## 2 Mathematical Representation
### CFD Simulation
Hence, the present study was set under the following specific conditions: First, the whole computational domain is 1000 m in length (\(x\)), 1000 m in width (\(y\)), and 100 m in depth (\(z\)). The ship is 160 m long, 30 m wide, and has a 14.5 m draft. In the cases of a side and bottom crack, the area was set to 0.25\(\pi\) m\({}^{2}\) located 80 m from the bow of the ship. In addition, the current scenario assumed that the accidental crack only occurred at the side and bottom surfaces of the ship. The side crack was located 10 m below sea level, while the bottom crack was located at the center of the ship. Besides this, we conducted grid-independent tests to ensure grid convergence. As a result, the selected number of grids was 293,162, which were applied for all computational cases. It is necessary to use the appropriate boundary and initial conditions for CFD simulations. In fact, the sea-current (or current) velocity is affected by tidal, wind-driven, and density-driven currents [16]. Due to the lack of information regarding this variation, it may be difficult to consider all factors accurately in simulation. Practically, the current velocity is assumed to be uniformly distributed with a mean value. The current velocity (\(v_{c}\)) in the range of 0.514-1.542 m/s was used as the inlet velocity. Accordingly, the Reynolds number was estimated to be larger than 0.51 \(\times\) 1010 with a scale length of 100 km, which has usually been used for large-scale ocean currents [17]. For simplicity, the free surface flow was not considered and the HNS propagation at sea level was assumed to not be affected by the interfacial motion. The slip condition was used at sea level, assuming the normal gradient of the tangential velocity component becomes zero. Similarly, a zero-gradient pressure condition was set to all the outlet surfaces, as the type of Neumann condition. As mentioned above, there is a lack of sufficient information on HNS properties. To examine the density effect on the propagation behavior, we set such artificial cases to show the influence of the HNS density on the propagation behavior. In fact, there are a lot of factors that can affect the HNS spill behavior. Thus, it is not possible to consider all possible scenarios on CFD simulations. Here, we took two important parameters such as current flow velocity and HNS density in CFD simulations for developing the metamodel. Styrene is one of the top 20 imports and exports of chemicals in Korea. Therefore, this study used styrene, which has a high possibility of spillage in case of spill accident. Styrene shows floating behavior and was used as the base material for the simulation. Its density change was considered to be in the range of 700-900 kg/m\({}^{3}\) without changes to other properties, such as the viscosity, and the diffusion coefficients were 0.00076 kg/m-s and 8\(\cdot\)10-8 m\({}^{2}\)/s. It was also assumed that the HNS initially spills with a mass flow rate of 900 kg/s at the crack of the ship. For the simulation, the timestep was fixed to 0.05 s and the averaged Courant number was estimated to be 0.1231 for a total flow time of 1000 s. Table 1 summarizes the different cases for this simulation. Figure 1: Schematic of the computational domain and accident ship.