## 1 Introduction  
Grouting technology offers several advantages, including safety, cost-effectiveness, simplicity in equipment, a short construction period, and minimal environmental impact. It demonstrates strong adaptability and significant effectiveness in complex environments, playing a crucial role in various fields such as civil engineering, water conservancy, transportation, mining, and environmental remediation [1]. Grouting involves injecting certain grouts that can effectively bond with the injected structure into the voids, pores, and micro-cracks of the structure, transforming it into a new structure with high strength, good impermeability, high stability, and good overall integrity. This process aims to improve the physical and mechanical properties of the injected structure [2]. Research on grouting mainly involves theoretical analysis, numerical simulation, and experimental studies. Grouting experiments can provide the basis for establishing grouting theoretical models and analyzing the diffusion patterns of grout [3]. Therefore, scholars have conducted a series of grouting experiments targeting different receiving structures and grouting conditions to study the transport and diffusion mechanisms of grout [4; 5]. The diffusion pattern of grout in soil and rock masses is influenced by various factors [6; 7], such as grout properties, grouting techniques, and soil and rock parameters. Grouting construction often encounters different soil and rock conditions. Some scholars have employed grouting model experiments to study the diffusion patterns and diffusion radii of grout in soil and rock masses. [PERSON] et al. [8] investigated the influence of fracture characteristics on grout diffusion in rock masses by introducing fracture roughness. They conducted grouting model experiments and numerical simulation studies on single fractures in rock masses. [PERSON] et al. [9] developed a visualization platform for grout diffusion and a three-dimensional grouting experimental system to study the diffusion of grout in fractured rock masses. [PERSON] et al. [10] and [PERSON] et al. [11] conducted studies on the diffusion patterns of grout in segmented grouting sections of clayey fault zones based on three-dimensional grouting diffusion model experiments. Grouting technology is also widely applied in the reinforcement of soil foundations, and thus, significant progress has been made in understanding the diffusion patterns of grout in clayey soils [12]. For weathered granite soil, [PERSON] et al. [13] conducted grouting model experiments to study the diffusion patterns of grout in soil with different particle size distributions. They defined permeation, compaction, fracturing, and mixed diffusion patterns and distinguished different diffusion patterns based on particle size distribution and grouting pressure. The diffusion of grout in sandy soil has also received widespread attention, with many scholars conducting numerous grouting model experiments on sand. For instance, [PERSON] et al. [14], based on the purpose of grouting reinforcement in sandy soil, studied the effects of clay content and initial moisture content on grout diffusion and soil reinforcement effectiveness. [PERSON] et al. [15] investigated the process of grouting in fractured sand and simplified the complex crack shapes in sand to geometric shapes. They established an analytical model for the diffusion of grout in fractured sand during grouting. [PERSON] et al. [16] analyzed the influencing factors of permeation grouting diffusion in sand layers and evaluated various parameters affecting the grouting performance in sand layers. They proposed a new method for predicting the diffusion of permeation grouting in sand layers. [PERSON] et al. [17] evaluated the reinforcement effectiveness of permeation grouting in sandy soil under different water-cement ratios, cement grouts, and curing times. They analyzed the diffusion patterns of permeation grouting in sandy soil. [PERSON] et al.