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.     

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.     

Spectral analysis of water level fluctuations in aquifers

 The water level of a seawater gauging station and 18 groundwater wells coupled with atmospheric pressure in southwestern Taiwan are analyzed by using spectral analysis in time and frequency domain. The semidiurnal component is found to be the most significant signal from the measurement of water level and atmospheric pressure, and the diurnal component is less distinctive in part of water level and atmospheric pressure record. Although auto-spectral and cross-spectral density functions are significant in atmospheric pressure and water level, the pressure variations do not significantly affect the seawater and the majority of groundwater level in the study area with amplitude of time series observations. The astronomical tidal components are likely the main factor causing seawater and groundwater level to fluctuate in Pingtung, Taiwan. Time lags are estimated from 20 min to a few hours in aquifers. It concludes that the disturbance on groundwater levels from the effect of oceanic astronomical tide is different from the varying hydrogeological characteristics of aquifer. In this study, the spectral analysis of water level in time and frequency domains gives strong indications of sensitive variations to water level fluctuation. This research is supported by the Institute of Nuclear Energy Research (INER), AEC, Taiwan, Republic of China, under a fund from Executive Yuan. The authors thank Taiwan Central Weather Bureau and Taiwan Provincial Government Water Resources Department for providing useful data.     

Analytical stochastic solutions of saltwater/freshwater interface in coastal aquifers

Three analyitcal solutions of saltwater intrusion under uncertain hydrologic, hydrogeologic, and pumping conditions are presented. These solutions extends the existing deterministic, sharp interface solutions to stochastic ones. The randomness is represented in terms of statistical measures of mean, standard deviation and covariance. The analysis is based on perturbation using Taylor series expansion. Simulations based on probability distribution are conducted for verification.     

Analytical stochastic solutions of saltwater/freshwater interface in coastal aquifers

Three analyitcal solutions of saltwater intrusion under uncertain hydrologic, hydrogeologic, and pumping conditions are presented. These solutions extends the existing deterministic, sharp interface solutions to stochastic ones. The randomness is represented in terms of statistical measures of mean, standard deviation and covariance. The analysis is based on perturbation using Taylor series expansion. Simulations based on probability distribution are conducted for verification.     

Effect of climate change on sea water intrusion in coastal aquifers

There is increasing debate these days on climate change and its possible consequences. Much of this debate has focused in the context of surface water systems. In many arid areas of the world, rainfall is scarce and so is surface runoff. These areas rely heavily on groundwater. The consequences of climate change on groundwater are long term and can be far reaching. One of the more apparent consequences is the increased migration of salt water inland in coastal aquifers. Using two coastal aquifers, one in Egypt and the other in India, this study investigates the effect of likely climate change on sea water intrusion. Three realistic scenarios mimicking climate change are considered. Under these scenarios, the Nile Delta aquifer is found to be more vulnerable to climate change and sea level rise. Copyright © 1999 John Wiley & Sons, Ltd.     

Biased monitoring of fresh water-salt water mixing zone in coastal aquifers

In coastal aquifers, significant vertical hydraulic gradients are formed where fresh water and underlying salt water discharge together upward to the seafloor. Monitoring boreholes may act as "short circuits" along these vertical gradients, connecting between the higher and the lower hydraulic head zones. When a sea tide is introduced, the fluctuations of both the water table and the depth of the mixing zone are also biased due to this effect. This problem is intensified in places of long-screen monitoring boreholes, which are common in many places in the world. For example, all approximately 500 boreholes of the fresh water-salt water mixing zone in the coastal aquifer of Israel are installed with 10 to 50 m long screens. We present field measurements of these fluctuations, along with a three-dimensional numerical model. We find that the in-well fluctuation magnitude of the mixing zone is an order of magnitude larger than that in the porous media of the actual aquifer. The primary parameters that affect the magnitude of this bias are the anisotropy of the aquifer conductivity and the borehole hydraulic parameters. With no sea tide, borehole interference is higher for the anisotropic case because the vertical hydraulic gradients are high. When tides are introduced, the amplitude of the mixing zone fluctuation is higher for the isotropic case because the overall effective hydraulic conductivity is greater than the conductivity in the anisotropic case. In the aquifer, the fresh water-salt water mixing zone fluctuations are dampened, and tens of meters inland from the shoreline, the fluctuations are on the order of few centimeters.     

