A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches

The Boussinesq equation appears as the zeroth-order term in the shallow water flow expansion of the non-linear equation describing the flow of fluid in an unconfined aquifer. One-dimensional models based on the Boussinesq equation have been used to analyse tide-induced water table fluctuations in coastal aquifers. Previous analytical solutions for a sloping beach are based on the perturbation parameter, _N=αcotβ (in which β is the beach slope, α is the amplitude parameter and is the shallow water parameter) and are limited to tan⁻¹(α)βπ/2. In this paper, a new higher-order solution to the non-linear boundary value problem is derived. The results demonstrate the significant influence of the higher-order components and beach slope on the water table fluctuations. The relative difference between the linear solution and the present solution increases as and α increase, and reaches 7% of the linear solution.     

Closed-form analytical solutions incorporating pumping and tidal effects in various coastal aquifer systems

Pumping wells are common in coastal aquifers affected by tides. Here we present analytical solutions of groundwater table or head variations during a constant rate pumping from a single, fully-penetrating well in coastal aquifer systems comprising an unconfined aquifer, a confined aquifer and semi-permeable layer between them. The unconfined aquifer terminates at the coastline (or river bank) and the other two layers extend under tidal water (sea or tidal river) for a certain distance L. Analytical solutions are derived for 11 reasonable combinations of different situations of the L-value (zero, finite, and infinite), of the middle layer’s permeability (semi-permeable and impermeable), of the boundary condition at the aquifer’s submarine terminal (Dirichlet describing direct connection with seawater and no-flow describing the existence of an impermeable capping), and of the tidal water body (sea and tidal river). Solutions are discussed with application examples in fitting field observations and parameter estimations.     

Tidal effects on groundwater dynamics in unconfined aquifers

The variation of seawater level resulting from tidal fluctuations is usually neglected in regional groundwater flow studies. Although the tidal oscillation is damped near the shoreline, there is a quasi‐steady‐state rise in the mean water‐table position, which may have an influence on regional groundwater flow. In this paper the effects of tidal fluctuations on groundwater hydraulics are investigated using a variably saturated numerical model that includes the effects of a realistic mild beach slope, seepage face and the unsaturated zone. In particular the impact of these factors on the velocity field in the aquifer is assessed. Simulations show that the tidal fluctuation has substantial consequences for the local velocity field in the vicinity of the exit face, which affects the nearshore migration of contaminant in coastal aquifers. An overheight in the water table as a result of the tidal fluctuation is observed and this has a significant effect on groundwater discharge to the sea when the landward boundary condition is a constant water level. The effect of beach slope is very significant and simplifying the problem by considering a vertical beach face causes serious errors in predicting the water‐table position and the groundwater flux. For media with a high effective capillary fringe, the moisture retained above the water table is important in determining the effects of the tidal fluctuations. Copyright © 2001 John Wiley & Sons, Ltd.     

Beach water table fluctuations due to spring–neap tides: moving boundary effects

Tidal water table fluctuations in a coastal aquifer are driven by tides on a moving boundary that varies with the beach slope. One-dimensional models based on the Boussinesq equation are often used to analyse tidal signals in coastal aquifers. The moving boundary condition hinders analytical solutions to even the linearised Boussinesq equation. This paper presents a new perturbation approach to the problem that maintains the simplicity of the linearised one-dimensional Boussinesq model. Our method involves transforming the Boussinesq equation to an ADE (advection–diffusion equation) with an oscillating velocity. The perturbation method is applied to the propagation of spring–neap tides (a bichromatic tidal system with the fundamental frequencies ω₁andω₂) in the aquifer. The results demonstrate analytically, for the first time, that the moving boundary induces interactions between the two primary tidal oscillations, generating a slowly damped water table fluctuation of frequency ω₁−ω₂, i.e., the spring–neap tidal water table fluctuation. The analytical predictions are found to be consistent with recently published field observations.     

