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 general solution for tide‐induced groundwater fluctuation in an estuarine‐coastal confined and unconfined aquifer system

This paper presents an analytical solution to tide‐induced head fluctuations in a two‐dimensional estuarine‐coastal aquifer system that consists of an unconfined aquifer and a heterogeneous confined aquifer extending under a tidal river with a semipermeable layer between them. This study considers the joint effects of tidal‐river leakage, inland leakage, dimensionless transmissivity between the tidal‐river and inland confined aquifer, and transmissivity anisotropic ratios. The analytical solution for this model is obtained via the separation of variables method. Three existing solutions related to head fluctuation in one‐ or two‐dimensional leaky confined aquifers are considered as special cases in the present solution. This study shows that there is a threshold of tidal‐river confined aquifer length. When the tidal‐river length is greater than the threshold length, the inland head fluctuations remain sensitive to the leakage effect but become insensitive to the tidal‐river width and dimensionless transmissivity. Considering leakage and transmissivity anisotropy, this study also demonstrates that at a location farther from the river–inland boundary, head fluctuations increase with increasing leakage and transmissivity anisotropy; the maximum head fluctuation occurs when leakage and transmissivity anisotropy are both at their maximum values. The combined action of the 3 effects of loading, tidal‐river aquifer leakage, and inland aquifer leakage differs significantly according to various aquifer parameters. The analytical solution in this paper can be applied to demonstrate the behaviours of the head fluctuations of an estuarine‐coastal aquifer system, and the head fluctuations can be clearly described when the tidal and hydrogeological parameters are derived from field measurement data or hypothetical cases.     

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.     

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.     

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.     

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.     

Groundwater response to tidal fluctuations in a leaky confined coastal aquifer with a finite length

The study on the hydraulic properties of coastal aquifers has significant implications both in hydrological sciences and environmental engineering. Although many analytical solutions are available, most of them are based on the same basic assumption that assumes aquifers extend landward semi‐infinitely, which does not necessarily reflect the reality. In this study, the general solutions for a leaky confined coastal aquifer have been developed that consider both finitely landward constant‐head and no‐flow boundaries. The newly developed solutions were then used to examine theoretically the joint effects of leakage and aquifer length on hydraulic head fluctuations within the leaky confined aquifer, and the validity of using the simplified solution, which assumes the aquifer is semi‐infinite. The results illustrated that the use of the simplified solution may cause significant errors, depending on joint effects of leakage and aquifer length. A dimensionless characteristic parameter was then proposed as an index for judging the applicability of the simplified solution. In addition, practical application of the general solution for the constant‐head inland boundary was used to characterize the hydraulic properties of a leaky confined aquifer using the data collected from a field site at the Seine River estuary, France, and the versatility of the general solution was further justified.     

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.     

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.     

Maximizing Net Extraction Using an Injection‐Extraction Well Pair in a Coastal Aquifer

In this study, we examine the maximum net extraction rate from the novel arrangement of an injection‐extraction well pair in a coastal aquifer, where fresh groundwater is reinjected through the injection well located between the interface toe and extraction well. Complex potential theory is employed to derive a new analytical solution for the maximum net extraction rate and corresponding stagnation‐point locations and recirculation ratio, assuming steady‐state, sharp‐interface conditions. The injection‐extraction well‐pair system outperforms a traditional single extraction well in terms of net extraction rate for a broad range of well placement and pumping rates, which is up to 50% higher for an aquifer with a thickness of 20 m, hydraulic conductivity of 10 m/d, and fresh water influx of 0.24 m²/d. Sensitivity analyses show that for a given fresh water discharge from an inland aquifer, a larger maximum net extraction is expected in cases with a smaller hydraulic conductivity or a smaller aquifer thickness, notwithstanding physical limits to drawdown at the pumping well that are not considered here. For an extraction well with a fixed location, the optimal net extraction rate linearly increases with the distance between the injection well and the sea, and the corresponding injection rate and recirculation ratio also increase. The analytical analysis in this study provides initial guidance for the design of well‐pair systems in coastal aquifers, and is therefore an extension beyond previous applications of analytical solutions of coastal pumping that apply only to extraction or injection wells.     

Flow to a well in a five-layer system with application to the Oxnard Basin

Nearly 40 years ago, Neuman (1968) developed an analytical solution for drawdown in a system of three aquifers separated by two aquitards when one of the aquifers is pumped at a constant rate. Whereas the simpler case of two aquifers separated by one aquitard has been presented by Neuman and Witherspoon (1969a), the full five-layer solution has not been previously evaluated. We do so here using numerical inversion of its Laplace-transformed version and present selected results graphically in dimensionless form. The solution demonstrates that the effect of pumping propagates across all five layers, adding emphasis to a question previously raised by Neuman and Witherspoon about the validity of leaky aquifer theories that disregard drawdowns in unpumped aquifers. A large-scale, long-term pumping test spanning three aquifers separated by two aquitards near Oxnard, California, has been conducted and analyzed by Neuman and Witherspoon (1972). They evaluated the vertical hydraulic diffusivities of the aquitards using the Neuman-Witherspoon ratio method and their specific storage values on the basis of laboratory consolidation tests. We reinterpret the Oxnard pumping test by coupling the five-layer analytical solution of Neuman (1968) with the parameter estimation code PEST (Doherty 2002) and validate our results against drawdowns from a subsequent pumping test at the site. Our parameter estimates compare favorably with those of Neuman and Witherspoon (1972).     

