Dimensionless Drawdown Research Articles

Regression-Based Interpretation Method for Confined Aquifer Pumping Tests

The constant-discharge pumping test is a widely used aquifer characterization tool for estimating hydrogeological parameters. Various graphical and numerical solution techniques have been devised for interpretation of pumping test results. This study proposes an alternative regression-based solution approach for analyzing pumping tests conducted in nonleaky, confined aquifers. A regression model was developed by making use of the similarity solution that describes the temporal drawdown behavior at a monitoring well within a pumping period. The field curve obtained from a pumping test (drawdown versus time data) and the theoretical type curve (dimensionless drawdown versus dimensionless time) exhibit identical behavior in logarithmic scale. Correspondingly, based on the equivalence of the area above these two curves, the proposed regression equation establishes a link between a selected drawdown value and its dimensionless counterpart. Once this drawdown value is associated with the corresponding dimensionless drawdown, the explored aquifer transmissivity and storativity can be retrieved easily with the similarity solution. The proposed fit equation enables users to retrieve hydrogeological parameters effortlessly by eradicating the difficulties encountered while implementing conventional graphical or optimization-based methods. The method was tested for a large number of hypothetical noise-free and noise-perturbed time drawdown data sets, each synthesized with different aquifer parameters within a wide range, and in all these attempts the proposed algorithm achieved providing highly accurate parameter estimates. In addition, two published test cases were employed to assess the estimation performance of the method in comparison with the existing techniques. The developed solution scheme was revealed to yield results in very good agreement with those reported in previous works. Considering the accuracy of the method and the ease of its implementation, the proposed approach can be a practical tool for analyzing constant-discharge tests performed in confined aquifers.

Semi-analytical solutions for transient flow to a partially penetrated well with variable discharge in a general three-layer aquifer system

A general semi-analytical model for transient flow in a three-layered aquifer system with a partial penetration well having a variable discharge of pumping is developed with the consideration of the interface flow on the adjacent layers. This general three-layer system includes the conventional aquitard-aquifer-aquitard system as a subset and does not require that the permeability contrasts of different layers must be greater than a few orders of magnitude and does not ignore any flow components (either vertical or horizontal) in any layer. The pumping well of infinitesimal radius is screened at any portion of the middle layer. Three widely used types of top and bottom boundary conditions are assigned prescribed head (Case 1), zero flux (Case 2), or prescribed head at top and zero flux at bottom (Case 3). Laplace domain solutions for dimensionless drawdown are obtained using Hankel transformation, and associated time-domain solutions are evaluated numerically. The newly obtained solutions include some available solutions for two- or single-layer aquifer systems as subsets. The drawdowns for individual layers caused by a well with an exponentially decreased discharge are explored as an example of illustration. The results indicate that the pumped layer drawdown close to the partially penetrated well is mainly influenced by the variable pumping rate. The late-time drawdown for all layers is remarkably affected by the chosen types of top and bottom boundary conditions, and the drawdown for Case 3 is greater than that for Case 1 and smaller than that for Case 2. Additionally, the effect of the pumped layer anisotropy on drawdowns in the three-layer system is significant, and the anisotropy of the unpumped layers significantly affects the drawdown in the whole aquifer system without large contrast of hydraulic conductivity between the unpumped layers and the pumped layer. The drawdowns in all three layers are greatly affected by the location and length of well screen, and a larger drawdown can be seen at the position that is closer to the middle point of the screen of the partially penetrating pumping well.

Groundwater flow to a well in a strip-shaped unconfined-fractured aquifer system with a transition zone

