contribution au développement de la résistivité complexe et à ses ... · we used cr1dmode code...
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THESE DE DOCTORAT DE L’UNIVERSITE PIERRE ET MARIE CURIE
Spécialité :
Géophysique Appliquée
Pour obtenir le grade de
DOCTEUR de L’UNIVERSITE PIERRE ET MARIE CURIE (PARIS 6)
Sujet de la thèse :
Contribution au développement de la
résistivité complexe
et à ses applications en environnement
Présentée par
Ahmad GHORBANI
Soutenue le 2 mars 2007
devant le jury composé de :
M. C. Camerlynck Maître de Conférences, Paris 6 Invité
M. Ph. Cosenza Maître de Conférences, Paris 6 Invité
M. N. Florsch Professeur, Université Paris 6 Directeur de thèse
M. A. Revil Chargé de recherche, CEREGE Examinateur
M. A. Tabbagh Professeur, Université Paris 6 Examinateur
M. K. Titov Professeur, Uni. St. Petersburg Rapporteur
Mme M. Zamora Professeur, Université Paris 7 Rapporteur
Résumé: Ce mémoire est divisé en deux chapitres. Le premier chapitre présente une synthèse bibliographique sur la méthode de Polarisation Provoquée (PP). Il passe en revue les derniers résultats théoriques et expérimentaux et insiste sur les applications environnementales de la méthode Polarisation Provoquée Spectrale (PPS ou Spectral Induced Polarization ou SIP en anglais), encore souvent appelée résistivité complexe.
Le deuxième chapitre comprend cinq sous-chapitres. Dans un premier sous-chapitre, l'approche bayésienne est développée pour inverser les paramètres du modèle de Cole-Cole dans les domaines temporel et fréquentiel. Nous avons évalué la pertinence et la robustesse de la méthode des moindres carrés pour l’inversion des données issues de la méthode PP. Les résultats montrent que l'approche classique dans le domaine temporel ne peut pas mener à une évaluation appropriée des paramètres Cole-Cole. L'emploi harmonique de signal carré injecté à fin d’obtenir un plus large spectre n’est pas non plus satisfaisant. À l'opposé, les paramètres Cole-Cole sont mieux inversés dans le domaine fréquentiel.
Dans un deuxième sous-chapitre, nous avons conçu un algorithme d'inversion 1D par homotopie pour récupérer les paramètres du modèle Cole-Cole à partir des données de PPS. Cet algorithme permet de découpler les effets associés à la seule PP de ceux issus de couplages électromagnétiques parasites se produisant dans une large bande de fréquences. Nous avons employé le code numérique, CR1Dmod, développé initialement par Ingeman-Nielsen et Baumgartner (2006), résout le problème direct. L’interface utilisateur graphique, que nous avons conçue, facilite les entrées des données et des paramètres du modèle a priori, ainsi que la définition de la configuration géométrique du dispositif (géométrie des câbles). Nous présentons deux exemples synthétiques pour illustrer la récupération des paramètres spectraux à partir de données de résistivités complexes.
Dans un troisième sous-chapitre, la PPS est appliquée pour le suivi de l'infiltration d'eau dans un sol limoneux-argileux. L’infiltration a été réalisée à partir d’une pluie artificielle (à débit de pluie quasi-constant) et au cours de deux expérimentations, l’une menée in situ et l’autre sur une colonne de sol. Les expériences ont été basées sur l'acquisition couplée de données de tensiomètres et de spectres de PPS durant l’infiltration. Cette approche a confirmé l'existence d'une baisse significative de la phase (ou de la conductivité imaginaire) dans la partie haute fréquence du spectre (typiquement, 1-12 kHz) qui a été corrélée avec le remplissage d’eau dans des pores de diamètres compris dans la gamme de [30-85] m. Ces baisses de phase sont interprétées comme un effet de type Maxwell-Wagner. Les résultats de cette étude suggèrent que la méthode PPS serait en mesure de fournir des informations sur le remplissage par l'eau des plus gros pores et ainsi d’estimer indirectement des propriétés structurales.
Dans un quatrième sous-chapitre, la mesure de la résistivité complexe est employée pour le suivi de la teneur en eau et des microfissures induites thermiquement sur des échantillons quasi-saturés d'argillite. Les mesures de PPS ont été enregistrées durant deux trajets consécutifs en dessiccation: (a) le premier trajet a consisté en une phase de désaturation à l'air ambiant et (b) durant le deuxième trajet, les échantillons ont été soumis à quatre paliers de chauffe (température égale à 70, 80, 90 et 105 °C). Durant la phase de désaturation, l'amplitude de la résistivité complexe a été extrêmement sensible au changement de teneur en eau (un facteur de 3 à 5). Pendant la phase de chauffage, la résistivité a augmenté de deux ordres de grandeur comparativement à l'état initial. Les modèles Cole-Cole et Cole-Cole Généralisé sont les meilleurs modèles pour inverser respectivement les données de PPS pour la phase de désaturation et la phase de chauffage. Cependant, les résultats prouvent que le développement de l'anisotropie et les changements par conséquent de texture peuvent induire différentes signatures spectrales et processus de polarisation.
Dans le dernier sous-chapitre, le modèle empirique de Cole-Cole qui est très largement utilisé dans les études PP est employé pour l'étude de la compaction par dissolution sous contrainte de formations sédimentaires. Notre contribution dans cette étude a consisté à écrire l'algorithme nécessaire à l’application du modèle de Cole-Cole dans le domaine temporel.
Abstract: After a review on induced polarization method, this work is divided in five
parts. The parts one to four deal with theory, laboratory and field studies of environmental applications
of the spectral induced polarization (SIP) method. Last part is devoted to Cole-Cole model parameters
that are applied to the compaction of quartz sands by pressure solution.
In a first part, Bayesian approach is developed to invert of the Cole-Cole model parameters in
the time or in the frequency domain. We evaluate whether standard least square methods are capable
to invert the induced polarization data. The Bayesian procedure results show that the classical time
domain approach cannot lead to a proper estimate of the Cole-Cole parameters. Using the harmonics
of square current signals to reach a broader spectrum than the classical time domain single current
injection cannot lead to estimate the Cole-Cole parameters. At the opposite, the Cole-Cole parameters
can be more correctly inverted in the frequency domain.
In a second part, we developed a homotopy inversion algorithm for recover the parameters of
Cole-Cole model from spectral induced polarization (SIP) data in a 1D earth. Both induced
polarization and electromagnetic coupling effects occur simultaneously in a certain frequency
bandwidth. We used CR1Dmode code published by Ingeman-Nielsen and Baumgartner (2006) as
forward modeling. The graphical user interfaces allows easy entering the data and the a priori model
and also cable configuration. We present two synthetic examples to illustrate that the spectral
parameters can be recovered from multifrequency complex resistivity data.
In a third part, SIP is used for monitoring the water infiltration in a silty clay loamy soil in
both in situ and soil column experiments. These experiences were based on the coupled acquisition of
tensiometer data and SIP spectra during a water infiltration achieved by a artificial constant rainfall.
This approach confirmed the existence of a significant phase angle (or imaginary conductivity of SIP
data) drop in the high-frequency domain during an infiltration cycle that were correlated with the
water filling of pores in the [30-85] m diameter range. These phase drops are qualitatively interpreted
as a Maxwell-Wagner effect. The results of this study suggest that the SIP method would be able to
give valuable information about the water filling of bigger pores and to estimate indirectly soil
structural properties.
In a fourth part, SIP is used for monitoring the evolution of the water content and the thermally-
induced microcracks measured on four nearly water-saturated argillite samples. The SIP
measurements recorded during two consecutive desiccation paths: (a) the first path consisted in a
desaturation phase under ambient air and (b) during the second path; the samples were heated in a
temperature range of 70 to up 105 °C. The amplitude of the complex resistivity was extremely
sensitive to water content change (a factor of 3 to 5). During the heating phase, the resistivity
increased by more than two orders of magnitude compared to the initial state. Cole-Cole model and
the general Cole-Cole model are the best models to invert the SIP data for the desaturation phase and
the heating phase, respectively. However, the results show that the development of the anisotropy and
consequently textural changes can induce different spectral signatures and polarization processes.
In the last part, empirical Cole-Cole model that is largely used in induced polarization studies is
used to study of compaction of quartz sands by pressure solution. Our contribution in this study
involves the forward algorithm created in first part to solve Cole-Cole model in time domain.
Table of contents
Preface.................................................................................................................................... 1
General introduction....................................................................................................... 3
Chapter 1: Spectral Induced Polarization: State of the art ......................... 7
1) Brief history of Spectral Induced Polarization....................................................... 9
2) Principles of Spectral Induced Polarization.......................................................... 12
3) Complex resistivity instrumentation....................................................................... 14
3-1) The SIP-FUCHS II laboratory and field equipment.............................................. 15
3-2) Sample holders and Electrodes ............................................................................... 16
4) Complex resistivity parameters ............................................................................... 21
5) Origins of Induced Polarization............................................................................... 22
5-1) Electrical double layer (EDL) ................................................................................. 22
5-2) Interfacial polarization (Maxwell-Wagner effects) ................................................ 24
6) IP models....................................................................................................................... 25
6-1) Empirical IP models ................................................................................................ 26
6-1-1) Cole-Cole model .................................................................................................. 26
6-1-2) Constant phase angle model (CPA or Drake model) ........................................... 29
6-1-3) Generalized Cole-Cole model (GCC) .................................................................. 30
6-2) Electrochemical (EDL)-based models .................................................................... 30
6-2-1) Electrode polarization in soils and rocks with mineral particles.......................... 31
6-2-2) Membrane polarization (Marshall and Madden, 1959) ....................................... 32
6-2-3) Vinegar and Waxman model (1984) .................................................................... 34
6-2-4)Granular models versus Capillary model (Titov et al., 2002) ............................... 35
7) Environmental IP applications ................................................................................ 38
7-1) Permeability estimation ........................................................................................... 38
7-2) SIP measurement in vadose zone............................................................................ 43
7-3) Detection of contaminations in vadose zone and in ground water ........................ 44
7-3-1) Organic contamination of ground water ............................................................. 44
7-3-2) Contamination and SIP studies ........................................................................... 46
7-3-3) SIP studies of the effects of microbial processes .................................................. 49
7-3-3-1) Effects of Microbial Processes ........................................................................... 49
7-3-3-2) SIP laboratory studies of the effects of microbial processes ................................... 53
7-3-3-3) SIP field studies of the effects of microbial processes ............................................ 54
8) Conclusion .................................................................................................................... 56
9) References ..................................................................................................................... 58
Chapter 2: Papers........................................................................................................... 67
Paper1: Ghorbani, A., Camerlynck, C., Florsch, N., Cosenza, P. and Revil, A., Bayesian
inference of the Cole-Cole parameters from time and frequency domain
induced polarization, Geophysical Prospecting, in press 2007....................................... 69
Paper 2: Ghorbani, A., Camerlynck, C., Florsch, N., CR1Dinv: A Matlab program to
invert 1D Spectral Induced Polarization data for Cole-Cole model include
electromagnetic effects, Submitted for Computers and Geosciences, Feb. 2007. ....... 107
Paper3: Ghorbani, A., Cosenza, P., Ruy, S., Doussan, C. and Florsch, N., Noninvasive
monitoring of water infiltration in a silty clay loamy soil using Spectral Induced
Polarization, submitted.for Water Resources Research, April 2007 .............................. 129
Paper4: Cosenza, P., Ghorbani, A., Florsch, N. and Revil, A., Effects of drying on the
low-frequency electrical properties of Tournemire argillites, Pure and Applied
Geophysics, in press, 2007. ................................................................................................ 175
Paper5: Revil, A., Leroy, P., Ghorbani, A., Florsch, N. and Niemeijer, A. R., Compaction
of quartz sands by pressure solution using a Cole-Cole distribution of
relaxation times, Journal of Geophysical Research, vol. 111, B09205,
doi:10.1029/2005JB004151, 2006. 209
General conclusion and perspectives .................................................................. 221
Preface
This work was supported by the INSU-ECCO program (Polaris II project, 2005-2007)
in France, a project in which I am participating. This project, which studies the application
and development of conventional induced polarization and spectral induced polarization, is a
collaboration between four laboratories: UMR Sisyphe (Department of Applied Geophysics at
University Pierre and Marie Curie), CEREGE (Department of Hydrogeophysics at the
University of Aix Marseille III), EGID (Department of Environment, Geo-engineering and
Development in the University of Bordeaux III) and EOST (University Louis Pasteur in
Strasbourg).
I would first like to express my gratitude to INSU (Institut National des Sciences de
l'Univers) for their continuing support in the Polaris II project. I would also like to thank the
Ministry of Science, Research and Technology in Iran and the SFERE (Société française
d'exportation des ressources éducatives) for granting me the PhD. scholarship that has made
my work possible.
I wish to acknowledge Professor Alain Tabbagh, Director of UMR Sisyphe, who has
created a scientific and friendly environment conducive for implementing research projects.
I thank my advisor Professor Nicolas Florsch for his continuous support in the Ph.D.
program. Nicolas was always there to listen and to give advice. He is responsible for
involving me in the Polaris-II project in the first place. He taught me how to formulate
questions and express my ideas. Nicolas taught me different ways of looking at inversion
problems.
I have greatly profited from the generous advice of Professor Albert Tarantola
regarding the inversion problem course and from his published works. My cordial thanks for
these and for the helpful discussions about inversion problems we had.
Special thanks go to assistant professor Philippe Cosenza who helped me to write this
dissertation as well as helping me with the challenging research that lies behind it. Philippe
has been a friend and mentor. He taught me how to write academic papers, made me a better
programmer, had confidence in me when I had doubts, and brought out the good ideas in me.
Without his encouragement and constant guidance, I could not have finished this dissertation.
He was always there to meet me and talk about ideas, to read proofs and mark up our papers,
and to ask me the right questions to steer me through my problems.
Thanks also to assistant professor Christian Camerlynk who guided my studies in
signal processing and inversion problems. He showed me different ways to approach a
research problem and the need to be persistent to accomplish any goal.
I would like to thank research director André Revil, coordinator of the Polaris-II
project, who helped me for scientific and linguistic editing of papers. I also extend my
gratitude to Claude Doussan and Stephane Ruy, researchers in INRA (Institut National de la
Recherche Agronomique) in Avignon.
I thank all my colleagues and friends in the UMR Sisyphe and Polaris-II project.
Finally, I sincerely thank my family, particular my wife Parisa, for her invaluable
support and patience throughout this project.
