Résumé
Upscaling hydrodynamic transport processes in porous media from laboratory scale characterization to modeling large-scale systems is challenging. The practice relies on the existence of a representative elementary volume (REV) which is often assessed from directly measurable medium properties such as porosity or permeability. Average transport properties such as dispersion are supposed to be constant on the REV scale. Transport at the REV scale then is classically described using Fickian transport based on hydrodynamic dispersion. However, it is well known that transport often displays marked non-Fickian behaviors at laboratory and field scales. The incomplete sampling of the flow heterogeneity by the solute at pre-asymptotic times is at the origin of non-Fickianity of the transport processes. The failure of Fickian transport model at preasymptotic times is sometimes equated to the failure of the REV concept for transport. In this work, we investigate transport upscaling in the continuous time random walk (CTRW) framework. While the CTRW framework can describe preasymptotic non-Fickian transport, it relies on statistical stationarity of an underlying transition time distribution or, for purely advective transport, stationarity of particle velocity statistics. This stationarity has a spatial notion because particle velocities evolve with travel distance. Thus, we propose a novel REV concept based on the distance after which the velocity statistics become stationary. This stationary distribution characterizes the spatial variability of particle velocities within the REV domain and is the center piece to parameterize particle based upscaled models such as CTRWs. This transport REV depends on the kinematics of particle transport and the Eulerian flow statistics. In addition, we investigate the impact of diffusion, and thus of the Peclet number, on the stationarity of the Lagrangian velocity statistics and on the correlation length.