Series solutions for saturated–unsaturated flow in multi-layer unconfined aquifers

A semi-analytical mesh-free series solution method is presented for modeling regional steady-state subsurface saturated–unsaturated flow in 2-D geometrically complex homogenous and stratified hill-slope cross sections. Continuous solutions for pressure in the saturated and unsaturated zone are determined iteratively, as is the location of the water table surface. Mass balance is satisfied exactly over the entire domain except along boundaries and interfaces between layers, where errors are in an acceptable range. The solutions are derived and demonstrated on multiple test cases. The errors for specific cases are assessed and discussed.     

An analytical solution for ground water transit time through unconfined aquifers

An exact, closed-form analytical solution is developed for calculating ground water transit times within Dupuit-type flow systems. The solution applies to steady-state, saturated flow through an unconfined, horizontal aquifer recharged by surface infiltration and discharging to a downgradient fixed-head boundary. The upgradient boundary can represent, using the same equation, a no-flow boundary or a fixed head. The approach is unique for calculating travel times because it makes no a priori assumptions regarding the limit of the water table rise with respect to the minimum saturated aquifer thickness. The computed travel times are verified against a numerical model, and examples are provided, which show that the predicted travel times can be on the order of nine times longer relative to existing analytical solutions.     

Scaling procedures for heterogeneous unconfined aquifers

Complex flows in heterogeneous confined and unconfined aquifers is a phenomenon that continues to present difficulties in flow mapping and modelling in the field, laboratory, and through numerical simulations. It is often the case with complicated phenomena that transformative scaling and reduction of the problem through symmetry is of great efficacy in the formation of predictive models in both the laboratory and computational settings. A detailed a study of the application of a broad class of Lie scaling transformations on a set of equations representing the groundwater flows in heterogeneous confined and unconfined aquifers has produced a set of scaling relationships between the spatial variables, hydrologic variables, and parameters. The set of scaling transformations preserve the structure of the equations in the sense that the scaling transformations leave the initial‐boundary value system representing the invariant groundwater flows. This theoretical approach elucidates not only the scaling relationships but also the properties that hydrologic variables and parameters must satisfy in order for calling to be possible. Validation of the theory developed is carried out through a series of four numerical simulations using the USGS modflow ‐2005 software package. The results of these experiments demonstrate that the derived scaling transformations can effectively form predictive models of large‐scale phenomena at small scales with negligible error in many cases. Comments on the limitations of the approach and directions for future research are made in the closing sections. Copyright © 2016 John Wiley & Sons, Ltd.     

Tidal effects on groundwater dynamics in unconfined aquifers

The variation of seawater level resulting from tidal fluctuations is usually neglected in regional groundwater flow studies. Although the tidal oscillation is damped near the shoreline, there is a quasi‐steady‐state rise in the mean water‐table position, which may have an influence on regional groundwater flow. In this paper the effects of tidal fluctuations on groundwater hydraulics are investigated using a variably saturated numerical model that includes the effects of a realistic mild beach slope, seepage face and the unsaturated zone. In particular the impact of these factors on the velocity field in the aquifer is assessed. Simulations show that the tidal fluctuation has substantial consequences for the local velocity field in the vicinity of the exit face, which affects the nearshore migration of contaminant in coastal aquifers. An overheight in the water table as a result of the tidal fluctuation is observed and this has a significant effect on groundwater discharge to the sea when the landward boundary condition is a constant water level. The effect of beach slope is very significant and simplifying the problem by considering a vertical beach face causes serious errors in predicting the water‐table position and the groundwater flux. For media with a high effective capillary fringe, the moisture retained above the water table is important in determining the effects of the tidal fluctuations. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of data from pumping tests in unconfined anisotropic aquifers

The errors involved in finding the coefficients of transmissibility and storage are briefly discussed. Distance-drawdown data are advocated as generally most suitable for finding the transmissibility.

Procedures based on the Author's previous papers are described, which combine distance-drawdown analysis with constant storage coefficient and time-drawdown analysis with delayed yield from storage. Anomalies which result from well-known methods of analysis, based on a constant coefficient of storage, are thus avoided. The discussion and analysis are illustrated by pumping test data reported by Wenzel.

The methods described assume a fully-penetrating pumped well and shallow observation wells, except in the case of a very deep aquifer. A computer is not required.

