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The paper deals with the numerical investigation and simulation of viscoelastic effects during polymer flooding. A possible pore structure is derived from high-resolution micro-CT images of a typical sandstone suitable for EOR applications. The simulation results achieved will allow for better understanding of micro-scale displacement efficiency and will help to optimize polymer flooding designs for enhanced oil recovery (EOR) projects. Numerical modelling of viscoelastic fluids is notoriously difficult. Standard Galerkin finite element method tends to have numerical oscillations. Therefore, a stabilised finite element method was used in this work, which contains stabilization terms for the constitutive equation. The model was validated using existing literature results. The viscoelasticity effect of polymer was simulated using the Oldroyd-B model. After validation this method was applied to several pore structures (contraction, expansion and expansion-contraction structure) and one real pore geometry based on SEM (Scanning Electron Microscopy). Flow effects and recovery effects were studied by changing the Reynolds number (Re) and Weissenberg number (We). Correspondingly, the streamline plots of the polymer solution flowing through the pore structures are shown. The plot of the first normal stress difference at the re-entrant profile of a contraction structure shows the cause of vortex enhancement. The fluid invades deeper into dead pores with higher elasticity. With a high Reynolds number, the fluid reaches into the dead pore even much deeper. The simulation results indicate the contribution of the viscoelasticity of polymer solutions to the micro-scale sweep efficiency. The non-dimensional model of simplified pore structures will contribute to the selection of suitable rheological parameters for polymer flooding processes.