Filtration is one of the most used technologies in chemical engineering. Development of computer technology and computational mathematics made it possible to explore such processes by mathematical modeling and computational methods. The article deals with the study of suspension filtration in a porous medium with modified deposition kinetics. It is suggested that deposition is formed in two types, reversible and irreversible. The model of suspension filtration in porous media consists of the mass balance equation and kinetic equations for each type of deposition. The model includes dynamic factors and multi-stage deposition kinetics. By using the symmetricity of porous media, the higher dimensional cases are reduced to the one-dimensional case. To solve the problem, a stable, effective and simple numerical algorithm is proposed based on the finite difference method. Sufficient conditions for stability of schemes are found. Based on numerical results, influences of dynamic factors on solid particle transport and deposition characteristics are analyzed. It is shown that the dynamic factors mainly affect the profiles of changes in the concentration of deposition of the active zone.

In the paper, a mathematical model for the filtration of two-component suspensions in a dual-zone porous medium is considered. The model consists of the mass balance equations, the kinetic equations for active and passive zones of porous medium for each component of the suspension and Darcy’s law. To solve the problem, a numerical algorithm for computer experimentation is developed on the basis of finite difference method. Based on numerical results, the main characteristics of suspension filtration in a porous medium are established. Influences of model parameters on transport and deposition of suspended particles of two-component suspension in porous media are analysed. It is shown that the polydispersity of suspension and multistage nature of the deposition kinetics can lead to various effects that are not characteristic for the transport of one-component suspensions with one-stage particle deposition kinetics. In particular, in distribution of the concentration of suspended particles in a moving fluid non-monotonic dynamics are obtained at individual points in the medium. It is shown that at the points of the medium near to the input section, the concentration of deposited particles can reach partial capacities in the passive zone.

A model of deep-bed filtration of a two-component suspension through a porous medium with formation of two types of the deposit which have different structures and properties is constructed. The influence of the parameters of fluid and particle flux densities which determine mass transfer between different components of the suspension and deposits on the filtration characteristics and properties of the resulting deposits is estimated on the basis of numerical experiments for the suspensions with contrast particle fractions.

The problem of suspension filtration in a porous medium consisting of active and passive zones is posed and numerically solved in the case of a multistage kinetics of particle deposition. Some mathematical model of the process is proposed that is based on the general conservation laws and additional phenomenological assumptions. The influence of the multistage kinetics of particle deposition on the filtration characteristics is considered. We establish that, as the parameter characterizing the duration of the stage of formation of the irreversible deposition increases, a region with complete saturation of the passive zone capacity appears near the filter inlet. No further increase of the deposition concentration in the passive zone is observed, while the process of particles movement in suspension and deposition of them in the active zone continues.

In this work, anomalous solute transport using adsorption effects and the decomposition of solute was studied. During the filtration of inhomogeneous liquids, a number of new phenomena arise, and this is very important for understanding the mechanisms of the filtration process. Recently, issues of mathematical modeling of substance transfer processes have been intensively discussed. Modeling approaches are based on the law of matter balance in a certain control volume using additional phenomenological relationships. The process of anomalous solute transport in a porous medium was modeled by differential equations with a fractional derivative. A new mobile—immobile model is proposed to describe anomalous solute transport with a scale-dependent dispersion in inhomogeneous porous media. The profiles of changes in the concentrations of suspended particles in the macropore and micropore were determined. The influence of the order of the derivative with respect to the coordinate and time, i.e., the fractal dimension of the medium, was estimated based on the characteristics of the solute transport in both zones. The hydrodynamic dispersion was set through various relations: constant, linear, and exponential. Based on the numerical results, the concentration fields were determined for different values of the initial data and different relations of hydrodynamic dispersion.

In this paper, a problem of anomalous solute transport in a coaxial cylindrical two-zone porous medium with fractal structure is posed and numerically solved. The porous medium is studied in the form of cylinder with two parts: macropore—with high permeability characteristics in the central part and micropore—with low permeability around it. Anomalous solute transport is modeled by differential equations with a fractional derivative. The solute concentration and pressure fields are determined. Based on numerical results, the influence of the fractional derivatives order on the solute transport process is analysed. It was shown that with a decrease in the order of the derivatives in the diffusion term of the transport equation in the macropore leads to a “fast diffusion” in both zones. Characteristics of the solute transport in both zones mainly depend on the concentration distribution and other hydrodynamic parameters in the macropore.