Two-phase immiscible fluids in a two-dimensional micro-channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow while the Oldroyd-B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro-channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level-set method is used and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two-phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one.