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.     

An analytical solution for non-Darcian flow in a confined aquifer using the power law function

We have developed a new method to analyze the power law based non-Darcian flow toward a well in a confined aquifer with and without wellbore storage. This method is based on a combination of the linearization approximation of the non-Darcian flow equation and the Laplace transform. Analytical solutions of steady-state and late time drawdowns are obtained. Semi-analytical solutions of the drawdowns at any distance and time are computed by using the Stehfest numerical inverse Laplace transform. The results of this study agree perfectly with previous Theis solution for an infinitesimal well and with the Papadopulos and Cooper’s solution for a finite-diameter well under the special case of Darcian flow. The Boltzmann transform, which is commonly employed for solving non-Darcian flow problems before, is problematic for studying radial non-Darcian flow. Comparison of drawdowns obtained by our proposed method and the Boltzmann transform method suggests that the Boltzmann transform method differs from the linearization method at early and moderate times, and it yields similar results as the linearization method at late times. If the power index n and the quasi hydraulic conductivity k get larger, drawdowns at late times will become less, regardless of the wellbore storage. When n is larger, flow approaches steady state earlier. The drawdown at steady state is approximately proportional to r¹⁻ⁿ, where r is the radial distance from the pumping well. The late time drawdown is a superposition of the steady-state solution and a negative time-dependent term that is proportional to t^{(1−n)/(3−n)}, where t is the time.     

Flowing Well—Time-Domain Solution and Inverse Problem Revisited

Time-domain analytical solution for groundwater flow to a fully penetrating flowing well is derived using the same substitution technique used to re-derive (Perina 2010) the Theis (1935) equation and the approximate solution by Mishra and Guyonnet (1992) is confirmed. The exponential integral-based flowing well function is a computationally effective alternative to the original Jacob and Lohman (1952) solution in integral form. For a constant drawdown test, the ratio of drawdown at an observation well to the flowrate is equivalent to drawdown response to pumping at unit constant rate; the transformed observations can be analyzed using the Theis (1935) function. Analysis of field test shows that simultaneous fitting to measurements of flow from the test well and drawdown at an observation well results in more accurate and better resolved estimates of aquifer properties than fitting to flow observations only.     

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.     

Field Testing the Hypothesis of Darcian Flow Through a Carbonate Aquifer

The acceptability of the hypothesis of Darcian flow through a semiconfined carbonate aquifer was tested prior to running a multiple-day aquifer test in Pinellas County, Florida. The approach used to test the hypothesis was to run a number of hour-long aquifer tests at different discharges with drawdown measured at the same time during each test in two observation wells, one at 35 feet and the other at 733 feet from the pumped well. If the hypothesis were acceptable, a plot of drawdown versus discharge should describe a straight line. A linear relation was described in 15-minute and 1-hour data plots of drawdown versus discharge for both observation wells. Thus, the hypothesis of Darcian flow through the semiconfined carbonate aquifer was deemed acceptable. Drawdown data from the distant observation well collected during the following multiple-day aquifer test were then analyzed for aquifer-property values of transmissivity, storage coefficient, and leakance coefficient using standard methods. Discharge for the multiple-day test was within the discharge range of the hour-long tests.     

A Method to Determine the Formation Constants of Leaky Aquifers, and Its Application to Pumping Test Data

Abstract. A method to calculate aquifer transmissivity, storage coefficient, and the leakage coefficient from pumping test data for a leaky aquifer is presented. The method is carried out by a computer program and is based on a minimization of the sum of squares of differences between drawdown in the observation well and the theoretical values from the Hantush and Jacob formula. No user defined starting points are necessary. Random error estimates for the parameters are given. Applications of the method are illustrated using data from pumping tests performed in leaky aquifers at the Cauca River Valley, Colombia.     

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.     

Use of Groundwater Level Fluctuations near an Operating Water Supply Well to Estimate Aquifer Transmissivity

We developed a method to estimate aquifer transmissivity from the hydraulic-head data associated with the normal cyclic operation of a water supply well thus avoiding the need for interrupting the water supply associated with a traditional aquifer test. The method is based on an analytical solution that relates the aquifer's transmissivity to the standard deviation of the hydraulic-head fluctuations in one or more observation wells that are due to the periodic pumping of the production well. We analyzed the resulting analytical solution and demonstrated that when the observation wells are located near the pumping well, the solution has a simple, Dupuit like form. Numerical analysis demonstrates that the analytical solution can also be used for a quasi-periodic pumping of the supply well. Simulation of cyclic pumping in a statistically heterogeneous medium confirms that the method is suitable for analyzing the transmissivity of weakly or moderately heterogeneous aquifers. If only one observation well is available, and the shift in the phase of hydraulic-head oscillations between the pumping well and the observation well is not identifiable. Prior knowledge of aquifer's hydraulic diffusivity is required to obtain the value of the aquifer transmissivity.     

