Global change from carbon-based to carbon-free economy has driven the development of a number of innovative technologies for sealing oil wells in mature fields. The common technology in use nowadays relies on cementing the borehole to permanently seal and abandon old wells. However, this procedure has a high cost and takes several weeks to be concluded, which constitutes a burden for offshore wells. This work investigates an innovative technology for plug and abandonment based on the release of large amounts of heat from thermite reactions at the sealing location in the borehole. Inserting two volumes of thermite generating different amounts of heat between the two steel pipes forming the oil well allows to melt the internal structure of the well to form a plug, while preserving its external structure to prevent leakage. Tubing and casing are melted, forming a plug after the cool down period. Transitory transport of heat generated by a thermite mixture is numerically investigated by using a new formulation for a phase change model based on the enthalpy-porosity idea. A general One-Energy Equation Model (1EEM) is extended to deal with melting and solidification of pure substances and alloys. Prior to melting and after solidification, the solid material is characterized as a porous medium with minimal porosity and extremely low permeability. As temperature surpasses the melting point, both viscous and form drags in the volume-averaged momentum equation are diminished. In the energy equation, latent heat is implicitly accounted for in the accumulation term, deviating from the explicit treatment commonly found in existing literature. The liquid fraction across the entire domain is adjusted following the computation of a new temperature field. The governing equations undergo discretization through the control-volume method. The algebraic equation sets are eased using the Simple Method, and inner iterations employ the Strong Implicit Procedure. Studies have been carried out to validate the methods used, by comparing numerical results with both analytical and experimental results. Problems involving pure thermal diffusion and diffusion associated with natural external convection were compared with analytical results, while a problem combining diffusion and convection was tested against experimental results. Next, the well closure problem was investigated, first for a diffusion problem without phase change, then for a diffusion problem with state change, and finally for a more complete model taking into account convection generated by melting materials. For all three cases, grid size and time step independence studies were carried out, and the heat generation required to run the process was determined. Finally, the importance of convection in this study was evaluated.