**4. Dynamic Response of the Seabed**

In this study, a new meshfree model is developed, based on Biot's "*u* − *p*" approximation to simulate the dynamic sandy seabed behaviour around an immersed tunnel under complex natural loading including wave and current. As shown in Figure 8, the external loadings are assumed to be propagating over the porous seabed in which the immersed tunnel is buried. The buried depth is defined as *b* adopting 0.5 m below the seabed surface in this case. The sandy seabed foundation is treated as an elastic two-phase medium above a rigid impermeable bottom with 200 m for seabed length (*Lx*) and 40 m for seabed thickness (*h*). The seawater depth is specified as *d*. The immersed tunnel is assumed to be placed on the trench dredged on the middle of the seabed of the computational domain. The trench is back-filled by the same type of loose sand with the seabed soil. The tunnel geometry, wave profile and seabed profile in this case are roughly the same as the actual conditions of the Hong Kong–Zhuhai–Macao bridge tunnel, which could provide a reference of the sandy seabed dynamic for such a large immersed tunnel under wave loading.

**Figure 8.** The sketch of the computational domain of wave-seabed-tunnel interaction problem.

The detailed dimensions of the immersed tunnel are given in the Figure 9. As shown in figure, the immersed tunnel in this study is considered as an elastic material comprising two traffic tubes of 30 m long and 9 m high (cross section). The boundary condition of the immersed tunnel is treated as impermeable with zero pore pressure gradient. No relative displacement is assumed between the seabed soil and the tunnel frame on the consequence of the high fraction exists between the concrete tunnel surface and seabed soil.

**Figure 9.** Cross section of an immersed tunnel element (Unit: mm).

The configuration of the fluid domain and seabed domain can be found in Table 1, as well as the wave characteristics, seabed soil properties and modelling parameters. The seabed foundation is assumed to be composed of relative dense deposited sand [53]. Figure 10 shows the applicability range of the different wave theories [54]. The wave characteristics adopted in this study are in the range of Stokes second-order wave (*H* = 4 m, *T* = 10 s, *d* = 30 m, shown as a red star in Figure 10), which is generated and simulated by fluid model. The node number of the local region (*K*) for local RBF method is 9, while a positive constant *c* which is known as the shape factor equals 6. The total number of nodes in this case is 771,140. The convergence test has been done to check the stability of the present parameter configuration, which is quite enough to obtain an accuracy and detailed result. The time step ∆*t* set in this case is 0.5 s, while 20 time steps are contained in one wave period.

The aim of this section is tracking the dynamic soil response during the wave propagating over the seabed around the immersed tunnel. During the wave propagation from the left to the right of the computational domain continuously, the effective stresses and pore water pressures show a correlation trend with the change of the water pressure acting on the surface of the seabed. As shown in Figure 11, the oscillatory wave-induced pore water pressures, horizontal displacements (*us*) and vertical displacement (*ws*) for the computational domain of the seabed at *t* = 13 s are presented, respectively. It is found that the seabed dynamic behaviour with immersed tunnel is not periodic symmetry any more under the cyclic wave loading. The figure shows that the existence of the immersed tunnel has an obvious influence on the dynamic behaviour of the sandy seabed soil nearby by comparing the region located on the leftward and the rightward of the tunnel. It can be seen in Figure 11 that the placement of a tunnel weakens the displacement change of local area in a way, while the fluctuation of the dynamic pore water pressure decrease around as well. In Figure 11c, the dynamic pore water pressure of the seabed soil beneath the tunnel bottom shows a different tendency from the surrounding that the positive oscillatory pore pressure occurs on the left corner while the negative occurs on the right corner. In order to figure out a more detailed dynamic soil response in the vicinity of the immersed tunnel, the results of dynamic pore water pressure of some typical locations are analysed.

**Figure 10.** Wave theories range of applicability [54]. The red star represents the wave characteristics used in this study.



As shown in Figure 8, points A to D are a set of symmetrical nodes about the tunnel at the depth of −5 m in the seabed (*x* = 60, 82, 118, 140 m, respectively), while points E to F located at the depth of −10 m (*x* = 60, 85, 115, 140 m respectively). Points F and G are set on 0.5 m below the two base corners of the immersed tunnel. Figure 12 depicts the time series of the dynamic pore water pressure generated by the two point sets. From the figure, it can be seen that the vibration of the pore water pressure is more violent on the remote seabed from the tunnel, which is consistent with the conclusion mentioned above. Furthermore, the amplitude reduced for point F and G below the tunnel are slightly less than the points B and C on two sides of the tunnel. The pore water pressure vibration of the soil

on points F and G is even larger than on points B and C, which indicates that the seabed foundation beneath the immersed tunnel is more likely to be unstable due to transient liquefaction. In addition, a phenomenon occurred which is that there is a phase difference of dynamic pore pressure brought out under the base corners of the tunnel (points F and G), as shown in Figure 12b.

**Figure 12.** Time series of wave-induced pore pressure in seabed foundation at (**a**) *z* = −5 m; (**b**) *z* = −10 m.
