Storage Coefficient and Transmissivity from Residual Drawdowns

A computationally simple method is proposed for the estimation of transmissivity and storage coefficient from only residual drawdowns at an observation well, the calculations for which can be performed using a calculator. The method does not require the last pumping drawdown, however, duration of pumping is required. Different estimates of storage coefficient during pumping and recovery can be obtained using the new method if applied on such data sets affected by the hysteresis in storage coefficient during pumping versus recovery. The new method may be suited for advanced analysis of pumping/residual drawdowns, such as storage coefficient increasing with recovery. It is able to identify the nonideal aquifer condition (other than infinite confined aquifer) from only residual drawdowns if applied on such data. It can yield reliable estimates of aquifer parameters, which are as good as that obtained using an optimization approach developed previously by the author.     

Simple Method for Quick Estimation of Leaky-Aquifer Parameters

Simple method and explicit equations are proposed for estimating the parameters of leaky aquifers from drawdown at an observation well, which avoid the curve matching or initial estimate of the parameter. The proposed method is computationally simple and the calculations can be performed even on a handheld calculator. The application of the methods is illustrated, using published data sets. The new method yields quick and accurate estimates of the leaky-aquifer parameters, if observed drawdowns do not contain large errors. The proposed method can also analyze the early drawdowns for accurate characteristics/parameters of a confined aquifer, if the conductance of the aquitard is assigned a zero value. It is hoped that the proposed method would be of help to field engineers and practitioners.     

Simple Method for Confined-Aquifer Parameter Estimation

Early drawdown data, for which the argument u of the well function is >0.01, have often been considered unimportant in evaluating aquifer parameters. This paper shows that these early drawdown data, especially in the neighborhood of u = 0.43, can yield accurate values of aquifer parameters. A simple method has been presented for explicit determination of aquifer parameters using early drawdown data. The method does not require curve matching, initial guess of the parameters, or special care to check for u < 0.01, and the computations involved can be performed on a calculator. Application of the method on published data sets shows that the estimates of the aquifer parameters using only a few initial drawdowns are as good as those obtained by Theis curve matching when all data, including the late drawdowns (u ≤ 0.01), are used. The new method converges with the Cooper-Jacob method when the late drawdown data are considered. Thus, the late drawdown data also can be analyzed using the new method.     

Estimating Aquifer Parameters from Early Drawdowns in Large-Diameter Wells

The existing equation applicable for large diameter wells in confined aquifers is transformed into a convenient form and a set of semilogarithmic diagnostic curves is developed for identifying the aquifer parameters (storage coefficient and transmissivity) from early drawdowns in large diameter wells. A scaled well function is proposed for the diagnostic curves. The aquifer parameters are estimated by matching the diagnostically plotted drawdowns to one of the diagnostic curves by a parallel shift of only one axis. The substantial curvature of the diagnostic curves and shifting of only one axis facilitate matching and reduce subjectivity. The proposed method is an improvement over the existing matching methods. The new method can reliably identify the aquifer parameters from only early drawdowns and would result in a 100-fold saving in time and money. It is hoped that this method would be helpful to field engineers and practitioners.     

Leaky One-Dimensional Flow with Storage and Skin Effect in Finite-Width Sink

One-dimensional horizontal flow in a semiinfinite confined aquifer can be described in terms of mathematical solutions that relate drawdown in the aquifer to aquifer parameters and flow into or out of a line sink. A new solution that considers the effects of a low-permeability skin along with storage in a finite-width sink is developed for the leaky-aquifer case. A coefficient Sk is defined to represent the skin effect for one-dimensional flow. The transient solution, which is obtained by inverting the Laplace-space solution using the Stehfest numerical algorithm, calculates drawdowns in the sink as well as in the aquifer. A nondimensional drain function D(u,x/B,A/x,Sk/x)q is defined based on the solution. Selected type curves for the drain function are plotted, and a match-point procedure is described that is based on matching observed drawdowns at observation wells to an appropriate type curve. The match-point procedure is illustrated by fitting simulated drawdown data to a type curve and determining the aquifer parameters. The drawdown solution is also represented by dimensional time–drawdown plots, which can be used to determine aquifer parameters by adjusting the parameters until model-calculated drawdowns match observed values. This new solution can be used to analyze drawdowns that result from a canal pumping test in which the discharge from the canal is derived from water stored in the canal and from a leaky aquifer and in which the drawdowns are affected by storage and a low-permeability skin in the canal.     

Simple Model for Analyzing Transient Pumping from Two Aquifers without Cross Flow

A computationally simple semianalytical model is proposed for calculating the drawdown due to pumping a well tapping two aquifers separated by an aquiclude with no cross flow. The new model can take into account the transient pumping discharge. Equations are proposed for calculating the transient contributions of the aquifers to the pumped discharge and drawdowns in aquifers. The residual drawdowns in the aquifers and the aquifer contributions during recovery period can also be obtained using the proposed model. Based upon a similar principle, another model is also developed that can consider the effect of the well storage. The proposed models can be used to calculate drawdowns neglecting or considering the well storage, in the case of transient pumping from two aquifers having different values of transmissivity and storage coefficient. It is hoped that the new models would be of help to the field engineers and practitioners.     

