Volume 12, Issue 7 p. 1099-1106
Research Article
Open Access

Efficient model based on genetic programming and spline functions to find modes of unconventional waveguides

Alexandre Ashade L. Cunha

Corresponding Author

Alexandre Ashade L. Cunha

Applied Computational Intelligence Lab (ICA), PUC-RJ, 615L/DEE, Rio de Janeiro, 22451–900 RJ, Brazil

Search for more papers by this author
Marco Aurélio Pacheco

Marco Aurélio Pacheco

Applied Computational Intelligence Lab (ICA), PUC-RJ, 615L/DEE, Rio de Janeiro, 22451–900 RJ, Brazil

Search for more papers by this author
First published: 07 March 2018

Abstract

The contribution of this work is twofold: the authors developed an accurate model to solve the vector wave equation of radially-layered inhomogeneous waveguides based on spline function expansions and automated grid construction by genetic programming, and thenemployed this model to analyse the propagation of electromagnetic waves within oil wells. The developed model uses a spline expansion of the fields to convert the wave equation into a quadratic eigenvalue problem where eigenvectors represent the coefficients of the splines and eigenvalues represent the propagation constant of the eigenmode. The present study compared the proposed model using the classical winding number technique. The results obtained for the first eigenmodes of a typical oil well geometry were more accurate than those obtained by the winding number method. Moreover, the authors model could find a larger amount of eigenmodes for a fixed azimuthal parameter than the standard approach.

1 Introduction

Predicting the physics of electromagnetic (EM) waves in oil fields has been a topic of interest. The applications include oil heating, drill stem testing, and more recently, wi-fi downhole sensors. While the first applications are more frequent in the industry, the last one is of particular interest in the present article because there is a growing interest in building a reliable communication system between the downhole sensors and the platform.

The literature has proposed some approaches for wireless downhole communication, most of them using the well as an antenna that radiates EM signals through rock formations [[1][5]]. In these approaches, the methods of propagation usually solve the radiation problem in a layered inhomogeneous medium for frequencies of few Hertz, which introduces an obstacle for duplex digital communication links. These approaches usually suffer from numerical issues, due to the discretisation of a metallic structure that is both thin and very long.

Another method is to propagate the EM signal within the region between the production string and the formation [[6]]. This area is usually known as the annular region of the well. In this region, the wave is guided using a transverse EM mode at a low frequency to minimise losses. However, this approach suffers from scattering of the fields on the equipment usually placed within this region of the well.

We are interested in predicting the wave modes and attenuation losses of an EM wave guided by the production string, as opposed to the propagation through rock formations or the annular region. Propagating through the production string is theoretically better because it should have less attenuation since the signal is confined to a metallic casing.

In our approach, the production string behaves like a circular waveguide, filled with oil. Usually, the oil has a non-constant conductivity in the direction of propagation due to temperature variations. Additionally, solid CaCO3 incrustations are common on the wall of the pipe string [[7]], causing a radial inhomogeneity on the waveguide (Fig. 1).

Details are in the caption following the image

Slice of the pipe string model at a fixed z coordinate. The oil flows into the outer tube, whereas the urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0002 incrustation stays adhered to the inner wall of the pipe

In [[8]], the authors studied propagating EM waves within the production string, considering only the oil inhomogeneity. In that paper, the authors divided the production string into equal segments and assumed that each segment is a homogeneous medium. After that, by applying boundary conditions between adjacent sections, the authors of [[8]] defined an approximated solution for the original inhomogeneous problem. This approach, however, does not consider the CaCO3 incrustations on the walls of the production string.

On the contrary, the present paper considers both the radial inhomogeneity and the axial inhomogeneity, that is, we assume that the oil is an inhomogeneous medium and there are two dielectrics radially separated: oil and incrustations. Therefore, we are interested in finding the modes of propagation and the attenuation of the waves in a radially layered and axially inhomogeneous waveguide, which models the production pipe of the oil well. This waveguide has two layers of the dielectric material in the radial direction, and these layers might be inhomogeneous in the axial direction.

