This paper investigates the self-similar analysis of a system of equations given by a diffusion process. The study focuses on a coupled system of nonlinear parabolic partial differential equations (PDEs) that describe mass conservation laws and filtration flows in heterogeneous media. To evaluate the qualitative properties of the solutions, a system of self-similar equations and the nonlinear splitting method were used. The findings contribute to understanding nonlinear filtration dynamics in various physical and engineering applications, such as groundwater flow, oil extraction, and heat transfer in porous media. Additionally, the study provides insights into the mathematical properties of such systems, including regularity, stability, and the qualitative behavior of solutions in multidimensional settings.