Radial groundwater flow to a finite diameter well in a leaky confined aquifer with a finite‐thickness skin

A mathematical model that describes the drawdown due to constant pumpage from a finite radius well in a two‐zone leaky confined aquifer system is presented. The aquifer system is overlain by an aquitard and underlain by an impermeable formation. A skin zone of constant thickness exists around the wellbore. A general solution to a two‐zone leaky confined aquifer system in Laplace domain is developed and inverted numerically to the time‐domain solution using the modified Crump (1976) algorithm. The results show that the drawdown distribution is significantly influenced by the properties and thickness of the skin zone and aquitard. The sensitivity analyses of parameters of the aquifer and aquitard are performed to illustrate their effects on drawdowns in a two‐zone leaky confined aquifer system. For the negative‐skin case, the drawdown is very sensitive to the relative change in the formation transmissivity. For the positive‐skin case, the drawdown is also sensitive to the relative changes in the skin thickness, and both the skin and formation transmissivities over the entire pumping period and the well radius and formation storage coefficient at early pumping time. Copyright © 2009 John Wiley & Sons, Ltd.     

Semi‐analytical solution for a slug test in partially penetrating wells including the effect of finite‐thickness skin

This paper presents a new semi‐analytical solution for a slug test in a well partially penetrating a confined aquifer, accounting for the skin effect. This solution is developed based on the solution for a constant‐flux pumping test and a formula given by Peres and co‐workers in 1989. The solution agrees with that of Cooper and co‐workers and the KGS model when the well is fully penetrating. The present solution can be applied to simulate the temporal and spatial head distributions in both the skin and formation zones. It can also be used to demonstrate the influences of skin type or skin thickness on the well water level and to estimate the hydraulic parameters of the skin and formation zones using a least‐squares approach. The results of this study indicate that the determination of hydraulic conductivity using a conventional slug‐test data analysis that neglects the presence of a skin zone will give an incorrect result if the aquifer has a skin zone. Copyright © 2008 John Wiley & Sons, Ltd.     

Effect of wellbore storage and finite thickness skin on flow to a partially penetrating well in a phreatic aquifer

An analytical model is presented for the analysis of constant flux tests conducted in a phreatic aquifer having a partially penetrating well with a finite thickness skin. The solution is derived in the Laplace transform domain for the drawdown in the pumping well, skin and formation regions. The time-domain solution in terms of the aquifer drawdown is then obtained from the numerical inversion of the Laplace transform and presented as dimensionless drawdown–time curves. The derived solution is used to investigate the effects of the hydraulic conductivity contrast between the skin and formation, in addition to wellbore storage, skin thickness, delayed yield, partial penetration and distance to the observation well. The results of the developed solution were compared with those from an existing solution for the case of an infinitesimally thin skin. The latter solution can never approximate that for the developed finite skin. Dimensionless drawdown–time curves were compared with the other published results for a confined aquifer. Positive skin effects are reflected in the early time and disappear in the intermediate and late time aquifer responses. But in the case of negative skin this is reversed and the negative skin also tends to disguise the wellbore storage effect. A thick negative skin lowers the overall drawdown in the aquifer and leads to more persistent delayed drainage. Partial penetration increases the drawdown in the case of a positive skin; however its effect is masked by the negative skin. The influence of a negative skin is pronounced over a broad range of radial distances. At distant observation points the influence of a positive skin is too small to be reflected in early and intermediate time pumping test data and consequently the type curve takes its asymptotic form.     

A semi‐analytical solution for groundwater responses to stream‐stage variations and tidal fluctuations in a coastal aquifer

A two‐dimensional semi‐analytical solution to analyse stream–aquifer interactions in a coastal aquifer where groundwater level responds to tidal effects is presented. The conceptual model considered is a two‐dimensional subsurface system with stream and coastline boundaries at right angles. The dimensional and non‐dimensional boundary value problems were solved for water level in the aquifer by successive application of Laplace and Fourier transform techniques, and the results were obtained by numerical inversion of the transformed solution. The solution was then verified by reducing the solutions to one‐dimensional known problems and comparing the results with those from previous studies. Hypothetical examples were used to examine the characteristics of water‐level variations due to the variations in stream stage and the fluctuations in tide level. Sensitivity analysis indicated that streambed leakance has no influence over the amplitude of groundwater fluctuations, but that the effect of stream stage increases with increasing leakance. Little difference was observed in the water level for different aquifer penetration ratios with narrow stream width. Increases in streambed leakance caused increases in the effect of aquifer penetration by the stream on the water level. An increased specific yield value resulted in decreased amplitude of water fluctuations and mean water level, and showed that water‐level variations due to stream and tidal boundaries are sensitive to specific yield. Copyright © 2006 John Wiley & Sons, Ltd.     

Wellbore flow‐rate solution for a constant‐head test in two‐zone finite confined aquifers

The solution describing the wellbore flow rate in a constant‐head test integrated with an optimization approach is commonly used to analyze observed wellbore flow‐rate data for estimating the hydrogeological parameters of low‐permeability aquifers. To our knowledge, the wellbore flow‐rate solution for the constant‐head test in a two‐zone finite‐extent confined aquifer has never been reported so far in the literature. This article is first to develop a mathematical model for describing the head distribution in the two‐zone aquifer. The Laplace domain solutions for the head distributions and wellbore flow rate in a two‐zone finite confined aquifer are derived using the Laplace transform, and their corresponding time domain solutions are then obtained using the Bromwich integral method and residue theorem. These new solutions are expressed in terms of an infinite series with Bessel functions and not straightforward to calculate numerically. A large‐time solution for the wellbore flow rate is therefore developed by employing the relationship of small Laplace variable versus large time variable and L'Hospital's rule. The result shows that the large‐time solution is identical to the steady‐state solution obtained after applying the Tauberian theorem into the Laplace domain solution. This large‐time solution can reduce to the Thiem equation in the case of no skin. Finally, the newly developed solution is used to investigate the effects of outer boundary distance and conductivity ratio on the wellbore flow rate. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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.     

