Water wave-driven seepage in marine sediments

Abstract

The action of water waves moving over a porous seabed drives a seepage flux into and out of the marine sediments. The volume of fluid exchange per wave cycle may affect the rate of contaminant transport in the sediments. In this paper, the dynamic response of the seabed to ocean waves is treated analytically on the basis of poro-elastic theory applied to a porous seabed. The seabed is modelled as a semi-infinite, isotropic, homogeneous material. Most previous investigations on the wave–seabed interaction problem have assumed quasi-static conditions within the seabed, although dynamic behaviour often occurs in natural environments. Furthermore, wave pressures used in the previous approaches were obtained from conventional ocean wave theories, which are based on the assumption of an impermeable rigid seabed. By introducing a complex wave number, we derive a new wave dispersion equation, which includes the seabed characteristics (such as soil permeability, shear modulus, etc.). Based on the new closed-form analytical solution, the relative differences of the wave-induced seabed response under dynamic and quasi-static conditions are examined. The effects of wave and soil parameters on the seepage flux per wave cycle are also discussed in detail.

Introduction

Sediments in bays, estuaries, and in the seabed near river inlets are often contaminated. Many inorganic contaminants (notably heavy metals) do not decompose. Under certain conditions, these accumulated substances can be released back into the receiving body of water through mass transfer processes at the seabed. The mass transfer rate is largely controlled by the seepage flux exchange between the sediment and the seawater [26]. Increased wave action and higher sediment hydraulic conductivity generally cause larger transfer rates. Clearly, quantification of the mass transfer rate is a key factor in water quality modelling.

Water wave effects on the marine sediment in shallow water have been studied intensively in the last few decades. Most earlier studies [1], [18], [19], [21], [23] were based on the assumption that the porous seabed was non-deformable, and that the pore water was incompressible. Numerous investigations have been carried out based on Biot's consolidation equation [2] and the assumptions of a compressible pore fluid and soil skeleton. The seabed has been modelled as being isotropic [9], [20], [22], [29], [30], anisotropic [10] and inhomogeneous [16]. The potential for extreme seabed instability (such as liquefaction) due to a generalised three-dimensional wave system has recently been explored [17]. Limitations and applications of previous investigations have been reviewed also [11]. Two major shortcomings of most previous approximations in the area of wave–seabed interaction are:

• Assumption of a quasi-static state: The dynamic terms generated from the acceleration of soil particles and the movement of pore fluid were not included in most previous solutions. Although the inertia effect generated from the acceleration of soil particles was recently considered in the wave–seabed interaction problem [15], the acceleration due to pore fluid was excluded from the analysis. Thus, it is not a complete dynamic solution.

• Wave pressures at the seabed surface obtained from conventional ocean wave theories: These wave pressures were based on the assumption of an impermeable seabed. However, the wave pressures were then applied to a porous seabed. This contradiction has been removed recently [12], with the assumption of a quasi-static state, not a dynamic state. Recently, Liu and Wen [19] derived a fully non-linear, diffusion and weakly dispersive wave equation for describing gravity surface wave propagation in a shallow porous medium. However, they only considered a rigid medium (such as rock). Thus, only soil permeability is included in their dispersion relation.

The objective of this paper is to overcome these two shortcomings and investigate water wave-driven seepage flux into and out of the seabed under more realistic conditions. Based on the governing equations presented by Mei [22], which were derived on the basis of poro-elastic theory [3], a closed-form analytical solution for dynamic wave–seabed interactions is derived. The model also includes a new wave–dispersion relation, which accounts for the characteristics (including soil permeability, shear modulus, porosity, etc.). A comparison of the present dynamic and previous quasi-static solution is performed. Then, the effects of wave and seabed characteristics on the seepage flux at the seabed surface are investigated in a parametric study.