## 1 Introduction  
Collapse and gushing water disasters often occur due to the concealment and unpredictability of the geological conditions during the tunnel construction process. Grouting, an effective control technique, can improve the strength of the surrounding rock and form a water blockage section to effectively improve and solve the abovementioned problems. This technique is quick, short in its time limit, simple in construction, and easy to control. Simultaneously, some considerable progress also has been made in the theoretical research of grouting diffusion. The grouting diffusion process and the final effect are the result of the joint action of the fracture's structural characteristics, grouting technology, and other factors. The study of the grouting diffusion process of a single rough fracture is the basis for investigating the grouting diffusion law in multi-dimensional space and practical engineering. Many scholars have carried out research on the law and factors influencing grouting diffusion in fractures, which is often modeled based on a Newtonian fluid and non-Newtonian fluid, such as a power-law fluid and Bingham fluid [1; 2; 3; 4]. In terms of theoretical research, the theory of grouting diffusion in fractures is mainly divided into two categories: the permeation diffusion theory and filling diffusion theory. The grouting diffusion process in fractures is characterized as a nonlinear Darcy permeation process. The Forchheimer permeation equation for grouting in fractures was developed by establishing the permeability coefficient equation for a slurry of a Bingham fluid in fractures [5]. A grouting diffusion model of a single parallel plate fracture can be obtained based on the solution of the continuity equation and Navier-Stokes (N-S) equation for the grout flow process. A grouting diffusion model based on Newtonian fluids is derived to reveal the relationship between the grouting pressure, flow rate, and diffusion radius during the grouting diffusion process [6]. A maximum radius exists during the grouting diffusion process of a Bingham-flow-based slurry when given a certain grouting pressure [7]. Based on a constant flow rate, a Bingham-flow-based model of grouting diffusion is derived, which reveals the spatiotemporal distribution of pressure during the grouting diffusion process [8]. In addition, modified grouting diffusion analytical models considering the temporal variability of the slurry viscosity or under the influence of temperature are obtained [9; 10]. In terms of experimental studies, the grouting diffusion process in a circular tube is also studied [11]. The diffusion process of a Bingham-fluid-type cement slurry in the fractures of artificial slabs is being observed and studied [12; 13; 14]. In addition, the grouting diffusion process influenced by the temperature and dynamic water environment has also been studied, revealing the variation of physical quantities, such as the velocity and viscosity, in the diffusion process of a slurry under different influencing factors [10; 15]. In respect to numerical simulation studies, the simulation of the grouting diffusion process is studied by a finite element method (FEM), finite volume method (FVM), and particle flow code (PFC). The FEM calculation software COMSOL Multiphysics (COMSOL Inc., Stockholm, Sweden) is used to establish a numerical simulation model to study the diffusion of different types of slurries under hydrostatic and dynamic water environments in plate fractures. The circular and U-shaped grouting diffusion patterns are revealed. Additionally, the pressure and velocity fields are discussed [16; 17]. Based on the FVM, the triangular collapse process of the concrete is simulated by using the Herschel-Bulkley rheological model, revealing the rheological behaviors of the Bingham fluid [18]. Three different approaches of gap grouting modeling are investigated with the FLAC3D software, paying special attention to the influence of the grout material hardening process [19].