Distribution of Excess Pore Water Pressure in Layered Seabed Induced by Internal Solitary Waves

Abstract

The research focuses on the complex stratification of the seabed, which is consistent with real marine geological conditions. This article presents the effects of four parameters—seabed shear modulus, permeability coefficient, porosity, and saturation—on the distribution of excess pore water pressure within the seabed by internal solitary waves (ISWs). Additionally, the changes in excess pore water pressure distribution in layered seabed are analyzed. The findings indicate that increases in saturation and permeability coefficient lead to deeper penetration of excess pore water pressure into the seabed by ISWs. Conversely, the effects of shear modulus and porosity are relatively minor and inversely related to the depth of influence of excess pore water pressure. When stratification occurs in the permeability coefficient and saturation of the seabed, significant alterations are observed in the downward propagation of excess pore water pressure. Saturation stratification exhibits similar effects, with soil layers exhibiting higher saturation levels being more conducive to the transmission of excess pore water pressure by ISWs. These research findings hold substantial implications for assessing seabed liquefaction and surface erosion processes induced by ISWs.

1. Introduction

The complex hydrodynamic environment of the ocean significantly impacts the seabed sedimentary environment, submarine safety, marine engineering structures, and seabed stability. This environment is influenced by various factors, including waves, storm surges, tides, ocean currents, and internal waves [1,2,3]. Internal solitary wave (ISW) is a specific type of nonlinear internal wave characterized by their solitary wave morphology, occurring in stably stratified seawater. ISWs are characterized by short periods, large amplitudes, high flow velocities, and rapid propagation speeds, primarily affecting deep-water environments [4,5]. ISWs can induce changes in water pressure at the seabed surface, which subsequently alters the excess pore water pressure within the seabed. Variations in excess pore water pressure are crucial indicators of the dynamic response of the seabed [6,7]. Due to the challenges and high costs associated with in situ observations of seabed responses, the existing research predominantly relies on laboratory flume experiments and numerical simulations.

Numerical simulation studies have primarily concentrated on the dynamic responses within a horizontal and uniform seabed. Qiao et al. (2016) developed a model of an infinite-depth horizontal seabed to calculate the forces exerted by ISWs and analyzed the excess pore water pressure generated in the seabed [8]. Rivera Rosario et al. (2017) utilized the Dubreil–Jacotin–Long equation to solve for the velocity and density fields produced by the propagation of ISWs over a horizontal seabed, discovering that the excess pore water pressure in the seabed can also be negative [9]. They noted that the gradient of excess pore water pressure would drive upward flow of excess pore water, potentially leading to seabed instability [9]. Tian et al. (2019) assessed the excess pore water pressure in sandy silt and clayey silt seabed and evaluated the depth of ISW influence on the seabed [10]. Additionally, Tian et al. (2023) analyzed the spatial and temporal variations in the dynamic response of the seabed induced by ISWs [11]. Collectively, these numerical studies provide a comprehensive analysis of the dynamic effects of ISWs on a horizontal seabed, as well as the distribution characteristics of excess pore water pressure within the seabed.

The dynamic effects of ISWs on a sloping seabed have primarily been investigated through laboratory flume experiments. Qiao et al. (2018) observed that both the pressure exerted by ISWs and the excess pore water pressure within the seabed exhibited negative pressure changes at the slope and at the summit of the slope [12]. They also noted that the amplitude of ISW pressure increased as the slope angle was raised from 0.071 to 0.16. In a subsequent study, Li et al. (2021) [13] varied the slope angles to 3°, 6°, and 9° and found that the excess pore water pressure in the seabed consistently increased or decreased at the base, mid-section, and top of the slope, revealing differences in the intensity of dynamic responses between these locations [13,14]. Tian et al. (2023) employed large eddy simulation to investigate four mechanisms of ISW breaking and, for the first time, analyzed the pressures and excess pore water pressures induced by ISWs on a sloping seabed [15]. The pressure generated by ISWs underwent a polarity reversal, with the rate of this reversal increasing as the slope angle increased [16,17,18]. Compared to a horizontal seabed, the changes in excess pore water pressure resulting from ISW propagation on a sloping seabed exhibited significant differences, indicating that the dynamic response on a sloping seabed is more complex. However, excess pore water pressure distribution in a layered seabed is ignored.

