Series solutions for flow in stratified aquifers with natural geometry

An analytical series solution method is presented for modeling regional steady-state groundwater flow in a two-dimensional stratified aquifer cross-section where the water table is well-characterized. The aquifer system may have any number of contiguous or non-contiguous layers and the geometry of each layer is restricted only by the requirement that the elevation of the stratigraphic unconformities between layers is a function of the x-coordinate alone. Various techniques may be used to handle pinching layers, faults, and other discontinuities. The solutions are obtained by minimizing head and flow continuity errors between layers and errors in the Dirichlet surface at a set of control points along these unconformities; the governing equation is met exactly. The solutions are derived and demonstrated on multiple test cases. The errors for some specific, geometrically challenging cases are assessed and discussed.     

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.     

Three-dimensional semi-analytical solution to groundwater flow in confined and unconfined wedge-shaped aquifers

The Laplace domain solutions have been obtained for three-dimensional groundwater flow to a well in confined and unconfined wedge-shaped aquifers. The solutions take into account partial penetration effects, instantaneous drainage or delayed yield, vertical anisotropy and the water table boundary condition. As a basis, the Laplace domain solutions for drawdown created by a point source in uniform, anisotropic confined and unconfined wedge-shaped aquifers are first derived. Then, by the principle of superposition the point source solutions are extended to the cases of partially and fully penetrating wells. Unlike the previous solution for the confined aquifer that contains improper integrals arising from the Hankel transform [Yeh HD, Chang YC. New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions. Adv Water Resour 2006;26:471–80], numerical evaluation of our solution is relatively easy using well known numerical Laplace inversion methods. The effects of wedge angle, pumping well location and observation point location on drawdown and the effects of partial penetration, screen location and delay index on the wedge boundary hydraulic gradient in unconfined aquifers have also been investigated. The results are presented in the form of dimensionless drawdown-time and boundary gradient-time type curves. The curves are useful for parameter identification, calculation of stream depletion rates and the assessment of water budgets in river basins.     

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.     

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].     

A new solution of transient confined–unconfined flow driven by a pumping well

A new solution of transient confined–unconfined flow driven by a pumping well is developed and compared to previous approximate solutions of Moench and Prickett [Moench AF, Prickett TA. Radial flow in an infinite aquifer undergoing conversion from artesian to water table conditions. Water Resour Res 1972;8:494–9] and Hu and Chen [Hu L, Chen C. Analytical methods for transient flow to a well in a confined–unconfined aquifer. Ground Water 2008;46(4):642–6]. The problem is rewritten in dimensionless form with the Boltzmann transform. The nonlinear equation for flow in the unconfined zone is solved with the Runge–Kutta method. Position of the conversion interface is determined with an iteration scheme. This study shows that the confined–unconfined flow depends on three dimensionless parameters that represent the confined–unconfined storativity ratio (a_D), the ratio of the initial hydraulic head over the aquifer thickness (f_i), and the dimensionless pumping rate (q_D). The rate of expansion of the unconfined zone increases with q_D, but decreases with a_D and f_i. Differences between the two previous approximate solutions and the new solution of this study are observable in the estimated position of the conversion interface and the drawdown–time curves. The new solution can be applied to estimate the time for confined–unconfined conversion to occur (critical conversion time), and the time when the pumping well becomes dry (critical drying time). The critical conversion time is found to be very sensitive to the initial hydraulic head. The critical drying time is often much larger than the critical conversion time and may never be observed during a finite pumping period.     

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.     

