A Leaky Aquifer below Champlain Sea Clay: Closed‐Form Solutions for Natural Seepage

Closed‐form solutions are proposed for natural seepage in semiconfined (leaky) aquifers such as those existing below the massive Champlain Sea clay layers in the Saint‐Lawrence River Valley. The solutions are for an ideal horizontal leaky aquifer below an ideal aquitard that may have either a constant thickness and a constant hydraulic head at its surface, or a variable thickness and a variable hydraulic head at its surface. A few simplifying assumptions were needed to obtain the closed‐form solutions. These have been verified using a finite element method, which did not make any of the assumptions but gave an excellent agreement for hydraulic heads and groundwater velocities. For example, the difference between the two solutions was smaller than 1 mm for variations in the 5 to 8 m range for the hydraulic head in the semiconfined aquifer. Note that fitting the hydraulic head data of monitoring wells to the theoretical solutions gives only the ratio of the aquifer and aquitard hydraulic conductivities, a clear case of multiple solutions for an inverse problem. Consequently, field permeability tests in the aquitard and the aquifer, and pumping tests in the aquifer, are still needed to determine the hydraulic conductivity values.     

An analytic benchmark test for karst-aquifer flow

A benchmark test for flow in karstic aquifers is presented in the form of an exact solution of the harmonic variations of water flux and head within a karst conduit that is imbedded within a three-dimensional porous matrix having a free surface. The variations are driven by a prescribed variation of head applied at one end of the conduit. The benchmark consists of expressions for the spring discharge as a function of time and the conduit head and flux as functions of distance along the conduit and time. These expressions contain three dimensionless parameters, permitting development of a wide range of specific benchmark tests. The expressions are particularly simple in the case of an infinitely deep aquifer. This limiting solution should provide the most severe test for two-dimensional models of karst aquifer flow. Another limiting case of interest is that in which the conduit diameter is equal to the water depth. This limiting solution should provide the easiest test for two-dimensional models.     

Cross-borehole flowmeter tests for transient heads in heterogeneous aquifers

Cross-borehole flowmeter tests have been proposed as an efficient method to investigate preferential flowpaths in heterogeneous aquifers, which is a major task in the characterization of fractured aquifers. Cross-borehole flowmeter tests are based on the idea that changing the pumping conditions in a given aquifer will modify the hydraulic head distribution in large-scale flowpaths, producing measurable changes in the vertical flow profiles in observation boreholes. However, inversion of flow measurements to derive flowpath geometry and connectivity and to characterize their hydraulic properties is still a subject of research. In this study, we propose a framework for cross-borehole flowmeter test interpretation that is based on a two-scale conceptual model: discrete fractures at the borehole scale and zones of interconnected fractures at the aquifer scale. We propose that the two problems may be solved independently. The first inverse problem consists of estimating the hydraulic head variations that drive the transient borehole flow observed in the cross-borehole flowmeter experiments. The second inverse problem is related to estimating the geometry and hydraulic properties of large-scale flowpaths in the region between pumping and observation wells that are compatible with the head variations deduced from the first problem. To solve the borehole-scale problem, we treat the transient flow data as a series of quasi-steady flow conditions and solve for the hydraulic head changes in individual fractures required to produce these data. The consistency of the method is verified using field experiments performed in a fractured-rock aquifer.     

Two-dimensional stochastic analysis of flow in leaky confined aquifers subject to spatial and periodic leakage

This paper addresses the issue of flow in heterogeneous leaky confined aquifers subject to leakage. The leakage into the confined aquifer is driven by spatial and periodic fluctuations of water table in an overlying phreatic aquifer. The introduction of leakage leads to non-uniformity in the mean head gradient and results in nonstationarity in hydraulic head and velocity fields. Therefore, a nonstationary spectral approach based on Fourier–Stieltjes representations for the perturbed quantities is adopted to account for the spatial variability of nonstationary head fields. Closed-form expressions for the variances of hydraulic head and specific discharge are developed in terms of statistical properties of hydraulic parameters. The results indicate that the spatiotemporal variations in leakage leads to enhanced variability of the hydraulic head and of the specific discharge, which increase with distance from any arbitrary reference point. The coefficient of leakage and the spatial structure of log transmissivity field and of the amplitude of water table fluctuation are critical in quantifying the variability of the hydraulic head and of the specific discharge.     

