Reservoir simulators often predict poor injectivity in polymer enhanced oil recovery (EOR) projects because of the high polymer viscosity, which is a deterrent for the project. However, field studies have shown much higher injectivity than predicted by the models. The objective of this work is to perform grain-scale, coupled fluid dynamics and geomechanics modeling to predict the injectivity of viscous, non-Newtonian polymers in wellbores. Fluid-rock interactions are modeled by coupling computational fluid dynamics (CFD) and the discrete element method (DEM). Fluid flow is determined using an open-source CFD software that solves the volume-averaged Navier-Stokes equation using the finite volume method on Eulerian grids. Grain-scale geomechanics (DEM) is used to explicitly solve the particle trajectories in a Lagrangian reference system. The simulation results confirm the hypothesis of fracture initiation and sand failure near the injector. The results show that the polymer-driven fracture initiation is associated with sand shear failure, while the fracture geometry is the result of the localization of sand shear failure and fluidization of unconsolidated sand at the fracture tip. The injection of a viscous fluid can create fractures in the direction perpendicular to the applied minimum principal stress. The presence of fractures increases the injectivity. The peak injection pressure is more than 3 times greater than the applied minimum principal stress. The viscosity increase of polymers promotes the initiation of fractures and results in a greater fracture aperture. The injection of polymer can promote the initiation of fractures, and therefore, increase the polymer injectivity. This work, for the first time, uses a grain-scale model to predict polymer-driven fracture initiation and demonstrates the improved injectivity observed in the field.