The main objective of this work is to understand the major peculiarities of the process of clean fracturing fluid being displaced by a flow of particulate suspension (slurry) in a fracture channel. A number of 2D and 3D numerical simulations (based on an Euler-Euler type model) of slurry flows in channels of complex geometry are carried out. The performance of the model is validated against experimental data by Gillies et al. Euler-Euler simulations of a laminar slurry flow in a horizontal pipeline showed good agreement between computed and measured particle concentration distributions across a pipe. Numerical studies of slurry flows in 2D and 3D channels show the formation of an M-shape profile of the solid phase across a channel at the initial fracture section. Our computations demonstrate the rapid growth of a dispersion zone separating a slurry and a clean fluid. It is shown that the dispersion zone is shorter in a tortuous channel than in a straight one. The computations illustrate a reduction in the dispersion zone length with an increase in the fracturing fluid viscosity. The results obtained show the importance of taking into account the previously ignored process of slurry/clean fluid boundary dispersion in prospective hydraulic fracturing simulators.