Wave dispersion equation in a porous seabed

The wave dispersion equation has played a very important role in the development of ocean surface wave theories. The evaluation of the length of a water wave is an essential example of solving the dispersion relation. Conventional ocean wave theories have been based on an assumption of a rigid impermeable seabed. Thus, the conventional wave dispersion equation can only be used in the case of a wave propagating over a rigid impermeable seabed. For waves propagating over a porous seabed (such as a sandy bed), the conventional dispersion relation is no longer valid because of the absence of the characteristics of the porous seabed. The objective of this study is to establish a new wave dispersion equation for waves propagating over a porous seabed. Based on the new relation, the effects of a porous seabed on wave characteristics (such as the wavelength and wave profile) are discussed in detail.     

Numerical study for waves propagating over a porous seabed around a submerged permeable breakwater: PORO-WSSI II model

The phenomenon of the wave, seabed and structure interactions has attracted great attentions from coastal geotechnical engineers in recent years. Most previous investigations have based on individual approaches, which focused on either flow region or seabed domain. In this study, an integrated model (PORO-WSSI II), based on the Volume-Averaged/Reynolds-Averaged Navier-Stokes (VARANS) equations and Biot's poro-elastic theory, is developed to investigate the mechanism of the wave-permeable structure-porous seabed interactions. The new model is verified with the previous experimental data. Based on the present model, parametric studies have been carried out to investigate the influences of wave, soil and structure parameters on the wave-induced pore pressure. Numerical results indicated: (i) longer wave period and larger wave height will obviously induce a higher magnitude of pore pressure at the leading edge of a breakwater; (ii) after a full wave-structure interaction, the magnitude of pore pressure below the lee side of a breakwater decreases with an increasing structure porosity while it varies dramatically with a change of structure height; and (iii) the seabed thickness, soil permeability and the degree of saturation can also significantly affect the dynamic soil behaviour.     

An integrated three-dimensional model of wave-induced pore pressure and effective stresses in a porous seabed: II. Breaking waves

The evaluation of the wave-induced liquefaction potential is particularly important for coastal engineers involved in the design of marine structures. Most previous investigations of the wave-induced liquefaction have been limited to two-dimensional non-breaking waves. In this paper, the integrated three-dimensional poro-elastic model for the wave-seabed interaction proposed by [Zhang, H., Jeng, D.-S., 2005. An integrated three-dimensional model of wave-induced pore pressure and effective stresses in a porous seabed: I. A sloping seabed. Ocean Engineering 32(5/6), 701–729.] is further extended to simulate the seabed liquefaction potential with breaking wave loading. Based on the parametric study, we conclude: (1) the liquefaction depth due to breaking waves is smaller than that of due to non-breaking waves; (2) the degree of saturation significantly affects the wave-induced liquefaction depth, and no liquefaction occurs in full saturated seabed, and (3) soil permeability does not only significantly affect the pore pressure, but also the shear stresses distribution.     

Unfluidized soil responses of a silty seabed to monochromatic waves

A flume experimental study on unfluidized responses of a silty bed (d₅₀=0.05 mm) to monochromatic water waves had shown that pore pressure variations were generally poro-elastic in the bulk body and displayed two other characteristic features not found in previous laboratory sand tests. They were an immediately fluidized thin surface layer induced by wave stresses inside the seabed's boundary layer and a porous skeleton with internally suspended sediments due to channeled flow motions. The analyses verified that on soils beneath the measurement points, both features resulted in relatively small-step pore pressure build-ups, while the former played a primary role. Besides, laboratory observations confirmed that there were some near-bed sediment suspensions during wave actions resulting in a flat bed form over a silty bed compared to small-scaled ripples over a sandy bed with no clearly identified suspended sediments. These characteristic silt responses suggest that sediment transport is critically associated with the internal soil responses and some field-observed sediment suspensions near above sandy beaches can further be approached in the laboratory by utilizing fine-grained soils.     

On waves propagating over poro-elastic seabed

In this study, an analytical solution is developed for the problem of periodic waves propagating over a poro-elastic seabed of infinite depth. Water waves above the seabed are described using the linear wave theory. The poro-elastic seabed is modelled based on the Biot theory in which the inertia effect and Darcy's friction are added. Continuity of dynamic pressure and flow flux at the interfacial seabed surface are considered. Adopting an approach similar to Hsu et al. (1993), the governing equations for the pore pressure and displacements of the poro-elastic medium are derived. The present analytic solution compares favorably well with experimental results by Yamamoto et al. (1978), and analytical results by Song (1993) for the case of fine sand. Using the present theory, variations of the wavelength and fluid pressure caused by coupling of waves and the poro-elastic seabed are discussed. Results show that higher elasticity of the poro-elastic seabed induces larger interface pressure, but higher permeability causes smaller pressure on the seabed interface. The wave length is affected by the poro-elastic seabed and becomes shorter for softer seabed and shallower water depth.     

