Analysis of seabed instability using element free Galerkin method

Transcription

1 ARTICLE IN PRESS Ocean Engineering 34 (27) Analysis of seabed instability using element free Galerkin method J.G. Wang a,, M.R. Karim a, P.. Lin b a Tropical Marine Science Institute b Department of Civil Engineering National University of Singapore 1 Kent Ridge Crescent, Singapore Singapore Received 6 September 25; accepted 11 January 26 Available online 19 April 26 Abstract Wave-induced seabed instability, either momentary liquefaction or shear failure, is an important topic in ocean and coastal engineering. Many factors, such as seabed properties and wave parameters, affect the seabed instability. A non-dimensional parameter is proposed in this paper to evaluate the occurrence of momentary liquefaction. This parameter includes the properties of the soil and the wave. The determination of the wave-induced liquefaction depth is also suggested based on this non-dimensional parameter. As an example, a two-dimensional seabed with finite thickness is numerically treated with the EFGM meshless method developed early for wave-induced seabed responses. Parametric study is carried out to investigate the effect of wavelength, compressibility of pore fluid, permeability and stiffness of porous media, and variable stiffness with depth on the seabed response with three criteria for liquefaction. It is found that this non-dimensional parameter is a good index for identifying the momentary liquefaction qualitatively, and the criterion of liquefaction with seepage force can be used to predict the deepest liquefaction depth. r 26 Elsevier Ltd. All rights reserved. Keywords: Momentary liquefaction; Shear failure; Wave loading; Pore water pressure; Element-free Galerkin method; Parametric study 1. Introduction In recent years, wave-induced seabed responses have engrossed growing interests not only in coastal engineering but also in geotechnical engineering. This is because seabed instability may have caused the damage and destruction of some coastal and offshore installations, such as breakwaters, piers, pipelines and so on (Sumer et al., 21; Jeng, 23). Two types of seabed instability may occur in a sandy seabed: momentary liquefaction and shear failure. Waveinduced momentary liquefaction was reported in laboratory tests (Sekiguchi et al., 1995; Sassa and Sekiguchi, 1999, 21) and in field observations (en and Yamazaki, 199; Sakai et al., 1992). Computational results (Madsen, 1978; Yamamoto et al., 1978; Okusa, 1985; Hsu and Jeng, 1994; Jeng and Seymour, 1997; Jeng, 21) also revealed the potential of momentary liquefaction. If momentary liquefaction occurs routinely in shallow waters, serious stability Corresponding author. Tel.: ; fax: addresses: tmswjg@nus.edu.sg, nuswjg@yahoo.com (J.G. Wang). problems have to be confronted for the structures laid on a cohesionless seabed. Because momentary liquefaction and shear failure are directly related to the excess pore pressure and effective stresses within seabed sediments, prediction of seabed responses and evaluation of seabed instability have become important issues in coastal and ocean engineering. Liquefaction and shear failure are produced by different mechanisms (en et al., 1998; Jeng, 21). The liquefaction is a state that effective stress in any direction becomes zero. For example, quick sand or boiling is closely related to vertical seepage flow. When water wave propagates over a seabed, the fluctuation of water pressure exerts on the seabed surface, causing pore fluid in the seabed to flow out or into and producing frictional force on soil skeletons (this frictional force is called as seepage force). The wave train generates the fluctuation of pore water pressure, and thus the transient fluctuation of effective stress in soil masses. If the effective stress momentarily becomes zero, soil skeleton loses its structural strength and the seabed becomes momentarily liquefied. In addition, shear stresses in a seabed may be big enough to overcome its shear resistance, resulting in another type of seabed instability, shear failure /$ - see front matter r 26 Elsevier Ltd. All rights reserved. doi:1.116/j.oceaneng

