## 1 Introduction  
With the rapid advancement of urban construction, a variety of deep foundation pit projects have proliferated, with an increasing number of such projects extending into sandy pebble strata [1, 2, 3, 4]. Soil nailing and anchor support are commonly employed in foundation pit engineering. However, the sandy pebble stratum is characterized by its susceptibility to hole collapse and challenges in hole formation, often leading to construction issues such as borehole instability and drill rod sticking. Consequently, casing wall protection drilling technology is frequently utilized in these strata, despite its low construction efficiency. In areas with a high content of large-sized pebbles, achieving a satisfactory drilling rate is often unfeasible, and drill bit wear becomes a significant concern. The self-drilling anchor bolt, which integrates drilling, grouting, and anchoring into a single system, offers an effective solution to the challenges of hole formation and collapse in sandy pebble strata. Currently, most indoor experimental studies on anchor bolts involve prefabricated anchor solids, which are subsequently embedded in soil for curing and pull-out testing [5, 6, 7, 8]. However, due to the unique construction technology of self-drilling anchor bolts, the shape of the anchorage body cannot be predetermined, making it impractical to conduct indoor tests using prefabricated rods. As a result, research on self-drilling anchor bolts predominantly relies on field tests [9]. To address this limitation, the three stages of drilling, grouting, and anchoring in the construction process of self-drilling anchor bolts are separated, and indoor tests are conducted in conjunction with soil grouting experiments [10; 11; 12]. For instance, [PERSON] conducted indoor grouting tests using custom-designed grouting equipment to determine the horizontal grouting diffusion patterns in shallow sandy pebble layers. The study examined the effects of grouting duration, grouting pressure, water-cement ratio, and permeability coefficient on grouting diffusion [13]. Similarly, [PERSON] performed indoor grouting tests using a self-designed grouting model system, analyzing the influence of parameters such as grouting volume and grouting pressure on tunnel mechanical deformation [14]. Additionally, [PERSON] utilized a self-developed grouting model test apparatus to simulate the crack grouting process, enhancing the understanding of crack propagation mechanisms during grouting [15]. In recent years, with the continuous advancement of the PFC numerical simulation system based on Mesomechanics theory, researchers have explored methods to analyze slurry diffusion and its coupling with the injected medium from a microscopic perspective. This approach has opened new avenues for studying the grouting process of self-drilling bolts [16; 17; 18]. The sandy pebble stratum, characterized by large pores, can be effectively simulated and analyzed using the \"pipe domain\" model derived from particle flow theory. Initially developed to study sandstone permeability, this model has been progressively refined and widely applied in fields such as hydraulic fracturing and fluid injection in granular media [19; 20]. For example, based on a loess fissure grouting model test, [PERSON] utilized PFC particle flow software to simulate the slurry diffusion process in loess [21]. [PERSON] employed PFC numerical simulations to investigate the reinforcement effects of cement slurry injection in soil, analyzing the influence of cement particle size distribution, soil particle size distribution, and parallel bond strength between particles on the mechanical properties of cement-reinforced soil [22]. [PERSON] established a discrete element fluidization gauge model using PFC, examining the diffusion and rock movement of fluidized gangue under varying grouting speeds, void ratios, and gangue particle sizes [23].