An exact equation, allowing for the vertical velocity-component of the flow and delayed yield from storage, is given in Appendix 3.     

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.     

Two-dimensional physical-based inversion of confined and unconfined aquifers under unknown boundary conditions

An inverse method is developed to simultaneously estimate multiple hydraulic conductivities, source/sink strengths, and boundary conditions, for two-dimensional confined and unconfined aquifers under non-pumping or pumping conditions. The method incorporates noisy observed data (hydraulic heads, groundwater fluxes, or well rates) at measurement locations. With a set of hybrid formulations, given sufficient measurement data, the method yields well-posed systems of equations that can be solved efficiently via nonlinear optimization. The solution is stable when measurement errors are increased. The method is successfully tested on problems with regular and irregular geometries, different heterogeneity patterns and variances (maximum K_max/K_min tested is 10,000), and error magnitudes. Under non-pumping conditions, when error-free observed data are used, the estimated conductivities and recharge rates are accurate within 8% of the true values. When data contain increasing errors, the estimated parameters become less accurate, as expected. For problems where the underlying parameter variation is unknown, equivalent conductivities and average recharge rates can be estimated. Under pumping (and/or injection) conditions, a hybrid formulation is developed to address these local source/sink effects, while different types of boundary conditions can also exert significant influences on drawdowns. Local grid refinement near wells is not needed to obtain accurate results, thus inversion is successful with coarse inverse grids, leading to high computation efficiency. Furthermore, flux measurements are not needed for the inversion to succeed; data requirement of the method is thus not much different from that of interpreting classic well tests. Finally, inversion accuracy is not sensitive to the degree of nonlinearity of the flow equations. Performance of the inverse method for confined and unconfined aquifer problems is similar in terms of the accuracy of the estimated parameters, the recovered head fields, and the solver speed.     

A grid generating algorithm for simulating a fluctuating water table boundary in heterogeneous unconfined aquifers

An algorithm is presented for generating finite element grids that can be used to calculate the position of a fluctuating water table and the formation of seepage faces within a heterogeneous unconfined aquifer. Our approach overcomes limitations with existing techniques by allowing the water table to rise or decline through hydrostratigraphic boundaries yet maintains numerical and conceptual accuracy with respect to hydrostratigraphic geometry. The algorithm involves (1) limited stretching or shrinking of elements along the water table if the change in the position of the water table is small with respect to the vertical grid spacing, and (2) the addition or removal of nodes and elements in the finite element mesh along the water table as the change becomes large with respect to the vertical grid spacing. This technique is applicable to any 2-D or 3-D finite element code that contains an automatic finite-element grid generator.     

Describing near surface, transient flow processes in unconfined aquifers below irrigated lands: A deforming finite element model for heterogeneous aquifers

Hydrologic models of irrigated lands generally adopt either a basin-scale or a root-zone perspective. While basin-wide macro-scale models rely on the aggregation of important spatial and temporal data across large areas, micro-scale root-zone models depend on the definition of rigid boundaries around the zone of plant–soil–water interaction. In reality, irrigation management decisions are made on a field by field basis and can interact across field boundaries. This paper first describes a shallow water table model, based on deforming finite element (DFE) framework, to characterize the near-surface field-to-field hydrologic response to various irrigation and drainage management regimes along a gently sloping alluvial fan. The model is then enhanced through changing geometry of a fluctuating water table below a series of irrigated fields. Such an enhancement also offers computational flexibility relative to the saturated–unsaturated models commonly used in micro-scale studies. The model is designed with the alluvial fan aquifers of California’s western San Joaquin Valley as reference systems.     

