A coupled mathematical model for accumulation of wave-induced pore water pressure and its application

Coastal Engineering 154 (2019) 103577

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Coastal Engineering journal homepage: http://www.elsevier.com/locate/coastaleng

A coupled mathematical model for accumulation of wave-induced pore water pressure and its application Xiao-li Liu a, b, Hao-nan Cui a, Dong-sheng Jeng c, Hong-yi Zhao d, * a

College of Environmental Science and Engineering, Ocean University of China, 266100, Qingdao, China Shandong Provincial Key Laboratory of Marine Environment and Geological Engineering (Ocean University of China), 266100, Qingdao, China c School of Engineering & Built Environment, Griffith University Gold Coast Campus, QLD, 4222, Australia d State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, 210098, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Cyclic wave loadings Accumulation of pore water pressure Coupling effect Liquefaction

Prediction of the wave-induced instability of seabed due to the build-up of pore water pressure is essential for coastal engineers involved in the foundation design of marine infrastructure. Most Previous studies have been limited to decoupled residual mechanism for the rise in excess pore water pressures. In this study, the existing decoupled approach has been improved to consider the coupling effect between the development of pore water pressures and evolution of seabed stresses. Comparisons with the existing wave flume tests and geotechnical centrifuge wave tests have demonstrated that the developed coupling approach is capable of reproducing the accumulation of pore water pressure under cyclic wave loadings, and shows more promising predictions compared to the existing decoupled model, especially for the case of standing waves. The validated framework is further extended to a field scale numerical model to investigate the development of pore water pressures and the corresponding liquefaction susceptibility of seabed to water waves. The numerical results have revealed the different mechanism for wave-induced pore pressure build-up in marine soils between the developed model and the existing decoupled model. The coupling effect of residual pore pressure and seabed stress could accelerate pore pressure accumulation, which implies that the existing decoupled model may underestimate the liquefac­ tion potential of seabed, particularly under standing wave loadings. Furthermore, results of the developed model have shown that the effect of wave nonlinearity on advancing seabed liquefaction is more noticeable for pro­ gressive waves than that for standing waves.

1. Introduction In the past few decades, a large number of infrastructures, such as mono-piles, breakwaters, pipelines, and oil platforms, have been con­ structed in offshore areas due to the ongoing demand of marine re­ sources. For design of offshore structures, seabed instability under ocean waves has become one of the main concerns of coastal engineers. Waveinduced excess pore water pressure may lead to liquefaction of seabed, which may further cause instability of offshore structures. Such evidence has been reported in previous literatures (Zhang and Ge, 1996; Chung et al., 2006; Sumer, 2014). Therefore, investigations of wave-induced soil response are significantly urgent and meaningful in the practice of coastal engineering. Two mechanisms for wave-induced soil liquefaction have been observed in laboratory and field measurements (Zen and Yamazaki,

1990a; Tzang, 1998; Sumer et al., 1999, 2012; Jeng, 2013, 2018), depending on the way how the excess pore pressure is generated. One is momentary liquefaction by oscillatory excess pore water pressure, associated with the amplitude damping and phase lag in pore pressure, which plays a more important role in partially-saturated marine soils under wave loadings. The other is residual liquefaction, associated with accumulation of pore water pressure, i.e., residual component of pore pressure resulting from volumetric contraction of the soil under cyclic shear stress (Seed and Rahman, 1978; Sumer and Fredsøe, 2002; Sumer, 2014). In this study, liquefaction due to the build-up of pore water pressure is concerned. Understanding the build-up of pore water pressure requires the development of elastoplastic model with capability of capturing the volumetric contraction of soil under cyclic shearing. Pastor et al. (1990) proposed a generalized plasticity model (Pastor-Zienkiweicz Mark III)

* Corresponding author. E-mail address: [email protected] (H.-y. Zhao). https://doi.org/10.1016/j.coastaleng.2019.103577 Received 22 January 2019; Received in revised form 27 September 2019; Accepted 12 October 2019 Available online 15 October 2019 0378-3839/© 2019 Elsevier B.V. All rights reserved.

