Capillary Rise in Micro-Channels Accounting for Boundary Layer Effect and Wall Roughness for Shale Oil

A new capillary imbibition model is presented for shale that uses the momentum theorem of meniscus, for calculating the imbibition rate and height. Results are compared with those from previously published capillary rise models and also obtained in this research. The new capillary rise model considers two main micro-level effects, viz. wall roughness and boundary layer. The imbibed fluid is divided into two parts in the calculation. One is the fluid close to the wall, affected by the wall roughness. The resistance of wall roughness is treated analogously to flow in porous media. The other part is the bulk fluid with a boundary layer. Also considered is the effect of pressure on the boundary layer during the imbibition process. The effects of wall roughness and boundary layer are discussed and compared. An analytical expression for the evolution of the height of capillary rise in a single capillary and in a bundle capillaries is obtained as functions of time, considering both wall roughness and the boundary layer. It was found that the boundary layer enhances the nonlinear characteristic of imbibition, while it has a smaller influence on imbibition height, due to the relatively slow rate of imbibition at nano-scale. In nano-tubes, the wall roughness effect can be equivalent to that of a solid wall, which reduces the effective radius. Although the driving force of capillary rise is enhanced at nano-scale because of the small radius of capillary, the imbibition height does not increase compared to capillaries of larger radii at the same time, since these two microscale effects lower the capillary rise. However, for large times, the equilibrium height of capillary rise at nano-scale is higher than that in the larger radius capillary. The novelty of the new model is used to understand capillary imbibition in porous media with nano-channels, which constitute the main mechanism for shale oil recovery after fracturing. It is believed that the proposed model will lead to improved shale oil production calculations.