A pore network model of drying with heat transfer is developed. The model is applied to Study the influence of surface tension gradients induced by thermal gradients on the phase distribution within a capillary porous medium. The numerical simulations show that surface tension gradients can lead to invasion percolation in a destabilizing gradient (IPDG) patterns or invasion percolation in a stabilizing gradient patterns depending on the sign of thermal gradient. The surface tension gradient effect is shown to be significant for sufficiently weakly disordered porous media. The results are summarized on a phase diagram delineating the various patterns that can be expected as functions of thermal gradient and disorder parameter. This diagram is pertinent to situations where occupation probability gradients induced by viscous or gravity effects are negligible. The results also indicate the possibility of a somewhat paradoxical convective drying situation when thermal gradients and disorder are such that a IPDG pattern develops. In this case, contrary to more conventional situations, it may be much more efficient to blow an air colder than the porous medium initial temperature.