The enhancing effect on tidal signals of a submarine spring connected to a semi-infinite confined aquifer

Abstract

Submarine springs play an important role in submarine groundwater discharge (SGD). To investigate the effects of these springs on the propagation of tidal signals in coastal confined aquifers, this paper considers a general coastal aquifer system with a submarine spring on the seabed where the length of the aquifer's offshore extent is finite and its submarine outlet is covered by an impermeable outlet-capping. An approximate analytical solution is obtained for describing the tidal head fluctuations in the aquifer. Solution analyses indicate that the error of the approximate analytical solution is negligible when both distances from the spring hole to the coastline and to the submarine outlet-capping are much greater than the radius of the spring hole. Sensitivity tests are conducted to investigate the effects of hydraulic properties, tidal and spring geometric configuration parameters on the tidal signal propagation in the inland aquifer. For aquifers with infinite offshore length, or without submarine springs, existing solutions in the literature are obtained. The comparison of groundwater head fluctuations for the cases with and without a submarine spring demonstrate the enhancing effect of the submarine spring on tidal signal propagation in the inland aquifer. Three situations that fit our model assumptions are given for future potential applications. A hypothetical example is used to show the possibility of identifying a spring's location using the present analytical solution together with tidal signals observed from inland wells.

Editor D. Koutsoyiannis; Associate editor Y. Guttmann

Citation Xia, Y.Q., Li, H.L., Yang, Y., and Huang, W., 2012. Enhancing effect on tidal signals of a submarine spring related to a semi-infinite confined aquifer. Hydrological Sciences Journal, 57 (6), 1231–1248.     

Internal sedimentary architecture and coastal dynamics as revealed by ground penetrating radar, Kachchh coast, western India

The coastline constitutes a very sensitive geomorphic domain which is constantly subjected to dynamic coastal processes and stores vital information regarding past sea level fluctuations. A ground-penetrating radar (GPR) survey was carried out along the northern coast of the Gulf of Kachchh which is one of the largest macrotidal inlets of the Arabian Sea, Western India. Our studies have delineated several radar surfaces and radar facies which reflect the internal architecture and sediment body geometry, which can be related to the processes acting along this coastline. Various radar facies, namely, beach ridge (Br), washover (Wo), coastal dune (Cd), swale (Sw), berm plain (Bp), and sandsheet facies (Ss) have been identified. The GPR studies successfully documented the subsurface presence of ancient beach ridge system towards the sea, and the coastal dunes towards the land side. The results are suggestive of signatures of changes in sea level and the coastline being prone to high energy events in the recent past. The GPR has been found to be an important non-invasive geophysical tool in the study of past coastal dynamics.     

Homotopy Perturbation Method‐Based Analytical Solution for Tide‐Induced Groundwater Fluctuations

The groundwater variations in unconfined aquifers are governed by the nonlinear Boussinesq's equation. Analytical solution for groundwater fluctuations in coastal aquifers under tidal forcing can be solved using perturbation methods. However, the perturbation parameters should be properly selected and predefined for traditional perturbation methods. In this study, a new dimensional, higher‐order analytical solution for groundwater fluctuations is proposed by using the homotopy perturbation method with a virtual perturbation parameter. Parameter‐expansion method is used to remove the secular terms generated during the solution process. The solution does not require any predefined perturbation parameter and valid for higher values of amplitude parameter A/D, where A is the amplitude of the tide and D is the aquifer thickness.     

An analytical solution for the head distribution in a tidal leaky confined aquifer extending an infinite distance under the sea

A mathematical model is developed to investigate the effects of tidal fluctuations and leakage on the groundwater head of leaky confined aquifer extending an infinite distance under the sea. The leakages of the offshore and inland aquitards are two dominant factors controlling the groundwater fluctuation. The tidal influence distance from the coast decreases significantly with the dimensionless leakage of the inland aquitard (u_i). The fluctuation of groundwater level in the inland part of the leaky confined aquifer increases significantly with the dimensionless leakage of the offshore aquitard (u_o). The influence of the tidal propagation parameter of an unconfined aquifer on the head fluctuation of the leaky confined aquifer is comparatively conspicuous when u_i is large and u_o is small. In other words, ignoring water table fluctuation of the unconfined aquifer will give large errors in predicting the fluctuation, time lag, and tidal influence distance of the leaky confined aquifer for large u_i and small u_o. On the contrary, the influence of the tidal propagation parameter of a leaky confined aquifer on the head fluctuation of the leaky confined aquifer is large for large u_o and small u_i.     