2 248 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) A shear failure refers to a state that stress level reaches to shear failure envelope which is usually described by Mohr Coulomb criterion. Wave-induced seabed response can be evaluated by Biot consolidation theory (Biot, 1941) and Verruijt s storage equation (Verruijt, 1969). Analytical and numerical methods have been employed to solve the Biot consolidation equation. Analytical solutions are usually available for those problems with simple boundary conditions. For example, Madsen (1978) investigated a hydraulically anisotropic and partially saturated seabed, whilst Yamamoto et al. (1978) studied an isotropic seabed with infinite thickness. Okusa (1985) used the compatibility equation under elastic conditions and reduced the sixth-order governing equation of Yamamoto et al. (1978) to a fourth-order differential linear equation. Yamamoto (1981) developed a semi-analytical solution for a nonhomogeneous layered porous seabed. Later, Hsu and Jeng (1994) and Jeng and Hsu (1996) further extended the framework to a finite-thickness seabed as well as a layered seabed (Hsu et al., 1995). Several numerical algorithms were proposed to accumulate complex geometry and physical conditions. For example, Thomas (1989) developed a semi-analytical onedimensional finite element model to simulate the waveinduced stresses and pore water pressure. This method was later extended to 2D and 3D wave-seabed interaction problems (Jeng and Lin, 1996; Jeng, 23). Gatmiri (199) developed a simplified finite element model for an isotropic and saturated permeable seabed. Recently, a meshless EFGM model was developed for the analysis of transient wave-induced soil responses (Karim et al., 22). In order to improve the computation efficiency, a radial PIM method (Wang and Liu, 22a, b; Wang et al., 22) was extended for wave-induced seabed responses by introducing repeatability conditions (Wang et al., 24). This radial PIM considers not only sinusoidal waves but also other nonlinear waves such as solitary wave. These numerical meshless methods provide a useful tool for further analysis of seabed instability. Seabed instability is a complicated topic in marine geotechnics (Poulos, 1988). Past studies have revealed that both soil characteristics and wave properties play a dominant role in the wave-induced seabed responses (Jeng, 23). Some parameters are proposed to assess the potential of momentary liquefaction. For example, Sakai et al. (1992) proposed two parameters to justify the occurrence of liquefaction based on a boundary layer theory (Mei and Foda, 1981). They concluded that the maximum depth of liquefaction is about half the wave height for surf conditions. However, no single parameter is available to evaluate the momentary liquefaction. Furthermore, the parameters so far are obtained for a homogeneous porous seabed. Is it suitable for a nonhomogeneous seabed? Seabed is usually non-homogeneous and its shear modulus of seabed increases with soil depth such as Gibson soil (Jeng, 23). This paper will also explore the effect of depth-variable modulus on seabed responses. This paper proposes a non-dimensional parameter to study the most critical condition governing seabed instability due to momentary liquefaction. This single parameter includes both wave characteristics and seabed properties. Numerical examples are studied to verify its effectiveness through a meshless EFGM model as well as three criteria of liquefaction. It is noted that this EFGM model has been critically examined against available analytical and/or semianalytical methods (Karim et al., 22). A parametric study, in the perspective of wave-induced soil instability, is carried out to examine the sensitivity of soil and wave properties on seabed responses. The shear failure in one-wave period is briefly discussed, too. The focus is mainly on the potential of wave-induced momentary liquefaction and liquefaction depth. This paper is organized as follows: The criteria of seabed instability, momentary liquefaction and shear failure, are first discussed in Section 2. Then Biot consolidation equation and its EFGM meshless method are briefly introduced in Section 3. A non-dimensional parameter which includes both soil and wave properties is proposed to identify the potential of momentary liquefaction in Section 4, and this parameter is validated with a finite-thickness seabed and three criteria for liquefaction in Section 5. Parametric study is carried out in Section 6. Parameters include wavelength, fluid compressibility or degree of saturation, soil permeability and Young s modulus, as well as variable shear modulus along depth. Mohr cycles of effective stress status within one wave period and shear failure mechanisms are discussed in Section 7. Finally, the conclusion and remarks are given. 2. Seabed instability under wave loading Wave-induced momentary liquefaction and shear failure are caused by different mechanisms. When a sandy seabed is subjected to cyclic wave loading, the effective stresses and pore water pressure fluctuate with the propagation of waves. When the effective stress attains to some critical value, momentary liquefaction or shear failure may occur. This section will discuss the criteria of liquefaction and the shear failure Liquefaction As discussed in the Introduction, liquefaction is a state where a soil loses its structural strength and behaves like a fluid, producing large deformation and the evolution of seabed such as ripples (Ourmieres and Chaplin, 24). Three criteria have been proposed in various reports to assess momentary liquefaction. Criterion 1: Soil is liquefied when vertical effective stress becomes zero (Yamamoto, 1981): s zz Xg bz, (1) where s zz stands for wave-induced vertical effective stress, g b ¼ðg s g w Þ is for the effective unit weight of soil and z

3 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) denotes the soil depth beneath seabed surface, in which g s and g w are the unit weights of soil and pore fluid, respectively. Eq. (1) indicates that liquefaction occurs when seepage force lifts above soil column and soil particles are no more in contact. Criterion 2: Liquefaction occurs when wave-induced effective volumetric stress in soil becomes identical or larger than the initial in situ effective volumetric stress (Okusa, 1985; Tsai, 1995): s vol Xs vol (2) where s vol is the wave-induced effective volumetric stress, and s vol the initial in-situ effective volumetric stress. These stresses are defined as s vol ¼ 1 3 ð1 þ nþðs zz þ s xx Þ; s vol ¼ 1 ð1 þ nþ 3 ð1 nþ g bz, (3) where s xx stands for wave-induced horizontal effective stress and n is the Poisson ratio. Criterion 3: Liquefaction may occur if upward seepage force is equal or larger than overburden load. This criterion is mathematically expressed as (en and Yamazaki, 199) ðp z P ÞXg b z, (4) where P z is the pore pressure at the depth z and P the pore pressure amplitude at the seabed surface Shear failure The ratio of shear stress to normal stress is an important parameter for investigating the potential of wave-induced shear failure. Shear failure occurs when stress angle f becomes equal to or greater than the angle of internal friction f u (Yamamoto, 1981): fxf u (5) For sandy soils, the stress angle f is calculated by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ððs f ¼ sin 1 xx s zz Þ=2Þ 2 þ s 2 A. (6) ððs xx þ s zz Þ=2Þ The effective stress has two components: initial in situ effective stress and wave-induced effective stress. That is, s zz ¼ s oz þ s zz, s xx ¼ s ox þ s xx. s xz is the shear stress. Initial in-situ effective vertical stress is s oz ¼ g bz, and initial in situ horizontal effective stress s ox is related to vertical effective stress s oz by s ox ¼ K s oz ¼ n g 1 n b z. (7) 3. Biot consolidation theory and variational formulations 3.1. Governing equations A seabed soil is assumed to be elastic, isotropic and homogeneous. Biot consolidation equation (Biot, 1941) and Verruijt s storage equation (Verruijt, 1969) have been applied to describe the wave-induced response of a mixture of compressible pore fluid and compressible porous seabed. These two equations are expressed as follows: Dq 2 u þ q s p þ b ¼, (8) q s qu qt nb qp qt k q 2 s p ¼, (9) g w where u is the displacement, p the pore water pressure, b the body force vector, n the porosity of soil skeleton, k the soil permeability, and g w the unit weight of pore fluid, and t as real time. The constitutive law of soils is given by s ¼ Dqu (1) For a plane strain problem, the material matrix D, and operators q s and q are given by n n E n 1 n D ¼ 6 7 ð1 þ nþð1 2nÞ 4 1 2n 5, q 2 3 qx q q q ¼ qx ; q 6 qz s ¼ q 5. ð11þ 6 4 q qz q qx 7 5 qz The compressibility of pore fluid b is a function of the degree of saturation, the bulk modulus of fluid K f, and the absolute fluid pressure P a, such as (Okusa, 1985) b ¼ 1 þ 1. (12) K f P a The repeatability conditions are used to implement periodic temporal and spatial conditions (Karim et al., 22; Wang et al., 24). Incorporating these virtual boundaries together with physical boundary conditions, a modified variational formulation was proposed for the EFGM model whose final discrete system equation is as ½RŠ½S tþ1 Š¼½FŠþ½QŠ½S t Š. (13) A Crank Nicholson scheme is adopted to discretize the time domain in Eq. (13). Appendix A gives the definitions of various coefficient matrices in Eq. (13) Moving least-square approximation Excess pore water pressure and displacements in the Biot consolidation theory are approximated by moving least square (MLS) approximants (Lancaster and Salkauskas, 1981). The MLS approximant u h ðxþ for a function uðxþ has the following form: u h ðxþ ¼ Xn f I ðxþu I, (14) I¼1

