This work presents a new discrete fracture model, namely the Projection-based Embedded Discrete Fracture Model (pEDFM). Similar to the existing EDFM approach, pEDFM constructs independent grids for the matrix and fracture domains, and delivers strictly conservative velocity fields. However, as a significant step forward, it is able to accurately model the effect of fractures with general conductivity contrasts relative to the matrix, including impermeable flow barriers. This is achieved by automatically adjusting the matrix transmissibility field, in accordance to the conductivity of neighboring fracture networks, alongside the introduction of additional matrix-fracture connections. The performance of pEDFM is investigated extensively for two- and three-dimensional scenarios involving single-phase as well as multiphase flows. These numerical experiments are targeted at determining the sensitivity of the model towards the fracture position within the matrix control volume, grid resolution and the conductivity contrast towards the matrix. The pEDFM significantly outperforms the original EDFM and produces results comparable to those obtained when using DFM on unstructured grids, therefore proving to be a flexible model for field-scale simulation of flow in naturally fractured reservoirs.@en