A semi‐analytical solution for groundwater responses to stream‐stage variations and tidal fluctuations in a coastal aquifer

A two‐dimensional semi‐analytical solution to analyse stream–aquifer interactions in a coastal aquifer where groundwater level responds to tidal effects is presented. The conceptual model considered is a two‐dimensional subsurface system with stream and coastline boundaries at right angles. The dimensional and non‐dimensional boundary value problems were solved for water level in the aquifer by successive application of Laplace and Fourier transform techniques, and the results were obtained by numerical inversion of the transformed solution. The solution was then verified by reducing the solutions to one‐dimensional known problems and comparing the results with those from previous studies. Hypothetical examples were used to examine the characteristics of water‐level variations due to the variations in stream stage and the fluctuations in tide level. Sensitivity analysis indicated that streambed leakance has no influence over the amplitude of groundwater fluctuations, but that the effect of stream stage increases with increasing leakance. Little difference was observed in the water level for different aquifer penetration ratios with narrow stream width. Increases in streambed leakance caused increases in the effect of aquifer penetration by the stream on the water level. An increased specific yield value resulted in decreased amplitude of water fluctuations and mean water level, and showed that water‐level variations due to stream and tidal boundaries are sensitive to specific yield. Copyright © 2006 John Wiley & Sons, Ltd.     

Two‐dimensional flow response to tidal fluctuation in a heterogeneous aquifer‐aquitard system

In alluvial coastal aquifers, finer sediments are preferentially deposited along the downstream direction, so the hydraulic conductivity is generally heterogeneous and changes with distance from the coastline. To investigate the influence of aquifer heterogeneity on seawater‐groundwater interaction, a new two‐dimensional model characterising groundwater flow in an aquifer‐aquitard system was developed assuming that the hydraulic conductivity of the aquifer linearly increases with the distance from the coastline along the inland direction. A closed‐form analytical solution was derived using the separation‐of‐variables method. Comparing the new solution with the numerical solution by comsol Multiphysics (Sweden) based on the finite‐element method, one can see that the new solution agreed with the numerical solution very well except at the early time. We found that both aquitard leakance and the heterogeneity factor (b) could result in the propagation bias. The propagation bias represents the inconsistency between the theoretical calculation and the observed strong attenuation and small time lag between the head and tide fluctuations. The attenuation decreased with perpendicular distance from the coastline (x‐axis), whereas the time lag increased with distance along the x‐axis. The relationship between the time lag and the distance along the x‐axis seemed to be linear when b was 0.001 m^?1, whereas it obeyed a power function when b was greater than 0.01 m^?1. Copyright © 2014 John Wiley & Sons, Ltd.     

A two-dimensional analytical solution of groundwater responses to tidal loading in an estuary and ocean

Previous studies on tidal dynamics of coastal aquifers have focussed on the inland propagation of oceanic tides in the cross-shore direction, a configuration that is essentially one-dimensional. Aquifers at natural coasts can also be influenced by tidal waves in nearby estuaries, resulting in a more complex behaviour of head fluctuations in the aquifers. We present an analytical solution to the two-dimensional depth-averaged groundwater flow equation for a semi-infinite aquifer subject to oscillating head conditions at the boundaries. The solution describes the tidal dynamics of a coastal aquifer that is adjacent to a cross-shore estuary. Both the effects of oceanic and estuarine tides on the aquifer are included in the solution. The analytical prediction of the head fluctuations is verified by comparison with numerical solutions computed using a standard finite-difference method. An essential feature of the present analytical solution is the interaction between the cross- and along-shore tidal waves in the aquifer area near the estuary’s entry. As the distance from the estuary or coastline increases, the wave interaction is weakened and the aquifer response is reduced, respectively, to the one-dimensional solution for oceanic tides or the solution of Sun (Sun H. A two-dimensional analytical solution of groundwater response to tidal loading in an estuary, Water Resour Res 1997;33:1429–35) for two-dimensional non-interacting tidal waves.     