An analytical solution for the head distribution in a tidal leaky confined aquifer extending an infinite distance under the sea

A mathematical model is developed to investigate the effects of tidal fluctuations and leakage on the groundwater head of leaky confined aquifer extending an infinite distance under the sea. The leakages of the offshore and inland aquitards are two dominant factors controlling the groundwater fluctuation. The tidal influence distance from the coast decreases significantly with the dimensionless leakage of the inland aquitard (u_i). The fluctuation of groundwater level in the inland part of the leaky confined aquifer increases significantly with the dimensionless leakage of the offshore aquitard (u_o). The influence of the tidal propagation parameter of an unconfined aquifer on the head fluctuation of the leaky confined aquifer is comparatively conspicuous when u_i is large and u_o is small. In other words, ignoring water table fluctuation of the unconfined aquifer will give large errors in predicting the fluctuation, time lag, and tidal influence distance of the leaky confined aquifer for large u_i and small u_o. On the contrary, the influence of the tidal propagation parameter of a leaky confined aquifer on the head fluctuation of the leaky confined aquifer is large for large u_o and small u_i.     

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.     

A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches

The Boussinesq equation appears as the zeroth-order term in the shallow water flow expansion of the non-linear equation describing the flow of fluid in an unconfined aquifer. One-dimensional models based on the Boussinesq equation have been used to analyse tide-induced water table fluctuations in coastal aquifers. Previous analytical solutions for a sloping beach are based on the perturbation parameter, _N=αcotβ (in which β is the beach slope, α is the amplitude parameter and is the shallow water parameter) and are limited to tan⁻¹(α)βπ/2. In this paper, a new higher-order solution to the non-linear boundary value problem is derived. The results demonstrate the significant influence of the higher-order components and beach slope on the water table fluctuations. The relative difference between the linear solution and the present solution increases as and α increase, and reaches 7% of the linear solution.     

Two-dimensional power series solution for non-axisymmetrical transport in a radially convergent tracer test with scale-dependent dispersion

It has been known for many years that dispersivity increases with solute travel distance in a subsurface environment. The increase of dispersivity with solute travel distance results from the significant variation of hydraulic properties of heterogeneous media and was identified in the literature as scale-dependent dispersion. This study presents an analytical solution for describing two-dimensional non-axisymmetrical solute transport in a radially convergent flow tracer test with scale-dependent dispersion. The power series technique coupling with the Laplace and finite Fourier cosine transform has been applied to yield the analytical solution to the two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates with variable-dependent coefficients. Comparison between the breakthrough curves of the power series solution and the numerical solutions shows excellent agreement at different observation points and for various ranges of scale-related transport parameters of interest. The developed power series solution facilitates fast prediction of the breakthrough curves at any observation point.     

Evaluation of longitudinal and transverse dispersivities/distance ratios for tracer test in a radially convergent flow field with scale-dependent dispersion

This study presents a novel mathematical model for analysis of non-axisymmetrical solute transport in a radially convergent flow field with scale-dependent dispersion. A two-dimensional, scale-dependent advection–dispersion equation in cylindrical coordinates is derived based on assuming that the longitudinal and transverse dispersivities increase linearly with the distance of the solute transported from its injected source. The Laplace transform finite difference technique is applied to solve the two-dimensional, scale-dependent advection–dispersion equation with variable-dependent coefficients. Concentration contours for different times, breakthrough curves of average concentration over concentric circles with a fixed radial distance, and breakthrough curves of concentration at a fixed observation point obtained using the scale-dependent dispersivity model are compared with those from the constant dispersivity model. The salient features of scale-dependent dispersion are illustrated during the non-axisymmetrical transport from the injection well into extraction well in a convergent flow field. Numerical tests show that the scale-dependent dispersivity model predicts smaller spreading than the constant-dispersivity model near the source. The results also show that the constant dispersivity model can produce breakthrough curves of averaged concentration over concentric circles with the same shape as those from the proposed scale-dependent dispersivity model at observation point near the extraction well. Far from the extracting well, the two models predict concentration contours with significantly different shapes. The breakthrough curves at observation point near the injection well from constant dispersivity model always produce lesser overall transverse dispersion than those from scale-dependent dispersivity model. Erroneous dimensionless transverse/longitudinal dispersivity ratio may result from parametric techniques which assume a constant dispersivity if the dispersion process is characterized by a distance-dependent dispersivity relationship. A curve-fitting method with an example is proposed to evaluate longitudinal and transverse scale-proportional factors of a field with scale-dependent dispersion.     

含有旅行时间的水平反射界面上P-SV转换波转换点坐标的唯一解析解

转换波转换点位置的求取一直是一个引人关注的问题.Schneider（2002）从一个关于转换点的三次方程中推导出了P-SV转换波转换点位置的一种解析解,这种解是偏移距、横纵波速度和P-SV转换波旅行时的函数.我们严格地论证了Schneider（2002）给出的解析解的唯一性.这个解析解可以被看作为水平反射层P-SV转换波叠前偏移的精确算子.