Approximate discharge for constant head test with recharging boundary

The calculation of the discharge to a constant drawdown well or tunnel in the presence of an infinite linear constant head boundary in an ideal confined aquifer usually relies on the numerical inversion of a Laplace transform solution. Such a solution is used to interpret constant head tests in wells or to roughly estimate ground water inflow into tunnels. In this paper, a simple approximate solution is proposed. Its maximum relative error is on the order of 2% as compared to the exact analytical solution. The approximation is a weighted mean between the early-time and late-time asymptotes.     

Analytical Solution for Modeling Discharge into a Tunnel Drilled in a Heterogeneous Unconfined Aquifer

Predicting transient inflow rates into a tunnel is an important issue faced by hydrogeologists. Most existing analytical solutions overestimate the initial discharge due to the assumption that drilling was instantaneous over the entire tunnel length. In addition, they assume a homogeneous system. An alternative model was recently developed for tunnels intersecting heterogeneous formations, but its application was reduced to the case of confined flow to deep tunnels in weakly diffusive aquifers. In this paper, we adapt existing analytical solutions for drainage systems to the specific case of a tunnel progressively drilled in a highly diffusive heterogeneous unconfined aquifer. The case of a tunnel overlying an impervious layer is analytically solved by applying the superposition principle, while the case of a tunnel constructed some distance above an impervious layer is solved by discretizing the tunnel length into subsectors. Both models can simulate transient discharge into a tunnel drilled at various speeds through a heterogeneous unconfined aquifer, and allow the prediction of discharge rates in shallow tunnels located in highly diffusive aquifers. We successfully applied this approach to a tunnel in heterogeneous volcanic rock.     

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.     

Analytical Interpretation of Slug Test in a Vertical Cutoff Wall

An analysis method for slug tests performed in a partially penetrating well within a vertical cutoff wall is presented. A steady‐state shape factor for evaluating hydraulic conductivity of the material within the wall was derived by applying the method of images to the previously developed analytical solution of Zlotnik et al. (2010) for an infinite aquifer. Two distinct boundary conditions were considered: constant‐head boundary for the case of direct contact between the wall and the aquifer, and no‐flux boundary representing an impermeable filter cake on the sides of the wall. The constant‐head and no‐flux boundary conditions yield significantly higher and lower shape factors, respectively, than those for the infinite aquifer. Consequently the conventional line‐fitting method for slug test analysis would yield an inaccurate estimate of the hydraulic conductivity of a vertical cutoff wall.     

Analytic solutions to transient groundwater flow under time-dependent sources in a heterogeneous aquifer bounded by fluctuating river stage

Analytical solutions for the water table and lateral discharge in a heterogeneous unconfined aquifer with time-dependent source and fluctuating river stage were derived and compared with those in an equivalent homogeneous aquifer. The heterogeneous aquifer considered consists of a number of sections of different hydraulic conductivity values. The source term and river stage were assumed to be time-dependent but spatially uniform. The solutions derived is useful in studying various groundwater flow problems in a horizontally heterogeneous aquifer since the spatially piecewise-constant hydraulic conductivity and temporally piecewise-constant recharge and lateral discharge can be used to quantify variations in these processes commonly observed in reality. Applying the solutions derived to an aquifer of three sections of different hydraulic conductivity values shown that (1) the aquifer heterogeneity significantly increases the spatial variation of the water table and thus its gradient but it has little effect on lateral discharge in the case of temporally and spatially uniform recharge, (2) the time-dependent but spatially uniform recharge increases the temporal variation of groundwater table over the entire aquifer but its effect on lateral discharge is limited in the zone near the river, and (3) the effect of river stage fluctuation on the water table and lateral discharge is limited in the zone near the river and the effect of the heterogeneity is to increase lateral discharge to or recharge from the river.     

