## 1 Introduction  
Hydraulic fracturing is an effective reservoir stimulation technique applied in the development of unconventional oil and gas reservoirs [1, 2]. The proppant is injected into the wellbore along with the fracturing fluid to prevent fracture closure prior to switching to the production stage [3]. The characteristics of proppant migration and distribution impose an important effect on the initiation and evolution of hydraulic fractures [4]. The propagation of fractures during the hydraulic treatment of shale reservoirs has been extensively investigated [5, 6]. In contrast, little concern has been given to the optimization of the injection system. Regarding such a system, the mixing device and delivery pump are two main components. The former is responsible for mixing the proppant (such as sand particles) and the fracturing liquid. Then, the mixture is transported into the delivery pump, which pressurizes the mixture to attain the required injection pressure. The configuration of such a system is subject to concerns of large size, high cost of maintenance, and low mobility. The solid-liquid two-phase flow pump has been applied in various industrial processes such as mining and dredging [7]. Generally, liquid and solid particles are pre-mixed, and then the mixture enters the pump via a single inlet [8]. From the perspective of the parametric design, the effects of geometric parameters, such as the blade wrap angle and blade outlet angle, have been comprehensively investigated to improve pump efficiency [9]. The pump structure has been optimized to minimize the radial and axial hydraulic forces acting on the pump components, which affect the operating stability of the whole pump unit [10]. Moreover, the blockage of the flow passage and wear of the hydraulic components are two major issues encountered in such a branch of pumps [11]. Although new schemes of the pump structure have been frequently devised, generalizable and validated design principles are scarce. In most cases, the designed pump adapts exclusively to some specific application. The operating parameters influence the operating efficiency and stability of the solid-liquid two-phase flow pump. For instance, the effect of the particle size has raised much concern [12; 13]. Generally, large particles are relatively difficult to deliver; however, this depends as well on the dimensions of the flow passage. In some industrial processes, particle sizes ranging from 0.1 mm to 50.0 mm have been adopted [14]. The reported solid volume fraction of the mixture is usually lower than 0.30. The combination of large particles and high solid volume fraction will impose a heavy load on the blades; moreover, the risk of blockage and wear is conceivably high. Certainly, other factors, such as the particle shape, are argued to be non-negligible. In this circumstance, a comprehensive consideration of the influential factors helps to minimize the discrepancy between research results and real situations. Essentially, the interaction between the transported medium and the hydraulic components determines the working ability of the pump. The available flow measurement techniques, intrusive or non-intrusive, have shortages when treating solid-liquid two-phase flows. Actually, even the flow rate of the solid-liquid mixture is difficult to measure, and some advanced techniques are exposed to the queries of cost and convenience [15]. The particle image velocimetry (PIV) can be used to measure the velocity distribution in the flow field. However, with the participation of solid particles, the incident light will be shielded by the particles, yielding spurious elements in the captured images [16]. Furthermore, the configuration of the optical system is often infeasible in the presence of complex pump structures. In this situation, the computational fluid dynamics (CFD) technique serves as a more desirable tool for probing into the solid-liquid two-phase flow [17; 18].