Кузьмина Людмила Ивановна

Bidisperse filtration problem with non-monotonic retention profiles

2022 · ARTICLE · en

During deep bed filtration of suspensions and colloids in a porous medium, some particles are retained in the pores and form a fixed deposit. A one-dimensional mathematical model of filtration with particles of two types is considered. Exact solution is derived. The existence and the uniqueness of the solution are proved by the method of characteristics, and a solution in the form of a traveling wave is obtained. The profiles of total and partial retained concentrations, showing the dependence of the retained particles concentrations on the coordinate at a fixed time, are studied. It is shown by Taylor expansions that the retained profiles of large particles decrease monotonically, while the retained profiles of small particles are non-monotonic. At a short time, the profile of small particles decreases monotonically; with increasing time, a maximum point appears on it, moving from the inlet to the outlet of the porous medium. When the maximum point reaches the outlet, the profile becomes monotonically increasing. The condition for the non-monotonicity of the total retained profile is obtained. © 2022, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

ФИЛЬТРАЦИЯ В ПОРИСТОЙ СРЕДЕ С ДВУМЯ МЕХАНИЗМАМИ ЗАХВАТА

2022 · ARTICLE · ru

Для создания водонепроницаемых стен в грунте в пористую породу закачивается жидкий раствор, который фильтруется в порах породы и, застывая, закупоривает их. При долговременной фильтрации некоторые частицы задерживаются на каркасе пористой среды и образуют неподвижный осадок. Рассматривается модель фильтрации с двумя механизмами захвата частиц, действующими одновременно (блокирование частиц на входе узких пор и прилипание к стенкам широких пор). Каждому механизму захвата отвечает своя функция фильтрации. Задача сводится к стандартной модели с одной агрегированной функцией фильтрации, заданной неявно. Исследуются приближенные модели с явными функциями фильтрации, позволяющие упростить вычисление решения. Линейно-постоянная функция фильтрации допускает решение в явном виде, однако она имеет излом и не оптимально приближена к агрегированной функции фильтрации. Гиперболическая функция фильтрации содержит свободный параметр, значение которого подбирается из условия наилучшего приближения к агрегированной функции. Показано, что решение модели с гиперболической функцией фильтрации ближе к точному решению, чем решение модели с линейно-постоянной функцией.

Filtration in Porous Medium with Particle Release

2022 · CHAPTER · en

Asymptotics for filtration of polydisperse suspension with small impurities

2021 · ARTICLE · en

A model for deep bed filtration of a polydisperse suspension with small impurities in a porous medium is considered. Different suspended particles move with the same velocity as the carrier water and get blocked in the pore throats due to the size-exclusion mechanism of particle retention. A solution of the model in the form of a traveling wave is obtained. The global exact solution for a multiparticle filtration with one high concentration and several low concentrations of suspended particles is obtained in an explicit form. The analytic solutions for a bidisperse suspension with large and small particles are constructed. The profiles of the retained small particles change monotony with time. The global asymptotics for the filtration of a polydisperse suspension with small kinetic rates is constructed in the whole filtration zone.

Filtration of 2-particles suspension in a porous medium

2021 · CHAPTER · en

During the construction of underground storage of hazardous waste, it is necessary to create waterproof walls in the ground. The grout is filtered in the rock, fills the pores and, when hardened, creates a reliable barrier to groundwater. A one-dimensional model of the flow of inhomogeneous particles in a porous medium is considered. The retained particles profiles formed during deep bed filtration are studied. It is shown that when filtering a 2-particle suspension, the deposit is distributed unevenly. The profile of large retained particles is always monotonous, and the profile of small retained particles is nonmonotonic. The monotonicity of the total deposit profile depends on the model parameters. The shape of non-monotonic profiles is time-dependent. At short times, the profile decreases monotonously. At some point, a maximum appears on the profile graph, which shifts from the inlet to the output with increasing time. When the maximum point reaches the outlet, the profile becomes monotonically increasing. With a further increase in time, the retention profiles remain monotonically increasing. Analytical solutions for a filtration model with particles of three or more different types are unknown. Analysis of the retention profiles of the polydisperse suspension requires further study.