Hydraulics of a partially penetrating well with skin zone in a confined aquifer

A steady/quasi-steady model is developed for predicting flow into a partially penetrating well with skin zone in a confined aquifer overlying an impervious layer. The model takes into account flow through the bottom of the wellbore, finite skin thickness and finite horizontal and vertical extent of the aquifer. Moreover, the solution can be easily extended to include the mixed-type boundary condition at the well face, where a Dirichlet in the form of a specified hydraulic head and a Neumann in the form of zero flux coexist at the same time at different portions of the well face. The validity of the proposed solution is tested by comparing a few results obtained from the developed model with corresponding results obtained by analytical and numerical means. The study shows that, among other factors remaining constant, both the horizontal and vertical extent of an artesian aquifer, thickness of the skin zone, bottom flow and conductivity contrast of the skin and formation zones, play an important part in deciding flow to a well dug in the aquifer, and hence these factors must be considered while analyzing the problem. The model proposed here can be used to estimate skin thickness as well as hydraulic conductivities of the skin and formation zones of a well with skin zone in an artesian aquifer underlain by an impervious layer by utilizing pumping test data falling in the steady or quasi-steady state of a typical pumping test. As the proposed solution is of a general nature in the sense that it can handle, apart from partial penetration and bottom flow, the finite size skin zone and finite horizontal and vertical extent of an artesian aquifer together with the mixed-type boundary condition at the well face, it is hoped that the predictions coming out of the model will be more realistic than those obtained using solutions developed with more stringent assumptions.     

Analysis of steady ground water flow toward wells in a confined-unconfined aquifer

A confined aquifer may become unconfined near the pumping wells when the water level falls below the confining unit in the case where the pumping rate is great and the excess hydraulic head over the top of the aquifer is small. Girinskii's potential function is applied to analyze the steady ground water flow induced by pumping wells with a constant-head boundary in a mixed confined-unconfined aquifer. The solution of the single-well problem is derived, and the critical radial distance at which the flow changes from confined to unconfined condition is obtained. Using image wells and the superposition method, an analytic solution is presented to study steady ground water flow induced by a group of pumping wells in an aquifer bounded by a river with constant head. A dimensionless function is introduced to determine whether a water table condition exists or not near the pumping wells. An example with three pumping wells is used to demonstrate the patterns of potentiometric surface and development of water table around the wells.     

General Steady-State Shape Factor for a Partially Penetrating Well

We present the closed form of a general steady-state shape factor for a partially penetrating well in a uniform anisotropic aquifer. Our simple analytical expression for the shape factor has a uniform representation for full range of parameters and meets or exceeds the accuracy of previous results obtained through semiempirical methods (e.g., Bouwer and Rice [1976] equations). This general shape factor pertains to the flow of fluids (water or air) in subsurface formations when the upper formation boundary has constant potential and the lower boundary is impermeable. The results of our investigation are directly applicable to analyses of (1) slug tests with falling or rising head and (2) injection/extraction tests with constant head, essential techniques for the characterization of hydraulic conductivity of aquifers, streambeds, or lakebeds as well as the design of aquifer and soil remediation systems.     

Static and dynamic conceptual model of a complexly fractured crystalline rock aquifer

A conceptual model of anisotropic and dynamic permeability is developed from hydrogeologic and hydromechanical characterization of a foliated, complexly fractured, crystalline rock aquifer at Gates Pond, Berlin, Massachusetts. Methods of investigation include aquifer‐pumping tests, long‐term hydrologic monitoring, fracture characterization, downhole heat‐pulse flow meter measurements, in situ extensometer testing, and earth tide analysis. A static conceptual model is developed from observations of depth‐dependent and anisotropic permeability that effectively compartmentalizes the aquifer as a function of foliation intensity. Superimposed on the static model is dynamic permeability as a function of hydraulic head in which transient bulk aquifer transmissivity is proportional to changes in hydraulic head due to hydromechanical coupling. The dynamic permeability concept is built on observations that fracture aperture changes as a function of hydraulic head, as measured during in situ extensometer testing of individual fractures, and observed changes in bulk aquifer transmissivity as determined from earth tides during seasonal changes in hydraulic head, with higher transmissivity during periods of high hydraulic head, and lower transmissivity during periods of relatively lower hydraulic head. A final conceptual model is presented that captures both the static and dynamic properties of the aquifer. The workflow presented here demonstrates development of a conceptual framework for building numerical models of complexly fractured, foliated, crystalline rock aquifers that includes both a static model to describe the spatial distribution of permeability as a function of fracture type and foliation intensity and a dynamic model that describes how hydromechanical coupling impacts permeability magnitude as a function of hydraulic head fluctuation. This model captures important geologic controls on permeability magnitude, anisotropy, and transience and therefor offers potentially more reliable history matching and forecasts of different water management strategies, such as resource evaluation, well placement, permeability prediction, and evaluating remediation strategies.     

