We derived a well-balanced scheme using a hydrostatic reconstruction of the variables at the interface of each cell to numerically simulate blood flow in arteries with a varying section and elasticity. The method has been then validated on examples taken from the literature, where an artery with a varying section at rest and a constant elasticity were considered. Asymptotic solutions were computed to highlight the effect of the viscous and viscoelastic source terms. Finally, a simulation of blood flow in a tapered artery was performed, where the section at rest and the elasticity were varying. In each test case, the well-balanced scheme showed good results when other simpler schemes behaved poorly, generating spurious reflections and nonphysical velocities.