The problem of finding modes of unconventional waveguides has been studied in the literature. Some authors [[9], [10]] found the modes of a radially layered circular waveguide using a semi-analytical procedure: the vector wave equation is solved analytically to obtain the general solution, and then boundary conditions are applied to find the particular parameters of the analytical solution. Usually, this procedure results in a transcendental equation on a single complex variable, and numerical methods [[11], [12]] become necessary to find a solution. We refer to it as the classical method.

Despite its analytical nature, the conventional method does not behave well numerically because finding a (possibly infinite) number of roots of a non-linear equation in the complex plane is not an easy task. Usually, given a fixed azimuthal dependence, five or more modes are required to do mode matching at adjacent waveguide sections, and the numerical methods to find roots of a non-linear equation might struggle to find this amount of roots accurately. Another drawback of the classical method is the need to specify a rectangular interval in the complex plane to look for the roots. This requirement forces the engineer to know the approximate position of the modes in the complex plane, which may be ineffective. Finally, the majority of the available methods to find the roots of a non-linear equation rely on the argument principle to count the number of roots within a closed domain in the complex plane. However, since the computation of the number of roots inside a domain involves a numerical integration, sometimes poor convergence may interfere on the number of roots computed and, as a consequence, the final result may have missing modes.

We propose a new method to calculate the modes, based on spline functions, which converts the problem of solving the vector wavefunction subject to boundary conditions into a problem to find the first eigenvalues of a sparse matrix. The eigenvalues are the modes of propagation for a given azimuthal dependence, and they are guaranteed to be sorted by increasing attenuation. Therefore, our method can determine the first N least-attenuated modes of a radially layered circular waveguide given the azimuthal parameter without any primary guess, which is very useful to mode-matching applications. In particular, this model is helpful herein to do mode matching of the adjacent sections of the production pipe, as we show in the next section.

This paper is organised as follows:
  • Section 2’ depicts both the classical method and the proposed method to compute the modes of a radially layered circular waveguide. Then, it shows how to split the inhomogeneous waveguide into smaller homogeneous segments, and describes how to perform mode matching between adjacent waveguide segments to obtain the final solution of the wave propagating within the inhomogeneous waveguide.

  • Section 3 compares the classical method and our proposed approach on a typical oil well scenario for a high wave frequency and a low wave frequency.

  • Section 4 outlines our findings and reveals possible applications of our method outside the oil and gas area.

2 Description and methodology

This section approaches the problem of solving the EM wave equation in inhomogeneous waveguides that are radially layered. We divide this section into three subsections: (1) the mathematical formulation of the problem, (2) the classical approach, and (3) our model.

Moreover, we also apply the models to the context of oil wells wireless telemetry, because they behave very similar to circular waveguides. Therefore, in this paper the oil and gas terminology is used to explain how parts of the oil wells are being modelled as waveguides.

2.1 Mathematical formulation

This work uses a cylindrical coordinate system urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0004 with unit vectors urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0006, respectively. We represent the production string of the oil wells as an infinite cylindrical waveguide with a lossless wall, filled with crude oil and CaCO3. The crude oil fills the space having urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0008, whereas the CaCO3 incrustation fills the space separating the petroleum and the pipe wall urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0010, where a is the radius of the production string (Fig. 1).

Consider the time-harmonic Maxwell equations for a linear, source-less, and isotropic medium:
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0012(1)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0014(2)
In (1) and (2), we assume a time dependence of the form urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0016, where urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0018, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0020, is the angular frequency, t, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0022, is the time variable, and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0024. Additionally, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0026 is the magnetic permeability of the vacuum, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0028 is the electrical permittivity, and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0030 is the static electrical conductivity. Finally, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0032 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0034 are, respectively, the divergence and curl operators.
This work is interested in inhomogeneities in the conductivity and the electrical permittivity of the medium. Therefore, all the parameters are complex constants for space and time, with exception of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0036 which is assumed to be a function of the z-coordinate. Hence, by applying the curl operator on both sides of (1), we have
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0038(3)
where
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0040(4)
The electric and magnetic fields must satisfy (2) and (3). The authors of [[13]] showed that
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0042(5)
are a pair of linearly independent solutions of the vector wave equation urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0044 having constant urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0046 provided that the scalar functions of space u and v satisfy the scalar Helmholtz differential equation in three dimensions
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0048(6)
Since urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0050 for every scalar field urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0052, any linear combination of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0054 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0056 has a vanishing divergence. Moreover, the authors of [[13]] argue that every solution of the wave equation featuring an identically vanishing divergence is a linear combination of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0058 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0060. Hence, we write the solutions of (3) as
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0062(7)
and we define the transverse magnetic and transverse electric vector fields as
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0064(8)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0066(9)
Note that equations in (9) are derived from (1).

