# 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 Flow Equation  
The shear rate of the Bingham fluid is not proportional to the shear stress, and the slurry starts to flow only when the shear stress exceeds the yield value \((\tau_{o})\). The rheological equation of the slurry is shown in Equation (1). \[\tau=\tau_{o}+\eta_{p}\frac{dv}{dr} \tag{1}\]  
The crack flow equation will be derived from the Bingham fluid rheology equation. Figure 1 shows the flow of the Bingham fluid in a crack. Take a quadrangular prism with width \(b=1\) in the crack, and crack width \(\delta\). Take a fluid column with the central axis of the tube as the axis of symmetry in the quadrangular prism; the length of the fluid column is \(dl\), the height is \(2r\), the width is \(b=1\), and the side of the fluid column is subject to uniformly distributed shear stress \((\tau)\). According to the force balance relationship of the fluid column (without considering gravity), we can obtain the following:  
\[\tau=-\frac{rdp}{dl} \tag{2}\]  
That is, the shear stress \((\tau)\) on the fluid column surface is proportional to the product of the fluid column height \((r)\) and the pressure gradient \((\frac{dp}{dl})\), but the sign is the opposite. Substituting Equation (2) into Equation (1), we obtain the following:  
\[\frac{dv}{dr}=\frac{-rdp}{\eta_{p}dl}-\frac{\tau_{0}}{\eta_{p}}=\frac{1}{\eta_ {p}}(\tau-\tau_{o}) \tag{3}\]  
According to Equation (3), it is found that the magnitude of the shear stress varies linearly with height; the further away from the center, the greater the shear stress, while at the centerline, the shear stress is minimal. When the shear stress is \(\tau=-\frac{rdp}{dl}\leq\tau_{o}\), the velocity gradient is \(0\); that is, the velocity does not change with \((r)\), indicating that the shear stress is \(0\) within this range. There is a height \((r_{m})\) in the quadrangular prism such that the fluid in the range \((0\leq r\leq r_{m})\) is relatively stationary, and its velocity is denoted as the velocity \((v_{m})\) at the height \((r_{m})\). When the fluid is in the range \((r_{m}\leq r\leq r^{\prime})\), the fluid is moving with respect to the adjacent fluid. Therefore, the fluid as a whole exhibits piston motion, which is divided into the shear zone \((r_{m}\leq r\leq r^{\prime})\) and piston zone \((0\leq r\leq r_{m})\). When \(r=r_{m}\), \(\tau=\tau_{0}\), according to Equation (3):  
\[r_{m}=-\frac{\tau_{0}dl}{dp} \tag{4}\]  
When boundary condition \(r=\frac{\delta}{2}\) is taken into account, velocity \(v_{\frac{\delta}{2}}=0\). By separating the variables in Equation (3) and integrating both sides' yields, we can obtain the following:  
\[v=\frac{dp}{8\eta_{p}dl}\Big{(}\delta^{2}-4r^{2}\Big{)}+\frac{\tau_{0}(\delta -2r)}{2\eta_{p}},\ \ \ r_{m}\leq r\leq r^{\prime} \tag{5}\]  
When \((0\leq r\leq r_{m})\), it follows that:  
\[v=v_{m}=\frac{dp}{8\eta_{p}dl}\Big{(}\delta^{2}-4r_{m}^{2}\Big{)}+\frac{\tau_ {0}(\delta-2r_{m})}{2\eta_{p}} \tag{6}\]  
Figure 1: Bingham fluid flow in a crack. The flow rate \((q)\) of the slurry in the fluid column can be found by integrating the following:  
\[q=2\int_{0}^{\frac{\delta}{2}}v_{r}bdr+2 bv_{m}r_{m} \tag{7}\]  
Substituting Equations (5) and (6) into Equation (7) gives:  
\[q=\frac{2b}{\eta_{p}}\bigg{(}\frac{dp}{8 dl}\bigg{(}\frac{8r_{m}^{3}-\delta^{3}} {3}\bigg{)}-\tau_{0}\bigg{(}\frac{\delta^{2}+4r_{m}^{2}}{8}\bigg{)}\bigg{)} \tag{8}\]  
The average flow velocity in the longitudinal section is \(\overline{v}\)  
\[\overline{v}=\frac{q}{\delta b}=\frac{2}{\eta_{p}\delta}\bigg{(}\frac{dp}{8 dl} \bigg{(}\frac{8r_{m}^{3}-\delta^{3}}{3}\bigg{)}-\tau_{0}\bigg{(}\frac{\delta^ {2}+4r_{m}^{2}}{8}\bigg{)}\bigg{)} \tag{9}\]  
Equations (8) and (9) are the crack flow equations of a Bingham fluid. When the shear stress is insufficient to produce a fluid flow rate, the quadrangular prism's flow rate is 0.