In this paper, the non-Newtonian immiscible two-phase polymer flow in a petroleum reservoir has been investigated numerically. The fluid flow in a porous medium is simulated as a compressible flow. The Carreau-Yasuda constitutive equation is employed as the model of non-Newtonian fluid. The IMPES method is used for numerical simulation, in which the pressure equation is discretized and solved by an implicit approach and the saturation equation is solved by an explicit method. Results reveal that zero-shear rate viscosity has a high impact on the sweep efficiency of the reservoir and also controls the channeling and viscous fingering effects. In addition increasing the viscosity of non-Newtonian fluid improves cumulative oil production and diminishes the viscous fingering phenomenon caused by injected fluid. The relaxation time of Carreau-Yasuda fluid, which is the elastic characteristic of the non-Newtonian fluid, for low permeability values cannot influence flow characteristics inside the reservoir, however for higher permeability values its effect becomes more sensible. Increasing the injection rates leads to the increase of fluid production, while the injection rate has an optimum range to reach the optimum oil production. In addition, the effect of variation of the injected fluid properties on the polymer breakthrough time has been investigated and results presented.