Two-region non-Darcian flow toward a well in a confined aquifer

We have derived an analytical solution for two-region flow toward a well in a confined aquifer based on a linearization method. The two-region flow includes Izbash non-Darcian flow near the well and Darcian flow in the rest of the aquifer. The wellbore storage is also considered. The type curves in the non-Darcian and Darcian flow domains are obtained by a numerical Laplace inversion method incorporated in MATLAB programs. We have compared our results with the one-region Darcian flow model (Theis). Our solutions agree with those of Sen [Sen Z. Type curves for two-region well flow. J Hydr Eng 1988;114(12):1461–84] which were obtained using the Boltzmann transform at late times for fully turbulent flow, while some difference has been found at early and moderate times. We have defined a dimensionless non-Darcian hydraulic conductivity term which is shown to be a key parameter for analyzing the two-region flow. A smaller dimensionless non-Darcian hydraulic conductivity results in a larger drawdown in the non-Darcian flow region at late times. However, the dimensionless non-Darcian hydraulic conductivity does not affect the slope of the dimensionless drawdown versus the logarithmic dimensionless time in the non-Darcian flow region at late times. The dimensionless non-Darcian hydraulic conductivity does not affect the late time drawdown in the Darcian flow region.     

Solutions for Non‐Darcian Flow to an Extended Well in Fractured Rock

We have investigated non‐Darcian flow to a vertical fracture represented as an extended well using a linearization procedure and a finite difference method in this study. Approximate analytical solutions have been obtained with and without the consideration of fracture storage based on the linearization procedure. A numerical solution for such a non‐Darcian flow case has also been obtained with a finite difference method. We have compared the numerical solution with the approximate analytical solutions obtained by the linearization method and the Boltzmann transform. The results indicate that the linearized solution agrees generally well with the numerical solution at late times, and underestimates the dimensionless drawdown at early times, no matter if the fracture storage is considered or not. When the fracture storage is excluded, the Boltzmann transform solution overestimates the dimensionless drawdown during the entire pumping period. The dimensionless drawdowns in the fracture with fracture storage for different values of dimensionless non‐Darcian hydraulic conductivity β approach the same asymptotic value at early times. A larger β value results in a smaller dimensionless drawdown in both the fracture and the aquifer when the fracture storage is included. The dimensionless drawdown is approximately proportional to the square root of the dimensionless time at late times.     

Non-Darcian flow to a well in an aquifer–aquitard system

In this study, we use a linearization procedure and a finite difference method to solve non-Darcian flow to a well in an aquifer–aquitard system. The leakage effect is considered. Flow in the aquifer is assumed to be non-Darcian and horizontal, whereas flow in the aquitard is assumed to be Darcian and vertical. The Izbash equation [Izbash SV. O filtracii V Kropnozernstom Materiale. USSR: Leningrad; 1931 [in Russian]] is employed to describe the non-Darcian flow. The wellbore storage is also considered in this study. An approximate semi-analytical solution has been obtained by the linearization procedure, and a numerical solution has been obtained by using a finite difference method. The previous solutions for Darcian flow case and non-Darcian flow case without leakage can be described as special cases of the new solutions. The error caused by the linearization procedure has also been analyzed. The relative error caused by the linearization procedure is nearly 100% at early times, and decreases to zero at late times. We have also compared the results in this study with Wen et al. [Wen Z, Huang G, Zhan H. A numerical solution for non-Darcian flow to a well in a confined aquifer using the power law function. J Hydrol, 2008d [in revision]] in which the leakage effect is not considered, and Hantush and Jacob [Hantush MS, Jacob CE. Non-steady radial flow in an infinite leaky aquifer. Trans Am Geophys Union 1955;36(1):95–100] who investigated a similar problem in Darcian flow case. The comparison of this study and Wen et al. (2008d) indicates the dimensionless drawdown in the aquifer with leakage is less than that without leakage, and the leakage has little effect at early times. The comparison between the results of this study and that of Hantush and Jacob (1955) indicates that the dimensionless drawdown in the aquifer for non-Darcian flow is larger at early times and smaller at late times, than their counterparts for Darcian flow. A larger dimensionless non-Darcian conductivity k_D results in a smaller dimensionless drawdown in the aquifer at late times, and leads to a larger dimensionless drawdown in the aquifer at early times. A smaller dimensionless leakage parameter B_D results in a smaller drawdown at late times, and the leakage does not affect the early-time drawdown. The analysis of the dimensionless drawdown inside the well has also been included in this study when the wellbore storage is considered.     