4 25 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) where n is the number of nodes I in the neighbourhood of x for which the weight function wðxþa, and u I is the nodal index of u at x ¼ x I. The shape function f I ðxþ is obtained as f I ðxþ ¼s T A 1 B I, (15) where A ¼ s T wðxþs and B ¼ s T wðxþ. For a linear basis, s j ðxþ is as s T ðxþ ¼½1 x zš in 2D space (16) Then the shape functions can be expressed as f I ðxþ ¼ Xm s j ðxþða 1 ðxþbðxþþ ji. (17) j¼ Weight function usually takes radial function as w I ðxþ wðx x I Þ¼w I ðd I Þ, where d I ¼kx x I k is the distance between two points x I and x. The size of the influence domain of x l is defined as d mi ¼ d max d I, where d max is known as support size factor (usually taken as 2.5). In this study, cubic spline is used to express the weight function for different ranges within an influence domain: wðx x I Þ d 2 I d 3 I d I þ 4 for p 1 d mi d mi d mi 2 ; >< 4 ¼ 3 4 d I d 2 I þ 4 4 d 3 I 1 for d mi d mi 3 d mi 2 o d I p1; d mi d I >: for 41: ð18þ Because the shape function in Eq. (17) has no property of Kronecker delta functions, Lagrange multiplier method is employed to implement essential boundary conditions (Belytschko et al., 1994). d mi 4. Non-dimensional parameter for the evaluation of liquefaction potential Wave-induced effective stresses at any point reach their extremes when wave crest or trough goes directly over. Momentary liquefaction is likely to occur under a wave trough due to uplift seepage force on the soil skeleton. We propose a non-dimensional parameter to assess the liquefaction potential for given soil and wave properties: k ¼ P nba 1 e 2ah, (19) g b ðm v þ nbþ 1 þ e 2ah rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ iog wðm v þ nbþ ; k m v ¼ ð1 þ nþð1 2nÞ, (2) Eð1 nþ where h is the thickness of seabed. o is the wave frequency. This non-dimensional parameter k is the ratio of the slopes for one-dimensional depth-wise effective stress profile to the initial in-situ effective stress. It can be derived from the analytical solutions of wave-induced soil responses for one-dimensional and two-dimensional problems (Karim et al., 22). In situ soil condition is presented by a and a vol as follows: a ¼ g bh ð1 þ nþ g and a vol ¼ b h. (21) P 3ð1 nþ P 5. Validation of parameter j for identifying liquefaction potential 5.1. Meshless model and computation parameters A 2-D wave-induced transient problem is defined in Fig. 1. The meshless model for soil domain is shown in Fig. 1. This domain is discretized with regular distributed nodes (441 nodes) for function approximation and regular background cells for integration. Domain integrals use 4 4 Gauss points in each background cell and 4 Gauss points in each boundary integral cell. The bottom of the soil domain is assumed to be rigid and impermeable: qp u ¼ ; ¼. (22) qn Ignoring the relative acceleration between water and soil skeleton, the boundary conditions at the seabed surface are: s z ¼ ; s xz ¼ and p ¼ P cosðwx otþ, (23) where W ¼ 2p=L is the wave number, o ¼ 2p=T the wave frequency, x the horizontal coordinate, and t is the time. L is the wavelength, and T the wave period. The amplitude P is obtained from the linear theory of a monochromatic wave (Madsen, 1976): P ¼ g wh, (24) 2coshWd w where H is the wave height and d w water depth. The dispersion equation determines the relationship among o, W and d w : o 2 ¼ Wg tanh Wd w. (25) Unless otherwise specified, parameters used in the computation are given in this section. In the fluid domain, the wave conditions for a 5 s wave with height H ¼.5 m in water depth d w ¼ 4:86 m; such given P ¼ 1:656 kn=m 2 and wavelength L ¼ 3 m. In the soil domain, the parameters of seabed soil are for soil thickness h ¼ 2 m, Young s modulus E ¼ 2:5 1 5 N=m 2, Poisson ratio n ¼ :3, porosity n ¼ :4, and isotropic permeability k ¼ 2:5 1 2 m=s. The density of the pore fluid and sea water is g w ¼ 1 kn=m 3 and the fluid compressibility of b ¼ m 2 =kn ð ¼ :97Þ. Time step size taken is 625 s for the whole wave period of 5 s in the computation. In order to better express the results, a linear normalization procedure as shown in Fig. 2 is carried out. The effective stresses and excess pore water pressure are normalized by P while the depth is normalized by the seabed thickness h. It is noted that the seabed condition is finite when hol (Hsu and Jeng, 1994).

