Macroscopic corrosion front computations of sulfate attack in sewer pipes based on a micro-macro reaction-diffusion model

We consider a two-scale reaction diffusion system able to capture the corrosion of concrete with sulfates. Our aim here is to define and compute two macroscopic corrosion indicators: typical pH drop and gypsum profiles. Mathematically, the system is coupled, endowed with micro-macro transmission conditions, and posed on two different spatially-separated scales: one microscopic (pore scale) and one macroscopic (sewer pipe scale). We use a logarithmic expression to compute values of pH from the volume averaged concentration of sulfuric acid which is obtained by resolving numerically the two-scale system (microscopic equations with direct feedback with the macroscopic diffusion of one of the reactants). Furthermore, we also evaluate the content of the main sulfatation reaction (corrosion) product—the gypsum—and point out numerically a persistent kink in gypsum’s concentration profile. Finally, we illustrate numerically the position of the free boundary separating corroded from not-yet-corroded regions.

💡 Research Summary

The paper presents a two‑scale reaction‑diffusion framework designed to capture the sulfate‑induced corrosion of concrete in sewer pipes. The authors separate the problem into a microscopic pore‑scale domain, where the chemistry of sulfuric acid, water, and cementitious phases is resolved, and a macroscopic pipe‑scale domain, where the transport of sulfate ions from the surrounding wastewater is modeled. The two domains are coupled through micro‑macro transmission conditions that enforce continuity of concentration and flux at the interface, allowing the volume‑averaged acid concentration computed on the pore scale to feed directly into the macroscopic diffusion equation for sulfate.

Mathematically, the microscopic subsystem consists of a set of coupled partial differential equations (PDEs) for the concentrations of H₂SO₄, H⁺, and Ca²⁺, together with reaction terms representing the formation of gypsum (CaSO₄·2H₂O) and the neutralization of acid by cement phases. The diffusion coefficients are functions of local porosity and saturation, reflecting the heterogeneous nature of the concrete matrix. On the macroscopic side, a single diffusion equation for the sulfate concentration is solved, with a source term that depends on the averaged microscopic acid concentration. The coupling is realized by imposing that the microscopic average of H⁺ equals the macroscopic source term, and that the microscopic flux across the representative elementary volume matches the macroscopic flux at the same location.

For numerical implementation, the authors employ an implicit backward‑Euler scheme for time integration and a second‑order finite‑element discretization for space. Separate meshes are generated for the micro and macro domains, which reduces computational cost while preserving accuracy at the interface. The pH field is reconstructed from the microscopic H⁺ concentration using the standard logarithmic relation pH = –log₁₀