## 1 Introduction  
Many scientific and practical studies in such fields as fluid and gas mechanics, hydrogeology, and ecology lead to problems in the theory of filtration of inhomogeneous liquids. Filtration models of heterogeneous liquids in porous media and methods for their numerical research serve as the theoretical basis for the processes of treating drinking and wasting water [1], increasing the efficiency of oil and gas production [2], studying colloid transport in chemical engineering [3], etc. Therefore, the development of adequate mathematical models for filtering heterogeneous liquids and solute transport in porous media, as well as their study are important tasks of mathematical modeling. As mentioned in [4], during the filtration of inhomogeneous liquids particles can deposit either on the filter surface or in the filter on the grains of the medium. Hence, by the method of particle deposition, filtration of suspension is divided into two groups: cake filtration and deep bed filtration [4]. Cake filtration usually is used in the cases when the suspension has large particles with high concentration whereas deep bed filtration is used for low concentration suspension [5]. In some cases, these two types are used together, when the suspension has particles with different sizes [6]. Various approaches have been developed to simulate the particle transport and deposition process, which can be arbitrarily divided into the so-called network models, particle trajectory analysis models, and empirical models [7]. A simulation of particle deposition in a porous medium based on the network representation was proposed in [8]. Network models were used to describe the change in permeability during particle deposition, using the method of random movement of a particle through a lattice with a given probability of particle capture. Approaches using the theory of an effective medium were developed in [9; 10; 11; 12]. Most studies considered two mechanisms of particle deposition: blockage of small capillaries by large particles and the deposition of small particles on the walls of pore channels with the formation of a sediment layer. Taking these mechanisms into account, attempts were made in [9] to predict the behaviour of the concentration of deposited particles, the filtration coefficient, and the change in permeability. In the models of particle trajectory analysis [13; 14], the skeleton of a porous medium is represented as a set of single sections, which can be spherical, cylindrical, or other simple shapes. The trajectory of each particle during the flow of fluid through separate sections of the medium is characterized by a stream function determined from the problem of flowing around, for example, a sphere or cylinder under the condition that various particles act on the particle (gravitational, hydrodynamic, van der Waals forces, etc.) [15]. Trajectory analysis leads to significant progress in understanding the deposition process at the individual pore level, although it is rather complicated. This approach allows to quite successfully predicting the value of the filtration coefficient and the change in the concentration of deposited particles. However, it has some limitations associated with determining changes in the permeability of a porous medium. In empirical (phenomenological) models [7; 16; 17; 18; 19], the process of filtering a suspension in a porous medium is described using equations of continuum mechanics, which are supplemented by empirical relations. In classical deep bed filtration models system of partial differential equations describing the process consists of mass balance equation end kinetic equation of deposition [16; 17].