Slow viscous flow through a circular orifice in a plane wall bounded by a fibrous medium is studied using an effective medium approach, based on the Brinkman equation, using an integral equation technique. The solution is of more general interest because it describes the transition in behavior from the classic Sampson solution for creeping flow through an orifice to a potential flow solution for Darcy flow as the permeability parameter increases. Asymptotic analytical results for the total flux through the orifice for a given pressure difference are obtained for both small and large values of the permeability parameter alpha defined by a/rootK(p), where a is the orifice radius and K-p the Darcy permeability. For intermediate values of alpha, the integral equation is solved numerically for the flux and velocity profile at the opening. For alpha much greater than O(1), the velocity profile at the opening has a minimum at the orifice center, rises dramatically near the edge of the orifice and then experiences a boundary-layer-like correction of thickness O(1/alpha) to satisfy the no-slip boundary condition. The close relation between pressure driven flow through a circular orifice and broadside translation of the complementary geometry, namely a circular disk in Stokes flow, is also discussed. The effect of the finite thickness of the orifice is taken into account using a simple model proposed by Dagan et al. (J. Fluid Mech. 115 (1982)) for Stokes flow through a pore of finite length. The present results are used to estimate the hydraulic conductance of orifice like pores in fenestrated capillaries and fenestral pores in the internal elastic lamina of the arterial intima.