## 1 Introduction  
In traditional offshore drilling processes, drilling mud needs to be recovered and treated using a riser pipe. However, as oil reservoirs go deeper, the lengths of riser pipes increase, posing a series of challenges and issues [1, 2, 3, 4]. The applications of the riser system are mostly limited to oil and gas drilling [5, 6, 7, 8] because of its high cost and complex structure and the demand for large-tonage drill platforms or drillships. In open-circuit drilling conditions, the direct discharge of mud into the sea induces marine environmental pollution and involves a substantial consumption of mud [9]. To address the mud-handling and recovery challenges in a downhole operation, riserless mud recovery (RMR) technology has been widely adopted in deepwater oil production. RMR technology has undergone continuous improvement and development. It has been successfully applied in different countries and regions. British Petroleum (BP) successfully implemented RMR technology in offshore waters near Egypt, effectively mitigating the effect of formation fractures and improving drilling efficiency [10]. RMR was also deployed near Australia to address the specific challenges associated with soft sandstone in the Browse Basin [11; 12; 13]. Its initial application in the Gulf of Mexico enhanced operational efficiency and improved overall wellbore integrity [14; 15]. Moreover, RMR was employed in the United Kingdom continental shelf (UKCS) field exploration project, specifically in top-hole operations on the UKCS [16]. Double-gradient technology was used in the Santos Basin in offshore waters near Brazil to drill extended-reach wells. It effectively solved the problems of formation erosion and wellbore collapse in the region [17]. RMR technology was also applied in Sabah, Malaysia, successfully cementing the surface casing for the Malikai deepwater oil field [18]. In the Zumba well in Norway, RMR was utilized in a unique manner to prevent drilling fluid and cuttings from being discharged into the seabed. Thereby, it protected a large number of corals near the well [19; 20; 21]. RMR was employed in the Dampier sub-basin in offshore waters near Australia, where it successfully stabilized the system's formation, reduced the system's vibration, and prevented leakage or loss [22; 23]. In Norway, RMR was utilized in three wells with shallow water flow issues. It led to complete subsea mud recovery and prevented the shallow flow [24]. RMR can also compensate for the lack of intermediate casing and the absence of a blowout preventer (BOP) when drilling in overpressured zones [25; 26; 27; 28; 29]. Thus, it has been successful in various offshore areas with loose formations and losses. The Integrated Ocean Drilling Program (IODP), in collaboration with companies such as AGR, conducted research on RMR technology for ultra-deepwater applications [30]. The Elvary Neftegaz Company conducted riserless drilling northeast of the Kola Peninsula shelf using the extensively developed Discflo pump and increasing efficiency by 50% [31]. The DEMO 1800 program, led by AGR Subsea, BP America, Shell, and others, pioneered deepwater mud-circulation systems [32]. [PERSON] et al. [33] [PERSON] controlled mud-level technology in the Barents Sea to prevent mud leakage and maintain pressure balance. [PERSON] et al. [34] established a physical model to analyze the heat transfer characteristics of drilling fluid under different RMR system configurations. They also developed a transient simulation model to temporally and spatially simulate the circulating fluid's temperature. Currently, research on RMR technology focuses mainly on its applications in different regions. During the drilling process, the stability of a mud-return circulation system in an RMR system can be affected by various factors[PERSON] For example, the actions of sea currents and vessel motion can pose a threat to operational safety.