Liquid permeability was measured on a range of porous samples, using a specially-constructed high pressure permeation cell. The samples consisted of compacted fine isotropic mineral pigment - calcium carbonate - compressed over a range of compaction pressures. These samples have pore diameters typically (number basis) finer than 0.1μm. The porosities and the pore size distributions of the samples have been determined by means of mercury porosimetry. The permeability of these fine-pore network systems is seen under certain conditions not to obey the well-known linearity of the Darcy relation as a function of applied liquid pressure differential. A considerable pressure-flux hysteresis is observed following saturation by imbibition. Furthermore, it is seen that there is no direct linear correlation between permeability and porosity despite the use of a constant pigment particle size distribution and, hence, skeletal size distribution. The measured permeability displays a local maximum at a fractional porosity of approximately 0.26 with a further distinct drop at around 0.27. Interestingly, this phenomenon is also confirmed using a computer-based porous network simulator which provides a relative permeability based on matching the mercury percolation data. Taking into account the effects of earlier-proposed mechanisms of preferred pathway flow and film flow during imbibition, it is postulated that some pores remain unfilled during imbibition and saturation prior to the permeability study, such that there remains entrapped air or vapour phase in microscopic ganglia. This air does not dissolve in the aliphatic mineral oil used in the experiments. Structures with a delineated pore-throat structure will probably lead generally to observed non-linear phenomena within the range of dimensions studied here, which can have implications, amongst others, for microscopic filtration, catalysis and absorption phenomena.