Groundwater ages in coastal aquifers

The interpretation of groundwater ages in coastal aquifers requires an improved understanding of relationships between age distributions and the processes accompanying dispersive, density-dependent flow and transport. This study uses numerical modeling to examine the influence of mixing and a selection of other hydrogeological factors on steady-state age distributions in coastal aquifers. Three methods of age estimation are compared: the piston flow age, the direct age, and the tracer-based age. These are applied to various forms of the Henry problem, as well as to three variants of a larger, hypothetical coastal aquifer. Circulation of water within the seawater wedge results in markedly higher ages in the transition zone than in the underlying saltwater or overlying freshwater. Piston flow ages show a sharp increase where the freshwater and saltwater systems meet, whereas direct- and tracer-based simulations result in a smoother age distribution, as expected. Greater degrees of mixing result in larger differences between piston flow and direct or tracer-based ages, and bring about lower ages in the saltwater wedge. Tracer-based ages are preferred over direct- and piston flow ages for comparison with field data, especially in cases with wide transition zones. Despite the relatively simple conditions used for the simulations, complex age distributions with depth were obtained. Hence, the assessment of ages in field cases will be difficult, particularly where accurate ages in the transition zone are sought.     

Analytic solutions for interface flow in combined confined and semi-confined,coastal aquifers

Analytic solutions are presented for steady interface flow in aquifers consisting of a confined and a semi-confined section. The total discharge is constant in the confined section and is directed towards the semi-confined section, which is bounded on top by a leaky layer that separates the aquifer from the sea. All solutions are based on the Dupuit approximation. The interface position is fully determined by two dimensionless parameters. The first parameter is the product of the uniform gradient towards the sea and the dimensionless leakage factor divided by the dimensionless density difference. The second dimensionless parameter is the length of the semi-confined section divided by the leakage factor. Four types of flow are distinguished. For cases I and II the interface does not reach the end of the semi-confined section, while it does for cases III and IV. For cases I and III the interface extends into the confined section, while for cases II and IV it remains entirely in the semi-confined section. Analytic solutions are presented for the position of the interface for all four cases. Diagrams are presented to determine the type of flow and the position of the interface toe based on the two dimensionless parameters. The pre-development position of the interface along the Georgia coast is computed as an illustration of the practical application of the presented formulas. Another practical application is the estimation of how far a numerical model of seawater intrusion should be extended into the sea for accurate simulations.     

An analytical solution for ground water transit time through unconfined aquifers

An exact, closed-form analytical solution is developed for calculating ground water transit times within Dupuit-type flow systems. The solution applies to steady-state, saturated flow through an unconfined, horizontal aquifer recharged by surface infiltration and discharging to a downgradient fixed-head boundary. The upgradient boundary can represent, using the same equation, a no-flow boundary or a fixed head. The approach is unique for calculating travel times because it makes no a priori assumptions regarding the limit of the water table rise with respect to the minimum saturated aquifer thickness. The computed travel times are verified against a numerical model, and examples are provided, which show that the predicted travel times can be on the order of nine times longer relative to existing analytical solutions.     

