## 1 Introduction  
Large quantities of waste rock are generated during coal mining, roadway driving, and coal separation [1], the production of which accounts for 15%-20% of total coal production [2]. The main method used to dispose of this waste rock is to stockpile it on the ground. However, stacking lots of waste rock on the ground (Figure 1a) not only occupies land resources, impairs the atmospheric environment, and damages the air environment and water resources, but may also cause potential disasters including landslides, debris flow, and spontaneous combustion [3, 4, 5]. In the meantime, the roof loses support from the lower coal seam with the mining of a coal seam. As a result, the roof subsides, breaks, and collapses under the dead load of strata and compression of the overlying strata [6, 7]. This process affects the overlying strata (Figure 1b) and ultimately causes surface subsidence, giving rise to problems such as surface breaks, water and soil erosion, and vegetation dieback [8, 9, 10]. To solve this series of problems, a disposal method involving the crushing and backfilling of solid waste rock into the goar has been proposed [11, 12]. As shown in Figure 2, waste rock is crushed to the required particle size at ground level and then transported underground through a vertical feeding system. It is then conveyed to the backfill panel on conveyor belts, where it is compacted by the compactor of the backfill support to form a backfill body, which is used to support the overlying strata. This disposal method preventsa series of geological disasters and addresses environmental issues arising from intensive coal mining and the accumulation of solid waste rock. Meanwhile, solid waste rock functions as the main load-bearing carrier to support overlying strata after being backfilled in the goat, and the compressive behavior of waste rock backfill materials (WRBM) plays a decisive role in roof control [13]. To predict the control effect on strata movement, it is necessary to test the compressive behaviors of backfill materials [14; 15]. However, specimen size also significantly influences the compressive deformation of WRBM [16; 17; 18]. Therefore, it is necessary to investigate the compressive deformation properties of WRBM with different specimen sizes. Scholars have studied the compressive behaviors of WRBM, focusing mainly on experimental tests and numerical simulation. [PERSON] and [PERSON] [19] investigated the mechanical behavior of rock blocks caved in the goat and revealed the relationships between the tangent and secant moduli and stress in the compaction process of rock blocks. They also used [PERSON]'s and [PERSON]'s formulas to describe the stress-strain relationship. By conducting compaction tests on broken rock blocks, [PERSON] et al. [20] obtained the relationships between the compressive modulus, axial stress, and axial strain and provided expressions linking the compressive modulus to the strength and porosity of rock blocks. [PERSON] et al. [21] used [PERSON]'s formula to express the stress-strain relationship of caved rock and proposed a numerical method to simulate the compaction of caved rock in the goat based on the double-yield model of FLAC\({}^{\text{3D}}\) software (ITASCA, Minneapolis, MN, USA, version V5.0, 2012). [PERSON] et al. [22] proposed an energy calculation method for broken  
Figure 1: Environmental problems caused by coal mining: (**a**) accumulation of waste rock; (**b**) surface breaks. Figure 2: Schematic diagram of the crushing and backfilling of waste rock into the goat. waste rock and utilized a self-designed compactor and a SANS-CMT5305 (Shenzhen Sansi Test Instrument Co., Ltd., Shenzhen, Guangdong Province, China) test system to explore the influences of particle size on the energy evolution of broken waste rock. [PERSON] [23] demonstrated a predictive approach that combines fractal scaling in a porous medium with principles of fluid flow to calculate the porosity and permeability distribution of caved rock in the goaf in the compression process.