# Theoretical Approach to Predicting the Diffusion Radius of Fracture Grouting in Soil-Rock Mixtures
## 2 Calculation Theory for Fracture Diffusion of Bingham Fluid
### Bingham Fluid Fracture Diffusion Equation  
The theoretical derivation in this paper is mainly based on the engineering geological and hydrogeological conditions of the relying project (Wangjiacheng Station to Shengjiabao Station of Chongqing Railway Line 4 Phase II Project), without considering the influence of groundwater. Since the relying project is located in Chongqing, China, which is a mountainous city with large topographic relief and low groundwater level, the water table line is generally located below the bottom of the tunnel of the relying project, and the groundwater pressure is low during the grouting process, the coupling of mechanical and hydraulic effects is not considered. It is assumed that the slurry diffuses radially in the crack surface of the soil-rock mixtures with the exit of the grouting pipe as the center. The soil-rock mixtures are considered to be mixed evenly. The slurry is injected into the soil-rock mixtures from the bottom of the grouting pipe. The theoretical model of slurry diffusion is shown in Figure 2.  
The slurry diffusion form is planar radiative diffusion, which can be described as a \"point source planar radiative diffusion model\". The fluid cross-section is not affected by the void ratio, so the amount of fracture grouting is:  
\[Q=\pi R^{2}\delta=\overline{v}At,\ A=2\pi r\delta \tag{11}\]  
\(h_{0}\) is the groundwater hydraulic head above the grouting pipe, \(h_{1}\) is the grouting pressure, and \(H\) is the sum of the groundwater hydraulic head and grouting pressure, namely, \(H=h_{0}+h_{1}\); \(H\) is equal to \(h_{1}\) when \(h_{1}\) is much larger than \(h_{0}\). The crack width is \(\delta\), the grouting time is \(t\), and the grouting pipe radius is \(r_{0}\). For a Bingham fluid, substituting Equation (9) into Equation (11) gives:  
\[\overline{v}=\frac{Q}{At}=\frac{2}{\eta_{p}\delta}\bigg{(}\frac{dp}{8 dl} \bigg{(}\frac{8r_{m}^{3}-\delta^{3}}{3}\bigg{)}-\tau_{0}\bigg{(}\frac{\delta^ {2}+4r_{m}^{2}}{8}\bigg{)}\bigg{)} \tag{12}\]For Equation (12), we separate the variables and consider the boundary conditions: let \(p_{k}\) be the initiation fracture pressure of soil when the grouting pressure \(p_{r}\) reaches the initiation fracture pressure \(p_{k}\), at which point the diffusion radius \(r=r_{0}\); after \(t\) time, when the diffusion radius reaches the maximum value (\(R\)), the grouting pressure is \(p_{0}\), then we have:  
\[p_{0}-p_{k}=\frac{12\eta_{p}\delta R^{2}}{\left(8r_{m}^{3}-\delta^{3}\right)t} \ln\frac{R}{r_{0}}+\frac{3\tau_{0}\left(\delta^{2}+4r_{m}^{2}\right)}{8r_{m}^{ 3}-\delta^{3}}(R-r_{0}) \tag{13}\]  
Since \(r_{0}\) is much smaller than \(R\), Equation (13) can be simplified, which in turn, gives the relationship between the grouting pressure (\(p_{0}\)) and the diffusion radius (\(R\)) and grouting time (\(t\)) as:  
\[p_{0}=p_{k}+\frac{12\eta_{p}\delta R^{2}}{\left(8r_{m}^{3}-\delta^{3}\right)t} \ln\frac{R}{r_{0}}+\frac{3\tau_{0}\left(\delta^{2}+4r_{m}^{2}\right)}{8r_{m}^ {3}-\delta^{3}}R \tag{14}\]  
There are many unknown quantities in Equation (14), among which the radius of the grouting pipe (\(r_{0}\)) is known, the grouting time (\(t\)) can be controlled according to the actual engineering situation and field experience, while the indoor grouting test in this paper records the grouting time through a timer. In addition, the static shear force (\(\tau_{0}\)) and plastic viscosity (\(\eta_{p}\)). of the slurry are measured using a rotating viscometer, and the rotational speed of the viscometer is changed to measure the shear stress readings at different shear rates, which are brought into the Bingham fluid model for calculation. The value of the parameter (\(r_{m}\)). can be calculated by Equation (4). The value of (\(p_{k}\)). was obtained by referring to the literature [43]. The mechanical mechanism of fracture grouting inevitably requires a thorough analysis of the aggregation and compaction process in the first stage.