Acidic fluid injection in rock formations may generate geochemical reactions that can modify the mineral assemblage of the rock and disturb thermodynamic equilibria. Numerical difficulties of reactive transport simulation are that geochemical reactions are at the pore scale, may appear in short time period and are very sensitive to the mesh size and/or time step. The classical approach for reservoir engineers consists in upscaling the high resolution petrophysical values to assign to a low-resolution model. For reactive transport modelling, the upscaling step will impact not only the mass fraction of each species but also the mineral dissolution and/or precipitation processes that highly depend on mass fractions. This paper recalls the Compositional Dual Mesh Method, an original algorithm for a compositional flow modelling in porous media with rock-fluid interactions using two different space and time discretization: one mesh, as usual for the pressure equation and a much finer one for the chemical reactions. The interest of this scheme is that the calculation of the flow on the high-resolution grid is done solving a local problem on each coarse cell. Two examples of CO2 injection in carbonate reservoirs illustrate this algorithm.