Carbonation reactions dynamically change the microstructure of a concrete matrix, dissolving calcium hydroxide and precipitating calcite, and require the presence of carbon dioxide and moisture at the pore or crack site. Consequently, carbonation reaction play an important role in the durability of concrete structures. On one hand, they change the local pH of the matrix, which may favour the corrosion of reinforcement bars. On the other hand, they take place in a natural way in young concrete and can favour the sealing of cracks of small width. In this work we model the concrete matrix as a partially saturated porous medium where Richards' equation is used to model the fluid flow. This equation is coupled to a set of conservation equations for the chemical species participating in the carbonation reaction (CO$_2$, Ca(OH)$_2$ and CaCO$_3$). The rate of the chemical reactions depends on the content of moisture and of the different reactants. Furthermore, the carbonation reaction produces water. Therefore, we obtain a flow and reactive transport model on which nonlinear couplings appear on the different reaction terms. Different linearization techniques (modified Picard, Newton and L-scheme) are combined with monolithic and splitting schemes. The performance of these schemes is discussed for experimental set ups of normal and accelerated carbonation obtained from the literature.