Abstract
Highly conductive thin casings pose a great challenge in the numerical simulation of well-logging instruments. Witty asymptotic models may replace the presence of casings by impedance transmission conditions in those numerical simulations. The accuracy of such numerical schemes can be tested against benchmark solutions computed semi-analytically in simple geometrical configurations. This paper provides a general approach to construct those benchmark solutions for three different models: one reference model that indeed considers the presence of the casing; one asymptotic model that avoids computations in the casing domain; and one asymptotic model that reduces the presence of the casing to an interface. Our technique uses a Fourier representation of the solutions, where special care has been taken in the analytical integration of singularities to avoid numerical instabilities.
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Notes
See https://www.mathworks.com/help/matlab/ref/polyfit.html for further reference.
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Funding
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS). G. Pinochet has received funding from the National Agency for Research and Development (ANID), Scholarship Program, Beca de Magíster Nacional 2021 - 22210496. A. Erdozain, V. Péron and I. Muga also have received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 644602 (GEAGAM). I. Muga acknowledges support from the project DI INVESTIGACIÓN INNOVADORA INTERDISCIPLINARIA PUCV 2021 N\(^\mathrm{o}\)039.409/2021. Nanoiónica: Un enfoque interdisciplinario.
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Appendices
A Explicit expressions for the linear systems defining the coefficients of the Fourier modes
As a consequence of the transmission conditions (12) over the solution (11), we observe that \({\mathbf {A}}^k_{\tiny \hbox {ref}}:={\mathbf {A}}^k_{\tiny \hbox {ref}}(\xi ):= (a_{ij}^k(\xi ))_{i,j=1,2,3,4}\) are defined by
and \(a_{14}^k, a_{24}^k, a_{31}^k, a_{41}^k\equiv 0.\) In a similar manner, \({\mathbf {b}}^k_{\tiny \hbox {ref}}:={\mathbf {b}}^k_{\tiny \hbox {ref}}(\xi ):=(b_i^k(\xi ))_{i=1,2,3,4},\) are defined by
and \(b_3,b_4\equiv 0.\)
As a consequence of the transmission conditions (15) over the solution (14), we observe that \({\mathbf {A}}^k_{\tiny \hbox {gap}}:={\mathbf {A}}^k_{\tiny \hbox {gap}}(\xi ):= (a_{ij}^k(\xi ))_{i,j=1,2}\) are defined by
In a similar fashion, \({\mathbf {b}}^k_{\tiny \hbox {gap}}:={\mathbf {b}}^k_{\tiny \hbox {gap}}(\xi ):=(b_i^k(\xi ))_{i=1,2},\) are defined by
Finally, as a consequence of the transmission conditions (17) over the solution (14), we observe that \({\mathbf {A}}^k_{\tiny \hbox {kau}}:={\mathbf {A}}^k_{\tiny \hbox {kau}}(\xi ):= (a_{ij}^k(\xi ))_{i,j=1,2}\) are defined by
and \({\mathbf {b}}^k_{\tiny \hbox {kau}}:={\mathbf {b}}^k_{\tiny \hbox {kau}}(\xi ):=(b_i^k(\xi ))_{i=1,2}\) are given by
B Bessel functions
We remind the definitions of the Bessel functions of first kind and second kind, and the modified Bessel functions.
Definition 1
Following (Abramowitz and Stegun 1964, Sect. 9.1), for \(k\in {\mathbb {Z}}\), the Bessel functions of first kind \(J_k\) and second kind \(Y_k\) are defined as two independent solutions to the Bessel equation
According to the Frobenius method, it is possible to obtain the following series expression for function \(J_k\)
where \(\varGamma \) represents the Gamma function. Function \(J_k\) can then be used to define \(Y_k\):
Definition 2
Following (Abramowitz and Stegun 1964, Sect. 9.6), for \(k\in {\mathbb {Z}}\), the modified Bessel functions of first kind \(I_k\) and second kind \(K_k\) are defined as two independent solutions to the modified Bessel equation
Finally, we can obtain the expressions of the modified Bessel functions \(I_k\) and \(K_k\) from the definitions of the Bessel functions given in Definition 1
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Erdozain, A., Muga, I., Péron, V. et al. Semi-analytical solutions for the problem of the electric potential set in a borehole with a highly conductive casing. Int J Geomath 13, 6 (2022). https://doi.org/10.1007/s13137-022-00197-3
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DOI: https://doi.org/10.1007/s13137-022-00197-3