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This paper studies diffusion processes at different scales in two-dimensional (2D) composite media (grain–pore porous micromodels). To model diffusion, this study uses the random walk particle tracking (RWPT) method based on 2D analytical solutions in finite and semi-infinite domains with zero-flux boundary conditions (BC) at grain boundaries. The analytical solutions are developed using a method of images. Then, an RWPT combined to a Stop&Go algorithm is derived from these analytical solutions. The developed RWPT algorithm is then used to model diffusion processes inside the pores at the “microscopic” level (microscale), while the grain elements are assumed inaccessible to diffusion (zero-flux BC condition at the interface grain/pore). The composite medium, where the diffusion occurs, is a numerical micromodel made of periodic motifs of rectangular grains. Particles are initially located at the center of a pore, and diffuses in the periodic motifs micromodel (infinite domain). First, the grains are chosen square with different sizes to vary the porosity. Second, for constant porosities, the grains are elongated to study the effect of the anisotropy ratio on diffusion processes. Effective macroscale properties (porosities, effective diffusion tensors, tortuosities) are then calculated using moments of particles positions. The results obtained fit well with theoretical expectations and are in very good agreement with results found in the literature.

Random walk (RW) methods are recurring Monte Carlo methods used to model convective and diffusive transport in complex heterogeneous media. Many applications can be found, including fluid mechanic, hydrology and chemical reactors modeling. These methods are easy to implement, very versatile and flexible enough to become appealing for many applications because they generally overlook or deeply simplify the building of explicit complex meshes required by deterministic methods. RW provides a good physical understanding of the interactions between the space scales of heterogeneities and the transport phenomena under consideration. In addition, they can result in efficient upscaling methods, especially in the context of flow and transport in fractured media. In the present study, we review the applications of RW to several situations that cope with diverse spatial scales and different insights into upscaling problems. The advantages and downsides of RW are also discussed, thus providing a few avenues for further works and applications. [ABSTRACT FROM AUTHOR]

We present in this Note a stochastic approach to the matrix-fracture exchange in a heterogeneous fractured porous medium. We introduce an intermediate scale, called the unit-scale, between the local-scale (fracture-scale) and the large-scale characteristic of the reservoir mesh (reservoir block). This paper focuses on the problem of upscaling fluid exchange phenomena from the unit scale to the reservoir mesh or block scale. Simplifying the Darcian flow terms enables us to obtain a probabilistic solution of the dual continuum problem, in continuous time, in the case of a purely random exchange coefficient. This is then used to develop several upscaling approaches to the fluid exchange problem, and to analyze the so-called ‘effective’ exchange coefficient. The results are a first contribution to the more general problem of upscaling multidimensional flow-exchange processes in space and time, in randomly heterogeneous dual continua. To cite this article: M. Kfoury et al., C. R. Mecanique 332 (2004). [Copyright &y& Elsevier]