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2017 | OriginalPaper | Chapter

5. Interpretations and Modeling

Author : Stéphane Sainson

Published in: Electromagnetic Seabed Logging

Publisher: Springer International Publishing

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Abstract

This chapter describes the various methods of interpretation of electromagnetic data. After a succinct recollection of the approach to solve the forward problem determined from the Maxwell equations and Ohm’s law, several methods of resolution are proposed. These involve either analytical 1D models considering some (quasistatic) approximations or numerical models for higher-order dimensions. Analytical equation resolutions have been favored for their educational value considering geological canonic models and relatively simple integration methods. Then we recall some data inversion techniques, which allow us to directly access specific resistivity values of the subsoil and therefore to detect the presence or absence of hydrocarbons. Finally we describe the analog models that concretely allow us to establish special detection devices or check some assumptions.

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previous chapter Instrumentation and Equipment

next chapter Geological Applications

As in previous chapters, we recall, throughout the presentation, a brief history of the interpretation methods, allowing us to follow the evolution of ideas and concepts that have led to current techniques.

Scope of action of transient methods, for example.

Only electric logs (wireline or MWD) are able to provide the “true” resistivity. This assumes, of course, that the geological layers are crossed by a drilling (borehole).

This corresponds, as discussed later, to the resolution of the inverse problem.

“To interpret is to establish relations of cause and effect between the structure of the subsoil and the highlighted physical phenomena, it is to deduce what are (or possibly what are not) the geological structures compatible with a specific anomaly. These relations of cause and effect between structure and anomaly are conceivable only in numerical form, in other words mathematics. Whether one likes it or not, the interpretation is only a mathematical problem ”(Cagniard 1950).

The mathematical problem applies above all to geological data, i.e., to facts of observation that fix reality at a certain moment in time and in space. We can thus arrive at valid conclusions only if we take into account this natural reality, which can be obtained by sufficient knowledge of the local geology. The more numerous and reliable the geological data are, the less theoretical and long the mathematical work of the interpretation will be.

It is easier for now to memorize a series of abacuses than to run numerical calculation codes on embarked units and more particularly on seabed sensors.

See the definition in Appendix 2.5.

The first to be put into practice and still used.

Not always easy to get.

In the 1950s, these models, locked in cabinets, were completely automated and controlled by a panel where it was possible to set different combinations with electromechanical switches.

As discussed later, cylindrical and spherical coordinates are also used.

At the frequencies, the propagation term (second term) disappears.

Weber’s law allows us to connect the images theory developed by Hummel to that of S. Stefanescu (Lasfargues 1957).

The integration of linear partial differential equations with a constant coefficient was built very naturally at the beginning (A. Cauchy) based on theorems of existence and uniqueness, then providing analytical integrals in the form of power series, where the coefficients were calculated by recurrence from the initial conditions and boundary conditions. According to their research, physicists realized pretty quickly that certain series of special functions (Bessel functions, Legendre polynomials, etc.) highlighted, better than the power series, the properties of integrals. However, these integrals such as the series converge only very slowly, which limits the calculation of apparent resistivity in practice.

It is necessary to know that these methods offer incomparable implementation flexibility with respect to the complexity of the considered media, but require on the other hand the monopolization of significant computational resources not always compatible with the data stream.

The reader will find in the works of Professors Parker (Parker 1994) and Scales (Scales et al. 2001) an excellent introduction to the resolution of the inverse problem in the field of geophysics. The works of Professor Tarantola, however, are more difficult to access (Tarantola 1981, 2005).

The resolution of the inverse problem must, according to the French mathematician J. Hadamard match to “a well-posed problem”—i.e., that its solution must imperatively respond to three separate criteria: to exist, be unique and stable with respect to measurement errors.

Cited by Wenner but not demonstrated by the author (see, more precisely, Appendix A2.2).

