International audienceRandom walk (RW) or Continuous Time Random Walk (CTRW) 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 of deeply simplify the building of explicit complex meshes required by deterministic methods. RW and CTRW provide a good physical understanding of the interactions between the space scales of het-erogeneities and the transport phenomena under consideration. In addition, they can result in efficient up-scaling methods, especially in context of flow and transport in fractured media. In the present study, we review the applications of RW or CTRW for several situations coping with various spatial scales, and different insights into up-scaling applications. RW and CTRW advantages and downsides are also discussed, thus providing a few avenues for further works and applications

[1] Fracture-matrix interactions can significantly affect solute transport in fractured porous media and rocks, even when fractures are major (or sole) conduits of flow. We develop a semi-analytical solution for transport of conservative solutes in a single fracture. Our solution accounts for two-dimensional dispersion in the fracture, two-dimensional diffusion in the ambient matrix, and fully coupled fracture-matrix exchange, without resorting to simplifying assumptions regarding any of these transport mechanisms. It also enables one to deal with arbitrary initial and boundary conditions, as well as with distributed and point sources. We investigate the impact of transverse dispersion in a fracture and longitudinal diffusion in the ambient matrix on the fracture-matrix exchange, both of which are neglected in standard models of transport in fractured media.Citation: Roubinet, D., J.-R. de Dreuzy, and D. M. Tartakovsky (2012), Semi-analytical solutions for solute transport and exchange in fractured porous media, Water Resour. Res., 48, W01542,

Self-potential signals can be generated by different sources and can be decomposed in various contributions. Streaming potential is the contribution due to the water flux in the subsurface and is of particular interest in hydrogeophysics and reservoir characterization. Being able to estimate water fluxes in porous and fractured media using streaming potential data relies on our understanding of the electrokinetic coupling at the mineral-solution interface and our capacity to understand, model, and upscale it. Two main approaches have been proposed to predict streaming potential generation in geological media. One of these approaches is based on determining the excess charge which is effectively dragged in the medium by water flow. In this chapter, we describe how to model the streaming potential by considering this effective excess charge density, how it can be defined, calculated and upscaled. We provide a short overview of the theoretical basis of this approach and we describe different applications to both water saturated and partially saturated soils and fractured media.