We develop a pore network model for the evaporative drying of macroporous media that accounts for the major pore-scale mechanisms experimentally identified to play an important role on the drying rates and phase distribution patterns. The model accounts for viscous flow through liquid films, gravity and mass transfer, both within the dry medium and also through a mass boundary layer over the external surface of the medium. Also accounted are the heterogeneity of the pore size distribution and pore wall microstructure effects expressed through the degree of corner roundness. The latter plays a major role on the extent of the film region. The model is then used to study capillary, gravity and external mass transfer effects through the variation of the appropriate dimensionless numbers. The effect of gravity is particularly analyzed for the two cases, when gravity is opposing and when it is enhancing drying, respectively. In the latter case, strong mass transfer and viscous forces compared to gravity can prevent instability of the receding evaporation front, leading to a two constant-rate-regime drying curve in agreement with the 1-D theory proposed earlier.