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2
General introduction
Induced polarization (IP) is an electrical geophysical method in which the electric field is
generated by feeding low-frequency current, generally 0.1 to 10 Hz, between a pair of
grounded electrodes and where the potential is then measured by another pair of grounded
electrodes. Mineral-electrolyte interfaces in the presence of an external electric field produce
electrode polarization of electrochemical origin. Non-mineralized soils and rocks also exhibit
a weaker background IP that is sometimes called membrane polarization.
The DC-electrical method is based on in-phase conductivity and is associated with the
conduction occurring in-phase with the reference electric current. However, the measured
electrical impedance is in general a complex quantity with both an in-phase and out-of-phase
or quadrature component (e.g., Keller and Frischknecht, 1982; Ward, 1990). In the frequency
domain, this out-of-phase conduction is related to a phase shift between the measured voltage
and the applied alternative current. Out-of-phase or quadrature conduction at low frequencies
(typically from 10 mHz to 10 kHz) is referred to in geophysics as complex resistivity. When
the spectra of these complex quantities are measured, the method is called Spectral Induced
Polarization (SIP). However, the amount of information obtained at each point in a SIP survey
(amplitude | | and phase shift ) compared to a DC-method (resistivity ) is very large.
Although data processing, data inversion and modeling of SIP data are difficult, SIP
measurement is very useful, for example, for discriminating the spatial variability of clay
from the spatial variability of salinity. Induced polarization phenomena increase with clay
content but decrease with salinity, whereas resistivity decreases with both clay content and
salinity (Slater and Sandberg, 2000).
IP was initially measured in the time domain (TD) – that is, the voltage decay after excitation
by a current pulse is measured. Recording spectra in the time domain is achieved by
measuring the voltage transient at a number of instants after the current pulse has been
switched off. This method was developed in the 1950s and has been extensively used in
mineral exploration. The frequency domain and time domain-induced polarization methods
are equivalent in a linear and causal system. Both of these methods are interrelated through
the Fourier transform (Tombs, 1981). Normally, in the conventional Time Domain IP method,
one hopes and expects to avoid the problem of EM coupling by using the “delay time”
(waiting for a suitable length of time, a few dozens of milliseconds) after the transmitter is
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switched off before starting to acquire useful data. During this delay it is expected that the EM
coupling will dissipate to negligible levels. However, IP measurements in the time domain do
not include wide-band spectral information. Duckworth and Calvert (1995) showed that time
domain integral chargeability that is obtained by integrating a voltage decay curve can be
misleading. Studies in mining geophysics surveying (e.g. Tombs, 1981) show that time
domain IP measurement is not capable of discriminating between graphite and sulfides or
between massive and disseminated sulfide deposits.
Moreover, electromagnetic (EM) coupling is a major impediment to the inversion of complex
resistivity and SIP data. EM coupling can be defined as the inductive and capacitive response
of the Earth, which manifests itself as a response over the IP signal. The effect is increased
when the survey is carried out over a conductive Earth or when the dipole length and dipole
spacings are large. However, EM coupling limits the SIP measurements in laboratory and/or
at least on small scales. EM coupling calculating corresponds to computing the mutual
impedance of the grounded transmitter and receiver circuits at the desired frequency in 1D,
2D or 3D. Usually, calculating mutual impedance is complicated and time-consuming.
More recently, IP and SIP have been applied in environmental problems, hydrogeophysics
(e.g., Kemna et al., 1999, Kemna et al., 2004). For example, in the case of water-saturated
sedimentary materials, SIP parameters are closely related to grain size distribution and to the
mean grain size of the medium (e.g., Chelidze et al., 1977; Chelidze and Guéguen, 1999;
Kemna, 2000). Moreover, the mean relaxation time of induced polarization is also known to
be closely related to the specific surface area of the porous medium (Binley et al., 2005).
When combined with the electrical formation factor, the mean relaxation time can be used to
determine the hydraulic conductivity of soils and rocks (Schön, 1996; Slater and Lesmes,
2002). In summary, although the SIP method has become increasingly popular in
environmental and groundwater investigations, its full potential has yet to be attained in
Environmental Geophysics.
The overall objective of this thesis is to develop the scientific basis for laboratory and field
measurements of IP effects. Three specific objectives have been defined:
1. To develop a specific inversion method in order to evaluate IP data acquisition
techniques in both the time and frequency domains.
2. To develop specific inversion algorithms that allow interpretation of field SIP data.
4
3. To provide physical hypotheses for understanding the mechanisms of IP in soil and
clayey rocks.
To reach the first objective, we developed a Bayesian inversion approach. This method
provides solutions in the form of probability densities of the parameters obtained from an
induced polarization model (see paper 1). Regarding the second objective and as mentioned
above, EM coupling is a major impediment to the interpretation of complex resistivity and IP
data. Therefore, we will use the 1D forward modeling code written by Ingeman-Nielsen and
Baumgartner (2006b) in order to propose new inversion algorithms that take into account EM
coupling (see paper 2).
The third objective centers on two experimental studies performed with two different
materials: agricultural, silty, clayey, loamy soil and argillaceous rocks (argillites). First, an
experimental investigation was undertaken to study the ability of the SIP method for
monitoring water infiltration in a clayey loamy soil. It was based on the combined acquisition
of tensiometric and SIP data during water infiltration modeled by artificial constant rainfall
(see paper 3). Second, SIP measurements were performed on nearly water-saturated argillite
samples which were cored from an underground facility located at Tournemire (Aveyron,
France). These samples were subjected to desiccation phases to simulate the thermal loading
and hydric disturbance associated with deep nuclear waste storage (see paper 4).
The thesis is divided into two chapters. The first chapter contains a bibliographical review of
the SIP method. We present a brief history of the method and focus on environmental
applications such as indirect determination of in situ permeability and detection of
contaminants in aquifers and in the vadose zone. The second chapter contains five papers.
Paper 1 is a development of the Bayesian inversion for the time and frequency domains of
induced polarization. The second paper describes an inversion algorithm to invert SIP
parameters in the presence of EM coupling. The third and fourth papers present experimental
investigations performed in two geomaterials that are involved in environmental applications:
agricultural, silty, clayey, loamy soil and argillaceous rocks (argillites). The fifth paper uses
the time-domain Cole-Cole model to study the problem of compaction of quartz sands by
pressure. A forward modeling routine for the time-domain Cole-Cole model is used in the
inversion process.
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6
Chapter 1: Spectral induced polarization: State of the art
7
8
1) Brief history of Spectral Induced Polarization
The Induced Polarization (IP) and Spectral Induced Polarization (SIP) developments before
1990 that are described in this section, are taken from Collett (1990). Electrical polarization
effects in soils and rocks were first recognized by Dr. Conrad Schlumberger in 1911 at Ecole
des Mines in Paris. Early in 1911, Schlumberger conducted a study of the electrical properties
of ore samples and rocks. Using alternating current in the audible frequency, he performed
experiments in watertight tanks holding water of various salinities. In 1912, he shifted from
the laboratory to actual field conditions at Val Richer, Normandy. In the laboratory the mutual
induction between the current and measuring circuits was negligible. However, in the field, he
found a strong interference between the lines. To overcome this problem he switched to direct
current and used a galvanometer as a potential measuring device. However, he experienced
another problem when using direct current – that is, the polarization effect occurring on the
copper potential electrodes. Since this imbalance made his measurements unreliable, he
designed and built a non-polarizing type of potential electrode, and the resistivity sounding
technique was born. In 1913, while prospecting for highly conductive metalliferous ore
bodies, Schlumberger noticed that after the current was switched off, small but measurable
potential differences continued for some time. It seemed to him that the ground was polarized
by the direct current and then discharged. He remembered that during his experimentation at
Val-Richer, while working above an iron water-supply pipe buried close to the surface, it had
affected his readings. From this observation, he reasoned that a metallic mass, or metal-like
conductive ore, could be distinguished from the surrounding rocks by this type of
polarization. By sending a direct current into the ground, interrupting the current and
measuring the polarization effect, he had developed a new prospection method for detecting
metallic conductors such as pyrite, pyrolusite and galena. In 1920 his first work describing the
electrical method of prospecting was published (Schlumberger, 1920). In this book, Chapter
VIII is entitled “Induced Polarization”.
Because of the measurement technology equipment available in those days, it was difficult to
measure IP signals accurately. During the 1930s and 1940s the IP method was tested widely
in oil exploration in USA and the former USSR. Time-domain IP studies continued in the
1940s in USA. In 1947 in Utah, scientists developed the long time-constant fluxmeter to filter
out the high frequency noise and allow visual observation of the integral of the decay. Low-
grade disseminated mineralization such as porphyry copper deposits were particularly
successful targets for IP surveys (Sumner, 1976). Seigel (1959) calculated a mathematical
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formula for overvoltage and normal IP response over layered media and over spheres for
several electrode arrays. The use of the term “chargeability”, referred to as “m”, could be
attributed to Seigel. Wait (1959) almost single-handedly laid the mathematical foundation for
the time and frequency domain for IP and electromagnetic (EM) prospecting.
Up to 1950, all IP measurements were in the time domain. In 1950, Seigel showed that the
apparent resistivity of a rock sample containing sulfides decreased markedly with increasing
frequency. The frequency domain technique was theoretically investigated by Wait but not
experimentally. because the power and sensitivity of the equipment available at that time were
inadequate. In 1956-1959, Madden and Marshall conducted a range of laboratory research and
basic theoretical studies of IP effects due to nonmetallic causes. They also defined the terms
“electrode polarization” and “membrane polarization”. Madden and Cantwell (1967) in the
equivalent circuit representation of the IP phenomenon used the Warburg impedance.
Vacquier et al., (1957) attempted to apply IP techniques to groundwater exploration. IP
effects were attributable to membrane polarization in “dirty” sands with a significant clay
content. Studies in the frequency domain began in 1951 in the former USSR. From 1963,
various prototype instruments and methods for making precision amplitude and phase
measurements of IP at very low frequencies were designed. After these research projects in
the 1950s, the IP method became widely established in the exploration for sulfide
mineralization.
In the 1960s, IP investigations began to look at complex resistivity (amplitude and phase)
measurements for discriminating between sulfides and other conductors such as graphite or
between economic sulfides and pyrite. Working on laboratory samples, Fraser et al. (1964)
measured different spectra for a discriminated texture such as a rich dissemination with ore
veins and shaly sandstone. Zonge and Wynn (1975) also observed a systematic change in the
shape of the Cole-Cole plot with the alteration and degree of mineralization of a rock. Van
Voorhis et al. (1973) measured in-situ phase spectra of porphyry copper mineralization, but
concluded that the spectra were similar and that the broad-band response could be exploited
by recognizing and eliminating the effect of inductive coupling. Pelton et al. (1978) followed
up on these investigations. They found that two spectral parameters, chargeability and the
time constant, showed a wide variation between types of mineralization in an applied Cole-
Cole impedance model. They made small-scale in-situ measurements in ore outcrops in order
to discriminate sulfide ore from graphite and porphyry copper mineralization by its ore-
mineral grade and texture. This model has been extensively used in the field and has proved
reliable.
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During the 1980s, SIP was increasingly used for (a) petroleum industry and (b)
environmental applications. Concerning the petroleum industry, the IP technique has been
used to detect geochemical anomalies created by the presence of hydrocarbons at depth. The
hypothesis is that the IP anomalies measured above oil reservoirs are related to fine-grained
pyrite disseminations originating from hydrocarbons that have seeped from the reservoir.
Shaly reservoirs have been studied and specific models have been developed (Bussian; 1983;
Vinegar and Waxman, 1984; de Lima and Sharma, 1992).
Concerning environmental applications, Klein and Sill (1982) studied the effect of grain size,
electrolyte conductivity and clay content on the resistivity spectrum of artificial samples
composed of clay, glass beads and water. They found that the time constants approximately
follow the grain size of the glass beads.
During the 1990s, new laboratory SIP instruments capable of measuring phase spectra became
available and wide-band frequency devices were constructed for environmental applications.
Olhoeft (1984, 1985) obtained different kinds of phase spectra for clean and organically
contaminated clay samples, while Vanhala et al. (1992) and Vanhala (1997b) measured
complex resistivity in an attempt to map oil contamination in glacial till with a very low clay
mineral content. That team also observed a time dependence of the signals in polluted
samples, whereas measurements in uncontaminated samples were stable (Vanhala, 1997b).
Indeed, these measurements showed that the SIP method is sensitive to changes in grain
surface properties corresponding to organic contaminants.
Studies from long ago (e.g., Madden and Marshall, 1959; Klein and Sill, 1982) had already
shown that SIP data depend on pore-space geometry or grain size and on the microstructure of
the internal rock boundary layer. For this reason, they contain information about rock
permeability and fluid properties in the pore space. In recent years, SIP measurement was
used to determine hydraulic conductivity in the laboratory (Börner and Schön, 1991) and in
the field (Börner et al., 1996).
In the last decade, certain authors have traced a link between SIP and the mechanical
properties of soils and rocks (e.g., Lockner and Byerlee, 1985, Glover et al., 1997, and 2000).
Glover and colleagues showed the frequency dependency of the electrical properties of water-
saturated sandstones during triaxial deformation.
Much knowledge has been acquired since then on IP phenomena in hydrogeologic
investigations. Ulrich and Slater (2004) measured the complex conductivity responses of
unconsolidated sediments as a function of water/air saturation. Titov (2004) presented results
of laboratory time domain measurements conducted on sieved sands with different degrees of
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saturation and proposed a conceptual model to explain the experimental data. Binley et al.
(2005) showed the possible link between SIP and hydraulic properties in saturated and
unsaturated laboratory samples of sandstone. They also showed that the Cole-Cole model
gives a better data fit than the constant phase angle model used by Vinegar and Waxman
(1984) and Börner et al. (1996).
Understanding of the organic contamination of aquifers has accelerated recently by the use of
SIP measurements in the laboratory. For example, it is known that an increase in the
biological activities of microbes corresponds with higher levels of organic materials in
aquifers (Balkwill and Boon, 1997). Recent studies have shown that SIP measurements are
capable of assessing microbial activities so the method can be used to detect organic
contamination (e.g., Ntarlagiannis et al., 2005b; Abdel Aal et al., 2006).
However, other studies have shown that wider ranges of frequencies are necessary in order to
develop efficient model descriptions. One of the factors contributing to the underutilization of
IP methods is the need for careful wiring in order to prevent unwanted electromagnetic
coupling effects. Recent developments of remove or calculation of EM coupling response are
published by Routh and Oldenburg (2001) or Ingeman-Nielsen and Baumgartner (2006a and
2006b).