ISWs predominantly influence deep-water environments and pose a risk to seabed stability [16,17,18,19,20,21]. Current research has not adequately addressed the complex stratification of the seabed, which diverges from real marine geological conditions. In the context of intricate geological formations and marine sedimentation, the seabed typically displays heterogeneous and stratified characteristics. The physical and mechanical properties of various soil layers within the seabed are not uniform, which subsequently influences the distribution of excess pore water pressure among these layers [22,23]. Zhang et al. (2024) demonstrated that under wave–current interactions, the permeability coefficient significantly affects the distribution of excess pore water pressure in a stratified seabed [23]. When an ISW interacts with a layered seabed, the resulting seabed response may differ from that observed in a homogeneous seabed. However, current research has yet to address the complexities associated with seabed stratification, which diverges from actual marine geological conditions. This article aims to analyze the distribution of excess pore water pressure in a layered seabed under the influence of ISWs through numerical simulations.

2. Numerical Model of ISW and Layered Seabed

Utilizing the Navier–Stokes equations, the Biot consolidation equation, the volume of fluid method for tracking layered fluid interfaces, and the method for generating ISWs, numerical models of a flume and seabed were developed. These models were subsequently coupled under conditions of continuous water pressure at the seabed surface to facilitate the numerical simulation of the interaction between ISWs and a sloping seabed. A detailed description of this process can be found in reference [15].

2.1. Seabed Model

In Biot consolidation theory, the dissipation of excess pore water pressure and the displacement of the soil skeleton in the porous media are coupled. We adopted the governing equations of Biot consolidation theory in references [10,15] to calculate the excess pore water pressure in the seabed induced by ISWs. The seepage continuous equation of excess pore water is

$$K_x{\displaystyle \frac{{\partial }^{2}p}{\partial x^2}}+K_Y{\displaystyle \frac{{\partial }^{2}p}{\partial z^2}}-{\gamma }_{\omega }n\beta {\displaystyle \frac{\partial p}{\partial t}}={\gamma }_{\omega }{\displaystyle \frac{\partial \epsilon }{\partial t}}$$

where p is the excess pore water pressure. Kx and K_Y are the permeability coefficients of sediment in the X-direction and Y-direction, respectively. In this study, the permeability of the seabed is isotropic. The γ_w is the bulk density of excess pore water, n is the porosity of sediment, β is the compression coefficient of pore fluid, and ε is the volumetric strain of water.

The boundary conditions of the governing equation are as follows. The normal stress $\sigma$ and shear stress $\tau$ on the seabed surface are generally negligible when the water viscosity and friction are ignored. The excess pore water pressure p on the seabed surface equals the wave-induced pressure ΔP [15]. Namely, on the seabed surface, we have

$$\sigma =0, \tau =0, p=\Delta P$$

The bottom of the seabed is considered impermeable, and the soil displacement at this section is zero. Therefore,

$$u=0, w=0, \partial p/\partial y=0$$

where u is the horizontal displacement of soil, w is the vertical displacement of soil, and y is the Cartesian coordinate.

Assuming that the horizontal displacement on the left and right sides of the seabed boundary is 0, and the normal flow of excess pore water is 0, we have

$$u=0, \partial p/\partial x=0$$

where x is the Cartesian coordinate.

Tian et al. (2023) reported that the excess pore water pressure in the seabed induced by ISWs propagates to a depth of 0.15 m [15]. Beyond this depth, the excess pore water pressure remains unaffected. According to the findings of Rivera Rosario et al. (2017) [9], it is established that the excess pore water pressure in the seabed due to ISWs is confined to a specific depth range. When the depth exceeds this range, the excess pore water pressure remains unchanged. Consequently, the stratification in the deep seabed exerts less influence on the seabed response compared to that in the shallow seabed. For this study, the seabed was divided into two layers, with the specific dimensions of the model illustrated in Figure 1. The upper layer of the seabed, depicted in purple, has a thickness of 0.1 m, while the lower layer, shown in gray, represents the remainder of the seabed. The slope of the seabed is set at 9°, and the ISWs are modeled as breaking waves in the form of rolling waves. The parameters associated with the ISWs in the model are detailed in

Table 1. The parameters of the ISW.

