Heat Flow in an Infinite Solid Bounded Internally by a Cylinder

Smith, Lloyd P.

doi:10.1063/1.1710319

Cited by 35 publications

(17 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, this is also known as a constant drawdown variable discharge test. This is analogous to the heat flow equation solved by Smith (1937) for heat-flow in an infinite solid bounded by an internal cylinder.…”

Section: Jacob-lohman Methods For a Flowing Well In A Confined Aquifermentioning

confidence: 99%

Documentation of spreadsheets for the analysis of aquifer-test and slug-test data

Halford¹,

Kuniansky²

2002

Open-File Report

View full text Add to dashboard Cite

PrefaceThis report documents several spreadsheets that have been developed for the analysis of aquifer-test and slug-test data. Each spreadsheet incorporates analytical solution(s) of the partial differential equation for ground-water flow to a well for a specific type of condition or aquifer. The spreadsheets were written in Microsoft Excel version 9.0. Use of trade names does not constitute endorsement by the U.S. Geological Survey (USGS). The spreadsheets have been tested for accuracy using datasets from different aquifer tests or generated from the analytical solution. If users find or suspect errors with these spreadsheets, please contact the USGS.Every effort has been made by the USGS or the United States Government to ensure the spreadsheets are error free. Despite our best efforts, the possibility exists that there are errors in the spreadsheets. The distribution of the spreadsheets does not constitute any warranty by the USGS, and no responsibility is assumed by the USGS in connection therewith.

show abstract

Section: Jacob-lohman Methods For a Flowing Well In A Confined Aquifermentioning

confidence: 99%

Documentation of spreadsheets for the analysis of aquifer-test and slug-test data

Halford¹,

Kuniansky²

2002

Open-File Report

View full text Add to dashboard Cite

show abstract

“…The earliest studies of the problem (1.1), such as Nicholson [1], Goldstein [2], Smith [3], Carslaw & Jaeger [4] and Titchmarsh [5, § 10.10], gave the solution for this problem and for situations with different boundary conditions, but did not investigate further. Jaeger [6] appears to have been the first to examine the integral in (1.4), and gave the asymptotic behaviour I ∼ 1/(ln t − 2γ) in order to calculate the heat flux from the inner boundary.…”

Section: S G Llewellyn Smithmentioning

confidence: 99%

The asymptotic behaviour of Ramanujan's integral and its application to two-dimensional diffusion-like equations

Smith

2000

Eur. J. Appl. Math

View full text Add to dashboard Cite

The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

show abstract

“…Assuming uniform hydrostatic initial heads and a sudden, constant drawdown at the well, an analytical expression for the resulting transient discharge rates was first published by Jacob and Lohman (1952) who applied the heat conduction solution of Smith (1937) to groundwater dynamics.…”

Section: Classical Solutionmentioning

confidence: 99%

A simple solution to tunnel or well discharge under constant drawdown

Perrochet

2004

Hydrogeol J

View full text Add to dashboard Cite

A simple analytical formula is developed to calculate transient discharge inflow rates into a tunnel or a well under constant drawdown. The agreement with the classical, but cumbersome diffusion-equation-based solution of Jacob and Lohman is excellent throughout the range of dimensionless times. By using only a straightforward logarithmic function, this explicit solution may therefore be used with great computational benefits in practice, and also when further mathematical manipulations such as differentiation or integration are required.Resumen Una fórmula analítica sencilla fue desarrollada para calcular el grado de descarga transitoria hacia un tfflnel, o un pozo, bajo condición de un abatimiento constante. Existe una concordancia excelente, desde el comienzo hasta el final, en el rango de los tiempos adimensionales, con la solución clµsica pero complicada, de Jacob y Lohman, basada esta fflltima en la ecuación de difusión. Solamente mediante el uso de una función logarítmica simple, esta solución explícita puede por tanto ser usada en la prµctica con grandes ventajas computacionales, y tambiØn cuando se necesitan manipulaciones matemµticas adicionales, tales como diferenciación o integración.RØsumØ On a dØveloppØ une formule analytique simple pour le calcul du flux infiltrØ en rØgime transitoire dans un tunnel ou un puits en supposant le rabattement constante. Les rØsultats sont en accord avec la solution plus compliquØe de Jacob-Lohman de l'Øquation de diffusion, sur tout l'intervalle de temps adimensionel considØrØ. En utilisant une fonction logarithmique ce solution explicite peut Þtre utilisØe sans un grand effort de calcul dans la pratique courante, ainsi que dans les situations o il est nØcessaire à dØriver ou intØgrer l'expression du rabattement. Keywords Well hydraulics · Tunnel discharge Classical SolutionConfined horizontal flow towards a fully penetrating well in a semi-infinite, homogeneous aquifer of constant thickness is classically described by the radial form of the diffusion equation. Assuming uniform hydrostatic initial heads and a sudden, constant drawdown at the well, an analytical expression for the resulting transient discharge rates was first published by Jacob and Lohman (1952) who applied the heat conduction solution of Smith (1937) to groundwater dynamics.Consider the radial form of the diffusion equationwith the initial and boundary conditions sðr; 0Þ ¼ 0;sðr o ; tÞ ¼ s o ;sð1;tÞ ¼ 0 ð2Þwhere the symbols stand for aquifer transmissivity (T), storage coefficient (S), time (t), radial coordinate (r), well radius (r o ), drawdown (s) and drawdown at the well (s o ). The above problem was analysed by Smith who resorted to Green's functions and integral transforms but could only find a solution for @s=@r at the origin. Using this result, Jacob and Lohman deduced the flow rate at the origin (i.e. into the well) aswithwhere a is dimensionless time, J o and Y o are first and second kind zero-order Bessel functions respectively, and u is a dummy variable. The above solution has bec...

show abstract

Heat Flow in an Infinite Solid Bounded Internally by a Cylinder

Cited by 35 publications

References 1 publication

Documentation of spreadsheets for the analysis of aquifer-test and slug-test data

Documentation of spreadsheets for the analysis of aquifer-test and slug-test data

The asymptotic behaviour of Ramanujan's integral and its application to two-dimensional diffusion-like equations

A simple solution to tunnel or well discharge under constant drawdown

Contact Info

Product

Resources

About