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with no predefined yield or plastic potential surface to capture the nonlinear stress-strain behavior of cohesionless soil under cyclic loading. This model has been widely applied for the prediction of wave-induced soil response in liquefiable seabed undergoing elasto­ plastic soil deformations (Zienkiweicz et al., 1999; Vun et al., 2003; Dunn et al., 2006; Ye et al., 2015; Yang and Ye, 2017). Sassa and Seki­ guchi, 2001, Zhu et al. (2019) further modified Pastor-Zienkiweicz Mark III model to account for the features of principal stress orientation that rotates continuously in the soil element under cyclic waves. It was found that the continuous principal stress rotation can accelerate the plastic soil deformations and build-up of pore water pressures during the pro­ cess of wave-seabed interactions. Despite these various developments, all these advanced elastoplastic models are very complex and requires many parameters to construct the constitutive relationship, therefore arises their difficulties to be applied in the practice of engineering. An alternative method for predicting the accumulation of pore water pressure was proposed firstly by Seed and Raman, (1978) by introducing a source term in Biot’s equations based on tests of undrained soils under cyclic shearing. This simplified method is advantageous in terms of simple calculation, fewer and more easily determined parameters. Many investigations have been carried out based on this kind of approach. Tsotsos and Georgiadis, (1989) presented a two-dimensional (2D) nu­ merical method for evaluating the build-up of pore water pressure, which could calculate the component of oscillatory and residual pore water pressures simultaneously. However, the seabed stresses were ob­ tained directly from the transient analytical solution for infinite seabed subjected to wave loadings, which means that the stresses are inde­ pendent of residual component of pore water pressures. By the maximum oscillatory shear stress of the soil over a wave period, McDougal et al. (1989) developed a linear expression of the source term for determining the development of residual pore water pressure, which was employed in one-dimensional (1D) consolidation equations to assess wave-induced liquefaction. Similarly, Sumer and Fredsøe (2002) derived an analytical solution for assessing the build-up of pore water pressure through Laplace’s transformation, in which the shear stress is determined based on the analytical solution provided by Hsu and Jeng (1994) for the seabed of finite thickness. Sumer et al. (2012) further applied this analytical model to replicate their conducted wave flume tests in terms of residual pore water pressures under progressive waves, obtaining qualitative agreement. Recently, Jeng and Zhao (2015) extended the 1D model (Sumer and Fredsøe, 2002) to a 2D expression, in which the source term of pore pressure build-up is re-defined with the phase-resolved oscillatory shear stress, not the maximum one. One limitation of the model is the lack of appropriate parameters to fit the source term of pore pressure accumulation with the phase-resolved oscillatory shear stress. The aforementioned studies have treated the residual pore pressure and seabed stress in a decoupled approach, i.e., the oscillatory seabed response involving the oscillatory pore water pressure and oscillatory seabed stresses is calculated first, then the oscillatory shear stress is used as a source to determine the residual pore water pressure, assuming that development of residual pore pressure has no effect on evolution of seabed shear stress. One limitation of the decoupled models is that the oscillatory and residual seabed response are calculated separately. The other is that the coupling effect between residual pore pressure and seabed stress is not taken into account. In fact, the oscillatory and re­ sidual seabed response often exist simultaneously (Sumer, 2014), and the growth of residual pore water pressures and development of stresses in marine soils may affect each other (Wang et al., 2014; Ye et al., 2015). To overcome the shortage of existing works, a mathematical model is developed in this study with the aim of considering the coupling of re­ sidual pore water pressures and seabed stresses in a simple but workable way. In the following contents, derivations of theoretical formulations are introduced first. Then, the developed model is validated through comparison with analytical solutions and experimental data available in literature and shows its promising predictions compared to the existing

decoupled model (Sumer and Fredsøe, 2002; Sumer et al., 2012; Jeng et al., 2007). Finally, the validated framework is extended to a field-scale numerical study on the development of pore water pressure and the corresponding liquefaction susceptibility of seabed to water waves. 2. Theoretical formulations The phenomenon of storm waves propagating over a porous seabed is shown in Fig. 1. In the model development, we assume that there is a rigid and impermeable bottom and the seabed deformations are not large enough to affect the wave-seabed interactions. Storm waves are simulated as nonlinear waves. For seabed response, the normal stress is positive for tension and the pore water pressure is positive for compression. 2.1. Wave model In this paper, IHFOAM solver developed by Higuera et al. (2013a, 2013b, 2014a, 2014b), based on open source CFD toolbox OpenFOAM®, is adopted for simulation of nonlinear water waves propagating over a porous seabed. IHFOAM is a robust two-phase numerical tool applicable for prac­ tical coastal engineering. In IHFOAM, Volume averaged Reynoldsaveraged Navier-Stokes (VARANS) equations, as proposed in del Jesus et al. (2012), are solved by the modified version of interFoam solver available in OpenFOAM®. By VARANS equations, the complex nonlin­ earity interactions among different processes could be considered, and two-phase incompressible flow through porous media could be simu­ lated by averaging properties of flow within the porous media along control volumes, including k-ε and k-ω SST turbulence closure models to work inside and outside the porous media (Higuera et al., 2014a). Active wave absorption theory is implemented in IHFOAM, which greatly en­ hances the stability of wave simulation by decreasing energy of the system and prevents the mean water level from rising on long simula­ tions (Higuera et al., 2013a). The basic governing equations of IHFOAM solver will not be outlined here, for detailed information and characteristics about IHFOAM, the reader is referred to Higuera et al. (2013a, 2013b, 2014a, 2014b). In this study, the appropriate wave theory selected based on Le M�ehaut�e (1976) is used for wave generation by IHFOAM. The validity and reliability of IHFOAM solver have been indicated through comparison against laboratory measurements involving wave breaking, run up, undertow currents, and fluid-porous structure in­ teractions Higuera et al. (2013a, 2013b, 2014a, 2014b). Moreover, verifications and applications of IHFOAM solver for various coastal

Fig. 1. Definition sketch for wave-seabed interaction. 2

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Coastal Engineering 154 (2019) 103577

engineering issues have been implemented by co-authors of this paper (Zhao et al., 2018; Duan et al., 2019).

the mass of pore fluid can be expressed as: � � K ∂2 p ∂2 p ∂p ∂ε þ 2 n’ βs ¼ 2 γ w ∂x ∂z ∂t ∂t

2.2. Seabed model

where γw ¼ unit weight of water in the pore; n’ ¼ soil porosity; K ¼ permeability coefficient; ε ¼ volumetric strain; t ¼ time; and βs ¼ compressibility of the pore fluid, which is defined as follows:

