## 2 Mathematical Model of Two-Component Suspension Filtration  
Let us consider a semi-infinite homogeneous porous media with initial porosity \(m_{0}\), filled with a homogeneous fluid. At the point \(x=0\) starting from \(t=0\) a two-component suspension with concentration \(c_{0}=c_{0}^{(1)}+c_{0}^{(2)}\) (\(c_{0}^{(1)}\), \(c_{0}^{(2)}\) are concentrations of first and second type of particles in inlet suspension, respectively) is injected into the porous media with the velocity \(v\left(t\right)=v=\text{const}\). The mathematical model of filtration of two-component suspensions dual-zone porous media is a generalization of the corresponding one-component model [34; 42; 43; 44]. The system of equations consists of balance equations for each component, [PERSON]'s law and kinetic equations for each zone of media. The balance equation with a given flow velocity regime, including the diffusion term is  
\[m\frac{\partial c^{(i)}}{\partial t}+v\frac{\partial c^{(i)}}{\partial x}+ \frac{\partial\rho_{a}^{(i)}}{\partial t}+\frac{\partial\rho_{p}^{(i)}}{ \partial t}=D^{(i)}\frac{\partial^{2}c^{(i)}}{\partial x^{2}}, \tag{7}\]  
where \(m\) is current porosity of media, \(c^{(i)}\) is the concentration of the \(i\)th component of the suspension \((m^{3}/m^{3})\), \(\rho_{a}^{(i)}\), \(\rho_{p}^{(i)}\) are respectively the concentration of deposition in the active and passive zones of the \(i\)th component of the suspension \((m^{3}/m^{3})\), \(D^{(i)}\) is diffusion coefficient for each component \((m^{2}/s)\), \(i=1\), \(2\) corresponds to the numbers of the components. These mass balance equations derived on the base of assumption \"accumulation rate = (flow in) \(-\) (flow out)\" [16] for each component. The first term on left hand side describes changes in concentration of suspended particles, the second is convective term, and the third and fourth terms describes changes in concentration of deposition in active and passive zone, respectively. The expression on right hand side describes diffusion term. For the changes of porosity and permeability of porous media, we use [PERSON]'s law and the Carman-Cozeny equation  
\[v =K\left(m\right)\left|\
abla p\right|, \tag{8}\] \[K\left(m\right) =k_{0}m^{3}\Big{/}\left(1-m\right)^{2},\] (9) \[m =m_{0}-\left(\rho_{a}^{(i)}+\rho_{p}^{(i)}\right), \tag{10}\]  
where \(K\left(m\right)\) is filtration coefficient, \(\left|\
abla p\right|\) is the modulus of pressure gradient, \(k_{0}=\text{const}\). We generalized the kinetic equation of the process of attachment and detachment of particles in the active zone for a two-component suspension reflecting the dynamic factors [18; 37] as follows  
\[\frac{\partial\rho_{a}^{(i)}}{\partial t}=\frac{\beta_{a}^{(i)}}{1+\gamma^{( i)}\left|\
abla p\right|}c^{(i)}-\beta_{a}^{(i)}\frac{\rho_{a}^{(i)}\left(1+ \omega^{(i)}\left|\
abla p\right|\right)}{\rho_{a0}^{(i)}}c_{0}^{(i)}, \tag{11}\]  
where \(i=1\), \(2\), \(\rho_{a0}^{(i)}\) is the capacity of the active zones for the \(i\)th component of the suspension, \(\beta_{a}^{(i)}\) is the coefficient characterizing the kinetics in the active zone of the \(i\)th component of the suspension, \(\gamma^{(i)}\), \(\omega^{(i)}\) are constant coefficients characterizing intensity of influence of pressure gradient on particle attachment and detachment process in active zone. The equations of kinetics of the formation of irreversible deposition in passive zone are as follows  
\[\frac{\partial\rho_{p}^{(i)}}{\partial t}=\beta_{p}^{(i)}\phi_{i}\left(\rho_ {p}^{(1)}\,,\rho_{p}^{(2)}\right)c^{(i)},i=1,\,2, \tag{12}\]  
where \(\beta_{p}^{(i)}\) is the coefficient characterizing the kinetics in the passive zone of the \(i\)th component of the suspension. We denote the partial capacity of the passive zones by \(\rho_{p0}^{(i)}\). As in [34], we can assume that the total filter capacity is mostly determined by the first component so that \(\rho_{p0}^{(1)}>\rho_{p0}^{(2)}\).