Epitaxial Laser Metal Forming (E-LMF) consists in impinging a jet of metallic powder onto a molten pool formed by controlled laser heating and thereby, generating epitaxially a single crystal deposit onto the damaged component. This new technique aims to be used for the repair and reshape single crystal gas turbine components. Because of the very localised melting pool, the high temperature gradients produced during the process must be carefully controlled in order to avoid both the columnar-to-equiaxed transition (CET) and the appearance of hot tears. To this end, heat flow modelling is required to establish the relationship between process parameters such as laser power, beam diameter and scanning speed, and the local solidification conditions. When modelling the heat transfer within the sample, it is necessary to include the liquid flow pattern generated by the surface tension driven convection known as the Marangoni effect. Indeed, the fluid flow in the liquid pool dictates the shape of the traces as shown by the measurements carried out at EPF-Lausanne in re-melting experiments. A three dimensional (3D) model is implemented in the finite element software calcosoft in order to model the development of the fluid convection within the liquid pool. It is shown that the velocities due to natural convection are of the order of 1 mm/sec whereas Marangoni convection produces velocities of the order of 1 m/sec. Moreover, at low scanning speeds, the liquid pool becomes larger than the beam diameter and the development of Marangoni eddies leads to a widening and deepening of the pool. The local solidification conditions such as the thermal gradient and the solidification speed can be extracted at both the solidus and liquidus temperatures to assess the risk of CET and hot cracking.