Depending on the characteristics of waves and marine soils, different formulations of wave-induced porous seabed response referred as fully dynamic (FD), partly dynamic (PD) and quasi-static (QS) are possible (Ulker et al., 2009). By analytical solutions, Ulker et al. (2009) proposed the regions indicating the domains applicable to the appropriate for­ mulations in terms of seabed permeability, water depth and wave period in the shallow water range. Based on field observations, it can be known that in real ocean environment, the dominant wave period is between 8s and 15s associ­ ated with water depth from about 4 m to 70 m (Yamamoto et al., 1978; Zen and Yamazaki, 1991; Rahman, 1991; Sassa et al., 2006), indicating the low-frequency wave loading for most cases of ocean waves. More­ over, both the field and laboratory measurements have implied that, pore water pressure hardly accumulates if the permeability of porous seabed is greater than the order of 1.0 � 10 4 m/s due to the pore pressure dissipation effects (Zen and Yamazaki, 1990b, 1991; Sassa et al., 2006; Chowdhury et al., 2006). The main concern in this study concentrates on accumulation of pore pressure in marine soil under ocean waves, which associates with rela­ tively low permeability of porous seabed less than the order of 1.0 � 10 4 m/s and low-frequency wave conditions for most wave cases. According to regions of applicability of FD, PD and QS formulations presented by Ulker et al. (2009), it can be identified that the QS formulation is applicable to the main purpose of this paper in most cases. Thus, governing equations of the proposed seabed model are derived based on the QS formulation as follows.

βs ¼

(1)

∂τxz ∂σ ’z ∂p þ ¼ ∂x ∂z ∂z

(2)

where

ðσ ’x ; σ’z Þ

2.2.2. Governing equations of the coupled mathematical model In this study, we assume that the saturated porous seabed is hy­ draulically isotropic with the same permeability, K, in all directions. According to Eqs. (3) and (4) and the principle of effective stress, the total horizontal and vertical normal stresses can be written respectively as follows: � � �� ~ ∂~u μ ∂~u ∂w σ x ¼ σ ’x p ¼ 2G þ þ p (8) ∂x 1 2μ ∂x ∂z �

σ z ¼ σ’z

�

~ ~ ∂w μ ∂~u ∂w þ þ ∂z 1 2μ ∂x ∂z

�� p

(9)

(10)

~ ) ¼ soil displace­ where p ¼ wave-induced pore water pressure; (~ u, w ments corresponding to variation of pore water pressure in x- and z-di­ rections, respectively; mv ¼ coefficient of volume compressibility, which is defined as follows: mv ¼

1

2μ G

(11)

Following the reasoning presented by Seed and Rahman, (1978) and Tsotsos and Georgiadis, (1989) for two-dimensional consolidation, change of the volumetric strains Δε yields the following: Δε ¼ Δðεx þ εz Þ ¼ mv ðΔp

(12)

ψ Δt þ Δσ m Þ

or �

∂ε ∂p ¼ mv ∂t ∂t

¼ effective normal stresses in the x- and z-directions;

�

τxz ¼ G

p ¼ 2G

The mean total normal stress can be expressed as: � � � � ~ u ∂w ~ σ þ σz G ∂~u ∂w 1 ∂~ σm ¼ x ¼ þ p¼ þ p 1 2μ ∂x ∂z mv ∂x ∂z 2

�

∂w μ ∂u ∂w þ þ ∂z 1 2μ ∂x ∂z

∂u ∂w þ ∂z ∂x

�

ψþ

∂σ m ∂ðp þ σ m Þ ¼ mv ∂t ∂t

mv ψ

(13)

where ψ is the source term, representing a function giving the rate of accumulated pore-pressure generation caused by the wave-induced cy­ clic shear stresses under undrained conditions; Δσ m ¼ the pore pressure change caused by the change in mean total normal stress and Δp ¼ the change in net pore pressure. Eqs. (10) and (13) are combined to the follows: � � � � � � u ∂w ~ ~ ∂ε ∂ 1 ∂~ ∂ ∂u~ ∂w (14) ¼ mv pþ þ p mv ψ ¼ þ mv ψ mv ∂x ∂z ∂t ∂t ∂t ∂x ∂z

According to linear geometric equation and elastic stress-strain relationship in plain strain state, the effective normal stress and shear stress can be expressed as follows, respectively: � � �� ∂u μ ∂u ∂w σ ’x ¼ 2G þ þ (3) ∂x 1 2μ ∂x ∂z

�

(7)

modulus of elasticity of water which is taken as 2 � 109 ​ N=m2 (Yama­ moto et al., 1978); Sr ¼ degree of saturation; and Pow ¼ absolute water pressure.

τxz ¼ shear stress; p ¼ wave-induced pore water pressure.

σ ’z ¼ 2G

1 1 1 Sr ¼ þ Pow K ’f Kf

where K’f ¼ apparent bulk modulus of elasticity of water;Kf ¼ true bulk

2.2.1. Biot’s consolidation equations Biot’s consolidation equations (Biot, 1941) provide a general theory coupling the deformation of soil skeleton and compressibility of pore fluid in a poro-elastic medium, which have been widely used to inves­ tigate wave-seabed interactions (Yamamoto et al., 1978; Hsu and Jeng, 1994). In this study, Biot’s consolidation equations are adopted for linking the soil skeleton-pore fluid interactions under plain strain idealization. The equations governing the force equilibrium in soils can be expressed as:

∂σ ’x ∂τxz ∂p þ ¼ ∂x ∂z ∂x

(6)

�� (4)

Combining Eqs. (14) and (6) yields the following: ! � K ∂2 p ∂2 p ∂ ∂ue ∂we ’ ∂p þ n βs ¼ þ mv ψ γ w ∂x2 ∂z2 ∂t ∂t ∂x ∂z

�

�

(5)

where (u, w) ¼ soil displacement in the x- and z-directions, respectively;

(15)

in ​ which; ​ ​ ψ can be expressed as follows (Sumer and Fredsøe, 2002; Sumer et al., 2012; Sumer, 2014; Jeng, 2013, 2018):

μ ¼ Poisson’s ratio and G ¼ shear modulus of the soil.

