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].     

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 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.     

Flowing partially penetrating well: solution to a mixed-type boundary value problem

A new semi-analytic solution to the mixed-type boundary value problem for a flowing partially penetrating well with infinitesimal skin situated in an anisotropic aquifer is developed. The solution is suited to aquifers having a semi-infinite vertical extent or to packer tests with aquifer horizontal boundaries far enough from the tested area. The problem reduces to a system of dual integral equations (DE) and further to a deconvolution problem. Unlike the analogous Dagan's steady-state solution [Water Resour. Res. 1978; 14:929–34], our DE solution does not suffer from numerical oscillations. The new solution is validated by matching the corresponding finite-difference solution and is computationally much more efficient. An automated (Newton–Raphson) parameter identification algorithm is proposed for field test inversion, utilizing the DE solution for the forward model. The procedure is computationally efficient and converges to correct parameter values. A solution for the partially penetrating flowing well with no skin and a drawdown–drawdown discontinuous boundary condition, analogous to that by Novakowski [Can. Geotech. J. 1993; 30:600–6], is compared to the DE solution. The D–D solution leads to physically inconsistent infinite total flow rate to the well, when no skin effect is considered. The DE solution, on the other hand, produces accurate results.     

Plants in water-controlled ecosystems: active role in hydrologic processes and response to water stress: I. Scope and general outline

This series of four papers studies the complex dynamics of water-controlled ecosystems from the hydro-ecological point of view [e.g., I. Rodriguez-Iturbe, Water Resour. Res. 36 (1) (2000) 3–9]. After this general outline, the role of climate, soil, and vegetation is modeled in Part II [F. Laio, A. Porporato, L. Ridolfi, I. Rodriguez-Iturbe, Adv. Water Res. 24 (7) (2001) 707–723] to investigate the probabilistic structure of soil moisture dynamics and the water balance. Particular attention is given to the impact of timing and amount of rainfall, plant physiology, and soil properties. From the statistical characterization of the crossing properties of arbitrary levels of soil moisture, Part III develops an expression for vegetation water stress [A. Porporato, F. Laio, L. Ridolfi, I. Rodriguez-Iturbe, Adv. Water Res. 24 (7) (2001) 725–744]. This measure of stress is then employed to quantify the response of plants to soil moisture deficit as well as to infer plant suitability to given environmental conditions and understand some of the reasons for possible coexistence of different species. Detailed applications of these concepts are developed in Part IV [F. Laio, A. Porporato, C.P. Fernandez-Illescas, I. Rodriguez-Iturbe, Adv. Water Res. 24 (7) (2001) 745–762], where we investigate the dynamics of three different water-controlled ecosystems.     

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.     

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.     

Analytical solutions for 2D topography-driven flow in stratified and syncline aquifers

Two analytical solution methods are presented for regional steady-state groundwater flow in a two-dimensional stratified aquifer cross section where the water table is approximated by the topographic surface. For the first solution, the surficial aquifer is represented as a set of dipping parallel layers with different, but piecewise constant, anisotropic hydraulic conductivities, where the anisotropy is aligned with the dip of the layered formation. The model may be viewed as a generalization of the solutions developed by [Tóth JA. A theoretical analysis of groundwater flows in small drainage basins. J Geophys Res 1963;68(16):4795–812; Freeze R, Witherspoon P. Theoretical analysis of regional groundwater flow 1) analytical and numerical solution to the mathematical model, water resources research. Water Resour Res 1966;2(4):641–56; Selim HM. Water flow through multilayered stratified hillside. Water Resour Res 1975;11:949–57] to an multi-layer aquifer with general anisotropy, layer orientation, and a topographic surface that may intersect multiple layers. The second solution presumes curved (syncline) layer stratification with layer-dependent anisotropy aligned with the polar coordinate system. Both solutions are exact everywhere in the domain except at the topographic surface, where a Dirichlet condition is met in a least-squared sense at a set of control points; the governing equation and no-flow/continuity conditions are met exactly. The solutions are derived and demonstrated on multiple test cases. The error incurred at the location where the layer boundaries intersect the surface is assessed.     

Tide-induced airflow in a two-layered coastal land with atmospheric pressure fluctuations

Tide-induced airflow is commonly seen in coastal lands and affects ground stability especially with a less permeable pavement on the ground surface. A tide-induced airflow model in a two-layered unsaturated zone consisting of a highly permeable layer underneath a less permeable layer was established by Li and Jiao [Li HL, JJ Jiao. One-dimensional airflow in unsaturated zone induced by periodic water table fluctuation. Water Resour Res 2005;41:W04007. doi:10.1029/2004WR003916] to describe the one-dimensional airflow with constant atmospheric pressure at the ground surface. In this study, we expand the Li and Jiao model by considering the realistic atmospheric pressure fluctuations and the initial condition. A new transient solution to the airflow model is developed for an initial boundary value problem (IBVP). The transient solution can be used not only to calculate the subsurface air pressure at a future time with a known initial condition, but also to evaluate the asymptotic air pressure variations when time becomes long. The amplitude ratio and phase lag of the subsurface air pressure relative to the tide-induced hydraulic head variations inside the unconfined aquifer below the unsaturated zone are investigated. The results reveal that effect on the subsurface pressure due to changes of atmospheric pressure amplitude depends on the configurations of air resistance in the less permeable layer and the air-filled porosity difference in the two layers. The introduction of atmospheric pressure fluctuations into the airflow model leads to insignificant influence on water table level. A field application of the new solution at Hong Kong International Airport in Hong Kong, China is demonstrated. It indicates that the new transient solution can be conveniently used to evaluate the subsurface air pressure with discrete atmospheric pressure data at the ground surface.     

