## 2 The Permeation Grouting Mechanism of Bingham Fluids under Various Particle Size Distributions
### Analysis of Spherical Permeation by Bingham Fluid  
From percolation mechanics, the laminar flow rate of Bingham fluid in a single pore channel of radius r in a porous medium is as follows [34]:  
\[ \begin{array}{l}q=\frac{\pi r^{4}}{8\mu}\Big{(}-\frac{dp}{dL}\Big{)}\left(1 -\frac{4}{3}\Big{(}\frac{2r_{0}/r}{-dp/dL}\Big{)}+\frac{1}{3}\Big{(}\frac{2r_ {0}/r}{-dp/dL}\Big{)}^{4}\right)\\ =\frac{\pi r^{4}}{8\mu}\Big{(}-\frac{dp}{dL}\Big{)}\bigg{[}\Big{(}1-\frac{2r_ {0}/r}{-dp/dL}\Big{]}-\frac{1}{3}\frac{2r_{0}/r}{-dp/dL}\bigg{(}1-\frac{(2r_{0 }/r)^{3}}{-|-dp/dL|^{3}}\Big{)}\bigg{]}\end{array}  \tag{6}\]  
where \(\mu\) and \(r_{0}\) are the plastic viscosity and yield stress of Bingham fluid, respectively; \(q\) is the flow rate of the pore channel with radius \(r\); and \(-dp/dL\) is the pressure gradient of the fluid in the direction of the diffusion path. The starting pressure gradient for [PERSON] fluid in the above equation is \(\lambda=2\tau_{0}/r\) (the value of the pressure gradient in the circular pipe at zero flow). The equation is simplified by omitting the second term in square brackets in Equation (6) following the approximation of the derivation of the generalized [PERSON]'s law for [PERSON]'s fluid, and Equation (6) simplifies to the following:  
Figure 2: \(A_{0}\) schematic diagram. \[q=\frac{\pi r^{4}}{8\mu}\left(-\frac{dp}{dL}\right)\left(1-\frac{2\tau_{0}/r}{|- dp/dL|}\right) \tag{7}\]  
In the capillary tube as the slurry diffusion distance increases, the grouting pressure is decreasing, so there is \(dp/dL<0\), then Equation (7) is simplified to the following:  
\[q=\frac{\pi r^{4}}{8\mu}\left(-\frac{dp}{dL}\right)\left(1+\frac{2\tau_{0}/r}{ dp/dL}\right) \tag{8}\]  
Combining Equation (2) with Equation (8) yields the total flow rate \(Q(r)\) of [PERSON] fluid slurry through all pore channels in a representative unit \(A_{0}\) as the following:  
\[Q(r)=-\left[\frac{D_{f}\pi r_{\max}^{4}}{8\mu\left(4-D_{f}\right)}\left[1- \left(\frac{r_{\min}}{r_{\max}}\right)^{4-D_{f}}\right]\frac{dp}{dL}+\frac{ \tau_{0}D_{f}\pi r_{\max}^{3}}{4\mu\left(3-D_{f}\right)}\left[1-\left(\frac{r _{\min}}{r_{\max}}\right)^{3-D_{f}}\right]\right] \tag{9}\]  
Usually, for a fractal porous medium, this satisfies \(r_{\min}/r_{\max}<10^{-2}\), and for a two-dimensional space \(0<D_{f}<2\), yields \(2<4-D_{f}<4\), \(1<3-D_{f}<3\), so that there are \(\left(r_{\min}/r_{\max}\right)^{4-D_{f}}\ll 1\), \(\left(r_{\min}/r_{\max}\right)^{3-D_{f}}\ll 1\). Thus, expression (9) reduces to the following:  
\[Q(r)=-\left[\frac{D_{f}\pi r_{\max}^{4}}{8\mu\left(4-D_{f}\right)}\frac{dp}{dL }+\frac{\tau_{0}D_{f}\pi r_{\max}^{3}}{4\mu\left(3-D_{f}\right)}\right] \tag{10}\]  
From Equations (5) and (10), the average flow rate \(V\) of [PERSON] fluid in representative cell \(A_{0}\) is as follows:  
\[V=-\left[\frac{\varphi_{A}r_{\max}^{2}\left(2-D_{f}\right)}{8\mu\left(1- \varphi\right)\left(4-D_{f}\right)}\frac{dp}{dL}+\frac{\tau_{0}\varphi r_{ \max}\left(2-D_{f}\right)}{4\mu\left(3-D_{f}\right)\left(3-D_{f}\right)}\right] \tag{11}\]  
During the grouting process, the total amount of grout is satisfied:  
\[Q=VAt \tag{12}\]  
where \(A\) is the area of the sphere through which the slurry passes at time \(t\), \(A=4\pi L^{2}\); \(t\) is the time of grouting, and \(V\) is the average flow rate of the Bingham fluid slurry.