Considering the compressibility of pore water, the conservation of 3

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�

ψ¼

∂ug σ’0 jτmax j ¼ ∂t T ασ’0

�

1=β

implementation of the hydrodynamic model IHFOAM which solves the VARANS and Volume of Fluid equations. Then, the wave pressures are applied at seabed surface as the boundary condition for simulation of seabed response. Unlike the previous decoupled seabed models, the proposed model takes into account the coupling effect between residual pore water pressures and seabed stresses by solving Eqs. (1)–(5) and (15) in an instantaneous approach, in which both the component of oscillatory and residual seabed responses can be obtained simultaneously. The numer­ ical simulation is implemented within Comsol Multiphysics environ­ ment (Comsol, 2016), which has been widely applied for wave-seabed interactions (Zhou et al., 2014; Liao et al., 2015; Zhao et al., 2016, 2017). By the PDE (Partial Differential Equation) module of Comsol Multiphysics, the mathematical model is constructed and solved using the finite element method with the second-order Lagrange elements. The mesh is refined until no evident discrepancy in the numerical calculation is achieved. For the numerical calculation of seabed response, the initial value of the maximum shear stress in the source term for the first wave period comes from the oscillatory seabed response, and then the value is updated for each associated new wave cycle according to results ob­ tained from the previous one. Other variables are also continuously updated for each wave cycle from the initial conditions. In this study, the wave-induced accumulation of pore water pressure up to occurrence of liquefaction is what we highly concerned but neglecting the post-liquefaction behavior of soil. On behalf of the ob­ servations that the residual pore pressure at a certain depth in seabed couldn’t exceed the initial, vertical effective stress at the corresponding location (Zen and Yamazaki, 1990a; Sassa and Sekiguchi, 1999; Sumer and Fredsøe, 2002; Sumer et al., 2012), magnitude of the residual pore pressure in liquefied soil will be retained the same as that initiating liquefaction, for both the present and existing models.

(16)

where ug ¼ generation of pore pressure;τmax ¼ the amplitude of cyclic shear stress in soil; and σ ’0 ¼ initial mean effective stress, which is defined as:

σ ’0 ¼

1 þ 2K0 ’ γ jzj 3

(17)

whereK0 ¼ coefficient of lateral earth pressure; and γ’ ¼ submerged unit weight of the soil. Base on the hypothesis of incompressible soil particles, Eq. (15) ex­ presses the continuity condition due to fluid mass balance, in which the left-hand side represents volume variation of pore fluid mass in unit time per unit volume soil (Biot, 1941), and the right-hand side represents volume change of soil skeleton in unit time per unit volume soil. The first term at the right-hand side of Eq. (15) is the volume change of soil skeleton caused by effective normal stress according to stress-strain relationship (Biot, 1941), and the second term is the volume change of soil skeleton caused by cyclic shear stress, first proposed by Seed and Rahman (1978) and then applied to wave-induced residual pore pres­ sure by McDougal et al. (1989), Sumer et al. (2012) and others. In Eq. (16), α and β are dimensionless curve-fitting coefficients related to soil type and relative density, which can be calculated from the following empirical expressions (Sumer et al., 2012), obtained based on the large-scale simple shear test data of De Alba et al. (1976):

α ¼ 0:34Dr þ 0:084; β ¼ 0:37Dr

0:46

(18)

where Dr is the relative density, expressed as: Dr ¼

emax e emax emin

(19)

where e ¼ void ratio; and ðemax ; emin Þ ¼ the maximum and minimum void ratios, respectively. Thus, Eq. (15) together with Eqs. (1)–(5) constitutes the governing equations of the present 2D coupled model, where (u, w) are the same as ~, w ~ ). Wave-induced pore pressure accumulation can be obtained by (u solving the governing equations with appropriate boundary conditions.

3. Validation In this section, we will validate the developed framework through comparison with analytical solutions for oscillatory soil response under waves (Ulker et al., 2009; Ulker, 2012) and the existing laboratory ex­ periments including the wave flume tests (Sumer et al., 2012) and geotechnical centrifuge wave tests (Sassa and Sekiguchi, 1999) for accumulated pore pressure, demonstrating its advantage compared to the existing decoupled model (Sumer and Fredsøe, 2002; Sumer et al., 2012; Jeng et al., 2007).