At this point, one should pay attention that the proposed solution has a vanishing divergence. Thus it does not satisfy (2). However, we assume, for now, that the wave travels a small distance in the z-direction, so it can be assumed that urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0068 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0070 are approximately constant and consequently, the divergence of the electric field vanishes. Thus, the forthcoming paragraphs adopt the solution of (7), that is, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0072 is not a function of the space variables. After that, we present how to use this solution to the case which urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0074 is a function of z.

Now, we define urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0076 and observe that urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0078 and, from (5), urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0080. Hence, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0082. Similarly, we define urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0084. Therefore, equations in (9) read
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0086(10)
After expansion, the electric field urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0088 reads
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0090(11)
Subscripts in [] represent derivatives. From (5), (6) and (8) and the expression of the Laplacian in cylindrical coordinates
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0092(12)
Similarly, we express the fields in (10) as
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0094(13)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0096(14)
Again, these solutions are true assuming constant urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0098. For solving the case in which urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0100 is a function of the z-coordinate, we divide the waveguide into small sections along the axial direction (see Fig 2), and solve for constant urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0102 in each section. Moreover, the value of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0104 of each section is computed from its value at the centre of the section so that adjacent sections may have different values of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0106. Therefore, this model embeds a piecewise approximation of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0108 that gets more accurate as the length of each section diminishes, at the expense of requiring more sections to represent the full waveguide extent.
Details are in the caption following the image

Inhomogeneous waveguide split into small segments. Each segment is assumed to be small enough to be considered a homogeneous medium. The EM wave propagates in the z-direction

The next subsections show how the classical and the proposed spline-based model solve the mathematical problem described in this subsection. In particular, they show how to apply the boundary conditions between adjacent materials to compute the modes of propagation and then between adjacent waveguide sections to do mode matching. As the following subsections show, the main difference between the classical and the proposed approaches is how the modes are computed. While the classical approach uses the exact analytical solution and computes the propagation constant of the modes by numerically finding the roots of a transcendental equation, the proposed method approximates the electric field using splines, thus converting the original problem to the well-known problem of finding the smallest eigenvalues of a sparse matrix.

2.2 Classical method

The presentation of the classical method is divided into three parts:
  1. Formulation of the transcendental equation whose solution is the set of modes of a particular homogeneous segment of the waveguide.

  2. Description of a numerical algorithm to solve this transcendental equation.

  3. Description of how to do mode matching between adjacent waveguide sections.

Therefore, each of the following sections depicts these topics in detail.

2.2.1 Formulation of the modes equation

The employed solution of (6) is given by [[14], p. 113]
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0110(15)
where
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0112(16)
Moreover, the coefficients of (15) are complex valued. Finally, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0114 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0116 are the Bessel functions of the first and second kind defined for any non-negative integer m.
Therefore, from (15) we deduce that urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0118, and by using (16) we rewrite (12) as
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0120(17)
Similarly, we express the magnetic fields as
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0122(18)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0124(19)
provided that u is similar to (15).

Recall that we are particularly interested in understanding the propagation of waves in an oil well, as shown in Fig. 1. Thus, we apply boundary conditions to the electric and magnetic fields in the oil, in the incrustation and the interface between them. We use ‘ o ’ and ‘ c ’ superscripts to denote coefficients for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0126 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0128, respectively.

