A Lattice–Boltzmann method is used to calculate grain-scale, low Reynolds number, single phase, interstitial porous melt flow in two independent igneous microstructures: 1) a microstructure simulated using a stochastic algorithm for progressive crystallization, and 2) a natural microstructure obtained through X-ray Computed Tomography (CT) of a partially melted basalt. In both cases, the error in the calculated flow field due to finite discretization and domain size has been estimated or demonstrated to be insignificant. Visually, the porous melt flow tends to localize into high flux channels, especially with increasing crystallinity, and this impression is quantitatively confirmed by increasing skewness of the velocity distribution in the direction parallel to the imposed pressure gradient driving the flow. A change from uniform to localized melt flow in naturally occurring situations may have a profound effect on the distribution of trace elements and, if the melt is reactive, the chemical and structural evolution of the igneous microstructure through localized phase change. Permeabilities of both microstructures are determined from the calculated steady flow field. The permeabilities are then fitted to two different correlation models which are based on the Rumpf–Gupte and Carman–Kozeny relations. In these models, we express permeability as functions of melt fraction and either the mean crystal length or specific surface area. Extrapolated permeability to higher melt fractions using both correlation models for the partially melted basalt is shown to be within a factor of four of experimentally determined permeability of a similar sample. We determine permeability estimates in the relatively unexplored melt fraction range of 30%–80% and find consistency with previous work for permeability estimates in the melt fraction range of 20%–30%.