2) Principles of spectral-induced polarization
Electrical resistivity is measured by imposing an electric current I and measuring the resulting
electric potential or voltage V. The electric field is generated between a pair of grounded
electrodes, and the potential is measured by another pair of grounded electrodes, similar to the
DC resistivity method. The excitation signal can be constant in time (known as direct current
or DC) or alternating in time (known as alternating current or AC). During a DC
measurement, the resistance R is defined by Ohm’s law as the ratio of the voltage to the
current, i.e. V/IR . The resistivity parameter is derived from the measured resistance by
applying a sample geometric correction. For an AC measurement, the capacitance properties
of the medium must be taken into account in addition to the resistance per se. In this case,
Ohm’s law has been generalized to allow for the resulting frequency-dependent effects, and
the ratio of the voltage to the current, V/IZ , is known as the impedance rather than the
resistance. However, resistivity is often used for both DC and AC measurements.
12
During an AC electrical resistivity measurement of soil or rock, the voltage measured at any
given frequency of excitation induces a phase lag in time relative to the excitation current
(Figure 1). To describe this effect, the output voltage can be expressed as the vector sum of
both in-phase (real) and out-of-phase (quadrature or imaginary) components. Therefore, the
electrical impedance Z of the medium can be expressed as a complex number where both the
magnitude and phase lag of the voltage relative to the input current depend on the specific
electrical properties of the sample. This is the origin of the term “complex resistivity”.
The resistivity and the amount of phase lag in the measured voltage signal is frequently a
function of the excitation frequency. In fact, in a first-order approach, the soil acts as a
capacitor in that it inhibits the passage of direct currents but passes alternating currents with
increased efficiency at higher frequencies.
Conventional induced polarization is an electrical geophysical method in which the electric
field is generated by feeding low-frequency current, generally 0.1-10 Hz. Induced polarization
can be measured in two ways. In the time-domain method (TD), the voltage decay after
excitation by a current pulse is measured (Figure 2), while in the frequency-domain method
(FD) it is the phase shift between injection current and voltage response that is measured. SIP
has been developed as an extension to conventional IP. In SIP the amplitude and phase
components of the earth’s resistivity are measured in a broad frequency range, typically 10-2
Hz to 104 Hz (Figure 1).
time
Am
pli
tud
e
Amplitude current I
measured voltage V
I
V0
log(freq.)
Am
pli
tud
e
Amplitude |
phase
-Ph
as
e
| =k*V/I
0
(a) (b)
Figure 1: Data acquisition in spectral-induced polarization. (a) A sinusoidal electric current I
of period T in a single frequency is imposed on a medium and the resulting voltage V is
measured. The measured voltage is characterized by a phase lag in time relative to the
excitation current. (b) Resistivity and phase spectrum. The ratio of voltage |V| to current |I|
multiplied by the geometric factor, K, expresses the magnitude (amplitude) of the complex
resistivity of the medium, where both its magnitude and the phase depend on the specific
electrical properties of the sample.
13
time
IVoltage signal (V)
Current signal (I)
V
Vp
V
V0
V0
Vp
Vp
t1t2
Figure 2: Data acquisition in the time-domain-induced polarization method (TD). The voltage
decay after excitation by a current pulse is measured. By definition, the ratio of voltage
immediately after the current is shut off (Vp) divided by the maximum potential difference
measured during current transmission (V0) is known as chargeability M. Integral
chargeability, Ma, is the area under the decay curve between t1 and t2 times that is normalized
by the maximum potential difference measured during current transmission and (t2 - t1).
3) Complex resistivity instrumentation
In the laboratory, three different measurement systems are needed to measure the broadband
electrical impedance response from 10-3
Hz to 109 Hz:
(a) Four electrode systems are used to measure the impedance response from 10-3
Hz to104 Hz
because of electrode polarization in low frequency;
(b) Two-electrode systems are used to measure the impedance response from 102 Hz to10
7
Hz;
(c) Transmission line systems (e.g. TDR) are used to measure the impedance response from
106 Hz to10
10 Hz.
The effective frequency range of each of these measurement systems is limited by systematic
errors intrinsic to the measurement configuration. For each measurement system, it is possible
to have large systematic errors that are stable and repeatable but give erroneous results. It is
important, therefore, to test and calibrate these systems with known standards (e.g. standard
resistors and brines) to determine the overall accuracy of the measurements. It is not possible
to measure the complete broadband impedance response in the field with currently available
instrumentation (Lesmes and Friedman, 2005).
%orV
Vm
V
VM
0
p
120
t
t
p
attV
dttV
M
2
1
14
3-1) The SIP-FUCHS II laboratory and field equipment
The SIP-FUCHS II instrument (Radic Research) operates in a frequency range from 1.4 mHz
to 12 kHz over 7 decades of frequency. To measure the whole spectrum, the SIP FUCHS-II
apparatus starts with the highest frequency, 12 kHz, and the N other decreasing frequencies
are obtained by the following division: 12 kHz/2N. It consists in two remote units that record
the current and voltage signal. The measured data are transferred to the base unit, where the
apparent resistivity and the phase shift are determined. The SIP-FUCHS-II is connected to a
computer, which enables real-time visualization of the results. The SIP-FUCHS-II can be also
used in the laboratory. For deep investigation, an external 600 W transmitter is preferred in
order to improve the signal-to-noise ratio (Figure 3). Optical fibers are used for data
transmission and system synchronization. This eliminates uncontrolled cross-coupling
between transmitter and receivers and increases the measuring accuracy because potential
cables can be shortened to the distance between the electrodes (Radic Research).
In laboratory experiments, we also used a complex resistivity system (1253 Gain-Phase
analyzer Solartron Schlumberger) to measure the frequency range from 1 mHz up to 20 kHz.
An example of SIP data acquisition is given in paper 3.
1 1
Figure 3: SIP-FUCHS-II: Laboratory and field equipment. 1) Base unit and PC for real-time
visualization of results. Built-in base unit 50 W transmitter, signal generation, data
management and synchronism. 2) Remote units for parallel current and voltage recording. 3)
External 600 W transmitter. 4) Generator.
4
1
3
2
15
3-2) Sample holders and Electrodes
The main difficulty in measuring complex resistivity is to avoid electrode effects. Electrode
effects are as follows:
(a) Electrode polarization: this effect appears when the frequency responses of the
electrode (typically lower than 100 Hz) and the geomaterial are superposed. Electrode
polarizations may occur at the boundary between the sample and the measuring
electrodes (e.g., Vinegar and Waxman, 1984, Garrouch and Sharma, 1994; Levitskaya
and Stenberg, 1996; Carrier and Soga, 1999; Chelidze et al., 1999, Vanhala and
Soininen, 1995).
(b) Electrode-sample contact: when the electrode impedance is a dominant one, the
sample impedance is a small part of the total impedance and hence cannot be
measured with accuracy. This effect generally occurs in laboratory low frequency
measurements when dry and rigid samples are used.
There are methods to decrease the electrode effects for impedance measurements (Chelidze et
al., 1999). A maximal increase in the electrode-specific surface will significantly decrease
electrode impedance (e.g. Substitution technique where platinized electrodes are separated
from the sample by a contact electrolyte solution, Figures 4b and 4d).
1) Carrying out measurements at a minimal possible current intensity reduces
electrochemical reactions in the system.
2) Increasing the length of the sample to a maximum decreases distortions of the field in
the sample.
3) Cells with variable electrode spacing may be used (Figure 4a). As the electrode
impedance is the same for samples of different lengths, it can be excluded using the
subtraction of impedances, Z. The accuracy becomes low as impedance Z=0.
4) The four-electrode method may be used in which the measuring electrodes can be kept
away from the source electrodes. (Figures 4c and 4d).
16
Direct Sample-electrode
contact
Substitution technique
Two-electrode
systems
(> 100 Hz)
(a) (b)
Four-electrode
systems
(c) (d)
Figure 4: simple schema of laboratory sample holders in complex resistivity measurement. (a)
Cell with varying sample length. 1-electrodes; 2-sample. (b) Substitution technique: 1-
platinized electrodes; 2-sample; 3-perforated diaphragm for holding sample; 4-contact
electrolyte solution. (c) Direct contact in four-electrode cell: 1- current electrodes; measuring
electrodes. (d) Four-electrode system with electrolytic cells preventing direct contact of
current electrodes with sample: 1-current electrodes; 2-measuring perforated electrodes; 3-
sample; 4-electrolytic compartments allowing convection of fluid; 5-sample holder (modified
from Chelidze et al., 1999).
Laboratory rock sample holders
The Binley et al. (2005) sample holder, which is a four-electrode system, consists in a 20-
mm diameter cylinder, copper disk current electrodes and Ag-AgCl non-polarizing electrodes
for potential measurements that are inserted into the holder arrangement. Binley et al. (2005)
filled the end cap reservoirs with the saturating fluid for water-saturated sandstone
measurements and filled it with a 4% agar-synthetic groundwater mix that prevented ingress
for unsaturated measurements (Figure 5).
17
Figure 5: Sample holder arrangement for SIP measurements constructed by Binley et al.
(2005).
To minimize electrode effects simply, we used a four-electrode device that had low-cost
medical electrodes. The potential (or measuring) electrodes and the current (or source)
electrodes were electrocardiogram (ECG) Ag/AgCl electrodes (Asept Co.) and thin carbon
films (Valutrode® electrodes from Axelgaard Manufacturing Co.), respectively. Ag-AgCl
electrodes, which are known to be stable and almost non-polarizable in comparison with
metal electrodes, are widely used in laboratory SIP measurements (e.g. Vinegar and Waxman,
1984; Vanhala and Soininen, 1995). ECG Ag/AgCl electrodes consist of a small round metal
piece (10 mm diameter) galvanized by silver and covered with a soft sponge imbibed with
AgCl gel. The carbon films (50 mm diameter, 1 mm thick) that are used in electrotherapy are
circular and covered with a conductive adhesive gel to ensure a good electrical contact
between the electrodes and the rock sample (i.e. no air gaps in between). This four-electrode
device was validated by using porous and electrically inert samples (explained in paper 4)
(Figure 6).
20 mm
Rock
sample
60 mm
End chamber
(fluid or gel filled)
C+
P+
P-
C-
18
Figure 6: Rock sample holder. 1) Potential electrodes: electrocardiogram (ECG) Ag-AgCl
electrodes. 2) Current electrodes: thin carbon films.
Laboratory soil sample holders
Olhoeft (1979a) used a four-electrode sample holder to measure the complex resistivity of soil
samples. The sample was divided into three parts each separated by through-going platinum
screen electrodes completely separating and covering the sample sections (Figure 7a).
Vanhala et al. (1992) used a four-electrode device where the sample was intact in a single
tube. The voltage electrodes were heavy-gage gold wires which penetrated only the outer skin
of the sample (Figure 7b).
Figure 7: Soil sample holders used by (a) Olhoeft (1979a) and (b) Vanhala et al. (1992)
(Figure taken from Brown et al., 2003).
1
2
19
We designed and produced non-polarizing Cu/CuSO4 electrodes for use in SIP laboratory soil
electrodes in column infiltration monitoring (Figure 8). The porous ceramic has air pressure
entry value of 1.5 Bar. This electrode cannot be used in solutions with very low conductivity
(e.g. 200 S/cm).
Figure 8: Cu-CuSO4 non-polarizing electrodes used in experiment of infiltration on soil
column. The electrodes consist in: 1) porous ceramic; 2) flexible tube that is filled by
saturated CuSO4 solution; 3) copper wire; and 4) stopper.
Field electrodes
Metallic electrodes can be used as current-bearing electrodes for IP field deployment.
However, potential electrodes must be non-polarizing and their spectral response must not
overlap those of the signals to be measured. Usually, non-polarizing Cu/CuSO4 electrodes are
used in field IP as potential measuring electrodes. We used commercial ceramic porous
manufactured by Aquasolo that are used for watering (www.aquasolo.fr) and produced our
own non-polarizing Cu/CuSO4 electrodes for use in SIP field applications (Figure 9).
Figure 9: Potential electrodes for SIP field survey. These electrodes consist in a porous
ceramic (permeability: 7 cl/day) filled with Cu-CuSO4. We used commercial ceramic porous
devices made by Aquasolo that are used for watering (www.aquasolo.fr).
Porous ceramic
CuSO4 solution
Copper wire
stopper
1
2
4
3
20
4) Complex resistivity parameters
The conductive and capacitive properties of a material can be represented by a complex
conductivity, , a complex resistivity, or a complex permittivity, , where
i1
(1)
and f2 is the angular frequency and 1i . Complex electrical parameters can be
expressed either in polar or rectangular form. For example, complex conductivity can be
expressed in terms of a magnitude, , and phase, , or real, and imaginary, ,
components:
iei
and (2)
1tan
The relationships between the real and imaginary components of complex resistivity and
conductivity are given by:
2
1 ,
and (3)
2
1
The other frequency domain-induced polarization parameters are percent frequency effect,
PFE and metal factor, MF defined by:
)(
)()(100
0
01PFE
and (4)
)()( 01aMF
and time-domain IP parameters are chargeability, M:
1
0
)(1
01max
t
t
dttVttV
M (5)
and normalized chargeability, MN (Lesmes and Frye, 2001):
MMN (6)
21
where 0 a lower frequency and 1 a higher frequency; a a dimensionless constant; V(t) the
potential difference measured at a time t after the current is shut off, Vmax the maximum
potential difference measured during current transmission and t0 and t1 define the time
window over which the voltage decay curve is integrated.
The proportionality between the complex resistivity phase, , the percent frequency effect,
PFE, and chargeability, M, is both theoretically and experimentally well established (e.g.,
Marshall and Madden, 1959). These field IP parameters effectively measure the ratio of the
capacitive-to-conductive properties of the material at low frequencies. The low frequency
capacitive component, , is primarily controlled by electrochemical polarization
mechanisms, whereas the low frequency conductive component, is primarily controlled by
electrolytic conduction in the bulk pore solution. Therefore, field IP parameters are sensitive
to the ratio of surface conductivity to bulk conductivity effects. Metal factor and normalized
chargeability, MN are more directly related to the surface chemical properties of the material
and are therefore useful for characterizing lithological and geochemical variability (Lesmes
and Frye, 2001).