Non-Darcian flow in a single confined vertical fracture toward a well

Non-Darcian flow towards a well which fully penetrates a confined single vertical fracture is presented in this paper on the basis of the Izbash equation. We have obtained semi-analytical solutions for non-Darcian flow by using the Boltzmann transform and developed the non-Darcian flow well functions for cases with and without the effect of wellbore storage. The results show that the non-Darcian flow type curves are more or less deviated from the Darcian flow type curve. The non-linear effect is mainly attributable to the turbulent factor, v, a dimensionless parameter related to the pumping rate, the fracture aperture, the fracture thickness, and two constants k′ and n used in the Izbash power-law. The non-linear effect appears to be less sensitive to the power-law index, n. When excluding wellbore storage, the well function at early times is proportional to v^−1/(n−1)u^−n/(n−1), where u is a dimensionless term inversely proportional to time; whereas the well function at late times is approximated as , where A₀(n) is a finite term depending on n. When considering wellbore storage, drawdowns inside the well with different v values approach the same asymptotic value at small times, and the effect of wellbore storage is only found at the early stage of pumping.     

Non-Darcian groundwater flow in leaky aquifers

Abstract

A simplified method has been developed for solving leaky aquifer non-Darcian flow hydraulics. The principle of volumetric approach is combined with the confined-aquifer, time-dependent drawdown equation in an observation well. The groundwater flow in the leaky aquifer is assumed to obey a non-Darcian flow law of exponential type. The results are obtained in the form of type-curve expressions from which the necessary bundles of curves are drawn for a set of selective non-Darcian flow aquifer parameters. Although application of the methodology appears as rather limited but it provides a scientific contribution and extension of leaky aquifer theory towards nonlinear flow conditions. The methodology developed herein is applied to some actual field data from the eastern sedimentary basin in the Kingdom of Saudi Arabia.     

Observed Drawdown Pattern Around a Well Partially Penetrating a Vertically Extensive Water-Table Aquifer

In order to understand the flow pattern around a pumping well partially penetrating a vertically extensive aquifer, a specially designed pumping test was carried out in Pakistan. In this paper salient features of the test have been described. The spatial distributions of drawdown have been shown graphically. Some of the preliminary conclusions made from the drawdown pattern include:

• • The distance beyond which the flow is likely to be horizontal increases with decrease in the degree of aquifer penetration.
• • In equidistant observation wells open at different depths, (1) the drawdowns tend to merge at larger times, provided the observation point is located within the screened section of the aquifer; (2) the less the depth of penetration is, the earlier the drawdowns start merging; and (3) the initial rate of drawdown near the aquifer top is slow but catches up with time to exceed those at deeper points.

     

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.     

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.     

A Novel Semi-Analytical Solution of Over-Damped Slug Test in a Three-Layered Aquifer System

The slug test has been commonly used to estimate aquifer parameters. Previous studies on the slug test mainly focused on a single-layer aquifer. However, understanding the interaction between layers is particularly important when assessing aquifer parameters under certain circumstances. In this study, a new semi-analytical model on transient flow in a three-layered aquifer system with a partially penetrating well was developed for the slug test. The proposed model was solved using the Laplace transform method and the Goldstein-Weber transform method, where the semi-analytical solution for the model was obtained. The drawdowns of the proposed model were analyzed to understand the impacts of the different parameters on the drawdowns in a three-layered aquifer system. The results indicated that groundwater interactions between the layers have a significant impact on the slug test. In addition, a shorter and deeper well screen as well as a greater permeability ratio between the layers creates a greater interface flow between them, leading to a higher drawdown in the slug test. Finally, a slug test in a three-layered aquifer system was conducted in our laboratory to validate the new model, which indicated that the proposed model performed better in the interpretation of the experimental data than a previous model proposed by Hyder et al. (1994). We also proposed an empirical relationship to qualitatively identify the errors in the application of single-layer model for the analysis of response data in a three-layered aquifer system.     

Analysis of Transient Flow to an Extended Fully Penetrating Well at Constant-Head Pumping

Vertical wells with radial extension at the well bottom can improve the rate of water production. No study has yet investigated the effects of the transient state and anisotropy in directional hydraulic conductivities on the wellbore flux rate for this type of well. This study derives a semianalytical transient drawdown solution for constant-head pumping at a fully penetrating well radially extended at the bottom of a confined, anisotropic aquifer by applying Laplace transform and separation of variables as well as conducting a Fourier analysis. The results of this new solution indicate that transient and steady-state wellbore flux rates can be increased by a factor of two for greater radial extension of the well. Compared with an isotropic aquifer (a ratio of vertical and horizontal hydraulic conductivities equal to one), an anisotropic aquifer with the ratio less than one may produce a higher transient wellbore flux rate and lower steady-state wellbore flux rate. Moreover, the time required to achieve the steady-state wellbore flux rate can be substantially affected by anisotropy of the aquifer.     

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.     

