Mathematical modeling of the three-dimensional evolution of the interface between fluids of different viscosities and densities in an inhomogeneous ground

The three-dimensional evolution of the interface between fluids of different viscosities and densities in the case of a piston displacement is considered. The problem is reduced to a system of integral and differential equations, which are solved numerically by the method of discrete singularities. The practical convergence of the numerical scheme is analyzed, and the numerical results are compared with an analytical solution.