Semi-infinite unconfined-fractured strip shaped aquifer systems are common in alluvial plain margins, but have received little attention in the hydrogeological community. Thus, the aim of this study is to present semi-analytical solutions of flow to a well in these aquifer systems. Two conceptual models are considered: 1-An unconfined aquifer with a lateral fractured aquifer and a pumping well installed in the unconfined aquifer (model I); 2-An unconfined aquifer with a lateral fractured aquifer and a pumping well installed in the fractured aquifer (model II). A transition zone is considered between two aquifers. Three-dimensional groundwater flows are considered in unconfined, fractured and transition zone aquifers. Homogeneous, anisotropic hydraulic conductivity and instantaneous drainage water table condition are assumed first but can be relaxed to accommodate delayed drainage water table condition if needed. The point sink/source solutions are obtained via finite and infinite Fourier transforms for space and Laplace transform for time. The line sink/source solutions are obtained via integration along the desired direction. The uniform flux and uniform head boundary conditions are considered for the pumping well. The vertical distribution of the flux toward the well screen is explored. The effects of inner well condition on the variation of the dimensionless drawdown and boundary depletion volume are investigated. We investigate the influences of the hydraulic parameters of the transition zone on the spatial and temporal variations of the sensitivity of the drawdown to hydraulic parameters of the aquifer system. Furthermore, the influences of the transition zone on the spatial distribution of the drawdown are explored. The results of this study can be utilized to evaluate head distribution in the aquifer system; to calculate the water budget of the alluvial aquifers near a fractured one; to analyze the influences of a transition zone on the head and flow distribution in the aquifer system.

Three-dimensional response to a partially penetration well pumping in a general anisotropic three-layer aquifer system

Three-dimensional response to well pumping in a multilayer aquifer system with rigorous treatment of interfacial flow attracts great attention among the subsurface hydrologists because of its theoretical importance and practical merits. In this study, a general theory of three-dimensional groundwater flow to a well of infinitesimal radius in an anisotropic three-layer aquifer system is developed with a constant-rate pumping in the lower layer, and associated semi-analytical solutions of drawdowns in individual layers are obtained. This theory discards many previous assumptions for studying a multilayer aquifer system such as simplified flow configuration in the aquitard, treatment of cross-formation flow as a volumetric averaged term adding to the governing equations of aquifer flows, and sometimes negligence of vertical flow components in the aquifers. Additionally, this theory considers three types of commonly used top boundary conditions: A water table with delayed gravity drainage boundary condition (Case 1), a constant-head top boundary condition (Case 2), and a no-flow top boundary condition (Case 3). The dimensionless drawdown solutions in Laplace domain are obtained and inverted numerically to calculate the time-domain solutions. The solution encompasses existing solutions for a two-layer or a single-layer (e.g. unconfined or confined) aquifer system as subsets. The developed solutions are tested extensively with a finite-element numerical solution using COMSOL and other existing solutions, and a sensitivity analysis is performed to prioritize the influences of different parameter groups. The results of this study can be used to predict the pumping induced drawdown at any position with an observation well, and to estimate the hydraulic parameters of the lower pumped layer and upper layer, as well as the middle layer. The developed theory provides a basis for other potential applications such as geotechnical engineering and groundwater management.

A New Method for the Evaluation of Well Rehabilitation from the Early Portion of a Pumping Test

This study analyzes the unsteady groundwater flow to a real well (with wellbore storage and the skin effect) that fully penetrates the confined aquifer. The well is located within an infinite system, so the effect of boundaries is not considered. The Laplace-domain solution for a partial differential equation is used to describe the unsteady radial flow to a well. The real space solution is obtained by means of the numerical inversion of the Laplace transform using the Stehfest algorithm 368. When wellbore storage and the skin effect dominate pumping test data and testing is conducted for long enough, two semilogarithmic straight lines are normally obtained. The first straight line can be identified readily as the line of the maximum slope. The correlation of the dimensionless drawdown for the intersection time of this first straight line, with the log time axis as a function of the dimensionless wellbore storage and the skin factor, is shown. This paper presents a new method for evaluating the skin factor from the early portion of a pumping test. This method can be used to evaluate the skin factor when the well-known Cooper–Jacob semilogarithmic method cannot be used due to the second straight line not being achieved in the semilogarithmic graph drawdown vs. the log time. A field example is presented to evaluate the well rehabilitation in Veselí nad Lužnicí by means of the new correlation.

