Low-velocity non-Darcy flow often occurs in low permeability reservoirs. In this paper, we discuss how to incorporate a non-Darcy flow correction into embedded discrete fracture modelling (EDFM). The resulting method is validated by reproducing type curves containing the characteristic no-flow segment, followed by a nonlinear and a linear segment. Simulation results are also validated against a direct simulation method that uses local grid refinement to (almost) resolve hydraulic fractures. Second, we discuss how grid-based flow diagnostics can be formulated on top of the new EDFM method and used to provide computationally inexpensive predictions of flow patterns, interwell connections, and measures of dynamic heterogeneity. We then demonstrate how such flow diagnostics can be used to optimize well placement and improve oil recovery.