspension retained in the pores, strengthen the loose soil and create watertight partitions. The aim of the study is to construct an explicit analytical solution of the filtration problem. A one-dimensional model of deep bed filtration of a monodisperse suspension in a homogeneous porous medium with size-exclusion mechanism of particles retention is considered. Solid particles are freely transferred by the carrier fluid through large pores and get stuck in the throats of small pores. The mathematical model of deep bed filtration includes the mass balance equation for suspended and retained particles and the kinetic equation for the deposit growth. The model describes the movement of concentrations front of suspended and retained particles in an empty porous medium. Behind the concentrations front, solid particles are transported by a carrier fluid, accompanied by the formation of a deposit. The complex model has no explicit exact solution. To construct the asymptotic solution in explicit form, methods of nonlinear asymptotic analysis are used. The new coordinate transformation allows to obtain a parameter that is small at all points of the porous sample at any time. In this paper, a global asymptotic solution of the filtration problem is constructed using a new small parameter. Numerical calculations are performed for a nonlinear filtration coefficient found experimentally. Calculations confirm the closeness of the asymptotics to the solution in the entire filtration domain. For a nonlinear filtration coefficient, the asymptotics is closer to the numerical solution than the exact solution of the problem with a linear coefficient. The analytical solution obtained in the paper can be used to analyze solutions of problems of underground fluid mechanics and fine-tune laboratory experiments.

Grout filtration in porous soil is used in construction industry to create underground waterproof walls. When the suspension flows through the pores, various forces act on the suspended particles, blocking them on the frame of the porous medium. A one-dimensional model of deep bed filtration for a monodisperse suspension in a porous medium with several particle capture mechanisms is considered. The mathematical model includes the equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth with a piecewise-smooth linear-constant filtration function and a nonlinear concentration function. The solution of the nonlinear model is obtained by the finite difference method using an explicit difference scheme with second-order approximation. To construct the asymptotics of a complex model, the solutions of simplified linear and semilinear models and their combination are used. In the zone of a smooth filtration function, the best approximation of the solution of a complex model is determined by a certain linear combination of simple solutions. In another area, solution of a simplified problem with a piecewise-smooth filtration function and a linear concentration function is closest to the solution of a nonlinear model. Calculations show that in the zone of a smooth filtration function, a combination of simple solutions defines a solution approximation with second-order of smallness. In the area where it is necessary to take into account the non-smoothness of the filtration function, the approximation of a solution has a first order of smallness.

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.

Deep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion of reservoir production zones, fines migration in oil fields, bacteria, viruses or contaminant transport in groundwater, industrial filtering, etc. The basic features of the process are particle capture by the porous medium and consequent permeability reduction.<br>Models for deep bed filtration contain two coefficients that represent rock and fluid properties: the filtration function, which is the fraction of captured particles per unit of particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities.<br>The coefficients cannot be measured directly in the laboratory or in the field; therefore, they must be calculated indirectly by solving inverse problems. The practical petroleum and environmental engineering purpose is to predict injectivity loss and particle penetration depth around wells. Reliable prediction requires precise knowledge of these two coefficients.<br><br>In this work we determine these coefficients from pressure drop and effluent concentration histories, measured in one-dimensional laboratory experiments. <br>The filtration function is recovered by optimizing a nonlinear functional with box constraints. The permeability reduction is recovered likewise, taking into account the filtration function already found.<br>The recovery method consists of optimizing Tikhonov's functionals in appropriate subdomains.<br>In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.

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.

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

Severe injectivity decline during the injection of sea/produced water is a serious problem in offshore waterfloodings. The permeability impairment occurs due to capture of particles from injected water by the rock. The reliable modelling-based prediction of this decline is important for the injected-water-treatment design, for injected water management (injection of sea- or produced water, their combinations, water filtering etc.). Particle transport in porous media is determined by advective flow of carrier water and by hydrodynamic dispersion in micro-heterogeneous media. Thus, the particle flux is the sum of advective and dispersive fluxes. Transport of particles in porous media is described by an advection-diffusion equation and by a kinetic equation of particle capture. Conventional models for deep bed filtration take into account hydrodynamic particle dispersion in the mass balance equation but do not consider the effect of dispersive flux on retention kinetics. In the present study, a model for deep bed filtration taking into account particle hydrodynamic dispersion in both the mass balance and retention kinetics equations is proposed. Analytical solutions are obtained for flows in infinite and semi-infinite reservoirs and in finite porous columns. The physical interpretation for the steady-state flows described by the proposed and the traditional models favours the former. Comparative matching of experimental data on particle transport in porous columns by the two models is performed for two sets of laboratory data.

