In this paper, a hybrid finite difference/finite element method for solving the saturation equation in the problem of two-phase non-equilibrium fluid flow in porous media is constructed. The model under consideration is obtained on the basis of the non-equilibrium fluid flow model by S. M. Hassanizadeh with the generalized global pressure concept. Due to the hyperbolic nature of the equation, it has several difficulties leading to the need for a careful choice of the solution method. The classical Galerkin method leads to the appearance of non-physical oscillations at phase interfaces. The paper investigates the application of stabilized finite element methods for their suppression. Three classical stabilized methods are compared: the streamline upwind Petrov-Galerkin (SUPG), the Galerkin least squares (GLS), and the unusual stabilized finite element method (USFEM), and several stabilizing parameters. The comparison of these methods and stabilization parameters is carried out on the basis of three computational experiments.