• A
  • A
  • A
  • АБВ
  • АБВ
  • АБВ
  • A
  • A
  • A
  • A
  • A
Обычная версия сайта
  • RU
  • EN
  • HSE University
  • Publications
  • Book chapter
  • Long-term filtration of particles in a porous medium
  • RU
  • EN
Расширенный поиск
Высшая школа экономики
Национальный исследовательский университет
Priority areas
  • business informatics
  • economics
  • engineering science
  • humanitarian
  • IT and mathematics
  • law
  • management
  • mathematics
  • sociology
  • state and public administration
by year
  • 2027
  • 2026
  • 2025
  • 2024
  • 2023
  • 2022
  • 2021
  • 2020
  • 2019
  • 2018
  • 2017
  • 2016
  • 2015
  • 2014
  • 2013
  • 2012
  • 2011
  • 2010
  • 2009
  • 2008
  • 2007
  • 2006
  • 2005
  • 2004
  • 2003
  • 2002
  • 2001
  • 2000
  • 1999
  • 1998
  • 1997
  • 1996
  • 1995
  • 1994
  • 1993
  • 1992
  • 1991
  • 1990
  • 1989
  • 1988
  • 1987
  • 1986
  • 1985
  • 1984
  • 1983
  • 1982
  • 1981
  • 1980
  • 1979
  • 1978
  • 1977
  • 1976
  • 1975
  • 1974
  • 1973
  • 1972
  • 1971
  • 1970
  • 1969
  • 1968
  • 1967
  • 1966
  • 1965
  • 1964
  • 1963
  • 1958
  • More
Subject
News
July 20, 2026
Scientists Create Open Dataset for Studying Concentration
A team of Russian researchers, including scientists from HSE University–St Petersburg, has developed the first open multimodal dataset containing recordings of brain activity, heart function, and video observations to help researchers understand what happens in the human brain during deep concentration. In the future, the dataset could accelerate the development of neural interfaces, rehabilitation technologies, and AI systems. The article has been published in Scientific Data.
July 20, 2026
‘Science Is Universal-It Knows No Borders
Fuad Aleskerov, Tenured Professor and Director of the International Centre of Decision Choice and Analysis at HSE University, together with his colleagues, has developed methods of network analysis in bibliometrics that have made it possible to identify patterns in the appearance and citation of publications in academic journals, as well as their influence on each other. When one or a number of studies are frequently cited by a wide range of journals, this is an indicator that the research is of high quality. By contrast, extensive cross-citation within a limited group of journals increases the likelihood of identifying a network of predatory publications.
July 20, 2026
Scientists Propose Method for More Efficient Resource Use in Machine Learning
An international group of researchers, including mathematicians from the AI and Digital Science Institute at the HSE Faculty of Computer Science, has provided a theoretical justification for a simple and computationally efficient method of estimating uncertainty in Stochastic Gradient Descent (SGD). The paper has been published on the scientific preprint server arXiv.org and presented at AISTATS 2026.

 

Have you spotted a typo?
Highlight it, click Ctrl+Enter and send us a message. Thank you for your help!

Publications
  • Books
  • Articles
  • Chapters of books
  • Working papers
  • Report a publication
  • Research at HSE

?

Long-term filtration of particles in a porous medium

Ch. 03003.
Liudmila Kuzmina, Osipov Y.

The formation of grout sediment in the pores of loose rock increases the water resistance of the soil and strengthens the foundation. A one-dimensional model of filtration in a porous medium considers the particles transport by the flow of a carrier fluid and the deposition of particles on the framework of a porous medium. The purpose of the work is to study the concentrations of suspended and settled particles of a suspension over a long time. Exact and asymptotic methods are used to obtain a solution to the model. The exact solution is presented in an implicit integral form. A set of solutions in the form of traveling waves with an arbitrary initial condition and their asymptotics are constructed. For the exact solution, an explicit second-order asymptotic solution for a long time is obtained as an expansion in decreasing exponents. Comparison of the asymptotic solution with the traveling waves makes it possible to choose a single traveling wave corresponding to the exact solution. The closeness of the traveling wave to the exact solution of the filtration model is verified numerically. The traveling wave found determines the explicit asymptotics of the concentration of deposited particles for a long time.

