## 2 Proposed Calculation Method
### Calculation of the Additional Stress Caused by Grouting
According to the basic equations of elastic mechanics, the formulas for calculating the strain and stress generated in the soil in the above process are defined as  
\[\varepsilon_{i}=\frac{\partial S_{i1}}{\partial i}+\frac{\partial S_{i2}}{ \partial i};i=x,y,z \tag{3}\]  
\[\gamma_{xx}=\frac{\partial\left(S_{x1}+S_{x2}\right)}{\partial z}+\frac{ \partial\left(S_{z1}+S_{z2}\right)}{\partial x} \tag{4}\]  
\[\gamma_{yz}=\frac{\partial\left(S_{y1}+S_{y2}\right)}{\partial z}+\frac{ \partial\left(S_{z1}+S_{z2}\right)}{\partial y} \tag{5}\]  
\[\sigma_{i}=\frac{E\mu}{(1+\mu)(1-2\mu)}\big{(}\varepsilon_{x}+\varepsilon_{y} +\varepsilon_{z}\big{)}+2G\varepsilon_{i};i=x,y,z \tag{6}\]  
where \(E\) is the elastic modulus of the soil, \(\mu\) represents the Poisson's ratio of the soil, and \(G\) stands for the shear modulus of the soil. By substituting Equations (1)-(5) into Equation (6), the formula for the calculation of the horizontal additional stress generated in step one and step two is expressed by  
\[ \begin{array}{l}\sigma_{x1-2}=a^{3}\frac{E}{3(1-2\mu)}\bigg{(}\frac{1}{r_{2 }^{3}}-\frac{1}{r_{1}^{3}}\bigg{)}+a^{3}\frac{E(1-\mu)}{(1+\mu)(1-2\mu)}\big{(} x-x_{0}{}^{2}\big{)}\bigg{(}\frac{1}{r_{1}^{3}}-\frac{1}{r_{2}^{3}}\bigg{)}\\ +a^{3}\frac{E\mu}{(1+\mu)(1-2\mu)}\bigg{[}\big{(}y-y_{0}\big{)}^{2}\bigg{(} \frac{1}{r_{1}^{3}}-\frac{1}{r_{2}^{3}}\bigg{)}+(z-z_{0})^{2}\frac{1}{r_{1}^{3 }}-(z+z_{0})^{2}\frac{1}{r_{2}^{3}}\bigg{]}\end{array}  \tag{7}\]  
As shown in Figure 3, when the model of uniform expansion of the grouting area is used, Equation (10) can be expressed by  
\[\sigma_{x}=\underset{V_{2}-V_{1}}{\iint}\sigma^{\prime}{}_{x}(x,y,z)\dd x\dd y \dd z=\int_{a_{2}}^{b_{2}}\int_{c_{2}}^{d_{2}}\int_{c_{2}}^{f_{2}}\sigma^{\prime }{}_{x}(x,y,z)\dd\xi d\xi-\int_{a_{1}}^{b_{1}}\int_{c_{1}}^{d_{1}}\int_{c_{1}}^ {f_{1}}\sigma^{\prime}{}_{x}(x,y,z)\dd\xi d\xi \tag{11}\]  
where \(a\) and \(b\) are the lower and upper \(x\) limits of the integral, respectively; \(c\) and \(d\) represent the lower and upper \(y\) limits of the integral, respectively; \(e\) and \(f\) denote the lower and upper \(z\) limits of the integral, respectively. Subscripts 1 and 2 indicate the state of the volume expansion of the soil before and after grouting, respectively. The formulas for calculating the lower and upper limits of the integral are defined as  
\[a_{1}=x_{1}-R_{1},\ b_{1}=x_{1}+R_{1},\ c_{1}=y_{1}-\sqrt{{R_{1}}^{2}-(\xi-x_ {1})^{2}},\ d_{1}=y_{1}+\sqrt{{R_{1}}^{2}-(\xi-x_{1})^{2}},\ e_{1}=h_{1},\ f_{1}=h_{2},\] \[a_{2}=x_{1}-R_{2},\ b_{2}=x_{1}+R_{2},\ c_{2}=y_{1}-\sqrt{{R_{2}} ^{2}-(\xi-x_{1})^{2}},\ d_{2}=y_{1}+\sqrt{{R_{2}}^{2}-(\xi-x_{1})^{2}},\ e_{2}=h_{1},\ f_{2}=h_{2}.\]  
The rate of volume expansion of the soil (\(Q\)) before and after grouting can be defined as  
\[Q=\frac{V_{2}}{V_{1}}-1= \frac{\pi R_{2}^{2}(h_{2}-h_{1})}{\pi R_{1}^{2}(h_{2}-h_{1})}-1= \left(\frac{R_{2}}{R_{1}}\right)^{2}-1 \tag{12}\]  
According to the work of [PERSON] et al. [6], grouting efficiency (\(\xi_{\text{inj}}\)) is related to the rate of the volume expansion of the grouting ring and the amount of grout (\(V_{\text{inj}}\)) as expressed by  
\[\xi_{\text{inj}}=\frac{V_{2}-V_{1}}{V_{\text{inj}}}=\frac{QV_{1}}{V_{\text{inj }}} \tag{13}\]  
The formula can be further transformed into  
\[V_{\text{inj}}=\left[\left(\frac{R_{2}}{R_{1}}\right)^{2}-1\right]\cdot\frac{ V_{1}}{\xi_{\text{inj}}} \tag{14}\]  
Assuming that \(\varepsilon\) is the expansion of the radius of the grouting zone, it can be expressed by  
\[\varepsilon=R_{2}-R_{1} \tag{15}\]  
In the above study, volume expansion of the soil adopts the uniform expansion model. Assuming that the grouting area of the soil expands evenly along the horizontal and radial directions, [PERSON] et al. [2] observed diffusion of the grout slurry during a field test. They believed that the main body of the hardened grout slurry has an approximate conical shape with a small bottom and a large top.