To make the fields non-singular everywhere, we should get rid of the coefficients of the Bessel function of the second kind in the expressions of u and v for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0130:
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0132(20)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0134(21)
Additionally, the perfectly conducting wall of the waveguide forces the tangential electric fields to vanish at urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0136. For urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0138, we use (11) to conclude that urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0140 must be null at urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0142. Similarly, from (17) we conclude that both v and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0144 should vanish at urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0146, so it suffices to let urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0148 at urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0150. Therefore, for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0152
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0154(22)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0156(23)
where ʹ indicates a differentiation w.r.t. the argument
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0158(24)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0160(25)
Finally, we should guarantee the continuity of the tangential electric and magnetic fields at the interface between the two media. To do so, we assume urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0162, and we enforce the equality at urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0164 of the components urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0166 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0168 of the fields of (7) in oil and incrustation
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0170(26)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0172(27)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0174(28)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0176(29)
where
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0178(30)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0180(31)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0182(32)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0184(33)
As a result, we have a linear and homogeneous system of equations in the variables urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0186 that should have non-trivial solutions. This requirement leads to the non-linear equation
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0188(34)
We are using urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0190 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0192. Additionally, we let urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0194 and define H and G as
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0196(35)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0198(36)
Since urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0200, we define urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0202 and then
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0204(37)
Therefore, by solving the non-linear system of (34) and (37), we find the values of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0206. In the next section, we address the problem of solving (34) numerically.

2.2.2 Numerical solution of the modes equation

There are numerous approaches to solving (34) in the complex plane. In [[12]], the authors developed a branch-and-bound method based on two root tests and the Newton–Raphson method to enumerate all isolated roots in a rectangular domain. However, our equation might have multiple roots, so we believe this approach might not be ideal.

In [[11]], the authors developed a method for enclosing all the zeros of a complex function using a combination of the argument principle technique and the Newton–Raphson algorithm. In [[10]], the authors used a similar approach to find the modes of cylindrical tunnels. The drawback of these approaches is the need for numerical integration in the complex plane during the algorithm execution. We use the method described in [[10], [11]].

Although it might seem straightforward, this approach has many drawbacks. For example, the argument principle is not true whenever a root lies in the contour of the integral. Sometimes, a domain splitting procedure may render sub-domains have roots in the contour, which requires special treatment. Another issue arises when a sub-domain has multiple roots because splitting this sub-domain does not reduce the number of roots in it, which may lead to infinite subdivisions. In this case, heuristics must be applied to detect root multiplicity and, therefore, stop splitting the domain. Again, for more information, please refer to [[10], [11]].

2.2.3 Mode matching between adjacent sections

After solving (34) for a fixed value of m, there is an enumerable set of values for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0208 such that the linear system of (26)–(29) has non-trivial solutions. Given a particular solution urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0210, and setting urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0212, the equations for u and v read
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0214(38)
Therefore, the fields are given by (11) and (17)–(19)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0216(39)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0218(40)
where
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0220(41)
Then, for a fixed value of m, the electric field and the magnetic field read
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0222(42)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0224(43)
where urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0226 is the urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0228 solution of the modes’ equation. The coefficients urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0230 multiply the waves that propagate in the urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0232 direction, whereas the coefficients urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0234 multiply the waves that propagate in the urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0236 direction. Herein, we assume urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0238, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0240 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0242.
To match the modes between two adjacent waveguide sections at a fixed urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0244, the tangential components of the fields should match every value of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0246 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0248. If the superscripts urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0250 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0252 denote, respectively, the regions urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0254 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0256, then the mode matching boundary condition reads
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0258(44)
Then, combining (43) and (44) yields
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0260(45)
for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0262, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0264, and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0266. We proceed considering only the electric field equation.
By applying the inner product
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0268(46)
In (47), we have
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0270(47)
where
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0272(48)
The system of (47) is solvable if we truncate the summations to a finite number of modes. By taking the first M modes, (47) read
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0274(49)
where urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0276 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0278 are matrices determined by the coefficients of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0280 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0282, and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0284. Therefore, by inverting urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0286, we have
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0288(50)
which gives the relationship between the waves of two adjacent sections of the waveguide.

Hence, by splitting the propagation direction into small uniform subsections, we can mode match adjacent subsections and, consequently, solve the inhomogeneous problem by reducing it to many homogeneous problems.