5 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) H/2 η(x, t) = H 2 cos(a x ωt) X Fluid domain d w Γ t Γ vl Soil domain Γ vr h Γ b L Fig. 1. Wave-induced seabed response problem and its meshless model. σ z z σ z z / P Liquefied zone Liquefied zone γ b γ b P = α z h Original expression Normalized expression Fig. 2. Normalized expression of effective stress along depth Non-dimensional parameter k When the line representing effective self-weight (called as a-line) intercrosses with the effective stress lines, the soil above the intercrossing point is regarded as liquefied and the depth of crossing point is the liquefaction depth because the liquefaction criterion of Eq. (1), (2) or (4) is satisfied in this zone. At this time, the kx1. The parameter k is obtained through changing fluid compressibility (degree of saturation), Young s modulus, and permeability of seabed soil, respectively, based on those parameters in Section 5.1. Fig. 3 shows that all liquefaction criteria

6 252 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) e e σ zz / P σ vol / P Non-dimensional κ 6.85 α = Non-dimensional κ α vol = e-4 Non-dimensional κ α = 5 (c) (P z -P )/P Fig. 3. Implication of non-dimensional parameter k. (Eqs. (1) (4)) are satisfied if the value of k is equal to or greater than unity. In other words, the seabed soil is liquefied if kx1. Ideally, when ko1, the liquefaction may not occur and precaution for protection may not be necessary for the seabed or offshore structure against possible damage induced by liquefaction. Fig. 4 and show the variations of soil responses with k at the depth of 1.5 m. The soil response enhances exponentially with k. Therefore, the condition kx1 can be used as a criterion to predict the vulnerability of momentary liquefaction without the details of the wave-induced soil response. Detail analysis reveals that the three criteria mentioned in Eq. (1) (4) perform slightly different in the assessment of liquefaction potential. Criterion 2 always predicts the occurrence of liquefaction at the seabed surface because it is expressed by volumetric effective stress including horizontal effective stress. The parameter k is obtained from a one-dimensional formulation, thus it does not take the effect of horizontal effective stress into account, consequently, the condition that kx1 might overestimate the liquefaction zone at seabed surface Verification of numerical algorithm with centrifuge test data Experimental results of two centrifuge tests 1 (Wang and Lin, 24) are used to verify the numerical algorithm in this 1 Centrifuge test was carried out by Cheng Chen for his Master thesis. His work is appreciated. paper. Table 1 lists the computational parameters obtained from experiments. These parameters are used for meshless method, and the analytical solutions. Fig. 5 is the comparison of excess pore water pressure predicted by meshless method, Madsen s solution (1978), and Hsu and Jeng s solution (1994). The experimental data obtained by Centrifuge tests are also plotted for comparison. They generally agree well in the whole seabed whether the seabed soil is fine sand or coarse sand. It is noted that the pore water pressure predicted by meshless method is between those of Madsen s solution and Hsu and Jeng s solution. 6. Parametric study for identifying liquefaction potential 6.1. Effect of wavelength This section reports the effect of wavelength on soil response. Wavelength varies from site to site. For example, Yamamoto et al. (1978) took the design wavelength for North Sea as 324 m, while Jeng (23) used the value of 2 m in his analysis. In this study, the wavelength is assumed to vary from 1 to 18 m which corresponds to a reasonable range of wave periods (Demirbilek and Vincent, 22) according to the dispersion equation of Eq. (25) at the water depth of 4.86 m. Typical seabed response for wavelengths of 4 1 m is shown in Fig. 6 for vertical effective stress, Fig. 6 for volumetric effective stress, and Fig. 6(c) for vertical seepage force. These curves have almost the same slopes before maximum values, and the slopes are not affected by the variation in wavelength L.

7 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) Fig. 7 further compares the vertical effective stress for onedimensional and two-dimensional conditions with three wavelengths (2, 4, and 16 m). Take vertical effective stress as an example for detailed analysis. The waveinduced maximum vertical effective stress occurs at 8 h for L ¼ 1 m,.143 h for L ¼ 2 m and 3 h for LX4 m. Thomas (1995) obtained the maximum soil response at.15l depth if the seabed is deep enough, i.e., h L. Yamamoto compared the North Sea data with his analytical solution for a seabed in infinite thickness. He concluded that the most unstable bed thickness varied between L (Yamamoto et al., 1978) and 5L (Yamamoto, 1981). For a finite seabed, our results show that maximum soil responses are more likely one-dimensional within some depth near surface. The depth increases with wavelength until some value. When wavelength exceeds this value, for example two times of seabed thickness in our study, the maximum response is independent of wavelength. As indicated in Fig. 2, the maximum liquefaction depth is determined by taking the intersection.8.7 At z = 1.5 m /P σ zz.5.4 Actual soil response A 2 + (A 1 -A 2 )/(1 + exp((x-x )/Δx)) A 1 = -9898; A 2 = Depth (m) -6-8 Meshless method Madsen's solution Hsu & Jeng's solution Centrifuge data.3 x = ; Δx = Liquefied one κ P/ P o.5 At z = 1.5 m -2.4 σ vol /P.3.1 Actual soil response A 2 + (A 1 -A 2 )/(1 + exp((x-x )/Δx)) A 1 = -955; A 2 =.4421 x = ; Δx = Depth (m) Meshless method Madsen's solution Hsu & Jeng's solution Centrifuge data -.1 Liquefied one κ Fig. 4. Variation of effective stresses versus non-dimensional parameter k P/ P o Fig. 5. Comparison of pore water pressure for different sandy seabeds. Fine sand, coarse sand. Table 1 Computation parameters for sand and wave Seabed type Parameters of sandy bed Wave parameters n n E (Pa) k (m/s) b h (m) T (s) d w (m) Fine sand Coarse sand