Language: English
Full text
DOI
Text on another site
Keywords: asymptoticsDeep bed filtration porous medium exact solutiontraveling wave solution

In book

BIO Web of Conferences: Volume 107:19th International Conference Water and Wastewater: Transportation, Treatment, Management “Yakovlev Readings” (YRC-2024)
Vol. 107. , EDP Sciences, 2024.
Similar publications
On the Electro-Hydrodynamical Boundary Value Problem for the Unit Cell of Cation-Exchange Membrane
Yulia O. Koroleva, Alexander V. Korolev, , in: Proceedings of the 9th International School-Seminar on Nonlinear Analysis and Extremal Problems (NLA-2026). Irkutsk, Russia, June 22–26, 2026.: Irkutsk: ISDCT SB RAS, 2026. P. 132–133.
We study a model problem on the filtration of a conducting fluid through a porous layer. A porous medium is presented as an assemblage of identical spherical cells. Each cell consists of a porous core and liquid shell. We derive apriori estimates for flow characteristics which show the specific behavior of the fluid. Our estimates are validated numerically. ...
Added: July 5, 2026
Proceedings of the 9th International School-Seminar on Nonlinear Analysis and Extremal Problems (NLA-2026). Irkutsk, Russia, June 22–26, 2026.
Irkutsk: ISDCT SB RAS, 2026.
We study a model problem on the filtration of a conducting fluid through a porous layer. A porous medium is presented as an assemblage of identical spherical cells. Each cell consists of a porous core and liquid shell. We derive apriori estimates for flow characteristics which show the specific behavior of the fluid. Our estimates are validated numerically. ...
Added: July 5, 2026
Phase-field system as a regularization of the free-boundary problem for incompressible immiscible two-component flow
Gaydukov R. K., Burov N. A., Russian Journal of Mathematical Physics 2026 Vol. 33 No. 2 P. 263–275
The spreading of a droplet of incompressible fluid over a substrate under the action of capillary forces and gravity is studied within a phase-field framework. An asymptotic analysis shows that the solution of the phase-field system (regularized problem) converges to the solution of the corresponding limiting problem when a regularization parameter tends to zero. The ...
Added: April 11, 2026
PARTICLE TRANSPORT WITH FINITE FILTRATION TIME
Kuzmina L., Osipov Y. V., International Journal for Computational Civil and Structural Engineering 2025 Vol. 21 No. 3 P. 135–147
Abstract: Particle transport by a fluid flow occurs in many applied construction problems, including pumping mortar into porous soil, creating watertight diaphragm walls, and constructing dams and underwater structures. A model of deep bed filtration of suspensions and colloids in a homogeneous porous medium with a finite number of vacancies for retained particles is considered. ...
Added: February 25, 2026
ESTIMATES TO THE WEAK SOLUTION OF THE ELECTRO-HYDRODYNAMICAL BOUNDARY VALUE PROBLEM FOR THE UNIT CELL OF CATION-EXCHANGE MEMBRANE. THE CASE OF NONZERO DEBYE RADIUS
Koroleva Y., https://arxiv.org/abs/2604.15410 2025 No. 1 P. 1–19
We study a model problem on the filtration of a conducting fluid through a porous layer. A porous medium is presented as an assemblage of identical spherical cells. Each cell consists of a porous core and liquid shell. We show the dependence of each flow parameter on the Debye radius which characterizes how far the influence of a charge ...
Added: February 11, 2026
Spring–mass behavior of solitons under the influence of an external force field within the modified Korteweg–de Vries equation
Flamarion M., Pelinovsky E., Melnikov I., Chaos, Solitons and Fractals 2025 Vol. 196 Article 116422
We investigate the interaction of solitons with an external periodic field within the framework of the modified Korteweg–de Vries (mKdV) equation. In the case of small perturbation a simple dynamical system is used to describe the soliton behavior. Equilibrium points of this dynamical system are computed when the external force travels at a constant speed. Assuming that ...
Added: April 22, 2025
Advective Flow of a Rotating Fluid Layer in a Vibrational Field
