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.     

Radial groundwater flow to a finite diameter well in a leaky confined aquifer with a finite‐thickness skin

A mathematical model that describes the drawdown due to constant pumpage from a finite radius well in a two‐zone leaky confined aquifer system is presented. The aquifer system is overlain by an aquitard and underlain by an impermeable formation. A skin zone of constant thickness exists around the wellbore. A general solution to a two‐zone leaky confined aquifer system in Laplace domain is developed and inverted numerically to the time‐domain solution using the modified Crump (1976) algorithm. The results show that the drawdown distribution is significantly influenced by the properties and thickness of the skin zone and aquitard. The sensitivity analyses of parameters of the aquifer and aquitard are performed to illustrate their effects on drawdowns in a two‐zone leaky confined aquifer system. For the negative‐skin case, the drawdown is very sensitive to the relative change in the formation transmissivity. For the positive‐skin case, the drawdown is also sensitive to the relative changes in the skin thickness, and both the skin and formation transmissivities over the entire pumping period and the well radius and formation storage coefficient at early pumping time. Copyright © 2009 John Wiley & Sons, Ltd.     

Seawater intrusion in continuous coastal aquifer-aquitard systems

The problem of seawater intrusion in a confined coastal aquifer is investigated. The aquifer is overlain by a leaky aquitard and both units extend continuously out under the sea. The problem is formulated in terms of the two governing differential equations, the fluid continuity equation written conveniently in terms of equivalent freshwater head, and the mass continuity equation. Use of linear rectangular finite elements, with direct integration and an iterative solution technique lead to an efficient numerical scheme that is capable of handling long simulation periods. The results, for a 300 m thick aquifer overlain by a 100 m thick aquitard, show that the aquitard has a controlling influence on the salt distribution. A zone of mixing in the aquifer is found to extend for several kilometres in the seaward as well as the landward direction. The time period required by the system to attain a state of dynamic equilibrium after a perturbation is applied may be of the order of centuries. The aquitard is found to cause a downward and seaward displacement of the average salt front.     

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.     

On the aquifer's integrated balance equations

A general methodology is presented for describing transport phenomena in porous media at a macroscopic level. Then, these macroscopic balance equations are integrated (or averaged) along the vertical for confined, leaky and phreatic aquifers.The results are employed to derive (averaged) aquifer equations for the flow of water and of a solute (hydrodynamic dispersion). It is shown that in all cases, the resulting equation is identical to that derived on the basis of an assumption of horizontal flow (the Dupuit assumption).Macrodispersion, occurring at the aquifer level, is discussed and appropriate coefficients are proposed.     

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.     

A general solution for groundwater flow in an estuarine's leaky aquifer system after considering aquifer anisotropy

This paper presents an analytical model for describing the tidal effects in a two‐dimensional leaky confined aquifer system in an estuarine delta where ocean and river meet. This system has an unconfined aquifer on top and a confined aquifer on the bottom with an aquitard in between the two. The unconfined and confined aquifers interact with each other through leakage. It was assumed that the aquitard storage was negligible and that the leakage was linearly proportional to the head difference between the unconfined and confined aquifers. This model's solution was based on the separation of variables method. Two existing solutions that deal with the head fluctuation in one‐dimensional or two‐dimensional leaky confined aquifers are shown as special cases in the present solution. Based on this new solution, the dynamic effect of the water table's fluctuations can be clearly explored, as well as the influence of leakage on the behaviour of fluctuations in groundwater levels in the leaky aquifer system. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary depletion rate and drawdown in leaky wedge‐shaped aquifers