Regression method for estimating rainfall recharge at unconfined sandy aquifers with an equatorial climate

A reliable estimate of rainfall recharge is essential for groundwater system managements. This study develops a method based on regression equations for estimating rainfall recharge at unconfined sandy aquifers with an equatorial climate. The developed method (GR-I method) is generally efficient for estimating long-term regional recharge, as the computational procedures could be formulated and executed easily using Microsoft's Excel spreadsheet. More importantly, its application could be extended to sand textures different from the sand texture used in developing the regression equations. To evaluate its reliability, the method was applied to estimate monthly gross recharge percentages at the Changi reclaimed land. When ignoring the effect of rainfall clusters, the GR-I method was found to underestimate the monthly gross recharge percentages for those months with high monthly rainfall depths. By integrating the effect of rainfall clusters, the GR-I method yields reliable estimates of monthly gross recharge percentages. By including daily potential evaporation as an additional input variable, the Extended GR-I method was found to be not superior to the GR-I method, implying that soil moisture availability is the major governing factor for actual soil evaporation in the highly porous sand medium, instead of atmospheric demand represented by the potential evaporation rate. Using the GR-I method, the mean annual net recharge percentage of the study site was found to fall between 56·9 and 69·9%, which corresponds to a net recharge depth of 1073·8–1745·8 mm. Although the developed method provides a good alternative to other widely used methods, its recharge estimates still needs to be collaborated with estimates from other methods, as multiple techniques are highly recommended in any groundwater recharge estimations. Copyright © 2007 John Wiley & Sons, Ltd.     

Laplace transform solutions for solute transport in fissured aquifers

The main processes affecting the migration of a solute in a fissured aquifer will be advection and dispersion in the fissures, diffusion into the porous matrix; and adsorption. This paper considers solute transport in an idealized fissured aquifer consisting of slabs of saturated rock-matrix separated by equally spaced, planar fissures. The solution of the transport equations is developed as far as Laplace transforms of the solute concentrations in the fissure and matrix water. Numerical inversion of the transforms is used to investigate characteristic behaviour of the model for a number of special cases.     

The influence of delayed drainage on data from pumping tests in unconfined aquifers

Equations are derived for the flow to a pumped well in an aquifer having uniform anisotropy and overlain by a low-permeability aquitard. The water-table is assumed to be located in the aquitard. Drainage from the capillary zone above the water-table is taken into account.The differential equation for the flow in the aquifer is identical with that derived in a previous paper. The formation constants may therefore be evaluated by using type curves as described in that paper.A well-known pumping test is reanalysed, using the equations in the present paper. It is shown that the time-drawdown curves can be explained only by the existence of a low-permeability stratum in the vicinity of the water-table. In this example the slow draining of the unsaturated zone above the water-table seems to be a significant factor in determining the shape of the time-drawdown curves.     

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 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.     

Assessing radial transmissivity variation in heterogeneous aquifers using analytical techniques

Pumping tests are one of the most commonly used in situ testing techniques for assessing aquifer hydraulic properties. Numerous researches have been conducted to predict the effects of aquifer heterogeneity on the groundwater levels during pumping tests. The objectives of the present work were as follows: (1) to predict drawdown conditions and to estimate aquifer properties during pumping tests undertaken in radially symmetric heterogeneous aquifers, and (2) to identify a method for assessing the transmissivity field along the radial coordinate in radially symmetric and fully heterogeneous transmissivity fields. The first objective was achieved by expanding an existing analytical drawdown formulation that was valid for a radially symmetric confined aquifer with two concentric zones around the pumping well to an N concentric zone confined aquifer having a constant transmissivity value within each zone. The formulation was evaluated for aquifers with three and four concentric zones to assess the effects of the transmissivity field on the drawdown conditions. The specific conditions under which aquifer properties could be identified using traditional methods of analysis were also evaluated. The second objective was achieved by implementing the inverse solution algorithm (ISA), which was developed for petroleum reservoirs to groundwater aquifer settings. The results showed that the drawdown values are influenced by a volumetric integral of a weighting function and the transmissivity field within the cone of depression. The weighting function migrates in tandem with the expanding cone of depression. The ability of the ISA to predict radially symmetric and log‐normally distributed transmissivity fields was assessed against analytical and numerical benchmarks. The results of this investigation indicated that the ISA method is a viable technique for evaluating the radial transmissivity variations of heterogeneous aquifer settings. Copyright © 2013 John Wiley & Sons, Ltd.     

Fractal travel time estimates for dispersive contaminants

Alternative fractional models of contaminant transport lead to a new travel time formula for arbitrary concentration levels. For an evolving contaminant plume in a highly heterogeneous aquifer, the new formula predicts much earlier arrival at low concentrations. Travel times of contaminant fronts and plumes are often obtained from Darcy's law calculations using estimates of average pore velocities. These estimates only provide information about the travel time of the average concentration (or peak, for contaminant pulses). Recently, it has been shown that finding the travel times of arbitrary concentration levels is a straightforward process, and equations were developed for other portions of the breakthrough curve for a nonreactive contaminant. In this paper, we generalize those equations to include alternative fractional models of contaminant transport.     

