## 1 Introduction  
Problems of solute transport and filtration of nonhomogeneous fluids are of great practical importance in many branches of engineering and technology. Oil reservoirs and aquifers may contain areas with low filtration characteristics where fluid filtration and solute transport occurs with arising of several specifics. Many works have been published where on the basis of experimental studies, several novel effects are revealed. In theoretical works, research is based mainly on phenomenological mathematical models. Conceptually, models can be divided into two large groups. In the first group of models, the process is described from a microscopic point of view. The solute transport is considered in a single pore or channel with a certain geometry or in a hollow media between aggregates of a certain type. The transfer from macropores to the environment is described by the diffusion equation. Such types of models were analyzed, in particular, in [1; 2; 3; 4; 5; 6; 7; 8]. In the second group of models, the geometry of macropores and their environment is not considered explicitly, but instead, channels of various sizes and surrounding rocks are considered to be whole and are studied from a macroscopic point of view. The medium is divided into two parts, in one of which the fluid is considered to be active, i.e., mobile, and in the other part--immobile or inactive. Mass transport between two parts (or zones) is usually described by a first order kinetic equation. Models of this type are commonly referred to as \"mobile-immobile\" type models. As one of the first models of this type, one can indicate the work [9], where the concepts of mobile and immobile parts (zones) of the environment were introduced. This approach was further developed in [10; 11; 12; 13; 14; 15; 16; 17]. In [9], a porous medium was divided into two parts (zones): with a mobile and immobile fluid. The diffusion flow between the zones is considered proportional to the difference in concentration in the zones. Sorption processes in both zones were considered. Sorption was considered to be in equilibrium with a linear isotherm. The model, presented in the work, well describes the well-known phenomenon of \"tailing\", which is characteristic for solute transport in aggregated media. In addition, an earlier manifestation of breakthrough curves has been established. In a more general form, the issues of solute transport using the second group of models were analyzed in [18]. The porous medium was divided into two parts (zones): with a mobile and immobile fluid. The diffusion flow between the zones was considered proportional to the difference in concentration in the zones. In [18], three cases were identified that can lead to \"tailing\". (1) Unsaturated conditions. At the same time, not all the void space of the medium is saturated with fluid, part of the space remains saturated with air. With a decrease in the water content of the media, significant tailing is observed. Part of the pores is excluded from the transport process and, accordingly, part of the fluid becomes immobile. This part of the media is called the \"dead\" zone or zone with a immobile fluid. (2) Aggregated media. Such media consist of well and poorly permeable zones, i.e., parts. The poorly permeable zone contains many micropores, where diffusion is the main transport mechanism of solute, while convective transport is insignificant and can be neglected. This leads to the restriction of fluid mixing and, consequently, the appearance of tailing, even in the case of complete saturation of the media. In fairly large aggregates with an immobile fluid, the diffusion path increases, which leads to tailing. (3) Filtration velocity. The results of some experimental studies show that tailing becomes more noticeable at low filtration velocity. In [19], theoretical calculations obtained in [20] were compared with the results of experimental studies on the transport of tritium in an unsaturated sorbing porous medium.