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.     

Strategies for offsetting seasonal impacts of pumping on a nearby stream

Ground water pumping from aquifer systems that are hydraulically connected to streams depletes streamflow. The amplitude and timing of stream depletion depend on the stream depletion factor (SDF(i)) of the pumping wells, which is a function of aquifer hydraulic characteristics and the distance from the wells to the stream. Wells located at different locations, but having the same SDF and the same rate and schedule of pumping, will deplete streamflow equally. Wells with small SDF(i) deplete streamflow approximately synchronously with pumping. Wells with large SDF(i) deplete streamflow at approximately a constant rate throughout the year, regardless of the pumping schedule. For large values of SDF(i), artificial recharge that occurs on a different schedule from pumping can offset streamflow depletion effectively. The requirements are (1) that the pumping and recharge wells both have the same SDF(i) and (2) that the annual total quantities of recharge and pumping be equal. At larger SDF(i) values, it takes longer for pumping to impact streamflow in a wide aquifer than it does in a narrow aquifer. In basins that are closed to further withdrawals because streamflow is fully allocated, water-use changes replace new allocations as the source of water for new developments. Ground water recharge can be managed to offset the impacts of new ground water developments, allowing for changes in the timing and source of withdrawals from a basin without injuring existing users or instream flows.     

Migration of induced-infiltrated stream water into nearby aquifers due to seasonal ground water withdrawal

Analysis of stream-aquifer interaction due to ground water extraction has traditionally focused on the determination of the amount of water depleted in the stream. Less attention has been paid to the movement of infiltrated stream water inside aquifer, particularly for agricultural areas. This paper presents a method of using particle-tracking techniques to evaluate the transport of the leaked stream water in the nearby aquifers. Simple stream-aquifer conditions are used to demonstrate the usefulness of the analysis. Travel times, pathlines, and influence zones of stream water were determined between a stream and nearby pumping wells for seasonal ground water extraction areas. When water quantity is a concern, the analyses provide additional information about stream depletion; when water quality is an issue, they offer information for wellhead protection. Analyses were conducted for transient conditions, and both pumping and nonpumping periods were considered. According to the results from the simulation examples, migration of infiltrated stream water into the nearby aquifers is generally slow and most infiltrated stream water does not arrive at the pumping well at the end of a 90-day irrigation season. Infiltrated stream water may remain in the aquifer for several years before arriving at the pumping well. For aquifers with a regional hydraulic gradient toward streams, part of the infiltrated stream water may discharge back to streams during a recovery period.     

Boundary depletion rate and drawdown in leaky wedge‐shaped aquifers

This study presents analytical solutions of the three‐dimensional groundwater flow to a well in leaky confined and leaky water table wedge‐shaped aquifers. Leaky wedge‐shaped aquifers with and without storage in the aquitard are considered, and both transient and steady‐state drawdown solutions are derived. Unlike the previous solutions of the wedge‐shaped aquifers, the leakages from aquitard are considered in these solutions and unlike similar previous work for leaky aquifers, leakage from aquitards and from the water table are treated as the lower and upper boundary conditions. A special form of finite Fourier transforms is used to transform the z‐coordinate in deriving the solutions. The leakage induced by a partially penetrating pumping well in a wedge‐shaped aquifer depends on aquitard hydraulic parameters, the wedge‐shaped aquifer parameters, as well as the pumping well parameters. We calculate lateral boundary dimensionless flux at a representative line and investigate its sensitivity to the aquitard hydraulic parameters. We also investigate the effects of wedge angle, partial penetration, screen location and piezometer location on the steady‐state dimensionless drawdown for different leakage parameters. Results of our study are presented in the form of dimensionless flux‐dimensionless time and dimensionless drawdown‐leakage parameter type curves. The results are useful for evaluating the relative role of lateral wedge boundaries and leakage source on flow in wedge‐shaped aquifers. This is very useful for water management problems and for assessing groundwater pollution. The presented analytical solutions can also be used in parameter identification and in calculating stream depletion rate and volume. Copyright © 2011 John Wiley & Sons, Ltd.     