Shvarts K.G., Russian Journal of Nonlinear Dynamics 2019 Vol. 15 No. 3 P. 261–270
This paper presents a derivation of new exact solutions to the Navier – Stokes equations in Boussinesq approximation describing two advective flows in a rotating thin horizontal fluid layer with no-slip or free boundaries in a vibrational field. The layer rotates at a constant angular velocity; the axis of rotation is aligned with the vertical axis of coordinates. ...
Added: November 24, 2024
Plane-Parallel Advective Flow in a Horizontal Layer of Incompressible Permeable Fluid
Shvarts K. G., Russian Journal of Nonlinear Dynamics 2023 Vol. 19 No. 2 P. 219–226
In this paper a new exact solution of the Navier – Stokes equations in the Boussinesq approximation describing advective flow in a horizontal liquid layer with free boundaries, where the vertical velocity component is a constant value, is obtained. The temperature is linear along the boundaries of the layer. Solutions of this kind are used ...
Added: November 16, 2024
Inverse Filtration Problem of a Bidisperse Suspension
Liudmila I. Kuzmina, Osipov Y., , in: Proceedings of the 10th International Conference on Civil Engineering.: Singapore: Springer, 2024. P. 585–593.
Filtration problems of suspensions and colloids in porous media are considered when designing tunnels and underground structures. To strengthen weak soil, a liquid solution is injected into the rock, the particles of which are filtered in the pores and distributed far from the well. A deep bed filtration model of 2-particle suspension in a porous ...
Added: October 31, 2024
Model Of Cake Filtration In Porous Medium
Liudmila I. Kuzmina, Osipov Y., International Journal for Computational Civil and Structural Engineering 2024 Vol. 20 No. 3 P. 116–124
Strengthening of loose soil and creation of water-resistant underground walls are associated with filtration of small particles in a porous medium. Liquid solution pumped into a well under pressure spreads through hollow channels and strengthens the soil upon hardening. Many porous filters retain particles near the entrance. The particles deposited on the filter surface form ...
Added: October 31, 2024
Non-linear filtration model with splitting front
Liudmila I. Kuzmina, Osipov, Y., Pesterev, A., International Journal of Non-Linear Mechanics 2024 Vol. 167 Article 104905
We consider filtration of a suspension in a homogeneous porous medium which is described by a macroscopic 1-D model including the mass exchange equation and the kinetic equation for deposit growth. The standard model assumes that suspended particles move at the same speed as the carrier fluid. However, in some experiments, a lag between the ...
Added: October 31, 2024
Retention profiles of multiparticle filtration in porous media
Kuzmina L., Osipov, Y. V., Astakhov, M. D., Mathematical Methods in the Applied Sciences 2024 Vol. 47 No. 11 P. 8319–8341
A one‐dimensional deep bed filtration model of a polydisperse suspension or colloid in a porous medium is considered. The model includes a quasilinear system of 2n equations for concentrations of suspended and retained particles of n types. The problem is reduced to a closed 3 × 3 system for total concentrations of suspended and retained ...
Added: June 27, 2024
Deep bed filtration model for cake filtration and erosion
L. I. Kuzmina, Osipov Y. V., Pesterev A. R., Applied Mathematics and Mechanics (English Edition) 2024 Vol. 45 No. 2 P. 355–372
Many phenomena in nature and technology are associated with the filtration of suspensions and colloids in porous media. Two main types of particle deposition, namely, cake filtration at the inlet and deep bed filtration throughout the entire porous medium, are studied by different models. A unified approach for the transport and deposition of particles based ...
Added: June 27, 2024
Traveling Wave Solution to Filtration Model in Porous Medium
Kuzmina L., Osipov Y., , in: Proceedings of the 8th International Technical Conference on Frontiers of HCET 2023. Advances in Transdisciplinary Engineering, Volume 43.: IOS Press, 2023. P. 450–455.