Flow to a partially penetrating well with variable discharge in an anisotropic two-layer aquifer system

A general analytical model for the problem of variable discharge of groundwater from a well in an anisotropic two-layered aquifer system is developed by taking account of the interface flow between the two layers. The well of infinitesimal radius partially penetrates the lower layer and is pumped at a variable discharge. The horizontal as well as vertical flows in both layers are included, and importantly, the effects of constant-head (Case 1) and no-flux (Case 2) boundaries at the top of the upper layer are considered. Laplace domain solutions for drawdown in dimensionless form are derived with the help of Hankel transform, and are inverted to the time domain numerically. The drawdown characteristics induced by an exponentially decayed rate of pumping are discussed, and sensitivity analysis is also performed to explore the influence of different parameters on drawdown characteristics. The results show that the drawdown around a partially penetrating well in the lower pumped layer induced by an exponential decreasing pumping rate usually contains three stages. The effect of anisotropy and well configuration have great influences on the drawdown distribution near the pumping well, and the horizontal flow and storativity of the upper (unpumped) layer cannot be ignored in determining the upper aquifer drawdown. The sensitivity analysis also illustrates that the dimensionless drawdown in the lower pumped layer is very sensitive to well configuration parameters all the time. It is only sensitive to the hydraulic parameters of the lower aquifer for Case 1 and to the hydraulic parameters of the lower layer and the specific storage of the upper layer at late times. The dimensionless drawdown of the upper layer is not sensitive to the variable well discharge parameters over the whole pumping period, and is not sensitive to the storage parameters of both layers for Case 1. However, the drawdown in the upper layer is sensitive to all the hydraulic parameters and well configuration parameters for Case 2 at late times.

Groundwater flow to a general well configuration in an unconfined aquifer overlying a fractured bedrock

Unconfined aquifers sometimes overlie fractured bedrocks, consisting of unconfined-fractured aquifer systems. Pumping induced hydrodynamics in such aquifer systems has not received much attention as in other types of aquifer systems. This paper aims to obtain semi-analytical solutions of flow to a general well configuration in such an unconfined-fractured two-layer aquifer system. The examples of general well configuration, specifically discussed here, include: 1) a horizontal well, 2) a vertical well, 3) a vertical well with one horizontal collector, and 4) a vertical well with two horizontal collectors which are perpendicular with each other. The methodology is general and can be extended to other types of wells with minor modifications of solutions developed in this study. The point sink/source solution is first obtained using double infinite Fourier and Laplace transformations (where sink is for pumping and source is for injection). The line sink/source solution is then obtained via integrating the point sink/source along the desired direction and following the principle of superposition. The de Hoog numerical inverse Laplace transform and Gaussian Quadrature are used to obtain time-domain dimensionless drawdown solutions. The instantaneous drainage of water table and the inter-porosity flow of the underlying fractured aquifer are taken into account. The effects of the length of horizontal well on dimensionless drawdown and dimensionless drawdown derivative, and scaled sensitivity of the hydraulic parameters of the underlying fractured aquifer are explored. The dimensionless drawdowns and drawdown derivatives of a vertical pumping well, a vertical well with a single horizontal collector and a vertical well with two perpendicular horizontal collectors are also compared. The scaled sensitivity of dimensionless drawdown to the hydraulic parameters of the underlying fractured aquifer of these different pumping well structures is explored. The results of this study can be used to evaluate the hydraulic parameters using the drawdown data collected during a pumping test performed in a general well configuration in a unconfined-fractured two-layer aquifer system. Furthermore, the presented solutions can be utilized to design the pumping well to minimize drawdown in low-permeability or thin aquifers. By eliminating the inter-porosity flow term from the underlying fractured aquifer, the solutions are applicable to groundwater flow to a pumping well in leaky aquifers.

A New Approach for Analyzing Drawdown Data from a Partially Penetrating Well

A pumping test conducted in an aquifer of large thickness is typically performed with partially penetrating wells. In this study, a novel, practicable method is proposed for the interpretation of drawdown data from a partially penetrating monitoring well drilled in a confined aquifer. The method is established on the assessment of the area below the curve passing through the measured drawdown versus time data points. The evaluated area can be matched with the identical area below the theoretical type curve to acquire the corresponding dimensionless time and drawdown variables, which ultimately yield the explored hydraulic conductivity and specific storage of the aquifer. The method eliminates the knotty and time consuming steps employed in the traditional curve matching procedure and proposes a simple solution algorithm with straightforward algebraic operations. The accuracy of the methodology was verified with hypothetical and real test cases. The attained results reveal the capability of the method in estimating the hydrogeological parameters at high accuracy with relatively little effort.