2.3 Proposed spline-based model

The proposed model depicts the radial dependency of the fields using spline functions. Therefore, instead of solving the wave equation analytically, we solve it numerically. More precisely, the original partial differential equation is converted to a discrete eigenvalue problem, whose eigenpairs represent the eigenmodes. To address inhomogeneous problems, we employ the split-and-mode-match approach presented above, where the waveguide is split into segments small enough to be considered homogeneous, and the propagation problem is solved for each segment. Then, we apply boundary conditions to mode match adjacent sections.

The following sections describe how the waves are represented by spline functions and show how to solve the resulting quadratic eigenvalue problem.

2.3.1 Spline function representation of the waves

In the context of this work, spline functions are piecewise polynomial functions having a specific degree of smoothness [[15], [16]]. The smoothness usually requires the continuity of the function and some of its derivatives at every point.

Since spline functions are piecewise functions, they are defined over a discrete grid of points within the domain of the approximated function. The choice of this grid is, a priori, an arbitrary decision, but the grid usually has a significant influence on the approximation accuracy. Therefore, we employ a computational intelligence methodology, known as genetic programming [[17]], to optimally define the grid points given the required number of points. The following paragraphs define the equations of the spline model, how to determine the spline coefficients and how to employ genetic programming to set the optimal grid.

In our model, the solution of (6) is represented by
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0290(51)
where the functions of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0292 are spline functions defined for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0294. To specify the spline functions, we use a set urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0296 of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0298 distinct values of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0300, and we interpolate cubic polynomials between adjacent values of this set. Additionally, we require continuity up to the second derivative on every point of U. Therefore, the function f reads
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0302(52)
and we require that
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0304(53)
Similarly, the definition of g uses a similar approach that is omitted herein for brevity. After defining f and g, (51) are plugged into the original partial differential equations in (6), resulting in the following ordinary differential equation for both f and g:
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0306(54)
Therefore, in addition to the spline requirements of (53), we also ensure (54) for every point of the set U. This forces the spline function to be an approximate solution for every point between adjacent grid points. The total number of equations is urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0308, because there are urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0310 equations to guarantee the continuity at urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0312 ((53)) and there are urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0314 Bessel equations ((54)) to satisfy at the points urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0316. Next, the same reasoning is applied to model g as a spline function.
The adopted model uses a pair of f, g functions for each medium. In our case study, we utilise urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0318 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0320 for oil urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0322 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0324 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0326 for incrustation urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0328. Then, we apply boundary conditions to enforce the continuity of the tangential fields at urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0330 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0332 (see Fig. 1). At urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0334, the perfectly conducting wall of the waveguide forces the components urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0336 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0338 of the electric field in the incrustation to vanish. Using (11), (12), (51) and (53), this requirement leads to
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0340(55)
where urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0342 is the number of points used in the incrustation. Moreover, we use in the incrustation urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0344 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0346. The superscript c indicates the grid points of the incrustation.
Similarly, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0348 is the number of grid points used to define the spline urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0350 for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0352. At urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0354, the continuity of the tangential components of the EM field yields
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0356(56)
As a result, there is a total of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0358 equations and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0360 coefficients to determine the four spline functions urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0362, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0364, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0366, and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0368. Therefore, we choose to let the coefficients of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0370 equal to 0 in both urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0372 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0374, which makes the number of equations equal to the number of variables. Hence, (53)–(56) can be rewritten as a quadratic eigenvalue problem
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0376(57)
where x is the vector of spline coefficients to determine (the eigenvector) and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0378 is the wave constant in the urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0380 direction (the eigenvalue). urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0382, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0384, and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0386 are square matrices having urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0388 rows.
To solve (57), we rewrite it as a simple eigenvalue problem
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0390(58)
wherein I and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0392 are, respectively, the identity matrix and the zero matrix with urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0394 rows. The total number of eigenvalues is urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0396, therefore the number of modes that can be found increases with the number of grid points.

The matrices involved in (58) are usually sparse matrices, that is, most of their elements are null. Therefore, there are methods available to find a predefined number smallest (or largest) eigenvalue. For example, the implicitly restarted Arnoldi method [[18]] is used in ScyPy [[19]] and Matlab [[20]] via the function eigs. In particular, we suggest finding the first eigenvalues with the smallest imaginary parts, which represent the attenuation of the mode.