8 254 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) α vol = L = 4 m L = 6 m L = 8 m L = 1 m α = 4 α = L = 4 m L = 6 m L = 8 m L = 1 m α vol = σ zz / P σ vol / P.1 α = L = 4 m L = 6 m L = 8 m L = 1 m α = (c) (P z -P )/ P Fig. 6. Effect of wavelength on wave-induced soil response. of the stress profile (s zz, s vol or (P z P )) with the initial effective stress line (a-line or a vol -line). Therefore, onedimensional analysis may suffice to identify seabed liquefaction, especially when Criteria 1 and 3 are considered. Because volumetric effective stress s vol includes the horizontal effective stress and is difficult to obtain accurately close to seabed surface, Criterion 2 always predicts the liquefied status at the surface. Within the zone close to seabed surface, both momentary liquefaction and shear failure may occur, and the later mechanism may turn out to be more important D (Independent of L) 2-D (L = 2 m) 2-D (L = 6 m) 2-D (L = 16 m) 6.2. Effect of fluid compressibility The degree of saturation has been recognized as a dominant factor for the wave-induced seabed response. Pore water in seabed soils is compressible due to gas bubbles (Okusa, 1985; Thomas, 1989; Jeng and Lin, 1996). The structure of an unsaturated marine soil can vary significantly depending on the relative size of gas bubbles to soil particles. The in-site degree of saturation of unsaturated marine sediments normally lies on the range of 85 1% (Esrig and Kirby, 1977; Pietruszczak and Pande, 1996). The compressibility b in Eq. (12) is assumed to vary from to m 2 /kn which corresponds to ¼ 1:2:9. The maxima of s zz, s vol and (P z P ) increase as the degree of saturation ( ) decreases. Fig. 8 indicates the effect of the degree of saturation on vertical seepage force. It reveals that unsaturated soil is more vulnerable to liquefaction. The mechanism for this fragility to liquefaction is complicated. Fluid compressibility increases the σ zz / P Fig. 7. Comparison of soil responses for 1-D and 2-D problems. absorbing rate of wave energy in this surface zone. This prevents the pore fluid pressure from infiltrating easily into subsurface layers and produces a phase lag in pore pressure response near the surface zone (see Fig. 9). This phase lag increases when degree of saturation decreases. Due to this phase difference, the soil response at any given time could be greater than the initial load, thus enhancing the possibility towards liquefaction. Therefore, wave-induced soil response is sensitive to the fluid compressibility or degree of saturation. Numerical results again reveal that Criterion 2 overestimates liquefaction potential, and that Criterion 3 or ðp z P ÞXg b z predicts deepest liquefaction zone. Therefore, Criterion 3 is the most critical one.

9 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) =.9 =.97 =.975 =.98.3 =.985 =.99 S.4 r =.995 = = α = 4 α = 6 (P z -P )/ P Fig. 8. Effect of fluid compressibility on normalized seepage force k = 2.5e-3 m/s k = 5e-3 m/s k = 7.5e-3 m/s k = 1e-2 m/s k = 2.5e-2 m/s k = 5e-2 m/s k = 7.5e-2 m/s k = 1e-1 m/s k = 1.5e-1 m/s α = 6 α = 4 k = 2.5e-1 m/s k = 9.5e-1 m/s (P z - P )/ P Fig. 1. Effect of soil permeability on normalized seepage force. P/ P At z =.5 m =.9 =.97 =.975 =.98 =.985 =.99 S =.995 r Applied surface pressure t / T Fig. 9. Phase shift due to variation in the degree of saturation..1 E = 2.5e8 N/m 2 E = 2.5e7 N/m 2.3 E = 1e7 N/m 2 E = 2.5e6 N/m 2.4 E = 1.5e6 N/m 2 E = 1e6 N/m 2.5 E = 7.5e5 N/m 2 E = 5e5 N/m 2.6 E = 2.5e5 N/m 2 α = 6 α = 4 E = 2.5e4 N/m (P z -P )/ P Fig. 11. Effect of Young s modulus on normalized seepage force Effect of soil permeability Soil permeability is assumed to vary between 2:5 1 1 m=s (gravel) and 2:5 1 7 m=s (clay). The effect of permeability on seepage force, (P z P ), is shown in Fig. 1. It indicates that the seabed response is sensitive to the permeability k. The seepage force decreases with permeability. When the permeability is low, the seabed is more vulnerable to liquefaction. The maximum response of pore water pressure occurs at deeper position when soil permeability is higher. Again, Criterion 3 predicts the deepest liquefaction zone Effect of Young s modulus Young s modulus is assumed to vary between and N/m 2 but keeps constant along depth. This range is suitable for a wide range of soil masses (Jeng and Lin, 1996). The seepage force is shown in Fig. 11. In the shallow zone, Young s modulus E has almost no effect on vertical effective stress s zz and vertical seepage force (P z P ). The maximum effective stress increases with Young s modulus E. This implies that the seepage force becomes higher and the soil mass may become more susceptible to liquefaction when E increases. When Young s modulus E is very large ( N=m 2 such as gravels), the maximum response (s zz, s vol or (P z P )) is not affected. At this stage, a seabed can be regarded as rigid porous medium for the analysis of liquefaction. Again, Criterion 3 is the most critical one because it predicts the deepest liquefaction zone Effect of variable shear modulus Variable shear modulus along depth is a feature of seabed soil in ocean engineering and has been studied by many researchers (Thomas, 1989; Lin and Jeng, 2). A typical distribution of shear modulus along depth is shown

