Totally fissured media in which the individual cells are isolated by the fissure system are effectively described by double porosity models with microstructure. Such models contain the geometry of the individual cells in the medium and the flux across their interface with the fissure system which surrounds them. We extend these results to a dual-permeability model which accounts for the secondary flux arising from direct cell-to-cell diffusion within the solid matrix. Homogenization techniques are used to construct a new macroscopic model for the flow of a single phase compressible fluid through a partially fissured medium from an exact but highly singular microscopic model, and it is shown that this macroscopic model is mathematically well posed. Preliminary numerical experiments illustrate differences in the behaviour of solutions to the partially fissured from that of the totally fissured case.