In this paper, we investigated a two-phase model of incompressible fluid filtration. We apply the irregular grid method and the variable direction scheme. This approach gives the effect of solution accuracy near discontinuities (wells) due to the use of an irregular grid. The model is built taking into account the influence of capillary pressure and gravitational forces. The results confirm that the amount of oil that came out of the layer in a certain time is equal to the volume of water injected, except for the amount of water in the outlet stream in the same time. The proposed solutions of the new approach are intended to improve the methods and schemes of numerical investigation of this model. A balanced monotonic finitedifference scheme was developed and an efficient algorithm for its implementation was proposed. From a practical point of view, numerical modeling allows early prediction of performance. Thus, the applied aspect of the use of the obtained scientific result is the possibility of improving the process by taking into account the distribution of water saturation in the layer.

The article defines the basic concepts of filtration theory and provides an overview of the existing mathematical models of inhomogeneous liquids in porous media. The paper considers the Stefan problem. The number of scientific papers devoted to the study of porous structures has recently increased. This is primarily due to the fact that the prob-lems of oil and uranium production have been identified, and the solution of environmental problems is overdue. Therefore, a new device is needed to develop models of liquid filtration. With the advent and development of computer technology, it has become easier to solve problems that require numerical methods for their solution. Understanding the movement of fluids and the mechanism of dissolution of rocks under the action of acids in heterogeneous porous media is of great importance for the extraction and production of oil and the effective management of these processes. The article examines the mathematical model of the theory of isothermal filtration. Possible variants of the solva-bility of the model are shown. The research scheme consists of the output of a mathematical model, the formulation of the problem, one variant of the solution of the problem, the algorithm of the numerical method of solving the problem.