Underdamped slug tests with unsaturated‐saturated flows by considering effects of wellbore skins

An analytical solution is presented for the slug tests conducted in a partially penetrating well in an unconfined aquifer affected from above by an unsaturated zone. The solution considers the effects of wellbore skin and oscillatory responses on underdamped slug tests. The flow in the saturated zone is described by a two‐dimensional, axially symmetric governing equation, and the flow in the unsaturated zone above the water table by a linearized one‐dimensional Richards' equation. The unsaturated medium properties are represented by the exponential constitutive relationships. A Laplace domain solution is derived using the Laplace and finite Fourier transform and the solution in the real‐time domain is evaluated using the numerical inverse Laplace transform method. The solution derived in this study is more general and reduces to the most commonly used solutions for slug tests in their specified conditions. It is found that the unsaturated flow has a significant impact on the slug test conducted in an unconfined aquifer. The impact of unsaturated flow on such a slug test is enhanced with a larger anisotropy ratio, a shorter well screen length, a shorter distance between the well screen and the water table, or a larger well screen radius. The impact of unsaturated flow on slug tests decreases as the degree of penetration (the length of well screen) increases. For a fixed well screen length, the impact of unsaturated flow on slug tests decreases as the distance between the centre of screen and the water table increases. A large dimensionless well screen radius (>0.01) leads to significant effects of unsaturated flow on slug tests. The unsaturated flow reduces the oscillatory responses to underdamped slug tests. The unsaturated zone has significant impact on slug test under high‐permeability wellbore skin.     

A theoretical assessment of casing storage effects when pump sampling a partially penetrating borehole

ABSTRACT

A borehole partially penetrating a confined aquifer and pumped at a constant rate is modelled, taking account of water stored within the casing of the borehole. A solution for drawdown in the Laplace transform domain is obtained. The proportion of aquifer water in well discharge is numerically evaluated, tabulated as a function of time and compared with results for a fully penetrating well. Modification of the fully penetrating well theory, for application to partially penetrating wells, was found to give comparable results to the more complete analysis for a partially penetrating well both at early and late times. A previous estimate of the time of pumping before sampling (t_s) to minimize casing storage effects, based on the fully penetrating well theory, was confirmed by the partially penetrating well analysis and in fact was shown to be a conservative estimate (or overestimate) of the pumping time required when sampling from a partially penetrating well.     

Transient analysis for fluid injection into a dome reservoir

A dome-shaped layer can be selected as a storage site for fluid injection. In this study, we develop a mathematical model for simulating transient head distribution in a heterogeneous and anisotropic dome-shaped layer due to a constant-head injection in a fully penetrating well. In the model, a form of step change is adopted to approximate the upper and lower boundaries of the dome and then the layer is split into two regions. The Laplace-domain solution of the model is developed using the Laplace transform and method of separation of variables. The transient injection rate at wellbore can then be obtained based on Darcy’s law and Bromwich integral method. The predicted head contours from the head solution show significant vertical flow components near the location of step change in the dome reservoir. The results of sensitivity analysis indicate that the hydraulic conductivity is the most sensitive parameter and the specific storage is the least sensitive one to the injection rate after a short period of injection time. In addition, the injection rate for a dome reservoir is also very sensitive to the change of the height for the reservoir near the injection well (first region) at a very early injection time. In contrast, the injection rate is more sensitive to the change of the height of the second region than that of the first region at late time. This analytical solution may be used as a primary tool to assess the capacity of fluid injection to various dome reservoirs.     

The impact of well drawdowns on the mixing process of river water and groundwater and water quality in a riverside well field,Northeast China

Riverbank filtration (RBF) has been widely used throughout the world as an effective means to regulate surface water and groundwater resources and pretreat raw water for municipal water supply. The quality of the water from a riverside well field and the mixing ratios of surface water and groundwater is primarily impacted by the hydrodynamic processes in the RBF system. The RBF system is largely controlled by the water exploiting system in addition to the natural hydrologic condition of the river–aquifer system. As one of the most important design parameters of the riverside well field, the drawdown of groundwater level greatly determines the water head differences between the river water and groundwater as well as the field flow net, which subsequently impacts the mixing of river water and groundwater and water quality significantly. This study aimed to improve the understanding of the mixing process between the surface water and groundwater and estimate the impact of the RBF on the mixing ratio of surface water–groundwater and water quality quantitatively. A set of field pumping tests with various groundwater level drawdowns were carried out independently and successively at a riverside field with a single pumping well near the Songhua River in Northeast China in August 2017. During these tests, the water levels and hydrochemical parameters of the Songhua River, the adjacent aquifer, and the pumping well were monitored. The dynamic mixing process of the river water and groundwater and water quality under various drawdown conditions were analysed systematically using analytical methods. The results obtained from Dupuit method and the Mirror Image method in conjunction with the Hydrochemical Tracing method showed that the pumping water directly from the river water reached 60% ± 10% after a steady flow net was established. The larger the proportion of the pumping water captured from the river, the better quality of the pumping water was, because the quality of the river water (only limited to some water quality parameters monitored which were minority) was better than that of the groundwater. The results also showed that total Fe, TDS, total hardness, COD_Mn, and K⁺ were relatively sensitive to the changes of groundwater drawdown, and their concentrations decreased with an increase in the groundwater drawdown. It can be concluded that both the mixing ratio of the surface water and the groundwater and the water quality of the riverside well field can be regulated through adjusting the designed drawdown of the groundwater level, which is helpful for the design and the optimization of the riverside well water intake engineering.     

