Diffusion in Composite Porous Media by Random Walks: Improvement to Local Mass Conservation Error

Lim, D. H.

Discontinuities in hydraulic and transport properties, such as diffusivity and porosity, exist in transport models consisting of different materials. Random-walk simulations without proper treatments for such discontinuities cause local mass conservation errors, i.e. a physically infeasible gathering of mass around the interface between different materials. To resolve the local mass conservation problem, random-walk schemes (e.g. interpolation [1], reflection [2], generalized Stochastic Differential Equation (SDE) [3]) have been developed. However, application of these schemes into composite porous media especially with different medium porosity is in question because (1) the accuracy of the interpolation scheme suffers unless one refines the space discretization step and the time step simultaneously, (2) the medium porosity was not taken into account in the reflection scheme, and (3) the generalized SDE scheme was not fully justified with respect to the medium porosity. In the current study, the validity of the generalized SDE scheme [3] is investigated, especially in terms of the porosity. A random-walk reflection scheme is (1) developed by deriving transition-probabilities at the interface between two different homogeneous porous media, and (2) verified with an analytical solution. Transition-probabilities at the interface between two different homogeneous media with respect to region 1 and region 2 are obtained in the current study, respectively, where region 1 represents the region containing the source. Preliminary numerical results demonstrate that the reflection scheme by using above P1 and P2 shows good agreement with the analytical solution, while the agreement between numerical results generated by the generalized SDE scheme [3] and the analytical solution becomes worse as the porosity ratio between two media increases.