**1. Introduction**

Hydraulic fracturing technology has been widely used in recent years to economically exploit unconventional resources, especially for shale oil and gas [1]. In fracturing process, high pressurized fluid is injected to initial the fracture and propagate it. Once the fracture is created, the high strength particle (proppant) was carried by fracturing fluid and injected in the fracture, to keep the fracture open and ensure an effective flow path for hydrocarbon flow [2]. The artificial fracture geometry is influenced by geological factors and fracturing treatment. Figure 1 shows the comparison of the fracture geometry between water fracturing for shale and conventional fracturing.

**Figure 1.** Schematic diagram of different artificial fracture geometry for conventional fracturing and water fracturing.

Different from conventional fracturing, the goal for shale fracturing is breaking the formation and generating larger stimulated reservoir volume (SRV) with more complex fracture, due to the extremely low permeability and porosity of shale rock [3]. Therefore, the low viscosity water is used as the primary fracturing fluid system used for the stimulation of shale formation [4,5]. However, the low viscosity of the fluid affects the proppant transport capability. To address this problem, engineers often pump water–proppant mixture into the fracture at a very high pumping rate. Due to the low-viscosity fluid and high pumping rate, the turbulence effects become an important factor affecting the transport behaviors of the proppant particles in fracture [6,7]. In general, Reynolds number (*Re*) is a dimensionless number that can be used to characterize fluid flow type. For the general channel flow field, *Re* = 2000 is a critical value for laminar flow and turbulent flow, and we defined high Reynolds number condition (HRNCs) when *Re* is larger than 2000. The behavior and mechanism of the proppant transport process and placement under that HRNCs in the shale fracturing process are different from conventional fracturing. Because of its complexity and significance, continuous research on this problem is conducted by many scholars.

Numerous laboratory experimental research contributes to the understanding of the transport and settling behaviors in low viscosity fracturing fluids. The first experiment about the sand-water mixture transport in the slot is carried by Kern et al. [8], in which they found the sand quickly settled to the slot bottom and formed a proppant dune near the inlet side because of the poor sand-carrying ability of water. Besides, once the equilibrium height reached, sand injected later moved and settled to the rear of the proppant dune. STIM-LAB has been studied on the proppant transport process for more than 20 years, and the effects of the proppant density, diameter, and volume concentration on the equilibrium height of the proppant bed are comprehensively investigated [9,10]. Liu et al. [11] conducted similar slot experiments to STIM-LAB, and the results showed that the initial position and the equilibrium height of the proppant bed changed with the perforation position and the slurry flow rates. Palisch et al. [12] concluded that as the equilibrium height reached the critical value, the mechanisms of the proppant particles transporting on the top of the proppant dune were dominated by fluidization and sedimentation. The turbulent flow suspended the proppant particles off the proppant dune, and then proppant settled again after being transported to some distances during fluidization.

Based on the experimental results above, some correlations for the proppant particle transport and settling have been built up. Liu et al. [11] developed a fitting equation of the height of the equilibrium gap and the injection flow rate. Patankar et al. [9] and Wang et al. [13] established the empirical correlations in the form of power and bi-power law, which were widely used in the industry. Since there is no accurate method for the proppant particles settling in slick-water fracturing treatments, some revised Stokes' Laws which considered the e ffects of the proppant volume concentration and the fracture wall are often used to predict the proppant particles settling [14,15].

Due to the ability to solve the flow patterns of liquid-solid two phases and their interaction simultaneously, Computational Fluid Dynamics (CFD) technology provides an alternative method to study the proppant transport and to settle in hydraulic fracture accurately. Patankar et al. [16,17] used a DNS model to study the lift-o ff of particles in plane Poiseuille flows. The results showed the interactions between the particles in the sedimentation process, ignoring the fracture propagation and the e ffect of fracturing fluid loss. Tsai et al. [18] employed a large-eddy model to simulate the flow field and tracked the proppant particles in Lagrangian coordinates, where the Wen-Yu drag model was used to couple the interaction of the fracturing fluid and proppant. The results showed that the pumping rate and the proppant parameters have an essential influence on the proppant settling and placement. A Computational Fluid Dynamics coupled with Discrete Element Method (CFD-DEM) was employed by Zeng et al. [6] to study the proppant transport process in a small-scale fracture. Although a representative particle model was used to reduce computational e fforts, the time consumed is still considerable [19].

Compared with the CFD method mentioned above, the Eulerian multiphase flow model has the advantages of high e fficiency and low computational cost. This method is the priority choice for the fluid–solid multiphase flow simulations of the engineering problems [20]. In the last three decades, the Eulerian multiphase flow model has gained considerable progress [21]. As described by Agrawal et al. [22], the solid phase governing equations was extended from the mono-disperse system to the poly-disperse system. Srivastava et al. [23] developed a solid phase frictional stress model in the dense solid-phase flow by considering the interactions between the particles as the solid volume concentration is greater than the critical concentration. Benyahia et al. [24] evaluated Jenkins-Louge and Johnson-Jackson's solid wall boundary conditions and pointed out their application scope. The simulation results of the fluidized bed and the slurry flow in the pipeline using the Eulerian multiphase flow model were highly consistent with experimental results [25,26]. Recently, this method is also used to investigate the transport and settling behavior of proppant in single fractures [27] and the cross fractures [28].

Although Eulerian multiphase flow has been widely used in proppant transportation research, the systematic study of proppant transport mechanisms and behavior under HRNCs has not been reported before. In this paper, a Eulerian two-fluid model considering turbulence e ffect and particle friction stress were used to study the proppant transport and settling behaviors under HRNCs. The mechanism during the transport process under HRNCs and the impact of such factors as the inlet velocity, the inlet position, the proppant parameters, and the volume concentration on proppant transport and settlement are comprehensively discussed. The equilibrium height of the proppant dune was also studied, and a modified equilibrium height prediction model was proposed.

#### **2. Eulerian Multiple Flow Model**