Cluster-based fuzzy models for groundwater flow in the unsaturated zone

In this paper fuzzy models are used as an alternative to describe groundwater flow in the unsaturated zone. The core of these models consists of a fuzzy rule-based model of the Takagi–Sugeno type. Various fuzzy clustering algorithms are compared in the data-driven identification of these Takagi–Sugeno models. The performance of the resulting fuzzy models is evaluated on the training surface on which they were identified, and on time series measurements of water content values obtained through an experiment carried out by the non-vegetated terrain (NVT) workgroup of the European Microwave Signature Laboratory (EMSL) (see [Mancini M, Hoeben R, Troch PA. Multifrequency radar observations of bare surface soil moisture content: a laboratory experiment. Water Resour Res 1999;35(6):1827–38] and [Hoeben R, Troch PA. Assimilation of active microwave observation data for soil moisture profile estimation. Water Resour Res 2000;36(10):2805–19]). Despite higher errors at the borders of high water content values in the training surface, good results are obtained on the simulation of the time series.     

Addendum to “Polynomial approximate solutions to the Boussinesq equation”

A simple and accurate cubic approximation to the solution of the Boussinesq equation is given in case of power-law flux boundary condition being imposed at the inlet of an initially dry aquifer. The new approximation overcomes the numerical intensity of the earlier cubic approximation of Telyakovskiy and Allen [Telyakovskiy AS, Allen MB. Polynomial approximate solutions to the Boussinesq equation. Adv Water Resour 2006;29(12):1767–79], while producing comparably accurate results.     

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.     

Short- and long-time behavior of aquifer drainage after slow and sudden recharge according to the linearized Laplace equation

During the last 25 years there has been a great interest in deriving aquifer characteristics from outflow data. This analysis has been mainly based of the drainage of a horizontal aquifer after sudden drawdown, using the Boussinesq approximation. Following the general approach of Brutsaert and Lopez [Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40], it was determined that for this geometry the aquifer behavior could be characterized by dQ/dt ∝ Q³ for small t and by dQ/dt ∝ Q^3/2 for large t. It was remarked that dQ/dt ∝ Q for large t is often observed. In practice, it is also difficult to determine if dQ/dt ∝ Q³ for small t because this behavior can only be observed over a very short period.Here, we present a similar analysis of aquifer behavior based on the more fundamental Laplace solution for penetrated aquifers. It has been shown that also when the drain does not fully penetrate the aquifer, the solution still produces good results [Szilagyi, J. Sensitivity analysis of aquifer parameter estimations based on the Laplace equation with linearized boundary conditions. Water Resour Res 2003;39(6)]. The Laplace solution quickly shows that dQ/dt ∝ Q for t → ∞ and dQ/dt ∝ Q^∞ for t → 0, after sudden drawdown. This analysis reconfirms previous findings concerning long-time behavior. More importantly, the analysis shows that the exponent B in dQ/dt ∝ Q^B does not have a fixed limited value for short times for the given geometry. Further analysis, however, shows that under certain conditions the relation dQ/dt ∝ Q³ is retained for 0 ≪ t < 1. Detailed examination of the Laplace solution also shows under which types of recharge dynamics a well-identifiable transition takes place between short- and long-term behavior. As long as such a clear transition exists, the aquifer characterization method proposed earlier by Brutsaert and Lopez [Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40] can be applied. It is shown that for a sharp pulse input, the Laplace solution gives similar results as presented by Brutsaert and Lopez [Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40]. For a smooth pulse, the transition becomes unclear. What is “smooth” and “sharp” depends on input and aquifer characteristics, whereby shallow aquifers give clearer transitions than deep aquifers for the same input. The analysis shows that when rain ceases suddenly after the aquifer has come into equilibrium with a steady rain input, a usable transition in the relation between dQ/dt and Q can be found as well. Researchers can use the present analysis to assess whether specific aquifers and recharge events can be used for the previously suggested characterization method.     