Upper Water Depth (m)	Density of Upper Water Depth (kg/m3)	Lower Water Depth (m)	Density of Lower Water Depth (kg/m3)	Amplitude (m)	Wave Length (m)
0.1	998	0.4	1025	0.112	1.232

.

2.2. ISW Model

In this paper, fluid was regarded as an ideal incompressible fluid. We investigated the change in flow field disturbed by ISWs by Large Eddy Simulations (LESs). The governing Navier–Stokes and density transport equations were solved. A standard lock-release setup was used to generate ISWs. The interface between the stratified fluids was captured by the volume of fluid (VOF) method. At the top of the flume, the effect of surface waves was ignored, and the rigid cover assumption was adopted; therefore, the symmetry boundary condition was selected. Wall boundary conditions were adopted on both sides and at the bottom of the flume and the inclined terrain.

This study established a total of eight experimental cases, with consistent physical model dimensions across all conditions. An identical ISW load was applied to each case, and the relevant parameters of the ISW are detailed in Table 1. The eight cases were categorized into four groups. The grouping and seabed parameters for each case are summarized in

Table 2. The parameters of the layered seabed.

Group	Case	Water Layer	Shear Modulus	Permeability Coefficient	Porosity	Saturation	Poisson’s Ratio
Group 1	Case 1	Upper	1 × 106	1.33 × 10−4	0.717	0.98	0.31
		Lower	1 × 107
	Case 2	Upper	1 × 107	1.33 × 10−4	0.717	0.98	0.31
		Lower	1 × 106
Group 2	Case 3	Upper	1 × 107	1 × 10−7	0.717	0.98	0.31
		Lower		1 × 10−4
	Case 4	Upper	1 × 107	1 × 10−4	0.717	0.98	0.31
		Lower		1 × 10−7
Group 3	Case 5	Upper	1 × 107	1.33 × 10−4	0.4	0.98	0.31
		Lower			0.8
	Case 6	Upper	1 × 107	1.33 × 10−4	0.8	0.98	0.31
		Lower			0.4
Group 4	Case 7	Upper	1 × 107	1.33 × 10−4	0.717	0.9	0.31
		Lower				1
	Case 8	Upper	1 × 107	1.33 × 10−4	0.717	1	0.31
		Lower				0.9

.

In Group 1, the permeability coefficient, porosity, saturation, and Poisson’s ratio of the seabed remain unchanged across layers, while only the shear modulus varies. In case 1, the shear modulus of the upper soil is relatively low (1 × 10⁶), while the shear modulus of the lower soil is significantly higher (1 × 10⁷). Conversely, in case 2, the results are compared to illustrate the influence of the shear modulus parameter in seabed layering.

In Group 2, only the permeability coefficient varies with depth, while all other parameters are held constant. In case 3, the permeability coefficient of the upper soil is relatively low (1 × 10⁻⁷), whereas the lower soil exhibits a higher permeability coefficient (1 × 10⁻⁴). The opposite scenario is observed in case 4.

In Group 3, the porosity of the seabed varies with depth, while all other parameters remain constant. In case 5, the porosity of the upper layer of the seabed is 0.4, and that of the lower layer is 0.8. In case 6, the porosity values are reversed.

In Group 4, we examine variations in saturation with depth, while all other parameters are unchanged. In case 7, the saturation of the upper layer of the seabed is relatively low (0.9), while the lower layer is fully saturated (saturation = 1). This relationship is inverted in case 8.

We calculated Biot consolidation equations using COMSOL Multiphysics 5.6. The generation, propagation, and breaking of ISWs were simulated by LES. The numerical model aligns with the model of reference [15] and was validated by the paper; further validation is not necessary in this manuscript. For additional details, please refer to reference [15].

