Abstract
In the paper, we apply the vanishing viscosity method for an approximate solution to the Riemann problem. This approach gives the effects of the accuracy of the solution and the speed of convergence by discrediting time and spatial variables. The obtained method ensures the smoothness of the solution without taking into account the capillary pressure. The results confirm the negligible influence of cross-link conditions compared to the classical Darcy approach. The proposed solutions of the new approach are intended to improve the methods and schemes of discretization both in space and in time. This is achieved by minimizing viscosity, and discretization in space and time. These factors are of paramount importance for studying phenomena with variable saturation in the transient mode and analyzing water/oil flows and migrations in real time, since discretization in space and time affects the accuracy and convergence of calculations. Our result in the form of obtaining viscous solutions of the filtration process is interesting from a theoretical point of view. From a practical point of view, numerical modeling allows early prediction of performance. Thus, the applied aspect of using the obtained scientific result is the possibility of improving the process by taking into account the influence of phases of fluid flows