CALCULATION OF LONG-TERM FILTRATION IN A POROUS MEDIUM

he filtration problem in a porous medium is an important part of underground hydromechanics. Filtration of suspensions and colloids determines the processes of strengthening the soil and creating waterproof walls in the ground while building the foundations of buildings and underground structures. It is assumed that the formation of a deposit is dominated by the size-exclusion mechanism of pore blocking: solid particles pass freely through large pores and get stuck at the inlet of pores smaller than the diameter of the particles. A one-dimensional mathematical model for the filtration of a monodisperse suspension includes the equation for the mass balance of suspended and retained particles and the kinetic equation for the growth of the deposit. For the blocking filtration coefficient with a double root, the exact solution is given implicitly. The asymptotics of the filtration problem is constructed for large time. The numerical calculation of the problem is carried out by the finite differences method. It is shown that asymptotic approximations rapidly converge to a solution with the increase of the expansion order.

Download Full-text

DETERMINING THE LENGMUR COEFFICIENT OF THE FILTRATION PROBLEM

International Journal for Computational Civil and Structural Engineering ◽

10.22337/2587-9618-2020-16-4-50-56 ◽

2020 ◽

Vol 16 (4) ◽

pp. 50-56

Author(s):

Liudmila Kuzmina ◽

Yuri Osipov

Keyword(s):

Exact Solution ◽

Porous Medium ◽

Least Squares Method ◽

Size Exclusion ◽

Underground Structures ◽

Particle Capture ◽

Filtration Problem ◽

Problem Finding ◽

Langmuir Coefficient ◽

Exclusion Mechanism

Filtration of suspension in a porous medium is actual in the construction of tunnels and underground structures. A model of deep bed filtration with size-exclusion mechanism of particle capture is considered. The inverse filtration problem - finding the Langmuir coefficient from a given concentration of suspended particles at the porous medium outlet is solved using the asymptotic solution near the concentrations front. The Langmuir coefficient constants are obtained by the least squares method from the condition of best approximation of the asymptotics to exact solution. It is shown that the calculated parameters are close to the coefficients of the model, and the asymptotics well approximates the exact solution

Download Full-text

GLOBAL ASYMPTOTICS OF THE FILTRATION PROBLEM IN A POROUS MEDIUM

International Journal for Computational Civil and Structural Engineering ◽

10.22337/2587-9618-2019-15-2-77-85 ◽

2019 ◽

Vol 15 (2) ◽

pp. 77-85 ◽

Cited By ~ 1

Author(s):

Lyudmila Kuzmina ◽

Yuri Osipov ◽

Yulia Zheglova

Keyword(s):

Porous Medium ◽

Asymptotic Solution ◽

Size Exclusion ◽

Underground Structures ◽

Fractional Flow ◽

Filtration Problem ◽

Variable Porosity ◽

Particle Retention ◽

Global Asymptotics ◽

Exclusion Mechanism

Filtration of the suspension in a porous medium is important when strengthening the soil and creating watertight partitions for the constructi on of tunnels and underground structures. A model of deep bed filtration with variable porosity and fractional flow, and a size-exclusion mechanism of particle retention are considered. A global asymptotic solution is constructed in the entire domain in which the filtering process takes place. The obtained asymptotics is close to the numerical solution.

Download Full-text

Calculation of the filtration problem by finite differences methods

MATEC Web of Conferences ◽

10.1051/matecconf/201825104021 ◽

2018 ◽

Vol 251 ◽

pp. 04021 ◽

Cited By ~ 2

Author(s):

Yuri Osipov ◽

Galina Safina ◽

Yuri Galaguz

Keyword(s):

Porous Medium ◽

Construction Industry ◽

Solid Particles ◽

Size Exclusion ◽

Tvd Scheme ◽

Filtration Problem ◽

Loose Soil ◽

Monodisperse Suspension ◽

Medium Change ◽

Filtration Model

The filtration problem of a suspension in a porous medium is relevant for the construction industry. In the design of hydraulic structures, construction of waterproof walls in the ground, grouting the loose soil, it is necessary to calculate the transfer and deposition of solid particles by the fluid flow. A one-dimensional filtration problem of a monodisperse suspension in a porous medium with a size-exclusion capture mechanism is considered. It is assumed that as the deposit grows, the porosity and admissible flow of particles through the porous medium change. The solution of the initial filtration model and the equivalent equations are calculated. For the numerical calculation of the problem, both standard first-order finite difference formulas and more accurate second-order schemes were used. The obtained solutions are compared with the results given by the TVD-scheme.

Download Full-text

Model of suspension displacement in a porous medium

MATEC Web of Conferences ◽

10.1051/matecconf/201825104016 ◽

2018 ◽

Vol 251 ◽

pp. 04016

Author(s):

Liudmila Kuzmina ◽

Yuri Osipov

Keyword(s):

Porous Medium ◽

Analytic Solution ◽

Initial Conditions ◽

Water Area ◽

Size Exclusion ◽

Clean Water ◽

Monodisperse Suspension ◽

Linear Filtration ◽

Exclusion Mechanism ◽

Pore Clogging

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.