Applicability of spectral analysis to determine hydraulic diffusivity

This study is to evaluate the applicability of estimating the one-dimensional horizontal hydraulic diffusivity of an unconfined aquifer with time-dependent fluctuation of lateral head and vertical recharge boundaries using observed water level spectra. Different models of boundary condition are imposed to evaluate the statistical significance between the calculated hydraulic diffusivity (ξ^′) with the given hydraulic diffusivity (ξ). The auto-spectra of the water level in observation wells tapping the same aquifer are closely related to those at the disturbed boundaries. For an aquifer with a constant hydraulic diffusivity, the water level fluctuation in the monitoring wells is linearly related to the water level spectra observed at the boundaries. The spectral density function of aquifer hydraulic head varies inversely with specific yield (S _y) and directly with recharge. Given small variation in water level spectra at the disturbed boundaries, the water level fluctuation in the aquifer is affected by the recharge condition and the aquifer spectral density function is sensitive to S _y. Using an iterative technique to estimate ξ from 1400 sets of given parameters, 99% of the ξ^′/ξ values deviated within only one order of magnitude with the model length (L) being equal to 1 km and 10 km. For L equal to 100 m, approximately 82% of the ξ^′/ξ population falls within two orders of magnitude. Therefore, spectral analysis of aquifer hydraulic head response can be used to estimate the hydraulic diffusivity of an unconfined aquifer which is affected by periodic variations in recharge and head at boundaries.     

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.     

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.     

日本秋田地震(1983年7.7级)引起我国应力场的震时调整变化

本文通过水平分层承压含水层模型,推导出含水层的水头和含水层应力变化之间的定量关系.利用我国在日本秋田地震时有水位阶变的52口井孔的观测资料,通过水井水位的体应变固体潮系数,估算了震时引起我国应力场的调整变化.讨论了这种应力调整时应力增加区和降低区交替出现的可能原因.      

利用时序InSAR技术反演邯郸平原区地表形变与含水层参数

邯郸平原区承压地下水长期处于严重超采状态, 导致大面积地面沉降, 直接威胁到该地区重大基础设施安全.因此, 查明地面沉降时空演变特征, 厘清承压地下水变化与地面沉降的耦合关系, 对预防和治理地面沉降地质灾害具有重要意义.本文利用Sentinel-1A数据进行时间序列分析, 获取了2015—2019年邯郸平原区地表形变时空分布结果, 并结合水头数据分析了含水系统对水头变化的不同响应.利用谐波函数分离了地表形变及水头变化的季节性变化, 并用其估算了邯郸平原区空间差异变化的弹性骨架释水系数; 基于顾及弱透水层延迟排水的一维水头变化-形变模型, 反演了非弹性骨架释水系数和时间常数.结果表明: (1)邯郸平原区以沉降为主, 最大沉降速率可达14 cm·a^-1; 通过与水头数据对比发现该地区地面沉降主要是由承压含水层水头下降及弱透水层的延迟排水引起.(2)水头的季节性变化引起了明显的季节性形变, 沉降区季节性形变幅度可达25 mm, 峰值时间为1—3月.(3)邯郸平原区弹性骨架释水系数介于1.51×10^-3~4.05×10^-3之间, 与抽水试验结果较为相符; 非弹性骨架释水系数为3.62×10^-2~4.57×10^-2, 时间常数为0.47~0.77 a.本研究表明时序InSAR技术在含水层参数估算, 地下水超采相关的地质灾害监测方面有良好的应用潜力.

     

Tide-induced groundwater head fluctuation in coastal multi-layered aquifer systems with a submarine outlet-capping

This paper considered the tide-induced head fluctuations in two coastal multi-layered aquifer systems. Model I comprises two semi-permeable layers and a confined aquifer between them. Model II is a four-layered aquifer system including an unconfined aquifer, an upper semi-permeable layer, a confined aquifer and a lower semi-permeable layer. In each model, the submarine outlet of the confined aquifer is covered with a skin layer (“outlet-capping”). Analytical solutions of the two models are derived. In both models, leakages of the semi-permeable layers decrease the tidal head fluctuations. The outlet-capping reduces the aquifer’s head fluctuation by a constant factor and shifts the phase by a positive constant. The solution to Model II explains the inconsistency between the relatively small lag time and the strong amplitude damping effect of the tidal head fluctuations reported by Trefry and Johnston [Ground Water 1998;36:427–33] near the Port Adelaide River, Australia.     

Effect of spatial variation in hydraulic conductivity on groundwater flow by alternate solution techniques

The behaviour of an aquifer in which there is spatial variation in hydraulic conductivity is simulated by means of a Monte Carlo procedure. The lognormal function is used to model the distribution of conductivities. Thus variations in hydraulic head and specific discharge are studied. As an alternate technique a finite-difference technique is adopted for comparative purposes. The standard deviation of two-dimensional flows is related to the standard deviation of conductivity. Hence, an empirical probability distribution of flows is obtained by specific values of the variance in conductivities.     