Stream-aquifer interactions: evaluation of depletion volume and residual effects from ground water pumping

Numerical modeling techniques were used to simulate stream-aquifer interactions from seasonal ground water pumping. We used stream-aquifer models in which a shallow stream penetrates the top of an aquifer that discharges ground water to the stream as base flow. Because of the pumping, the volume of base flow discharged to the stream was reduced, and as the pumping continued, infiltration from the stream to the aquifer was induced. Both base-flow reduction and stream infiltration contributed to total stream depletion. We analyzed the depletion rates and volumes of the reduced base flow and induced stream infiltration during pumping and postpumping periods. Our results suggested that for a shallow penetrating stream with a low streambed conductance, base-flow reduction accounts for a significant percentage of the total stream depletion. Its residual effects in postpumping can last very long and may continue into the next pumping season for areas where recharge is nominal. In contrast, the contribution of the induced stream infiltration to the total stream depletion is much smaller, and its effects often become negligible shortly after pumping was stopped. For areas where surface recharge replenishes the aquifer, the residual effects of base-flow reduction and thus its depletion volume will be significantly reduced. A stream of large conductance has a high hydraulic connection to the aquifer, but the relationship between stream conductance and stream depletion is not linear.     

Aquifer response to stream-stage and recharge variations. I. Analytical step-response functions

Laplace transform step-response functions are presented for various homogeneous confined and leaky aquifer types and for anisotropic, homogeneous unconfined aquifers interacting with perennial streams. Flow is one-dimensional, perpendicular to the stream in the confined and leaky aquifers, and two-dimensional in a plane perpendicular to the stream in the water-table aquifers. The stream is assumed to penetrate the full thickness of the aquifer. The aquifers may be semi-infinite or finite in width and may or may not be bounded at the stream by a semipervious streambank. The solutions are presented in a unified manner so that mathematical relations among the various aquifer configurations are clearly demonstrated. The Laplace transform solutions are inverted numerically to obtain the real-time step-response functions for use in the convolution (or superposition) integral. To maintain linearity in the case of unconfined aquifers, fluctuations in the elevation of the water table are assumed to be small relative to the saturated thickness, and vertical flow into or out of the zone above the water table is assumed to occur instantaneously. Effects of hysteresis in the moisture distribution above the water table are therefore neglected. Graphical comparisons of the new solutions are made with known closed-form solutions.     

Application of a Discrete-Continuum Model to Karst Aquifers in North China

A generalized discrete-continuum model is developed to simulate ground water flow in the karst aquifers of North China. The model is a hybrid numerical flow model, which takes into account both quick conduit flow and diffusive fissure flow. The conduit flow is represented by a discrete network model, and the fissure flow is modeled by a continuum approach. The developed model strongly emphasizes the function of the conduits in the flow fields. They control the general drainage pattern, as demonstrated in the simulation of a complex karst aquifer in North China. The model reproduces reasonably well the flow field in response to an unanticipated discharge of ground water from the karst aquifer into an underground mine based on the aquifer parameters that are manually calibrated from a multiple-well pumping test. Sensitivity of the model to the aquifer parameters was evaluated in the context of the case study.     

Surface flow boundary conditions in modeling land subsidence due to fluid withdrawal

Land subsidence due to subsurface fluid (water, gas, oil) withdrawal is often predicted by either finite element or finite difference numerical models based on coupled poroelastic theory, where the soil is represented as a semi-infinite medium bounded by the traction-free (ground) surface. One of the variables playing a most important role on the final outcome is the flow condition used on the traction-free boundary, which may be assumed as either permeable or impermeable. Although occasionally justified, the assumption of no-flow surface seems to be in general rather unrealistic. A permeable boundary where the fluid pressure is fixed to the external atmospheric pressure appears to be more appropriate. This paper addresses the response, in terms of land subsidence, obtained with a coupled poroelastic finite element model that simulates a distributed pumping from a horizontal aquifer confined between two relatively impervious layers, and takes either a permeable boundary surface, i.e., constant hydraulic potential, or an impermeable boundary, i.e., a zero Neumann flow condition. The analysis reveals that land subsidence is rather sensitive to the flow condition implemented on the traction-free boundary. In general, the no-flow condition leads to an overestimate of the predicted ground surface settlement, which could even be 1 order of magnitude larger than that obtained with the permeable boundary.     