波浪在局部可渗透水平海床上传播的解析解

本文建立了波浪在局部可渗透水平海床上传播的解析解,并研究了波浪在局部可渗透海床上的透射、反射问题。研究中将计算域划分为4个区域,中间区域为流域,海底可渗透,其下区域为多孔介质海床,左右两个区域也为流域,但海底不可渗透。应用线性波浪理论,建立了各流域包含非传播模态的速度势表达式,给出了海床内部的压强表达式,并利用交界面上匹配条件,求解了表达式中的待定系数。基于该解析模型,探讨了海床渗透系数、相对水深、渗透海床长度对波浪传播变形的影响。结果表明,波高沿程衰减,强度随渗透系数、渗透海床长度的增加以及相对水深的减小而变大;局部可渗透海床会引起波浪的反射和透射,随着海床长度的增加,反射系数振荡变化,并最终趋于常数,透射系数指数衰减,并最终趋于0。     

Response of a porous seabed around breakwater heads

The evaluation of wave-induced pore pressures and effective stresses in a porous seabed near a breakwater head is important for coastal engineers involved in the design of marine structures. Most previous studies have been limited to two-dimensional (2D) or three-dimensional (3D) cases in front of a breakwater. In this study, we focus on the problem near breakwater heads that consists of incident, reflected and diffracted waves. Both wave-induced oscillatory and residual liquefactions will be considered in our new models. The mistake in the previous work [Jeng, D.-S., 1996. Wave-induced liquefaction potential at the tip of a breakwater. Applied Ocean Research 18(5), 229–241] for oscillatory mechanism is corrected, while a new 3D boundary value problem describing residual mechanism is established. A parametric study is conducted to investigate the influences of several wave and soil parameters on wave-induced oscillatory and residual liquefactions around breakwater heads.     

Analysis on pore pressure in an anisotropic seabed in the vicinity of a caisson

This paper presents an analysis of pore pressure around a caisson-type breakwater subjected to dynamic wave loading. Unlike previous investigations for wave-seabed-caisson interaction, cross-anisotropic soil behaviour is considered in this paper. Based on a linear poro-elastic theory, a finite element model is developed. A parametric study related to the effects of wave parameters, soil characteristics and geometry of caisson and rubble mound base on the pore pressure around a caisson is performed. The numerical results indicate that the effects of anisotropic soil behaviour on the wave-induced pore pressure in a sandy bed beneath a caisson are not negligible.     

An experimental study of seabed responses around a marine pipeline under wave and current conditions

Experiments on three types of soil (d₅₀=0.287, 0.057 and 0.034 mm) with pipeline(D=4 cm) either half buried or resting on the seabed under regular wave or combined with current actions were conducted in a large wave flume to investigate characteristics of soil responses. The pore pressures were measured through the soil depth and across the pipeline. When pipeline is present the measured pore pressures in sandy soil nearby the pipeline deviate considerably from that predicted by the poro-elasticity theory. The buried pipeline seems to provide a degree of resistance to soil liquefaction in the two finer soil seabeds. In the silt bed, a negative power relationship was found between maximum values of excess pore pressure p_max and test intervals under the same wave conditions due to soil densification and dissipation of the pore pressure. In the case of wave combined with current, pore pressures in sandy soil show slightly decrease with time, whereas in silt soil, the current causes an increase in the excess pore pressure build-up, especially at the deeper depth. Comparing liquefaction depth with scour depth underneath the pipeline indicates that the occurrence of liquefaction is accompanied with larger scour depth under the same pipeline-bed configuration.     

A semi-analytical solution for random wave-induced soil response and seabed liquefaction in marine sediments

In this study, unlike most previous investigations for wave-induced soil response, a simple semi-analytical model for the random wave-induced soil response is established for an unsaturated seabed of finite thickness. Two different wave spectra, the B-M and JONSWAP spectra, are considered in the new model. The influence of random wave loading on the soil response is investigated by comparing with the corresponding representative regular wave results through a parametric study, which includes the effect of the degree of saturation, soil permeability, wave height, wave period and seabed thickness. The maximum liquefaction depth under the random waves is also examined. The difference on the soil response under the two random wave types, B-M and JONSWAP frequency spectra, is also discussed in the present work.     

Analysis of seabed instability using element free Galerkin method

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.     