New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions

The hydraulic head distribution in a wedge-shaped aquifer depends on the wedge angle and the topographic and hydrogeological boundary conditions. In addition, an equation in terms of the radial distance with trigonometric functions along the boundary may be suitable to describe the water level configuration for a valley flank with a gentle sloping and rolling topography. This paper develops a general mathematical model including the governing equation and a variety of boundary conditions for the groundwater flow within a wedge-shaped aquifer. Based on the model, a new closed-form solution for transient flow in the wedge-shaped aquifer is derived via the finite sine transform and Hankel transform. In addition, a numerical approach, including the roots search scheme, the Gaussian quadrature, and Shanks’ method, is proposed for efficiently evaluating the infinite series and the infinite integral presented in the solution. This solution may be used to describe the head distribution for wedges that image theory is inapplicable, and to explore the effects of the recharge from various topographic boundaries on the groundwater flow system within a wedge-shaped aquifer.     

A grid generating algorithm for simulating a fluctuating water table boundary in heterogeneous unconfined aquifers

An algorithm is presented for generating finite element grids that can be used to calculate the position of a fluctuating water table and the formation of seepage faces within a heterogeneous unconfined aquifer. Our approach overcomes limitations with existing techniques by allowing the water table to rise or decline through hydrostratigraphic boundaries yet maintains numerical and conceptual accuracy with respect to hydrostratigraphic geometry. The algorithm involves (1) limited stretching or shrinking of elements along the water table if the change in the position of the water table is small with respect to the vertical grid spacing, and (2) the addition or removal of nodes and elements in the finite element mesh along the water table as the change becomes large with respect to the vertical grid spacing. This technique is applicable to any 2-D or 3-D finite element code that contains an automatic finite-element grid generator.     

Analytical models of contaminant transport in coastal aquifers

The Henry formulation, which couples subsurface flow and salt transport via a variable-density flow formulation, can be used to evaluate the extent of sea water intrusion into coastal aquifers. The coupling gives rise to nontrivial flow patterns that are very different from those observed in inland aquifers. We investigate the influence of these flow patterns on the transport of conservative contaminants in a coastal aquifer. The flow is characterized by two dimensionless parameters: the Péclet number, which compares the relative effects of advective and dispersive transport mechanisms, and a coupling parameter, which describes the importance of the salt water boundary on the flow. We focus our attention on two regimes – low and intermediate Péclet number flows. Two transport scenarios are solved analytically by means of a perturbation analysis. The first, a natural attenuation scenario, describes the flushing of a contaminant from a coastal aquifer by clean fresh water, while the second, a contaminant spill scenario, considers an isolated point source.     

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.     

Optimal and sustainable extraction of groundwater in coastal aquifers

Four examples are investigated for the optimal and sustainable extraction of groundwater from a coastal aquifer under the threat of seawater intrusion. The objectives and constraints of these management scenarios include maximizing the total volume of water pumped, maximizing the profit of selling water, minimizing the operational and water treatment costs, minimizing the salt concentration of the pumped water, and controlling the drawdown limits. The physical model is based on the density-dependent advective-dispersive solute transport model. Genetic algorithm is used as the optimization tool. The models are tested on a hypothetical confined aquifer with four pumping wells located at various depths. These solutions establish the feasibility of simulating various management scenarios under complex three-dimensional flow and transport processes in coastal aquifers for the optimal and sustainable use of groundwater.     

Geophysical monitoring and evaluation of coastal plain aquifers

We use time domain electromagnetic (TDEM) soundings to monitor ground water conditions beneath the coastal plain in eastern North Carolina. The TDEM method measures the earth's response to an induced electromagnetic field. The resulting signal is converted, through a complex inversion process, to apparent resistivity values, which can be directly correlated to borehole resistivity logs. TDEM soundings are used to map the interface between fresh and salt water within coastal aquifers, and estimate depth to basement when siting new monitoring wells. Focused TDEM surveys have identified areas of salt water encroachment caused by high volumes of discharge from local supply wells. Electromagnetic sounding, when used in tandem with the state's network of monitoring wells, is an accurate and inexpensive tool for evaluating fresh water/salt water relationships on both local and regional scales within coastal plain aquifers.     

