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Abstract

This paper develops a new analytical solution for the aquifer system, which comprises an unconfined aquifer on the top, a semi-confined aquifer at the bottom and an aquitard between them. This new solution is derived from the Boussinesq equation for the unconfined aquifer and one-dimensional leaky confined flow equation for the lower aquifer using the perturbation method, considering the water table over-height at the remote boundary. The head fluctuation predicted from this solution is generally greater than the one solved from the linearized Boussinesq equation when the ratio of the tidal amplitude to the thickness of unconfined aquifer is large. It is found that both submarine groundwater discharges from upper and lower aquifers increase with tidal amplitude–aquifer thickness ratio and may be underestimated if the discharge is calculated based on the average head fluctuation. The effects of the aquifer parameters and linearization of the Boussinesq equation on the normalized head fluctuation are also investigated.

Editor D. Koutsoyiannis; Associate editor J. Simunek

Citation Chuang, M.-H., Mahdi, A.-A. and Yeh, H.-D., 2012. A perturbation solution for head fluctuations in a coastal leaky aquifer system considering water table over-height. Hydrological Sciences Journal, 57 (1), 162–172.     

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.     

A new analytical solution of tidal water table fluctuations in a coastal unconfined aquifer

This paper presents a new perturbation solution of the non-linear Boussinesq equation for one-dimensional tidal groundwater flow in a coastal unconfined aquifer. Built upon the work of Parlange et al. [Parlange, J.-Y., Stagnitti, F., Starr, J.L., Braddock, R.D., 1984. Free-surface flow in porous media and periodic solution of the shallow-flow approximation, J. Hydrol., 70, 251–263], the solution adopts a new perturbation parameter that is by definition less than unit, and thus is applicable to a wider range of physical conditions within the constraint of the Boussinesq approximation. This approach avoids a secular term in the third-order perturbation equation of Parlange et al. (1984), enabling the derivation of the third- and higher-order solutions. In comparison with a numerical (“exact”) solution, the new perturbation solution is shown to be slightly more accurate than that of Parlange et al. (1984) with the second-order approximation. The obtained third-order solution exhibits considerable improvement in accuracy. In relatively simple analytical forms, the present perturbation solution will help to understand better the non-linear characteristics of tidal water table fluctuations in as modeled by the non-linear Boussinesq equation coastal unconfined aquifers.     

Observations of the spring–neap modulation of the gravitational circulation in a partially mixed estuary

The long-term variability of the non-tidal circulation in Southampton Water, a partially mixed estuary, was investigated using 71-day acoustic Doppler current profiler (ADCP) time series. The data show evidence that the spring–neap tidal variability of the turbulent mixing modulates the strength of the non-tidal residual circulation, with subtidal neap tide surface flows reaching 0.12 m s^–1 compared to <0.05 m s^–1 at spring tides. The amplitude of the neap-tide events in this non-tidal circulation is shown to be related to a critical value of the tidal currents, illustrating the strong dependence on tidal mixing. The results suggest that the dominant mechanism for generating these neap-tide circulation events is the baroclinic forcing of the horizontal density gradient, rather than barotropic forcing associated with ebb-induced periodic stratification. While tidal turbulence is thought to be the dominant control on this gravitational circulation, there is evidence of the additional effect of wind-driven mixing, including the effects of wind fetch and possibly wave development with along-estuary winds being more efficient at mixing the estuary than across-estuary winds. Rapid changes in atmospheric pressure also coincided with fluctuations in the gravitational circulation. The observed subtidal flows are shown to be capable of rapidly flushing buoyant material out of the estuary and into the coastal sea at neap tides.Responsible Editor: Iris Grabemann     

Flood–ebb and spring–neap variations of mixing,stratification and circulation in Chesapeake Bay

