Tracer dispersion in rough channels: A two‐dimensional numerical study

Mass Transfer from Mobile to Immobile Regions in Irregularly Shaped Micro-Channels at Low Reynolds Number

Transient mass transfer in rough-walled micro-channels was investigated experimentally. We conducted experiments using rough-walled channels with various irregularities at small Reynolds number conditions. Mass transfer in the mainstream (mobile region) and dead water region (immobile region) were quantified using an image analysis technique based on absorption photometry. The experimental results showed that the solute dispersion in the mobile region was influenced by the irregular shape of the channel wall complicatedly. In contrast, mass transfer in the immobile region occurred by molecular diffusion independently on the wall roughness in our experimental conditions. The irregular shape of channel wall may enhance the mass transfer in mobile region by distorting the velocity distribution (Togi et al., 2020), while the solute redistribution to immobile region may suppress it in streamwise direction, just on a longer time scale. We developed a mass transfer model analogous to Mobile-Immobile model (MIM model) proposed by previous studies. The concept of the model is the same as the previous study (Zhou et al., 2019) and the coefficients of the model describing mass transfer in each region were quantified from the experimental results as functions of geometric characteristics of the rough-walled channel. In addition, mass transfer coefficient from mobile to immobile regions were derived mathematically based on the experimental results. The MIM model with the coefficients derived in this study well describes solute dispersion in variously shaped irregular channels quantitatively.

Transport in rough self-affine fractures

Transport properties of three-dimensional self-affine rough fractures are studied by means of an effective-medium analysis and numerical simulations using the Lattice-Boltzmann method. The numerical results show that the effective-medium approximation predicts the right scaling behavior of the permeability and of the velocity fluctuations, in terms of the aperture of the fracture, the roughness exponent, and the characteristic length of the fracture surfaces, in the limit of small separation between surfaces. The permeability of the fractures is also investigated as a function of the normal and lateral relative displacements between surfaces, and it is shown that it can be bounded by the permeability of two-dimensional fractures. The development of channel-like structures in the velocity field is also numerically investigated for different relative displacements between surfaces. Finally, the dispersion of tracer particles in the velocity field of the fractures is investigated by analytic and numerical methods. The asymptotic dominant role of the geometric dispersion, due to velocity fluctuations and their spatial correlations, is shown in the limit of very small separation between fracture surfaces.

Experimental study of miscible displacement fronts in rough self-affine fractures

Miscible fluid displacements are studied experimentally in a radial flow between two complementary replica of a self-affine rough granite fracture surface. The displacement front between a dyed fluid and a transparent (but otherwise identical) one is followed optically through one face of the cell. The evolution of its geometry is studied as a function of time, flow-rate, and normal and lateral relative displacements between the two surfaces. For a purely normal displacement, the front is globally smooth, due to the constant local distance between surfaces. For a finite lateral displacement, the front is rough due to spatial variations of this distance; its geometry is fractal and its dimension is directly related to the Hurst exponent H approximately 0.8 of the surface. The fractal regime is observed only above a lower cut-off scale that depends on the normal spacing of the surfaces and an upper one that increases with the injected volume and with the amplitude of the lateral displacement.

Tracer dispersion in two-dimensional rough fractures

Tracer diffusion and hydrodynamic dispersion in two-dimensional fractures with self-affine roughness are studied by analytic and numerical methods. Numerical simulations were performed via the lattice-Boltzmann approach, using a boundary condition for tracer particles that improves the accuracy of the method. The reduction in the diffusive transport, due to the fractal geometry of the fracture surfaces, is analyzed for different fracture apertures. In the limit of small aperture fluctuations we derive the correction to the diffusive coefficient in terms of the tortuosity, which accounts for the irregular geometry of the fractures. Dispersion is studied when the two fracture surfaces are simply displaced normally to the mean fracture plane and when there is a lateral shift as well. Numerical results are analyzed using the Lambda parameter, related to convective transport within the fracture, and simple arguments based on lubrication approximation. At very low Péclet number, in the case where fracture surfaces are laterally shifted, we show using several different methods that convective transport reduces dispersion.