We consider a model of combustion in hydraulically resistant porous media. There are several reductions of this systems that can be used to understand the evolution of the combustion fronts. One reduction is based on the assumption that the ratio of pressure and molecular diffusivities is close to zero, a different reduction is obtained when the Lewis number chosen in a specific way. Fronts exists in both reduced systems. For the stability analysis of the fronts, we, first, consider initial conditions of a specific form, then show that the stability results extend to the fronts in the full system with the same Lewis number. The fronts are either absolutely unstable or convectively unstable.