Effects of Wave Non-Linearity on Residual Pore Pressures in Marine Sediments

A better understanding of the wave-induced pore pressure accumulations (i.e., residual pore pressure) is a key factor in the analysis of the wave-induced liquefaction in marine sediments. In this paper, the residual mechanism of nonlinear wave-induced pore water pressure accumulation in marine sediments is examined. Unlike previous investigations, the second-order Stokes wave theory is considered in this study. The new model is verified with experimental data and provides a better prediction of pore pressure accumulation than the previous solution with linear wave theory. The parametric study concludes that the influences of wave non-linearity increase as the following parameters increase (i) wave steepness (H/L) and (ii) residual parameter ( ). However, an opposite trend is found for (i) wave period (T), (ii) relative water depth (d/L), (iii) seabed thickness (h/L) and (iv) another residual parameter ( ). Furthermore, the effect of wave non-linearity becomes more significant in soft seabed (Soil C). INTRODUCTION The evaluation of the wave-induced soil response in marine sediments is particularly important for coastal engineers involved in the design of foundation of many marine installations, e.g., offshore mono-piles, breakwaters, pipelines and platforms etc. The prediction of the wave-induced excess pore pressure is a key procedure in the analysis of seabed instability such as liquefaction and scour. Therefore, it is necessary to have a better understanding of the mechanism of the wave-induce pore pressure in marine sediments. Two mechanisms for wave-induced pore pressure have been observed in the previous field measurements and laboratory experiments [1], as shown in Fig. (1), The first mechanism is resulted from the transient or oscillatory excess pore pressure and is accompanied by attenuation of the amplitude and phase lag in the pore pressure changes [2, 3]. The second mechanism is termed the residual pore pressure, which is the build-up of excess pore pressure caused by contraction of the soil under the action of cyclic loading [4]. Numerous studies for wave-induced momentary liquefaction, caused by oscillatory pore pressure, have been carried out since the 1970s. Among these, Yamamoto et al. [4] proposed a closed-form analytical solution for an infinite seabed. Mei and Foda [5] proposed a boundary-layer approximation to derive a rather simplified formulation for the wave-induced transient pore pressure, which is only valid for coarse sand [6, 7]. Jeng [3, 6-9] derived a series of analytical solutions for the oscillatory pore pressure within marine sediments. His models considered various soil behaviors, such as cross-anisotropic soil behaviors and variable permeability. Later, Kianoto and Mase [10] and Yuhi and Ishida *Address correspondence to this author at the Division of Civil Engineering, School of Engineering, Physics and Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK; Email: [email protected] [11] further suggested a new and simplified formulation for the wave-induced pore pressure in a cross-anisotropic seabed. An intensive review of the previous research for the wave-induced oscillatory pore pressure can be found in [12]. Fig. (1). Mechanisms of oscillatory and residual pore pressures in marine sediments. The mechanism of pore pressure build-up due to ocean waves has been studied by many researchers since the 1970s. Seed and Rahman [4] established a simple one-dimensional finite element model by taking into account the distribution of cyclic shear stresses in the soil profile, and pore-pressure dissipation. Sekiguchi et al. [13] proposed an elasto-plastic model for the standing wave-induced liquefaction using a Laplace transformation. Later, some numerical models for post-liquefaction and progressive liquefaction and densification in marine sediments were developed [14]. In addition to numerical modeling, McDougal et al. [15] proposed a set of analytical solutions for wave-induced pore pressure build-up in a uniform layer of soil, based on the assumption of an incompressible soil. In their approach, the source term in the modified Biot's consolidation equation is derived using a linear relationship between pore pressure 64 The Open Civil Engineering Journal, 2008, Volume 2 Dong-Sheng Jeng ratio ( 0 / g u ) and cyclic ratio ( 0 / ) [16, 17]. To provide a convenient practical result for engineers, McDougal et al. [15] proposed three solutions for the cases of shallow, finite and deep soil depths, respectively. These analytical solutions are useful for both engineers and researchers, as they can be used for either the investigation of qualitative behaviors of complicated engineering problems or the validation of numerical methods. Recently, using a similar approach, Cheng et al. [18] re-examined the analytical solution of McDougal et al. [15] and proposed a numerical model to investigate the same problem. As pointed out by Cheng et al. [18], the analytical solution proposed by McDougal et al. [15] revealed some errors in the formulations. However, after a close examination of both the McDougal et al. [15] and Cheng et al. [18] solutions, the author found numerous errors in both publications, as reported in [19-21]. Recently, a series of analytical approximations for the wave-induced accumulated pore pressure in marine sediments have been proposed by the author [19-21]. Both cases of infinite and finite soil layers were considered in the models. A simplified universal formula was proposed for the case of infinite seabed [19]. However, all these approximations were limited to linear regular wave loadings, although it should be non-linear waves in natural environments. In this paper, the models developed by the author [19-21] are further extended to non-linear wave loadings. The new non-linear wave model will be verified by comparing the previous experimental data, together with the previous linear wave model. Based on the present model, we will examine the effects of wave steepness and residual parameters on the wave-induced accumulated pore pressure in a porous seabed. THEORETICAL FORMULATIONS Non-Linear Wave Theory Herein, a soil matrix subject to a two-dimensional progressive waves system is considered, as depicted in Fig. (2). The wave crest propagates in the positive x-direction, while the z-axis is positive upward from the seabed surface, as shown in Fig. (2). Fig. (2). Sketch of wave-seabed interaction. Referring to a wave theory to the second-order [22], the velocity potential ( ) is given as 2 cosh sin( ) 2 cosh 3 cosh 2 sin 2( ) 32 cosh 2 gH kz kx t kd H kd kx t kd = (1) where H is the wave height, d is water depth, g is the gravitational acceleration and t is the time. The wave number (k) and wave frequency ( ) can be determined with the following wave dispersion relation: 2 tanh gk kd = (2) The water surface displacement to the second-order ( ) can be expressed as