Semi-analytical solutions for the problem of the electric potential set in a borehole with a highly conductive casing

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.

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

$$\begin{aligned} a_{11}^k(\xi )&:= I_k(|\xi | r_{\tiny \hbox {int}}),&a_{12}^k(\xi )&:= - I_k(|\xi | r_{\tiny \hbox {int}}),&a_{13}^k(\xi )&:= - K_k(|\xi | r_{\tiny \hbox {int}}),\\&\\ a_{21}^k(\xi )&:= \sigma _{\tiny \hbox {int}}\ I_k'(|\xi | r_{\tiny \hbox {int}})|\xi |,&a_{22}^k(\xi )&:= - \sigma _{\tiny \hbox {lay}}\ I_k'(|\xi | r_{\tiny \hbox {int}})|\xi |,&a_{23}^k(\xi )&:= - \sigma _{\tiny \hbox {lay}}\ K_k'(|\xi | r_{\tiny \hbox {int}})|\xi |,\\&\\ a_{32}^k(\xi )&:= I_k(|\xi | r_{\tiny \hbox {ext}}),&a_{33}^k(\xi )&:= K_k(|\xi | r_{\tiny \hbox {ext}}),&a_{34}^k(\xi )&:= - K_k(|\xi | r_{\tiny \hbox {ext}}),\\&\\ a_{42}^k(\xi )&:= \sigma _{\tiny \hbox {lay}}\ I_k'(|\xi | r_{\tiny \hbox {ext}})|\xi |,&a_{43}^k(\xi )&:= \sigma _{\tiny \hbox {lay}}\ K_k'(|\xi | r_{\tiny \hbox {ext}})|\xi |,&a_{44}^k(\xi )&:= - \sigma _{\tiny \hbox {ext}}\ K_k'(|\xi | r_{\tiny \hbox {ext}})|\xi |, \end{aligned}$$

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

$$\begin{aligned} b_1^k(\xi )&:= \dfrac{A}{2\pi \ \sigma _{\tiny \hbox {int}}}\ I_k(|\xi | r_\mathrm {t}) K_k(|\xi | r_{\tiny \hbox {int}}),&b_2^k(\xi )&:= \dfrac{A}{2\pi }\ I_k(|\xi | r_\mathrm {t}) K_k'(|\xi | r_{\tiny \hbox {int}})|\xi |, \end{aligned}$$

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

$$\begin{aligned} a_{11}^k(\xi )&:= I_k(|\xi | r_{\tiny \hbox {int}}),\\&\\ a_{12}^k(\xi )&:= - K_k(|\xi | r_{\tiny \hbox {ext}}),\\&\\ a_{21}^k(\xi )&:= \dfrac{\sigma _{\tiny \hbox {int}}}{\sigma _0} \left( \dfrac{\varepsilon ^3}{2r_0}-\varepsilon ^2 \right) I'_k(|\xi | r_{\tiny \hbox {int}})\ |\xi | - \dfrac{1}{2}\left( \xi ^2 + \dfrac{k^2}{r_0^2}\right) I_k(|\xi | r_{\tiny \hbox {int}}),\\&\\ a_{22}^k(\xi )&:= \dfrac{\sigma _{\tiny \hbox {ext}}}{\sigma _0} \left( \dfrac{\varepsilon ^3}{2r_0}+\varepsilon ^2 \right) K'_k(|\xi | r_{\tiny \hbox {ext}})\ |\xi | - \dfrac{1}{2}\left( \xi ^2 + \dfrac{k^2}{r_0^2}\right) K_k(|\xi | r_{\tiny \hbox {ext}}), \end{aligned}$$

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

$$\begin{aligned} b_1^k(\xi )&:= \dfrac{A}{2\pi \ \sigma _{\tiny \hbox {int}}}\ I_k(|\xi | r_\mathrm {t}) K_k(|\xi | r_{\tiny \hbox {int}}), \\ b_2^k(\xi )&:= \dfrac{A}{2\pi \ \sigma _{\tiny \hbox {int}}}\ I_k(|\xi | r_\mathrm {t})\\&\quad \times \left( \dfrac{\sigma _{\tiny \hbox {int}}}{\sigma _0}\left( \dfrac{\varepsilon ^3}{2r_0}-\varepsilon ^2\right) K'_k(|\xi | r_{\tiny \hbox {int}})\ |\xi | - \dfrac{1}{2} \left( \xi ^2+\dfrac{k^2}{r_0^2} \right) K_k(|\xi | r_{\tiny \hbox {int}}) \right) . \end{aligned}$$

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

$$\begin{aligned} {a}_{11}^k(\xi )&:= I_k(|\xi | r_0),&{a}_{12}^k(\xi )&:= - \dfrac{\sigma _{\tiny \hbox {int}}}{\sigma _{\tiny \hbox {ext}}}\ K_k(|\xi | r_0),\\ {a}_{21}^k(\xi )&:= I'_k(|\xi | r_0)|\xi |,&{a}_{22}^k(\xi )&:= - K'_k(|\xi | r_0)|\xi | + \dfrac{1}{\varepsilon ^2}\dfrac{\sigma _0}{\sigma _{\tiny \hbox {ext}}}\left( \xi ^2 + \dfrac{k^2}{r_0^2}\right) \ K_k(|\xi | r_0), \end{aligned}$$

and ${\mathbf {b}}^k_{\tiny \hbox {kau}}:={\mathbf {b}}^k_{\tiny \hbox {kau}}(\xi ):=(b_i^k(\xi ))_{i=1,2}$ are given by

$$\begin{aligned} {b}_1^k(\xi )&:= \dfrac{A}{2\pi \ \sigma _{\tiny \hbox {int}}}\ I_k(|\xi | r_\mathrm {t})K_k(|\xi | r_0),&{b}_2^k(\xi )&:= \dfrac{A}{2\pi \ \sigma _{\tiny \hbox {int}}}\ I_k(|\xi | r_\mathrm {t})K_k'(|\xi | r_0) |\xi |. \end{aligned}$$

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

$$\begin{aligned} x^2 \frac{d^2 y}{dx^2} + x \frac{\mathop {}\!\mathrm {d}y}{\mathop {}\!\mathrm {d}x} + (x^2-k^2) y = 0. \end{aligned}$$

According to the Frobenius method, it is possible to obtain the following series expression for function $J_k$

$$\begin{aligned} J_k (x) = \sum _{j=0}^\infty \frac{(-1)^j}{j! \varGamma (j+k+1)} \left( \frac{x}{2} \right) ^{2j+k}, \end{aligned}$$

where $\varGamma$ represents the Gamma function. Function $J_k$ can then be used to define $Y_k$:

$$\begin{aligned} Y_k (x) = \lim _{l \rightarrow k} \frac{J_l(x) \cos (l \pi ) - J_{-l} (x)}{\sin (l\pi )}. \end{aligned}$$

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

$$\begin{aligned} x^2 \frac{\mathop {}\!\mathrm {d}^2 y}{\mathop {}\!\mathrm {d}x^2} + x \frac{\mathop {}\!\mathrm {d}y}{\mathop {}\!\mathrm {d}x} - (x^2+k^2) y = 0. \end{aligned}$$

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

$$\begin{aligned}&I_k (x) = \lim _{l \rightarrow k} i^{-l} J_l(ix),\\&K_k (x) = \lim _{l \rightarrow k} \frac{\pi }{2} \frac{I_{-l} (x)-I_{l} (x)}{\sin (l\pi )}. \end{aligned}$$