A three-dimensional hydrodynamic model is used to investigate intra-tidal and spring–neap variations of turbulent mixing, stratification and residual circulation in the Chesapeake Bay estuary. Vertical profiles of salinity, velocity and eddy diffusivity show a marked asymmetry between the flood and ebb tides. Tidal mixing in the bottom boundary layer is stronger and penetrates higher on flood than on ebb. This flood–ebb asymmetry results in a north–south asymmetry in turbulent mixing because tidal currents vary out of phase between the lower and upper regions of Chesapeake Bay. The asymmetric tidal mixing causes significant variation of salinity distribution over the flood–ebb tidal cycle but insignificant changes in the residual circulation. Due to the modulation of tidal currents over the spring–neap cycle, turbulent mixing and vertical stratification show large fortnightly and monthly fluctuations. The stratification is not a linear function of the tidal-current amplitude. Strong stratification is only established during those neap tides when low turbulence intensity persists for several days. Residual circulation also shows large variations over the spring–neap cycle. The tidally averaged residual currents are about 50% stronger during the neap tides than during the spring tides.     

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.     

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.     

Analytical solution of water table fluctuations above an inclined leaky layer due to ditch recharge

Ditch recharge is a method used for raising a water table. However, if the aquifer lies above a leaky layer, then the leaky bed will have an effect on the height of the water table. This paper attempts to provide an analytical solution of the transient state to the linearized Boussinesq equation. The solution proposed in this paper is intended for a generalized form of the leaky layer, with the boundary condition of the water level moving according to an exponential form. The results of theoretical analysis and experimental studies can provide a much better explanation of ditch recharge on an inclined leaky layer. Copyright © 2006 John Wiley & Sons, Ltd.     

Modelling of water table variation in response to time-varying recharge from multiple basins using the linearised Boussinesq equation

A mathematical model is presented to describe the variations of the water table in an unconfined aquifer due to time-varying recharge applied from four rectangular basins. The model is developed by solving the linearised Boussinesq equation using the extended finite Fourier cosine transform. The time-varying recharge rate is approximated by a number of piecewise linear elements of different lengths and slopes depending on the nature of the variation in recharge rate. Application of this model for the prediction of water table fluctuations and in the sensitivity analysis of various controlling parameters on the aquifer response is demonstrated in an example.     

Polynomial approximate solutions to the Boussinesq equation

We provide closed-form approximate solutions to models of horizontal infiltration described by the Boussinesq equation in a semi-infinite aquifer that is initially dry. The approximations preserve such important qualitative properties as scaling and wetting fronts. They are applicable to four types of boundary conditions, two on head and two on flux, enumerated in the paper. All the considered problems admit self-similar variables that allow reduction to boundary value problems for a nonlinear ordinary differential equation. This work extends recent results by Lockington et al. [Lockington DA, Parlange J-Y, Parlange MB, Selker J. Similarity solution of the Boussinesq equation. Adv Water Resour 2000;23(7):725–9] and Telyakovskiy et al. [Telyakovskiy AS, Braga GA, Furtado F. Approximate similarity solutions to the Boussinesq equation. Adv Water Resour 2002;25(2):191–4], with new approximations developed for two of the four cases and a new extension of a previously existing method for a third case. Numerical results extending the work of Shampine [Shampine LF. Some singular concentration dependent diffusion problems. ZAMM 1973;53:421–2] provide a basis for assessing the accuracy of the new methods.     

Subtidal flow and its variability at the entrance to a subtropical lagoon

Spatial and temporal variability of the subtidal exchange flow at West Pass, an inlet at the entrance to a subtropical lagoon (St. Andrew Bay, Florida), was studied using moored and towed current velocity profiles and hydrographic data. Towed and hydrographic measurements were captured over one diurnal tidal cycle to determine intratidal and spatial changes in flow. Hydrographic profiles over the tidal cycle showed that tidal straining modified density stratification asymmetrically, thus setting up the observed mean flow within the inlet. During the towed survey, the inlet's mean flow had a two-layer exchange structure that was moderately frictional and weakly influenced by Coriolis accelerations. Moored current profiles revealed the additional contribution to the dynamics from centrifugal accelerations. Along channel residual flows changed between unidirectional and exchange flow, depending on the forcing from the along-estuary wind stress and, to a lesser extent, the spring–neap tidal cycle. Increases in vertical shear in the along channel subtidal flow coincided with neap tides and rain pulses. Lateral subtidal flows showed the influence on the dynamics of centrifugal accelerations through a well-developed two-layer structure modulated in magnitude by the spring–neap tidal cycle.     

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.     