This study presents analytical solutions of the three‐dimensional groundwater flow to a well in leaky confined and leaky water table wedge‐shaped aquifers. Leaky wedge‐shaped aquifers with and without storage in the aquitard are considered, and both transient and steady‐state drawdown solutions are derived. Unlike the previous solutions of the wedge‐shaped aquifers, the leakages from aquitard are considered in these solutions and unlike similar previous work for leaky aquifers, leakage from aquitards and from the water table are treated as the lower and upper boundary conditions. A special form of finite Fourier transforms is used to transform the z‐coordinate in deriving the solutions. The leakage induced by a partially penetrating pumping well in a wedge‐shaped aquifer depends on aquitard hydraulic parameters, the wedge‐shaped aquifer parameters, as well as the pumping well parameters. We calculate lateral boundary dimensionless flux at a representative line and investigate its sensitivity to the aquitard hydraulic parameters. We also investigate the effects of wedge angle, partial penetration, screen location and piezometer location on the steady‐state dimensionless drawdown for different leakage parameters. Results of our study are presented in the form of dimensionless flux‐dimensionless time and dimensionless drawdown‐leakage parameter type curves. The results are useful for evaluating the relative role of lateral wedge boundaries and leakage source on flow in wedge‐shaped aquifers. This is very useful for water management problems and for assessing groundwater pollution. The presented analytical solutions can also be used in parameter identification and in calculating stream depletion rate and volume. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

General well function for soil vapor extraction

This paper develops a well function applicable to extraction of groundwater or soil vapor from a well under the most common field test conditions. The general well function (Perina and Lee, 2006) [12] is adapted to soil vapor extraction and constant head boundary at the top. For groundwater flow, the general well function now applies to an extraction well of finite diameter with uniform drawdown along the screen, finite-thickness skin, and partially penetrating an unconfined, confined, and leaky aquifer, or an aquifer underneath a reservoir. With a change of arguments, the model applies to soil vapor extraction from a vadose zone with no cover or with leaky cover at the ground surface. The extraction well can operate in specified drawdown (pressure for soil vapor) or specified flowrate mode. Frictional well loss is computed as flow-only dependent component of the drawdown inside the extraction well. In general case, the calculated flow distribution is not proportional to screen length for a multiscreen well.     

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.     

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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.     

Hydraulic modeling of riverbank filtration systems with curved boundaries using analytic elements and series solutions

A new analytic solution approach is presented for the modeling of steady flow to pumping wells near rivers in strip aquifers; all boundaries of the river and strip aquifer may be curved. The river penetrates the aquifer only partially and has a leaky stream bed. The water level in the river may vary spatially. Flow in the aquifer below the river is semi-confined while flow in the aquifer adjacent to the river is confined or unconfined and may be subject to areal recharge. Analytic solutions are obtained through superposition of analytic elements and Fourier series. Boundary conditions are specified at collocation points along the boundaries. The number of collocation points is larger than the number of coefficients in the Fourier series and a solution is obtained in the least squares sense. The solution is analytic while boundary conditions are met approximately. Very accurate solutions are obtained when enough terms are used in the series. Several examples are presented for domains with straight and curved boundaries, including a well pumping near a meandering river with a varying water level. The area of the river bottom where water infiltrates into the aquifer is delineated and the fraction of river water in the well water is computed for several cases.     

Non-Darcian groundwater flow in leaky aquifers

Abstract

A simplified method has been developed for solving leaky aquifer non-Darcian flow hydraulics. The principle of volumetric approach is combined with the confined-aquifer, time-dependent drawdown equation in an observation well. The groundwater flow in the leaky aquifer is assumed to obey a non-Darcian flow law of exponential type. The results are obtained in the form of type-curve expressions from which the necessary bundles of curves are drawn for a set of selective non-Darcian flow aquifer parameters. Although application of the methodology appears as rather limited but it provides a scientific contribution and extension of leaky aquifer theory towards nonlinear flow conditions. The methodology developed herein is applied to some actual field data from the eastern sedimentary basin in the Kingdom of Saudi Arabia.     