10 256 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) Depth, z (m) Variable G Constant G Approx. variable G Shear Modulus, G (kn/m 2 ) Variable G Constant G (P z - P )/ P Fig. 12. Effect of variable shear modulus on normalized seepage force. in Fig. 12. Here the effect of variation of shear modulus on seabed response is studied. For comparison, an equivalent constant modulus (called constant modulus), which has the same area over the entire thickness, is also used. The meshless method approximates this variable shear modulus with stepwise constants over background cells. Typical response is shown in Fig. 12 for excess pore water pressure. The contours of effective stresses are compared in Fig. 13 (c), where solid lines are for the variable modulus and dashed lines are for constant modulus. Vertical effective stress is larger for variable modulus than for constant modulus, and horizontal effective stress is more sensitive than vertical effective stress. The maximum response occurs at deeper zone for variable modulus, and the liquefaction depth is larger for variable modulus. 7. Stress angle under wave loading Shear failure may occur in the seabed. The stress angle f is used to describe the mobilization of soil shear strength. Seabed is only stable when fof u. It is noted that the angle of internal friction f u is between 21 and 31 for sandy seabed (Poulos, 1988). The stress angle at each node is computed with Eq. (5). Typical Mohr circles within a wave period are shown in Fig. 14. The line AOB (f u line) passes through the crown point of the Mohr circle, and f u of u. When the crown envelope crosses the line AOB, part of the seabed may be subject to shear failure. However, a soil is liquefied when the instantaneous stress at the horizontal plane reaches the point O or the stress crown is on the line COD, i.e. s 3 ¼ for Criterion 1 and :5ðs 1 þ s 3Þ¼ for Criterion 2. Because a liquefied soil behaves like fluid, the stress status cannot be obtained by Biot s consolidation equation. Theoretically, the crown envelope cannot go beyond the line COD as indicated in the current elastic analysis of Fig. 14. The wave-induced seabed instability may be induced by a complex coupled process combining shear failure with momentary liquefaction. Once shear failure occurs, seabed soil becomes highly nonlinear and the current theory is inappropriate to deal with the situation. Mohr circles are also drawn at two particular depths for different degrees of saturation as shown in Fig. 14 (c). The wave and seabed parameters are the same as those in Section 6.3 with a ¼ 4. These Mohr circles correspond to the maximum vertical effective stress at that point. If Criterion 1 is used, a soil is liquefied because the minor principal stress is zero or negative. Fig. 15 shows a typical relationship of shear failure depth and liquefaction depth when a ¼ 4. It can be seen that shear failure occurs at the surface and is shallower than that for the liquefaction (en et al., 1998). Shear failure occurs before liquefaction if internal frictional angle is f u ¼ 31. According to Criterion 2, a seabed soil is always liquefied near seabed surface, and thus protection work for seabed surface, such as covering the seabed by a layer of concrete blocks or rubble, is necessary (Jeng, 21). 8. Concluding remarks Wave-induced seabed instability, both momentary liquefaction and shear failure, is studied under various soil and wave properties. A non-dimensional parameter is proposed to evaluate liquefaction potential. The response of a seabed with finite thickness is numerically studied when a twodimensional progressive wave is applied on the surface of seabed. Parametric study on soil and wave properties is carried out and their effects on the seabed responses and liquefaction potential are analyzed. From these studies, following conclusions can be made. Momentary liquefaction may occur within the shallow zone of a seabed and the non-dimensional parameter k can be used to identify the momentary liquefaction. The seabed is likely liquefied if kx1 for any one of the three criteria of liquefaction. Three criteria of liquefaction, which are based on vertical effective stress, effective volumetric stress and dynamic excess pore pressure or seepage force, respectively, are discussed for the identification of soil liquefaction. For the same soil and wave properties, Criterion 3 (for seepage force) predicts the deepest liquefaction zone and Criterion

