This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 + x(2))(mu), where mu is a constant and x is the distance along the surface. It is shown that for mu > -1/2 the solution develops from a similarity solution which is valid for small values of x to one which is valid for large values of x. However, when mu less than or equal to -1/2 no similarity solutions exist for large values of x and it is found that there are two cases to consider, namely mu < -1/2 and mu = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of mu.