scholarly journals GLOBAL ASYMPTOTICS OF THE FILTRATION PROBLEM IN A POROUS MEDIUM

2019 ◽  
Vol 15 (2) ◽  
pp. 77-85 ◽  
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.

scholarly journals DETERMINING THE LENGMUR COEFFICIENT OF THE FILTRATION PROBLEM

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

scholarly journals CALCULATION OF LONG-TERM FILTRATION IN A POROUS MEDIUM

2018 ◽  
Vol 14 (1) ◽  
pp. 92-101
Author(s):  
Ludmila I. Kuzmina ◽  
Yuri V. Osipov ◽  
Nikita V. Vetoshkin
Keyword(s):  
Porous Medium ◽  
Large Time ◽  
Solid Particles ◽  
Size Exclusion ◽  
Underground Structures ◽  
Filtration Problem ◽  
Finite Differences Method ◽  
Monodisperse Suspension ◽  
Exclusion Mechanism

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.

scholarly journals Filtration in porous rock with initial deposit

2019 ◽  
Vol 97 ◽  
pp. 05005
Author(s):  
Yuri Osipov ◽  
Yuri Galaguz
Keyword(s):  
Porous Medium ◽  
Numerical Calculation ◽  
Asymptotic Solution ◽  
Suspended Particles ◽  
Size Exclusion ◽  
Tvd Scheme ◽  
Time Interval ◽  
Fractional Flow ◽  
Porous Rock ◽  
Curvilinear Boundary

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.

scholarly journals NUMERICAL AND ASYMPTOTIC MODELING OF A FILTRATION PROBLEM WITH THE INITIAL DEPOSIT

2017 ◽  
Vol 13 (3) ◽  
pp. 70-76 ◽  
Author(s):  
Ludmila I. Kuzmina ◽  
Yuri V. Osipov ◽  
Yuri P. Galaguz
Keyword(s):  
Porous Medium ◽  
Numerical Calculation ◽  
Asymptotic Solution ◽  
Hydraulic Structures ◽  
Filtration Problem ◽  
Deep Bed Filtration ◽  
Asymptotic Modeling ◽  
Variable Porosity ◽  
Porosity And Permeability ◽  
Initial Deposit

The study of filtration as one of the problems of underground hydromechanics is necessary for the design and construction of tunnels, underground and hydraulic structures. Deep bed filtration of suspension in a porous medium with variable porosity and permeability and with an initial deposit is considered. An asymptotic solution to a model with small limit deposit is constructed; the asymptotics is compared with numerical calculation.

Modelling Of Transport And Capture Of Particles In A Porous Medium

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.

scholarly journals Deep bed filtration with multiple pore-blocking mechanisms

2018 ◽  
Vol 196 ◽  
pp. 04003 ◽  
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.

scholarly journals Model of suspension displacement in a porous medium

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.

scholarly journals CALCULATION OF THE FILTRATION PROBLEM AT THE FILTER INLET

2017 ◽  
Vol 13 (1) ◽  
pp. 63-68
Author(s):  
Ludmila I. Kuzmina ◽  
Yuri V. Osipov
Keyword(s):  
Porous Medium ◽  
Asymptotic Solution ◽  
Nonlinear Model ◽  
Particle Capture ◽  
Filtration Problem ◽  
Deep Bed Filtration ◽  
Capture Mechanism ◽  
Porosity And Permeability ◽  
Two Phases ◽  
Initial Deposit

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.

scholarly journals Asymptotic solution of the filtration equation

Vestnik MGSU ◽  
2016 ◽  
pp. 49-61 ◽  
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.

scholarly journals Numerical solution of filtration in porous rock

2019 ◽  
Vol 97 ◽  
pp. 05016 ◽  
Author(s):  
Galina Safina
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Numerical Solution ◽  
Laboratory Research ◽  
Careful Study ◽  
Fractional Flow ◽  
Porous Rock ◽  
Filtration Problem ◽  
Geological Processes ◽  
Explicit Analytical Solution

The filtration problem is one of the most relevant in the design of retaining hydraulic structures, water supply channels, drainage systems, in the drainage of the soil foundation, etc. Construction of transport tunnels and underground structures requires careful study of the soil properties and special work to prevent dangerous geological processes. The model of particle transport in the porous rock, which is based on the mechanical-geometric interaction of particles with a porous medium, is considered in the paper. The suspension particles pass freely through large pores and get stuck in small pores. The deposit concentration increases, the porosity and the permissible flow of particles through large pores changes. The model of one-dimensional filtration of a monodisperse suspension in a porous medium with variable porosity and fractional flow through accessible pores is determined by the quasi-linear equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth. This complex system of differential equations has no explicit analytical solution. An equivalent differential equation is used in the paper. The solution of this equation by the characteristics method yields a system of integral equations. Integration of the resulting equations leads to a cumbersome system of transcendental equations, which has no explicit solution. The system is solved numerically at the nodes of a rectangular grid. All calculations are performed for non-linear filtration coefficients obtained experimentally. It is shown that the solution of the transcendental system of equations and the numerical solution of the original hyperbolic system of partial differential equations by the finite difference method are very close. The obtained solution can be used to analyze the results of laboratory research and to optimize the grout composition pumped into the porous soil.

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