Triple Integral Equations

Cooke, J. C.

doi:10.1093/qjmam/16.2.193

Cited by 95 publications

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“…These problems are often solved by transforming the governing differential equations into the integral Fredholm equations of the first kind which are commonly ill-posed [see, e.g., Dagan, 1978;Goggin et al, 1988;Cole and Zlotnik, 1994;Cassiani and Kabala, 1998]. Alternatively, one can transform the governing equations into a system of the well-posed Fredholm equations of the second kind [Cooke, 1963;Ufliand, 1977]. In turn, the auxiliary functions given by the solutions of these Fredholm equations are used to define the solution of the original differential equation.…”

Section: Paper Number 2000wr900178mentioning

confidence: 99%

Kinematic structure of minipermeameter flow

Tartakovsky¹,

Moulton²,

Zlotnik³

2000

Water Resources Research

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Abstract. Minipermeameters are rapidly becoming a popular tool for collecting localized measurements of permeability in both laboratory and field studies. While one of the main advantages of minipermeameters is their ability to collect data on various support volumes, there have been only limited attempts to analyze their size and geometry. We define the support volume of minipermeameter measurements as a region containing 90% of the total gas flow, i.e., a region bounded by the 10% streamline. Using our new semianalytical solutions for the Stokes' stream function, we demonstrate that the support volume has a shape of the semitoroid adjacent to the sample surface. Hence there is a blind spot directly below the minipermeameter, which is not probed by the measurement. We demonstrate that the support volume of the minipermeameter measurements decreases with the tip-seal's ratio (a ratio of the inner tip-seal radius to the outer tip-seal radius), while the size of the corresponding blind spot increases. IntroductionDelineation of the spatial distribution of permeability in water-and oil-bearing formations is one of the major challenges in hydrogeology and petroleum engineering. Specifically, this is an ill-posed inverse problem, and hence it is inherently difficult to solve. Mathematical models that provide a means to extract permeability data indirectly from experimental measurements of dependent quantities (e.g., pressure head and flow rates) do so by defining a related well-posed problem through some form of regularization. The necessary presence of this regularization, which may not be stated explicitly, is likely a critical factor in the recent debate over the scale dependence of permeability measurements. Consequently, there is a growing interest in experimental procedures that possess well-defined regions of investigation or support volumes.Minipermeameters seem well suited for this purpose because they induce a localized flow by injecting gas into a sample through a small tip seal. Although these devices were first described by Dykstra and Parsons [1950], it was not until recently that Goggin et al. [1988] proposed a mathematical model for the application of the minipermeameter to localized permeability measurements. In particular, for the case of steady state gas flow, Goggin et al. [1988] introduced a coefficient of proportionality into an integral form of Darcy's law. Dubbed the geometric factor, this coefficient allowed the permeability to be inferred from the injection rate and the corresponding gas pressure. The experimental aspect of this work focused on measuring the permeability of core samples; thus the support volume was defined by the sensitivity of the geometric factor to the sample size. Specifically, the support volume was deter-

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Section: Paper Number 2000wr900178mentioning

confidence: 99%

Kinematic structure of minipermeameter flow

Tartakovsky¹,

Moulton²,

Zlotnik³

2000

Water Resources Research

View full text Add to dashboard Cite

show abstract

“…Making use of the work discussed by Gubenko and Mossakovskii [1], Collins [2] first solved the triple series equations and replaced the triple series equations by two sets of dual series equations which could be easily solved. Later on by using the method of Cooke [3], Srivastava [5] and Lowndes [6] solved the triple series equations involving Jacobi polynomials. The general solutions given by [5] and [6] have less application in physical problems.…”

Section: Introductionmentioning

confidence: 99%

“…Solution of eqs. (1) to (3) given by the authors [1--3] in different forms were complicated. The purpose of this paper is to obtain solution of eqs.…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this paper is to obtain solution of eqs. (1) to (3) in a simplified form and to get a power series solution for small parameters a and b 1 .…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we have solved the triple series equations (1) to (3) by modifying the method of Cooke [3] for solving triple integral equations of Bessel functions. We reduce the solution of the triple series equations to two simultaneous Fredholm integral equations with kernels in simple forms.…”

Section: Introductionmentioning

confidence: 99%

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The Elementary Solution of Triple Series Equations Involving Series of Legendre Polynomials and their Application to an Electrostatic Problem

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Hankel Transforms

Rahman¹

1999

Wiley Encyclopedia of Electrical and Electronics Engineering

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The sections in this article are The Hankel Transform Some Elementary Properties of Hankel Transforms The Hankel Transforms of Derivatives of a Function Relation Between Fourier and Hankel Transforms Parseval's Relation For Hankel Transforms The Hankel Operator The Erdelyi–Kober Operators of Fractional Integration Beltrami‐Type Relations Dual Integral Equations Involving Hankel Transforms Triple Integral Equations Involving Hankel Transforms Quadruple Integral Equations Involving Hankel Transforms Miscellaneous Compendium of Basic Formulas Suggested Further Reading Acknowledgments

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Triple Integral Equations

Cited by 95 publications

References 0 publications

Kinematic structure of minipermeameter flow

Kinematic structure of minipermeameter flow

The Elementary Solution of Triple Series Equations Involving Series of Legendre Polynomials and their Application to an Electrostatic Problem

Hankel Transforms

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