The paper studies the coupled diffusion-dissolution process in reactive porous media, separated by a fracture channel. An aggressive solute, corresponding for e.g., to a complete demineralization that dissolves the solid skeleton of the surrounding porous material, is prescribed at the inlet of the fracture. By means of asymptotic dimensional analysis it is shown that for large times the diffusion length in the fracture develops with the quadratic root of time. In comparison with the 1D-Stefan Problem, in which the dissolution front evolves with the square root of time, this indicates that the overall solute evacuation through the structure slows down in time. This phenomenon is referred to as a diffusive solute congestion in the fracture. This asymptotic behavior is confirmed by means of model-based simulation, and the relevant material parameters, related to only the chemical equilibrium condition, are identified. The obtained results suggest that the presence of a small crack does not significantly increase the propagation of the dissolution front in the porous bulk, and hence the overall chemical degradation of the porous material. The same applies to other diffusion induced demineralization, mineralization, sorption and melting processes, provided that the convective transport of the solute in the crack is small in comparison with the solute diffusion. The result is relevant for several problems in durability mechanics: calcium leaching of concrete in nuclear waste containment, mineralization and demineralization in bone remodeling, chloride penetration, etc.