We consider the problem of underground flow simulations in fractured media. This is a large scale, heterogeneous multi-scale phenomenon involving very complex geological configurations. Within the Discrete Fracture Network (DFN) model, we focus on the resolution of the steady-state flow in large fracture networks. Exploiting the peculiarity of the Virtual Element Method (VEM), which allows the use of rather general polygonal mesh elements, we propose an approach for building a suitable mesh for representing the solution on DFNs. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]