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.     

Flow Depletion Induced by Pumping Well from Stream Perpendicularly Intersecting Impermeable/Recharge Boundary

Analytical solutions for rate and volume of flow depletion induced by pumping a well from a stream that intersects an impermeable or a recharge boundary at right angles are derived using the basic flow depletion factor defined earlier by the author. A new concept of directly obtaining stream flow depletion using the method of images is proposed. The solutions are derived for five different management cases of a stream and boundary intersecting at right-angles, assuming the aquifer to be confined with semi-infinite areal extent. A computationally simple function is proposed for accurately approximating the error function. The existing analytical solution in the case of a right-angle bend of stream given by Hantush was obtained for unconfined aquifers using a linearization of the governing partial differential equation. The solution for this case obtained using the proposed method for confined aquifer is the same as obtained by Hantush for unconfined aquifers, which shows that the linearization adopted by Hantush does not actually solve this problem for unconfined aquifers.     

Flow Depletion Induced by Pumping Well from Finite Length of Stream

A procedure for calculating the flow depletion from a finite length of a stream induced by a pumping well in an adjacent aquifer is developed. Four management cases of finite length of the stream including a basic case are considered. A “basic flow depletion factor” is defined, in terms of which the flow depletion factors for all cases are expressed. The basic flow depletion factor is twice the Hantush M function. A computationally simple and accurate practical approximation of the basic flow depletion factor is presented that encompasses the full practical range of the solutions. Using this approximation, an optimization method is proposed for the estimation of the aquifer hydraulic diffusivity and effective distance from the pumping well to the line of recharge from the measured temporal variation of stream flow depletion between two sections. During optimization, repeated computation of stream flow depletion is required; use of the proposed approximation simplifies the computation.     

Flow Depletion of Semipervious Streams Due to Pumping

Expressions for the rate and volume of flow depletion of semipervious streams due to pumping are presented in computationally simple forms. Analytical expressions have been proposed to take into account both partial penetration and semipervious bed and banks of the stream. Graphs suitable for engineering applications are presented for siting wells, and the effect of an intermittent pumping cycle on the rate and volume of stream flow depletion has also been discussed. The exclusive volume of flow depletion during a cycle is shown to vary with the selection of the end of the cycle.     

Rate and Volume of Stream Depletion due to Pumping

The two analytical solutions researched by Theis, and also Glover and Balmer, for the rate of stream depletion are shown to be the same in the mathematical sense by solving the integral appearing in the Theis solution. Expressions for the volume of stream depletion expressed as a fraction of total volume pumped relative to the rate of depletion are obtained for pumping and recovery. The optimum location of a well near the stream and the time to reach equilibrium are suggested.     

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.     

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.     

Explicit Estimation of Aquifer Diffusivity from Linear Stream Stage

Different approaches are available for estimation of aquifer hydraulic diffusivity from a linear stream-stage variation and corresponding groundwater heads. These approaches require interpolation from tabulated values or computation of hydraulic gradient at the stream aquifer interface. Certain methods use approximation or interpolation of tabulated values for an infinite series. These methods are prone to errors in the estimated aquifer hydraulic diffusivity. An alternative approach is to use a closed-form solution of the problem and develop an explicit expression for the aquifer hydraulic diffusivity, which is free from errors of using infinite series, interpolation, and computation of hydraulic gradient. Such an alternative method is developed for a linearly varying stream stage. The new method can yield the estimate of hydraulic diffusivity even from a single observation. The proposed method would be applicable to practical hydraulic engineering problems keeping in view that most of a rising part of a stream stage hydrograph can be approximated by a linear rise and certain rivers may show a linearly varying stream stage. Use of the new method is demonstrated on published and field data, which shows that the estimates obtained using the new method are comparable to that obtained using an optimization approach.     

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.     

Solution for Flow Rates across the Wellbore in a Two-Zone Confined Aquifer

A closed-form solution for transient flow rates across the wellbore in a confined aquifer is derived from a two-zone radial ground-water flow equation subject to the boundary condition of keeping a constant head at the well radius. An aquifer may be considered as a two-zone system if the formation properties near the wellbore are significantly changed due to the well construction and/or well development. An efficient numerical approach is used to evaluate this newly derived solution. Values of the transient flow rate are provided in a tabular form and compared with those obtained by numerical inversion for the Laplace-domain solution. The results show that the two solutions are in good agreement. This newly derived solution can be used not only for predicting the transient flow rate across the wellbore but also for identifying the effects of a skin with a finite thickness on the estimation of transient flow rates in a ground-water system with two different formation properties.     