Three-dimensional flow in the storative semiconfining layers of a leaky aquifer

An analytical solution for three-dimensional (3D) flow in the storative semiconfining layers of a leaky aquifer fully penetrated by a production well is developed in this article to provide a method from which accurate hydraulic parameters in the semiconfining layers can be derived from aquifer test data. The analysis of synthetic aquifer test data with the 3D analytical solution in the semiconfining layers provided more accurate optimal hydraulic parameters than those derived using the available quasi-two-dimensional (2D) solution. Differences between the 3D and 2D flow solutions in the semiconfining layers become larger when a no flow boundary condition is imposed at either at the top of the upper semiconfining layer or at the bottom of the lower semiconfining layer or when the hydraulic conductivity ratio of the semiconfining layer to the aquifer is larger than 0.001. In addition, differences between the 3D and 2D flow solutions in the semiconfining layers are illustrated when the thickness ratio of the semiconfining layer to the aquifer is changed. Analysis of water level data from two hypothetical and one real aquifer test showed that the 3D solution in the semiconfining layers provides lower correlation coefficients among hydraulic parameters than the 2D solution.     

Depth–energy and depth–force relationships in open channel flows. II: Analytical findings for power-law cross-sections

In a recent work [Valiani A, Caleffi V. Depth–energy and depth–force relationships in open channel flows: analytical findings. Adv Water Resour 2008;31(3):447–54], the authors analytically inverted the depth–specific energy and depth–total force relationships for flows in open channels with wide rectangular cross-sections.     

Characterization of mixing and spreading in a bounded stratified medium

Matheron and de Marsily [Matheron M, de Marsily G. Is the transport in porous media always diffusive? A counter-example. Water Resour Res 1980;16:901–17] studied transport in a perfectly stratified infinite medium as an idealized aquifer model. They observed superdiffusive solute spreading quantified by anomalous increase of the apparent longitudinal dispersion coefficient with the square root of time. Here, we investigate solute transport in a vertically bounded stratified random medium. Unlike for the infinite medium at asymptotically long times, disorder-induced mixing and spreading is uniquely quantified by a constant Taylor dispersion coefficient. Using a stochastic modeling approach we study the effective mixing and spreading dynamics at pre-asymptotic times in terms of effective average transport coefficients. The latter are defined on the basis of local moments, i.e., moments of the transport Green function. We investigate the impact of the position of the initial plume and the initial plume size on the (highly anomalous) pre-asymptotic effective spreading and mixing dynamics for single realizations and in average. Effectively, the system “remembers” its initial state, the effective transport coefficients show so-called memory effects, which disappear after the solute has sampled the full vertical extent of the medium. We study the impact of the intrinsic non-ergodicity of the confined medium on the validity of the stochastic modeling approach and study in this context the transition from the finite to the infinite medium.     

Modelling coastal ground‐ and surface‐water interactions using an integrated approach

Beach water table fluctuations have an impact on the transport of beach sediments and the exchange of solute and mass between coastal aquifer and nearby water bodies. Details are given of the refinement of a dynamically integrated ground‐ and surface‐water model, and its application to study ground‐ and surface‐water interactions in coastal regions. The depth‐integrated shallow‐water equations are used to represent the surface‐water flow, and the extended Darcy's equation is used to represent the groundwater flow, with a hydrostatic pressure distribution being assumed to apply for both these two types of flows. At the intertidal region, the model has two layers, with the surface‐water layer being located on the top of the groundwater layer. The governing equations for these two types of flows are discretized in a similar manner and they are combined to give one set of linear algebraic equations that can be solved efficiently. The model is used to predict water level distributions across sloping beaches, where the water table in the aquifer may or may not decouple from the free water surface. Five cases are used to test the model for simulating beach water table fluctuations induced by tides, with the model predictions being compared with existing analytical solutions and laboratory and field data published in the literature. The numerical model results show that the integrated model is capable of simulating the combined ground‐ and surface‐water flows in coastal areas. Detailed analysis is undertaken to investigate the capability of the model. Copyright © 2009 John Wiley & Sons, Ltd.     

