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.     

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.     

Simplified Kernel Method for Flow to Large Diameter Wells

A computationally simple kernel method is proposed for obtaining drawdowns due to unsteady pumping of large diameter wells. The kernels can be worked out even on a hand-held calculator. The new method can also be used to obtain residual drawdowns. The new method yields results as good as those obtained using earlier methods.     

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.     

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.     

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.     

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.     

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.     

Modeling the Transient Pumping from Two Aquifers Using MODFLOW

A procedure is proposed for calculating the spatial and temporal variation of drawdown due to pumping a well tapping two aquifers separated by an aquitard, using convolution and MODFLOW. It can take into account the unsteady pumping discharge and cross flow through the intervening aquitard. A discrete pulse kernel method based on superposition/convolution is used to account for the unsteady pumping discharge. The discrete pulse kernels are calculated using MODFLOW. The contributions of the aquifers to the pumped discharge are accounted implicitly and not required to be specified explicitly. Available numerical models (e.g., MODFLOW) require the aquifer contributions that are implicitly controlled, to be specified explicitly. The use of the suggested procedure is illustrated using examples. The contributions of the aquifers are found not in proportion to their transmissivities but vary with time, when the diffusivities of the aquifers are not equal. Applying the new procedure, the numerical models, such as MODFLOW can be used to correctly model the transient pumping from two aquifers with cross flow; thus, it opens up the possibility of numerically accounting for the aquifer heterogeneity while dealing with the flow to a well tapping two aquifers under a transient pumping, which would be otherwise difficult to account for analytically.     

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.     

Rational Criterion for Designing Opening of Slit-Check Dam

This paper presents a theoretical approach to the problem of designing the opening in a slit-check dam. The approach is based on the conservation of the mass of water and sediments and on the energy balance under steady conditions. It leads to a relationship among opening width, sediment characteristics, mountain river geometry, and water and sediment discharge. The final relationship can be simplified to make it suitable for practical applications. Also, the problem of unsteadiness of both water and sediment is considered, as well as the possibility of treating the unsteady flow as a sequence of steady states. The results of the theory were checked in a laboratory investigation using a scale model. Different opening widths were tested under conditions of steady and unsteady supplies of water and sediment. The mean grain size of the sediment, as well as the rates of sediment and water discharge, were changed in the experiments. The results of the experiments confirm the theory quite well.     

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.     

Discussion of “Parametric Study of One-Dimensional Solute Transport in Deformable Porous Media” by Akram N. Alshawabkeh and Nima Rahbar

An approach for incorporating large-diameter wells into MODFLOW is presented. A transformation of transmissivity in the flow field is devised to simulate the well function for a large-diameter wells. Transformations for accounting for the well radius and well storage are also developed. Using these transformations, MODFLOW is used to obtain the drawdowns. It is observed that the flow to a large diameter well can be accurately modeled using the proposed transformations and MODFLOW.     

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.     

Simple Approximation of Well Function for Constant Drawdown Variable Discharge Artesian Wells

A simple approximation of the well function for a constant drawdown is developed. This approximation is used to estimate the storage coefficient and transmissivity of the aquifer from observed unsteady discharge under a constant drawdown condition, using an optimization method. Another simple approximation for calculating the total production volume during a time span is also developed. The developed approximations for the well function and production function are accurate within a maximum error of 0.7% for the practical range of the argument.     

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.     

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.     

Kernel Method for Transient Rate and Volume of Well Discharge under Constant Drawdown

A kernel method is proposed for calculating transient rate and cumulative volume of well discharge under constant drawdown. The new method can also be used for obtaining the drawdown (in pressure head) in the aquifer at some distance away from the well. Employing the new method, an optimization method is used to estimate the aquifer parameters from transient well discharge or drawdown in the aquifer pressure head. The proposed method can also be used to model the recovery of drawdown (in aquifer pressure head) after the plug-in of the well.     

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.     

Local Balance Unsteady Friction Model

The paper proposes the evaluation of unsteady friction by a one-dimensional local balance model. The model is applied for the case of water hammer in a single pipeline for both the downstream end and upstream end valve, and for both rapid valve closure and opening. The model is based on local balance of the friction force. Comparisons with experimental results show that the model correctly predicts the extreme values of pressure head oscillation, as well as its shape for both rapid valve closure and opening, and then overcomes the limits of previous unsteady friction models based on instantaneous acceleration. As the comparisons with experimental results can be made easily only for pressure oscillations and can be affected by dissipation mechanisms other than friction, the performance of the model is examined also by comparison with the results of a two-dimensional low-Reynolds number k–ε model.