Effective velocity and effective dispersion coefficient for finite-sized particles flowing in a uniform fracture

In this work we derive expressions for the effective velocity and effective dispersion coefficient for finite-sized spherical particles with neutral buoyancy flowing within a water saturated fracture. We considered the miscible displacement of a fluid initially free of particles by another fluid containing particles of finite size in suspension within a fracture formed by two semi-infinite parallel plates. Particle spreading occurs due to the combined actions of molecular diffusion and the dispersive effect of the Poiseuille velocity profile. Unlike Taylor dispersion, here the finite size of the particles is taken into account. It is shown that because the finite size of a particle excludes it from the slowest moving portion of the velocity profile, the effective particle velocity is increased, while the overall particle dispersion is reduced. A similar derivation applied to particles flowing in uniform tubes yields analogous results. The effective velocity and dispersion coefficient derived in this work for particle transport in fractures with uniform aperture are unique and ideally suited for use in particle tracking models.