2.3. Boundary conditions Appropriate boundary conditions are necessary for accurate nu­ merical simulations. For the wave model, no slip boundary condition is imposed for velocities at the fluid-seabed interface. The incident waves are generated by the corresponding wave theories on the inlet wave­ maker boundary. At the outlet, an active wave absorption theory is used. Details of the wave generation and absorption can be found in Higuera et al. (2013a, 2013b, 2014a). At the seabed surface (z ¼ 0), the pore pressure equals to the wave pressure obtained from the wave model, and the vertical effective normal stress and shear stress vanish: pðx; 0; tÞ ¼ Pb ðx; tÞ; ​ σ ’z ¼ τzx ¼ 0

3.1. Analytical solution Concerning the oscillatory seabed response under harmonic waves, Ulker et al. (2009) developed a set of generalized analytical solution for wave-induced saturated seabed response under plane strain condition for different formulations (fully dynamic, partly dynamic, quasi-static). And then a one-dimensional analytical solution for two-layer porous soil in full dynamic form was presented by Ulker in 2012. Neglecting all the inertial terms, the corresponding analytical solution in quasi-static (QS) form can be obtained. The QS solutions are used to validate results of the numerical model developed in this paper. For comparison with analytical solution in Ulker (2012), the soil characteristics are as follows: soil permeability (K) ¼ 1.0 � 10 5 m/s; soil porosity (n’ ) ¼ 0.35; Poisson’s ratio (μ) ¼ 0.3; degree of saturation (Sr) ¼ 0.98; and the shear modulus (G) ¼ 1.15 � 107 Pa. At seabed sur­ face, wave pressure P0 cosðωtÞ is applied as boundary condition, in which P0 ¼ ðγ w HÞ=½2 coshðkdÞ � with wave height H, wave number k, and water depth d. While for comparison with analytical solution in Ulker et al. (2009), the soil properties are: soil permeability (K) ¼ 1.0 � 10 5 m/s; soil porosity (n’ ) ¼ 0.333; Poisson’s ratio (μ) ¼ 0.35; degree of saturation (Sr) ¼ 1.0; and the shear modulus (G) ¼ 5.2 � 106 Pa.

(20)

where Pb ¼ dynamic wave pressure at seabed surface determined from the hydrodynamic model IHFOAM. At the impermeable seabed bottom (z ¼ -h), no vertical flow and zero displacement are assumed:

∂p ¼ u~ ¼ w ~ ¼0 ∂z

(21)

Periodic boundary conditions are applied on the lateral boundaries. 2.4. Numerical scheme Firstly, wave pressures at seabed surface are obtained by 4

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Fig. 2 shows comparison results of the numerical model with the analytical solution by Ulker (2012) and Ulker et al. (2009), respectively, for the maximum wave-induced pore pressure, effective and shear stress. It is obvious that results of the numerical model match well with the theoretical ones overall. This proves that the finite element solution of the present soil model is reliable.

Table 1 Input data for the first validation. Characteristics

Value

Wave Water depth (d) Wave period (T) Wave height (H) Seabed Poisson’s ratio (μ)

3.2. Wave flume tests Sumer et al. (2012) experimentally investigated the development of pore water pressure in silty seabed through a series of wave flume tests. The developed numerical model is used to reproduce the wave flume tests with identical parameters tabulated in Table 1, as stated by Sumer et al. (2012). The comparison results have been shown in Fig. 3 for the time series of the pore water pressures including both oscillatory and residual components, whereas the value from the existing decoupled model is merely the period-averaged pore water pressure of total component (Sumer and Fredsøe, 2002). As seen in the figure, both of the simulated results from the existing decoupled model and 9 the present model are able to capture the trend of the experimental data. However, the present model enables a faster development of pore water pressure which fits better to the experi­ mental findings compared to the existing decoupled model, especially for those in deeper locations. This is because the coupling effect of re­ sidual pore pressures and seabed stresses is considered in the developed model, which results in larger volumetric contraction. The detailed discussions related to this phenomenon will be discussed in later sections.

0.55 (m) 1.6 (s) 0.18 (m) 0.29

Shear modulus (G)

1:92 � 106 (N/m2)

Soil permeability (K)

1:5 � 10

Soil porosity (n )

0.51

Degree of saturation (Sr) Relative density (Dr) Coefficient of lateral earth pressure (K0) Submerged unit weight of soil (γ’ )

1.0 0.28 0.42 8 140 (N/m3)

Thickness (h)

0.4 (m)

’

5

(m/s)

1.0 � 106 Pa (Sassa and Sekiguchi, 2001; Sumer et al., 2012). Fig. 4 shows the simulated results and their comparison with the experimental data in the case of progressive wave. As a comparison, the results obtained from the decoupled model have been also incorporated into the analysis. As indicated in Fig. 4, at the shallower depth of 0.75 m, both of the results from the developed model and the existing decoupled one are in good agreements with the experimental data. Whereas as the location becomes deeper, i.e. 2.5 m, the developed model gives a much better prediction in terms of the development rate of the pore water pressures which promisingly reproduce the experimental findings in comparison with the existing decoupled model. Fig. 5 shows the comparison between the simulated results and the experimental data for standing wave-induced pore water pressures. It can be seen that the development of pore pressure predicted by the present model overall fits well to the experimental data. While the re­ sidual pore pressure observed in laboratory tests is about two times of those predicted by the existing decoupled model. This phenomenon is also attributable to the consideration of the coupling between the re­ sidual pore water pressures and seabed stresses in the developed model, which leads to an increase of cyclic shear stress. Further explanations can be found in later sections. Overall, result of the developed model agrees well with the experi­ mental data, regardless of progressive or standing waves. Apparently, the proposed model promisingly replicates the development of pore water pressure in the laboratory tests, in comparison with the existing decoupled model, especially for the case of standing waves. In the following contents, the validated model will be extended to a field scale to further investigate the features of wave-induced marine soil response