Short- and long-time behavior of aquifer drainage after slow and sudden recharge according to the linearized Laplace equation

During the last 25 years there has been a great interest in deriving aquifer characteristics from outflow data. This analysis has been mainly based of the drainage of a horizontal aquifer after sudden drawdown, using the Boussinesq approximation. Following the general approach of Brutsaert and Lopez [Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40], it was determined that for this geometry the aquifer behavior could be characterized by dQ/dt ∝ Q³ for small t and by dQ/dt ∝ Q^3/2 for large t. It was remarked that dQ/dt ∝ Q for large t is often observed. In practice, it is also difficult to determine if dQ/dt ∝ Q³ for small t because this behavior can only be observed over a very short period.Here, we present a similar analysis of aquifer behavior based on the more fundamental Laplace solution for penetrated aquifers. It has been shown that also when the drain does not fully penetrate the aquifer, the solution still produces good results [Szilagyi, J. Sensitivity analysis of aquifer parameter estimations based on the Laplace equation with linearized boundary conditions. Water Resour Res 2003;39(6)]. The Laplace solution quickly shows that dQ/dt ∝ Q for t → ∞ and dQ/dt ∝ Q^∞ for t → 0, after sudden drawdown. This analysis reconfirms previous findings concerning long-time behavior. More importantly, the analysis shows that the exponent B in dQ/dt ∝ Q^B does not have a fixed limited value for short times for the given geometry. Further analysis, however, shows that under certain conditions the relation dQ/dt ∝ Q³ is retained for 0 ≪ t < 1. Detailed examination of the Laplace solution also shows under which types of recharge dynamics a well-identifiable transition takes place between short- and long-term behavior. As long as such a clear transition exists, the aquifer characterization method proposed earlier by Brutsaert and Lopez [Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40] can be applied. It is shown that for a sharp pulse input, the Laplace solution gives similar results as presented by Brutsaert and Lopez [Brutsaert W, Lopez, JP. Basin-scale geohydrologic drought flow features of riparian aquifers in the southern Great Plains. Water Resour Res 1998;34(2):233–40]. For a smooth pulse, the transition becomes unclear. What is “smooth” and “sharp” depends on input and aquifer characteristics, whereby shallow aquifers give clearer transitions than deep aquifers for the same input. The analysis shows that when rain ceases suddenly after the aquifer has come into equilibrium with a steady rain input, a usable transition in the relation between dQ/dt and Q can be found as well. Researchers can use the present analysis to assess whether specific aquifers and recharge events can be used for the previously suggested characterization method.     

A new analytical solution solved by triple series equations method for constant-head tests in confined aquifers

The constant-head pumping tests are usually employed to determine the aquifer parameters and they can be performed in fully or partially penetrating wells. Generally, the Dirichlet condition is prescribed along the well screen and the Neumann type no-flow condition is specified over the unscreened part of the test well. The mathematical model describing the aquifer response to a constant-head test performed in a fully penetrating well can be easily solved by the conventional integral transform technique under the uniform Dirichlet-type condition along the rim of wellbore. However, the boundary condition for a test well with partial penetration should be considered as a mixed-type condition. This mixed boundary value problem in a confined aquifer system of infinite radial extent and finite vertical extent is solved by the Laplace and finite Fourier transforms in conjunction with the triple series equations method. This approach provides analytical results for the drawdown in a partially penetrating well for arbitrary location of the well screen in a finite thickness aquifer. The semi-analytical solutions are particularly useful for the practical applications from the computational point of view.     

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.     

Evaluation of Pumping Induced Flow in Observation Wells During Aquifer Testing

The vertical variation of drawdown around pumping wells generates an induced flow in the observation wells. A set of governing equations is presented to couple the drawdown variation and the vertical flux distribution in observation wells. A numerical example is performed to justify the governing equations and to verify the solution methods used by the simulation software WT. The example analyzes the effect of skin loss, wellbore storage, and vertical segmentation on the drawdown and induced flow in observation well during pumping. The evaluation of the Fairborn pumping test involves a vertically homogeneous and anisotropic water table aquifer, uniform well‐face drawdown conditions in the pumping well and simulation of the drawdown evolution in the observation well with and without the effect of induced flow. The computer calibrations resulted in small differences between the measured and simulated drawdown curves.