5) Origins of Induced Polarization
5-1) Electrical double layer (EDL)
When there is contact between solid and electrolyte media, the contact creates an ion
concentration gradient in the solution in the vicinity of the interface. Owing to the presence of
surface defects in their crystalline structure, the grains carry an excess often negative,
electrical charge compared to that inside the crystal. The interface thus attracts compensation
ions to restore electroneutrality. Under the effect of physical forces (Van der Waals) or
chemical (covalent) bonds, these compensation ions are adsorbed on the surface. The
effective charge of the ion atmosphere compensates the present surface charge and in
equilibrium between diffusive and electrostatic forces, the electrical double layer (EDL) is
formed at the interface between the solid and liquid phases (Figure 10) (Comparon, 2005).
Water and salt molecules bound to the rock surface constitute the Helmholtz or Stern layer,
which is sometimes divided into the inner Helmholtz plane (IHP) and the outer Helmholtz
plane (OHP) in order to include surface conductivity effects. In the IHP, salt and water
molecules are directly bound to the mineral structure. In the OHP, solvated salts are more
weakly bound to the mineral structure through higher-order interactions. In the OHP, it is
22
assumed that ghee charges are free to migrate perpendicular to the particle surface and that
they may contribute to the DC conductivity of the mixture. However, it is impossible to
measure the amount of charge in the inner and outer layers (Ishido and Mizutani, 1981;
Lesmes and Morgan, 2001).
Ions in the electrolyte are affected by this electrical structure in a volume called the diffuse
layer, which by definition starts at a plane called the Stern plane. Far from the mineral
surface, the electrolyte can be considered to be unaffected and is therefore called the free
electrolyte. The charge densities in the diffuse layer are constrained by the surface charge
density in the Stern layer. The simplest theory of the EDL consists in neglecting the width of
the Stern layer and to model its charge density by a surface charge density concentrated on the
Stern plane.
Figure 10: Sketch of the structure of the electrical double layer in the three-layer model
(modified from Lorne et al., 1999; Chelidze and Gueguen, 1999). IHP, internal Helmholtz
plane; OHP, outer Helmholtz plane. The potential is defined as the electric potential at the
shear plane, located at a distance s from the Stern plane. Surface potential versus distance
from the solid surface x. 0 is the total potential drop between the surface of the solid and the
bulk of the electrolyte, s and are respectively potential drops in the Stern layer and in the
mobile part of the EDL and k-1
is the Debye screening length or thickness of the diffuse
double layer.
23
The potential within the EDL is shown in figure 10, where the total potential drop 0
between the solid and the bulk of the electrolyte includes that in the diffuse, mobile part of the
EDL, which is usually called the potential and that in the dense, immobile part of the EDL,
which in turn includes the Stern potential s and the potential drop ( 0- s) due to chemically
bound non-solvated ions, which cannot be exchanged with the electrolyte. It is expected that
all ions that are at a potential field < s can be exchanged with the bulk of the electrolyte.
The Debye screening length or thickness of the diffuse double layer, k-1
, depends on various
parameters: electrolyte concentration, exchange cation valence, hydration ion size, solution
permittivity and pH. Increasing the electrolyte concentration and exchange cation valence
reduces the Debye length while increasing the hydration ion size, solution permittivity and pH
increases it.
5-2) Interfacial polarization (Maxwell-Wagner effects)
The second main origin of induced polarization in earth materials is geometrical and
interfacial. Spatial polarization results from differences in conductivity and polarizability
among components in a mixture, producing charge accumulation at the interface. The first
calculation concerning interfacial polarization was by Maxwell (1891) in his work on layered
materials. As Wagner (1924) solved the complex permittivity of a dilute suspension of
conductive spheres, interfacial polarization is also known as the Maxwell-Wagner effect. This
mechanism, which can be seen as a bulk effect, has a pure macroscopic definition and does
not require any comprehensive understanding of the physical process of charge accumulation
at the molecular level.
The best known formulation for modeling the Maxwell-Wagner effect is the Maxwell-
Wagner-Hanai-Bruggeman (MWHB) equation (e.g., Chelidze and Gueguen, 1999; Lesme and
Morgan, 2001), which corresponds to a Differential Effective Medium (DEM) theory (e.g.,
Cosenza et al., 2003). Consider a mixture of two components: spheroidal inclusions with an
effective complex permittivity i are embedded in a matrix characterized by an effective
complex permittivity m. The effective complex permittivity of the mixture mix is given by
the MWHB equation:
i
1/m
mix
m
mi
mixi d1 (7)
24
where di is the volume fraction of spheroidal inclusions and m is a particle shape factor related
to the eccentricity of the spheroidal inclusions. It is also called the “cementation exponent”
(m=3 for spherical inclusions). Figure 11 shows an example of DEM theory for an initial
material with two phases. Moreover, it should be emphasized that the interfacial polarization
and the electrochemical models are not incompatible, since they operate at different scales.
For instance, Lesmes and Morgan (2001) proposed a granular model for the electrical
properties of saturated sedimentary rocks that combines both approaches. According to their
model, the MWHB equation was considered as a mixture formula in which the
electromagnetic properties of components, especially the clay fraction, are governed by
microscopic and physico-chemical laws.
Figure 11: An example of differential effective medium (DEM) theory. Homogenization
process in DEM theory: the initial material “m” corresponding to a relative permittivity km in
a volume V0. Phase “i” is embedded in phase “m” in such a way that the volume remains
fixed at V0. This is an iterative process that is continued until the volumetric fraction is
reached.
6) IP models
In this section, we review the empirical, theoretical and experimental models for complex
resistivity responses of rocks and soils and try to establish connections between the empirical
and theoretical parameters.
25
6-1) Empirical IP models
6-1-1) Cole-Cole model
One of the most popular relaxation models is the Cole-Cole (C-C) model. It was first used in
geophysical prospecting by Pelton et al. (1978), but was originally developed by Cole and
Cole (1941) to describe complex permittivity behavior. In applied geophysics, Cole-Cole
resistivity dispersion is usually expressed in terms of complex resistivity (Pelton et al., 1978):
cim
)(1
111)( 0 (8)
Or as complex conductivity:
)1()(1
)(1)(* 0
mi
im
c
c
(9)
where 0 and 0 are respectively the resistivity and conductivity in the DC limit,
/1/ 000m is the chargeability, and are respectively the resistivity
and conductivity at high frequency, is the time constant, and c is the frequency dependence.
Pelton et al. (1978) showed that this model could be used to discriminate different types of
mineralization as well as the grain size of mineralized inclusions. The parameters 0, 0, m, c
and are obtained by fitting the model into the complex data. The parameter m describes the
magnitude of the polarization effect and is related to the relaxation time and indicates the
position (frequency) of the phase peak c
peak mf5.0
12/1 . The Cole-Cole phase spectrum
is symmetric and parameter c gives the slope of the phase spectrum in a double logarithmic
plot (Figure 12) (Major and Silic, 1981). The advantage in describing the dispersion of
resistivity in mineralized rocks lies in the similarity between the equivalent circuit (Figure 13)
exhibiting the impedance model and a model simulating the current paths in a mineralized
rock. In the equivalent circuit, resistance R0 corresponds to the resistance of the ohmic current
paths through the rock. The term c
Xi in series with resistance R1 describes the current
paths blocked by ore minerals. The term c
Xi is the complex impedance at the interface
between the ore minerals and the electrolyte, and resistance R1 is the resistance in the blocked
path.
In the Cole-Cole model for the equivalent circuit, parameters m and can be written (Pelton et
al., 1978) as 01 /1/1 RRm andc
RRX/1
10 . The characteristic frequency fpeak shifts
26
towards higher frequencies when the value of m increases. Figure 14, that gives a plot of fpeak
versus c for different values of m, shows that if c is small and m large, the fundamental
relationship between the time constant and the phase maximum, e.g., peakf2/1 , disappears
totally (Vanhala, 1997a).
The Argand plot is a complex plane plot where the real and imaginary parts are usually
normalized by the real part of the measurement at the lowest frequency. Conventionally,
phases and imaginary parts are plotted with the sign reversed, yielding a positive phase peak.
The Argand diagram for C-C model is the arc of a circle (Figure 12c).
10-3
10-2
10-1
100
101
102
103
104
Frequency (Hz)
100
101
102
103
Am
plit
ud
e (
m)
0=1000 m
=0.05 sm=0.7c=0.7
100
101
102
103
-ph
ase
(m
rad
)
10-3
10-2
10-1
100
101
102
103
104
Frequency (Hz)
10-6
10-5
10-4
10-3
10-2
' (S
/m)
0=1000 m
=0.05 sm=0.7c=0.7
10-6
10-5
10-4
10-3
10-2
" (S
/m)
(a) (b)
10-3
2x10-3
2x10-3
3x10-3
3x10-3
4x10-3
' (S/m)
0x100
2x10-4
4x10-4
6x10-4
8x10-4
" (S
/m)
0=1000 m
=0.05 sm=0.7c=0.7
(c)
Figure 12: Different ways to display SIP results. (a) amplitude and phase shifts of spectra
when complex resistivity is used (C-C model), (b) real and imaginary conductivity of spectra
when complex conductivity is used (C-C model) and (c) Argand diagram of complex
conductivity.
27
Figure 13: Mineralized rock and electrical equivalent circuit of Cole-Cole model connected.
R0 corresponds to the resistance of the ohmic current paths through the rock. Resistance R1 is
the resistance in the blocked path and cXi is the complex impedance at the interface
between the ore minerals and the electrolyte.
Figure 14: Dependence of the peak frequency fpeak, on c at different values of m in the Cole-
Cole model (Vanhala, 1997a).
The time domain analogue of the Cole-Cole equation (Pelton et al., 1978) describes the
transient IP characteristic during switching-off of the polarizing current at time:
0 )1(
)/()1()(
n
ncn
nc
tmtM (10)
28
where 0/)()( VtVtM is time domain chargeability, V(t) and V0 are the potential value of the
decay curve at time t after shutting off the current and the potential just before shutting off the
current, respectively. (x) is the Gamma function. Parameters m, c and are C-C model
parameters.
However, in mineralized rocks, C-C parameters m and generally depend on the quantity of
polarizable elements and their size, respectively (Pelton et al., 1978; Luo and Zhang, 1998).
The exponent c depends on the size distribution of the polarizable elements (Vanhala, 1997;
Luo and Zhang, 1998). Binley et al. (2005) showed that there is a strong relationship between
C-C parameters determined in the frequency domain and the hydraulic conductivity of
saturated and unsaturated sandstone cores.
6-1-2) Constant phase angle model (CPA or Drake model)
Experimental investigations on rock samples have shown in several cases (porphyry copper
deposits, some sedimentary rocks) that complex resistivity data can appear as a straight line in
the Argand plot (Van Voohris et al., 1973; Vinegar and Waxman, 1984; Börner et al., 1993,
1996; Weller et al., 1996): the real and imaginary parts obey identical power laws of
frequency (Figure 15). Such data are represented in the so-called Drake model, also called the
Constant-Phase-Angle (CPA) model. This can be expressed by the equation (Pelton et al.,
1983):
ai )1(
1)(* 0 (11)
At high frequencies ( t >> 1), the amplitude is given by a)()(* 0 and the phase is
given by a2
.
Binley et al. (2005) examined the spectral IP response of samples taken from a UK sandstone
aquifer and compared the measured parameters with physical and hydraulic properties. They
showed that their SIP data were superposed with the C-C model.
29
10-3
10-2
10-1
100
101
102
103
104
Frequency (Hz)
10-1
100
101
102
103
Am
plit
ude (
m)
0=1000 m
=5 sa=0.005
100
101
102
103
-phase (
mra
d)
920 940 960 980 1000
' ( m)
0
2
4
6
8
" (
m)
0=1000 m
=5 sa=0.005
(a) (b)
Figure 15: (a) An example of amplitude and phase shift spectra for the CPA model. (b) The
same example in an Argand diagram (CPA model).
6-1-3) Generalized Cole-Cole model (GCC)
To model an asymmetrical circular arc in the Cole-Cole representation, the Davidson-Cole
(D-C) model was proposed by Davidson and Cole (Davidson and Cole, 1950):
aim
)1(
111)(* 0 (12)
The C-C, CPA, and D-C models can also be generalized into a single formulation – that is, the
so-called generalized Cole-Cole model (GCC) (Pelton et al., 1983)
acim
))(1(
111)(* 0 (13)
When a=1 or c=1, one obtains the C-C model or the D-C model, respectively. When c=1 and
m=1, the CPA model is obtained. Klein and Sill (1982) showed that the GCC model is the
best model to fit the experimental data obtained from artificial clay-bearing sandstones.
6-2) Electrochemical (EDL)-based models
These type of models have been derived from a variety of diffusive polarization mechanisms
that can contribute to the complex conductivity response of water-bearing rocks and soils (e.g.
Marshall and Madden, 1959; Schwartz, 1962; Wong, 1979; Ward, 1990; de Lima and
Sharma, 1992) or of metal particle-electrolyte interface in rocks (Wong, 1979). In this
approach, the system consisting in an electrolyte-solid grain is characterized by a relaxation
time ( ) which is given by (e.g., Schwarz, 1962) =L2/2D , where D is the ion diffusion
coefficient and L is a characteristic length such as the mean grain size, pore size, the spacing
30
between clay blockages of pore throats, or surface roughness (Lesmes and Friedman, 2005).
For earth materials, a relaxation time distribution must be taken into consideration.
6-2-1) Electrode polarization in soils and rocks with mineral particles
Wong (1979) attributes the polarization that occurs at a metal-electrolyte interface to two
mechanisms:
1) The polarization that results from the flow of inactive ions in the mobile part of the
EDL normal to the metal surface, resulting in the accumulation of inactive charge
excess or deficit near the metal surface.
2) The polarization that results from a minor concentration of redox-active ions in
solutions engaging in electrochemical reactions and allowing charge transfer across
the metal-electrolyte interface.
These mechanisms are illustrated in the conceptual model presented in Figure 16. In this
figure, the thickness of the EDL is grossly exaggerated for illustration; EDL electrochemical
theory assumes that the EDL thickness (1/ ) << grain radius r (Slater et al., 2005).
Figure 16: Conceptual model for possible polarization mechanisms in metal-particle-
containing soils such as sand-Fe0 mixtures. (a) No electric field; (b) application of electric
field. IP1 represents diffusion inactive ions normal to the metal surface; IP2 represents an
electrochemical reaction between metal (Mes) and metal ions in solution (Mex+
). Symbols are
r = metallic particle radius; 1/ = EDL thickness; )/1( kfrreff is the postulated
polarization sphere radius sensed with IP. EDL thickness is vastly exaggerated for the
purposes of illustration (EDL theory assumes 1/ <<r) (Slater et al., 2005).
