## 1 Introduction
In one-dimensional case, mass balance equation takes the form [20; 21; 22; 23; 24]  
\[\frac{\partial c}{\partial t}+v\frac{\partial c}{\partial x}+\frac{\partial \rho}{\partial t}=D\frac{\partial^{2}c}{\partial x^{2}}, \tag{1}\]  
where \(c\) is the concentration of suspended particles in the suspension, \(\rho\) is the concentration of deposition, \(D\) is diffusion (hydrodynamic dispersion) coefficient, \(v\) is the velocity, \(t\) is time, \(x\) is coordinate. In some cases the diffusive (dispersive) fluxes are ignored for large particles and in large scale approximation, since they are insignificantly smaller than the advective particle fluxes [25; 26], so mass balance negligible diffusion takes the form [9; 10; 25]  
\[\frac{\partial c}{\partial t}+v\frac{\partial c}{\partial x}+\frac{\partial \rho}{\partial t}=0. \tag{2}\]  
Mainly, kinetic equations can be divided into two groups: irreversible and reversible deposition according to their characteristics. In irreversible deposition [16; 17], deposited particles never get back to flow again and kinetic equation is in the form  
\[\frac{\partial\rho}{\partial t}=\beta c, \tag{3}\]  
where \(\beta\) is the coefficient characterizing intensity of deposition, which may be changed during the filtration. In reversible deposition [18; 19; 20; 21; 22; 23; 24; 25; 26], both attachment and detachment of suspended particles may occur in parallel and kinetic equation has the form  
\[\frac{\partial\rho}{\partial t}=\beta_{att}c-\beta_{det}\rho, \tag{4}\]  
where \(\beta_{att}\) and \(\beta_{det}\) are empirical attachment and detachment coefficients. In the literature, we can find various types of kinetic equations, because there are many factors affecting the process of suspended particles transportation and deposition in porous medium such as particle size distribution [24; 27; 28; 29], grain size distribution [28; 29; 30], hydrodynamic forces [30; 31], temperature [32; 33] and others. In [7; 19], a multistage deposition kinetics is used, with ripening, transient and breakthrough stages. Studies of the flow in deep bed filters revealed the existence of two components: the jet stream and stagnant zones in the areas of grain contact [34]. The deposition formed in the pores depends on the place of formation. It was noted [34] that on the frontal parts of grains, that is, in the areas washed by the jet component, the deposition has a dynamic character: the attached particles quite often they break down in a stream and again reattach. Deposition in stagnant zones is more passive. Its formation occurs slowly in comparison with deposition on the frontal parts of grains, but reattachment from stagnant zones is not observed either. According to this, in [34; 35; 36; 37], the porous media is assumed to have two types of zones: active and passive. It is assumed that active and passive zones evenly distributed in porous medium and at each point of the porous medium portion of active and passive zones proportional to their capacity. In active zones, the deposition is formed in reversible form, whereas in passive zones it is formed in irreversible form. The kinetic equations are given by the equations, respectively, for each zone,  
\[\frac{\partial\rho_{a}}{\partial t}=\beta_{a}\left(c-\frac{\rho_{a}}{\rho_{ a0}}c_{0}\right), \tag{5}\]  
\[\frac{\partial\rho_{p}}{\partial t}=\left\{ \begin{array}{ll}\beta_{p0}c& \quad\text{for}\qquad\quad\rho_{p}\leq\rho_{p1},\\ \beta_{p0}c\rho_{p1}/\rho_{p}&\quad\text{for}\qquad\quad\rho_{p1}<\rho_{p}< \rho_{p0}\,,\\ 0&\quad\text{for}\qquad\quad\rho_{p}=\rho_{p0}\,,\end{array} \right.