Inertial effects in channels with periodically varying aperture and impact on solute dispersion

International audience; Contaminant transport in heterogeneous aquifers occurs mostly in the networks of intersecting channels. The sinusoidalchannel geometry is relevant to transport moderately-disordered porous media, as well as (to some extent)to flow and transport in fractures with varying apertures. Here we investigate the spreading of a finite amount ofsolute entering such a channel of periodically-varying aperture.In channels of uniform apertures (parallel plate), when solute buoyancy is negligible, the advection and diffusionprocesses eventually lead to the well-known asymptotic Taylor-Aris dispersion regime. After this asymptoticregime has been reached, the solute progresses along the fracture at the average fluid velocity, according to aone-dimensional longitudinal advection-diffusion process. The corresponding diffusive term features an apparentdispersion coefficient instead of the molecular diffusion coefficient. In many real applications the relevant channelsdo not have constant aperture. Prior works have shown that deviation from the parallel plate geometry can significantlyalter the behavior, leading to relative increases or even decreases in the apparent dispersion. These studiesassume small Reynolds number and thus that flow is governed by the Stokes equation. While this is very often areasonable assumption, the Reynolds number can sometimes be of order unity or larger such that inertial effectsare no longer negligible. Increased inertial effects lead to the presence of recirculation zones, which represent immobileregions that can have a significant impact on solute transport and in particular on the asymptotic dispersion.We address flow and geometry configurations for which inertial effects affect flow and transport. In these conditions,flow can no longer be predicted by analytical solutions. We compute the stationary velocity fields based onNavier-Stokes equations, using a numerical scheme based on a finite element analysis. The transport problem isthen solved numerically by Lagrangian particle random walk simulations based on the Langevin equation. Dependingon the geometry parameters (ie the aspect ratio of a cell and the relative amplitude of the aperture fluctuations)and on the Reynolds number, the size and the position of the recirculation zones vary. This phenomenon is not onlyresponsible for longer residence times of the solute in these zones but also for higher velocities in the middle ofthe channel. At short times, this leads to unusual spreading patterns and oscillations in the velocity and apparentdispersion evolutions over time. At longer time, solute trapping is clearly visible in the recirculation zones. Wecharacterize the asymptotic dispersion coefficient as a function of the geometry parameters, the Péclet numberand of the Reynolds number. In our parameters range, a higher Reynolds number value systematically leads toan increase in the asymptotic dispersion coefficient. In tracer tests, higher apparent dispersion coefficient are ofteninterpreted from the parallel plate model. This study shows that, at sufficiently high Reynolds number values,the channel uniform aperture inferred in that manner would be larger than the real mean channel aperture, thediscrepancy arising from the effect of aperture variations.