Source speciation resolving hydrochemical complexity of coastal aquifers

There is a growing concern of seawater intrusion to freshwater aquifers due to groundwater overexploitation in the eastern coastal belt of Southern India. The problem becomes complex in the regions where industrial effluents are also contaminating the freshwater aquifers. In order to understand the hydrochemical complexity of the system, topographic elevation, static water level measurements, major ion chemistry, ionic cross plots, water type contours and factor analysis were applied for 144 groundwater samples of shallow and deep sources from Quaternary and Tertiary coastal aquifers, located within the industrial zone of 25 km² area near Cuddalore, Southern India. The ionic cross plots indicates dissolution of halite minerals from marine sources and seawater mixing into inland aquifers up to the level of 9.3%. The factor analysis explains three significant factors totaling 86.3% of cumulative sample variance which includes varying contribution from marine, industrial effluent and freshwater sources.     

Tidal influence on BTEX biodegradation in sandy coastal aquifers

A numerical study was conducted to investigate the influence of tides on the fate of terrestrially derived BTEX discharging through an unconfined aquifer to coastal waters. Previous studies have revealed that tide-induced seawater circulations create an active salt–freshwater mixing zone in the near-shore aquifer and alter the specific subsurface pathway for contaminants discharging to the coastal environment. Here the coupled density-dependent flow and multi-species reactive transport code PHWAT was used to examine the impact of these tidal effects on the aerobic biodegradation of BTEX released in a coastal aquifer and its subsequent loading to coastal waters. Simulations indicated that tides significantly enhance BTEX attenuation in the near-shore aquifer. They also reduce the rate of chemical transfer from the aquifer to the ocean and exit concentrations at the beach face. For the base case consisting of toluene transport and biodegradation, 79% of toluene initially released in the aquifer was attenuated prior to discharge with tides present, compared to only 1.8% for the non-tidal case. The magnitude of tidal forcing relative to the fresh groundwater flow rate was shown to influence significantly the extent of biodegradation as it controls the intensity of salt–freshwater mixing, period of exposure of the contaminant to the mixing zone and rate of oxygen delivery to the aquifer. The oxygen available for biodegradation also depends on the rate at which oxygen is consumed by natural processes such as organic matter decomposition. While simulations conducted with heterogeneous conductivity fields highlighted the uncertainties associated with predicting contaminant loadings, the study revealed overall that BTEX may undergo significant attenuation in tidally influenced aquifers prior to discharge.     

Assessing radial transmissivity variation in heterogeneous aquifers using analytical techniques

Pumping tests are one of the most commonly used in situ testing techniques for assessing aquifer hydraulic properties. Numerous researches have been conducted to predict the effects of aquifer heterogeneity on the groundwater levels during pumping tests. The objectives of the present work were as follows: (1) to predict drawdown conditions and to estimate aquifer properties during pumping tests undertaken in radially symmetric heterogeneous aquifers, and (2) to identify a method for assessing the transmissivity field along the radial coordinate in radially symmetric and fully heterogeneous transmissivity fields. The first objective was achieved by expanding an existing analytical drawdown formulation that was valid for a radially symmetric confined aquifer with two concentric zones around the pumping well to an N concentric zone confined aquifer having a constant transmissivity value within each zone. The formulation was evaluated for aquifers with three and four concentric zones to assess the effects of the transmissivity field on the drawdown conditions. The specific conditions under which aquifer properties could be identified using traditional methods of analysis were also evaluated. The second objective was achieved by implementing the inverse solution algorithm (ISA), which was developed for petroleum reservoirs to groundwater aquifer settings. The results showed that the drawdown values are influenced by a volumetric integral of a weighting function and the transmissivity field within the cone of depression. The weighting function migrates in tandem with the expanding cone of depression. The ability of the ISA to predict radially symmetric and log‐normally distributed transmissivity fields was assessed against analytical and numerical benchmarks. The results of this investigation indicated that the ISA method is a viable technique for evaluating the radial transmissivity variations of heterogeneous aquifer settings. Copyright © 2013 John Wiley & Sons, Ltd.