Polynomial approximate solutions to the Boussinesq equation

We provide closed-form approximate solutions to models of horizontal infiltration described by the Boussinesq equation in a semi-infinite aquifer that is initially dry. The approximations preserve such important qualitative properties as scaling and wetting fronts. They are applicable to four types of boundary conditions, two on head and two on flux, enumerated in the paper. All the considered problems admit self-similar variables that allow reduction to boundary value problems for a nonlinear ordinary differential equation. This work extends recent results by Lockington et al. [Lockington DA, Parlange J-Y, Parlange MB, Selker J. Similarity solution of the Boussinesq equation. Adv Water Resour 2000;23(7):725–9] and Telyakovskiy et al. [Telyakovskiy AS, Braga GA, Furtado F. Approximate similarity solutions to the Boussinesq equation. Adv Water Resour 2002;25(2):191–4], with new approximations developed for two of the four cases and a new extension of a previously existing method for a third case. Numerical results extending the work of Shampine [Shampine LF. Some singular concentration dependent diffusion problems. ZAMM 1973;53:421–2] provide a basis for assessing the accuracy of the new methods.     

Analytical solutions to the transient,unsaturated transport of water and contaminants through horizontal porous media

We present a range of analytical solutions to the combined transient water and solute transport for horizontal flow. We adopt the concept of a scale and time dependent dispersivity used for contaminant transport in aquifers and apply it to transient, unsaturated horizontal flow to develop similarity solutions for both constant solute concentration and solute flux boundary conditions. Through the use of a specific form of the water profile as used by Brutsaert [Water Resour Res 1968:4;785], the solute profiles can be reduced to a simple quadrature. We also derive a solution for the instantaneous injection of water and solute into a horizontal media for an arbitrary dispersivity. It is found that the solute concentration remains constant in both space and time as the water redistributes, suggesting that the solute does not disperse relative to the water.     

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.     

New equations for binary gas transport in porous media,Part 2: experimental validation

A previous study [Water Resour Res 39 (3) (2003) doi:10.1029/2002WR001338] questioned the validity of the traditional advection–dispersion equation for describing gas flow in porous media. In an original mathematical derivation presented in Part 1 [Adv Water Resour, this issue] we have demonstrated the theoretical existence of two novel physical phenomena which govern the macroscopic transport of gases in porous media. In this work we utilize laboratory experiments and numerical modeling in order to ascertain the importance of these novel theoretical terms. Numerical modeling results indicate that the newly derived sorptive velocity, arising from closure level coupling effects, does not contribute noticeably to the overall flux, under the conditions explored in this work. We demonstrate that the newly discovered “slip coupling” phenomenon in the mass conservation equation plays an important role in describing the physics of gas flow through porous solids for flow regimes of both environmental and industrial interest.     

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.     

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.     

Application of a mixing-ratios based formulation to model mixing-driven dissolution experiments

We address the question of how one can combine theoretical and numerical modeling approaches with limited measurements from laboratory flow cell experiments to realistically quantify salient features of complex mixing-driven multicomponent reactive transport problems in porous media. Flow cells are commonly used to examine processes affecting reactive transport through porous media, under controlled conditions. An advantage of flow cells is their suitability for relatively fast and reliable experiments, although measuring spatial distributions of a state variable within the cell is often difficult. In general, fluid is sampled only at the flow cell outlet, and concentration measurements are usually interpreted in terms of integrated reaction rates. In reactive transport problems, however, the spatial distribution of the reaction rates within the cell might be more important than the bulk integrated value. Recent advances in theoretical and numerical modeling of complex reactive transport problems [De Simoni M, Carrera J, Sanchez-Vila X, Guadagnini A. A procedure for the solution of multicomponent reactive transport problems. Water Resour Res 2005;41:W11410. doi: 10.1029/2005WR004056, De Simoni M, Sanchez-Vila X, Carrera J, Saaltink MW. A mixing ratios-based formulation for multicomponent reactive transport. Water Resour Res 2007;43:W07419. doi: 10.1029/2006WR005256] result in a methodology conducive to a simple exact expression for the space–time distribution of reaction rates in the presence of homogeneous or heterogeneous reactions in chemical equilibrium. The key points of the methodology are that a general reactive transport problem, involving a relatively high number of chemical species, can be formulated in terms of a set of decoupled partial differential equations, and the amount of reactants evolving into products depends on the rate at which solutions mix. The main objective of the current study is to show how this methodology can be used in conjunction with laboratory experiments to properly describe the key processes that occur in a complex, geochemically-active system under chemical equilibrium conditions. We model three CaCO₃ dissolution experiments reported in Singurindy et al. [Singurindy O, Berkowitz B, Lowell RP. Carbonate dissolution and precipitation in coastal environments: Laboratory analysis and theoretical consideration. Water Resour Res 2004;40:W04401. doi: 10.1029/2003WR002651, Singurindy O, Berkowitz B, Lowell RP. Correction to Carbonate dissolution and precipitation in coastal environments: laboratory analysis and theoretical consideration. Water Resour Res 2005;41:W11701. doi: 10.1029/2005WR004433], in which saltwater and freshwater were mixed in different proportions. The integrated reaction rate within the cell estimated from the experiments are modeled independently by means of (a) a state-of-the-art reactive transport code, and (b) the uncoupled methodology of [12, 13], both of which use dispersivity as a single, adjustable parameter. The good agreement between the results from both methodologies demonstrates the feasibility of using simple solutions to design and analyze laboratory experiments involving complex geochemical problems.     

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.