Flow to a well in an unconfined-fractured and leaky wedge-shaped aquifer system

Wedge-shaped aquifers are usually formed where two linear boundaries intersect. These types of aquifers are commonly seen in river basins and alluvial fans, but have received much less attention in the hydrological sciences community. Fundamental understanding of the effects of lateral boundaries on the hydrodynamic process in such aquifers is quite poor, particularly for unconfined-fractured and leaky wedge-shaped aquifer systems, which will be the concern of this study. Solutions of flow to a well in an unconfined-fractured wedge-shaped two aquifer system are obtained using Laplace, Hankel and Fourier transforms. An infinitesimal radius pumping well partially penetrates the overlying unconfined aquifer. Both aquifer layers are homogeneous and anisotropic bounded by two intersecting linear lateral boundaries of constant-head or barrier types and the aquifer system is semi-infinite in radial direction. The time domain drawdowns are obtained via the Stehfest algorithm and Gaussian Quadrature. The inter-porosity flow in the underlying fracture aquifer, the instantaneous water table drainage, the horizontal and vertical hydraulic conductivity anisotropy and the wedge angle are taken into account. Eliminating the inter-porosity flow in the underlying fractured aquifer, the presented solution reduces to a leaky wedge-shaped aquifer case. The influences of the underlying fractured aquifer hydraulic parameters on the overlying unconfined aquifer for different wedge angles and linear lateral boundaries types are explored using dimensionless drawdown and scaled sensitivity curves. The results of this study can be utilized to obtain hydraulic parameters of an alluvial and river basin unconfined-fractured wedge-shaped aquifer systems. The presented solution can also be used for water budget assessment of river basin aquifers.

A Semianalytical Model for Pumping Tests in Finite Heterogeneous Confined Aquifers With Arbitrarily Shaped Boundary

AbstractThis study presents a Laplace‐transform‐based boundary element method to model the groundwater flow in a heterogeneous confined finite aquifer with arbitrarily shaped boundaries. The boundary condition can be Dirichlet, Neumann or Robin‐type. The derived solution is analytical since it is obtained through the Green's function method within the domain. However, the numerical approximation is required on the boundaries, which essentially renders it a semi‐analytical solution. The proposed method can provide a general framework to derive solutions for zoned heterogeneous confined aquifers with arbitrarily shaped boundary. The requirement of the boundary element method presented here is that the Green function must exist for a specific PDE equation. In this study, the linear equations for the two‐zone and three‐zone confined aquifers with arbitrarily shaped boundary is established in Laplace space, and the solution can be obtained by using any linear solver. Stehfest inversion algorithm can be used to transform it back into time domain to obtain the transient solution. The presented solution is validated in the two‐zone cases by reducing the arbitrarily shaped boundaries to circular ones and comparing it with the solution in Lin et al. (2016, https://doi.org/10.1016/j.jhydrol.2016.07.028). The effect of boundary shape and well location on dimensionless drawdown in two‐zone aquifers is investigated. Finally the drawdown distribution in three‐zone aquifers with arbitrarily shaped boundary for constant‐rate tests (CRT) and flow rate distribution for constant‐head tests (CHT) are analyzed.

