## 3 Column Permeation Diffusion Model of Bingham Fluid  
As mentioned in [5; 29], the continuity equation of steady column penetration can be written as:  
\[\frac{1}{r}\frac{\partial}{\partial r}(r\cdot V)=0 \tag{7}\]  
Substituting Equation (6) into Equation (7), and in \(\frac{\partial p}{\partial r}\) rather than \(\
abla p\), the differential equation of column penetration of Bingham fluid can be expressed as:  
\[\frac{\partial^{2}p}{\partial r^{2}}+\frac{1}{r}\frac{\partial p}{\partial r} -\frac{\lambda}{r}=0,\left|\frac{\partial p}{\partial r}\right|>\lambda \tag{8}\]  
It is assumed that the injected grout is Bingham fluid, while the groundwater is Newtonian fluid. The gravity and compressibility of the fluid are ignored. The penetration differential equation and boundary conditions of this model are given by:  
\[ \begin{array}{ccc}\frac{\partial^{2}p_{1}}{\partial r^{2}}+\frac{1}{r}\frac {\partial p_{1}}{\partial r}-\frac{\lambda}{r}=0,&r_{0}<r\leq r(t)\\ \frac{1}{r}\frac{\partial}{\partial r}(\frac{r\partial^{2}p}{\partial r})=0,&r( t)\leq r<r_{w}\\ p_{1}=p_{0},&r=r_{0}\\ p_{2}=p_{w},&r=r_{w}\end{array}  \tag{9}\]  
where \(p_{1}\) is the pressure of Bingham fluid in the penetration zone, \(p_{2}\) is the pressure in the area of groundwater, \(r_{0}\) is the radius of capillary channel, \(p_{0}\) is the grouting pressure, \(r_{w}\) is the grouting disturbance range and \(p_{w}\) is the groundwater pressure. The pressure and the velocity (flow rate) are continuous at the grout front \(r=r(t)\):  
\[ \begin{array}{ccc}2\pi rh\left[-\frac{K_{1}}{\mu_{1}}\frac{\partial p_{1}}{ \partial r}\left(1-\lambda\left(\frac{\partial p_{1}}{\partial r}\right)^{-1 }\right)\right]=-2\pi rh\frac{K_{2}}{\mu_{2}}\frac{\partial p_{2}}{\partial r} \\ p_{1}=p_{2}\end{array}  \tag{10}\]  
where \(\mu_{1}\) and \(\mu_{2}\) are, respectively, the viscosity coefficient of grout and water, \(K_{1}\) and \(K_{2}\) are, respectively, the absolute penetration of grout penetration zone and groundwater penetration zone and \(h\) is the grouting height. Assuming that \(\partial p_{1}/\partial r=Y_{1}\), \(\partial p_{2}/\partial r=Y_{2}\), Equation (9) can be reduced to the following equations:  
\[ \begin{array}{ccc}\frac{\partial Y_{1}}{\partial r}+\frac{1}{r}Y_{1}-\frac {\lambda}{r}=0\\ \frac{\partial Y_{2}}{\partial r}+\frac{1}{r}Y_{2}=0\end{array}  \tag{11}\]  
The solution can be resolved as \(Y_{1}=\lambda+C_{1}/r,Y_{2}=C_{2}/r\), and \(C_{1}\) and \(C_{2}\) are the coefficients of the general solution. The expression of \(p_{1}\) and \(p_{2}\) is given by:  
\[ \begin{array}{ccc}p_{1}=C_{1}lnr+\lambda r+C_{3}\\ p_{2}=C_{2}lnr+C_{4}\end{array}  \tag{12}\]  
\(C_{3}\) and \(C_{4}\) are constants of the general solution. By considering Equations (9), (10) and (12), the following can be obtained:  
\[ \begin{array}{ccc}p_{0}=\lambda r_{0}+C_{1}lnr_{0}+C_{3}\\ p_{w}=C_{2}lnr_{w}+C_{4}\\ \lambda r(t)+C_{1}lnr(t)+C_{3}=C_{2}lnr(t)+C_{4}\\ -\frac{K_{1}}{\mu_{1}}\left(2\lambda+\frac{C_{1}}{r(t)}\right)=-\frac{K_{2}}{ \mu_{2}}\frac{C_{2}}{r(t)}\end{array}  \tag{13}\]  
Considering \((K_{1}/\mu_{1})/(K_{2}/\mu_{2})=M\), when the absolute penetration of the stratum of grout flowing zone is similar to that of the groundwater flowing zone,is the ratio of the viscosity of water to the viscosity of grout.