## 1 Introduction  
Coal gauge is one of the most significant industrial waste residues in China [1; 2; 3; 4]. Statistics indicate that by 2022, China's coal gangue stock had reached approximately 7.5 billion tons and continues to grow at an annual rate of 500-800 million tons. Consequently, 1500 to 1700 gangue hills have formed, covering an area of more than 200,000 acres [5; 6; 7]. In addition to a small amount of comprehensive utilization, the current treatment methods still rely mainly on ground emissions, which not only occupy a large amount of valuable land resources but also inevitably cause air, groundwater, and soil pollution and ultimately lead to severe ecological and environmental problems [8; 9; 10; 11]. In order to effectively deal with both environmental and resource pressures caused by coal gangue, during the 14 th Five-Year Plan period, China vigorously promoted the use of coal gangue to carry out green filling and resource utilization [12]. The implementation of these policies not only provides technical guidance for the green development of the coal industry but also provides an effective way of solving the harm of coal gangue. Among these methods, gangue slurry filling technology, as a green and efficient solid waste resource utilization method, has been widely used in the mining industry in recent years due to its advantage of facilitating the large-scale disposal of coal gangue [13; 14; 15]. The process of gangue slurry filling often requires long-distance pipeline transportation [16; 17; 18]. During the filling operation, there have been many accidents due to slurry segregation and pipe plugging [19; 20]. During the backfill mixture transportation process, the critical flow rate is defined as the minimum flow rate required to prevent particle settlement and ensure continuous pipeline transportation, which ensures the possibility of backfill slurry transportation and is particularly important for non-Newtonian slurry transportation, such as in the case of coal gangue. These speeds depend on factors such as yield stress, viscosity, and particle size distribution [21]. The flow and sedimentation performance of gangue slurry is the key factor for its safe transportation. It is essential to evaluate the flow and sedimentation performance of gangue slurry accurately. Relevant scholars have carried out much research on the flow and settlement performance of filling slurry. For example, [PERSON] et al. [22] evaluated the fluidity of filling slurry through a slump test for mine-filling mining, ensured that it was suitable for long-distance pipeline transportation, and determined a reasonable transportation length. [PERSON] et al. [23] established the ideal filling slurry ratio by testing the slump and fluidity of several coal-based solid waste filling slurries and by combining this with the response surface analysis method. [PERSON] et al. [24] investigated the influence of polycarboxylate superplasticizer dosage, as well as solid and binder content, on the rheological behavior, fluidity, and flow properties of filling slurry. Their findings indicated that the fluidity of the filling slurry exhibited an exponential increase with the rising dosage of polycarboxylate superplasticizer. [PERSON] et al. [25] developed a mechanical model to describe the suspension state of gangue particles in filling slurry, examined the factors influencing this state, and highlighted that incorporating a suspension agent effectively prevents the settling of gangue particles. [PERSON] et al. [26] evaluated the effects of solid content and the cement-to-tailings ratio on the rheological behavior and sedimentation performance of filling slurry through sedimentation rate and rheological parameter experiments. Existing research on the flow and settlement performance of gangue slurry mainly focuses on adjusting the ratios of cementing materials, aggregates, and other components to evaluate and optimize its flow and settlement performance.