## 5 Conclusions  
In this study, a three-dimensional DEM-based numerical model of ship oil-spill movement in Arctic icy waters was established. The trajectory model of oil film coupled with marine loading including wind, current and waves utilized the multiple tracking method of oil drift and ice floe drift. The accident scenario was simulated with reference to a real experiment. The interaction details of oil and ice floes were expanded by a 6 DOF model to describe the features of oil spreading. Subsequently, the correlation of parameters on the behaviour of oil in cold water were examined by sensitivity analysis, and the area of the oil spill was predicted using an RSM model in order to conduct rapid emergency treatment. The RSM model was evaluated by reliability analysis to indicate a high degree of credibility.  
This model was presumably based on the unique characteristics of the Arctic Ocean and combined effects of wind, wave, current and ice. The wind was regarded as a steady wind of winter; the wave was simulated as a fifth-order Stokes wave in non-linear motion over time; the current was the relatively stable polar snap coupled with the wave; the ice floe was sprayed for 100 s before the oil's spraying and stopped immediately when the oil started to spray. The oil spill was assumed as a continuous spill in a static point and the amount of oil spill was assumed to be 3 tons. The following conclusions were drawn:  
1. The presence of ice floes had a significant suppression effect on the movement of the oil spill. The maximum drift distance of oil film and the maximum radius of oil diffusion in icy waters were found to be much smaller than those in open waters. With the combined action of wind, wave, current and ice, the oil film was observed to expand to a larger area in ice-free waters but form a long and narrow belt in icy waters.
2. The oil film diffusion and drift processes occurred simultaneously. The oil film in waters with medium-density ice was observed to be distributed on both the water surface and underwater. The underwater oil gradually rose until it came into contact with the ice, and then spread to the surroundings. The changes in the horizontal drift distance of the oil spill were more evident than those in oil-film thickness. Thus, oil-film drift constituted the primary behavior of oil-spill expansion.
3. The regression function of time \(t\), the centroid coordinates (\(X\), \(Y\)), perimeter \(C\)', horizontal maximum distance \(L\) and vertical maximum distance \(D\) were generated on the base of the approximate model method. This function exhibited a relatively high degree of credibility. It could predict the extent of oil spill in icy waters scientifically and reasonably, which could provide support for establishing a rapid emergency oil-spill cleanup system in icy waters, such as those found in the increasingly accessible Arctic Ocean.  
Formal analysis, W.L., X.L. and Z.D.; Methodology, W.L., X.L. and J.L. ([PERSON]); Supervision, J.L. ([PERSON]); Writing--original draft, W.L., P.G. and Q.M.; Writing--review and editing, J.L. ([PERSON]), H.W. and Z.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Engineering Laboratory for Mobile Source Emission Control Technology (NELMS2018A06) and Fundamental Research Funds for the Central Universities of China (3132017097, 3132018177, 3132019304).  
The authors declare no conflict of interest.  
DPM Discrete Phase Model  
DEM Discrete Element Method  
RSM Response Surface Method  
VOF Volume of Fluid Method  
**6 DOF** Six Degrees of Freedom