With few exceptions (mise à la masse, spontaneous polarization), electrical methods have never been used for the detection of a local anomaly, this type of investigation being reserved for the field of electromagnetic methods. The reader will find details on the development of these techniques in general works on electrical prospecting (Keller and Frischknecht 1966).

DC marine models (calculations) are not restricted as in situ investigations (measurements) and may in some cases be useful for the interpretation of low frequency data.

Under these conditions, we introduce the time factor, which must be strictly considered.

In these cases, the devices then undergo phenomena of induction (a temporal change of flux, creating secondary currents).

It was in 1932 that Maxwell equations were introduced (for the first time) in the interpretation of geophysical data (Peters and Bardeen 1932).

The theory is recalled in the Appendix to Chap.2 (cf. Eq. A2.1.4).

Infinitely resistant layer, with also infinite thickness, being under the sedimentary substratum. This case answers the question of the presence or of the absence of hydrocarbons. It doesn’t give any information about the layer thickness.

Provided that the depth of water does not exceed a priori, according to some authors, 15 m (Lagabrielle et al. 2001).

It is possible to acquire this information by performing crossword profiles.

Theory introduced by Lord Kelvin in 1845 for electrostatics (Thomson 1884; Ramsey 1937) and more generally in electrodynamics by Maxwell in 1848, for which a definition of the electrical image is given on page 246 of his treatise on electricity and magnetism (Maxwell 1892).

This was the first interpretation method employed in the 1930s by the German Hummel (series expansion), showing that under the reciprocity theorem (see Chap. 2, Appendix A2.2) the question to be addressed was similar to an optical reflection problem. This technique was proposed for the first time by the physicist Thomson (1845–1884). It preceded the methods of the Schlumberger school and particularly those developed by Sabba Stefanescu using, in particular, integration methods for a half-space:

V(r) = RI/2πr [1 + 2r∫ A(t)J _o(rt)dt].

Although less elegant than the latter, the Hummel series is more suitable for numerical calculations. We do not detail here the Stefanescu/Schlumberger method, which the reader will find in all books on applied geophysics more particularly about electrical methods (Kunetz 1966; Telford et al. 1978) or more generally about wave propagation in a stratified medium (Bannister 1968).

The problem of calculating the response of a laminate subsoil to electromagnetic solicitation was resolved early in the twentieth century (Sommerfeld 1909; Weyl 1919).

Reminder: we can distinguish by electrical sounding only sets powerful enough to affect diagrams in a measurable way. The most favorable structures in this regard are series formed by alternating resistant and conductive sedimentary layers whose thickness increases with depth. The diagram is then formed by a sequence of hollows and bumps that allow us to read:

The horizontal conductances (h/ρ) of the conductor sets
The transverse resistances (hρ) of the resistant sets

Each survey thus provides:

At one point, valuable information on the conductance of the subsoil at this point
From one point to another, the changes (deformation of the diagram) in conductors and resistants in space, i.e., in the thicknesses if the facies do not vary

First (in the 1960s) implemented in big systems (IBM 1620), mathematical models were gradually installed (in the 1970s) on portable microcomputers of the HP85 type (64KB of memory, expandable to 128KB using a specific programming language: HP basic) and then the compatible PC type (in the 1980s).

The theoretical response of a spherical anomaly in an alternative regime is extremely complex to carry out. Therefore, as a first approximation (low frequency approximation), we prefer calculation in a continuous regime (Telford et al. 1978). Under these conditions, the equation of diffusion is reduced to the Laplace equation.

It may be assumed that the injection of an alternating current whose injection poles (EM source) are opposed in phase, for example, is similar to the injection of a polarized DC (+/−).

This can be modeled by arranging some subtleties (see Appendix A5.4 ).

See Professor Tai’s work (Tai 1971).

Also called a dyadic Green’s function, this function then connects the scalar equations with vector equations. The Green functions, sensu stricto, are used among others in the methods of numerical resolution by integrals (Roach 1970; Eskola 1992). Their definition is given in Footnote 28 of Chap. 3: Metrology.