3. Results and Discussion

3.1. Distribution of Excess Excess Pore Water Pressure of Homogeneous Seabed

Under the influence of wave action, the variation in excess pore water pressure distribution within the seabed, attributed to stratification, is primarily a result of changes in the relevant physical and mechanical parameters among the seabed layers. Consequently, prior to investigating the impact of seabed stratification on excess pore water pressure distribution, this section first examines the effects of four key parameters—shear modulus, permeability coefficient, porosity, and saturation—on excess pore water pressure distribution within a homogeneous seabed.

Figure 2 illustrates a schematic representation of the numerical model utilized for parameter analysis, where the dimensions of the seabed and the parameters of ISW are consistent with case 3 of reference [15]. The analysis involves modifying the relevant parameters of the seabed soil to facilitate comparison of the results. The analysis is conducted at a time point of t = 27 s, during which the entire ISW interacts with the horizontal seabed. A vertical intercept is established at a cross-section located at x = 4.5 m on the seabed to monitor the distribution characteristics of excess pore water pressure along the intercept, with a length of 1 m.

The permeability coefficient of the seabed soil on the northern continental slope of the South China Sea ranges approximately from 10⁻⁵ to 10⁻⁷ m/s. The shear modulus typically varies around 10⁷ Pa. Porosity values are generally high, predominantly ranging between 0.5 and 0.7, with extreme cases reaching approximately 0.8. The saturation level is close to complete saturation; however, some gas-bearing seabeds may not be fully saturated, yet their saturation remains above 0.9. To comprehensively assess the impact of these parameters without introducing excessive distortion, the range of parameter values is appropriately expanded. The permeability coefficient is varied to include medium permeability values of 10⁻⁴, 10⁻⁵, 10⁻⁶, and 10⁻⁷ m/s. Considering the presence of a gas-bearing seabed, saturation values of 0.90, 0.93, 0.96, and 1.00 are employed. Porosity values are set at 0.5, 0.6, 0.7, and 0.8, respectively. The shear modulus is varied around 10⁷ Pa, with values of 1 × 10⁶, 5 × 10⁶, 1 × 10⁷, and 5 × 10⁷ Pa sequentially applied.

Figure 3 illustrates the distribution of excess pore water pressure with depth by ISW, within a 1 m range on the seabed. The distribution of excess pore water pressure demonstrates consistent characteristics. At the surface of the seabed, the continuous condition of water pressure being equal to the pressure of ISWs, along with the negative force exerted by ISWs, results in a negative excess pore water pressure. As depth increases, the excess pore water pressure gradually rises at varying rates, influenced by differences in parameters, and approaches zero. Specific cases may exhibit unique conditions, which will be elaborated upon in subsequent sections. The absolute value of excess pore water pressure decreases with depth, indicating that the influence of ISWs on the seabed diminishes as depth increases. These results are consistent with reference [10].

Figure 3a compares the effects of varying shear modulus on excess pore water pressure in the seabed. Within the depth range of 0 to 0.2 m, the results across the four cases show minimal differences. Below a depth of 0.2 m, the results for shear moduli G2, G3, and G4 become more similar and approach zero at a depth of 0.5 m (Figure 3a). In contrast, the excess pore water pressure under shear modulus G1 tends towards −0.3 Pa at 0.5 m. Overall, at any given depth, a smaller seabed shear modulus correlates with a greater change in excess pore water pressure induced by ISWs.

Figure 3b examines the impact of differing permeability coefficients on excess pore water pressure in the seabed. Variations in permeability coefficients significantly influence the distribution of excess pore water pressure (Figure 3b). Under the permeability coefficient K1, the excess pore water pressure approaches zero at a depth of 0.4 m. When the permeability coefficient is K2, this pressure reaches zero at a depth of 0.2 m. For permeability coefficient K3, the rate of change in excess pore water pressure with depth accelerates, reaching zero at approximately 0.5 m. In the case of permeability coefficient K4, the excess pore water pressure caused by ISWs is limited to a depth of about 0.2 m at the seabed surface, with no further influence detected below this depth. Comparative analysis indicates that permeability coefficients significantly affect the distribution of excess pore water pressure in the seabed. Higher permeability coefficients allow ISWs to exert influence at greater depths, while lower coefficients result in a more rapid decrease in excess pore water pressure with depth, thereby reducing the depth of ISW influence on the seabed.

