# A Calculation Model of Grout Migration Height for Post-Grouting Technology
## 2 Solution of Grout Migration Height
### Governing Equation  
Figure 2 shows the strained condition of grout in an annular space, where a small annular cement slurry element of size \(\mathrm{d}r\) by \(\mathrm{d}z\) is selected. The cement slurry is considered as a steady flow, and thus, the component of [PERSON]'s second law along the pile length direction, z, can be written as:  
\[\sum dF_{z}=2\pi r\mathrm{d}r\cdot p+2\pi r\mathrm{d}z\cdot\tau_{z}-2\pi(r+ \mathrm{d}r)\cdot\left(\tau_{z}+\frac{\mathrm{d}\tau_{z}}{\mathrm{d}r}\mathrm{ d}r\right)\mathrm{d}z-2\pi r\mathrm{d}r\left(p+\frac{\mathrm{d}p}{\mathrm{d}z} \mathrm{d}z\right)-2\pi r\mathrm{d}r\mathrm{d}z\cdot\rho_{c}g \tag{1}\]  
where \(\sum dF_{z}\) is the sum of the z components of all forces acting on the small annular cement slurry element, which has mass \(dm=2\pi r\mathrm{d}rz\rho_{c}\); \(p\) is the pressure at the bottom surface of annular element; \(\tau_{z}\) is the shear stress along the pile length direction; and \(\rho_{c}\) is the cement slurry density. Let \(\sum dF_{z}=0\), ignore third order infinitesimal, Equation (1) can be rearranged as:  
\[\frac{\mathrm{d}p}{\mathrm{d}z}+\frac{1}{r}\tau_{z}+\frac{\mathrm{d}\tau_{z}}{ \mathrm{d}r}+\rho_{c}g=0 \tag{2}\]  
Figure 1: Calculation model of the post-grouted pile foundation. Figure 2: Strained condition of grout in an annular space. The rheological equation of Bingham fluid can be expressed as:  
\[\tau=\tau_{0}+\eta\frac{\mathrm{d}v}{\mathrm{d}r} \tag{3}\]  
where \(v\) is the shear rate; and \(\tau_{0}\) is the yield shear strength (Pa). By integrating Equation (2) with respect to \(r\), the following equation can be obtained:  
\[\tau=-\frac{r}{2}\bigg{(}\frac{\mathrm{d}p}{\mathrm{d}z}+\rho_{c}g\bigg{)}+ \frac{c_{1}}{r} \tag{4}\]  
Then, the boundary conditions can be written as follows:  
\[\tau|_{r=r_{0}+b/2} =0 \tag{5}\] \[v|_{r=r_{0}} =0\] (6) \[v|_{r=r_{0}+b} =0 \tag{7}\]  
By means of (5) and suitable manipulation, Equation (4) can be rewritten as:  
\[\tau=c_{1}\bigg{[}\frac{1}{r}-\frac{r}{\left(r_{0}+b/2\right)^{2}}\bigg{]} \tag{8}\]  
Substituting Equation (4) to (5) yields:  
\[\frac{\mathrm{d}p}{\mathrm{d}z}+\frac{1}{r}\tau_{z}+\frac{\mathrm{d}\tau_{z}}{ \mathrm{d}r}+\rho_{c}g=0 \tag{2}\]0  
The radius ratio of the weak layer zone to the bored pile can be described as \(R_{h}=\frac{b}{r_{0}}\). By means of Equations (6) and (7), then Substituting Equation (8) into Equation (3) gives:  
\[\frac{\mathrm{d}p}{\mathrm{d}z}+\frac{1}{r}\tau_{z}+\frac{\mathrm{d}\tau_{z}}{ \mathrm{d}r}+\rho_{c}g=0 \tag{2}\]1  
Change the form of Equation (9):  
\[\frac{\mathrm{d}p}{\mathrm{d}z}+\frac{1}{r}\tau_{z}+\frac{\mathrm{d}\tau_{z}}{ \mathrm{d}r}+\rho_{c}g=0 \tag{2}\]2  
where \(\gamma_{c}\) is the unit weight of cement slurry (kN/m\({}^{3}\)), \(\gamma_{c}=\rho_{c}g\). Defining \(m=\frac{2c_{1}}{\left(r_{0}+\frac{b}{2}\right)^{2}}-\gamma_{c}\) and integrating Equation (10) with respect to \(z\), the following relationship is obtained:  
\[\frac{\mathrm{d}p}{\mathrm{d}z}+\frac{1}{r}\tau_{z}+\frac{\mathrm{d}\tau_{z}}{ \mathrm{d}r}+\rho_{c}g=0 \tag{2}\]3  
where \(p_{0}\) is the grouting outlet pressure (kPa); and \(p\) is the pressure of cement slurry at depth of \(z\) (kPa). The start-crack pressure \(p_{c}\) at depth of \(z\) can be expressed as:  
\[\frac{\mathrm{d}p}{\mathrm{d}z}+\frac{1}{r}\tau_{z}+\frac{\mathrm{d}\tau_{z}}{ \mathrm{d}r}+\rho_{c}g=0 \tag{2}\]4  
where \(k_{0}\) denotes lateral earth pressure coefficient of soil around bored pile; \(\gamma\) is the unit weight of soil (kN/m\({}^{3}\)); \(H\) is the depth of grouting outlet point in ground; \(z\) is the migration height of cement slurry; \(\sigma_{cl}\) is the tensile stress of the two weak interfaces (kPa). According to the results reported by [PERSON] et al.