In this paper, the melting process inside an irregular geometry filled with high thermal conductivity porous matrix saturated with phase change material PCM is investigated numerically. The numerical model is resting on solving the volume averaged conservation equations for mass, momentum and energy with phase change (melting) in the porous medium. The convection motion of the liquid phase inside the porous matrix is solved considering the Darcy, Brinkman and Forchiemer effects. A local thermal non-equilibrium assumption is considered due to the large difference in thermal properties between the solid matrix and PCM by applying a two energy equation model. The numerical code shows good agreement for pure PCM melting with another published numerical work. Through this study it is found that the presence of the porous matrix has a great effect on the heat transfer and melting rate of the PCM energy storage. Decreasing the porosity of the matrix increases the melting rate, but it also damps the convection motion. It is also found that the best technique to enhance the response of the PCM storage is to use a solid matrix with high porosity and high thermal conductivity.