11 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) Vertical effective stress Horizontal distance (m) Vertical distance (m) Horizontal effective stress Horizontal distance (m) Shear stress (c) Horizontal distance (m) Vertical distance (m) Vertical distance (m) Fig. 13. Comparison of stress contours for variable and constant shear modulus. P (k N/m 2 ) P (k N/m 2 ) P (k N/m 2 ) 2 is the least critical one. Criterion 2 always predicts soil liquefaction at the seabed surface. Therefore, Criterion 3 becomes the most critical condition for liquefaction. The sensitivity of wave and seabed properties is different in the evaluation of liquefaction potential. Within the shallow zone, wavelength has almost no effect on the maximum seabed response. Seabed response is similar to that in one-dimensional case within the shallow depth near seabed surface. As an approximation, one-dimensional analysis suffices for the identification of soil liquefaction. However, seabed characteristics have dominant effects on wave-induced seabed response. Among all the soil parameters described, compressibility of pore fluid (degree of saturation) is the most critical one. The higher the fluid compressibility is, the more vulnerable condition for the occurrence of soil liquefaction. The coefficient of permeability also plays an important role. The lower the permeability is, the more vulnerable to soil liquefaction. In the shallow zone near the seabed surface, Young s modulus of soil skeleton has almost no effect on vertical effective stress and excess pore pressure, but has some effect on effective volumetric stress. If Young s modulus is very high ( N=m 2 ), the effect on soil response may be ignored and the seabed can be regarded as a rigid one. Such simplification can predict vertical effective stress and excess pore pressure with reasonable accuracy. However, the predicted effective volumetric stress is slightly larger. Variable shear modulus predicts bigger maximum response and deeper liquefaction zone. Therefore, variable shear modulus along depth has to be considered.

12 258 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) Envelop of crown of stresses as time passes D B z = 1 m, α = 8, * δ =.4135, κ = φ u = 3.1 =.9 =.97 S =.975 r S =.98 r =.985 =.99 S =.995 r z = 1 m, α = 4 φ u = 3 σ xz / P t = T/2 A C O In-situ stress conditions t = σ xz / P σ / P σ / P z = 2 m, α = 4 φ u = 3.1 σ xz / P (c) =.98 S =.9 r S =.985 r -.3 =.97 S =.99 r S =.975 r -.4 = σ / P Fig. 14. Shear failure status by Mohr Coulomb failure criterion Maximum liquefaction depth for α = 4 φ u = 2 φ u = 3 φ u = 4 L = 4 m L = 6 m L = 8 m L = 1 m φ (degrees) Fig. 15. Vertical distribution of stress angle. Stress angle is another important parameter leading to seabed instability due o shear failure. Shear failure may occur near and at the surface. The stress angle of soil has nothing to do with the liquefaction except for causing shear failure. Elastic analysis indicates that shear failure takes place before momentary liquefaction. Once a soil failed in shear, soil deformation becomes highly nonlinear. Therefore, the present linear theory would not be appropriate for the prediction of further failure. It is then necessary to employ the transition mechanism from shear failure to liquefaction as a progressive process. Acknowledgement This work is financially supported by the US Office of Navy Research under grant number N Appendix A 2 K L G ðg vl G vr 3 Þ L T ðm ydthþ ydtg ðg vl G vr ÞDty G T ½RŠ ¼ G T, (A.1) 6 4 ðg vlt G vrt Þ 7 5 ðg vlt G vrt Þ

13 ARTICLE IN PRESS 2 3 L T ðm þ Dtð1 yþhþ Dtð1 yþg DtðG vl G vr Þð1 yþ ½QŠ ¼, (A.2) h i ½S tþ1 Š T ¼ u tþ1 p tþ1 l tþ1 1 l tþ1 2 l tþ1 3 l tþ1 4, (A.3) ½S t Š T ¼ u t p t l t 1 l t 2 l t 3 l t 4, (A.4) h i ½FŠ T ¼ f u Dtf p f l1 f l2. (A.5) The superscript ðt þ 1Þ denotes the current time ðt þ DtÞ. The repeatability conditions create two virtual boundaries at both ends, as denoted by G nl and G nr (u nl ¼ u nr and p nl ¼ p nr ). Other notations in Eqs. (A.1) (A.5) are given by K IJ ¼ B T I DB J do; L IJ ¼ f I A J do; M IJ ¼ nb f I :f J do; H IJ ¼ k A T I g :A J do, w O G IK ¼ N K f I dg; G u G vl IK ¼ N K f I dg; G vl f ui ¼ t:f I dg þ G s f l1 I ¼ N K u dg; G u f I;z O O G IK ¼ N Kf I dg; G p G vl O G vl IK ¼ IK G ¼ N K f I dg, vr b:f I do; f pi ¼ j:f I dg, G j f l2 I ¼ N K p dg, G p f I;x G vl N K f I dg; 2 3 f I;x " 6 B I ¼ qðf I Þ¼4 f 7 I;z 5; A I ¼ f # " I;x ; N f K ¼ N # k ; N K I;z N ¼½N kš. k O G vr IK ¼ G vr N K f I dg, Following boundary conditions are also used during the variational formulation: uðx; tþ ¼uðx; tþ on G u and pðx; tþ ¼pðx; tþ on G p, s: ^nðx; tþ ¼tðx; tþ on G s and k g w qp q ^n ðx; tþ ¼jðx; tþ on G j, J.G. Wang et al. / Ocean Engineering 34 (27) j and t indicate pore water flux and traction, respectively. ^n is the unit normal to boundary G s, G u, G p, G s and G j are the boundaries where displacement, pore water pressure, total stress and flux of pore water are prescribed. Obviously, they satisfy the following relations: G u [ G s ¼ G and G u \ G s ¼ +; G p [ G j ¼ G and G p \ G j ¼ +. References Belytschko, T., Lu, Y.Y., Gu, L., Element-free Galerkin methods. Internationational Journal for Numererical Methods in Engineering 3, Biot, M.A., General theory of three-dimensional consolidation. Journal Applied Physics 12, Demirbilek,., Vincent, L., 22. Water Wave Mechanics. usace.army.mil/inet/usace-docs/eng-manuals/em.htm, No: EM , Part II. Esrig, M.I., Kirby, R.C., Implications of gas content for predicting the stability of submarine slopes. Marine Geotechnology 17, Gatmiri, B., 199. A simplified finite element analysis of wave-induced effective stress and pore pressures in permeable sea beds. Geotechnique 4 (1), Hsu, J.R.C., Jeng, D.S., Wave-induced soil responses in an unsaturated anisotropic seabed of finite thickness. International Journal of Numerical Analytical Methods in Geomechanics 18,