Non-Darcian flow to a well in an aquifer–aquitard system

In this study, we use a linearization procedure and a finite difference method to solve non-Darcian flow to a well in an aquifer–aquitard system. The leakage effect is considered. Flow in the aquifer is assumed to be non-Darcian and horizontal, whereas flow in the aquitard is assumed to be Darcian and vertical. The Izbash equation [Izbash SV. O filtracii V Kropnozernstom Materiale. USSR: Leningrad; 1931 [in Russian]] is employed to describe the non-Darcian flow. The wellbore storage is also considered in this study. An approximate semi-analytical solution has been obtained by the linearization procedure, and a numerical solution has been obtained by using a finite difference method. The previous solutions for Darcian flow case and non-Darcian flow case without leakage can be described as special cases of the new solutions. The error caused by the linearization procedure has also been analyzed. The relative error caused by the linearization procedure is nearly 100% at early times, and decreases to zero at late times. We have also compared the results in this study with Wen et al. [Wen Z, Huang G, Zhan H. A numerical solution for non-Darcian flow to a well in a confined aquifer using the power law function. J Hydrol, 2008d [in revision]] in which the leakage effect is not considered, and Hantush and Jacob [Hantush MS, Jacob CE. Non-steady radial flow in an infinite leaky aquifer. Trans Am Geophys Union 1955;36(1):95–100] who investigated a similar problem in Darcian flow case. The comparison of this study and Wen et al. (2008d) indicates the dimensionless drawdown in the aquifer with leakage is less than that without leakage, and the leakage has little effect at early times. The comparison between the results of this study and that of Hantush and Jacob (1955) indicates that the dimensionless drawdown in the aquifer for non-Darcian flow is larger at early times and smaller at late times, than their counterparts for Darcian flow. A larger dimensionless non-Darcian conductivity k_D results in a smaller dimensionless drawdown in the aquifer at late times, and leads to a larger dimensionless drawdown in the aquifer at early times. A smaller dimensionless leakage parameter B_D results in a smaller drawdown at late times, and the leakage does not affect the early-time drawdown. The analysis of the dimensionless drawdown inside the well has also been included in this study when the wellbore storage is considered.     

A new method for aquifer system identification and parameter estimation

The standard practice for assessing aquifer parameters is to match groundwater drawdown data obtained during pumping tests against theoretical well function curves specific to the aquifer system being tested. The shape of the curve derived from the logarithmic time derivative of the drawdown data is also very frequently used as a diagnostic tool to identify the aquifer system in which the pumping test is being conducted. The present study investigates the incremental area method (IAM) to serve as an alternative diagnostic tool for the aquifer system identification as well as a supplement to the aquifer parameter estimation procedure. The IAM based diagnostic curves for ideal confined, leaky, bounded and unconfined aquifers have been derived as part of this study, and individual features of the plots have been identified. These features were noted to be unique to each aquifer setting, which could be used for rapid evaluation of the aquifer system. The effectiveness of the IAM methodology was investigated by analyzing field data for various aquifer settings including leaky, unconfined, bounded and heterogeneous conditions. The results showed that the proposed approach is a viable method for use as a diagnostic tool to identify the aquifer system characteristics as well as to support the estimation of the hydraulic parameters obtained from standard curve matching procedures. Copyright © 2012 John Wiley & Sons, Ltd.     

A double-porosity model for a fractured aquifer with non-Darcian flow in fractures

Abstract

Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model.     

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.     

A New Package in MODFLOW to Simulate Unconfined Groundwater Flow in Sloping Aquifers

The nonhorizontal‐model‐layer (NHML) grid system is more accurate than the horizontal‐model‐layer grid system to describe groundwater flow in an unconfined sloping aquifer on the basis of MODFLOW‐2000. However, the finite‐difference scheme of NHML was based on the Dupuit‐Forchheimer assumption that the streamlines were horizontal, which was acceptable for slope less than 0.10. In this study, we presented a new finite‐difference scheme of NHML based on the Boussinesq assumption and developed a new package SLOPE which was incorporated into MODFLOW‐2000 to become the MODFLOW‐SP model. The accuracy of MODFLOW‐SP was tested against solution of Mac Cormack (1969). The differences between the solutions of MODFLOW‐2000 and MODFLOW‐SP were nearly negligible when the slope was less than 0.27, and they were noticeable during the transient flow stage and vanished in steady state when the slope increased above 0.27. We established a model considering the vertical flow using COMSOL Multiphysics to test the robustness of constrains used in MODFLOW‐SP. The results showed that streamlines quickly became parallel with the aquifer base except in the narrow regions near the boundaries when the initial flow was not parallel to the aquifer base. MODFLOW‐SP can be used to predict the hydraulic head of an unconfined aquifer along the profile perpendicular to the aquifer base when the slope was smaller than 0.50. The errors associated with constrains used in MODFLOW‐SP were small but noticeable when the slope increased to 0.75, and became significant for the slope of 1.0.     

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.