Hydraulic subsurface measurements and hydrodynamic modelling as indicators for groundwater flow systems in the Rotondo granite,Central Alps (Switzerland)

Regional groundwater flow in high mountainous terrain is governed by a multitude of factors such as geology, topography, recharge conditions, structural elements such as fracturation and regional fault zones as well as man‐made underground structures. By means of a numerical groundwater flow model, we consider the impact of deep underground tunnels and of an idealized major fault zone on the groundwater flow systems within the fractured Rotondo granite. The position of the free groundwater table as response to the above subsurface structures and, in particular, with regard to the influence of spatial distributed groundwater recharge rates is addressed. The model results show significant unsaturated zones below the mountain ridges in the study area with a thickness of up to several hundred metres. The subsurface galleries are shown to have a strong effect on the head distribution in the model domain, causing locally a reversal of natural head gradients. With respect to the position of the catchment areas to the tunnel and the corresponding type of recharge source for the tunnel inflows (i.e. glaciers or recent precipitation), as well as water table elevation, the influence of spatial distributed recharge rates is compared to uniform recharge rates. Water table elevations below the well exposed high‐relief mountain ridges are observed to be more sensitive to changes in groundwater recharge rates and permeability than below ridges with less topographic relief. In the conceptual framework of the numerical simulations, the model fault zone has less influence on the groundwater table position, but more importantly acts as fast flow path for recharge from glaciated areas towards the subsurface galleries. This is in agreement with a previous study, where the imprint of glacial recharge was observed in the environmental isotope composition of groundwater sampled in the subsurface galleries. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient conceptual model for simulating the effect of aquifer heterogeneity on natural groundwater discharge to rivers

When linearity can be assumed (linear response of heads to stresses), stream–aquifer flow exchange can be simulated as the drainage of a number of independent linear reservoirs. This conceptual model, which can be mathematically deduced in a univocal way from an eigenvalue solution of the linear groundwater flow problem, facilitates the understanding of the physical phenomenon and the analysis of influencing factors. The number of reservoirs required to simulate stream depletion in some ideal homogeneous cases of stream–aquifer connection was analyzed in detail in a previous investigation using analytical eigenvalue solutions [16]. However, most aquifers are heterogeneous in nature and numerical solutions must be employed to analyze whether they could also be simulated using few reservoirs. This paper presents a stochastic analysis of the influence of heterogeneity on the simulation of natural groundwater discharges in aquifers connected to rivers, as a series of linear reservoirs. A Monte-Carlo approach was employed to perform this study. The results show that, on a monthly time scale, many cases (even heterogeneous aquifers) can be simulated using just a few reservoirs with sufficient accuracy and at minimum computational cost. Therefore, this modeling technique can be useful to efficiently simulate the integrated management of complex water resources systems at the basin scale (with many aquifers, reservoirs, demands, etc.) that need to simultaneously consider surface and groundwater flow and stream–aquifer interaction.     

Spatial moments for colloid-enhanced radionuclide transport in heterogeneous aquifers

We consider colloid facilitated radionuclide transport by steady groundwater flow in a heterogeneous porous formation. Radionuclide binding on colloids and soil-matrix is assumed to be kinetically/equilibrium controlled. All reactive parameters are regarded as uniform, whereas the hydraulic log-conductivity is modelled as a stationary random space function (RSF). Colloid-enhanced radionuclide transport is studied by means of spatial moments pertaining to both the dissolved and colloid-bounded concentration. The general expressions of spatial moments for a colloid-bounded plume are presented for the first time, and are discussed in order to show the combined impact of sorption processes as well as aquifer heterogeneity upon the plume migration. For the general case, spatial moments are defined by the aid of two characteristic reaction functions which cannot be expressed analytically. By adopting the approximation for the longitudinal fluid trajectory covariance valid for a flow parallel to the formation bedding suggested by Dagan and Cvetkovic [Dagan G, Cvetkovic V. Spatial Moments of Kinetically Sorbing Plume in a Heterogeneous Aquifers. Water Resour Res 1993;29:4053], we obtain closed form solutions.