Penetration depth of the dynamic response of seabed induced by internal solitary waves

We use flume experiments and numerical modeling to examine the penetration depth of internal solitary waves (ISWs) on partially saturated porous sandy silt and clayey silt seabed. The results of the experiment and model showed that the instantaneous excess pore water pressure in both the sandy silt and clayey silt seabed followed the same trend of decreasing with the seabed depth. In general, the excess pore water pressure generated by the sandy silt was bigger than that by clayey silt at the same depth. The ISW-induced excess pore water pressure greatly influenced the surface seabed and showed a linear relationship. The penetration depth was approximately one order of magnitude smaller than the half-wavelength of the ISWs, which might be larger than the penetration depth induced by surface waves. Our study results are helpful for understanding the damage that ISWs inflict upon the seabed and for informing future field experiments designed to directly measure the interaction between ISWs and seabed sediments.     

Effects of dynamic soil behavior and wave non-linearity on the wave-induced pore pressure and effective stresses in porous seabed

Most previous investigations for the wave-induced soil response have only considered the quasi-static soil behavior under linear wave loading. However, it is expected that the dynamic soil behavior and wave non-linearity will play an important role in the evaluation of wave-induced seabed response. In this paper, we include dynamic soil behavior and wave non-linearity into new analytical models. Based on the analytical solution derived, the effects of wave non-linearity on the wave-induced seabed response with dynamic soil behavior are examined. Numerical results demonstrate the significant effects of wave non-linearity and dynamic soil behavior on the wave-induced effective stresses. The applicable range of dynamic and quasi-static approximations is also clarified for engineering practice.     

Comparison of existing poroelastic models for wave damping in a porous seabed

Mechanism of wave–seabed interaction has been extensively studied by coastal geotechnical engineers in recent years. Numerous poro-elastic models have been proposed to investigate the mechanism of wave propagation on a seabed in the past. The existing poro-elastic models include drained model, consolidation model, Coulomb-damping model, and full dynamic model. However, to date, the difference between the existing models is unclear. In this paper, the fully dynamic poro-elastic model for the wave–seabed interaction will be derived first. Then, the existing models will be reduced from the proposed fully dynamic model. Based on the numerical comparisons, the applicable range of each model is also clarified for the engineering practice.     

Stability analysis on a porous seabed under wave and current loadings

The stability of a porous seabed under wave and current loadings is particularly important for engineers to design marine structures such as submarine pipelines, breakwaters, and offshore platform foundations. Most previous investigations of dynamic response of marine structures and seabed have only considered the influence of wave loading, but the important influence of current is ignored. Even if the influence of current is considered, the interaction mechanism of both loadings has not been clearly elaborated. Based on the Biot’s dynamic theory and combined two-dimensional nonlinear progressive wave and uniform current theory, the interaction mechanism of wave and current loadings and the influence of current on wave characteristic are analyzed by numerical computations. The influence of current velocity, different permeability, and stratification in seabed on the effective stresses and pore pressures of seabed is discussed in detail. Further, the stability of seabed is evaluated through the liquefaction analysis of seabed, which will provide important reference frames to improve the design and construction of marine structures.     

Mechanism of the wave-induced seabed instability in the vicinity of a breakwater: a review

A detailed knowledge of the wave-induced seabed instability is particularly important for engineers involved in the design procedure of many marine structures and offshore installations. In this paper, the basic aspects of such instability will be examined. The current understanding of the mechanism of the wave–seabed interaction phenomenon and available approaches will be reviewed. Based on the framework of simplified analysis, the potential for such instability will be formulated that will help engineers to identify potential unstable sediments in the vicinity of a marine structure.     

A new approximation for ocean waves propagating over a beach with variable depth

In this paper, the phenomenon of ocean waves propagating over a beach with variable water depth is re-examined. Based on the assumption of shallow water, a linearised shallow water equation is solved with an arbitrary beach profile. These irregular beach profiles form a set of partial differential equation with variable coefficient as the governing equation, which is the main obstacle in obtaining analytical solutions. In this paper, two families of beach profile are used as examples. A parametric study is conducted to investigate the influence of the beach profiles on the water surface elevation (η) and velocities (u).     

The elemination of re-reflected waves by a porous medium of finite thickness

The elimination of re-reflected waves in a wave channel by installing a porous medium in front of the wavemaker is investigated. The thickness of the porous wall required to eliminate the re-reflected waves is shown to be related to th porosity, friction coefficient, and wave period, as well as to both the positions of the porous medium and the test structure. However, this study indicates that the goal of eliminating re-reflected waves can be achieved by simply varying the thickness of the porous medium according to the wave period, with all the other factors arbitrarily selected.Assuming that the oscillation amplitude of the wavemaker board is constant, the primitive wave amplitude, before reaching the porous medium, becomes smaller as the wave period is increased. In addition, the study found that the required thickness of the porous medium for eliminating the re-reflected wave becomes larger as the wave period is increased. This results in a trend which further reduces the wave amplitude after the wave passes through the porous medium. In consequence, the oscillation amplitude of a wavemaker board has to be adjusted in a larger scale if the wave period is to be increased.