We propose a new section-averaged one-dimensional model for blood flows in deformable arteries. The model is derived from the three-dimensional Navier-Stokes equations, written in cylindrical coordinates, under the "thin-artery" assumption (similar to the "shallow-water" assumption for free surface models). The blood flow/artery interaction is taken into account through suitable boundary conditions. The obtained equations enter the scope of the non-linear convection-diffusion problems. We show that the resulting model is energetically consistent. The proposed model extends most extant models by adding more scope depending on an additional viscous term. We compare both models computationally based on an Incomplete Interior Penalty Galerkin (IIPG) method for the parabolic part, and on a Runge Kutta Discontinuous Galerkin (RKDG) method for the hyperbolic. The time discretization explicit/implicit is based on the well-known Additive Runge-Kutta (ARK) method. Moreover, through a suitable change of variables, by construction, we show that the numerical scheme is well-balanced, i.e., it preserves exactly still-steady states solutions. To end, we numerically investigate its efficiency through several test cases with a confrontation to an exact solution.