# Theoretical Approach to Predicting the Diffusion Radius of Fracture Grouting in Soil-Rock Mixtures
## 1 Introduction  
The soil-rock mixture is an inhomogeneous loose geotechnical media system composed of a particular engineering scale, high-strength block, fine-grained soil, and void space with a certain stone content [1]. This type of soil-rock mixture has a loose structure, large void ratio, weak cementation ability, poor stability, and strong permeability, which often causes cavity deformation, uneven settlement, collapse, water seepage, and other problems when tunneling through deeply buried soil-rock mixture strata, bringing great difficulties to construction. Therefore, when the tunnel passes through a deeply buried soil-rock mixture, the tunnel surface geotechnical body and the surrounding rock need to be reinforced with grout to form a ring-shaped bearing arch around the perimeter of the tunnel to ensure the stability of the perimeter of the tunnel and the palm face and to improve the permeability of the surrounding rock tunnel. In the design of rock and soil mass grouting parameters, the slurry diffusion radius should be determined first, according to the diffusion radius to determine the grouting hole row distance, hole spacing, single hole grouting amount, grouting pressure, and other parameters, so the determination of the diffusion radius is the key to the success of the grouting project. Grouting diffusion can be divided into fracture diffusion, permeation diffusion, and compression diffusion, among which fracture grouting is widely used in tunnels, mines, road foundations, soft ground, and embankment reinforcement. Conventional soil-rock mixtures have a stone content ranging from 20 to 50%, and their grouting reinforcement is often in the form of fracture grouting [2]. The basic principle of fracture grouting is that when a continuously increasing grouting pressure is applied in a weaklypermeable foundation to cause hydraulic fracture of the soil, the pressure exerted by the slurry on the fracture surface will push the cracks to open rapidly and form grout cracks. The slurry veins extrude soil, and the skeleton effect of the slurry veins will reinforce the soil [3]. Due to the complexity of the process, the current research on the theory of fracture grouting lags far behind engineering practice. Moreover, research results on the diffusion mechanism of fracture grouting have only emerged in the 21 st century. Based on the statistics of domestic and foreign scholars, reference [4] took a particle size of 5 mm as the soil and stone boundary particle size of the soil-rock mixtures. Reference [5] established a flat plate fracture model with a power-law-type slurry and derived a new power-law fluid fracture grouting diffusion equation while giving a parameter determination method. Reference [6] used a model test of loess grouting to divide the process into three stages: primary, expansion and fracturing, and diffusion, and they simulated the process with PFC2D granular flow software. Reference [7] used a combination of FEM and VOF to simulate the fracture grouting process and studied the flow and diffusion law of the slurry. Reference [8] investigated the relationship between low-pressure permeation grouting and the starting pressure of fracture grouting based on engineering examples and proposed corresponding engineering empirical criteria. Reference [9] applied silica-sol material to simulate the grouting process, obtained an expression for the diffusion radius of the slurry within the fissure, and verified the correctness of the expression based on field measurement results. [PERSON] and [PERSON] [10] investigated the main factors affecting the expansion of hydraulic fractures in continuous media soils through numerical simulations and experiments. Reference [11] proposed an analytical model for fracture grouting in sandy soils, simplified the fracture channels into pipe and flat plate modes, and described the fracture grouting process, the diffusion radius of the slurry, and the width of the slurry veins.