3.3. Geotechnical centrifuge wave tests Sassa and Sekiguchi (1999) carried out a series of geotechnical centrifuge wave tests to investigate the liquefaction susceptibility of loose sand deposits to progressive and standing waves. The experiments were carried out using a balanced-type beam centrifuge under acceler­ ation of 50 g, where g is the acceleration of gravity. In the numerical simulation, the model setup is similar to that in the experiments, with a scaling factor of 50. The identical parameters suggested by Sassa and Sekiguchi (1999, 2001) were used here: water depth (d) ¼ 4.5 m; wave period (T) ¼ 4.5 s; seabed thickness (h) ¼ 5 m; amplitude of pore pres­ sure (P0) ¼ 6.0 kPa for progressive wave, and 7.7 kPa for standing wave; soil permeability (K) ¼ 1.5 � 10 4 m/s; soil porosity (n’ ) ¼ 0.45; relative density (Dr) ¼ 0.437 for progressive wave, and 0.412 for standing wave; Poisson’s ratio (μ) ¼ 0.3; coefficient of lateral earth pressure (K0) ¼ 0.52; and submerged unit weight of soil (γ’ ) ¼ 8 600 N/m3. For loosely packed, fine-grained sand, the shear modulus (G) is taken as

Fig. 2. Comparison of analytical solutions with present numerical model (a) Comparison with Ulker (2012) (b) Comparison with Ulker et al. (2009). 5

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Fig. 3. Comparison of the calculated data and experimental data (Sumer et al., 2012) under progressive waves (a) At depth of 8.5 cm (b) At depth of 24 cm.

Fig. 4. Comparison of the calculated data and experimental data (Sassa and Sekiguchi, 1999) under progressive waves (a) At depth of 0.75 m (b) At depth of 2.5 m.

and to explore the corresponding physical mechanisms.

shows the time series of pore water pressures at three different depths, i. e., z ¼ -1 m, -5 m and -8 m respectively. To make the comparison become more clearly, the residual component of pore water pressures from the developed model is also plotted in the figure by taking the period-averaged value of the total pore water pressures. It is found that the residual pore water pressure of the developed model increases faster compared with that of the previous decoupled model. Therefore, pore pressure of the present model at a certain depth will attain the maximum first, and may cause liquefaction at that depth. As indicated in Fig. 6, it takes about 480 s for pore pressure of the present coupled model to reach the maximum and induce liquefaction at depth of z ¼ 1 m; while pore pressure of the existing model continues to increase until catching up that of the present model. The physical mechanism of this phenomenon can be explained in the following contents. It has been well recognized that the irreversible compressive volu­ metric strain is essential for inducing the accumulation of pore water pressures (Seed and Rahman, 1978；; Sassa and Sekiguchi, 1999). The irreversible compressive volumetric strain can be caused not only by the cyclic shear stress, but also by the effective normal stress (Wang et al., 2014). Under progressive wave loadings, no additional shear stress is induced by build-up of pore pressure because of its uniformed

4. Model applications In this section, the validated model is further extended to a fieldscale to investigate the wave-induced soil response in sandy or silty seabed. A wave length Lw ¼ 109.01 m, is taken as the horizontal domain of the computational model with a thickness of h ¼ 40 m. Origin of the coordinates locates at the upper left corner of the seabed, with positive x-direction being right and z-direction being upward as shown in Fig. 1. The wave and soil parameters for the wave-seabed interactions in the numerical analysis are listed in Table 2. The geometric model and input parameters are same for both progressive and standing waves to facili­ tate comparisons. 4.1. Progressive wave-induced pore pressure accumulation Firstly, the progressive wave-induced accumulation of pore water pressure in marine soils is investigated, using simulated results from both the developed model and the previous decoupled model. For progressive waves, the accumulated pore pressures are almost horizontally uniformed (Sumer et al., 2012; Jeng and Zhao, 2015). Fig. 6 6

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Fig. 6. Accumulating process of progressive wave-induced pore water pressure at different depth.

pore pressures is obviously non-uniformed in z-direction, as indicated in Fig. 6, which leads to increase of compressive effective normal stress, resulting in a larger compressive volumetric strain and further enabling the pore pressure to accumulate faster. In other words, besides the shear stress, the normal stress can also contribute to the development of re­ sidual pore water pressure. Most of the previous models (Seed and Rahman, 1978; Sumer et al., 2012; Jeng and Zhao, 2015) have neglected the effects of normal stress on accumulation of pore water pressure. To demonstrate this mechanism more clearly, Fig. 7 illustrates the difference of mean effective normal stress between the two models (magnitude of the present model minus that of the existing decoupled model) with repetitive wave loadings, at three depths of z ¼ -1 m, -5 m and -8 m, respectively. The result has shown that the mean effective stress of the present model is smaller than that of the existing decoupled model. In view of the sign of stress (positive for tensile stress), it can be inferred that the compressive volumetric strain of the present model is greater than that of the existing decoupled model, which in turn accel­ erates development of pore water pressure. Therefore, the predicted residual pore water pressure by the present model accumulates faster than that of the existing decoupled model. From Fig. 7, it also can be

Fig. 5. Comparison of the calculated data and experimental data (Sassa and Sekiguchi, 1999) under standing waves (a) At depth of 1.5 m(b) At depth of 3.3 m. Table 2 Wave characteristics and soil properties used in numerical examples. Characteristics Wave Water depth (d) Wave period (T) Wave height (H) Seabed Poisson’s ratio (μ) Shear modulus (G)

