## 1 Introduction
[28] combined numerical simulation and laboratory experiments and proposed a grouting method for preventing and controlling safety hazards in a single coal seam goat by using pre-drilling, full-hole intubation, extended sealing, and multi-stage drilling for grouting. [PERSON] et al. [19] used numerical simulation software to simulate the reinforcement effects of two grouting methods--full-range grouting and strip grouting--in a single coal seam goat. They found that a reasonable grouting plan can effectively reduce residual deformation of the goat and its overlying rock mass, improve adverse stress conditions, and achieve the goal of effective grouting. Based on a theory of mining subsidence, [PERSON] et al. [29] studied grouting filling technology to control the residual subsidence of a single coal seam goat and concluded that under a certain mining depth, goat grouting can be filled with banded grouting. The grouting hole should be arranged within 20 m of the edge of the working face, in the direction of the steeply inclined coal seam and the fault development area to fill the voids and under-compacted areas at the edge of the goat. To explore the grouting scheme of multi-coal seam goats, [PERSON] et al. [30] studied the grouting treatment technology of multi-coal seam goats by combining various geophysical exploration methods and drilling methods. According to the spacing of each coal seam goat, the grouting horizon was divided, and the grouting technology and scheme of \"pressure less gravity flow and pressurized diffusion\" were used to reinforce the goat. [PERSON] et al. [31] used methods such as on-site investigation, laboratory testing, and numerical simulation to study the deformation and failure modes and stress distribution characteristics of multi-seam goats at close range under double-thick coal seam mining conditions. A zoning grouting treatment method centered on \"high pressure water jet, asymmetric high strength cable beam net, three hole anchor cable group and roof grouting\" was proposed. Most of the above studies focused on the treatment of single-coal seam grouting. Due to the large burial depth and multiple layers of goat in multi-coal seam goats, if grouting is carried out according to the recommended standard [32] and the grouting range is determined based on the rock movement angle, then it causes the grouting range to be too large, which affects the ecological protection and restoration of the grouting area and may not necessarily achieve the expected subsidence reduction effect. The selection of grouting treatment opportunity also directly affects the residual subsidence level of multi-seam goats. The goal of this study was to further clarify the control of the effects of different grouting schemes on the deformation of multi-coal seam goafs. A model of a multi-coal seam goaf with different spacing conditions was designed using a numerical simulation method. By changing the grouting range and opportunity, the long-term deformation of multi-coal seam goafs with different spacings after different grouting schemes was calculated, and the subsidence reduction effect of different grouting schemes on the deformation of multi-coal seam goafs was analyzed. The results can provide more scientific and reliable guidance for the treatment of multi-seam goafs.