Abstract.  In Part I of this paper, we introduce the Mason-Morrow shape factor, i.e., a dimensionless hydraulic radius, and corner half-angles to capture the geometry of noncircular capillaries pertinent to a physically adequate pore network description of porous media in single-phase flow.  We give analytic expressions for random corner half angles that satisfy a given shape factor calculated from the microscopic images of pore space. We demonstrate that use of the shape factor leads to particularly simple expressions for the hydraulic conductance in single-phase flow through noncircular capillaries.  In particular, we obtain the hydraulic conductances of triangular ducts semi-analytically, using conformal mapping.  The conductances of rectangular and elliptic ducts are calculated analytically. In Part II, we use the capillary shape factor to derive universal expressions for the hydraulic conductance in two-phase flow with no slip and perfect slip boundary conditions at the water corner filament-oil interfaces.