Download Full-text

Modelling Of Transport And Capture Of Particles In A Porous Medium

Promyshlennoe i Grazhdanskoe Stroitel stvo ◽

10.33622/0869-7019.2019.11.56-60 ◽

2019 ◽

pp. 56-60

Keyword(s):

Porous Medium ◽

Asymptotic Solution ◽

Retention Mechanism ◽

Inverse Filtering ◽

Underground Structures ◽

Loose Soil ◽

Monodisperse Suspension ◽

Homogeneous Porous Medium ◽

Model Equations ◽

Pass Through

The study of the transport and capture of particles moving in a fluid flow in a porous medium is an important problem of underground hydromechanics, which occurs when strengthening loose soil and creating watertight partitions for building tunnels and underground structures. A one-dimensional mathematical model of long-term deep filtration of a monodisperse suspension in a homogeneous porous medium with a dimensional particle retention mechanism is considered. It is assumed that the particles freely pass through large pores and get stuck at the inlet of small pores whose diameter is smaller than the particle size. The model takes into account the change in the permeability of the porous medium and the permissible flow through the pores with increasing concentration of retained particles. A new spatial variable obtained by a special coordinate transformation in model equations is small at any time at each point of the porous medium. A global asymptotic solution of the model equations is constructed by the method of series expansion in a small parameter. The asymptotics found is everywhere close to a numerical solution. Global asymptotic solution can be used to solve the inverse filtering problem and when planning laboratory experiments.

Download Full-text

Deep bed filtration with multiple pore-blocking mechanisms

MATEC Web of Conferences ◽

10.1051/matecconf/201819604003 ◽

2018 ◽

Vol 196 ◽

pp. 04003 ◽

Cited By ~ 5

Author(s):

Liudmila Kuzmina ◽

Yuri Osipov

Keyword(s):

Porous Medium ◽

Size Exclusion ◽

Medium Size ◽

Deep Bed Filtration ◽

Pore Blocking ◽

Variable Porosity ◽

Monodisperse Suspension ◽

Porosity And Permeability ◽

Constant Coefficients ◽

Blocking Mechanisms

A one-dimensional model for the deep bed filtration of a monodisperse suspension in a porous medium with variable porosity and permeability and multiple pore-blocking mechanisms is considered. It is assumed that the small pores are clogged by separate particles; pores of medium size, exceeding the diameter of the particles, can be blocked by arched bridges, forming stable structures at the pore throats. These pore-blocking mechanisms - size-exclusion and different types of bridging act simultaneously. Exact solutions are obtained for constant coefficients, on the concentrations front and at the porous medium inlet.

Download Full-text

Asymptotic solution of the filtration equation

Vestnik MGSU ◽

10.22227/1997-0935.2016.2.49-61 ◽

2016 ◽

pp. 49-61 ◽

Cited By ~ 1

Author(s):

Ludmila Ivanovna Kuzmina ◽

Yuri Viktorovich Osipov

Keyword(s):

Porous Medium ◽

Fluid Flow ◽

Differential Equations ◽

Asymptotic Solution ◽

Geometric Model ◽

Solid Particles ◽

Initial Moment ◽

Filtration Problem ◽

Filtration Equation ◽

Small Pore

The problem of filtering a suspension of tiny solid particles in a porous medium is considered. The suspension with constant concentration of suspended particles at the filter inlet moves through the empty filter at a constant speed. There are no particles ahead of the front; behind the front of the fluid flow solid particles interact with the porous medium. The geometric model of filtration without effects caused by viscosity and electrostatic forces is considered. Solid particles in the suspension pass freely through large pores together with the fluid flow and are stuck in the pores that are smaller than the size of the particles. It is considered that one particle can clog only one small pore and vice versa. The precipitated particles form a fixed deposit increasing over time. The filtration problem is formed by the system of two quasi-linear differential equations in partial derivatives with respect to the concentrations of suspended and retained particles. The boundary conditions are set at the filter inlet and at the initial moment. At the concentration front the solution of the problem is discontinuous. By the method of potential the system of equations of the filtration problem is reduced to one equation with respect to the concentration of deposit with a boundary condition in integral form. An asymptotic solution of the filtration equation is constructed near the concentration front. The terms of the asymptotic expansions satisfy linear ordinary differential equations of the first order and are determined successively in an explicit form. For verification of the asymptotics the comparison with the known exact solutions is performed.

Download Full-text

Global asymptotics of filtration in porous media

E3S Web of Conferences ◽

10.1051/e3sconf/20199705002 ◽

2019 ◽

Vol 97 ◽

pp. 05002 ◽

Cited By ~ 1

Author(s):

Liudmila Kuzmina ◽

Yuri Osipov ◽

Yulia Zheglova

Keyword(s):

Exact Solution ◽

Porous Medium ◽

Analytical Solution ◽

Asymptotic Solution ◽

Solid Particles ◽

Nonlinear Filtration ◽

Porous Rock ◽

Filtration Problem ◽

Carrier Fluid ◽

Loose Soil

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.

Download Full-text