A method for evaluating horizontal well pumping tests

Predicting the future performance of horizontal wells under varying pumping conditions requires estimates of basic aquifer parameters, notably transmissivity and storativity. For vertical wells, there are well-established methods for estimating these parameters, typically based on either the recovery from induced head changes in a well or from the head response in observation wells to pumping in a test well. Comparable aquifer parameter estimation methods for horizontal wells have not been presented in the ground water literature. Formation parameter estimation methods based on measurements of pressure in horizontal wells have been presented in the petroleum industry literature, but these methods have limited applicability for ground water evaluation and are based on pressure measurements in only the horizontal well borehole, rather than in observation wells. This paper presents a simple and versatile method by which pumping test procedures developed for vertical wells can be applied to horizontal well pumping tests. The method presented here uses the principle of superposition to represent the horizontal well as a series of partially penetrating vertical wells. This concept is used to estimate a distance from an observation well at which a vertical well that has the same total pumping rate as the horizontal well will produce the same drawdown as the horizontal well. This equivalent distance may then be associated with an observation well for use in pumping test algorithms and type curves developed for vertical wells. The method is shown to produce good results for confined aquifers and unconfined aquifers in the absence of delayed yield response. For unconfined aquifers, the presence of delayed yield response increases the method error.     

Effects of sea-level rise on ground water flow in a coastal aquifer system

The effects of sea-level rise on the depth to the fresh water/salt water interface were simulated by using a density-dependent, three-dimensional numerical ground water flow model for a simplified hypothetical fresh water lens that is similar to shallow, coastal aquifers found along the Atlantic coast of the United States. Simulations of sea-level rise of 2.65 mm/year from 1929 to 2050 resulted in an increase in water levels relative to a fixed datum, yet a net decrease in water levels relative to the increased sea-level position. The net decrease in water levels was much greater near a gaining stream than farther from the stream. The difference in the change in water levels is attributed to the dampening effect of the stream on water level changes in response to sea-level rise. In response to the decreased water level altitudes relative to local sea level, the depth to the fresh water/salt water interface decreased. This reduction in the thickness of the fresh water lens varied throughout the aquifer and was greatly affected by proximity to a ground water fed stream and whether the stream was tidally influenced. Away from the stream, the thickness of the fresh water lens decreased by about 2% from 1929 to 2050, whereas the fresh water lens thickness decreased by about 22% to 31% for the same period near the stream, depending on whether the stream was tidally influenced. The difference in the change in the fresh water/salt water interface position is controlled by the difference in the net decline in water levels relative to local sea level.     

Michigan basin regional ground water flow discharge to three Great Lakes

Ground water discharge to the Great Lakes around the Lower Peninsula of Michigan is primarily from recharge in riparian basins and proximal upland areas that are especially important to the northern half of the Lake Michigan shoreline. A steady-state finite-difference model was developed to simulate ground water flow in four regional aquifers in Michigan's Lower Peninsula: the Glaciofluvial, Saginaw, Parma-Bayport, and Marshall aquifers interlayered with the Till/"red beds," Saginaw, and Michigan confining units, respectively. The model domain was laterally bound by a continuous specified-head boundary, formed from lakes Michigan, Huron, St. Clair, and Erie, with the St. Clair and Detroit River connecting channels. The model was developed to quantify regional ground water flow in the aquifer systems using independently determined recharge estimates. According to the flow model, local stream stages and discharges account for 95% of the overall model water budget; only 50% enters the lakes directly from the ground water system. Direct ground water discharge to the Great Lakes' shorelines was calculated at 36 m3/sec, accounting for 5% of the overall model water budget. Lowland areas contribute far less ground water discharge to the Great Lakes than upland areas. The model indicates that Saginaw Bay receives only approximately 1.13 m3/sec ground water; the southern half of the Lake Michigan shoreline receives only approximately 2.83 m3/sec. In contrast, the northern half of the Lake Michigan shoreline receives more than 17 m3/sec from upland areas.     