The energy of the source must disperse to infinity (no return).

Before the inversion operation, the data undergo, if necessary, a migration (see Sect. 4.5.5.1), the results of which can then be compared to the inverted data (Zhdanov et al. 1996; Zhdanov and Portniaguine 1997; Mittet et al. 2005).

It is therefore important to remove the data that do not check this distribution, thereby improving the variance of the solution. For a series of noisy data (p = 1 or less), other laws are applied.

We get an idea of the theoretical model, thanks to external elements supplied by seismic or geological data.

The methods of Levenberg–Macquardt (see below) or those using Gauss/Newton-type optimization techniques, for example, can also be used (Chave and Jones 2012; Amaya 2015). These algorithms (Marquardt 1963) are often complemented by regulation and smoothing techniques to stabilize the inversion process. The latter are based on calculation of the sensitivity matrix at the measurement point. This matrix is calculated numerically by methods such as those for the moments, the finite differences or finite elements.

The credibility of the model with regard to the measurements is determined by the mean square error or RMS (root mean square).

We can say, as Professor Claerbout pointed out in his book, that there are as many inversion methods as there are article writers (Claerbout 1992).

We refer the reader to the specialized literature (Hohmann and Raiche 1987; Glasko 1988; Parker 1994; Colton and Kress 1992; Buland et al. 1996; Zhdanov 2002a, b, c; Hoversten et al. 2006; Stefano and Colombo 2006; Abubakar et al. 2007, 2008; Gribenko and Zhdanov 2007; Bornatici et al. 2007; Roth and Zach 2007; Plessix and Van der Sman 2007; Zeng et al. 2007; Mittet et al. 2007a, 2008; Carrazone et al. 2008; Jing et al. 2008; Plessix and Van der Sman 2008; Price et al. 2008; Zach et al. 2008a, b, c, d; Troyan and Kiselev 2009; Nguyen and Roth 2010; Kumar 2010).

The first inversion method to be used for electrical soundings.

Developed by Legendre (1805) and Gauss (1809).

Software sold by the French company IMAGIR (Brest).

A similar “holographic” process and details of the calculations are described, for example, in a US patent (Zhdanov 2001).

For more information, the general principles of migration are treated in various articles and books to which the reader can refer (Zhdanov et al. 1996; Zhdanov 2009). Firstly, as in seismic prospecting, the migration technique was used to correct seabed relief effect and bathemetry raw data errors.

The principle of Occam’s razor states that if two hypotheses have the same degree of probability, then we favor the simplest hypothesis (the parsimony principle). The procedure is a direct application of Bayes’ theorem, where the simplest hypothesis then presents the higher probability (see Sect. 5.3.1). The predictive techniques that use the principle of Occam do not guarantee the accuracy of the model.

Characterizes the evolution of computer power. Probably changed in the future by the appearance of quantum microprocessors.

Free software originally developed by Apple.

Technique using the graphic processor unit (GPU) to perform all the calculations instead of the central processor unit (CPU), which is slower.

Formerly called an electrolytic tank or, more commonly, a bathtub in memory of the first experiments performed in 1911 by Conrad Schlumberger at the Ecole des Mines in Paris (Allaud and Martin 1976; Gruner Schlumberger 1982).

At this level, it important to take into account the wavelength in order to choose the transmission frequency (tank), while meeting the above criteria.

In this case, it is recommended to choose not too small a tank and to perform the measurement at its center. One can also choose walls in a conductive material so that they are at the same potential: this was the case with C. Schlumberger’s bath.

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Title: Interpretations and Modeling
Author: Stéphane Sainson
Publisher: Springer International Publishing
Book: Electromagnetic Seabed Logging
Print ISBN: 978-3-319-45353-8

Electronic ISBN: 978-3-319-45355-2

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-45355-2_5