Hence, our model consists of finding the eigenvalues of (58) having the smallest imaginary part (the less attenuated modes). Then, again we apply mode matching as described in Section 2.2.3 between adjacent sections to understand how the waves propagate in the inhomogeneous medium.

We are now in a position to describe how to define the set of points urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0398 in the oil and in the incrustation using genetic programming.

Genetic programming is a technique that mimics the Darwin theory of evolution, where individuals compete for resources and the more suitable ones are selected for crossover and mutation to generate children. As the algorithm evolves, the individuals become more adapted, and the best of them approaches the optimal solution.

The degree of adaptation of an individual is measured by a fitness function, which is designed such that the best grid of points yields the smallest fitness value. Therefore, by defining a minimisation problem using a suitable fitness function, we find the best grid of points. By doing so, we expect to use considerably fewer grid points than a uniform grid while keeping the same accuracy. Using fewer grid points requires less memory and less running time, and simplifies the computation of the integrals used in mode matching, as defined in (48). Additionally, this technique becomes very useful in multidimensional grids, where the number of points are high and uniform grids are impractical. For instance, a uniform grid having 100 points per dimension and three dimensions would have a total of 1 million points, which might pose a severe computational problem.

In our model, we minimise the total error of the solution of the fundamental mode, given a fixed amount of points in the grid. The total error is defined as the Euclidean 2-norm of the l.h.s. of the wave equation (3) plus the Euclidean 2-norm of the boundary condition equations. Since the solution is written as combinations of degree-3 polynomials, the integrals involved in the Euclidean norm are straightforward and analytically computable. The solution of this problem is an optimal grid of points to find the fundamental mode. Then, we use this grid to solve for any amount of modes. Since our fitness function calculates only the first eigenvalue of (57), it is computed very quickly, which makes the whole optimisation fast and meaningful.

The next section discusses experiments and comparisons between our model and the classical model and presents results for typical configurations of oil wells.

3 Experiments and results

This section presents the comparison between the proposed spline model and the classical model to find the modes of a radially layered inhomogeneous oil well. The oil parameters follow the Cole–Cole model presented in [[21]] for the oil of type D. Additionally, from [[22], p. 323] we chose for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0400 and zero conductivity. We used urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0402 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0404.

To model the oil inhomogeneity, we considered the expression proposed in [[23]] that relates the conductivity of oil and its temperature
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0406(59)
where urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0408 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0410 are constants depending on the oil properties and T is the oil temperature. Then, by defining a temperature profile, we can determine a conductivity profile for the oil along the production pipe. In this work, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0412 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0414.

The experiments used a linear temperature profile, with oil temperature ranging from 323 K in the reservoir to 288 K in the well head. Then, we split the oil well along the z-direction into A segments (see Fig. 2), and we considered the temperature in the middle point of the segment to use in (59).

Additionally, we fixed urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0416 and simulated many values of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0418 to compare the classical and the proposed method.

This section is divided into two subsections. The first subsection presents a comparison between the classical and the proposed model to find the modes of propagation within a particular segment of the oil well. The experiments include a range of incrustation amounts and two ranges of frequencies: ultra-high frequency (UHF) and ultra-low frequency (ULF).

The second subsection is concerned with analysing the non-conventional propagation medium we defined in this study. We are interested in determining the attenuation of the fundamental mode, which is close to the fundamental TE11 mode in an ideal circular waveguide.

3.1 Mode determination of a particular well segment

In the following experiments, we determined the modes of propagation using the classical and the proposed model. To compare them quantitatively, we defined two measures: the total running time and the relative accuracy.

To define the relative accuracy, we first define the method error as the Euclidean 2-norm of the l.h.s. of the wave equation (3) plus the Euclidean 2-norm of the boundary condition equations. Then, the relative accuracy is defined as the ratio between the error of the classical method and the error of the proposed method. Therefore, if the relative accuracy is >1, then the proposed method is more accurate than the classical one. The subsequent paragraphs present the results of these experiments at UHF and ULF.

3.1.1 Mode determination at UHF

Tables 1 and 2 show the results of the wave number urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0420 computed at the frequency of 1 GHz using both the classical and the proposed spline model for an incrustation length of 5% a. Moreover, the last line of each table shows the total time spent to compute the modes. As seen, the spline model is about 35% faster than the classical model. These experiments were conducted using a fixed temperature of 298 K in (59).

