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

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Lim, D. H.

Lim, D. H.

複合材料の物質移行モデルでは、拡散係数や空隙率などの水理・物質移行特性に不連続性が存在するため、ランダムウォークシミュレーションを実施する際、取り扱い方法によっては局部的に大きな質量保存誤差を生じることがある。この論文では、特に空隙率に関して確率微分方程式の検証を実施した。一次元拡散に関する理論解をもとに複合材料(異なる拡散係数と異なる空隙率)の接点で非定常確率関数を定義し、複合材料における局部的な質量保存を説明できる新しいスキームを開発した。

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.