Value 15 (m) 10 (s) 3.0 (m) 0.33 8 � 106 (N/m2)

Soil permeability (K)

1 � 10

Soil porosity (n’ )

0.42

Degree of saturation (Sr) Relative density (Dr) Coefficient of lateral earth pressure (K0) Submerged unit weight of soil (γ’ )

1.0 0.3 0.41 9 730 (N/m3)

Thickness (h)

40 (m)

5

(m/s)

distribution along horizontal direction of seabed, which means the volumetric strain rate of soil skeleton caused by cyclic shear stress of the two models are identical and constant under progressive waves. The faster development of residual pore pressure of the coupled model couldn’t be attributed to shear stress variation. However, the build-up of

Fig. 7. Difference of the mean effective stress. Magnitude of the present model minus that of the existing decoupled model. 7

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known that difference of the mean effective stress doesn’t monotonously change with depth. The difference is much greater at depth of z ¼ -5 m, which implies that at this location the pore pressure of the coupled model accumulates relatively much faster than that of the decoupled model.

the existing decoupled one. This phenomenon can be observed further by the vertical distributions of residual pore water pressures of the two models at wave cycle t/T ¼ 100, as shown in Fig. 9. It can be seen that the relative difference attains maximum at antinode, and decreases from antinode to node. The reason for this significant difference in residual pore pressure of the two models is explored as follows. As stated previously, the compressive volumetric strains induced by cyclic shear stress and normal stress are essential for inducing pore pressure to build-up. The notably non-uniform distribution of residual pore pressure in horizontal direction under standing waves leads to in­ crease of cyclic shear stress in areas between node and antinode, which results in larger compressive volumetric stain, causing a more significant build-up of pore water pressure. Since the dissipated rate of pore pres­ sure is smaller than the generated one, the pore pressure will build-up more rapidly. Therefore, besides the oscillatory cyclic shear stress at the beginning of wave loading, the rapid development of residual pore pressure also attributes to the later increase of cyclic shear stress in soils, which is different from the case of progressive waves. In addition, variation of normal stress in seabed also has a certain influence on development of residual pore pressure, as analyzed in section 4.1 for the case of progressive waves. The present model considers the coupling effect between development of residual pore pressures and evolution of soil stresses, while the existing decoupled model neglects it. Therefore, a significant difference in pore pressure accumulation can be found be­ tween the two models under standing waves. To demonstrate the mechanism above in a more obvious manner, Fig. 10 gives the time series of soil shear strains along the horizontal direction of seabed under standing wave loadings, in which the sign represents direction of shear strains. In the figure, the curve at t ¼ 0 not only represents shear strains of the present model at the beginning, but also the shear strains of the decoupled model, the amplitude of which remains unchanged during all the loading time. For the present model, the amplitude of shear strains in the region between a node and the nearest antinode has increased to some extent along with loading time from the beginning t ¼ 0, indicating increment of the maximum value of cyclic shear stress. As shown in Fig. 11, increase of the maximum value of cyclic shear stress leads to increment of the corresponding volumetric strain rate of soil skeleton, which further accelerates the development of residual pore water pressure in the associated area until the maximum pore pressure. Therefore, it can be clearly seen in Fig. 8 that at midpoint, the re­ sidual pore water pressure of the present model increased more signif­ icantly compared to the existing decoupled model. At node, although the cyclic shear strain has no increment, due to the relatively large residual

4.2. Standing wave-induced pore pressure accumulation Standing waves often occur when incident waves are completely or partially reflected by vertical walls or breakwaters (Dean and Dalrym­ ple, 1991). This wave form doesn’t propagate in any directions. Within a standing wave system, the maximum amplitude of the wave pressure appears at the anti-nodes (x/Lw ¼ 0, 12, 1, …), corresponding to wave crests or troughs, while the minimum at the usual nodal positions (x/Lw ¼ 1434, …). It should be noted that the wave pressure at x/Lw ¼ 14 and 3 4,

referred as nodes in a standing wave by linear wave theory, is not zero by nonlinear wave theory (Tsai et al., 2000). Besides nodes and anti-nodes, the points at half-way between an anti-node and its neigh­ bouring nodes (x/Lw ¼ 1838, 5878, …), are referred as midpoints in this paper for simplicity of analysis. Compared to progressive waves, residual pore pressure under standing waves shows more obviously 2D characteristics, where the residual pore pressure around node is transported toward antinode (Sassa and Sekiguchi, 2001; Kirca et al., 2013). In order to examine performance of the proposed coupled model in standing waves, pore water pressure developments at three typical locations, i.e., node, antinode and midpoint are investigated. Fig. 8 displays time series of pore water pressure at the three typical points with a fixed depth of z ¼ -5 m. It should be noted that there exist oscillatory pore pressures at nodal point because the wave pressure acting at this point is not zero for nonlinear waves (Tsai et al., 2000). As seen, not surprisingly, the development of pore water pressure in case of standing waves is significantly different along the horizontal direction of the seabed foundation (Kirca et al., 2013; Jeng and Zhao, 2015). Pore water pressures of the present model are obviously larger than those of

Fig. 9. Vertical distributions of residual pore water pressure of the two models at the 100th standing wave cycle. Results from the present model in solid lines, the existing decoupled model in dashed lines.