Analysis of an unconfined aquifer subject to asynchronous dual-tide propagation

Most published solutions for aquifer responses to ocean tides focus on the one-sided attenuation of the signal as it propagates inland. However, island aquifers experience periodic forcing from the entire coast, which can lead to integrated effects of different tidal signals, especially on narrow high-permeability islands. In general, studies disregard a potential time lag as the tidal wave sweeps around the island. We present a one-dimensional analytical solution to the ground water flow equation subject to asynchronous and asymmetric oscillating head conditions on opposite boundaries and test it on data from an unconfined volcanic aquifer in Maui. The solution considers sediment-damping effects at the coastline. The response of Maui Aquifers indicate that water table elevations near the center of the aquifer are influenced by a combination of tides from opposite coasts. A better match between the observed ground water head and the theoretical response can be obtained with the proposed dual-tide solution than with single-sided solutions. Hydraulic diffusivity was estimated to be 2.3 × 10⁷ m²/d. This translates into a hydraulic conductivity of 500 m/d, assuming a specific yield of 0.04 and an aquifer thickness of 1.8 km. A numerical experiment confirmed the hydraulic diffusivity value and showed that the y -intercepts of the modal attenuation and phase differences estimated by regression can approximate damping factors caused by low-permeability units at the boundary.     

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.     

Patterns and Age Distribution of Ground-Water Flow to Streams

Simulations of ground-water flow in a generic aquifer system were made to characterize the topology of ground-water flow in the stream subsystem and to evaluate its relation to deeper ground-water flow. The flow models are patterned after hydraulic characteristics of aquifers of the Atlantic Coastal Plain and are based on numerical solutions to three-dimensional, steady-state, unconfined flow. The models were used to evaluate the effects of aquifer horizontal-to-vertical hydraulic conductivity ratios, aquifer thickness, and areal recharge rates on flow in the stream subsystem. A particle tracker was used to determine flow paths in a stream subsystem, to establish the relation between ground-water seepage to points along a simulated stream and its source area of flow, and to determine ground-water residence time in stream subsystems. In a geometrically simple aquifer system with accretion, the source area of flow to streams resembles an elongated ellipse that tapers in the down gradient direction. Increased recharge causes an expansion of the stream subsystem. The source area of flow to the stream expands predominantly toward the stream headwaters. Base flow gain is also increased along the reach of the stream. A thin aquifer restricts ground-water flow and causes the source area of flow to expand near stream headwaters and also shifts the start-of-flow to the drainage basin divide. Increased aquifer anisotropy causes a lateral expansion of the source area of flow to streams. Ground-water seepage to the stream channel originates both from near- and far-recharge locations. The range in the lengths of flow paths that terminate at a point on a stream increase in the downstream direction. Consequently, the age distribution of ground water that seeps into the stream is skewed progressively older with distance downstream. Base flow ia an integration of ground water with varying age and potentially different water quality, depending on the source within the drainage basin. The quantitative results presented indicate that this integration can have a wide and complex residence time range and source distribution.     

Pumping-induced drawdown and stream depletion in a leaky aquifer system

The impact of ground water pumping on nearby streams is often estimated using analytic models of the interconnected stream-aquifer system. A common assumption of these models is that the pumped aquifer is underlain by an impermeable formation. A new semianalytic solution for drawdown and stream depletion has been developed that does not require this assumption. This solution shows that pumping-induced flow (leakage) through an underlying aquitard can be an important recharge mechanism in many stream-aquifer systems. The relative importance of this source of recharge increases with the distance between the pumping well and the stream. The distance at which leakage becomes the primary component of the pumping-induced recharge depends on the specific properties of the aquifer, aquitard, and streambed. Even when the aquitard is orders of magnitude less transmissive than the aquifer, leakage can be an important recharge mechanism because of the large surface area over which it occurs. Failure to consider aquitard leakage can lead to large overestimations of both the drawdown produced by pumping and the contribution of stream depletion to the pumping-induced recharge. The ramifications for water resources management and water rights adjudication can be significant. A hypothetical example helps illustrate these points and demonstrates that more attention should be given to estimating the properties of aquitards underlying stream-aquifer systems. The solution presented here should serve as a relatively simple but versatile tool for practical assessments of pumping-induced stream-aquifer interactions. However, this solution should not be used for such assessments without site-specific data that indicate pumping has induced leakage through the aquitard.     