Table 1. Comparison of the values of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0422 obtained using the classical and the spline methods for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0424 and f = 1 GHz. The ‘rel. acc.’ column is the relative accuracy. The bigger the relative accuracy, the more accurate the spline-based model is when compared with the classical model
Mode urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0426 (urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0428) (classical) urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0430 (urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0432) (spline model) rel. acc.
1 20.653–0.10465urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0434 20.6521–0.10465urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0436 1.01
2 0.090337–22.2483urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0438 0.092429–19.292urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0440 1.33
3 0.048623–47.301urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0442 0.048603–47.3277urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0444 1.02
4 0.031924–62.7733urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0446 0.023235–59.8709urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0448 1.10
5 0.03082–83.7544urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0450 0.030807–83.8719urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0452 1.07
6 0.020626–96.7284urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0454 0.011018–93.5797urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0456 0.99
7 0.025621–117.8677urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0458 0.025638–118.1887urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0460 1.03
8 0.01533–129.3947urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0462 0.006350–126.4481urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0464 1.46
time (s): 1.33 1.20
Table 2. Comparison of the values of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0466 obtained using the classical and the spline methods for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0468 and f = 1 GHz. The ‘rel. acc.’ column is the relative accuracy. The bigger the relative accuracy, the more accurate the spline-based model is when compared with the classical model
Mode urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0470, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0472 (classical) urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0474, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0476(proposed model) rel. acc.
1 0.039601–54.8998urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0478 0.0218–48.3901urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0480 1.21
2 0.030432–85.0917urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0482 0.030395–85.1215urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0484 1.08
3 0.020152–100.2463 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0486 0.010074–96.7512urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0488 1.13
4 0.02505–123.7617urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0490 0.025056–123.9048urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0492 1.02
5 0.014633–136.1614urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0494 0.0053842–132.9063urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0496 1.22
6 0.023956–159.2651urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0498 0.02402–159.6573urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0500 1.04
7 0.011595–170.0997urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0502 0.0038932–167.4623urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0504 1.36
8 0.024778–193.3162urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0506 0.024943–194.127urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0508 1.10
time, s 1.52 1.45

The tables also display the relative accuracy of each method. From 1 and 2, we check that the relative accuracy is usually >1, which means that the classical approach is less accurate than the spline-based method.

At higher frequencies, the proposed approach still outperforms the classical algorithm. For instance, Table 3 depicts the modes for f = 2 GHZ and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0510. The running time is nearly half the running time of the classical method, and the relative accuracy is >1.

Table 3. Comparison of the values of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0512 obtained using the classical and the spline methods for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0514 and f = 2 GHz. The ‘rel. acc.’ column is the relative accuracy. The bigger the relative accuracy, the more accurate the spline-based model is when compared with the classical model
Mode urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0516, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0518 (classical) urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0520, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0522 (proposed model) rel. acc.
1 60.5959–0.10659urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0524 60.5957–0.10659urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0526 1.01
2 49.1996–0.12058urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0528 50.4937–0.11398urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0530 1.08
3 30.5181–0.20903urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0532 30.48–0.20931urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0534 1.08
4 0.18716–31.5881urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0536 0.21699–25.1077urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0538 1.35
5 0.10645–61.8881urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0540 0.10623–62.0408urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0542 1.02
6 0.073677–79.8339urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0544 0.068221–75.9555urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0546 1.11
7 0.066933–103.3163urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0548 0.066789–103.6724urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0550 1.00
8 0.049918–117.0884urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0552 0.044129–113.8942urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0554 1.04
time, s 2.11 1.87

Table 4 shows the relative accuracy for the first eight modes computed for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0556. According to the results, the proposed technique is more accurate than the classical one. It happens because we used an optimisation approach to find the best grid of points, thus minimising the error.

