## 2 Penetration and Diffusion Equation of Bingham Fluid
### Penetration Motion Equation  
The theoretical diffusion model used in the derivation of the column penetration and diffusion mechanism of Bingham fluid, considering the displacement effects, is shown in Figure 1. The grouting disturbance range of this experiment is 5 m, and the boundary of the experimental box verified through the experiment is 0.3 m [28].  
As mentioned in [29], the capillary group theory is used in the derivation of the mechanism. In Figure 2, the radius of a single capillary channel is assumed to be \(r_{s}\). In addition, the central axis of the capillary is considered as the axis of symmetry. The force conditions of a micro unit with a length of \(dl\) and a radius of \(r\) are shown in Figure 2, where \(p\) and \(p+dp\) are, respectively, the pressures on both sides of the column micro unit, \(r_{p}\) is  
Figure 1: Schematic diagram of the column diffusion of Bingham fluid.  
the radius of the flow core of Bingham fluid, \(\tau\) is the shear stress of grout when it permeates and diffuses in porous media, \(v_{p}\) is the flow velocity of the grout within the radius of the flow core and \(v\) is the flow velocity of the grout outside the radius of the flow core.  
The flow rate in a single capillary can be computed as [29]:  
\[q=\pi r_{p}^{2}v_{p}+\int_{r_{p}}^{r_{s}}2\pi rvdv=-\frac{\pi r_{s}^{4}}{8\mu} \left(\frac{dp}{dl}\right)\times\left[1-\frac{4}{3}\bigg{(}\frac{2\tau_{0}/r_{ s}}{-dp/dl}\bigg{)}+\frac{1}{3}\bigg{(}\frac{2\tau_{0}/r_{s}}{-dp/dl} \bigg{)}^{4}\right] \tag{1}\]  
\(\tau_{0}\) is the yield stress in the constitutive equation of Bingham-type grout and \(\mu\) is the viscosity of Bingham-type grout. The equilibrium flow velocity in the capillary channel is \(\widetilde{v}=q/\pi r_{s}^{2}\), and \(\widetilde{v}\) can be rewritten as:  
\[\widetilde{v}=-\frac{r_{s}^{2}}{8\mu}\frac{dr}{dl}\times\left[1-\frac{4}{3} \bigg{(}\frac{2\tau_{0}/r_{s}}{-dp/dl}\bigg{)}+\frac{1}{3}\bigg{(}\frac{2\tau _{0}/\tau_{s}}{-dp/dl}\bigg{)}^{4}\right] \tag{2}\]  
When the flow rate in the capillary channel is zero, the starting pressure gradient can be obtained:  
\[\lambda=\frac{2\tau_{0}}{r_{s}}=-\frac{dp}{dl} \tag{3}\]  
Based on the Dupuit-Forchheimer relation, namely \(v=\phi\widetilde{v}\), the penetration motion equation of Bingham fluid can be obtained, where \(v\) is the penetration velocity, \(\phi\) is the porosity of the stratum and \(K=\phi r_{s}^{2}/8\) is the absolute penetration of the stratum [29]:  
\[V=-\frac{K}{\mu}\bigg{(}\frac{dp}{dl}\bigg{)}\times\left[1-\frac{4}{3}\bigg{(} \frac{\lambda}{-dp/dl}\bigg{)}+\frac{1}{3}\bigg{(}\frac{\lambda}{-dp/dl}\bigg{)}\right] \tag{4}\]  
By extending Equation (4) to the three-dimensional case, the generalized Darcy law of Bingham fluid can be written as:  
\[V=-\frac{k}{\mu}\bigtriangledown P\bigg{[}\bigg{(}1-\frac{\lambda}{\left| \bigtriangledown p\big{|}}\bigg{)}\bigg{]}-\frac{1}{3}\bigg{[}\frac{\lambda}{ \left|\bigtriangledown p\big{|}}\bigg{(}1-\frac{\lambda^{3}}{\left|\bigtriangledown p \big{|}^{3}}\bigg{)}\bigg{]} \tag{5}\]  
In Equation (5), \(\bigtriangledown\) is the Hamilton operator, which is \(\bigtriangledown=\bigg{[}\frac{\partial}{\partial x},\frac{\partial}{ \partial y},\frac{\partial}{\partial z}\bigg{]}\). To simplify the calculation, the higher-order term in brackets in Equation (5) is omitted, and therefore, the simplified generalized [PERSON]'s law can be obtained [23; 29]:  
\[V=-\frac{K}{\mu}\bigtriangledown P\bigg{[}1-\frac{\lambda}{\left|\bigtriangledown p \big{|}}\bigg{]} \tag{6}\]  
Figure 2: Force analysis of grout in the capillary channel.