Suspension and colloid filtration in porous rocks is encountered in problems of underground hydromechanics associated with construction. The formation of grout sediment in the pores of loose rock increases the water resistance of the soil and strengthens the foundation. A one-dimensional model of filtration in a porous medium considers the particles transport by the flow ...
Added: June 27, 2024
Modelling of Filtration with Two Capture Mechanisms
Liudmila Kuzmina, Osipov Y., Sosedka M., , in: Construction The Formation of Living Environment. Proceedings of FORM 2022. (LNCE, volume 282).: Cham: Springer, 2024. P. 21–31.
Filtering fluids with tiny particles in porous media occurs in many technological processes of environmental and civil engineering. To strengthen the foundation, a slurry is pumped into loose rock and, when hardened, strengthens it. Deep bed filtration is accompanied by the retention of particles. Various forces act on the particles and cause them to settle ...
Added: June 27, 2024
Modeling of fluid flow along a small heated irregularity on the plate surface in the framework of double-deck boundary layer structure
Gaydukov R. K., Indian Journal of Pure and Applied Mathematics 2026 Vol. 57 No. 3 P. 808–825
The paper is deals with the heat transfer in the laminar flow of a viscous incompressible fluid along a plate with a small heated localized irregularity. It is well known that double- and triple-deck structures of the boundary layer arise in such flow problems. These structures make it possible, among other things, to describe the phenomenon of ...
Added: May 21, 2024
Асимптотика фундаментальных решений параболических задач
Danilov V., Математические заметки 2024 Т. 115 № 2 С. 219–229
We present several methods for constructing the asymptotics of the fundamental solution of Fokker–Planck–Kolmogorov-type parabolic equations with a small parameter both for small and finite positive times. ...
Added: March 21, 2024
Modeling the Ice–Water Phase Transition in a Tube with Small Ice Buildups on the Wall
R. K. Gaydukov, V. G. Danilov, Computational Mathematics and Mathematical Physics 2024 Vol. 64 No. 6 P. 1317–1325
Mathematical modeling of the ice–water phase transition during liquid flow inside a pipe with a small ice buildup on the wall at high Reynolds numbers is considered. As a mathematical model describing the dynamics of the phase transition, a double-deck boundary layer model and a phase field system are used. Results of numerical simulation are presented. ...
Added: February 17, 2024
On the Weak Solution of the Electro-Hydrodynamical Boundary Value Problem for the Unit Cell of Cation-exchange Membrane
Yulia O. Koroleva, , in: Proceedings of the 7th International Conference on Nonlinear Analysis and Extremal Problems (NLA-2022), Irkutsk, Russia, July 15–22, 2022.: ISDCT SB RAS, 2022. P. 54–55.
We study a model problem on the filtration of a conducting fluid through a porous layer. A porous medium is presented as an assemblage of identical spherical cells. Each cell consists of a porous core and liquid shell. The common case of finite Debye radius in comparison to the cell radius is analyzed. We derive ...
Added: February 2, 2024
  • About
  • About
  • Key Figures & Facts
  • Sustainability at HSE University
  • Faculties & Departments
  • International Partnerships
  • Faculty & Staff
  • HSE Buildings
  • HSE University for Persons with Disabilities
  • Public Enquiries
  • Studies
  • Admissions
  • Programme Catalogue
  • Undergraduate
  • Graduate
  • Exchange Programmes
  • Summer University
  • Summer Schools
  • Semester in Moscow
  • Business Internship
  • Research
  • International Laboratories
  • Research Centres
  • Research Projects
  • Monitoring Studies
  • Conferences & Seminars
  • Academic Jobs
  • Yasin (April) International Academic Conference on Economic and Social Development
  • Media & Resources
  • Publications by staff
  • HSE Journals
  • Publishing House
  • iq.hse.ru: commentary by HSE experts
  • Library
  • Economic & Social Data Archive
  • Video
  • HSE Repository of Socio-Economic Information
  • HSE1993–2026
  • Contacts
  • Copyright
  • Privacy Policy
  • Site Map
Edit