Evaluation of the MODFLOW-2005 Conduit Flow Process

The recent development of the Conduit Flow Process (CFP) by the U.S. Geological Survey (USGS) provides hydrogeologic modelers with a new tool that incorporates the non-Darcian, multiporosity components of flow characteristic of karst aquifers. CFP introduces new parameters extending beyond those of traditional Darcian groundwater flow codes. We characterize a karst aquifer to collect data useful for evaluating this new tool at a test site in west-central Florida, where the spatial distribution and cross-sectional area of the conduit network are available. Specifically, we characterize: (1) the potential for Darcian/non-Darcian flow using estimates of specific discharge vs. observed hydraulic gradients, and (2) the temporal variation for the direction and magnitude of fluid exchange between the matrix and conduit network during extreme hydrologic events. We evaluate the performance of CFP Mode 1 using a site-scale dual-porosity model and compare its performance with a comparable laminar equivalent continuum model (ECM) using MODFLOW-2005. Based on our preliminary analyses, hydraulic conductivity coupled with conduit wall conductance improved the match between observed and simulated discharges by 12% to 40% over turbulent flow alone (less than 1%).     

Analytical and Semi‐Analytical Tools for the Design of Oscillatory Pumping Tests

Oscillatory pumping tests—in which flow is varied in a periodic fashion—provide a method for understanding aquifer heterogeneity that is complementary to strategies such as slug testing and constant‐rate pumping tests. During oscillatory testing, pressure data collected at non‐pumping wells can be processed to extract metrics, such as signal amplitude and phase lag, from a time series. These metrics are robust against common sensor problems (including drift and noise) and have been shown to provide information about aquifer heterogeneity. Field implementations of oscillatory pumping tests for characterization, however, are not common and thus there are few guidelines for their design and implementation. Here, we use available analytical solutions from the literature to develop design guidelines for oscillatory pumping tests, while considering practical field constraints. We present two key analytical results for design and analysis of oscillatory pumping tests. First, we provide methods for choosing testing frequencies and flow rates which maximize the signal amplitude that can be expected at a distance from an oscillating pumping well, given design constraints such as maximum/minimum oscillator frequency and maximum volume cycled. Preliminary data from field testing helps to validate the methodology. Second, we develop a semi‐analytical method for computing the sensitivity of oscillatory signals to spatially distributed aquifer flow parameters. This method can be quickly applied to understand the “sensed” extent of an aquifer at a given testing frequency. Both results can be applied given only bulk aquifer parameter estimates, and can help to optimize design of oscillatory pumping test campaigns.     

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.     

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.     

How to Find Aquifer Statistics Utilizing Pumping Tests? Two Field Studies Using welltestpy

We present a workflow to estimate geostatistical aquifer parameters from pumping test data using the Python package welltestpy . The procedure of pumping test analysis is exemplified for two data sets from the Horkheimer Insel site and from the Lauswiesen site, Germany. The analysis is based on a semi-analytical drawdown solution from the upscaling approach Radial Coarse Graining, which enables to infer log-transmissivity variance and horizontal correlation length, beside mean transmissivity, and storativity, from pumping test data. We estimate these parameters of aquifer heterogeneity from type-curve analysis and determine their sensitivity. This procedure, implemented in welltestpy , is a template for analyzing any pumping test. It goes beyond the possibilities of standard methods, for example, based on Theis' equation, which are limited to mean transmissivity and storativity. A sensitivity study showed the impact of observation well positions on the parameter estimation quality. The insights of this study help to optimize future test setups for geostatistical aquifer analysis and provides guidance for investigating pumping tests with regard to aquifer statistics using the open-source software package welltestpy .     

Air and water flows induced by pumping tests in unconfined aquifers with low‐permeability zones

Air flows from the atmosphere into an unconfined aquifer when the water table falls during pumping tests. Pumping test results in unconfined aquifers may be significantly affected by low‐permeability zones (LPZs) near the initial water table position, because they restrict the downward movement of air. A transient, three‐dimensional air–water two‐phase flow model is employed to investigate numerically the effects of local heterogeneity on pumping test results in unconfined aquifers. Two cases of local heterogeneities are considered herein: a LPZ around the pumping well and on one side of the pumping well. Results show that the drawdown with the LPZ is significantly greater than that of the homogeneous aquifer. The differences in drawdown are the most significant at intermediate times and gradually diminish at later times. The LPZ significantly reduces air flow from the atmosphere to the aquifer. The pore air velocity in the LPZ is very low. The air pressure at the observation point under the LPZ when air begins to enter is significantly lower than the air pressure of the homogeneous aquifer at the same point. After that, the air pressure increases quickly and then increases slowly. The time for the air pressure to reach the atmospheric pressure is significantly longer. Copyright © 2013 John Wiley & Sons, Ltd.     