Confined Aquifer Parameters from Temporal Derivative of Drawdowns

A simple method that uses the time derivative of drawdowns is proposed for the evaluation of confined aquifer parameters. Explicit expressions are proposed for evaluation of the aquifer parameters as well as a graphical procedure. A reliable and accurate scheme to calculate the numerical derivative of drawdowns is developed based upon an analytical approach. The method requires early drawdown data (u > 0.01, where u is the argument of well function), and is shown to converge to the Cooper-Jacob method for late drawdowns (u ≤ 0.01). It does not require curve matching or an initial guess for the parameters. Calculations for the method can be performed on a hand-held calculator. The method has been applied to published data sets and the results have been compared with those obtained using traditional methods. The method accurately estimates the aquifer parameters using only early drawdown data, thereby indicating savings in time and money.     

Semianalytical Model for Drawdown due to Pumping a Partially Penetrating Large Diameter Well

A semianalytical model is developed for computing drawdowns in and around a partially penetrating large diameter well. The new model can take into account an unsteady pumping discharge and thus drawdowns during recovery can be computed. This model can also yield the unsteady contributions from well and aquifer storages to the pumped discharge. While developing the model, the flow from the bottom of the well is also accounted.     

Approximation of Well Function for Large Diameter Wells

A computationally simple and accurate algebraic approximation of the well function for large diameter wells is proposed. This approximation can be used to calculate drawdown in a large diameter well due to steady pumping. Using this approximation, a method is proposed for calculating drawdowns in large diameter wells due to unsteady pumping discharge. The proposed method is also applicable for calculating residual drawdowns in large diameter wells. In principle, the proposed method that uses the approximation of the well function should yield more accurate results than the previously proposed kernel methods.     

Diagnostic Curve for Confined Aquifer Parameters from Early Drawdowns

A diagnostic curve of unimodal shape is developed for identifying the confined aquifer parameters from early drawdowns. A scaled well function is proposed for the diagnostic curve and computationally simple functions are developed for its accurate approximation. The diagnostic curve may be viewed as an alteration of the Theis’ curve or as the generalization of a previous approach proposed by the writer. Plotting the pumping test data in a convenient form and matching it to the diagnostic curve with a parallel shift of axes identify the aquifer parameters. The unimodal shape of the diagnostic curve facilitates matching and reduces the personal errors. The proposed method is simple, easy to apply, and yields accurate estimates of aquifer parameters from only early drawdowns, which would save considerable time and money involved in conducting a long-duration pumping test. The estimates obtained using the new method are as good as those obtained from much more complex methods. The new method does not require either the initial guess for the parameter values or repetitive evaluation of the well function.     

Simple Equations for Aquifer Parameters from Drawdowns in Large-Diameter Wells

Simple equations are proposed for estimating storage coefficient and transmissivity of an aquifer from drawdowns in large- diameter wells. The proposed method requires determination of the peak and time to peak of a unimodal curve. Using these values and utilizing the provided set of equations, the aquifer parameters are estimated through an iterative procedure. The proposed method is void of subjectivity involved in the previously proposed curve matching methods. Also, the new method can be used when the conventional curve matching methods cannot be applied to estimate the aquifer parameters. The new method can be used to estimate the aquifer parameters from the drawdown data observed only up to a time so that the peak could be determined.     

Identifying Head Loss from Early Drawdowns

A method is proposed for identifying the head loss from the early drawdowns at the pumped well and an observation well. The new method is computationally simple and the calculations can be performed even on a hand-held calculator. The previously proposed methods for identifying the head loss requires a long duration pumping test, while the new method requires only a short duration pumping test. The use of the proposed method would save time and money and would be of help to field and practicing engineers. A nondimensional coefficient of head loss is also proposed, which permits a meaningful comparison of the efficiency conditions of different wells.     

Optimizing Aquifer Parameters from Drawdowns in Large Diameter Wells

An optimization method is proposed for estimating the storage coefficient and transmissivity of an aquifer from drawdowns in large diameter wells consequent to an unsteady pumping. The concept of an optimal time step size is propagated in the proposed method. The estimate of the aquifer parameters corresponding to the optimal time step size is termed final estimate. The estimates for any other time step size are not reliable. The proposed method can also take into account the residual drawdowns. The application of the method is illustrated using an example.     

Diagnostic Curves for Identifying Leaky Aquifer Parameters with or without Aquitard Storage

Two sets of unimodal diagnostic curves, one set assumes no aquitard storage and the other set assumes aquitard storage, are developed for identifying the parameter of leaky aquifers from early drawdowns, which yields accurate estimates of the parameters and lessens the subjectivity due to personal errors. The proposed diagnostic curve method is simple, easy to apply, and is based on matching of the diagnostically plotted observed drawdowns to an appropriate diagnostic curve. The new method is simple, easy to apply, does not require either the initial guess for the parameter values or repetitive evaluation of the leaky aquifer well function, and outperforms the conventional curve-matching, optimization, extended Kalman filter, and artificial neural network methods. The proposed set of diagnostic curves has a good diagnostic property and is able to easily identify nonideal conditions. The new method suggests a shorter duration pumping test, which would save time, money, and water. It is hoped that the proposed method would be useful to the field engineers and practitioners.     