Boundary condition effects on maximum groundwater withdrawal in coastal aquifers

Prevention of sea water intrusion in coastal aquifers subject to groundwater withdrawal requires optimization of well pumping rates to maximize the water supply while avoiding sea water intrusion. Boundary conditions and the aquifer domain size have significant influences on simulating flow and concentration fields and estimating maximum pumping rates. In this study, an analytical solution is derived based on the potential-flow theory for evaluating maximum groundwater pumping rates in a domain with a constant hydraulic head landward boundary. An empirical correction factor, which was introduced by Pool and Carrera (2011) to account for mixing in the case with a constant recharge rate boundary condition, is found also applicable for the case with a constant hydraulic head boundary condition, and therefore greatly improves the usefulness of the sharp-interface analytical solution. Comparing with the solution for a constant recharge rate boundary, we find that a constant hydraulic head boundary often yields larger estimations of the maximum pumping rate and when the domain size is five times greater than the distance between the well and the coastline, the effect of setting different landward boundary conditions becomes insignificant with a relative difference between two solutions less than 2.5%. These findings can serve as a preliminary guidance for conducting numerical simulations and designing tank-scale laboratory experiments for studying groundwater withdrawal problems in coastal aquifers with minimized boundary condition effects.     

A Correction on Coastal Heads for Groundwater Flow Models

We introduce a simple correction to coastal heads for constant‐density groundwater flow models that contain a coastal boundary, based on previous analytical solutions for interface flow. The results demonstrate that accurate discharge to the sea in confined aquifers can be obtained by direct application of Darcy's law (for constant‐density flow) if the coastal heads are corrected to ((α + 1)/α)h_s ? B/2α, in which h_s is the mean sea level above the aquifer base, B is the aquifer thickness, and α is the density factor. For unconfined aquifers, the coastal head should be assigned the value . The accuracy of using these corrections is demonstrated by consistency between constant‐density Darcy's solution and variable‐density flow numerical simulations. The errors introduced by adopting two previous approaches (i.e., no correction and using the equivalent fresh water head at the middle position of the aquifer to represent the hydraulic head at the coastal boundary) are evaluated. Sensitivity analysis shows that errors in discharge to the sea could be larger than 100% for typical coastal aquifer parameter ranges. The location of observation wells relative to the toe is a key factor controlling the estimation error, as it determines the relative aquifer length of constant‐density flow relative to variable‐density flow. The coastal head correction method introduced in this study facilitates the rapid and accurate estimation of the fresh water flux from a given hydraulic head measurement and allows for an improved representation of the coastal boundary condition in regional constant‐density groundwater flow models.     

A novel analytical solution for a slug test conducted in a well with a finite-thickness skin

An aquifer containing a skin zone is considered as a two-zone system. A mathematical model describing the head distribution is presented for a slug test performed in a two-zone confined aquifer system. A closed-form solution for the model is derived by Laplace transforms and Bromwich integral. This new solution is used to investigate the effects of skin type, skin thickness, and the contrast of skin transmissivity to formation transmissivity on the distributions of dimensionless hydraulic head. The results indicate that the effect of skin type is marked if the slug-test data is obtained from a radial two-zone aquifer system. The dimensionless well water level increases with the dimensionless positive skin thickness and decreases as the dimensionless negative skin thickness increases. In addition, the distribution of dimensionless well water level due to the slug test depends on the hydraulic properties of both the wellbore skin and formation zones.     

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.     

A general analytical solution for flow to a single horizontal well by Fourier and Laplace transforms

The objective of this paper is to present an analytical solution for describing the head distribution in an unconfined aquifer with a single pumping horizontal well parallel to a fully penetrating stream. The Laplace-domain solution is developed by applying Fourier sine, Fourier and Laplace transforms to the governing equation as well as the associated initial and boundary conditions. The time-domain solution is obtained after taking the inverse Laplace transform along with the Bromwich integral method and inverse Fourier and Fourier sine transforms. The upper boundary condition of the aquifer is represented by the free surface equation in which the second-order slope terms are neglected. Based on the solution and Darcy’s law, the equation representing the stream depletion rate is then derived. The solution can simulate head distributions in an aquifer infinitely extending in horizontal direction if the well is located far away from the stream. In addition, the solution can also simulate head distributions in confined aquifers if specific yield is set zero. It is shown that the solution can be applied practically to evaluate flow to a horizontal well.     

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.     

