Flow Depletion of Semipervious Streams Due to Unsteady Pumping Discharge

Analytical expressions for rate and volume of flow depletion of semipervious streams due to sinusoidal variation in pumping rate are obtained. An analytical but approximate method is developed for obtaining the rate and volume of stream flow depletion due to arbitrary unsteady pumping discharge. The method uses the ramp kernel and convolution. The use of ramp kernels permits linear interpolation between two consecutive discretized discharge values. The analytical equations for the ramp kernels for the rate and volume of stream flow depletion are derived. The proposed method is applicable for homogeneous and isotropic aquifers that are hydraulically connected to streams.     

Rate and Volume of Stream Flow Depletion due to Unsteady Pumping

Analytical but approximate methods are developed for obtaining pumping induced rate and volume of stream flow depletion, which can account for unsteady (any variation) pumping discharge and are also applicable for intermittent pumping and recovery. Exact analytical solutions for a sinusoidal variation in the pumping discharge are proposed; the proposed methods are verified using these solutions. The proposed methods use ramp kernels that give results superior to the conventional convolution. These ramp kernels assume the linear variation in pumping discharge between the two consecutive discretized points as opposed to the uniform variation assumed in the conventional convolution. The proposed solutions are applicable for homogeneous and isotropic aquifers hydraulically connected to streams.     

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.     

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.     

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.     

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.     

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.     

Simple Model for Analyzing Transient Pumping from Two Aquifers with Cross Flow

A simple semianalytical model is proposed for calculating the drawdown due to pumping a well tapping two aquifers. The new model can take into account the transient pumping discharge and cross flow between the aquifers. The transient contributions of the aquifers to the pumped discharge can also be implicitly obtained using the model.     

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.     

Ramp Kernels for Aquifer Responses to Arbitrary Stream Stage

Analytical expressions for ramp kernels (new kernels) for an improved convolution for obtaining aquifer responses, viz, groundwater head, rate, and cumulative volume of groundwater flow, to an arbitrary stage, are obtained. The use of the ramp kernels gives accurate aquifer responses and is superior to the conventional convolution in which numerical integration or pulse kernels are used. The extent of improvement in the results with the use of the ramp kernels is discussed and quantified for three examples, where the results are compared to analytical solutions. For the comparisons, the analytical solutions for linear and sinusoidal stream stages are derived. The use of the ramp kernels reproduces accurately the analytical solutions. The concept of ramp kernels can also be used for obtaining an accurate solution of convolution integrals observed in other fields.     

Aquifer Response to Sinusoidal or Arbitrary Stage of Semipervious Stream

Analytical expressions for the aquifer responses, viz., groundwater head, rate of flow and cumulative volume of flow, to a generalized sinusoidal stage of semipervious streams considering the stream boundary resistance, are derived. The analytical aquifer responses to a linear stream stage and to a typical analytical flood wave that was used by Cooper and Rorabaugh, are also derived. For a zero-stream resistance, the aquifer responses converge to those for a fully penetrating stream. Also, two analytical methods, a “ramp kernel method” and a “Fourier series method,” for obtaining the aquifer responses to an arbitrary temporal stage of sempervious stream, are developed. The analytical expressions of the ramp kernels for different aquifer responses are developed. The ramp kernel method is found superior to the conventional convolution that uses numerical integration or pulse kernels for obtaining the convolution integral. In the Fourier series method, the aquifer responses to sinusoidal stage are used along with Fourier series. The results obtained using both methods are in close agreement. The new methods are also applicable to fully penetrating streams by assigning a zero value to the stream resistance.     

Experiments on Discharge Equations of Compound Broad-Crested Weirs

A linear combination of traditional discharge equations for simple rectangular and/or triangular weirs is proposed to describe the discharge equations of compound broad-crested (CBC) weirs. The CBC weirs are composed of rectangular, triangular, and/or truncated triangular weirs. Dimensionless discharge equations have been also derived. Laboratory experiments on discharge relations for flows over four CBC weirs were conducted in this study in order to calibrate the proposed discharge equations. The experiments were carried out under the conditions of the H1/H2-ratio of water heads above upper and lower crests less than 0.54, and a dimensionless discharge less than 2.174. The result shows that the differences between the calculated discharges by the proposed equations and the measured ones are less than 3% for flows over these CBC weirs under the present experimental conditions.     

Measurement of Small Discharges in Open Channels by Slit Weir

A rectangular slit weir is designed for measurement of small (<0.005 m3/s) discharges. The discharge coefficient is determined experimentally using the measured discharges and the corresponding heads over the weir. The relationship between the discharge coefficient and all relevant dimensionless parameters is investigated. It is concluded that the discharge coefficient can be represented solely as a function of Reynolds number.     

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.     

Simple Approximation for Flowing Well Problem

The discharge and yield of a flowing well can be computed by existing complex solutions or by Duhamel's technique using kernel coefficients. The mathematical complexities of these methods can be very much simplified through numerical methods without loss of accuracy. In the present study simple equations for both the well discharge and well production functions are presented. The equation for the well discharge function has been used to find aquifer constants for known well discharge and drawdown through the error minimization method. The results are compared in order to demonstrate the relative simplicity of the proposed equations.     

Incipient motion of gravel in a bottomless arch culvert

Incipient motion was investigated for four gravel substrate materials in a bottomless arch culvert and a rectangular flume. Different methods for calculating Shields parameter at incipient motion (θc)based upon local flow parameters were explored. An incipient motion region for bottomless arch culverts in fully turbulent flow was defined with two bounding curves on Shields diagram. The variation of θc as a function of relative roughness was examined. Also, a method that utilizes measured flow velocities to determine stable substrate particle diameters for bottomless arched culverts is presented as an alternative to the Shields diagram.     

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.     

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.