Use of Steady-State Pump Head-Discharge Curve for Unsteady Pipe Flow Applications

An experimental pipeline system with a multistage centrifugal pump was used to study the effect of transient operations on the hydrodynamic performance of a centrifugal pump. Several transient flow operations were considered, ranging from very mild to severe transients. The dynamic relationship of total pressure rise across the pump to the flow rate was compared with that of the steady state. Deviation between the dynamic pump head and the value given by the steady-state curve at the same instantaneous discharge was established and found to be a function of the severity of the transient. It was found that severe flow conditions could cause this deviation to exceed 30% of the steady-state value. The use of the steady-state pump head-discharge relationship in the solution of transient pipe flow by the method of characteristics (MOC) is discussed. It was found that the steady-state pump head-discharge curve was not accurate enough to support the solution of unsteady pipe flow application by the MOC.     

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.     

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.     

Drawdown due to Temporally Varying Pumping Discharge: Inversely Estimating Aquifer Parameters

A ramp kernel method is proposed for accurately calculating the drawdown due to any temporal variation in pumping discharge. The use of the ramp kernels assumes the linear variation between the two consecutive measured pumping discharges. The prior studies assume a rectangular variation between the two consecutive measured pumping discharges. In the rectangular variation, a uniform pumping rate is assumed during a time span. An analytical equation for calculating the ramp kernel is derived. An optimization method is used with the proposed ramp kernels for inversely estimating the aquifer parameters from drawdown due to an arbitrary unsteady pumping discharge. Unlike the prior methods, the proposed method accurately identifies the parameters even when the sampling interval for the drawdown and pumping discharge is longer than that needed for assuming a linear variation. The proposed method outperforms the prior method. Application of the proposed method is illustrated using examples.     

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.     

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.     

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.     

Steady Streaming around a Circular Cylinder near a Plane Boundary due to Oscillatory Flow

Steady streaming due to an oscillatory flow around a circular cylinder close to and sitting on a plane boundary is investigated numerically. Two-dimensional (2D) Reynolds-averaged Navier-Stokes equations are solved using a finite element method with a k-ω turbulent model. The flow direction is perpendicular to the axis of the cylinder. The steady streaming around a circular cylinder is investigated for Keulegan-Carpenter (KC) number of 2 ≤ KC ≤ 30 with a constant value of Stokes number (β) of 196. The gap (between the cylinder and the plane boundary) to diameter ratio (e/D) investigated is in the range of 0.0–3.0. The steady streaming structures and velocity distribution around the cylinder are analyzed in detail. It is found that the structures of steady streaming are closely correlated to KC regimes. The gap to diameter ratio (e/D) has a significant effect on the steady streaming structure when e/D<1.0. The magnitude of the steady streaming velocity around the cylinder can be up to about 70% of the velocity amplitude of the oscillatory flow. One three-dimensional (3D) simulation (KC = 10, β = 196, and e/D = ∞) is carried out to examine the effect of three dimensionality of the flow on the steady streaming. Although strong 3D vortices are found around the cylinder, the steady streaming in a cross section of the cylinder span is in good agreement with the 2D results.     

Modeling Unsteady Flow Characteristics of Hydropeaking Operations and Their Implications on Fish Habitat

Reservoir releases associated with energy production and flood mitigation need to be reconciled with efforts to maintain healthy ecosystems in regulated rivers. Unsteady flow phenomena caused by hydropeaking operations typically affect riverbed erosion and fish displacement. A three-dimensional hydrodynamic model is used to simulate the flow characteristics during the passage of the rising limb of an observed hydropeaking event in a gravel-bed reach of Smith River, Virginia. The calculated time-dependent water surface elevations, velocities, and shear stresses are compared with field measurements. Further, comparison based on numerical simulations of this historical and a hypothetical “staggering” hydropeaking event reveals that the latter has the capability of reducing the area subject to erosion and prolonging refugia availability for juvenile brown trout. Issues related to the adoption of either a truly dynamic modeling approach or a quasi-steady methodology for simulating unsteady flows are examined through a proposed unsteadiness flow parameter. The insights obtained from this study can assist in properly accounting for the impact of hydropeaking operations on fish habitat and instream flow management.