Unphysical Moment-Method Solutions of an Approximate Integral Equation of Electrostatics [Open Problems in Computational EM]

2017 ◽  
Vol 59 (3) ◽  
pp. 142-153 ◽  
Author(s):  
George Fikioris ◽  
Ioannis Tastsoglou ◽  
Georgios D. Kolezas ◽  
Telemachos Hatziafratis
Keyword(s):  
Integral Equation ◽  
Moment Method ◽  
Open Problems ◽  
Approximate Integral

scholarly journals Some axially symmetric potential problems

1972 ◽  
Vol 18 (1) ◽  
pp. 55-76 ◽  
Author(s):  
F. G. Leppington ◽  
H. Levine
Keyword(s):  
Integral Equation ◽  
Asymptotic Expansion ◽  
Integral Equations ◽  
Asymptotic Estimate ◽  
Axially Symmetric ◽  
Primary Interest ◽  
Symmetric Potential ◽  
Approximate Integral ◽  
Potential Problems ◽  
Higher Order Terms

AbstractSome axially symmetric boundary value problems of potential theory are formulated as integral equations of the first kind. In each case the kernel admits an expansion, for small values of a parameter of the problem, that leads to an approximate integral equation whose solution provides a direct asymptotic estimate for the physical quantity of primary interest. A manipulation of the original and modified integral equations provides an efficient formula for calculating higher order terms in the asymptotic expansion.

A goal-oriented framework for rapid integral-equation-based simulation of borehole resistivity measurements of 3D hydraulic fractures

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. D123-D133 ◽  
Author(s):  
Kai Yang ◽  
Ali E. Yılmaz ◽  
Carlos Torres-Verdín
Keyword(s):  
Integral Equation ◽  
Method Of Moments ◽  
Low Frequency ◽  
Hydraulic Fractures ◽  
Approximate Methods ◽  
Electrically Conductive ◽  
Resistivity Measurements ◽  
Integral Equation Methods ◽  
Approximate Integral ◽  
Quantities Of Interest

We have developed a goal-oriented framework for fast integral-equation-based simulation of low-frequency borehole resistivity measurements of 3D arbitrarily shaped hydraulic fractures. The framework explores the possibility of applying various approximate integral-equation methods to simulate borehole electromagnetic (EM) measurements acquired in the vicinity of 3D hydraulic fractures generated with electrically conductive proppant. It includes four approximate methods that are progressively more accurate, costly, and rigorous. Each method is used to approximate the method-of-moments solution of the integral equation and to evaluate/extract quantities of interest, e.g., bucked signals detected at receivers. When compared with rigorous fast Fourier transform (FFT)-accelerated method-of-moments solutions, the numerical results obtained with the four methods indicate the following (1) All of the approximate methods capture the main features of the quantities of interest, e.g., the shape of detected signals. (2) Different approximate methods exhibit different accuracies and efficiencies in the simulation of EM scattering from various 3D fractures. (3) The identified approximate method achieves accurate results (error [Formula: see text]) while reducing the simulation time by a factor of 2–1000 compared with the FFT-accelerated rigorous method. Thus, our approximate simulation framework is a promising candidate for evaluating the Jacobian matrix in the fast inversion of borehole EM measurements to detect and assess the geometry of 3D hydraulic fractures generated with electrically conductive proppant.

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