31
6-2-2) Membrane polarization (Marshall and Madden, 1959)
Membrane polarization is caused by the presence of clay particles (negative surface charge)
attracting positive cations from the pore water onto the negative surface, thereby forming
EDL. In the presence of pore constrictions, cationic clouds act as semi-permeable
electronegative membranes between adjacent sand grains. During application of an electric
current, these membranes locally enhance the transport of cations relative to anions, as shown
in Figure 17a. When the applied current is stopped, diffusion voltages that result from these
impressed concentration gradients decrease with time as the ions redistribute themselves to
position of equilibrium (Vinegar Waxman, 1984). In the membrane polarization model of
Marshall and Madden (1959), the sediment is described by two zones (selective and non-
selective) connected in series. The zones have different mobilities of cations and anions in the
electrolyte which, under applied external voltage, produce local charge accumulations and
charge density gradients causing the frequency dependence of the resistivity. In order for
polarization effects to occur, however, restrictive zones for anions are required (membrane
zones or selective zones), which are to be alternated with non-selective zones. A geological
analogy of this model is the shaly sand in which rock grains (quartz and other minerals) are
coated by a thin clay film; the water-filled interspaces form the non-selective zones while the
grain contacts are the selective zones.
Evidence now exists to show that membrane polarization affects are also significant and
measurable even in clay-free unconsolidated material when a low concentrated electrolyte is
used (Vanhala, 1997b, Titov, 2002).
This model predicts the following features:
The time constant increases as the length of the non-selective zone increases.
The model consequently predicts an increased time constant with increasing grain
size.
Membrane polarization decreases as pore-fluid salinity increases. This decrease can be
related to a decreasing thickness of the EDL with increasing fluid salinity.
Theoretically, this model provides the best understanding of IP phenomena in soils and rocks
but it has several parameters that limit its practical use. Klein and Sill (1982) showed that the
Davidson-Cole model fits with the Madden and Marshall model (Figure 18). Consequently,
better agreement is obtained when the empirical Davidson-Cole model is used.
Figure 17b shows an equivalent electric circuit for a membrane polarization model (Ward and
Fraser, 1967). The resistor Rw represents unblocked ionic sections of pore paths. The Warburg
impedance W represents the frequency-dependent diffusion impedance resulting from the
32
membrane zones. Phenomenologically, the series resistance represents the electrolyte path in
series with the cationic clouds in Figure 17a and the bypass resistance represents leakage of
charge carriers around these zones.
(a)
(b)
Figure 17: (a) Schema of membrane polarization. (b) Equivalent circuit for membrane
polarization model. Rw is ionic solution resistance that represents unblocked ionic sections of
pore paths. W is Warburg impedance that represents the frequency-dependent diffusion
impedance resulting from the membrane zones (Slater and Sandberg, 2000).
10-3
10-2
10-1
100
101
102
103
1
2
3
0.9
0.8
0.7
0.6
0.5
Am
pli
tud
e (
m)
Madden-Marshall
Davidson-Cole
10-3
10-2
10-1
100
101
102
103
Frequency (Hz)
100
101
102
103
Ph
ase
(m
rad
)
Davidson-Cole parameters
=0.13 secm=0.24 a=0.5
R0=1 m
Amplitude
Figure 18: Comparison of Davidson-Cole model with the Madden and Marshall model where
L1=3.3*10-5
m, A=1, B=1, s1=1 and D1=2*10-9
m2/sec where L1 is length of non-selective
(clay free) zones, A is the ratio of zone lengths (length of non-selective zones divided by
length cation-selective zones), B=D1/D2 , D1 and D2 are diffusivity in non-selective and
selective (clay-bound) zones, respectively, and s1 are transference numbers in the selective
zones (modified from Klein and Sill, 1982).
Rw
Rw
W
33
6-2-3) Vinegar and Waxman model (1984)
In DC resistivity studies, shaly sandstone samples show the significant surface conductivity
effects that result in a non-linear relationship between rock saturated conductivity and pore-
solution conductivity. Waxman and Smits (1968) modified Archie’s law to include a surface
conduction term in parallel with a bulk conduction term in saturation state:
surfaceweffF
1 (14)
where F is the electrical formation factor and w is the specific conductivity of the
equilibrating electrolyte solution (S/m). The formation factor can be estimated from the slope
of the linear portion of the eff versus w plot at high solution conductivity. Waxman and Smits
(1968) explained the effects of lithology and solution chemistry on the surface conductivity
term in the following model:
F
BQvsurface
and as: (15)
vweff BQF
1
where Qv is the cation exchange capacity of the rock divided by unit pore volume (meq.ml-1
)
and B is the equivalent ionic conductance of clay exchange cations as a function of w
(S.m2.meq
-1). Waxman and Smits (1968) empirically obtained the following formula for the
dependence of B on the solution conductivity:
wB exp1 (16)
where the fitting parameters , and depend upon the solution type.
Vinegar and Waxman (1984) measured the complex conductivity response of a suite of 21
shaly sandstone cores as a function of pore water conductivity. They used this data set to
develop the following complex form of the Waxman and Smits (1968) surface conductivity
model:
nF
QiBQ
Fi v
vweffeff.
1 (17)
where n is the fractional porosity. The parameter in Equation 17 represents an effective
quadrature conductance for these surface polarization mechanisms. Vinegar and Waxman
empirically determined to be slightly dependent on salinity. The polarization was assumed
to increase with decreasing porosity, as more pores become blocked.
34
The real part of Equation 17 is the Waxman and Smits (1968) model, in which the bulk
and surface conduction terms are assumed to add up in parallel. The imaginary conductivity
results from displacement currents that are 90 degrees out-of-phase with the applied field
(Figure 19). Vinegar and Waxman assumed that the displacement currents were caused by the
following two polarization mechanisms:
1) The blockage of ions by clay minerals at pore throats (membrane polarization)
2) The accumulation of counter-ions migrating along the grain/pore surface.
Although these polarization mechanisms are intrinsically frequency-dependent, Vinegar and
Waxman showed that over the low frequency range of their measurements (3 Hz to 1 kHz),
the quadrature conductivity response was essentially independent of frequency. They assumed
that both of these polarization mechanisms were proportional to the effective clay content or
specific surface area, represented by the cation exchange capacity of the rock per unit pore
volume (Qv).
However, they also assumed that the phase angle is not dependent on using the frequency
range (CPA model) whereas consolidated rocks may display a distinct peak in the phase of the
conductivity spectra Cole-Cole model (Binley et al., 2005).
Figure 19: Complex conductance model for shaly sands in saturation state where F the
electrical formation factor, n the fractional porosity, w specific conductivity of the
equilibrating electrolyte solution (S/m), Qv cation exchange capacity of the rock divided by
unit pore volume (meq.ml-1
) and B equivalent ionic conductance of clay exchange cations
(modified from Vinegar and Waxman, 1984).
6-2-4) Granular models versus capillary model (Titov et al., 2002)
In induced polarization studies, the dominant relaxation time of the polarization has usually
been associated to grain size (namely Granular models) as shown in Figure 20a (e.g., Klein
and Sill, 1982; de Lima and Sharma, 1992; Chelidze and Guegen, 1999; Lesmes and Morgan,
2001). In the granular model (Figure 20a), a solid grain surrounded by its own ion atmosphere
35
produces a local ion concentration gradient under the influence of electric current. The
concentration gradients give rise to ion flows, mostly represented by cations in the EDL. A
grain surrounded by its own ion atmosphere can be considered as the polarizing cell. The
relaxation time ( ) characterizing polarization of this cell is related to the ion diffusion
coefficient (D) and the grain radius (R), =R2/2D (Schwarz, 1962).
Titov et al. (2002) presented a theoretical relationship between ion-selective pore throats and
larger pores, again relating the IP mechanism to pore-throat size. They compared the behavior
of the model with experimental data obtained on natural sands (namely capillary models,
Figure 20b). Scott and Barker (2003) linked laboratory SIP measurements to pore-throat size
distributions obtained from mercury injection capillary pressure (MICP) tests on sandstone
samples. They suggested that the relaxation time is primarily associated with localized charge
blockage caused by constrictions in the fluid-filled pore space. Indeed, their experience
confirms the validity of the capillary model.
In the model by Titov et al. (2002) (Figure 20b), electric current produces local ion
concentration gradients in areas where the pore radii vary. The concentration gradients give
rise to ion flows, mostly represented by cations in throats, which produce a secondary out-of-
phase voltage. The sequence of large and narrow pores can be considered as the polarizing
cell. In this model, contacts of sand grains and inter-grain solution-filled space are considered
as electrical current passages of varying thickness, which differ in ion transport values. In the
model, sediments are likened to serial connection of active (ion-selective) and passive
(nonselective) zones. The model describes spectral IP characteristics for the medium where
the length of passive zones is much greater than that of active ones (Figure 21). The model is
called the short narrow pore (SNP) model. The authors used time-domain equations but
applied a Laplace-Carson transform and obtained the SNP model in the frequency domain as
follows (Figure 22):
5.0
5.0
0)(2
2exp111)(
i
im (18)
The SNP model predicts an increase in IP time constant as the length of ion-selective zone
increases. Its advantages are as follows. First the model has fewer parameters than the Cole-
Cole model, so the number of 2D and 3D SIP calculations that may be obtained is greater.
Second, the SNP model is a conceptual physical model whereas the Cole-Cole model is an
empirical model. However, the SNP model has to be validated for different type of soils and
rocks.
36
Figure 20: schematic view of IP in ion-conductive rocks (modified from Titov, 2004). (a)
Scheme of electrical double layer at a negatively charged particle surface; (b) Schematic
distribution of cation and anion concentration outside the EDL; (c) Excess and deficiency in
ion concentration around a polarized changed spherical particle; (d) Excess and deficiency in
ion concentration along a polarized throat; diffusion flows and electric field same as in part
(c); (e) Schematic distribution of the ion concentration around a polarized spherical particle or
along a polarized pore throat; C* denotes the excess concentration produced by the polarizing
electric field.
Figure 21: Geometry of the electrical current pathway in sandy-clay sediments. 1l and 1l are
large-pore lengths and l2 is narrow- pore length (Titov et al., 2002).
37
10-1
100
101
Frequency (Hz)
0
0.001
0.002
0.003
Ph
ase
(rad
)
SNP:=1 s, m=0.5%
Cole-Cole:=1 s, m=0.5%, c=0.62
SNP:=1 s, m=1%
Cole-Cole:=1 s, m=1%, c=0.62
Figure 22: Comparison between Cole-Cole and SNP models in frequency domain (modified
from Titov et al., 2002).
7) Environmental IP applications
7-1) Permeability estimation
SIP measurements depend on the pore-space geometry and the microstructure of the internal
rock boundary layer. For this reason, they contain information about rock permeability and
fluid properties in the pore space. The Kozeny-Carman (K-C) permeability model forms the
basis of many permeability prediction equations (e.g. Lesmes and Friedman, 2005):
2
p
sFS
aK (19)
where a is a tube shape factor, F is formation factor and Sp is specific surface area (total
mineral surface area per unit pore volume or surface area-to-porosity ratio). This equation is
useful since parameters F and Sp are often measured. The other permeability prediction
method is the Hazen model:
b
s CdK 10 (20)
where the effective grain size d10 (in cm) corresponds to the grain size for which 10% of the
sample is finer, C depends upon grain sorting and b depends upon soil texture. The Hazen
model is applicable to sediments where d10 ranges between 0.1 and 0.3 mm. Börner and Schön
38
(1991) established the linear relationship between the quadrature component of electrical
conductivity ( ) and the specific surface area (Sp) of sandstones at a frequency of 1 Hz
(Figure 23):
)( Hz1p constS (21)
where Sp is in m-1
, Hz1 is in S/m and const=10-11
S-1
. Using this relationship in a modified
Kozeny-Carman type model, they obtained the following expression for hydraulic
conductivity:
c
Hz
c
p
s
F
a
FS
aK
1
510 (22)
where Ks is in m/s. Application of the fractal-based PARIS model of Pape et al. (1982) to a
range of rock types results in estimates of a as 10-5
and c as ranging between 2.8 and 4.6.
The constant phase angle (CPA) model of conductivity forms the basis for applying the
Börner and Schön (1991) algorithm. The formation factor F is obtained from the separated
volume conductivity ’. Börner et al. (1996) used the SIP method to estimate hydraulic
conductivity in the field, e.g. estimates of Sp in unconsolidated sands and gravels. De Lima
and Niwas (2000) also adopted a Kozeny-Carman type of petrophysical model linking IP to
hydraulic conductivity in shaly sandstones. They defined a “lithoporosity factor” which
accounts for the clay content, porosity and hydraulic tortuosity by using parameters derived
from geoelectrical measurements.
10-1
100
101
102
Sp (1/ m)
0.001
0.01
0.1
1
"(m
S/m
)
Figure 23: Linear correlation between imaginary conductivity of saturation sandstone samples
and the specific surface area (in log-log scale). The samples are saturated with 0.01 M NaCl
and measured at a frequency of 1 Hz (Börner and Schön, 1991).
39
Slater and Lesmes (2002b) measured the complex conductivity response and hydraulic
properties of sandy-clay mixtures and glacial tills. On the basis of electrical measurements in
a set of samples with a hydraulic conductivity magnitude range of five orders, they
demonstrated limitations in the practical application of the Börner model and proposed a
simpler model using grain size (an easier measurement to obtain than Sp for model calibration
purposes), based on a Hazen-type empirical model. They observed a power law relationship
between Sp and Hz1 :
p
HzpS )(2000 1 (23)
where 2.05.0p %)95,53.0( 2 CIR . In their samples, however, they found that was
better correlated with effective grain size d10 (Figure 24):
b
Hz d101 0005.0 (24)
where d10 is in m, is in S/m and 1.00.1b %)95,83.0( 2 CIR . Using a Hazen type
of grain permeability model, they obtained the following expression for hydraulic
conductivity in terms of :
b
Hzs CK )( 1 (25)
where 1200800C and 2.01.1b ( is in S/m, Ks in m/s, R2=0.7).
Binley et al. (2005) examined the spectral IP response of samples taken from a UK
sandstone aquifer and compared measured parameters with physical and hydraulic properties.
They showed:
- the Cole-Cole mean relaxation time to be a more appropriate measure of IP response
than imaginary conductivity for these sediments;
- the clear link between Cole-Cole relaxation time and some measure of hydraulic
length scale (e.g. pore-throat size);
- the significant linear log-log correlation between Cole-Cole relaxation time and
hydraulic conductivity (Figure 25).
However, they did not observe the linear relationship between imaginary conductivity, ,
and Sp (Figure 26) that was demonstrated by Börner and Schön (1991). These results confirm
the validity of capillary models.