Rate Dependence of Bilinear Flow in Unconventional Gas Reservoirs

Summary Unconventional fractured gas wells occasionally exhibit bilinear flow during early production. When this flow regime is observed, important fracture properties can be estimated such as conductivity and/or spacing. However, the application of present-day rate-transient-analysis (RTA) methods to bilinear flow results in up to 18% error. This error is rate-dependent, and occurs because of nonlinearities in the gas-diffusivity equation. This paper presents a new correction factor that handles the rate dependency of bilinear flow, and enables more-accurate reservoir characterization. The diffusivity equation of gas in porous media has some nonlinearities that cannot be addressed by only using the real-gas pseudopressure. Even though neglecting these nonlinearities results in a model that is sufficiently accurate for the radial-flow case, errors arise in the boundary-dominated-flow (BDF) (Fraim and Wattenbarger 1987), the linear-flow (Ibrahim and Wattenbarger 2006; Nobakht and Clarkson 2012), and, as shown in this article, the bilinear-flow cases. Currently, no research studies provide an accurate bilinear-flow analytical model for gas reservoirs. To bridge this gap and to provide rate-transient analysts with a practical tool, this work presents a correction factor that can be incorporated into the existing bilinear-flow models. The correction factor was developed following an approach similar to that of Ibrahim and Wattenbarger (2006). The first step was to simulate a number of reservoir and well conditions. Second, fracture properties were back calculated with the current bilinear-flow models. Finally, errors between the back calculated and the simulated properties were correlated to dimensionless drawdown. After these steps, a correction factor was obtained that applied to various bilinear-flow cases. The obtained numerical results indicate that errors in bilinear-flow models are correlated with rate, or more precisely, dimensionless drawdown [a similar behavior is observed in the analytical model of linear flow, as demonstrated by Ibrahim and Wattenbarger (2006)]. This correlation can be used to reduce error to less than ±3%. Because the simulated cases cover an extensive range of unconventional-reservoir conditions, the empirical correction is robust and applicable to a wide variety of reservoir conditions. Most importantly, the correction is practical and can be readily incorporated into existing bilinear-flow models. The application of the new correction is demonstrated in this article with synthetic and field examples.

Semi-analytical solutions for flow to a well in an unconfined-fractured aquifer system

Semi-analytical solutions of flow to a well in an unconfined single porosity aquifer underlain by a fractured double porosity aquifer, both of infinite radial extent, are obtained. The upper aquifer is pumped at a constant rate from a pumping well of infinitesimal radius. The solutions are obtained via Laplace and Hankel transforms and are then numerically inverted to time domain solutions using the de Hoog et al. algorithm and Gaussian quadrature. The results are presented in the form of dimensionless type curves. The solution takes into account the effects of pumping well partial penetration, water table with instantaneous drainage, leakage with storage in the lower aquifer into the upper aquifer, and storativity and hydraulic conductivity of both fractures and matrix blocks. Both spheres and slab-shaped matrix blocks are considered. The effects of the underlying fractured aquifer hydraulic parameters on the dimensionless drawdown produced by the pumping well in the overlying unconfined aquifer are examined. The presented solution can be used to estimate hydraulic parameters of the unconfined and the underlying fractured aquifer by type curve matching techniques or with automated optimization algorithms. Errors arising from ignoring the underlying fractured aquifer in the drawdown distribution in the unconfined aquifer are also investigated.

Improvement on the estimation of constant-rate drawdown in large-diameter wells

Papadopulos and Cooper's (PC) solution can be used to describe the drawdown resulting from pumping with a constant rate at a large-diameter well. However, this solution is too complicated to be accurately evaluated due to the oscillatory nature of the Bessel functions. The PC approach resulted in tabulated values of dimensionless drawdown at the well with an accuracy of four or fewer digits for selected values of dimensionless storage coefficient versus dimensionless time. Some researchers have fitted the tabulated values with interpolation formulas that are easy to use in engineering applications. Those formulas may be more accurate if the tabulated values are computed with greater accuracy. In this study, we propose an efficient numerical procedure, including a root search scheme, to find the roots of the integrand, Gaussian quadrature for numerical integration, and Shanks transform to accelerate convergence of infinite series. The proposed procedure can evaluate the dimensionless drawdown with greater accuracy and is useful in practice if there is a need for high accuracy for the observation either at the well or in the aquifer at some distance from the pumping well Editor D. Koutsoyiannis Citation Chang, Y.C., Yeh, H.D., and Wang, C.T., 2013. Improvement on the estimation of constant-rate drawdown in large-diameter wells. Hydrological Sciences Journal, 58 (3), 716–727.