Two-dimensional approximation for tide-induced watertable fluctuations in a sloping sandy beach

The prediction of watertable fluctuations in a coastal aquifer is important for coastal management. However, most previous approaches have based on the one-dimensional Boussinesq equation, neglecting variations in the coastline and beach slope. In this paper, a closed-form analytical solution for a two-dimensional unconfined coastal aquifer bounded by a rhythmic coastline is derived. In the new model, the effect of beach slope is also included, a feature that has not been considered in previous two-dimensional approximations. Three small parameters, the shallow water parameter (ε), the amplitude parameter (α) and coastline parameter (β) are used in the perturbation approximation. The numerical results demonstrate the significant influence of both the coastline shape and beach slopes on tide-driven coastal groundwater fluctuations.     

Scales and structure of frontal adjustment and freshwater export in a region of freshwater influence

Sea surface temperature satellite imagery and a regional hydrodynamic model are used to investigate the variability and structure of the Liverpool Bay thermohaline front. A statistically based water mass classification technique is used to locate the front in both data sets. The front moves between 5 and 35 km in response to spring–neap changes in tidal mixing, an adjustment that is much greater than at other shelf-sea fronts. Superimposed on top of this fortnightly cycle are semi-diurnal movements of 5–10 km driven by flood and ebb tidal currents. Seasonal variability in the freshwater discharge and the density difference between buoyant inflow and more saline Irish Sea water give rise to two different dynamical regimes. During winter, when cold inflow reduces the buoyancy of the plume, a bottom-advected front develops. Over the summer, when warm river water provides additional buoyancy, a surface-advected plume detaches from the bottom and propagates much larger distances across the bay. Decoupled from near-bed processes, the position of the surface front is more variable. Fortnightly stratification and re-mixing over large areas of Liverpool Bay is a potentially important mechanism by which freshwater, and its nutrient and pollutant loads, are exported from the coastal plume system. Based on length scales estimated from model and satellite data, the erosion of post-neap stratification is estimated to be responsible for exporting approximately 19% of the fresh estuarine discharge annually entering the system. Although the baroclinic residual circulation makes a more significant contribution to freshwater fluxes, the episodic nature of the spring–neap cycle may have important implications for biogeochemical cycles within the bay.     

Analysis of an unconfined aquifer subject to asynchronous dual-tide propagation

Most published solutions for aquifer responses to ocean tides focus on the one-sided attenuation of the signal as it propagates inland. However, island aquifers experience periodic forcing from the entire coast, which can lead to integrated effects of different tidal signals, especially on narrow high-permeability islands. In general, studies disregard a potential time lag as the tidal wave sweeps around the island. We present a one-dimensional analytical solution to the ground water flow equation subject to asynchronous and asymmetric oscillating head conditions on opposite boundaries and test it on data from an unconfined volcanic aquifer in Maui. The solution considers sediment-damping effects at the coastline. The response of Maui Aquifers indicate that water table elevations near the center of the aquifer are influenced by a combination of tides from opposite coasts. A better match between the observed ground water head and the theoretical response can be obtained with the proposed dual-tide solution than with single-sided solutions. Hydraulic diffusivity was estimated to be 2.3 × 10⁷ m²/d. This translates into a hydraulic conductivity of 500 m/d, assuming a specific yield of 0.04 and an aquifer thickness of 1.8 km. A numerical experiment confirmed the hydraulic diffusivity value and showed that the y -intercepts of the modal attenuation and phase differences estimated by regression can approximate damping factors caused by low-permeability units at the boundary.     

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.     

Soil saturation index of salt marshes subjected to spring‐neap tides: a new variable for describing marsh soil aeration condition