40
10-4
10-2
100
d10 (mm)
0.001
0.01
0.1
1
10
100
" (
S/c
m)
Silts and Sands
Tills
Figure 24: Imaginary conductivity of unconsolidated sand samples plotted as a function of the
effective grain size d10 (modified from Slater and Lesmes, 2002b). Complex conductivity is
measured at 1 Hz and the samples are saturated with 0.01 M NaCl.
Figure 25: Variation in Cole-Cole model relaxation time, , with vertical hydraulic
conductivity, K (Binley et al., 2005).
41
Figure 26: Variation in IP parameters with surface area per unit pore volume, Sp. (a)
Imaginary electrical conductivity at 1.4 Hz, ; (b) C-C model relaxation time, (Binley et
al., 2005).
The abovementioned studies showed that:
- various spectral IP parameters can be used to estimate hydraulic conductivity but it is
unclear how universally they may be applied;
- consolidated rocks may display a distinct peak in the phase of the conductivity spectra
whereas the constant phase angle model (CPA) is valid for unconsolidated sediments;
- normalized IP parameters such as the imaginary part can vary with solution
chemistry, thereby complicating the predictive relationships between IP parameters
and the desired lithologic variables.
42
7-2) SIP measurement in vadose zone
Induced polarization studies are often used in saturated media. The first IP study to investigate
the degree of water saturation was by Vinegar and Waxman (1984). They measured the
complex conductivity responses of their samples as a function of water/oil saturation.
Waxman and Smits (1968) also studied the effects of water saturation, S, on the electrical
conductivity of oil-bearing shaly sandstone. They proposed the equation:
SBQF
Svw
d
eff / (26)
where d is the saturation index for the real conductivity. The surface conductivity term
SBQv / increases with decreasing saturation, a phenomenon that Waxman and Smits
attributed to an increase in the volume concentration of clay exchange cations at low
saturation. Imaginary conductivity therefore depended upon saturation q
sat SS , where
sat is the imaginary conductivity of the fully saturated sample and the exponent q given by
1dq where d is the saturation index for the real conductivity. Then:
v
d
vw
d
eff QnF
SiSBQ
F
S
./
1
(27)
Ulrich and Slater (2004) reported the results of laboratory IP measurements on sandy
unconsolidated samples as a function of water content and saturation history. They showed
that IP parameters (phase shift and ”) evolve with water saturation and demonstrated the
existence of hysteresis in terms of both electrolytic conduction and polarization. They also
observed that polarization is only weakly frequency-dependent over the measurement range
they used (0.1–1000 Hz). Titov et al. (2004) confirmed these results and proposed an IP
model for unsaturated sands. Their model is able to detect the critical water saturation which
corresponds to the change in pore-water geometry from bulk water to an adsorbed film
existing on the quartz grains.
Binley et al. (2005) showed a relationship between relaxation time with the Cole-Cole model
and dominant pore-throat size, and they postulated a parallel relationship between spectral IP
behavior and unsaturated hydraulic characteristics. They showed that relaxation time is
affected by the degree of fluid saturation (Figure 27). However, in agreement with Binley et
al. (2005), we think that saturation levels must be taken into account if the SIP method is to be
applied in vadose zone studies.
43
Figure 27: Variation in Cole-Cole model relaxation time with saturation for core sandstone
samples VEC7-5 and VEC16-1 (Binley et al., 2005).
7-3) Detection of contaminations in vadose zone and in ground water
7-3-1) Organic contamination of ground water
During recent decades, a wide range of complex organic wastes have entered the urban
environment. A serious risk to groundwater quality has arisen from accidental spillage or
leakage from petroleum tanks and pipelines. Some of these substances are water-soluble but
many are only slightly soluble. The latter are referred to as non-aqueous phase liquids (or
NAPLs) and are divided into light and dense NAPLs according to whether they are less dense
or denser than water.
Light NAPLs are mainly petroleum products while the dense varieties include the chlorinated
hydrocarbons widely used as industrial solvents. Concentrations of these compounds
authorized in drinking water are in the parts per billion range. As NAPLs are only slightly
soluble and highly mobile, assessing their concentrations over time and their subsurface
distribution can be extremely difficult, particularly in complex environments (Cassidy, 2004).
As the NAPL moves down through the unsaturated zone, it also spreads laterally, because
capillary forces as well as gravity control its migration. Progress may be much more rapid in
fractured rocks with the contaminant following the most permeable route. Pollution of an
aquifer by NAPLs can increase by slow dissolution in groundwater. Because of the slow rate
of dissolution, NAPLs can persist as a source of pollution for decades. Light NAPLs are more
easily controlled as they float on the water table and occur at relatively shallow depths (Figure
28a). On the other hand, dense NAPLs move under the influence of their density and can
penetrate to considerable depths (Figure 28b) (Fetter, 1993).
44
(a)
(b)
Figure 28: Pollution of aquifers by solvents: (a) a light NAPL (Delage, 2005) and (b) a dense
NAPL. Pollutant dissolved in groundwater is migrating through the aquifer in the direction of
groundwater flow.
Groundwater scientists and engineers have developed a number of techniques for containing
and remediating soil and ground water contamination. In general, the remediation of a site
45
must address two issues (Fetter, 1993). If there is an ongoing source of contamination, source
control will be necessary to prevent the continuing release of contaminants to the subsurface
or from the subsurface to ground water. The second aspect of remediation is the treatment of
contaminated ground water and/or soil to remove or greatly reduce the concentrations of the
contaminants (e.g., pumping treat systems, bioremediation).
7-3-2) Contamination and SIP studies
As all polarization mechanisms show, the surface properties of solid mineral grains and
adsorption are very important to the IP effect. Take, for example, the polarization that may be
produced by membrane zones in clay materials (Marshall and Madden, 1959). When
contaminants enter the rock matrix, surface reactions between the grain surface and surface-
reactive contaminant solute affect the grain surface electrical characteristics of the soils and
rocks. Such surface reactions include:
- Adsorption (solute sticks to a solid surface by physical or chemical means and solute
diffuses into the sediment). Groundwater adsorption is a very important and common
sorptive process (Drever, 1997). For example, the dissolved components of aromatic
and chlorinated hydrocarbons, which are usually hydrophobic, are strongly adsorbed
by the solid phases in the subsurface. The resulting adsorption, which is usually
hydrophobic-driven, has been recognized as probably the most important mechanism
for groundwater contaminant removal of aromatic and chlorinated substances.
- Ion exchange (attraction between ions in the pore fluid and clay minerals in the soil).
It is very common for natural rock minerals, including clay, to be coated by organic
matter, which often has a high ion exchange capacity (Jenne, 1977).
- Chemisorption (solute is incorporated in the structure of a sediment/rock).
These processes tend to remove the contaminants from the free phase in the soil and are
affected by the saturation of the soil (Sogade et al., 2006).
The link between the IP response and adsorbed species has been recognized since the studies
of Olhoeft (1985 and 1986) and Olhoeft and King (1991). They catalogued many of these
effects and show that oxidation-reduction reactions, ion-exchange reactions, and clay-organic
solvent interaction can be distinguished from one another and from the properties of
chemically inert rocks. Olhoeft and King (1991) illustrate the ability of complex resistivity to
identify and track the extent of pollution by organic compounds. Figure 29 shows complex
46
resistivity measurements of uncontaminated soil as compared to the same soil contaminated
with various organic compounds, including petrochemical wastes and oil-field brines. The
samples come from two separate wells at the waste disposal facility in Louisiana. The
uncontaminated sample shows relatively little frequency dependence in both magnitude and
phase. There is a large anomaly in phase angle for the contaminated sample centered around a
frequency of 100 Hz, indicating a strong chemically-reacting mixture. In subsequent
laboratory measurements, contaminants were added to samples with clays. Olhoeft suggested
that the IP effects were due to the polymerization of toluene to bibenzyl in the Lewis Acid
environment on the surface of montmorillonite. In effect, the low frequency is affected by a
process whereby organic molecules coat the surfaces of clay particles, inhibiting the cation-
exchange process.
4
5
6
7
8
9
(m
)
Contaminated
Uncontaminated
10-3
100
103
106
Frequency (Hz)
0
1
2
3
log
(ph
as
e-m
rad
)
Contaminated
Uncontaminated
Figure 29: Complex resistivity laboratory measurements for core samples at the waste
disposal facility in Louisiana (modified from Olhoeft and King, 1991). Measurements from
two separate wells of uncontaminated soil as compared to the same soil contaminated with
various organic compounds, including petrochemical wastes and oil-field brines.
Ward (1990) suggested that contamination would affect the effective length of the EDL. If
contamination increases the length of the EDL, then the membrane effect is more prominent
in contaminated areas and can lead to enhanced polarization. Börner et al. (1993) performed
experiments on a range of organic pollutants. They found that while the presence of
contaminants could be detected, the effects were much more subtle. A range of contaminants
47
in a single clay-rich sample mixture tended to result in only simple level shifts or slope
changes in the real and imaginary parts of the resistivity (Figure 30).
10-3
10-2
10-1
100
101
102
103
Frequency (Hz)
10-6
10-5
10-4
10-3
10-2
Co
mp
lex
co
nd
ucti
vit
y (
S/m
)
Clean
Oil
hexane
Benzene
Dichlormeth.
Imaginary Conductivity
Real Conductivity
Figure 30: Frequency dependence of the real and imaginary parts of complex conductivity.
Shaly sandstone sample prepared with different organic contaminants ( water=0.01 S/m)
(Börner et al. 1993).
Sites contaminated with DNAPLs, especially the chlorinated solvents such as
perchloroethylene (PCE) and tetrachloroethylene (TCE), are often amongst the most common
and the most difficult to characterize. Ramirez et al. (1996) tried to produce images of
pollution location by combining electrical resistance tomography and complex resistivity. The
tomograms are taken at only two fixed frequencies (1 Hz and 64 Hz). they had only limited
success at producing meaningful images of a controlled release of PCE. Vanhala et al. (1992)
and Vanhala (1997b) measured complex resistivity in both the laboratory and field in an
attempt to map oil contamination in glacial till with a very low clay mineral content. They
found measurable differences between contaminated and uncontaminated experiments in both
settings. Laboratory and field results on the same materials were similar in only a qualitative
sense. They also observed a major time dependence of the signals in polluted samples,
whereas measurements in uncontaminated samples were stable. They suggested that time
48
dependence alone could be an indicator of pollution. Vanhala (1997b) also investigated the
amplitude and phase (IP effect) decrease in complex resistivity after oil contamination of till
and sand samples. Resistivity increased until an oil content of 10% by volume was reached.
He explained the increase in amplitude by the fact that oil can detach ions from the surface of
rocks, thereby increasing the number of totally dissolved solids. The ion concentration in the
electrolyte increased and the thickness of EDL decreased. In some artificially contaminated
samples and gravel and sand samples from waste sites, he also observed that the phase shift
increased with prolonged maturation time.
Despite the studies mentioned in this section, however, the bio-physicochemical interactions
that impact the polarization behavior of in-situ contaminated rocks/soils have not been
investigated to date.
7-3-3) SIP studies of the effects of microbial processes
7-3-3-1) Effects of Microbial Processes
Microbe distribution in the near subsurface environment varies and is closely related to
substrate (soil, rock), water and nutrient (organic matter such as humic and fulvic acids,
organic debris) availability. Owing to the abundance and diversity of organic matter, the soil
zone has large numbers of microorganisms ranging from 106 to 10
9 organisms per gram of
soil for bacteria and 104 to 10
6 organisms per gram of soil for fungi (Balkwill and Boon,
1997). Microbial population numbers decline with depth below the soil zone, thereby
mirroring the decrease in organic matter with increasing depth. Given the low amount of
organic carbon substrate in the lower unsaturated and saturated zone, most of the microbes are
dormant, although they are viable. This viable population can quickly adapt and grow with the
introduction of organic carbon substrate and nutrients. Increasing the supply of nutrients or
organic carbon substrate stimulates the growth of microbes (Fetter, 1993).
Figure 31 shows a schematic representation of bio-physicochemical changes induced in the
subsurface environments by microbes (Atekwana et al., 2006). Microbes provided with an
organic carbon source and nutrients will grow, proliferate and develop biofilms (Figure 32).
Biofilms are organized microbial systems consisting of microbial cells, microbial by-
products, nutrients, substrates and solid surfaces. Biofilms grow on the surfaces of the
substrate and between pore openings in rocks and sediments which induce:
49
- Physical changes in the rock environment. During microbial growth and biofilm
formation, the pore environment is altered in many ways, as shown in Figure 33.
Physical modifications include sediment texture, pore size, pore shape and grain
surface roughness. Biofilm growth has been found to reduce significantly the porosity
and permeability of porous media. This increased biofilm production alters pore-throat
geometry.
- Chemical changes in the rock environment. The chemistry of the fluid within the
pores where microbial growth occurs is significantly different compared to that of the
fluid in pores where microbial growth is absent. For example, the pH of the pore fluid
decreases as the carbon dioxide dissolves to form carbonic acid. Carbonic acid and
organic acids enhance the dissolution of sedimentary minerals. In addition, some of
the metabolic byproducts such as carbon dioxide and organic acids chemically interact
with the rock or sediment matrix, further altering the chemistry of the pore fluid. The
bio-chemical weathering processes in subsurface sediments potentially induce
physical changes (e.g., formation factor, tortuosity and pore geometry) in the
subsurface environment.
- Direct changes in electrical properties of the rock environment. Microbial adhesion
to mineral surfaces affects electrical interactions between the mineral particles and the
microbial cell surfaces (Figure 34). The important parameters that govern microbial
cell attachment to mineral surfaces are electrostatic attraction, hydrophobic interaction
and Van der Waals forces. This attachment of microbes to surfaces of clays alters the
EDL. Since the EDLs of microbes and clays repel each other, the thinner the EDL
layers, the less the repulsion between the clays and the cell surfaces.
50
Figure 31: schematic representation of microbial-mineral interactions and how the resulting
changes in the subsurface environment impact different geophysical responses (Atekwana et
al., 2006).
51
Figure 32: schematic representation of biofilm development and attachment to substrate
(Atekwana et al., 2006).
Figure 33: schematic representation of how changes in microbial growth and proliferation
impact pore structure (Atekwana et al., 2006).
Figure 34: Schematic representation of microbial interactions with clay surfaces and
attachment to clay surfaces through cation bridging (Atekwana et al., 2006).
