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.

An efficient method for numerically solving the one-dimensional Stefan problem is proposed herein. A computational algorithm for solving free boundary problems has been developed. This provides a means of solving problems with an arbitrary and variable number of phases, both in terms of thermal conductivity and diffusion. The solution algorithm is based on the application of the finite element method. The calculations are performed according to a homogeneous scheme. This makes the method universal and renders it possible to be referred to the class of shock-capturing methods. Accurate tracking of the position of the boundaries is carried out, in the same manner as in the methods with edge detection, which makes it possible to solve problems with high accuracy, inherent in methods of this type.