A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches
Introduction
Groundwater dynamics within a sandy beach influence erosion control [4], saltwater intrusion [3], chemical transformation [10] and biological activities [6], [9]. In particular, accurate prediction of dynamic groundwater hydraulics in coastal zones is required to improve coastal management. Most studies of coastal aquifers are based on the Boussinesq equation together with the Dupuit assumption [1], [2]. The non-linear governing equation was derived by Dagan [2] and subsequently approximated by expanding in terms of a perturbation parameter representing the shallow water approximation. Dagan [2] showed that higher-order approximations are significant for fine sand with lower hydraulic conductivity. These solutions are only applicable when the amplitude of the motion is small compared to the mean water depth. Parlange et al. [8] extended the work of Dagan [2] to a higher-order solution to describe the free surface elevation of the groundwater flow.
To simplify the problem, most previous investigations for water table fluctuations in coastal aquifers have considered the case of a vertical beach instead of the more realistic case of a sloping beach. Nielsen [7] was the first to derive an analytical solution where the assumption of a fixed location of the shoreline boundary condition is relaxed. However, his solution contains only an approximation to the boundary condition at the intersection of the beach and the ocean. Later, Li et al. [5] proposed the concept of a moving boundary to re-examine the problem using the same perturbation parameter as Nielsen [7]. The Li et al. model [5] overcame the inconsistency of the boundary condition in Nielsen’s model [7]. However, in both models [5], [7], the slope of the beach was included in the perturbation parameter, limiting the applicability of their models to a certain range of the beach slope. In addition, both models only provided incomplete solutions of the second harmonic oscillations. To date, only the zeroth-order boundary value problem (i.e., the Boussinesq equation) has been solved for a sloping beach, so a higher-order solution would be a useful comparison. It is expected that the higher-order correction to the linear solution will be particularly important under certain combination of wave and soil characteristics in coastal aquifers.
The aim of this paper is to derive a higher-order solution for the tide-induced water table fluctuations in coastal aquifers adjacent to a sloping beach using a perturbation technique. First, the shallow water expansion is used to derive the higher-order boundary value problems. Second, a complete analytical solution is derived through two perturbation parameters (shallow water parameter ε and amplitude parameter α). A comprehensive comparison with previous solutions is made, and the effects of the higher-order components and beach slope are examined.
Section snippets
Problem set-up
The flow is assumed to be homogeneous, isothermal and incompressible in a rigid porous medium. The configuration of the tidal forced dynamic groundwater flow is shown in Fig. 1. In the figure, h(x,t) is the total tide-induced water table height and D is the still water table height. The condition that the water table heights at the boundary of the ocean and coast (i.e., x=x0(t)) are equal to the specified tidal variation, iswhere α=A/D is a dimensionless amplitude
Previous solutions
Nielsen [7] presented the first analytical investigation where the assumption of a fixed location of the shoreline boundary condition is relaxed. His solution is, in our notation:where εN=αεcot(β) is a perturbation parameter.
Using the same perturbation parameter, εN, Li et al. [5] applied the concept of a moving boundary to Nielsen’s approach. Their solution is re-organised in non-dimensional form as:
Comparisons with previous solutions
To investigate the difference among Nielsen [7], Li et al. [5] and the present solution, graphs of water table fluctuations for various beach slopes are illustrated in Fig. 2. In general, the results of Nielsen and Li et al. [5], [7] lie between the linear solution (α) and the second-order solution (εα2). This is because the previous solutions only contain part of the higher-order components. As shown in the figure, Nielsen’s solution [7] does not match the boundary condition at X=X0(t) with β=π
Conclusions
A new analytical solution for tide-induced water table fluctuations in coastal aquifers is derived. Based on the examples considered, the following conclusions can be drawn.
- (1)
In the new model, two perturbation parameters, the shallow water parameter (ε) and the amplitude parameter (α), were introduced. With these two parameters, the boundary condition at the intersection of ocean and inland is satisfied consistently, and complete higher-order solutions were derived. Furthermore, the new model
Acknowledgements
The authors are grateful for the support from Australian Research Council Linkage-International # LX0345715 (2002–2003), Griffith University Research Development Grants (2002 and 2003), and NSERC Grant # A9117. The authors are also appreciative of the helpful suggestions of the reviewers, which led to improvements in the manuscript.
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