Table 4. Comparison of the relative accuracy for the first eight modes computed for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0558 and f = 1 GHz. The bigger the relative accuracy, the more accurate the spline-based model is when compared with the classical model
Mode urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0560 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0562 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0564 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0566 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0568 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0570
1 1.01 1.13 1.24 1.16 0.98 1.21
2 1.33 1.10 1.07 1.22 1.01 1.08
3 1.02 0.99 1.09 1.03 1.82 1.13
4 1.10 1.00 1.16 0.96 1.13 1.02
5 1.07 1.04 1.17 1.14 1.14 1.22
6 0.99 1.12 1.05 1.19 1.26 1.04
7 1.03 1.21 0.91 1.11 1.20 1.36
8 1.46 1.17 1.14 1.36 1.22 1.10

Therefore, we concluded that the proposed method is more accurate and faster than the classical method to compute the modes of this type of oil well.

3.1.2 Mode determination at ULF

At ULF, no modes propagate, because the chosen frequency is below the cut-off frequency of any mode. However, these modes are useful for mode matching and finding them demonstrates the power of the proposed model in comparison with the classical one.

Table 5 shows the first three modes computed using both the conventional model and the proposed model. The results are similar to the UHF case: the proposed method exhibits more accuracy and performs faster than the classical one.

Table 5. Comparison of the values of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0572 obtained using the classical and the spline methods for urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0574 and f = 1 kHz. The ‘rel. acc.’ column is the relative accuracy. The bigger the relative accuracy, the more accurate the Spline-based model is when compared to the classical model.
Mode urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0576, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0578 (classical) urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0580, urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0582 (proposed model) rel. acc.
1 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0584 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0586 1.77
2 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0588 urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0590 1.82
3 8.6459–6.5313urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0592
time, s 7.01 5.91

3.2 Analysis of the non-conventional medium ‘oil well’

The next experiment aims to understand how the waves propagate in the inhomogeneous well. We consider a good length of 1000 m and we divided it into V small segments (see Fig. 2) of length 1000/V. The wave propagates from urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0594 to urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0596. Additionally, we assumed that the first segment (which contains urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0598) had only the plane wave corresponding to the fundamental mode, which is the first mode having urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0600. Moreover, we used a linear temperature profile with equation urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0602 and sampled it at the centre of each segment. Then, the conductivity of the segment was calculated using (59).

Next, we defined the measure of attenuation urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0604 of a mode as the ratio of the electric field intensity of this mode measured in urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0606 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0608. Hence, for a fixed m, using (39)
urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0610(60)
where urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0612 is the wave number of the mode in the segment that contains the plane urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0614 and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0616 is a real function of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0618.

Fig. 3 shows the function urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0620 for the fundamental mode having urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0622, and a fixed urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0624 in the oil. This function was obtained using the spline-based model and urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0626. The inclination of the curve increases slightly as the value of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0628 increases. It happens because as urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0630 increases, the temperature of the oil increases and, consequently, the conductivity of the oil gets bigger. Moreover, the higher the conductivity, the higher the losses due to the Joule effect.

Details are in the caption following the image

Attenuation of the first mode having azimuthal dependency urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0632. The well is inhomogeneous, and the proposed method divided the well into ten segments to account for the conductivity variation along the urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0634-direction

Moreover, according to Fig. 3, the fundamental mode is highly attenuated in the well, reaching a 100 dB attenuation in 300 m. This happens due to the high equivalent conductivity of the oil: urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0636. At high frequencies, the value of urn:x-wiley:17518725:media:mia2bf00748:mia2bf00748-math-0638 is non-negligible, thus paying an important burden in the total attenuation.

4 Conclusions

This work presented a new method to find the modes of unconventional waveguides based on spline functions and automatic grid building by Genetic programming. We applied our method to the problem of EM propagation within oil wells, comparing its results to the classical winding number approach. The proposed method featured higher accuracy and smaller running time than the conventional winding number method, both at high frequency (UHF range) and low frequency (ULF range). The results also showed that the typical oil well presents high attenuation. Therefore signal repeaters are required to build links within the oil having more than 300 m of length. Finally, we consider that the proposed method is useful for any radially layered waveguide problem, possibly inhomogeneous, and we recommend the proposed method over the classical winding number technique whenever a significant number of modes is required, especially in the ULF range.