14 26 ARTICLE IN PRESS J.G. Wang et al. / Ocean Engineering 34 (27) Hsu, J.R.C., Jeng, D.S., Lee, C.P., Oscillatory soil response and liquefaction in an unsaturated layered seabed. International Journal of Numerical Analytical Methods in Geomechanics 19 (12), Jeng, D.S., Hsu, J.R.C., Wave-induced soil response in a nearly saturated seabed of finite thickness. Geotechnique 46 (3), Jeng, D.S., Lin, Y.S., Finite element modeling for water waves-soil interaction. Soil Dynamics and Earthquake Engineering 15, Jeng, D.S., Seymour, B.R., Response in seabed of finite depth with variable permeability. Journal of Geotechnical and Geoenvironmental Engineering, ASCE 123 (1), Jeng, D.S., 21. Mechanism of the wave-induced seabed instability in the vicinity of a breakwater: a review. Ocean Engineering 28, Jeng, D.S., 23. Wave-induced sea floor dynamics. Applied Mechanics Review 56 (4), Karim, M.R., Nogami, T., Wang, J.G., 22. Analysis of transient response of saturated porous elastic soil under cyclic loading using element-free Galerkin method. International Journal of Solids and Structures 39 (6), Lancaster, P., Salkauskas, K., Surfaces generated by moving least squares methods. Mathamatics of Computing 37, Lin, Y.S., Jeng, D.S., 2. Short-crested wave-induced liquefaction in porous seabed. Journal of Geotechnical and Geoenvironmental Engineering, ASCE 126 (5), Madsen, O.S., Wave climate of the continental margin: elements of its mathematical description. Marine Sediment Transport and Environmental Management. Wiley, New York, pp Madsen, O.S., Wave-induced pore pressures and effective stresses in a porous bed. Geotechnique 28 (4), Mei, C.C., Foda, M.A., Wave-induced responses in a fluid-filled poroelastic solid with a free surface a boundary layer theory. Geophysical Journal Royal of Astronomical Society 66, Okusa, S., Wave-induced stresses in unsaturated submarine sediments. Geotechnique 35 (4), Ourmieres, Y., Chaplin, J.R., 24. Visualizations of the disturbedlaminar wave-induced flow above a rippled bed. Experiments in Fluids 36, Pietruszczak, S., Pande, G.N., Constitutive relations for partially saturated soils containing gas inclusions. Journal of Geotechnical Engineering, ASCE 122 (1), Poulos, H.G., Marine Geotechnics. Unwin Hyman Ltd, London. Sakai, T., Hatanaka, K., Mase, H., Wave-induced effective stresses in seabed and its momentary liquefaction. Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE 118 (2), Sassa, S., Sekiguchi, H., Wave-induced liquefaction of beds of sands in a centrifuge. Geotechnique 49 (5), Sassa, S., Sekiguchi, H., 21. Analysis of wave-induced liquefaction of sand beds. Geotechnique 51 (2), Sekiguchi, H., Kita, K., Okamoto, O., Response of poreelastoplastic beds to standing waves. Soils and Foundations 35 (3), Sumer, B.M., Whitehouse, J.S., Torum, A., 21. Scour around coastal structures: A summary of recent research. Coastal Engineering 44, Thomas, SD., A finite element model for the analysis of waveinduced stresses, displacements and pore pressures in an unsaturated seabed, I: Theory. Computers and Geotechnics 8, Thomas, S.D., A finite element model for the analysis of waveinduced stresses, displacements and pore pressures in an unsaturated seabed, II: Model verification. Computers and Geotechnics 17, Tsai, C.P., Wave-induced liquefaction potential in a porous seabed in front of a breakwater. Ocean Engineering 22 (1), Verruijt, A., Elastic storage of aquifers. Flow through porous media. Academic Press, New York, pp Wang, J.G., Liu, G.R., 22a. A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering 54 (11), Wang, J.G., Liu, G.R., 22b. On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Computer Methods in Applied Mechanics and Engineering 191 (23 24), Wang, J.G., Liu, G.R., Lin, P., 22. Numerical analysis of Biot s consolidation process by radial point interpolation method. International Journal of Solids and Structures 39 (6), Wang, J.G., hang, B.Y., Nogami, T., 24. Wave-induced seabed response analysis by radial point interpolation meshless method. Ocean Engineering 31 (1), Wang, J.G., Lin,.P., 24. Wave-induced mine burial into seabed: Part 1 Cohesionless seabed in cyclic liquefaction state. Final report on ONR Project, National University of Singapore. Yamamoto, T., Koning, H.L., Sellmeijer, H., Hijum, EV., On the response of a poro-elastic bed to water waves. Journal of Fluid of Mechanics 87 (part 1), Yamamoto, T., Wave-induced pore pressures and effective stresses in inhomogeneous seabed foundations. Ocean Engineering 8, en, K., Yamazaki, H., 199. Mechanism of wave-induced liquefaction and densification in seabed. Soils and Foundations JSCE 3 (4), en, K., Jeng, D.S., Hsu, J.R.C., Ohyama, T., Wave-induced seabed instability: difference between liquefaction and shear failure. Soils and Foundations, JSCE 38 (2),