Vulnerability indicators of sea water intrusion

In this paper, simple indicators of the propensity for sea water intrusion (SWI) to occur (referred to as "SWI vulnerability indicators") are devised. The analysis is based on an existing analytical solution for the steady-state position of a sharp fresh water-salt water interface. Interface characteristics, that is, the wedge toe location and sea water volume, are used in quantifying SWI in both confined and unconfined aquifers. Rates-of-change (partial derivatives of the analytical solution) in the wedge toe or sea water volume are used to quantify the aquifer vulnerability to various stress situations, including (1) sea-level rise; (2) change in recharge (e.g., due to climate change); and (3) change in seaward discharge. A selection of coastal aquifer cases is used to apply the SWI vulnerability indicators, and the proposed methodology produces interpretations of SWI vulnerability that are broadly consistent with more comprehensive investigations. Several inferences regarding SWI vulnerability arise from the analysis, including: (1) sea-level rise impacts are more extensive in aquifers with head-controlled rather than flux-controlled inland boundaries, whereas the opposite is true for recharge change impacts; (2) sea-level rise does not induce SWI in constant-discharge confined aquifers; (3) SWI vulnerability varies depending on the causal factor, and therefore vulnerability composites are needed that differentiate vulnerability to such threats as sea-level rise, climate change, and changes in seaward groundwater discharge. We contend that the approach is an improvement over existing methods for characterizing SWI vulnerability, because the method has theoretical underpinnings and yet calculations are simple, although the coastal aquifer conceptualization is highly idealized.     

Finite layer method for flow in layered radial two-zone aquifer systems

A new finite layer method (FLM) is presented in this paper for transient flow analysis in layered radial two-zone aquifer systems. A radial two-zone system is an aquifer configuration in which a circular aquifer with finite radius is surrounded by a matrix possessing different permeability and storage properties. The aquifers can be pumped from fully or partially penetrating wells of infinitesimal radius. The trial function for drawdown is obtained through the use of piecewise linear correction functions in the present method. The trial function can satisfy the continuity conditions of flow and possess an appropriate continuity of C(0) at the two-zone interface. On the basis of Galerkin's method and the continuity condition of flow, the finite layer formulation is derived. The proposed method can cope with the anisotropy and layered heterogeneity in radial two-zone aquifer systems. Several numerical examples are presented to verify the validity of the present method through comparison with the analytical solution and the numerical results based on the finite difference method, in which a test of three-dimensional (3D) flow to a partially penetrating well in anisotropic two-zone aquifers is included. Furthermore, an additional application in simulating the two-zone flow in aquitard-aquifer systems is presented to demonstrate the applicability of FLM in modeling flow in more complex aquifer systems.     

Rainfall infiltration‐derived submarine groundwater discharge from multi‐layered aquifer system terminating at the coastline

This article investigates the quantity of submarine groundwater discharge (SGD) from a coastal multi‐layered aquifer system in response to constant rainfall infiltration. The system comprises an unconfined aquifer, a leaky confined aquifer and an aquitard between them and terminates at the coastline. An approximate analytical solution is derived based on the following assumptions: (i) flow is horizontal in the aquifers and vertical in the aquitard, and (ii) flow in the unconfined aquifer is described by nonlinear Boussinesq equation. The analytical solution is compared with numerical solutions of the strictly two‐dimensional nonlinear model to validate the model assumptions used for the analytical solution. The SGD from the leaky confined aquifer increases with the inland rainfall infiltration recharge and the specific leakage of aquitard. The maximum SGD ranges from 1·87 to 10·37 m³ per day per meter of shoreline when rainfall infiltration ranges from 18·2 to 182 mm/year and the specific leakage of aquitard varies from 10^?9 to 10^?1 l/day. Copyright © 2011 John Wiley & Sons, Ltd.     

A general analytical solution for flow to a single horizontal well by Fourier and Laplace transforms

The objective of this paper is to present an analytical solution for describing the head distribution in an unconfined aquifer with a single pumping horizontal well parallel to a fully penetrating stream. The Laplace-domain solution is developed by applying Fourier sine, Fourier and Laplace transforms to the governing equation as well as the associated initial and boundary conditions. The time-domain solution is obtained after taking the inverse Laplace transform along with the Bromwich integral method and inverse Fourier and Fourier sine transforms. The upper boundary condition of the aquifer is represented by the free surface equation in which the second-order slope terms are neglected. Based on the solution and Darcy’s law, the equation representing the stream depletion rate is then derived. The solution can simulate head distributions in an aquifer infinitely extending in horizontal direction if the well is located far away from the stream. In addition, the solution can also simulate head distributions in confined aquifers if specific yield is set zero. It is shown that the solution can be applied practically to evaluate flow to a horizontal well.     

