Collection and burial of carbon dioxide (CO2) is one of the practical options for carbon dioxide disposal. Carbon dioxide injected into the well at very high pressure mixes with the remaining oil, thus increasing oil volume and reducing its viscosity. Thanks to this, it is possible to increase oil production. Wells are tested before the injection to ensure the proper geological conditions. It is necessary to find a permeable rock that is stable enough to contain CO2 in a liquid state. Therefore, it is a problem to find a proper injection point to reliably entrap CO2 in microscopic pores of rocks similarly to the way that entraps oil, gas, and natural CO2 for millions of years. Such processes are described by mathematical filtration models for liquids and gases in poroelastic media. Up to date, there is no generally accepted approach for modeling the movements of multiphase systems. Some well-known models are quite complex to use in applied and engineering tasks. The mathematical theory to substantiate the initial boundary value problems is poorly developed, and there are insufficient results on the solvability and properties of solutions.