Figure 3c illustrates the relationship between porosity and excess pore water pressure in the seabed. At a constant depth, an increase in porosity corresponds to a reduction in the absolute value of excess pore water pressure, indicating a diminished impact of ISWs on the seabed (Figure 3c). Conversely, the depth of ISW influence on the seabed is inversely related to porosity.

Figure 3d examines the effect of saturation on excess pore water pressure within the seabed. Variations in saturation lead to notable differences in the distribution of excess pore water pressure (Figure 3d). In cases where the seabed is not fully saturated (S1, S2, S3), excess pore water pressure approaches zero at a specific depth, which increases with rising saturation levels. In contrast, when the seabed is fully saturated (saturation S4), the change in excess pore water pressure is gradual within a depth range of 1 m and does not stabilize. At a depth of 1.0 m, the excess pore water pressure is approximately −2.8 Pa.

From the analysis of these four parameters, it can be concluded that the saturation and permeability coefficient of the seabed significantly affect excess pore water pressure by ISW, while the effects of shear modulus and porosity are comparatively minor. The seabed’s response to ISWs is directly proportional to both saturation and permeability coefficient and inversely proportional to shear modulus and porosity.

3.2. Influence of Shear Modulus Stratification

In cases 1 and 2, while all other parameters were maintained constant, only the shear modulus of the seabed was varied, with its specific layered distribution being depicted in Figure 4. The response of excess pore water pressure in the seabed induced by ISWs is illustrated in Figure 4. This figure highlights several critical moments, including the propagation of ISWs over the horizontal seabed, their interaction with the slope at the foot of the slope, the point of breaking upon contact with the slope, and the subsequent phase after breaking. The dynamics of shoaling ISWs and the associated changes in excess pore water pressure within the seabed have been previously discussed by reference [15].

Upon comparison, no significant differences were observed in the results between the two cases, indicating that the interlayer was excessively smooth. Furthermore, no considerable impact of shear modulus stratification on the distribution of excess pore water pressure was detected (Figure 5). Consequently, it can be concluded that variations in shear modulus do not play a critical role in influencing the seabed response in a layered seabed (see Figure 5).

3.3. Influence of Permeability Coefficient Stratification

In cases 3 and 4, the permeability coefficient of the seabed varies by layer. In case 3, the upper layer of the seabed consists of soil with a low permeability coefficient, while the lower layer exhibits a high permeability coefficient. Conversely, the characteristics are reversed in case 4, as illustrated in Figure 6.

Figure 7 depicts the location of ISW action and the distribution of excess pore water pressure within the seabed. The layered differences in permeability coefficients result in significant variations in excess pore water pressure distribution induced by ISWs. In case 3, throughout the duration of ISW action, no excess pore water pressure was generated within the seabed; instead, excess pore water pressure was confined to the shallow surface of the seabed, rendering it difficult to discern in the cloud map (Figure 7). This observation indicates that when the permeability of the surface soil is poor, the effects of ISWs are limited to a shallow depth range, without propagating into the deeper layers of the seabed. Consequently, the excess pore water pressure within the seabed remains unchanged, and the original equilibrium state is preserved.

When the seabed soil is homogeneous and exhibits uniform permeability, the maximum (absolute value) of excess pore water pressure is observed in the surface layer of the seabed. And as it extends deeper, the pressure exhibits a gradual decline, ultimately approaching zero, with no further influence at greater depths. In case 4 (Figure 7b), the upper soil layer demonstrates strong permeability; thus, under ISW action, excess pore water pressure develops within this layer. However, due to the poor permeability of the lower soil layer, which behaves similarly to an impermeable layer, the transmission of excess pore water pressure is completely halted at the interface between the seabed layers. This results in the excess pore water pressure exhibiting minimal variation within the upper soil layer, where permeability is greater, leading to a more uniform distribution that abruptly reaches zero upon encountering the “impermeable layer.”

A comparison of cases 3 and 4 reveals that soil with a low permeability coefficient and poor permeability effectively functions as an “impermeable layer,” significantly influencing the transmission of excess pore water pressure induced by ISWs into the deeper strata of the seabed. Therefore, in real seabed environments, it is crucial to consider the permeability stratification of seabed soils.

