The present work describes the development of a two-parameter non-uniform capillary model to describe kinetics of flow in porous solids with complex tortuous varying paths. Experimentally, the rate of fluid flow in such a non-uniform capillary is found to be orders of magnitude slower compared with a corresponding average uniform capillary. This slow rate is explained in terms of an extremely small 'effective' hydrodynamic radius. The origin of such an 'unphysical' radius is rationalized based on geometrical considerations and effective driving forces for flow through a stepped capillary. Infiltration rate parameters are derived from the geometry of the porous medium for both wetting and non-wetting conditions.