Analytical Solutions for Characterizing Fluid Flow through Sand-Pack in Pipes

When fluid flows through a pipe that is packed with sand particles, the fluid will bear the resistance from the sand-pack, as well as the viscous shear from the pipe wall. If the viscous shear from the pipe wall can be neglected, the fluid flow will obey Darcy's law, and one can think the equivalent permeability of the packed-pipe equals the permeability of the sand-pack. However, if the viscous shear from the pipe wall cannot be neglected, the fluid flow will obey the Brinkman equation, and the permeability of the packed-pipe will be less than that of the sand-pack due to the additional viscous drag. In this work, on the basis of the Brinkman equation, we derived a series of analytical solutions for characterizing the fluid flow in packed-pipes. These solutions can be used to depict the velocity profiles, estimate the flux rate, and calculate the equivalent permeability of a packed-pipe. On the basis of these analytical solutions, we found that Poiseuille's law is a special form of the derived equivalent permeability solution. We further divided the fluid flow in a packed-pipe into three regimes, including N-S flow, Brinkman flow, and Darcy flow. During the regime of Brinkman flow, the dimensionless flow velocity at the pipe center is 1, and the dimensionless flow velocity is gradually decreased to 0 at the pipe wall. We also investigated the effects of sorting, sand particle size, and sand-pack porosity on the packed-pipe permeability. The calculated results show that a more uniform size of the sand particles or a smaller mean particle diameter can lead to lower packed-pipe permeability. Compared to the sorting and mean particle diameter, the sand-pack porosity exerts a more significant effect on the packed-pipe permeability.