When a porous medium is bounded by an impermeable surface it is well known that the porosity and hence the permeability increases near that surface. In this paper, we examine how this phenomenon affects the flow and heat transfer from a uniform temperature heated surface. In particular, we assume that the region of varying permeability has constant thickness, and we present detailed numerical and asymptotic solutions for the resulting nonsimilar flow. Near the leading edge the flow is enhanced and the rate of heat transfer is much higher than in the uniform permeability case. Further downstream the region of varying permeability is embedded well within the boundary layer, and in this case the flow and heat transfer is only slightly different from that of the uniform case.