In the Reactive Vapor Infiltration process a powder of molybdenum is compressed and exposed to a hot silicon-tetrachloride gas which reacts on external surfaces of the powder pellet and releases silicon atoms which in turn are absorbed into the solid pellet, diffuse, and react with molybdenum to form a thin layer of $M_5 Si_ 3$. Later on during the process, the Mo_5 Si_3 is transformed into the final material $Mo Si_2$. These reactions are accompanied by a 158% volume expansion which causes the material to flow to fill in the pores and possibly produce cracks in the material. We modeled the chemical reactions and volume expansion using a viscous model. We coupled a set of reactions-diffusion equations for species mass fractions with the Navier-Stokes equations for the material flow. The system is discretized in space using stablized Galerkin finite element method using the Artificial Incompressibility method with the adaptive mesh software. Our numerical results are in good agreement with experiments and predict diffusivities of silicon in different mixtures. We also showed that the dominant transport of reactant is through solid-state diffusion of silicon and not through gas infiltration. This is due to large volume expansions causing the pores to close and preventing gas diffusion in the solid. In order to reduce the effect of volume expansion, we considered different initial mixtures of $ Mo $ and $ Mo Si_2$ and porosities, again our results are in full agreement with experiments. This work can be extended by including fibers in the powder and an visco-elastic mechanical model to compute stress distributions during the process and residual stresses. We also need to perform different parameter studies with varying fiber sizes and distributions and different initial mixtures and porosities.