52
7-3-3-2) SIP laboratory studies of the effects of microbial processes
Sauck (2000) attributes the higher bulk electrical conductivity at hydrocarbon-contaminated
sites to biodegradation. At the time of Sauck’s (2000) study, there were no controlled studies
showing that higher conductivity resulted from biodegradation and that microbial degradation
in hydrocarbon-contaminated sediments actually caused increases in bulk electrical
properties. However, the laboratory experiments of Cassidy et al. (2001) showed a direct
relationship between the aerobic microbial mineralization of a petroleum hydrocarbon and an
increase in pore water conductivity and volatile organic acids. Atekwana et al. (2004) used DC
resistivity to investigate changes in electrical conductivity during biodegradation in a column
experiment to simulate a water saturated zone, a transition zone and a vadose zone. They
demonstrated that the microbes degraded the diesel fuel (LNAPL) in the contaminated column.
Abdel Aal et al. (2004) used the SIP method to measure the effects of microbial processes on
sediments in laboratory sand columns. The column treatments included a biotic column
(containing mineral nutrients, diesel fuel and bacteria) and two abiotic columns with one
containing only nutrients and the other containing nutrients and diesel fuel. Microbes affected
both the electrolytic and interface properties of unconsolidated sediments contaminated with
diesel (LNAPL). After a sufficient delay (about 4 months) for the microbial activity to
become effective, an increase in polarization response (imaginary conductivity) was found in
contaminated areas with microbial presence whereas none was found in the contaminated
areas without microbes (Figure 35). This increase was thought to be due either to the biofilms
formed by the bacteria because of their large surface area or to the byproducts of redox
activities.
Figure 35: Percent change in real ( ’) and imaginary ( ’’) conductivity in biotic and abiotic
sands columns over time (Abdel Aal et al., 2004).
53
Ntarlagiannis et al. (2005a) measured the low frequency electrical signature of a microbial-
mediated metal sulfide precipitation process occurring in sand columns packed with Ottawa II
sand for a period of 70 days under anaerobic conditions. Nutrients (lactate and sulfate) and a
metal solution consisting of iron and zinc were introduced into the columns. Biostimulation
resulted in the precipitation of Fe and Zn sulfide. Polarization anomalies were coincident with
regions of lactate consumption, metal reduction and higher microbial population numbers.
This study demonstrates the potential for using IP parameters to monitor microbial-induced
mineral precipitation. Subsequently, Ntarlagiannis et al. (2005b) detected the presence of
microbial cells in sand columns using the SIP method. A high concentration of bacterial cells
in the sand column resulted in a higher imaginary conductivity response when compared to
sand control columns with no bacteria. They suggested that the higher imaginary conductivity
response in the bacteria columns was due to the higher total available surface area and to
alterations in the EDL of mineral grains and bacteria cells.
Davis et al. (2006) used complex conductivity measurements (0.1-1000 Hz) in biostimulated
sand-packed columns to investigate the effect of microbial growth and biofilm formation on
the electrical properties of porous media. They proposed that the observed polarization
(imaginary conductivity) response arose from the direct interaction of the attachment of
microbial cells and biofilm development on mineral grain surfaces (sand surfaces). They did
not observe any discernible relationship between microbial cell concentration and the real
conductivity component.
However, these promising studies showed that the SIP method could discriminate between the
relative contributions of microbial activity to changes in the pore fluid chemistry and changes
in the physical and chemical properties at interfaces.
7-3-3-3) SIP field studies of the effects of microbial processes
Abdel Aal et al. (2006) investigated the effects of microbial activity on IP measurements (0.1
to 1000 Hz) obtained from field samples. The measurements were acquired on sediment cores
retrieved from an aged-hydrocarbon contaminated site where intrinsic biodegradation was
occurring. In general, they recorded higher real and imaginary conductivity for samples
obtained from the contaminated locations compared to uncontaminated locations (Figure 36a).
As expected, the imaginary conductivity of the sediments at the latter was higher in the
saturated zone compared to the unsaturated zone (Figure 36b). The imaginary conductivity
54
values for samples from residual phase hydrocarbon was higher compared to values obtained
for samples contaminated with dissolved phase hydrocarbons below the water table.
Figure 36: induced polarization data obtained on core samples from a hydrocarbon
contaminated site: (a) real ( ’) and imaginary ( ’’) conductivity and (b) bar plots 1 Hz
imaginary conductivity (Atekwana et al., 2006).
The real conductivity did not show a strong correlation with contamination but the imaginary
conductivity was able to better distinguish between contaminated and uncontaminated
samples. They suggested that the high imaginary conductivity response was due to an increase
in the ionic strength of the pore fluid and to an increase in the surface area of the mineral
55
surfaces. These increases were attributed to the attachment of microbes which were enhanced
due to mineral weathering from metabolic byproducts such as organic and carbonic acids.
8) Conclusion
In this part, we summarize the SIP method in terms of IP models and SIP applications. In the
IP models section, the various empirical and physical models of IP were presented and
explained. The physical models based on electrochemical phenomena (EDL) usually contain
numerous parameters that reduce their potential applications. On the otehr hand, the
interfacial models are simpler to use. However, all these types of models tend to oversimplify
the real medium and they are also often derived from the literature on colloid science. In the
SIP literature, empirical models such as the constant phase angle (CPA) and Cole-Cole
models are widely used for 1D to 3D SIP modeling. In the part concerning environmental IP
applications, we presented two applications of the SIP method. Laboratory studies show that
SIP measurements depend on the pore-space geometry and the microstructure of the internal
rock boundary layer. These measurements can be used to determine the permeability and fluid
properties in the pore space in saturated samples of sandy unconsolidated sediments and
sandstones. The dependence of SIP parameters (e.g. imaginary conductivity, relaxation time)
on structural parameters (e.g. pore-throat size) led to the prediction of hydraulic conductivity
using SIP measurements, but it is unclear how universally they may be applied. When
contaminants enter the rock matrix, surface reactions between the rock and soil grain surface
and surface reactive contaminant solute (adsorption, ion exchange and chemisorption) affect
the grain surface electrical characteristics of the rock and soils.
Microbes provided with an organic carbon source and nutrients will grow, proliferate and
develop biofilms. During microbial growth, consumption of the carbon substrate, utilization
of nutrients, and the production of byproducts considerably alter the chemistry of the pore
fluids. In addition, some of the metabolic byproducts such as carbon dioxide and organic
acids chemically interact with the rock or sediment matrix, further modifying the chemistry of
the pore fluid. Hence, these physicochemical and bio-physicochemical factors alter the grain
surface electrical properties of rock and soils, and it is thought that SIP measurements would
be a powerful geophysical method to detect organic contamination in the vadose zone and in
aquifers.
56
Upscaling from laboratory experiments with the SIP method to large-scale field investigations
is now a major challenge.
57
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66
Chapter 2: Papers
67
68
Bayesian inference of the Cole-Cole parameters from time and frequency
domain induced polarization
A. Ghorbani*, C. Camerlynck
*, N. Florsch
*, P. Cosenza
* and A. Revil
**
* UMR 7619 « Sisyphe », Université Pierre et Marie Curie, Paris, France.
**CNRS-CEREGE, Université d’Aix-Marseille III, Aix-en-Provence, France.
Summary. Inverting induced polarisation (IP) parameters is important to characterize the
frequency electrical response of porous rocks. A Bayesian approach is developed to invert
these parameters assuming the electrical response is described by a Cole-Cole model in the
time or in the frequency domains. We show that the Bayesian approach provides a better
analysis of the uncertainty associated with the parameters of the Cole-Cole model by
comparison with more conventional methods based on the minimization of a cost function
using the least-square criteria. This is due to the strong non-linearity of the inverse problem
and non-uniqueness of the solution in the time domain. The Bayesian approach consists in
propagating the information provided by the measurements through the model, and to
combine this information with an a priori knowledge of the data. Our analysis demonstrates
that the uncertainty in estimating the Cole-Cole model parameters from IP data is much
higher when the measurements are performed in the time domain than in the frequency
domain. Our conclusion is that it is very difficult, if not impossible, to retrieve the correct
value of the Cole-Cole parameters from time-domain IP data using standard least-squares
methods. At the opposite, the Cole-Cole parameters can be more correctly inverted in the
frequency domain. These results are also valid for other models describing the IP spectral
response like the Cole-Davidson or power law models.
Running title: Bayesian Inference of IP Parameters
69
1. Introduction
When a constant electrical current is injected through a water-saturated porous rock or
at the ground surface of a porous soil, part of the electrical power is stored in the medium.
When the current is stopped, this stored energy dissipates. This dissipation can be followed by
monitoring the voltage through the core sample or at the ground surface of the Earth
preferentially using non-polarizing electrodes (e.g., Schlumberger, 1920; Seigel, 1959). This
voltage exhibits a typical relaxation with a mean relaxation time ranging usually from few
milliseconds to thousands of seconds (e.g., Seigel et al., 1997). This phenomenon is called
“induced polarization” (abbreviated by IP hereinafter) and can be also observed in the
frequency domain. In the frequency domain, IP is characterized by a phase shift between the
injected current and the measured voltage. In the frequency domain, the apparent conductivity
of the porous material is therefore written as a complex number *( ),
)(")(')(*
1)(* i , (1)
where *( ) is the complex resistivity of the material, is the frequency of the excitation
current, i2 = -1, and ’ and ” are the measured real and imaginary components of the
conductivity, respectively.
Historically, induced polarization has been extensively used to locate ore deposits
(e.g., Madden and Cantwell, 1967). More recently, it has been applied in hydrogeophysics
(e.g., Kemna et al., 1999, Kemna et al., 2004). For example, in the case of water-saturated
sedimentary and granular materials and soils, the Cole-Cole parameters are closely related to
the grain size distribution and mean grain size of the medium (see Chelidze et al., 1977;
Chelidze and Guéguen, 1999; Kemna, 2000). Therefore, a 3D estimation of the grain size
distribution is possible using inversion of the induced polarization data obtained with an array
on non-polarizing electrodes. It was also found that the mean relaxation time of induced
polarization is also closely related to the specific surface area of the porous medium. When
combined with the electrical formation factor, the mean relaxation time can be used to
determine the hydraulic conductivity of soils and rocks (Schön, 1996; Slater and Lesmes,
2002). This opens exciting perspectives in determining non-invasively the distribution of the
hydraulic conductivity of aquifers with numerous potential applications in the domain of
water resources. The Cole-Cole function has also being used recently to model the
deformation, in the time domain, of porous sandstones (Revil et al., 2006).
70
The simplest model to represent IP phenomena is the Debye model, that is
characterized by a single relaxation time. However, the range of observed relaxation times of
water-saturated rocks is relatively wide. Consequently, there were many attempts to
generalize the Debye model in the literature (Wait, 1959; Dias 1972, 2000; Pelton 1977;
Pelton et al., 1978; Wong, 1979). Among the existing models, the Cole-Cole model (Cole and
Cole, 1941) is the most successful (see Taherian et al., 1990 for a statistical comparison of the
merits of different models). The Cole-Cole model can be described by the equivalent model
with resistors and capacitances (Figure 1). According to this model, the resistivity and the
conductivity of a porous rock are,
cim
)(1
111)(* 0 , (2)
)1()(1
)(1)(* 0
mi
im
c
c
, (3)
respectively. The Cole-Cole model depends on four fundamental parameters. They are the
DC-resistivity 00 /1 ( 0 is the DC-conductivity), the chargeability m, the mean
relaxation time , and the Cole-Cole exponent c, respectively. In mineralized rocks, generally,
m and depend on the quantity of polarizable elements and their size, respectively (Pelton et
al., 1978; Luo and Zhang, 1998). The exponent c depends on the size distribution of the
polarizable elements (Vanhala, 1997; Luo and Zhang, 1998). Klein and Sill (1982) found that
the time constants were approximately to the grain size of the glass beads. Binley et al. (2005)
showed that there is a strong relationship between Cole Cole parameters determined in the
frequency domain and the hydraulic conductivity of saturated and unsaturated sandstone
cores.
Several papers have been published in the last decade to invert the Cole-Cole
parameters both in the time and frequency domains. Examples include the works by Routh, et
al. (1998), Luo and Zhang (1998), Kemna ( 2000) and Xiang et al. (2001) just to cite few
examples. These classical methods are all based on a minimization of a cost function and the
determination of a single value for each of the four Cole-Cole parameters and, in some cases
only, an estimate of the uncertainty associated with the values of the inverted parameters. If
the inverse problem is non-linear, the solution non-unique, and if the a posteriori probability
distributions of the Cole-Cole parameters does not follow Gaussian distributions, optimization
approaches can not yield the correct result. The goal of our paper is to use a Bayesian
71
approach to invert the four Cole-Cole parameters and to determine the probability densities in
estimating these parameters. The inversion is performed both in the time and in the frequency
domains to compare the merits of inverting the parameters in these two domains. Finally, we
will discuss the limitation of the least-square methods by comparison with our novel
approach.
2. The Bayesian Inference Approach
Since the formulation of the Bayesian inversion by Tarantola and Valette (1982a, b)
(see also Tarantola, 1987; Mosegaard and Tarantola, 1995; and Scales and Sneider, 1997), the
Bayes theorem has been widely used to invert geophysical or hydrogeological data for a
variety of applications. The philosophy of this approach is closely related to the notion of
“information theory”. It consists actually in propagating the “information” (or knowledge)
provided by the measurements through the involved physical law (perfectly or
probabilistically known), and to include the a priori knowledge of the model parameters. Both
the data and the model parameters are described with probability distributions. The Bayesian
approach preserves the full knowledge provided by the data combined with the physical law
and the a priori information on the data and model parameters. Therefore, it is the most
suitable method to perform the inversion of non-linear problems (Tarantola and Valette,
1982a, b).
The validity of some assumptions reduces the Bayesian method to the use of simple
equations and rules (e.g., Florsch et al., 2000; Robain et al., 2001). These assumptions are the
following. (1) All the measured data and the unknown model parameters are assumed to be
independent parameters. The law connecting the data to the model parameters is written as,
d = G( ) (4)
where d is the vector of data, is the vector of the unknown model parameters, and G the
function describing the forward problem. In our case, G is based on the Cole-Cole model used
to characterize the induced polarization problem. (2) The physical law is supposed to be exact
and therefore the probability distribution of the physical law is a Dirac function. An
alternative choice would be to consider that the Cole-Cole model is not entirely appropriate to
describe induced polarization data. In such a case, a probability density, taking for example
72
the form of a Gaussian distribution, could be considered with a standard deviation reflecting
the confidence given to such a model. This standard deviation could be determined using a
large data base of IP measurements performed on a wide spectrum of frequency and
determining how well the Cole-Cole model performs to fit the data (e.g., Taherian et al.,
1990). In the following, we assume however that the Cole-Cole model is correct. (3) The
measurements are supposed to follow a Gaussian distribution. This is generally true as long as
the noise affecting the data is random and results from applying the superposition principle
(Tarantola and Valette, 1982a, b).
