Shear Dispersion in a Rough-Walled Fracture

An expression is analytically presented for the shear dispersion, or Taylor (1953) and Aris (1956) dispersion, of a solute transporting in a coupled system, which consists of a matrix and a rough-walled fracture. To derive a shear-dispersion coefficient in a fracture with rough and porous walls, the continuities of solute concentrations and their fluxes are imposed at the fracture walls. The dispersion coefficient for the coupled system is obtained as a function of the Peclet number and relative roughness, where the latter parameter is defined as the ratio of the maximum height of the roughness to the minimum half-aperture of the fracture. Several models for fracture-roughness geometry, including periodically and randomly shaped roughness models, are applied to study the effect of fracture-aperture variation on dispersion. The dispersion coefficient for all rough-walled fractures identifies three different regions in terms of the degree of relative roughness. The results show that for small values of the relative roughness (0 < epsilon <= 0.1), the dispersion coefficient is at maximum for bell-shaped geometry and at minimum for triangular-shaped and randomly shaped geometries. When the relative roughness is within 0.1 < epsilon < 10, the dispersion is observed to be at maximum for rectangular-wailed and at minimum liar triangular-walled fractures. The results also reveal that for high values of the relative roughness (epsilon >= 10), the dispersion is higher for bell-shaped roughness, whereas the triangular-walled fracture results in the lowest dispersion. It is found that for all roughness geometries an increase in either the Peclet number or relative roughness leads to an increase in the dispersion.