Comment on "Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients" by E. M. LaBolle, J. Quastel, G. E. Fogg, and J. Gravner

Lim, D. H.

An extended random-walk reflection scheme taking into account the local mass conservation error around the interface between different materials is developed for the molecular diffusion in composite porous media with different porosity as well as with different diffusivity. The local mass conservation error is a physically infeasible gathering of mass around the interface, and is caused by random-walk simulations without proper treatments for discontinuity in hydraulic and transport properties such as porosity and diffusivity. In the random-walk reflection scheme, in order to conserve the mass at the interface, once particles reach the interface for a unit time step, the further displacements of particles for remaining time step are determined based on transition-probabilities whether these particles can be reflected at the interface or entered into another region. In the current study, the transition-probabilities are derived from the analytical solutions for the diffusion in the composite media. An extended random-walk reflection scheme incorporating the transition-probabilities is developed and compared with the analytical solutions. A selected previous study, i.e. generalized SDE (Stochastic Differential Equations) scheme, is also compared with the analytical solutions especially for the media with different porosity. The current study shows that i) the newly developed extended random-walk reflection scheme has a good agreement with the analytical solutions for the composite media with different porosity as well as with different diffusivity, while the generalized SDE scheme has a good agreement for the composite media with the same porosity or with the porosity ratio less than 1, ii) a random-walk scheme without completely solving the local mass conservation error causes numerical error such as an overestimation of the maximum release rate of mass.