Semi-analytical solution of groundwater flow in a leaky aquifer system subject to bending effect

Summary The bending of aquitard like a plate due to aquifer pumping and compression is often encountered in many practical problems of subsurface flow. This reaction will have large influence on the release of the volume of water from the aquifer, which is essential for the planning and management of groundwater resources in aquifers. However, the groundwater flow induced by pumping in a leaky aquifer system is often assumed that the total stress of aquifer maintains constant all the time and the mechanical behavior of the aquitard formation is negligible. Therefore, this paper devotes to the investigation of the effect of aquitard bending on the drawdown distribution in a leaky aquifer system, which is obviously of interest in groundwater hydrology. Based on the work of Wang et al. (2004) this study develops a mathematical model for investigating the impacts of aquitard bending and leakage rate on the drawdown of the confined aquifer due to a constant-rate pumping in the leaky aquifer system. This model contains three equations; two flow equations delineate the transient drawdown distributions in the aquitard and the confined aquifer, while the other describes the vertical displacement in response to the aquitard bending. For the case of no aquitard bending, this new solution can reduce to the Hantush Laplace-domain solution ( Hantush, 1960 ). On the other hand, this solution without the leakage effect can reduce to the time domain solution of Wang et al. (2004) . The results show that the aquifer drawdown is influenced by the bending effect at early time and by the leakage effect at late time. The results of sensitivity analysis indicate that the aquifer compaction is sensitive only at early time, causing less amount of water released from the pumped aquifer than that predicted by the traditional groundwater theory. The dimensionless drawdown is rather sensitive to aquitard’s hydraulic conductivity at late time. Additionally, both the hydraulic conductivity and thickness of the aquifer are the most sensitive parameters in influencing the predicted dimensionless drawdown.

Boundary depletion rate and drawdown in leaky wedge‐shaped aquifers

AbstractThis 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.

A drawdown solution for constant-flux pumping in a confined anticline aquifer

An anticline, known as a convex-upward fold in layers of rock, commonly is formed during lateral compression, which may be elected as a potential site for carbon sequestration. A mathematical model is developed in this study for describing the steady-state drawdown distribution in an anticline aquifer in response to the constant-flux pumping. The topographical shape of the anticline is mimicked by three successive blocks. The solution is obtained by applying the infinite Fourier transform and the finite Fourier cosine transform in each blocks and acquiring the hydraulic continuities between the blocks. Simulated results reveal that the introduction of a thin-limbs or narrow-ridged anticline would produce a much greater head drop in the ridge zone. For a well of constant pumping rate, the dimensionless drawdown around the well increases with decreasing well screen length or/and aquifer anisotropy ratio. An examination of the effect of well location on the drawdown reveals that the partially penetrating well located at the top-middle of the ridge zone produces the largest drawdown. The simulation of the flow in an anticline aquifer based on MODFLOW results in slightly smaller drawdown values in most regions when compared with those predicted by the present solution. The present solution can also be used to simulate the flow in a slab-shaped aquifer or a hillslope aquifer. It can be applied to determine the aquifer parameters if coupled with an optimization scheme and to provide the basis for selecting a potential site for carbon sequestration in the future as well.

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.

A numerical solution for non-Darcian flow to a well in a confined aquifer using the power law function

In this study, we have obtained numerical solutions for non-Darcian flow to a well with the finite difference method on the basis of the Izbash equation, which states that the hydraulic gradient is a power function of the specific discharge. The comparisons between the numerical solutions and the Boltzmann solutions and linearization solutions have also been done in this study. The results indicated that the linearization solutions for both the infinitesimal-diameter well and the finite-diameter well agree very well with the numerical solution at late times, while the linearization method underestimates the dimensionless drawdown at early and moderate times. The Boltzmann method works well as an approximate analytical solution for the infinitesimal-diameter well. Significant differences have been found between the Boltzmann solution for a finite-diameter well and the numerical solution during the entire pumping period. The analysis of the numerical solution implies that all the type curves inside the well for different dimensionless non-Darcian conductivity kD values approach the same asymptotic value at early times, while a larger kD leads to a smaller drawdown inside the well at late times. A larger kD results in a larger drawdown in the aquifer at early times and a smaller drawdown in the aquifer at late times. Flow approaches steady-state earlier when kD is larger.

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.

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.