This is a study of thermal plumes in a permeable fluid-saturated slab of a porous medium (of finite uniform thickness but otherwise infinite extent) that is heated by either an instantaneous or a steady line source embedded in the medium. The slab, which may be horizontal or sloping, is initially at ambient conditions; the impervious upper surface remains at the ambient temperature, while the impervious base is thermally insulated. The transient temperature distribution, the surface heat flux and the convective flow velocity are calculated for small instantaneous line heat energy sources. They show how the flow develops, spreads and slows as time progresses, and an estimate of the time to decay is given. Steady-state temperature and velocity profiles are calculated for embedded line sources that provide heat energy at a small constant rate, and the surface heat flux distribution is calculated.