New approximation for free surface flow of groundwater: capillarity correction

An existing capillarity correction for free surface groundwater flow as modelled by the Boussinesq equation is re-investigated. Existing solutions, based on the shallow flow expansion, have considered only the zeroth-order approximation. Here, a second-order capillarity correction to tide-induced watertable fluctuations in a coastal aquifer adjacent to a sloping beach is derived. A new definition of the capillarity correction is proposed for small capillary fringes, and a simplified solution is derived. Comparisons of the two models show that the simplified model can be used in most cases. The significant effects of higher-order capillarity corrections on tidal fluctuations in a sloping beach are also demonstrated.     

A Correction on Coastal Heads for Groundwater Flow Models

We introduce a simple correction to coastal heads for constant‐density groundwater flow models that contain a coastal boundary, based on previous analytical solutions for interface flow. The results demonstrate that accurate discharge to the sea in confined aquifers can be obtained by direct application of Darcy's law (for constant‐density flow) if the coastal heads are corrected to ((α + 1)/α)h_s ? B/2α, in which h_s is the mean sea level above the aquifer base, B is the aquifer thickness, and α is the density factor. For unconfined aquifers, the coastal head should be assigned the value . The accuracy of using these corrections is demonstrated by consistency between constant‐density Darcy's solution and variable‐density flow numerical simulations. The errors introduced by adopting two previous approaches (i.e., no correction and using the equivalent fresh water head at the middle position of the aquifer to represent the hydraulic head at the coastal boundary) are evaluated. Sensitivity analysis shows that errors in discharge to the sea could be larger than 100% for typical coastal aquifer parameter ranges. The location of observation wells relative to the toe is a key factor controlling the estimation error, as it determines the relative aquifer length of constant‐density flow relative to variable‐density flow. The coastal head correction method introduced in this study facilitates the rapid and accurate estimation of the fresh water flux from a given hydraulic head measurement and allows for an improved representation of the coastal boundary condition in regional constant‐density groundwater flow models.     

Hydrodynamic characterization of a Sahelian coastal aquifer using the ocean tide effect (Dridrate Aquifer,Morocco)

Abstract

In wells tapping coastal aquifers, piezometric fluctuations can be observed in response to the ocean tide. Simultaneous recordings of the ocean tide and of the piezometric variations may provide a basis for characterizing the hydrodynamics of the aquifer. This approach was attempted to characterize the Dridrate aquifer, located on the Atlantic coast of Morocco. This aquifer accounts for most of the regional drinking water resources. However, its hydrodynamic characteristics are very poorly known. The study compares observed and simulated piezometric fluctuations, under various assumptions (confined, semi-confined aquifer). The model, which best explains the hydrodynamic behaviour of this aquifer is a semi-confined and strongly heterogeneous aquifer model (calculated hydraulic diffusivity values vary over several orders of magnitude). This result is new and rather surprising, since to date this aquifer was considered confined in view of its geological setting. Consequently, new questions are raised regarding the protection and management of the groundwater resources of this aquifer.     