New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions

The hydraulic head distribution in a wedge-shaped aquifer depends on the wedge angle and the topographic and hydrogeological boundary conditions. In addition, an equation in terms of the radial distance with trigonometric functions along the boundary may be suitable to describe the water level configuration for a valley flank with a gentle sloping and rolling topography. This paper develops a general mathematical model including the governing equation and a variety of boundary conditions for the groundwater flow within a wedge-shaped aquifer. Based on the model, a new closed-form solution for transient flow in the wedge-shaped aquifer is derived via the finite sine transform and Hankel transform. In addition, a numerical approach, including the roots search scheme, the Gaussian quadrature, and Shanks’ method, is proposed for efficiently evaluating the infinite series and the infinite integral presented in the solution. This solution may be used to describe the head distribution for wedges that image theory is inapplicable, and to explore the effects of the recharge from various topographic boundaries on the groundwater flow system within a wedge-shaped aquifer.     

Steady flow to a well near a stream with a leaky bed

We present an explicit analytic solution for steady, two-dimensional ground water flow to a well near a leaky streambed that penetrates the aquifer partially. Leakage from the stream is approximated as occurring along the centerline of the stream. The problem domain is infinite and pumping on one side of the stream induces flow on the other side. The solution includes the effects of uniform flow in the far field and a sloping hydraulic head in the stream. We use the solution to investigate the interaction between ground water and surface water in the stream, the effects of pumping on the opposite side of the stream, and the effects of the leaky streambed on the capture zone envelope of the well. We develop a relationship between parameters such that the pumping well will not capture water from the stream, or from the opposite side of the stream. When the discharge of the well is large enough to capture water from the stream, the shape of the capture zone envelope depends on flow conditions on the side of the stream opposite the well.     

Studying the Flow Dynamics of a Karst Aquifer System with an Equivalent Porous Medium Model

The modeling of groundwater flow in karst aquifers is a challenge due to the extreme heterogeneity of its hydraulic parameters and the duality in their discharge behavior, that is, rapid response of highly conductive karst conduits and delayed drainage of the low‐permeability fractured matrix after recharge events. There are a number of different modeling approaches for the simulation of the karst groundwater dynamics, applicable to different aquifer as well as modeling problem types, ranging from continuum models to double continuum models to discrete and hybrid models. This study presents the application of an equivalent porous model approach (EPM, single continuum model) to construct a steady‐state numerical flow model for an important karst aquifer, that is, the Western Mountain Aquifer Basin (WMAB), shared by Israel and the West‐Bank, using MODFLOW2000. The WMAB was used as a catchment since it is a well‐constrained catchment with well‐defined recharge and discharge components and therefore allows a control on the modeling approach, a very rare opportunity for karst aquifer modeling. The model demonstrates the applicability of equivalent porous medium models for the simulation of karst systems, despite their large contrast in hydraulic conductivities. As long as the simulated saturated volume is large enough to average out the local influence of karst conduits and as long as transport velocities are not an issue, EPM models excellently simulate the observed head distribution. The model serves as a starting basis that will be used as a reference for developing a long‐term dynamic model for the WMAB, starting from the pre‐development period (i.e., 1940s) up to date.     

Constraints on hydraulic parameters and implications for groundwater flux across the upland–estuary interface

Analyses of independent laboratory- and field-scale measurements from two sites on Sapelo Island, Georgia reveal heterogeneity in hydraulic parameters across the upland–estuary interface. Regardless of the method used (short-duration pumping tests, amplitude attenuation of tidal pumping data, sediment grain size distributions, and falling head permeameter tests), we obtain hydraulic conductivity of 10⁻⁴ m s⁻¹ for the fine-grained, well-sorted, clean sands that make up the upland areas. Proximal to the upland–estuary boundary, the tidal pumping analyses and permeameter tests suggest that hydraulic conductivities decrease by more than two orders of magnitude, a result consistent with the presence of a clogging layer. Such a clogging layer may arise due to a variety of physical, chemical, or biological processes. The extent and orientation of the layers of reduced hydraulic conductivity near the upland–estuary boundary influence the nature of the aquifer's response to tidal forcing. Where the lower conductivity layer forms a relatively flat creek bank, tidal pumping produces a primarily mechanical response in the adjacent aquifer. Where the creek bank is nearly vertical, there is a more direct hydraulic connection between the tidal creek and the adjacent aquifer. The clogging layer likely contributes to the development of complicated flow pathways across the upland–estuary boundary. Effective flow paths calculated from tidal pumping data terminate within the marsh, beyond the boundary of the upland aquifer, suggesting a diffuse regime of groundwater discharge in the marsh. We postulate that, in many settings, submarsh flow may be as important as seepage faces for groundwater discharge into the marsh–estuary complex.