Fig. 8. Accumulating process of standing wave-induced pore water pressure at different horizontal position at depth of z ¼ 5 m. 8

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Fig. 10. Horizontal distributions of shear strains at different times at depth of z ¼ -5 m under standing waves by the present model.

Fig. 12. Liquefaction zones under progressive and standing waves at different times. Results from the present model in solid lines, the existing decoupled model in dashed lines(a) Progressive waves(b) Standing waves.

main liquefied area is around the nodes and midpoints, and the mini­ mum liquefaction depth occurs at antinode. In case of both progressive and standing waves, the liquefaction depth of the present coupled model is larger than that of the existing decoupled model due to the faster and larger predicted pore water pressure of the present model. For progressive waves, the predicted liquefaction depth is about 1.4 times of the decoupled model at the 25th wave cycle, and about 1.1 times at the 100th wave cycle. While for standing waves, the liquefaction depth of the present model is up to 2.3 times that of the decoupled model at the 100th wave cycle.

Fig. 11. Time evolution of the volumetric strain rate caused by cyclic shear stress in standing waves.

pore pressures in areas between node and antinode, lateral transport of the residual pore pressure from node to antinode is significantly reduced, and thus the residual pore pressure at node by the present model is obviously larger than that of the decoupled model. At antinode, the cyclic shear strain is zero all the time, so it can be inferred that the accumulated pore water pressure at this location is mainly caused by the lateral transport of residual pore water pressures from both sides of this position (Kirca et al., 2013). With the developed model, the residual pore pressures on both sides of the antinode increase significantly, resulting in faster and larger transport of pore pressures heading to antinode.

4.4. Effect of nonlinear component of wave loading In general, the use of linear wave theory can only give a first-hand approximation of wave characteristics for a limited range of wave pa­ rameters. In order to understand the influence of wave nonlinearity on residual pore pressure development, the seabed response under linear progressive and standing waves are also calculated by the present model. The wave and soil parameters used in simulation for linear waves are the same as that for nonlinear waves. Fig. 13 presents the compar­ ison of pore pressure developments between linear and nonlinear wave loadings for progressive and standing waves, respectively. It is obvious that the pore pressure accumulates faster under nonlinear waves for both progressive and standing waves, which is attributed to the greater shear stress induced by the nonlinear terms of wave pressures. For the case of progressive waves, the residual pore pressure under linear waves hasn’t caught up the one under nonlinear waves in 100 wave periods. For standing waves, the effect of wave nonlinearity on process of pore pressure build-up is more pronounced at nodal zones. Due to the faster accumulation of residual pore pressure under nonlinear wave loadings, the liquefaction depth under nonlinear waves are greater than that under linear waves for the same duration time of waves, as indicated in Fig. 14. From the figure, it can be seen that the

4.3. Liquefaction of seabed under progressive and standing waves Different process of pore pressure accumulation will lead to different effect on seabed liquefaction. It is generally accepted that if the excess pore water pressure at a certain depth in seabed exceeds the overlying effective stress, liquefaction may occur at that depth (Zen and Yamazaki, 1990a). On basis of this liquefaction criterion, liquefied zones of the present and existing decoupled model at different times under pro­ gressive waves have been shown in Fig. 12 (a). The liquefaction zone shows a 2D pattern under the oscillatory and residual excess pore water pressures. It can be clearly seen that the maximum liquefaction depth appears under wave trough due to oscillatory pore pressure. The liquefaction occurs initially at the seabed surface, and then extends downward gradually. Compared with liquefaction zone under progres­ sive waves, the standing wave-induced liquefaction zone exhibits more notably 2D characteristics, as shown in Fig. 12 (b). It is found that the 9

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Fig. 13. Comparison of pore pressure accumulation between linear and nonlinear wave loadings (a) Under progressive waves (b) Under standing waves (z ¼ -5 m).

wave nonlinearity has much more influence on development of lique­ faction for the case of progressive waves than that for the case of standing waves. At the 100th wave cycle, the difference of liquefaction depth between nonlinear and linear waves is about 2.0 m for progressive waves, while it is about 0.6 m for standing waves.

induced build-up of pore pressure in sandy or silty seabed, in which the coupling effect between the generation of pore water pressure and evolution of seabed stress is considered. The developed framework is validated through comparison with experimental data and shows better predictions compared to the existing decoupled model. Comparisons between the proposed model and previous decoupled model indicate that both the cyclic shear stress and normal stress contribute to the volumetric contraction of soil which are essential for build-up of pore pressure under water waves. Compared with the developed model, the decoupled model may significantly underestimate the development of residual pore pressures and the resulting potential of liquefaction in seabed foundations. For progressive waves, the discrep­ ancy of residual pore pressure between the two models comes mainly from the difference of normal stress. In the case of standing waves, the discrepancy of residual pore pressure is maximum at antinode, decreasing from antinode to node, mainly due to the increase of cyclic shear stress in areas between node and neighbouring antinode by the coupled model. In general, wave nonlinearity may accelerate the progress of lique­ faction in marine soils. Numerical results of the developed coupled model have shown that the nonlinear component of wave loading has more apparent influence on liquefaction depth for the case of progres­ sive waves than that for the case of standing waves.

5. Conclusions Based on Biot’s consolidation equations and the empirical relation­ ship for residual pore water pressure accumulation caused by cyclic wave loadings, a numerical model is proposed for calculating the wave-

Fig. 14. Comparison of liquefaction depth between linear and nonlinear waves. 10

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Acknowledgement

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