## 2 Predicting Soil Displacement Induced by Capsule Grouting  
The expansion of the capsule is realized through grouting inside the capsule, and the mechanism of capsule expansion on the displacement field of the surrounding soil can be addressed utilizing the cavity expansion theory. However, the calculation method of cylindrical cavity expansion is based on the assumption of infinite cavity height, whereas the height of the capsule is finite in engineering applications. As a consequence, the results of calculating the cylindrical cavity expansion tend to overestimate the displacement field of the soil, but underestimate the dosage of the cement paste, making it impossible to achieve the desired deformation control effect. The image source method [29] provides a calculation method for the displacement field of soil induced by cavity expansion in a semi-infinite space, assuming that the soil is incompressible. Various analytical formulas derived from the image source method are widely adopted in the calculation of soil displacement fields caused by shield tunneling [30; 31; 32], deep excavation [33; 34], etc. Although it is based on the assumption that the soil is incompressible, its practicality has been confirmed in a wide range of applications. Therefore, virtual image technology is employed to equate the expansion of a cylindrical cavity with finite height to the expansion of multiple spherical cavities. The displacement field of the surrounding soil caused by the expansion of the capsule is then determined by superposing the displacement field of the soil caused by the expansion of multiple spherical cavities. The following assumptions are made to simplify the theoretical analysis:  
1. The pile is simplified as an Euler-Bernoulli beam acting on a Winkler foundation. 2. The soil is assumed to be incompressible. For saturated soil, the grouting period for the capsule is relatively short, allowing it to be considered under non-drainage conditions, which reach the incompressible conditions. Subsequently, soil displacement will recover to a certain extent with the dissipation of the pore water pressure. Hence, the proposed method in this paper is applicable to predicting the soil displacement at the initial stage after the completion of capsule grouting, while the calculation results will overestimate the soil displacement generated by the expansion of capsule grouting in the long-term prediction. Firstly, the soil displacement field induced by the expansion of a cylindrical cavity with infinite height is derived, as shown in Figure 2. The lateral soil displacement \(\delta\) can be calculated by the following equation as the radius of the cavity increases from 0 to \(R_{\mathrm{c}}\):  
\[\pi{R_{\mathrm{c}}}^{2}=\pi(R+\delta)^{2}-\pi R^{2} \tag{1}\]  
where \(R\) is the lateral distance from the calculation point to the axis of the capsule. The lateral displacement \((\delta)_{\mathrm{cy}}\) of soil can be calculated according to Equation (1), neglecting the higher-order infinitesimal quantities:  
\[(\delta)_{\mathrm{cy}}={R_{\mathrm{c}}}^{2}/2R \tag{2}\]The soil displacement field due to the expansion of a cylindrical cavity with finite height is then derived based on Figure 3, assuming that the cylindrical cavity of height \(H\) consists of a series of spherical cavities with neighboring spherical cavities spaced at \(S\). In infinite space, for a single spherical cavity expansion, the lateral displacement of soil \((\delta)_{\text{sp}}\) is given as  
\[\left(\delta\right)_{\text{sp}}={R_{\text{c}}}^{3}/3{R_{1}}^{2} \tag{3}\]  
where \(R_{1}=\sqrt{\left(x_{1}-x\right)^{2}+\left(y_{1}-y\right)^{2}+\left(z_{1}-z \right)^{2}}\) is the distance from the cavity to the calculation point. The image source method [29] was employed to solve the infinite half-space solution, considering the free surface, as depicted in Figure 4. Taking the free surface as the symmetry plane, a virtual expansion cavity is set up at the symmetry position of the expansion cavity.