The movement of a viscous fluid when injected into another with lower viscosity results in the formation of the Saffman-Taylor instability. In this phenomenon, the injected fluid penetrates the surrounding fluid, forming viscous fingers that compromise displacement efficiency. Understanding and controlling this mechanism are crucial for enhanced oil recovery. To investigate the radial displacement of viscous fluids in a confined environment, such as a Hele-Shaw cell, we conducted a linear stability analysis and nonlinear simulations based on contour integral method. The linear analysis predicts the mode of fastest growth, related to the expected number of fingers. On the other hand, the non-linear regime of this displacement results in tip-splitting events, ultimately determining the number of fingers. Thus, high-resolution numerical simulations were conducted, varying dimensionless parameters controlling the dynamics, the viscosity contrast and effective surface tension. These simulations examined non-linear stages in cases where the linear analysis predicts the same temporal evolution for the mode of fastest growth. Results of non-linear simulations differ significantly from linear expectations due to different time scales required for each case. Additionally, the interfacial patterns produced capture morphological details of the structure of viscous instabilities.