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.     

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.     

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.     

Stream Multiaquifer Well Interactions

In this paper, unsteady flow into a multiaquifer well due to stream stage changes and varying pumping rate is analyzed. The well is located at such a distance that the radius of influence touches the stream boundary; hence, pumping induces seepage from the stream to the aquifer. The discrete kernel approach, which is based on Duhamel’s principle, has been applied to find the interaction among stream, aquifers, and pumping well for constant as well as varying stream stage. The analytical expression for a damped sinusoidal flood wave passing in a fully penetrating stream has been used for obtaining the aquifer response. By applying image-well theory, the finite aquifer and well system has been transformed into an infinite aquifer and well system. The principle of superposition, which is applicable to a linear system, has been used to analyze the interaction processes among the three components of the system. The interaction of the stream, aquifers, and well with each other are analyzed during pumping, after stoppage of pumping, as well as during passage of a flood wave in the stream.     

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.     

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.     

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.     

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.     

Accounting for the variation in collision kerma-to-terma ratio in polyenergetic photon beam convolution

In photon beam convolution, the distribution of energy deposition about a primary photon interaction site due to charged particles set in motion at that site is represented by the primary kernel. Energy deposited due to scattered photons, bremsstrahlung, and annihilation photons is represented by the scatter kernel. As the energy deposited in each kernel voxel is normalized to the energy imparted at the interaction site, it is known as a fractional energy distribution. In terma-based convolution, where kernels are normalized to total energy imparted at the interaction site and are convolved with the terma in the dose calculation process, the sum of fractional energies contained in the primary kernel is equal to the ratio of collision kerma (Kc) to terma (T) corresponding to the energy spectrum used to generate the kernel. Since the ratio of collision kerma to terma increases with depth as the beam hardens, the integral fractional energy in a primary kernel formed for the spectrum at the surface is less than the ratio Kc/T at depth. This causes primary dose to be increasingly underestimated with depth and scatter dose to be increasingly overestimated. Single polyenergetic convolution (using polyenergetic primary and scatter kernels formed using a polyenergetic primary photon spectrum) is thus not as rigorous as if a separate convolution is performed for each energy component. The ratio of true primary dose to single polyenergetic primary dose increases almost linearly with depth and is almost equal to the Kc/T ratio. Primary and scatter dose are calculated correctly if a single polyenergetic convolution is performed in terms of Kc (for primary) and T-Kc (for scatter), where the kernels are weighted sums of monoenergetic kernels normalized to Kc and T-Kc. With this method, it is ensured that total primary energy deposited due to primary photon interactions in a unit mass at a point is equal to Kc at that point.     

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.     

Dispersion Coefficient of Streams from Tracer Experiment Data

The paper deals with a method for optimal identification of the dispersion coefficient for streams from observed concentration profiles at downstream sections (at least two sections are required) following injection of an environmentally safe tracer at an upstream section. The method makes use of the exact solution of the one-dimensional advection-dispersion equation. A reliable and accurate procedure is proposed for routing an arbitrary concentration variation further downstream using the convolution equation and pulse kernels. The new method does not require the frozen cloud approximation and avoids the error due to numerical integration of the convolution integral used for routing the concentration. It employs the objective criterion of minimum integral squared errors between observed and computed concentrations. Application of the method to field data sets shows that reliably accurate estimates of the dispersion coefficient are obtained.     

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.     

Analytical Solution for Conservative Solute Transport in One-Dimensional Homogeneous Porous Formations with Time-Dependent Velocity

The space-time variation in contaminant concentration in unsteady flow in a homogeneous finite aquifer subjected to point source contamination is analytically derived under two conditions: (1) the flow velocity in the aquifer is of sinusoidal form; and (2) the flow velocity is an exponentially decreasing function. The analytical solution is illustrated using an example. Analytical solutions are perhaps most useful for benchmarking numerical codes and solutions.     

Unsteady Solution for Well Recharge in a Low Diffusive Aquifer

Finite aquifer solution exists for the constant head in a fully penetrating well. Their use for well recharge is limited, as they do not permit simultaneous computation of unsteady wellhead pressure and variable recharge rate. In the present paper semianalytical solutions are presented for well recharge under variable head boundary condition. These solutions were developed using the method of separation of variables and Duhamel’s convolution theorem. The solution developed in the paper was verified with the Jacob-Lohman solution and subsequently validated using field data pertinent to constant-head boundary conditions. Subsequently for variable head boundary condition such an appropriate background was found missing in the literature.     

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.     

Modeling Laboratory Observations on Stream-Aquifer Interaction

A head loss concept, new to the stream-aquifer interaction, is introduced for modeling laboratory observations. Using this concept, the equation applicable to fully penetrating streams is modified to account for the head loss at the entrance. The modified model is able to explain and fit the observed laboratory data correctly. Equations are proposed to calculate the hydraulic diffusivity of aquifer and head loss at the entrance from the estimate of diffusivity obtained without considering the head loss. An optimization approach is also proposed to estimate the hydraulic diffusivity of the aquifer and head loss from the observed groundwater heads at different sections.     

Analytical Solutions for Constant-Flux and Constant-Head Tests at a Finite-Diameter Well in a Wedge-Shaped Aquifer

Neglecting the effect of well radius may lead to a significant error in the predicted drawdown distribution near the pumping well area. New analytical solutions describing aquifer responses to a constant pumping or a constant head maintained at a finite-diameter well in a wedge-shaped aquifer are derived based on the image-well method and applicable to an arbitrarily located well in the system. The solutions are useful for quantifying groundwater exploitation from a wedge-shaped aquifer and for determining the hydrogeological parameters of a wedge-shaped aquifer in inverse problems.     

Rapid Calculation of Oxygen in Streams: Approximate Delta Method

The “approximate delta method” is a simple procedure for simultaneous calculation of the stream reaeration coefficient, primary production rate, and respiration rate from a single-station stream diurnal profile of dissolved oxygen (DO). It approximates the exact graphs of results for the “delta method” reported in 1991 by Chapra and Di Toro by means of simple logistic curve-fitting approximations. The necessity of reading graphs or of obtaining numerical solutions is thereby avoided, so making it suitable for inclusion in a decision support system, particularly for streams reaeration coefficients less than 10?day?1 and for moderate photoperiods (10–14?h). Worked examples are given for streams in the USA and in New Zealand. Results are used to show that the constellation of parameters for the three fundamental processes is much more important than their individual values in calculating diurnal DO profiles. Independent measurement of the reaeration coefficient enhances the utility of the method, by enabling separate calculation of production and respiration rates.     

Approximate Analytical Solution for the Temperature Field in Welding

To determine the temperature fields associated with welding, significant efforts have been made to establish the relative merits of numerical approaches with variable material properties and the analytical approaches with constant material properties. Currently, analytical solutions are either based on the temperature field generated by a point source of heat or are developed for a finite domain derived approximately by using an infinite or semi-infinite heat kernel. Furthermore, the heat kernel applied in these solutions is derived from the Image method (for example, Nguyen’s book (Thermal Analysis of Welds, 2004)). The main problem with the heat kernels obtained from Image method is that they face the problem of singularity at and around the point where the heat source is located, and they do not satisfy the boundary condition accurately. That is why the Laplace transform method has been applied here instead of using the Image method to formulate a heat kernel that (1) converges rapidly, (2) avoids the problem of singularity, and (3) gives a good and robust approximation of the real analytic solution for the temperature field. The results obtained from the analytical solutions were compared with the results obtained from finite element method. The current work is believed to make a considerable contribution to the avoidance of previously mentioned problems by deriving a new approximate analytical solution for the temperature field on a three-dimensional finite body.