Tide-induced head fluctuations in coastal aquifers of variable thickness

In this work, a new analytical solution to describe tide-induced head fluctuations in aquifers of variable thickness is presented. The proposed model assumes a finite and confined aquifer with a thickness that increases or decreases quadratically with the distance to the coast. A closed-form analytical solution is obtained by solving a boundary-value problem with both a separation of variables method and a change of variables method. This solution is a generalization of the solution obtained by Cuello et al., Hydrogeological Journal, 2017, 25, 1509–1515. The analytical solution is expressed in terms of the wedging parameter, a parameter that depends on the length and thicknesses at the coast and at the inland edge of the aquifer. Positive values of the wedging parameter describe aquifers with increasing thickness towards land and negative values describe aquifers with a decreasing thickness in the inland direction. The comparison of the new solution and the solution for a finite aquifer with constant thickness indicates that the sign of the wedging parameter enhances or decreases the amplitude of the tide-induced signal. However, the differences in time-lag between both solutions are negligible near the coast. The slope factor, which quantifies the inconsistencies between aquifer diffusivities estimated from attenuation and time-lag data, is computed and analysed. Near the coast, slope factor values greater than one are obtained for negative wedging parameters while slope factor values less than one are obtained for positive wedging parameters. The analysis of the new solution also indicates that more reliable estimates of the hydraulic diffusivity can be obtained from time-lag data.     

Modelling tidal signals enhanced by a submarine spring in a coastal confined aquifer extending under the sea

Submarine springs discharge offshore groundwater from confined aquifers extending under the sea. The effects of these springs on the propagation of tidal oscillations in coastal confined aquifers are not known. This paper presents an approximate analytical solution of tidal head fluctuations in a confined aquifer with one submarine spring. The aquifer is assumed to extend in all directions infinitely. The spring is represented by a permeable round column on the seabed, which penetrates completely the impermeable layer overlying the confined aquifer. The error of the approximate solution is negligible if the distance from the spring to the coastline is much greater than the radius of the permeable column representing the spring. Through a hypothetical example, we demonstrate that it is possible to identify the spring’s location using tidal signals observed from inland wells. Tidal groundwater head fluctuations from three inland observation wells at least are needed to determine the 5 model parameters, including the location (2 parameters), the radius of the permeable column representing the spring, the diffusivity of the aquifer, and the tidal loading efficiency of the system.     

Modelling coastal ground‐ and surface‐water interactions using an integrated approach

Beach water table fluctuations have an impact on the transport of beach sediments and the exchange of solute and mass between coastal aquifer and nearby water bodies. Details are given of the refinement of a dynamically integrated ground‐ and surface‐water model, and its application to study ground‐ and surface‐water interactions in coastal regions. The depth‐integrated shallow‐water equations are used to represent the surface‐water flow, and the extended Darcy's equation is used to represent the groundwater flow, with a hydrostatic pressure distribution being assumed to apply for both these two types of flows. At the intertidal region, the model has two layers, with the surface‐water layer being located on the top of the groundwater layer. The governing equations for these two types of flows are discretized in a similar manner and they are combined to give one set of linear algebraic equations that can be solved efficiently. The model is used to predict water level distributions across sloping beaches, where the water table in the aquifer may or may not decouple from the free water surface. Five cases are used to test the model for simulating beach water table fluctuations induced by tides, with the model predictions being compared with existing analytical solutions and laboratory and field data published in the literature. The numerical model results show that the integrated model is capable of simulating the combined ground‐ and surface‐water flows in coastal areas. Detailed analysis is undertaken to investigate the capability of the model. Copyright © 2009 John Wiley & Sons, Ltd.     