Groundwater level monitoring by precision measurements

The paper presents the results of processing the data on variations of groundwater level in an aquifer opened by a well in a depth interval of 76–115 m. The obtained series are compared with the volume deformation of the aquifer, calculated theoretically using the ETERNA 3.0 tidal analysis program. Two components are detected in the long-period area: the one coherent to the luni-solar tides and the one incoherent to them. It is established that the phase shift between the variations of volume deformation of the aquifer and the corresponding variations of water level in the well depends not only on the stressed-deformed state of the surrounding rock massif, but on the position of the piezometric surface of the aquifer as well. Some peculiarities of the effects of luni-solar tides on variations of groundwater level are considered. The amplitudes of groundwater level and volume deformation of the aquifer, as well as the phase shift between these parameters are estimated for the detected tidal waves. The value of the amplitude factor lies in the range of 0.04–0.058 cm/nanostrain.     

Stochastic analysis of groundwater flow in aquifers with leakage

 Stochastic analysis of one- and two-dimensional flow through a shallow semi-confined aquifer with spatially variable hydraulic conductivity K represented by a stationary (statistically homogeneous) random process is carried out by using the spectral technique. The hydraulic head covariance functions for flows in a semi-confined aquifer bounded by a leaky layer above and an impervious stratum below are derived by assuming that the randomness forcing the head variation to originate from the hydraulic conductivity field of the aquifer. The head covariance functions are studied using two convenient forms of the logarithmic hydraulic conductivity process. The results demonstrate the significant reduction in the head variances and covariances due to the presence of a leaky layer. The hydraulic head correlation distance is also reduced greatly due to the presence of the leaky layer.     

A coupled one‐dimensional viscoelastic–plastic model for aquitard consolidation caused by hydraulic head variations in aquifers

Accurate and practical calculation of aquitard consolidation is required for a reliable analysis of land subsidence caused by groundwater overexploitation in a multilayered aquifer system because aquitards are generally more compressible than aquifers are. This study proposes a coupled one‐dimensional viscoelastic–plastic consolidation model that considers the combined effect of changes in soil parameters and body force to simulate aquitard consolidation caused by hydraulic head variations in neighbouring aquifers. The proposed model uses variable total stress and simultaneously solves hydraulic head and vertical soil displacement. The constitutive relation based on the Voigt model with different elastic moduli of the spring in normally consolidated and overconsolidated soils is used to describe the viscoelastic–plastic deformation mechanism of aquitards. In addition, the proposed model considers the combined effect of variations in hydraulic conductivity, elastic moduli, and body force on the calculation of aquitard consolidation. Three hypothetical scenarios with various hydraulic head variations in aquifers are used to examine the coupled one‐dimensional viscoelastic–plastic consolidation model. The results show that neglecting plasticity and viscosity of soil causes aquitard consolidation to be respectively underestimated and overestimated. In addition, ignoring body force variation underestimates aquitard consolidation, whereas neglecting soil parameters variation overestimates aquitard consolidation. Two real case scenarios are also studied to further demonstrate the applicability of the coupled one‐dimensional viscoelastic–plastic consolidation model. Copyright © 2015 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.     

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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.     

Factors Influencing the Stream‐Aquifer Flow Exchange Coefficient

Knowledge of river gain from or loss to a hydraulically connected water table aquifer is crucial in issues of water rights and also when attempting to optimize conjunctive use of surface and ground waters. Typically in groundwater models this exchange flow is related to a difference in head between the river and some point in the aquifer, through a “coefficient.” This coefficient has been defined differently as well as the location for the head in the aquifer. This paper proposes a new coefficient, analytically derived, and a specific location for the point where the aquifer head is used in the difference. The dimensionless part of the coefficient is referred to as the SAFE (stream‐aquifer flow exchange) dimensionless conductance. The paper investigates the factors that influence the value of this new conductance. Among these factors are (1) the wetted perimeter of the cross‐section, (2) the degree of penetration of the cross‐section, and (3) the shape of the cross‐section. The study shows that these factors just listed are indeed ordered in their respective level of importance. In addition the study verifies that the analytical correct value of the coefficient is matched by finite difference simulation only if the grid system is sufficiently fine. Thus the use of the analytical value of the coefficient is an accurate and efficient alternative to ad hoc estimates for the coefficient typically used in finite difference and finite element methods.     

Discrete stability of the Differential System Method evaluated with geostatistical techniques

The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques. We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated measurement errors and interpolation errors. The numerical results show that the DSM is stable.     

Discrete stability of the Differential System Method evaluated with geostatistical techniques

The Differential System Method (DSM) permits identification of the physical parameters of finite-difference groundwater flow models in a confined aquifer when piezometric head and source terms are known at each point of the finite-difference lattice for at least two independent flow situations for which the hydraulic gradients are not parallel. Since piezometric head data are usually few and sparse, interpolation of the measured data onto a regular grid can be performed with geostatistical techniques. We apply kriging to the sparse data of a synthetic aquifer to evaluate the stability of the DSM with respect to uncorrelated measurement errors and interpolation errors. The numerical results show that the DSM is stable.     

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.