## 2 Proposed Calculation Method
### Calculation of the Additional Stress Caused by Grouting  
The method developed in this paper is based on derivation of the mirror method proposed by [PERSON] (1999), which was originally chiefly used to calculate surface settlement and soil deformation caused by loss of the soil layer. It was later developed by [PERSON] (1999) to be used for calculating the compensation for displacement of soil layers caused by grout injection. [PERSON] and [PERSON] (2015) employed the mirror method to calculate soil displacement caused by tunnel construction. [PERSON] et al. (2010) also used the mirror method to derive formulas for calculating the additional stress along the \(x\)-axis, \(y\)-axis, and \(z\)-axis at any point in the soil layer caused by the loss of soil due to tunnel excavation. The traditional mirror method first transforms a semi-infinite body into an infinite body. It then uses the ground as the boundary and imagines an equal volume expansion at the original void mirror position in the infinite body. The displacement and stress at any point in the original semi-infinite body are derived by solving the normal stress and the shear stress caused by the void point and the expansion point, respectively. Based on the same idea, we just need to swap the position of the gap point with that of the expansion  
Figure 1: A schematic diagram of the corrective action of grouting. point in the traditional mirror method, and it can then be used to solve the additional stress at any point in the soil layer caused by expansion of the soil due to grouting. The specific analysis steps are as follows. First, the existence of the ground is ignored, and the problem of semi-infinite bodies in actual engineering is transformed into the problem of infinite bodies. The existing volume expansion point produces normal stress \(\sigma_{0}\) and shear stress \(\tau_{0}\) at the original ground position. Second, by considering the ground as the boundary, an inexistent gap point of equal size is imagined at the mirror position of the original expansion point in the infinite body. The void point produces normal stress \(\sigma_{0}\) and shear stress \(\tau_{0}\) at the original ground position. Third, the normal stresses on the original ground generated by the above two steps balance each other out; the total shear stress is equal to \(2\tau_{0}\). In order to comply with actual free boundary conditions, the additional shear stress generated is reversely applied to the surface of the semi-infinite body. The sum of the stresses generated by the above steps is the solution to the additional stress caused by the volume expansion point. Based on the abovementioned mirror image method and combined with the research results of [PERSON] et al. [10], the spherical expansion zone with radius \(a\) at point \(x_{0}\), \(y_{0}\), and \(z_{0}\) produces displacement components at point \(x\), \(y\), and \(z\), as expressed by  
\[S_{i1}=\frac{a^{3}(i-i_{0})}{3r_{1}^{3}};i=x,y,z \tag{1}\]  
A gap point of equal size at its mirror position at \(x_{0}\), \(y_{0}\), and \(z_{0}\) produces a displacement component at point \(x\), \(y\), and \(z\), respectively:  
\[S_{i2}=-\frac{a^{3}(i-i_{0})}{3r_{2}^{3}};i=x,y,-z \tag{2}\]  
where \(r_{1}=\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0} \right)^{2}}\) and  
\(r_{2}=\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z+z_{0} \right)^{2}}\).