3.4. Influence of Porosity Stratification

In cases 5 and 6, all seabed parameters remain consistent and do not vary, with the exception of porosity. The stratification of porosity in these two cases is illustrated in Figure 8. In case 5, the porosity of the upper layer of the seabed soil is relatively low, at 0.4, while the porosity of the lower layer is comparatively high, at 0.8. In contrast, case 6 exhibits the opposite porosity distribution.

Figure 9 presents the cases characterized by porosity stratification. In cases 5 and 6, the variation in excess pore water pressure induced by ISWs at the stratification interface is relatively gradual, lacking the abrupt changes observed in case 4. The difference in porosity of the upper soil layers between cases 5 and 6 results in a slight variation in the extent of influence exerted by excess pore water pressure (Figure 9). Specifically, in case 6, the excess pore water pressure at the seabed is transmitted to greater depths compared to case 5. This phenomenon is evident throughout the entire duration of ISW activity, including instances of negative excess pore water pressure during valley action and positive excess pore water pressure near the slope crest during the ISW breaking phase.

3.5. Impact of Saturation Stratification

We compare the response of ISWs to excess pore water pressure in the seabed during saturation stratification by controlling variables for cases 7 and 8, as illustrated in Figure 10. Figure 11 demonstrates that the stratification of saturation significantly influences the distribution of excess pore pressure.

In case 7, the saturation level of the upper soil layer is set at 0.9, while the lower soil layer is fully saturated. Under the influence of ISWs, excess pore water pressure generated in the upper soil layer is concentrated beneath the trough of the ISWs. As this pressure propagates downward and encounters the stratification, the increased saturation in the soil leads to a pronounced diffusion of excess pore water pressure to the lower left side (Figure 11). This indicates that higher saturation enhances the diffusion and transmission of excess pore water pressure. As the ISW approaches the foot of the slope, the excess pore water pressure in the seabed also diffuses downward due to the increased saturation of the lower soil layer. Furthermore, the excess pore water pressure (positive values) resulting from the breaking of ISWs near the top of the slope exhibits significant downward propagation (Figure 11).

In contrast, the saturation stratification for case 8 is reversed compared to that of case 7, leading to markedly different results. In this case, seabed saturation decreases from 1 at the surface to 0.9 at depth, which hinders the transmission of excess pore water pressure in the lower soil layer. Consequently, the distribution of excess pore pressure in case 8 is shallower than that in case 7 at the same time. The reduction in soil saturation suppresses the deeper diffusion of pore pressure (Figure 11).

4. Conclusions

The seabed response induced by ISWs and their shoaling in the presence of stratification within the seabed soil is investigated by numerical simulations. The research first examines the effects of four key parameters: seabed shear modulus, permeability coefficient, porosity, and saturation, on the distribution of excess pore water pressure in the seabed. Subsequently, a layered seabed model is developed to analyze the variations in excess pore water pressure distribution resulting from stratified seabed parameters. Our results indicate that the saturation and permeability coefficient of the seabed significantly influence the excess pore water pressure generated by ISWs. As both saturation and permeability increase, the excess pore water pressure induced by ISWs penetrates deeper into the seabed. In contrast, the effects of shear modulus and porosity are relatively minor and inversely related to the depth of influence of excess pore water pressure. When stratification occurs in the permeability coefficient and saturation of the seabed, notable changes arise in the downward propagation of excess pore water pressure. Low-permeability soil layers impede the transmission of excess pore water pressure. If such layers are situated in the upper portion of the seabed, excess pore water pressure is confined to the surface layer. Conversely, if low-permeability layers are located deeper within the seabed, they function as impermeable barriers, obstructing the downward movement of excess pore water pressure. Saturation stratification exhibits similar effects, with soil layers exhibiting higher saturation being more conducive to the transmission of excess pore water pressure.

Ocean engineering is developing rapidly all over the world and is constantly moving towards deep water. This study provides a basis for evaluating seabed liquefaction, erosion processes, stability, and risk assessment by ISWs and contributes to researching seabed topography induced by ISWs [24,25]. The research results provide scientific guidance for the exploitation of submarine resources and the prevention and control of deep-sea engineering geological disasters.