Polymer flooding plays an important role in enhanced oil recovery (EOR), particularly in China, where partially hydrolyzed polyacrylamide (HPAM) and hydrophobically associating water-soluble polymers (HAWP) are used in onshore and offshore reservoirs, respectively. Many researchers have highlighted the elasticity of HPAM, which can be used to improve the sweep efficiency, i.e., the ratio of the area swept by an injected fluid to the oil area. On the other hand, fewer studies exist on the elasticity of HAWP. In this study, we investigate the flow of HAWP and Xanthan solutions with identical viscosities in core experiments in terms of elasticity; results reveal that the HAWP can produce shear thickening in the core. The constitutive equation for the HAWP can be obtained using the simulation results matched with the experimental data. On the basis of these experiments, we established a two-phase flow model of a polymer and oil, including the continuity, momentum, constitutive, and phase equations. The volume-of-fluid (VOF) method was used to track the interface between the two phases. A complex pore model was established based on the glass-etched model used in the experiment. We used the OpenFOAM platform to solve the mathematical model. The saturation, pressure, and stress tensor distributions were obtained. The results show that the displacement efficiency increased as the elasticity of the polymer increased; accordingly, the elasticity can enlarge the sweep area and decrease the residual oil saturation. As the elasticity increases, the stresses (the first normal stress, second normal stress, and shear stress) increase. Finally, the results obtained in this study can be used as a guideline in polymer design, screening, and optimization in the polymer flooding oilfields.