Asymptotics of inverse filtration problem in porous media

2021 · CHAPTER · en

Filtration problems arise in the design of tunnels and underground structures. A onedimensional filtration model of a monodisperse suspension in a homogeneous porous medium is considered. For a general nonlinear filtration function, an asymptotic solution is constructed behind the concentrations front of suspended and retained particles. It is shown that the asymptotics is close to the numerical solution. Comparison of the asymptotics with the suspended particles concentration at the outlet of the porous medium allows solving the inverse filtration problem on finding the nonlinear filtration function. The proposed method allows to obtain the filtration function based on the results of standard laboratory experiments.

ASYMPTOTICS OF THE FILTRATION PROBLEM WITH ALMOST CONSTANT COEFFICIENTS

2021 · ARTICLE · en

During the construction of hydraulic and underground structures, a grout solution is pumped into the ground to create waterproof partitions. The liquid grout is filtered in the porous rock and clogs the pores when hardened. The mathematical model of deep bed filtration describes the transfer of suspension particles and colloids by a fluid flow through the pores of a rock. For a one-dimensional filtration problem in a homogeneous porous medium with almost constant coefficients, an asymptotic solution is constructed. The asymptotics is compared with the numerical solution.

Exact solution for 1D deep bed filtration with particle capture by advection and dispersion

2021 · ARTICLE · en

We consider a one-dimensional (1D) non-linear 2x2 hyperbolic problem of suspension-colloidal-nano transport in porous media, so-called deep bed filtration. The particle capture rate is proportional to the overall particle flux, which includes dispersion. The captured-particle concentration dependency on the proportionality coefficient, which is typical for large captured concentrations, causes the non-linearity of the problem. However, the 1D initial–boundary value problem allows for an exact solution. The introduction of the Riemann invariant converts the 2x2 system into a scalar quasi-linear hyperbolic equation, which is implicitly solved using the method of characteristics. The non-linear equation for particle capture rate yields a limited travelling wave that approximates the exact solution for the initial–boundary problem.

Динамика частиц в пористой среде

2021 · ARTICLE · ru

При проектировании тоннелей и подземных сооружений необходимо рассматривать фильтрацию частиц в пористой породе. Долговременная глубинная фильтрация суспензий и коллоидов в пористой среде приводит к образованию осадка в порах и изменению структуры каркаса пористой породы. Модель фильтрации включает в себя уравнение баланса концентраций взвешенных и осажденных частиц, а также кинетическое уравнение роста осадка. Процесс фильтрации определяется функцией фильтрации, задающей зависимость скорости роста осадка от концентрации осажденных частиц. Вид функции фильтрации связан со свойствами частиц, жидкости и пористой среды. Исследуется динамика (зависимость от времени) концентраций взвешенных и осажденных частиц для различных функций фильтрации в точке выхода частиц из пористой среды. Показано, что при больших интервалах времени скорость роста обеих концентраций уменьшается и стремится к нулю. При малом временном интервале скорости роста концентраций осадка убывают для частиц малого размера и увеличиваются для больших частиц, скорости роста концентраций взвешенных частиц растут или убывают в зависимости от вида функции фильтрации. Найдены условия существования точек перегиба на графиках динамики концентраций в точке выхода частиц из пористой среды.

GLOBAL ASYMPTOTICS OF PARTICLE TRANSPORT IN POROUS MEDIUM

2020 · CHAPTER · en

Particle transport in a porous medium occurs in environmental, chemical and industrial technologies. The transport of suspended concrete grains in a liquid grout through porous soil is used in construction industry to strengthen foundations. When particles are transported by a fluid flow in a porous medium, some particles are retained in the pores and form a deposit. The aim of the work is the construction and study of a one-dimensional mathematical model of particle transport and retention in the porous medium, taking into account the simultaneous action of several particle capture mechanisms. The model consists of mass balance equation and the kinetic equation of deposit growth. The deposit growth rate is proportional to the filtration function, which depends on the retained particles concentration, and the nonlinear concentration function, which depends on the concentration of suspended particles. The use of a new parameter, depending on the distance to the porous medium inlet allows to construct a global asymptotic solution in the entire area of the mathematical model. An explicit analytical solution is obtained as a series in two small parameters. The global asymptotics is close to the numerical solution at all points of the porous medium at any time.