Analytical studies of groundwater-head fluctuation in a coastal confined aquifer overlain by a semi-permeable layer with storage

Analytical studies are carried out to investigate groundwater-head changes in a coastal aquifer system in response to tidal fluctuations. The system consists of an unconfined aquifer, a semi-confined aquifer and a semi-permeable confining unit between them. An exact analytical solution is derived to investigate the influences of both leakage and storage of the semi-permeable layer on the tide-induced groundwater-head fluctuation in the semi-confined aquifer. This solution is a generalization of the solution obtained by Jiao and Tang (Water Resource Research 35 (1999) 747–751) which ignored the storage of the semi-confining unit. The analytical solution indicates that both storage and leakage of the semi-permeable layer play an important role in the groundwater-head fluctuation in the confined aquifer. While leakage is generally more important than storage, the impact of storage on groundwater-head fluctuations changes with leakage. With the increase of leakage the fluctuation of groundwater-head in the confined aquifer will be controlled mainly by leakage. The study also demonstrates that the influence of storativity of the semi-permeable layer on groundwater-head fluctuation is negligible only when the storativity of the semi-permeable layer is comparable to or smaller than that of the confined aquifer. However, for aquifer systems with semi-permeable layer composed of thick, soft sedimentary materials, the storativity of the semi-permeable layer is usually much greater than that of the aquifer and its influence should be considered.     

Estimation of the inland extending length of the freshwater–saltwater interface in coastal unconfined aquifers

ABSTRACT

The inland extending length of the freshwater–saltwater interface toe is useful in studies of seawater intrusion in coastal areas. The submarine fresh groundwater discharge in coastal zones is affected not only by hydraulic conductivity and hydraulic gradient of the aquifer, but also by the position of the interface. Two observation wells at different distances from the coast are required to calculate the fresh groundwater flow rate in coastal unconfined aquifers. By considering that the submarine groundwater discharge is equal to the groundwater flow rate, the length of the interface toe extending inland can be estimated when the groundwater flow is at a steady-flow state. Aquifers with horizontal and sloping confined beds and without/with unique surface vertical infiltration are considered. Examples used to illustrate the application of these methods indicate that the inland extending lengths of the interface toe in aquifers with vertical surface infiltration are much shorter than those in aquifers without vertical surface infiltration, and the length of the interface in aquifers with a horizontal confining lower bed are smaller than those in aquifers with a confining lower bed sloping towards the sea. The extent of the interface on the northwestern coast near the city of Beihai in southern Guangxi, China, on 18 January 2013 was estimated as 471–478 m.

Editor M.C. Acreman Associate editor not assigned     

Aquifer response to stream-stage and recharge variations. I. Analytical step-response functions

Laplace transform step-response functions are presented for various homogeneous confined and leaky aquifer types and for anisotropic, homogeneous unconfined aquifers interacting with perennial streams. Flow is one-dimensional, perpendicular to the stream in the confined and leaky aquifers, and two-dimensional in a plane perpendicular to the stream in the water-table aquifers. The stream is assumed to penetrate the full thickness of the aquifer. The aquifers may be semi-infinite or finite in width and may or may not be bounded at the stream by a semipervious streambank. The solutions are presented in a unified manner so that mathematical relations among the various aquifer configurations are clearly demonstrated. The Laplace transform solutions are inverted numerically to obtain the real-time step-response functions for use in the convolution (or superposition) integral. To maintain linearity in the case of unconfined aquifers, fluctuations in the elevation of the water table are assumed to be small relative to the saturated thickness, and vertical flow into or out of the zone above the water table is assumed to occur instantaneously. Effects of hysteresis in the moisture distribution above the water table are therefore neglected. Graphical comparisons of the new solutions are made with known closed-form solutions.     

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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.     

Analytical solutions of tidal groundwater flow in coastal two-aquifer system

This paper presents a complete analytical solution to describe tidal groundwater level fluctuations in a coastal subsurface system. The system consists of two aquifers and a leaky layer between them. Previous solutions of Jacob [Flow of groundwater, in: H. Rouse (Ed.), Engineering Hydraulics, Wiley, New York, 1950, pp. 321–386], Jiao and Tang [Water Resour. Res. 35 (3) (1999) 747], Li and Jiao [Adv. Water Resour. 24 (5) (2001a) 565], Li et al. [Water Resour. Res. 37 (2001) 1095] and Jeng et al. [Adv. Water Resour. (in press)] are special cases of the new solution. The present solution differs from previous work in that both the effects of the leaky layer's elastic storage and the tidal wave interference between the two aquifers are considered. If the upper and lower aquifers have the same storativities and transimissivities, the system can be simplified into an equivalent double-layered, aquifer–aquitard system bounded by impermeable layers from up and down. It is found that the leaky layer's elastic storage behaves as a buffer to the tidal wave interference between the two aquifers. The buffer capacity increases with the leaky layer's thickness, specific storage, and decreases with the leaky layer's vertical permeability. Great buffer capacity can result in negligible tidal wave interference between the upper and lower aquifers so that the Li and Jiao (loc. cit.) solution applies.