## 2 Numerical Simulation
### Material Models for Water and HBS  
The material of MAT_NULL in LS-DYNA (971, Livermore Software Technology Corporation, Livermore, CA, USA, 2007) [34] was adopted for the water jet, and the MAT_NULL constitutive model should be associated with a state equation to define the relationship between density, pressure, and internal energy. In this paper, the following GRUNEISEN equation of state for water was adopted [25]:  
\[p=\frac{\rho C^{2}\mu\left[1+\left(1-\frac{\gamma_{0}}{2}\right)\mu-\frac{ \alpha}{2}\mu^{2}\right]}{\left[1-(S_{1}-1)\mu-S_{2}\frac{\mu^{2}}{\mu+1}-S_{3 }\frac{\mu^{3}}{\left(\mu+1\right)^{2}}\right]}+(\gamma_{0}+\alpha\mu)E, \tag{2}\]  
where \(\rho\)\(\left(\text{kg}/\text{m}^{3}\right)\) is the density; \(E\)\(()\) is the unit volume internal energy; \(C\)\(m\)\(\cdot\)s\({}^{-1}\)) is the intercept of the curve \(\mu_{s}-\mu_{p}\), which is the relationship between the wave velocity and the particle velocity; and \(S_{1}\), \(S_{2}\), and \(S_{3}\) are the slopes of the curve \(\mu_{s}-\mu_{p}\). \(\gamma_{0}\) is the GRUNEISEN coefficient, and \(\alpha\) is the correction coefficient for the relationship between the GRUNEISEN coefficient and volume. The parameters of the state equation were derived from the previous literature on water jets [35].  
The hydrates and marine sediments in the Shenhu area of the South China Sea are cemented together, and the HBS is, thus, simplified as the soil body with improved mechanical properties. The \"MAT_FHWA_SOIL is an isotropic material model commonly used to describe soil [34], and the mechanical parameters of HBS mainly include cohesion, internal friction angle, bulk modulus, and shear modulus [36; 37]. The \"MAT_FHWA_SOIL material model in LS-DYNA was adopted to describe the HBS in the paper. The parameters of HBS from the South China Sea are shown in Table 1.  
In addition, some explanations deal with the parameters in Table 1. As the mechanical properties tests of in-situ HBS in the South China Sea have not been reported, the parameters in Table 1 are mainly referenced from [PERSON] [27], [PERSON] [36], and [PERSON] [38]. Based on in-situ sonic well logging data, the mechanical properties of HBS in the South China Sea were investigated by [PERSON] [27]. According to the change of longitudinal wave velocity of HBS, the parameters were obtained through calculation as follows: The bulk modulus is 6.27 to 9.55 GPa, the friction angle is around 0.218 rad, and the cohesion is 6 to 500 kPa. Through laboratory testing of HBS with 36% saturation, the bulk modulus of 9.4 GPa and the shear modulus of 4.4 GPa were obtained by [PERSON] [36]. Additionally, [PERSON] [38] acquired the cohesion of 45 kPa based on triaxial loading experiments of the remolSea, in which the hydrate saturation is 32% [38]. In particular, the mean hydrate saturation of HBS in the Shenhu area of the South China Sea is 33% [39], and the hydrate saturation in the experiments of [PERSON] [36] and [PERSON] [38] is almost equal to that. In this paper, the HBS with a hydrate saturation of 33% in the Shenhu area of the South China Sea is taken as the object of water jet erosion. As a consequence, the bulk modulus of 9.4 GPa, the shear modulus of 4.4 GPa, and the cohesion of 45 kPa were adopted for the material model of HBS.