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.     

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.     

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.     

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 Effective Distance to a Recharge Boundary

The solution to the problem of induced recharge where pumping is planned near a stream, requires determining effective distance to the line of recharge. This is generally accomplished by analyzing the drawdowns observed during the pumping test conducted near a recharge boundary. Traditional methods of estimating the distance between observation well and image well either use the concept of fully developed straight lines or make use of the type curves for matching. Hantush's method requires the locating of the inflection point on the drawdown curve. The curve matching or the Hantush method is subjective and hence involves errors due to personal judgment. A long-duration pumping test is required to be conducted in order to obtain a fully developed second straight line. In many situations, such a long-duration pumping test is not feasible. In this paper, a robust optimization method is presented that allows the use of shorter-duration pumping test data, for estimating aquifer parameters, and distance to the effective line of recharge. Thus, it saves time and costs involved in conducting a long-duration pumping test. Application of the method on published data sets shows that the new method yields reliably accurate estimates of the parameters.     

Identifying Impervious Boundary and Aquifer Parameters from Pump-Test Data

The common field problem of identifying and locating impervious aquifer boundaries is generally solved by analyzing the pump-test data. The estimate of the distance between the image well and the observation well (r′) is required for locating the boundary. Popular methods of estimating r′ either use the concept of fully developed two straight lines or rely on curve matching. Curve-matching methods are subjective and involve errors due to personal judgment. A very long duration pump test is required in order that the second straight line is developed. In many cases, the first straight line is not developed at all, especially when the observation well is at a greater distance from the pumped well. In this paper, a derivative-based robust optimization method has been proposed for the identification of aquifer parameters (transmissivity, T and storage coefficient, S) and r′ from the drawdowns observed at an observation well. It returns unbiased estimates of the parameters (T, S, and r′) even with a poor initial guess. The method can yield the estimates of the parameters from shorter duration pump-test data, thus saving time and money involved in conducting the pump test with the purpose of identifying r′. It is observed that the hydraulic diffusivity (T∕S) of the aquifer and r′ can be estimated even when the constant rate of pumping is not known.     

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.     

Diagnostic Curve Methods for Consolidation Coefficient

Two new diagnostic curve methods, which do not require the implicit or explicit determination of initial or final dial gauge readings for primary settlement, are proposed for the identification of the consolidation coefficient. Besides the estimate of the consolidation coefficient, these methods also yield an estimate of the range of primary settlement. In these methods, the diagnostically plotted points are matched to the diagnostic curves with a parallel shift of axes to yield the estimates of the consolidation coefficient and range of primary settlement. The new methods do not require dial gauge readings up to 100% of primary consolidation and can also be used even when very scarce data are available. Using the proposed methods, a nonideal condition can also be identified.     

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 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.     

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.     

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.     

Estimation of Aquifer Parameters from Pumping Test Data by Genetic Algorithm Optimization Technique

Adequate and reliable estimates of aquifer parameters are of utmost importance for proper management of vital groundwater resources. The pumping (aquifer) test is the standard technique for estimating various hydraulic properties of aquifer systems, viz., transmissivity (T), hydraulic conductivity (K), storage coefficient (S), and leakance (L), for which the graphical method is widely used. In the present study, the efficacy of the genetic algorithm (GA) optimization technique is assessed in estimating aquifer parameters from the time-drawdown pumping test data. Computer codes were developed to optimize various aquifer parameters under different hydrogeologic conditions by using the GA technique. Applicability, adequacy, and robustness of the developed codes were tested using 12 sets of the published and unpublished aquifer test data. The aquifer parameters were also estimated by the graphical method using AquiferTest software, and were compared with those obtained by the GA technique. The GA technique yielded significantly low values of the sum of square errors (SSE) for almost all the datasets under study. The results revealed that the GA technique is an efficient and reliable method for estimating various aquifer parameters, especially in the situation when the graphical matching is poor. Also, it was found that because of its inherent characteristics, GA avoids the subjectivity, long computation time and ill-posedness often associated with conventional optimization techniques. Furthermore, the performance evaluation of the developed GA-based computer codes showed that the fitness value (SSE) of the best point in a population reduces with increasing generation number and population size. The analysis of the sensitivity of the parameters during the performance of GA indicated that a unique set of aquifer parameters was obtained for all three aquifer systems. The GA-based computer programs with interactive windows developed in this study are user-friendly and can serve as a teaching and research tool, which could also be useful for practicing hydrologists and hydrogeologists.     

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.     

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.     

Search for Critical Slip Surface in Slope Stability Analysis by Spline-Based GA Method

The stability of a soil slope is usually analyzed by limit equilibrium methods, in which the identification of the critical slip surface is of principal importance. In this study the spline curve in conjunction with a genetic algorithm is used to search the critical slip surface, and Spencer’s method is employed to calculate the factor of safety. Three examples are presented to illustrate the reliability and efficiency of the method. Slip surfaces defined by a series of straight lines are compared with those defined by spline curves, and the results indicate that use of spline curves renders better results for a given number of slip surface nodal points comparing with the approximation using straight line segments.     

Aquifer Diffusivity and Stream Resistance from Varying Stream Stage

An efficient method that uses discrete ramp kernel is proposed for obtaining the piezometric head in an aquifer due to an arbitrary variation in stream stage considering stream resistance. The method assumes straight line variation between two consecutive points in representing the arbitrary stream stage variation. Expression for the ramp kernel is derived for homogeneous and isotropic aquifer conditions. Using the method, the stream resistance and hydraulic diffusivity of the aquifer are estimated for a set of published data. It is observed that the hydraulic diffusivity should be estimated along with the stream resistance for a better estimation of aquifer diffusivity.     

Moment-Based Calculation of Parameters for the Storage Zone Model for River Dispersion

Theoretical methods have been developed to calculate values of parameters of the storage zone model for river mixing. Analytical solutions of the Laplace-transformed equations of the storage zone model are related to the observed concentration distribution in order to determine model parameters in both the moment matching method and the maximum likelihood method, which were developed in this study. The results obtained by comparison with experimental data show that the parameters calculated by the moment matching method are in good agreement with the observed values of storage zone model parameters, whereas results from the maximum likelihood method and several existing methods are not in good agreement with the experimentally observed values. Dispersion data from natural streams show that the calculated concentration curves from the numerical solutions of the storage zone model with the parameters calculated by the moment matching method fit the observed concentration curves very well. It can be concluded that parameters of the storage zone model calculated using the moment matching method can properly explain the natural dispersion processes in real streams.     

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.     

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.