Nonlinear diffusive surface waves in porous media

Abstract

A fully nonlinear, diffusive, and weakly dispersive wave equation is derived for describing gravity surface wave propagation in a shallow porous medium. Darcy's flow is assumed in a homogeneous and isotropic porous medium. In deriving the general equation, the depth of the porous medium is assumed to be small in comparison with the horizontal length scale, i.e. O(μ2) =O(h0/L)2[Lt ]1. The order of magnitude of accuracy of the general equation is O(μ4). Simplified governing equations are also obtained for the situation where the magnitude of the free-surface fluctuations is also small, O(ε)=O(a/h0)[Lt ]1, and is of the same order of magnitude as O(μ2). The resulting equation is of O(μ4, ε2) and is equivalent to the Boussinesq equations for water waves. Because of the dissipative nature of the porous medium flow, the damping rate of the surface wave is of the same order magnitude as the wavenumber. The tide-induced groundwater fluctuations are investigated by using the newly derived equation. Perturbation solutions as well as numerical solutions are obtained. These solutions compare very well with experimental data. The interactions between a solitary wave and a rectangular porous breakwater are then examined by solving the Boussinesq equations and the newly derived equations together. Numerical solutions for transmitted waves for different porous breakwaters are obtained and compared with experimental data. Excellent agreement is observed.

Modelling and simulation of waves in three-layer porous media

“…Apart from wave reduction, we clearly observe the increase of mean water level in the far right end of the porous medium. This nonlinear phenomenon is derived in Liu and Wen (1997) as the analytical asymptotic solution of the single-layer Boussinesq model (27), in which η → εR 31 4h 3 for x → ∞. This well-known but surprising result was observed by Nielsen (1990) in the field.…”

“…Under the same assumption, Liu and Wen (1997) implement the asymptotic expansion method for the case of a single layer. In this section we extend this approach for the case of three-layer porous media.…”

“…The set of Eqs. (27) and (28) is equivalent to the Boussinesq equations for flow in one-layer porous media as derived in Liu and Wen (1997) using a slightly different asymptotic expansion.…”

“…These authors use the modified shallow water fluid flow equations to take into account frictional dissipation. Other authors, Chwang and Chan (1998), and Liu and Wen (1997), use porous media flow equations as described by Darcy's Law to describe flow through the structures. Since for thick groves, viscous effects dominate inertial effects, for our study here we adopt Darcy's assumption, and we develop an asymptotic model for gravity waves in a rigid isotropic three-layer porous structure.…”

An Exact Equation for the Free Surface of a Fluid in a Porous Medium

“…Extensions of the Boussinesq equation have been proposed, in the same perturbation theoretic spirit, by several authors, encompassing higher order expansions in the longness parameter [5,6,7] together with a small α [8]. Still other works introduce a new perturbative parameter, the steepness α √ β [9].…”