## 1 Introduction  
On the one hand, subway tunnels must be highly safe, but on the other hand, they can easily experience a horizontal displacement due to the excavation of the side foundation pits. Therefore, the sleeve valve pipe grouting technology is often employed for microdisturbance grouting and deviation correction. The compensatory grouting method is a mature corrective technology that is commonly used in tunnel settlement control and in the elevation of existing buildings. Furthermore, in recent years, there have been more and more projects using compensatory grouting to rectify the horizontal displacement of existing tunnels [1; 2; 3]. Therefore, studying the effect of compensatory grouting on rectification of the horizontal displacement of existing tunnels, as well as its influence law, is of great significance. The main methods for investigating the compensation for tunnel deformation caused by grouting include numerical simulation, measured data analysis, and model testing. In this context, [PERSON] et al. [2] studied the influence of sleeve valve pipe grouting on the horizontal displacement of existing tunnels through measured data analysis and numerical simulation. [PERSON] et al. [4] also investigated the development law of tunnel deformation under different grouting pressures and grouting positions through indoor model testing and numerical simulation. In another work, [PERSON] et al. [5] studied the influence of lateral tunnel grouting on the transverse deformation of a tunnel through numerical simulation and analyzed the impact of the grouting volume and range on grouting reinforcement. Because compensation grouting involves the penetration, compaction, and splitting of the slurry, the mechanism of its action is relatively complicated, and it is difficult to achieve direct research and analysis results. In this regard, [PERSON] et al. [6] pointed out that the effect of slurry injection in the surrounding soil can be examined by volume expansion of the soil in the slurry injection area. Calculation of displacement of the surrounding soil and additional stress caused by unit volume expansion requires other calculation methods. At present, the random medium theory is generally used. In this context, [PERSON] et al. [7] regarded surface displacement caused by grouting as a random process and assumed that the uplifting effect is the sum of the deformation effects caused by a series of radial expansions of slurry bubbles in the formation. According to the stochastic medium theory, they derived a theoretical calculation formula for surface lifting deformation and its simplified form. According to the random medium theory, [PERSON] et al. [8] considered that the influence of bubble expansion on displacement of ground surface can be equivalent to the sum of the effects of the expansion of an infinite number of tiny cells. In summary, the random medium theory is generally used to study displacement of the upper soil layer and the ground surface caused by the expansion of slurry bubbles. It can only calculate displacement of and additional stress on the soil in the upper part of the shrinkage point or the expansion point but not at the calculation point at a similar or lower depth; thus, the calculation method based on the random medium theory cannot analyze the corrective effect of grouting in the horizontal direction. Moreover, at present, there is no work on calculating the compensated level of horizontal displacement of tunnels that sleeve valve pipe grouting can achieve; hence, conducting research in this area can be of considerable importance. This study simulated the grouting effect by calculating the volume expansion of the grouting area. To this end, a mechanical model of volume expansion of the grouting area based on the characteristics of sleeve valve pipe grouting is proposed. Based on the mirror image method [9] and combined with the research results of [PERSON] et al.