Correction of basic equations for deep bed filtration with dispersion

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.

Models of transport and filtration of small particles in porous media are used in the construction industry when designing foundations and underground structures. A liquid with particles moves through the channels of a porous soil. When particles are transported, some of them are locked in the pores and form a sediment. If the fluid flows slowly, the sedimented particles, retained on the walls of wide pores or in the throats of narrow pores, remain motionless. The liquid and suspension cannot tear the sedimented particles away from the sedimentation sites. When the flow rate of the suspension or colloid increases, part of the sediment is washed out by the carrier fluid and is transferred through the pores. A one-dimensional model of particle transport in a homogeneous porous medium is considered, taking into account the sedimentation of suspended particles on the framework and the erosion of the sediment. The model specifies the relationship between suspended and sedimented particles and the balance of sedimentation and erosion of the sediment. At low suspension concentration, the intensity of sediment formation and rise depends on the filtration function and the concentration of suspended particles; sediment erosion is determined by the number of particles deposited on the framework of the porous medium. Analytical solutions to the model and asymptotics in the form of a traveling wave are obtained. The maximum concentration of sediment with simultaneous action of particle retention and lifting is found.

Filtration problems in porous media are important for studying the movement of groundwater in porous formations and the spreading of liquid concrete injected into porous soil. Deep bed filtration of a monodisperse suspension in a homogeneous porous medium with two simultaneously acting particle capture mechanisms is considered. A mathematical model of suspension flow through porous medium with pore blocking by size-exclusion and arched bridging is developed. Exact solutions are obtained on the concentration front and at the porous medium inlet. For the linear filtration function, exact and asymptotic solutions are constructed.

The problems of underground fluid mechanics play an important role in the design and preparation for the construction of tunnels and underground structures. To strengthen the insecure soil a grout solution is pumped under pressure in the porous rock. The liquid solution filters in the pores of the rock and strengthens the soil after hardening. A macroscopic model of deep bed filtration of a monodisperse suspension in a porous medium with a size-exclusion mechanism for the suspended particles capture in the absence of mobilization of retained particles is considered. The solids are transported by the carrier fluid through large pores and get stuck at the inlet of small pores. It is assumed that the accessibility factor of pores and the fractional flow of particles depend on the concentration of the retained particles, and at the initial moment the porous medium contains an unevenly distributed deposit. The latter assumption leads to inhomogeneity of the porous medium. A quasilinear hyperbolic system of two first-order equations serves as a mathematical model of the problem. The aim of the work is to obtain the asymptotic solution near the moving curvilinear boundary - the concentration front of suspended particles of the suspension. To obtain a solution to the problem, methods of nonlinear asymptotic analysis are used. The asymptotic solution is based on a small-time parameter, measured from the moment of the concentration front passage at each point of the porous medium. The terms of the asymptotics are determined explicitly from a recurrent system of ordinary differential and algebraic equations. The numerical calculation is performed by the finite difference method using an explicit TVD scheme. Calculations for three types of microscopic suspended particles show that the asymptotics is close to the solution of the problem. The time interval of applicability of the asymptotic solution is determined on the basis of numerical calculation. The constructed asymptotics, which explicitly determines the dependence on the parameters of the system, allows to plan experiments and reduce the amount of laboratory research.

Deep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion of reservoir production zones, fines migration in oil fields, industrial filtering, bacteria, viruses or contaminants transport in groundwater etc. The basic features of the process are particle capture by the porous medium and consequent permeability reduction. Models for deep bed filtration contain two quantities that represent rock and fluid properties: the filtration function, which is the fraction of particles captured per unit particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities. These quantities cannot be measured directly in the laboratory or in the field; therefore, they must be calculated indirectly by solving inverse problems. The practical petroleum and environmental engineering purpose is to predict injectivity loss and particle penetration depth around wells. Reliable prediction requires precise knowledge of these two coefficients. In this work we determine these quantities from pressure drop and effluent concentration histories measured in one-dimensional laboratory experiments. The recovery method consists of optimizing deviation functionals in appropriate subdomains; if necessary, a Tikhonov regularization term is added to the functional. The filtration function is recovered by optimizing a non-linear functional with box constraints; this functional involves the effluent concentration history. The permeability reduction is recovered likewise, taking into account the filtration function already found, and the functional involves the pressure drop history. In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.

Exact solution to non-linear filtration in heterogeneous porous media

Asymptotic model for deep bed filtration