Bridging the gap between numerical solutions of travel time distributions and analytical storage selection functions

Recent advancements in analytical solutions to quantify water and solute travel time distributions (TTDs) and the related StorAge Selection (SAS) functions synthesize catchment complexity into a simplified, lumped representation. Although these analytical approaches are efficient in application, they require rarely available long‐term and high‐frequency hydrochemical data for parameter estimation. Alternatively, integrated hydrologic models coupled to Lagrangian particle‐tracking approaches can directly simulate age under different catchment geometries and complexity, but at a greater computational expense. Here, we bridge the two approaches, using a physically based model to explore the uncertainty in the estimation of the SAS function shape. In particular, we study the influence of subsurface heterogeneity, interactions between distinct flow domains (i.e., the vadose zone and saturated groundwater), diversity of flow pathways, and recharge rate on the shape of TTDs and the SAS functions. We use an integrated hydrology model, ParFlow, linked with a particle‐tracking model, SLIM, to compute transient residence times (or ages) at every cell in the domain, facilitating a direct characterization of the SAS function. Steady‐state results reveal that the SAS function shape shows a wide range of variation with respect to the variability in the structure of subsurface heterogeneity. Ensembles of spatially correlated realizations of hydraulic conductivity indicate that the SAS functions in the saturated groundwater have an overall weak tendency toward sampling younger ages, whereas the vadose zone gives a strong preference for older ages. We further show that the influence of recharge rate on the TTD is tightly dependent on the variability of subsurface hydraulic conductivity.     

Importance of the vadose zone in analyses of unconfined aquifer tests

Analytical models commonly used to interpret unconfined aquifer tests have been based on upper-boundary (water table) conditions that do not adequately address effects of time-varying drainage from the vadose zone. As a result, measured and simulated drawdown data may not agree and hydraulic parameters may be inaccurately estimated. A 72-hour aquifer test conducted in Cape Cod, Massachusetts, in a slightly heterogeneous, coarse-grained, glacial outwash deposit was found to be a good candidate for testing models with different upper-boundary conditions. In general, under the commonly invoked assumption of instantaneous drainage, measured and simulated drawdowns were found to agree with one another only at late time and early time. In the intermediate-time range, because of delayed drainage, measured drawdowns always exceeded simulated values, most noticeably in piezometers located near the water table. To reduce these discrepancies, an analytical model was developed that can fully account for time-varying drainage given that the aquifer is not strongly heterogeneous. The approach is flexible as the model, which makes use of empirical relations, does not constrain drainage to follow any particular functional relation. By this approach, measured and simulated drawdowns agree over the complete time range, and the estimated parameters are consistent with prior studies and with what is known about the aquifer geometry, stratigraphy, and composition. By properly accounting for vadose zone drainage, it was found that realistic estimates of all hydraulic parameters, including specific yield, could be obtained with or without the use of late-time data.     

Air and water flows induced by pumping tests in unconfined aquifers with low‐permeability zones

Air flows from the atmosphere into an unconfined aquifer when the water table falls during pumping tests. Pumping test results in unconfined aquifers may be significantly affected by low‐permeability zones (LPZs) near the initial water table position, because they restrict the downward movement of air. A transient, three‐dimensional air–water two‐phase flow model is employed to investigate numerically the effects of local heterogeneity on pumping test results in unconfined aquifers. Two cases of local heterogeneities are considered herein: a LPZ around the pumping well and on one side of the pumping well. Results show that the drawdown with the LPZ is significantly greater than that of the homogeneous aquifer. The differences in drawdown are the most significant at intermediate times and gradually diminish at later times. The LPZ significantly reduces air flow from the atmosphere to the aquifer. The pore air velocity in the LPZ is very low. The air pressure at the observation point under the LPZ when air begins to enter is significantly lower than the air pressure of the homogeneous aquifer at the same point. After that, the air pressure increases quickly and then increases slowly. The time for the air pressure to reach the atmospheric pressure is significantly longer. Copyright © 2013 John Wiley & Sons, Ltd.