Organized spatial distribution of plants (plant zonation) in salt marshes has been linked to the soil aeration condition in the rhizosphere through simplistic tidal inundation parameters. Here, a soil saturation index (ratio of saturation period to tidal period at a soil depth) is introduced to describe the soil aeration condition. This new index captures the effects of not only the tidal inundation period and frequency but also the flow dynamics of groundwater in the marsh soil. One‐dimensional numerical models based on saturated flow with the Boussinesq approximations and a two‐dimensional variably saturated flow model were developed to explore the behaviour of this new soil aeration variable under the influence of spring‐neap tides. Simulations revealed two characteristic zones of soil aeration across the salt marsh: a relatively well aerated near‐creek zone and a poorly aerated interior zone. In the near‐creek zone, soils undergo periodic wetting and drying as the groundwater table fluctuates throughout the spring‐neap cycle. In the interior, the soil remains largely water saturated except for neap tide periods when limited drainage occurs. Although such a change of soil aeration condition has been observed in previous numerical simulations, the soil saturation index provides a clear delineation of the zones that are separated by an ‘inflexion point’ on the averaged index curve. The results show how the saturation index represents the effects of soil properties, tidal parameters and marsh platform elevation on marsh soil aeration. Simulations of these combined effects have not been possible with traditional tidal inundation parameters. The saturation index can be easily derived using relatively simple models based on five non‐dimensional variables. Copyright © 2010 John Wiley & Sons, Ltd.     

Storage in confined aquifer: Spectral analysis of groundwater in responses to Earth tides and barometric effect

Dilatation of aquifer and associated water level fluctuation in groundwater well is known to be driven periodically from lunar, solar, or other tidal forces. Time‐dependent variables in groundwater system, such as water level, can be converted to power spectra in the frequency domain using Fourier transform to evaluate significant fluctuation. The major innovation of this research is to develop spectral representation in frequency domain for the groundwater system that the storage in confined aquifer can be determined considering dilatation affected by Earth tides and barometric effect. In order to verify applicability of the evolved method, time series of Earth tides and barograph are collected; aquifer storage is then determined inversely by selecting significant semidiurnal and diurnal components in spectra computation. It suggests that to discover groundwater storage using groundwater level with barograph and tidal potential of Earth in frequency domain becomes accessible and feasible.     

A two‐dimensional analytical solution for groundwater flow in a leaky confined aquifer system near open tidal water

Groundwater in coastal areas is commonly disturbed by tidal fluctuations. A two‐dimensional analytical solution is derived to describe the groundwater fluctuation in a leaky confined aquifer system near open tidal water under the assumption that the groundwater head in the confined aquifer fluctuates in response to sea tide whereas that of the overlying unconfined aquifer remains constant. The analytical solution presented here is an extension of the solution by Sun for two‐dimensional groundwater flow in a confined aquifer and the solution by Jiao and Tang for one‐dimensional groundwater flow in a leaky confined aquifer. The analytical solution is compared with a two‐dimensional finite difference solution. On the basis of the analytical solution, the groundwater head distribution in a leaky confined aquifer in response to tidal boundaries is examined and the influence of leakage on groundwater fluctuation is discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

The predictability of near-coastal currents using a baroclinic unstructured grid model

A limited domain, coastal ocean forecast system consisting of an unstructured grid model, a meteorological model, a regional ocean model, and a global tidal database is designed to be globally relocatable. For such a system to be viable, the predictability of coastal currents must be well understood with error sources clearly identified. To this end, the coastal forecast system is applied at the mouth of Chesapeake Bay in response to a Navy exercise. Two-day forecasts are produced for a 10-day period from 4 to 14 June 2010 and compared to real-time observations. Interplay between the temporal frequency of the regional model boundary forcing and the application of external tides to the coastal model impacts the tidal characteristics of the coastal current, even contributing a small phase error. Frequencies of at least 3 h are needed to resolve the tidal signal within the regional model; otherwise, externally applied tides from a database are needed to capture the tidal variability. Spatial resolution of the regional model (3 vs 1 km) does not impact skill of the current prediction. Tidal response of the system indicates excellent representation of the dominant M ₂ tide for water level and currents. Diurnal tides, especially K ₁, are amplified unrealistically with the application of coarse 27-km winds. Higher-resolution winds reduce current forecast error with the exception of wind originating from the SSW, SSE, and E. These winds run shore parallel and are subject to strong interaction with the shoreline that is poorly represented even by the 3-km wind fields. The vertical distribution of currents is also well predicted by the coastal model. Spatial and temporal resolution of the wind forcing including areas close to the shoreline is the most critical component for accurate current forecasts. Additionally, it is demonstrated that wind resolution plays a large role in establishing realistic thermal and density structures in upwelling prone regions.