The role of beach morphology on coastal cliff erosion under extreme waves

Erosion of hard‐rock coastal cliffs is understood to be caused by a combination of both marine and sub‐aerial processes. Beach morphology, tidal elevation and significant wave heights, especially under extreme storm conditions, can lead to variability in wave energy flux to the cliff‐toe. Wave and water level measurements in the nearshore under energetic conditions are difficult to obtain and in situ observations are rare. Here we use monthly cliff‐face volume changes detected using terrestrial laser scanning alongside beach morphological changes and modelled nearshore hydrodynamics to examine how exposed cliffs respond to changes in extreme wave conditions and beach morphology. The measurements cover the North Atlantic storms of 2013 to 2014 and consider two exposed stretches of coastline (Porthleven and Godrevy, UK) with contrasting beach morphology fronting the cliffs; a flat dissipative sandy beach at Godrevy and a steep reflective gravel beach at Porthleven. Beach slope and the elevation of the beach–cliff junction were found to influence the frequency of cliff inundation and the power of wave–cliff impacts. Numerical modelling (XBeach‐G) showed that under highly energetic wave conditions, i.e. those that occurred in the North Atlantic during winter 2013–2014, with H_s = 5.5 m (dissipative site) and 8 m (reflective site), the combination of greater wave height and steeper beach at the reflective site led to amplified wave run‐up, subjecting these cliffs to waves over four times as powerful as those impacting the cliffs at the dissipative site (39 kWm^‐1 compared with 9 kWm^‐1). This study highlighted the sensitivity of cliff erosion to extreme wave conditions, where the majority (over 90% of the annual value) of cliff‐face erosion ensued during the winter. The significance of these short‐term erosion rates in the context of long‐term retreat illustrates the importance of incorporating short‐term beach and wave dynamics into geomorphological studies of coastal cliff change. © 2017 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.     

Tidal effects on variations of fresh–saltwater interface and groundwater flow in a multilayered coastal aquifer on a volcanic island (Jeju Island, Korea)

We conducted various field studies at the seawater intrusion monitoring wells located in the eastern part of Jeju Island, Korea, to observe the tidal effect on groundwater–seawater flow in the coastal aquifer. Studies included monitoring the fluctuations of groundwater and tide levels, electrical and temperature logging, and 2-D heat-pulse flowmeter tests. According to time-series analysis, tidal effects on groundwater level reached up to 3 km inland from the coastline. Water-level variation was more sensitive to tidal fluctuations near the coast, and more related to rainfall toward inland areas. Temporal and spatial variations in the shape and location of the freshwater–saltwater interface were analyzed using data from nine monitoring wells. The results indicated that the interface toe is located at a distance of 6–8 km from the coastline and its location was related to geological layers present. Long-term seasonal variations revealed no major changes in the interface; minor variations were due to moving boundary conditions induced by tidal fluctuations. Using the two-dimensional heat-pulse flowmeter, groundwater flow directions and velocities at four tidal stages were measured on three monitoring wells drilled into the multilayered aquifers. This direct measurement enabled us to relate the differences of flow velocities and directions with geology and tidal fluctuations. Combining the results of EC logging and flowmeter tests, we found a zone where freshwater and saltwater moved alternately in opposite directions, as influenced by the tidal fluctuations. Integrating various physical logging and flowmeter data with water-level fluctuations improved our understanding of the behavior of fresh and seawater flow in the coastal aquifers.     

Effects of beach slope breaks on nearshore groundwater dynamics

Interactions between fresh groundwater and seawater affect significantly the nearshore pore water flow, which in turn influences the fate of nutrients and contaminants in coastal aquifers prior to discharge to the marine environment. Field investigations and numerical simulations were carried out to examine the groundwater dynamics in the intertidal zone of a carbonate sandy aquifer on the tropical island of Rarotonga, Cook Islands. The study site was featured by distinct cross‐shore slope breaks on the beach surface. Measured pore water salinities revealed different distributions under the influences of different beach profiles, inland heads, and tidal oscillations. Fresh groundwater was found to discharge around a beach slope break located in the middle area of the intertidal zone. The results indicate a strong interplay between the slope break beach morphology and tidal force in controlling the nearshore groundwater flow and solute transport. The fresh groundwater discharge location was largely determined by the beach morphology in combination with the tidal force. The nearshore groundwater flow can be very sensitive to beach slope breaks, which induce local circulation and flow instabilities. As slope breaks are a common feature of beaches around the world, these results have important, general implications for future studies of nutrients transport and transformations in nearshore aquifers and associated fluxes via submarine groundwater discharge.