Adjoint simulation of stream depletion due to aquifer pumping

If an aquifer is hydraulically connected to an adjacent stream, a pumping well operating in the aquifer will draw some water from aquifer storage and some water from the stream, causing stream depletion. Several analytical, semi-analytical, and numerical approaches have been developed to estimate stream depletion due to pumping. These approaches are effective if the well location is known. If a new well is to be installed, it may be desirable to install the well at a location where stream depletion is minimal. If several possible locations are considered for the location of a new well, stream depletion would have to be estimated for all possible well locations, which can be computationally inefficient. The adjoint approach for estimating stream depletion is a more efficient alternative because with one simulation of the adjoint model, stream depletion can be estimated for pumping at a well at any location. We derive the adjoint equations for a coupled system with a confined aquifer, an overlying unconfined aquifer, and a river that is hydraulically connected to the unconfined aquifer. We assume that the stage in the river is known, and is independent of the stream depletion, consistent with the assumptions of the MODFLOW river package. We describe how the adjoint equations can be solved using MODFLOW. In an illustrative example, we show that for this scenario, the adjoint approach is as accurate as standard forward numerical simulation methods, and requires substantially less computational effort.     

Effects of Improper Characterization of Aquifer Thickness on Estimates of Hydraulic Conductivity from Pumping Tests

Pumping test data for surficial aquifers are commonly analyzed under the assumption that the base of the aquifer corresponds to the bottom of the test wells (i.e., the aquifer is truncated). This practice can lead to inaccurate hydraulic conductivity estimates, resulting from the use of low saturated thickness values with transmissivity estimates, and not accounting for the effects of partially penetrating wells. Theoretical time-drawdown data were generated at an observation well in a hypothetical unconfined aquifer for various values of saturated thickness and were analyzed by standard curve-matching techniques. The base of the aquifer was assumed to be the bottom of the pumping and observation wells. The overestimation of horizontal hydraulic conductivity was found to be directly proportional to the error in assumed saturated thickness, and to the (actual) ratio of vertical to horizontal hydraulic conductivity (K_v/K_h). Inaccurately high estimates of hydraulic conductivity obtained by aquifer truncation can lead to overestimates of ground water velocity and contaminant plume spreading, narrow capture zone configuration estimates, and overestimates of available ground water resources.     

A double-porosity model for a fractured aquifer with non-Darcian flow in fractures

Abstract

Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model.     

Aquifer parameter estimation using an incremental area method

Theoretical well functions have been derived over the years to predict ground water level behaviour in aquifer systems under stress owing to groundwater extraction. The drawdown data collected during pump tests are typically analysed using graphical curve‐matching procedures to estimate aquifer parameters based on these well functions. Difficulty in aquifer characteristic identification and parameter estimation may arise when the field data do not perfectly match the drawdown curves obtained from the well functions. The present study provides a new method for the interpretation of aquifer pump tests which supplements the existing curve‐matching procedures in case ideal conditions do not exist; the proposed method provides a greater degree of flexibility in the data analysis for diagnostic tool purposes. The method, referred to as the Incremental Area Method (IAM) is based on integrating the logarithmic‐based drawdown curves within a discrete time and matching the results with a corresponding time integral of the Theis ( 1935 ) Well Function which governs ideal confined aquifers. The application of the proposed method to synthetically generated data and field data showed that IAM represents a viable method which yields information on potential non‐idealness of the aquifer and provides aquifer parameter estimates thus potentially overcoming drawdown data curve‐matching difficulties. Copyright © 2011 John Wiley & Sons, Ltd.     

Transient‐state groundwater flow in various geometries with wells at arbitrary positions: analytical methods

We have developed a method for analytically solving the porous medium flow equation in many different geometries for horizontal (two‐dimensional), homogeneous and isotropic aquifers containing impermeable boundaries and any number of pumping or injection wells located at arbitrary positions within the system. Solutions and results are presented for rectangular and circular aquifers but the method presented here is easily extendible to many geometries. Results are also presented for systems where constant head boundary conditions can be emulated internal to the aquifer boundary. Recommendations for extensions of the present work are briefly discussed. Copyright © 2003 John Wiley & Sons, Ltd.