Applications of the time-naught term in the Cooper and Jacob (1946) equation

The ability to manipulate analytical expressions for aquifer drawdown can provide insights into groundwater flow processes and assist with assessing strengths and weaknesses of aquifer parameter estimation methods. In the Cooper and Jacob (1946) parameter estimation method, the antilog of the horizontal-axis intercept in a plot of drawdown vs. log(time) is referred to as time naught (t(0)), which is used for estimating storativity. This article briefly reviews traditional uses of the time-naught concept and then spends time introducing new insights and applications involving (1) time-naught/distance relationships, including ways to compensate for certain missing data; (2) use of time naught in a simple method providing a quick visual check of which data in a Cooper-Jacob plot are suitable for use in linear regression; (3) application of time naught, as determined for one well, in estimating the later minimum time for which data from a distant well can be used in the Cooper-Jacob method; (4) development of relationships between drawdown and time naught; (5) use of time naught in a simple algebraic equation to estimate drawdown at smaller times than feasible using the Cooper-Jacob method; and (6) employment of time naught and a vertical-axis intercept on a plot of drawdown vs. log(time) for evaluating storativity. This information may be useful to new hydrogeologists or others interested in further developing their analytical well hydraulics skills.     

New graphical methods for estimating aquifer hydraulic parameters using pumping tests with exponentially decreasing rates

Actual pumping tests may involve continuously decreasing rates over a certain period of time, and the hydraulic conductivity (K) and specific storage (S_s) of the tested confined aquifer cannot be interpreted from the classical constant‐rate test model. In this study, we revisit the aquifer drawdown characteristics of a pumping test with an exponentially decreasing rate using the dimensionless analytical solution for such a variable‐rate model. The drawdown may decrease with time for a short period of time at intermediate pumping times for such pumping tests. A larger ratio of initial to final pumping rate and a smaller radial distance of the observation well will enhance the decreasing feature. A larger decay constant results in an earlier decrease, but it weakens the extent of such a decrease. Based on the proposed dimensionless transformation, we have proposed two graphical methods for estimating K and S_s of the tested aquifer. The first is a new type curve method that does not employ the well function as commonly done in standard type curve analysis. Another is a new analytic method that takes advantage of the decreasing features of aquifer drawdown during the intermediate pumping stage. We have demonstrated the applicability and robustness of the two new graphical methods for aquifer characterization through a synthetic pumping test.     

Modeling of Contaminant Movement Near Pumping Wells: Saturated-Unsaturated Flow with Particle Tracking

A transient axisymmetric saturated-unsaturated numerical flow model was coupled with a particle tracking model to investigate the movement of contaminants when a shallow unconfined aquifer is pumped at a constant rate. The particle tracking model keeps track of locations and masses of solutes in the aquifer, and the time of capture by the well. At the end of each time-step the flow model solves the Richard's equation for the hydraulic head distribution from which elemental velocities are calculated. Solutes are then displaced for a period equivalent to the time-step using both the magnitude and direction of the elemental velocities. Numerical experiments were performed to investigate effluent concentrations in wells with screens of different length and in different positions relative to zones of stratified contamination. At early times of pumping the effluent concentrations were similar to the concentrations adjacent to the well screen, but at late times, the concentrations approached the vertically averaged concentration in the aquifer. Time to attain the vertically averaged concentration was determined by the well geometry, initial location of the contaminant plume in relation to the well screen, and hydraulic properties of the aquifer. The results are consistent with the hydraulics of flow to a pumping well and of particular importance, they demonstrate that short-term pump tests could give erroneous design concentrations for pump-and-treat systems. The model provides a means of quantifying arrival times and mixing ratios. It could therefore provide a useful means of designing production wells in aquifers with stratified contamination and more efficient recovery systems for aquifer remediation.     

Improved characterization of small "u" for Jacob pumping test analysis methods

Numerous refinements have been proposed to traditional pumping test analyses, yet many hydrogeologists continue to use the Jacob method due to its simplicity. Recent research favors hydraulic tomography and inverse numerical modeling of pumping test data. However, at sites with few wells, or relatively short screens, the data requirements of these methods may be impractical within physical and fiscal constraints. Alternatively, an improved understanding of the assumptions and limitations of Theis and, due to their widespread usage, Jacob analyses, leads to improved interpretations in data-poor environments. A fundamental requirement of Jacob is a "small" value of u = f(r(2)/t), with radial distance, r, and pumping time, t. However, selection of a too stringent (i.e., too low) maximum permissible u-value, u(max), results in rejection of usable data from wells beyond a maximum radius, r(max). Conversely, data from small radii, less than r(min), where turbulent- and vertical-flow components arise, can result in acceptance of inappropriate data. Usage of drawdown data from wells too close to the pumping well, and exclusion of data from wells deemed too far, can cause unrealistic aquifer transmissivity, permeability, and storativity determinations. Here, data from an extensive well field in a glacial-outwash aquifer in north-central Minnesota, USA, are used to develop a new estimate for u(max). Traditionally quoted values for u(max) range from 0.01 to 0.05. Our proposed value for Jacob distance-drawdown analyses is significantly higher with u(max) up to 0.2, resulting in larger allowable r(max)-values and a higher likelihood of inclusion of additional wells in such pumping test analyses.