## 1 Introduction  
Hydraulic fracturing with variable fluid viscosity and proppant density is critical for the development of tight oil and gas reservoirs. This type of fracturing adjusts the viscosity of the fracturing fluid and the proppant density to improve the proppant placement in fractures, which determines the final effect of fracturing measures [1; 2; 3; 4]. The proppant transport law in fractures is essential in affecting the morphology of proppant placement in fractures [5; 6; 7]. The flow state of the fluid in its fractures, the properties of the fracturing fluid and proppant, and the interaction between the proppant particles all seriously affect proppant transport and sedimentation [8; 9; 10]. Researchers at home and abroad have conducted a significant amount of research on the proppant transport law via experimental tests and numerical simulation methods [11; 12]. [PERSON] et al. [13] conducted the first vertical fracture experiment, injecting water and sand at a constant rate. [PERSON] et al. [14] established a small-slit experimental device to study the effect of proppant concentration on proppant convection. [PERSON] et al. [15] established a small fracture device with branch fractures and studied the proppant transport behavior in complex fractures. The results show that if the flow velocity is low, then the proppant only accumulates in the main fracture. When the flow velocity is greater than a specified threshold, the proppant can enter the secondary fracture from the main fracture. [PERSON] [16] carried out large-scale experimental tests in the laboratory to evaluate proppant transport in complex fracture geometries with primary and secondary fracture slots. [PERSON] et al. [17] found through experiments that as the fracture plane angle becomes closer to the horizontal plane, the proppant withstands greater friction along the wall, and the proppant placement area is larger. Several numerical simulation studies focused on the proppant transport mechanism have been conducted. [PERSON] et al. [18] took the lead in developing correlations using the \"equilibrium velocity\" and \"bank-buildup constant\", which can predict the proppant settling in both Newtonian and non-Newtonian fluids. [PERSON] et al. [19] established a proppant transport model, coupled it with a three-dimensional fracture propagation model, and solved the model using the finite element method. Their results show that the proppant settlement has a significant effect on the length of a fracture, considering turbulent flow. [PERSON] et al. [20] used the mixture model to simulate the movement of proppants carried by high viscosity fracturing fluids in simple rectangular fractures. [PERSON] et al. [21] established a Euler-Euler dual-fluid model of fluid-structure coupling that simulated the behavior of proppant transported by water in a single slot, and studied the effects of parameters such as the perforation position and injection speed on the shape of the proppant dune. [PERSON] et al. [22] adopted a representative particle model (RPM) to scale up proppant particles to study the transport mechanism of proppants in fractures. [PERSON] et al. [23] simplified the shale complex fracture network into a complex fracture model in which the primary and secondary fractures are orthogonal to each other. [PERSON] et al. [24] used the CFD-DEM method to study proppant transport in complex fractures, indicating that the proppant concentration and the ratio between the secondary fracture aperture and proppant diameter are the key factors affecting the proppant transport during fracture interactions. However, very few studies have focused on the effects of fracturing fluid viscosity and proppant density on the proppant transport law. Based on computational fluid dynamics and the discrete element method, the effects of proppant and proppant, proppant and fluid, and proppant and fracture wall surfaces are considered to establish a fluid-particle coupled proppant transport model.