*Article* **Numerical Investigation on Proppant–Water Mixture Transport in Slot under High Reynolds Number Conditions**

#### **Tao Zhang 1,\*, Ruoyu Yang 1, Jianchun Guo 1,\* and Jie Zeng 2**


Received: 29 September 2020; Accepted: 27 October 2020; Published: 29 October 2020

**Abstract:** Water hydraulic fracturing involves pumping low viscosity fluid and proppant mixture into the artificial fracture under a high pumping rate. In that high Reynolds number conditions (HRNCs, *Re* > 2000), the turbulence effect is one of the key factors affecting proppant transportation and placement. In this paper, a Eulerian multiphase model was used to simulate the proppant particle transport in a parallel slot under HRNCs. Turbulence effects in high pumping rates and frictional stress among the proppant particles were taken into consideration, and the Johnson-Jackson wall boundary conditions were used to describe the particle-wall interaction. The numerical simulation result was validated with laboratory-scale slot experiment results. The simulation results demonstrate that the pattern of the proppant bank is significantly affected by the vortex near the wellbore, and the whole proppant transport process can be divided into four stages under HRNCs. Furthermore, the proppant placement structure and the equilibrium height of proppant dune under HRNCs are comprehensively discussed by a parametrical study, including injection position, velocity, proppant density, concentration, and diameter. As the injection position changes from the lower one to the top one, the unpropped area near the entrance decrease by 7.1 times, and the equilibrium height for the primary dune increase by 5.3%. As the velocity of the slurry jet increases from 2 m/s to 5 m/s (*Re* = 2000–5000), the vortex becomes stronger, so the non-propped area near the inlet increase by 5.3 times, and the equilibrium height decrease by 5.2%. The change of proppant properties does not significantly change the vortex; however, the equilibrium height is affected by the high-speed flush. Thus, the conventional equilibrium height prediction correlation is not suitable for the HRNCs. Therefore, a modified bi-power law prediction correlation was proposed based on the simulation data, which can be used to accurately predict the equilibrium height of the proppant bank under HRNCs (mean deviation = 3.8%).

**Keywords:** water fracturing; turbulence effect; Eulerian multiphase modeling; proppant transport mechanism; equilibrium height prediction model