Drawdowns due to Intermittent-Pumping Cycles

This paper offers solutions for drawdowns due to intermittent pumping cycles or cyclic pumping, which are high accuracy approximations of the series of Theis functions superimposed in time. The proposed approximation formulas are an improvement over the earlier works. The earlier approximations are valid only if the number of pumping cycles is greater than 10 and involve gamma functions that are less convenient to evaluate than the rational approximation formulas offered in this paper. The proposed approximations are valid for any number of pumping cycles and involve simple functions that can be computed even using a calculator. The drawdown functions are defined for the drawdowns at the end of pumping or shutoff periods. The proposed expressions for these functions are also suitable for the estimation of aquifer parameters by plotting the observed drawdowns on semilogarithmic paper. Procedures for estimation of storage coefficient and head loss at the well from cyclic pumping drawdowns are not available. This paper also offers procedures for the estimation of transmissivity, storage coefficient, and head loss at the pumped well from the observed intermittent (cyclic) pumping drawdowns.     

Well Loss Estimation: Variable Pumping Replacing Step Drawdown Test

An optimization method is presented for simultaneous estimation of aquifer parameters and well loss parameters utilizing all the drawdowns observed during a variable rate pumping or multiple step pumping test. The proposed method does not require any graphical analysis. It is shown that a variable rate pumping test is a better substitute for the conventional step drawdown test to estimate well loss parameters. It suggests that the pumping rate may be changed frequently without waiting for a near steady state to be reached (or a selected duration, say 60 min) in each step of a conventional step drawdown test. This can result in a substantial saving of time and money involved in conducting a step drawdown test with a view to estimate well loss parameters. This gives a greater number of distinct discharges, which improves the estimates of the well loss parameters. Application of the method is demonstrated on published data sets, the results of which show that the parameters estimated using the new method are more reliable as compared to those obtained using prior methods.     

Approximation of M Function for Partially Penetrating Wells

A computationally simple approximation of the Hantush M function is proposed. Using this approximation, an optimization method is proposed for identifying the aquifer parameters from early drawdowns around partially penetrating wells. The aquifer parameters are hydraulic conductivity and specific storage. The application of the method is illustrated using a field example. The proposed method results in a 120-fold saving in time when compared to the prior method. It is hoped that the proposed approximation and method will be of help to the field engineers and practitioners.     

Drawdown due to Pumping a Partially Penetrating Large-Diameter Well Using MODFLOW

The approach by the author for modeling the large-diameter wells using MODFLOW is extended to the partially penetrating large-diameter wells. The temporal variation of drawdown due to a steady pumping is presented in the form of diagnostic curves for different penetration. These diagnostic curves can also be used to estimate the aquifer parameters from the observed drawdowns in a partially penetrating large-diameter well.     

Aquifer Boundaries and Parameter Identification Simplified

A simple method is proposed for simultaneous and explicit identification of confined aquifer parameters and boundaries from drawdowns measured at an observation well during a constant rate pumping test. The method requires the determination of peaks of unimodal curves. Only a little subjectivity is involved in the method as the peak is a well defined point. The method is applicable even when the first and second straight lines on the semilogarithmic characteristic drawdown curve are not fully developed. The calculations involved can be performed on a calculator. Results have also been presented for small arguments of the well functions defining the development of the straight lines. The times for initiation and development of straight lines are quantified. The minimum duration of a pumping test for a reliable identification of the aquifer parameters and boundaries using the proposed method is also quantified. Use of the new method suggests a much shorter duration pumping test for the accurate identification of aquifer parameters and boundaries. This would save considerable time and money. At least a 100-fold savings in time and money involved in a pumping test to locate a boundary is observed when compared to the use of the law of times in the Cooper-Jacob method. Application of the method to published data sets shows that reliable estimates of the aquifer parameters and distance to a boundary are obtained.     

Approximation of Well Function and Identification of Leaky Aquifer Parameters

The existing equation for leaky aquifers is transformed into a nondimensional form using new parameters and a scaled well function for leaky aquifers is proposed. A computationally simple function is developed for accurately approximating the scaled well function for the practical range of the parameters. Utilizing this function (approximation), an optimization method is proposed for identifying the leaky-aquifer parameters from observed drawdowns. The new function has an enhanced utility when a repetitive numerical evaluation of the well function for leaky aquifers is needed, e.g., while estimating the aquifer parameters using optimization or Kalman filter or artificial neural network methods. The application of the proposed method is illustrated using a few sets of published data. The proposed method outperforms the extended Kalman filter method, based on the reported results in the literature.