Tracer dispersion in two-dimensional rough fractures

Tracer diffusion and hydrodynamic dispersion in two-dimensional fractures with self-affine roughness is studied by analytic and numerical methods. Numerical simulations were performed via the lattice-Boltzmann approach, using a new 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 simple 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 P\'eclet number, in the case where fracture surfaces are laterally shifted, we show using several different methods that convective transport reduces dispersion.