The a posteriori probability density function p( ) combines the information related to
the Gaussian data, the a priori probability density function of the model parameters, and the
forward model d = G( ). We consider that this probability distribution is given by (Tarantola
and Valette, 1982a, b):
xCx*.exp)()(
1
dd
T50p , (5)
where Gdx , the superscript T denotes transpose of the vector, d is a vector of N
measurements, is the vector of unknown Cole-Cole model parameters to invert, ddC is the
N x N data covariance matrix, is the a priori probability density of the model parameters,
and is the homogeneous probability measure. As p( ) is a probability density, one must
have,
1)( dp . (6)
The a priori density, , is used to incorporate an a priori knowledge in the model
parameters. This probability density could be a Gaussian distribution, a multi-modal
distribution, or any probabilistic description of the parameter. A locally uniform law (the
probability distribution is constant over an interval 21 , and vanishes elsewhere) is often
used to describe this a priori density .
73
The interpretation of is more challenging, especially in the present case. It has
been introduced in the general Bayesian approach to generalize Shannon’s definition of
information entropy (Tarantola and Valette, 1982a). It represents the density probability (or
measure) a given parameter that follows naturally when no a priori knowledge about this
model parameter exists. Tarantola (2005) termed this term the “homogeneous probability
measure”.
Numerous physical parameters that are positive (like distances, volumes, densities,
electrical conductivities, absolute temperatures...) usually follow non-informative laws of the
form /a where a is constant. These types of parameters are called “Jeffrey’s
parameters” (Jeffreys, 1939). These laws show scale-invariance properties: any multiplication
or power of Jeffrey’s parameters is a Jeffrey’s parameter, and a fractal transform also
conserves their forms. Considering the four Cole-Cole parameters ( 0, m, , c), only 0 and
are Jeffrey’s parameters.
Since all the four parameters involved in the Cole-Cole model are independent, the
homogeneous probability measure of the Cole-Cole parameters vector = [ 0, m, , c]T is,
0)1()(
mm
a, (7)
Hence, the Bayesian probability distribution is,
GGmma dd dCdT 1exp)1(
0, (8)
It is not easy to plot and interpret the four parameter Bayesian probability distributions
given by Eq. (8). Instead, we compute the marginal probability density functions for one or
two parameters. This approach is fundamentally different and richer than plotting the
objective function, cmGcmG dd ,,,,,, 1 dCdT
. Indeed, it is impossible to
derive full parameter information by slicing the objective function. Marginal laws preserve
the full “probability density function” (pdf) because they involve the integration of probability
over parameters selected as integrating variables. For example, the marginal pdf integrated
with respect of 0 and c is defined by,
74
dcdcmpmp 00 ,,,, . (9)
However, special attention must be paid when performing this integration. In the space
parameter, dcd 0 must be a volume dV that preserves the information within intervals of
ranges 0d and dc . Therefore any change of variables must involve the corresponding
Jacobian determinant encapsulated within the volume element. It follows that Eq. (9) is
correct (in this form) only when a constant homogeneous probability measure is used. For
instance, when considering Jeffrey’s parameters, Eq. (9) can be used only if suitable variables
are used, like s’=log(s) instead of s. Tarantola (2004) strongly recommend to use “volumetric
probabilities” instead of “probability density”. It follows that the simplest way to use
“natural” variables (those for which the homogeneous probability is constant) is to make
change of variables so that all the homogeneous probabilities become constant. Therefore, we
performed the following change of variables:
)log('
)log('
))1/(log('
000
mmmm
(10)
and c is kept unchanged. The change affecting m is explained in details in Appendix A.
Another possibility that yield the same result is to consider that the transform,
')()'('
m
mmm , (11)
is constant. (m) and ’(m’) are homogeneous probability measures of m and m’, respectively
and '/ mm represents the absolute value of the Jacobian of the transformation (Tarantola
and Valette, 1982b). Equation (11) yields a differential equation to solve and
))1/(log(' mmm is a solution of this equation. Consequently, like in electrical resistance
tomography where )log( is a better parameterization than , ))1/(log( mm is a better
parametrization than m.
The same type of approach was previously applied to chemical concentrations
(Tarantola, 2005). Indeed, a chemical concentration (mass fraction) C is comprised between 0
and 1. Consequently, this is not a Jeffrey’s parameter. A concentration parameter like iC is
the ratio between a given product mass iw with respect to the total mass M, i.e., MwC ii / .
Tarantola (2005) introduced an “eigenconcentration” (in his terminology) defined by the ratio
75
)/(' iii wMwC , so that 'iC is now a Jeffrey’s parameter and aCi )'log( . We adopted a
similar point of view here. In addition, the chargeability m can be seen as a “concentration of
polarizable elements” (Pelton et al., 1978), with the case m = 0 corresponding to the absence
of polarisable elements and m = 1 corresponding to the saturation of the medium in
polarizable elements. Therefore, we point out that the chargeability m shows accumulation
effects at m = 0 and m = 1. Figure 2 shows the correspondence between the familiar m and
more meaningful parameter ))1/(log(' mmm that illustrates this accumulation effect (ill-
posed correspondence between m and m’).
In conclusions, to perform our computations, we used the parameter transform
described by Eq. (10) and the integrations required in estimating the marginal pdf can be just
obtained by summing the sampled probability laws over a regular grid.
3. Inversion in the Time Domain
3.1. The Forward Problem
In the time domain, the chargeability m is determined from the residual voltage
measured immediately after the shut-down of an infinitely long, impressed current, divided by
the observed voltage just before the shut-down of the current (Seigel et al. 1959, 1997). For
non-metallic media, m ranges between 0 and 0.1. The time constant determines the rate of
decay of the residual voltage over time. In practice, this parameter has a very broad range,
from few milliseconds to thousands of seconds. The exponent c controls the curvature of the
decaying voltage in a log-log space voltage versus time (Seigel et al., 1997).
Solving the parametric inversion problem in the time-domain requires a numerical
solution for the forward problem. Several methods can be used to compute the residual
voltage assuming a Cole-Cole model. In the time domain, the following transmitted current
cycle (I0, 0, -I0, 0), with current amplitude I0 and characteristic duration T, is used. This
transmitted current cycle may be expressed in terms of Fourier series as (Tombs, 1981),
T
tnnn
nItI
n 2sin
4
3cos
4cos
2)(
1
0 . (12)
Assuming a Cole-Cole impedance model, the voltage drop measured at a pair of electrodes is
(Tombs, 1981),
76
tinn
nItV nn
n
exp)(4
3cos
4cos
2Im)(
1
0 , (13)
where Im(a) corresponds to the imaginary part of the complex parameter a and Tnn 2/
are characteristic frequencies. Note that Eq. (13) converges slowly. An alternative expression
of the potential drop V(t) can be derived by applying the Laplace transform to the Cole-Cole
equation. This leads to a series of positive or negative powers of (i ). Pelton et al. (1978)
determined the voltage response corresponding to a unit positive step of applied current. This
yields
0
0)1(
)/()1()(
n
ncn
nc
tmtV , (14)
where (x) is the Gamma function. This expression of V(t) has an extremely slow
convergence for t/ > 10 and values of c less than 1. For a better convergence, Equation (14)
can be replaced by the following expression (Hilfer, 2002),
1
1
0)1(
)/()1()(
n
ncn
nc
ctmtV . (15)
Guptasarma (1982) introduced a digital linear filter to transform the frequency-domain
response of polarized ground into the time domain. The computation is easy to perform, very
fast, and the relative error of this approach is below 1%. We used separately Eq. (15) and the
Guptasarma’s approach to perform the forward model. No differences between these two
approaches were observed.
3.2. Classical inversion in the time domain
Yuval and Oldenburg (1997) solved the parametric inverse problem by using the
approach of Guptasarma (1982). As explained above, we consider a sequence of (I0, 0, -I0, 0)
current flows. The infinite pulse train )(tS response can be represented as a sequence of step
functions, each delayed by multiples of the switching time T < 4 (T is the period of the full
waveform). This sequence is given by:
77
...4
5
4
3
24)()(
TtVTtV
TtV
TtV
TtVtVtS (16)
Consequently, using the superposition principle, the forward modelling of this pulse train
response requires a superposition of the series of positive and negative step responses
(Madden and Cantwell, 1967). To accelerate the convergence of the infinite pulse train
response, we apply an alternating series by using Euler transformation (Press et al., 1992).
For conventional time-domain IP receivers, it is common to sample the decay through
a sequence of N slices or windows. The value recorded for each slice is given by (Johnson,
1984, 1990),
1
)()(
10
1
3 i
i
t
tiip
i dttVttV
M . (17)
We determine numerically the chargeability from the forward modelling approach described
above. The relaxation time curve is recorded with a minimum delay time of 20 ms; the width
of each partial window lasts at least 20 ms and 20 windows are used (Table 1). This
corresponds to classical values used for field investigations. The rate of sampling on the delay
curve is 5 ms. We note Vp the potential at the cut-off of the electrical current, that is normally
close to the initial value of the decay curve V(t). Since the chargeability is given by the
normalized ratio of )()( tdtV with respect to Vp(ti+1-ti). This ratio does not depend on 0.
Therefore, in the inversion process, we used the normalized transient potential V(t) / Vp.
When using Fourier’s series, at least 104 harmonics should be used to ensure the
convergence of the series. This involves frequencies from 0.125 Hz to 1250 Hz. However, in
order to reproduce the acquisition of data in the field, we consider a sampling rate of 5 ms like
in most field acquisitions. It is certain that this choice affects the recorded voltage signal (see
Table 1).
3.3. Bayesian Inversion in the Time Domain.
The computation of the Bayesian solution of the inverse problem is based on Eq. (8).
We note d the vector of the N sampled normalized residual voltage data supposed to be
78
independent of each other while is the vector of three remaining unknown parameters (m’,
c, ’). All the a priori distributions where taken uniform on the changed variables within given
intervals. The parameter m’ is in the range -2.3 to +2.3 and consequently m, given by
)101/(10 '' mmm , is in the range 0.005 to 0.995. The parameter c is in the range 0 to 1
while log is in the range -4 to 4 or alternatively is in the range 10-4
to 104 s.
To represent the 3-D a posteriori pdfs, we use 3-D plots in the space of the model
parameters (m’, c, ’). We plot the two-by-two marginal laws in the space (m’, c), (m’, ’), and
(c, ’), respectively. To provide a good insight about the results of the inversion, we study a
set of eight examples, each of the three Cole-Cole parameters m, c, and can take two types
of distinct values, which are m = (0.2 and 0.8), c = (0.25 and 0.75), and = (0.01 and 10).
However, because some of these eight cases present similar results in terms of marginal laws
in the space (m’, c), (m’, ’), and (c, ’), we choose to present four cases exhibiting very
distinct behaviours. They are displayed on Figures 3 to 7. These results are discussed in the
following section.
3.4. Presentation of the Results
Case 1. This case corresponds to the following set of parameters (m = 0.8, c = 0.25,
and = 0.01 s). In this first case m is large but both c and are small. Figure 3 shows that the
solution crosses the entire investigated domain. The marginal plots are shown on Figures 3c
to 3e. The marginal plots are not merely slices of the full solution domain. Figure 3b provides
the marginal pdf in the ( , c) plane (after integration, the diagrams are plotted by using the
familiar parameter m instead of m’).
The “equivalence domain” shown by Figure 3c is rather complex and very far from
those that would be given by Gaussian distributions. Several remarks can be made from
Figure 3. First, one must remember that the marginal laws are built to provide probabilistic
estimations of the inverted parameters. Let us consider, for instance, Figure 3e. This Figure
shows the pdf of m and log , that we note 'mp . If this probability is properly
normalized, we have,
1',
4
4
1
0
dmdmp . (18)
It can be used to determine the following probability:
79
',)',(
1
1
D
dmdmpDmp . (19)
The maximum of the pdf must be considered carefully. Indeed, considering the domains D2
and D3 of Figure 3e, it is possible that )',()',( 23 DmpDmp due to the fact that it is
the pdf integrals that are meaningful instead of the pdf themselves.
Figure 3e shows the marginal law (m’, ’) for which several maxima exist. This
explains the failure of the classical least-square methods to determine the Cole-Cole
parameters from time domain IP data. It is also important to realize that in such a case, the a
posteriori conditional pdf, when integrated within a given box, can have a maximum in a
region where the pdf has no maxima. This depends on the sharpness of the pdf in the box. We
will discuss this point below.
Case 2. This case corresponds to the following set of parameters m = 0.8, c = 0.75,
and = 10 s. In this case, the equivalence domain is simple (Figure 4). The consideration of
the values of the model parameters (m, ’, and c) following the relations between them that are
represented on these diagrams will all permit to fit the data equally well. For example, if we
take = 1000 s ( ’ = 3), m = 0.97, and c=0.6, it fits the data as well as taking m = 0.8, c =
0.75, and = 10 s.
Case 3. This case corresponds to the following set of parameters m = 0.2, c = 0.25,
and = 0.01 s. In this case, the marginal laws present an equivalence between the parameters
that is more pronounced than in the previous cases (Figure 5). For instance, there are more
than one values of for one values of m that fit the data.
Case 4. This case corresponds to the following set of parameters m = 0.2, c = 0.75,
and = 0.01 s. In the marginal law of (m, c), Figure 6d, two branches appear corresponding to
two possible solution areas. From this pdf, we can compute the marginal law of parameter c
alone, as shown on Figure 6b. It appears that the left part of the (m, c) pdf has a smaller
volume than the large “mount” on the right side of the plot, although the left part reach a
higher pdf in two or three dimensions. On Figure 6a, the pdf has a narrow shape. In such a
case, the least-square methods or the use of simulated annealing algorithms reach the absolute
maximum of the pdf in the 3-D parameter space domain but these methods would still fail in
retrieving the correct values of the model parameters. Indeed, the best fit will be found on the
left ridge, even though the initial synthetic parameter belongs to the right side of the plot (c =
0.75). This illustrates very well the suitability of the Bayesian approach for strongly non-
80
linear inverse problems. There are equivalence laws like for the one exhibited by the case
corresponding to m = 0.8, c = 0.75, and = 0.01 s.
Moreover, these diagrams show that the chargeability cannot be correctly recovered.
Indeed, any value of m comprised between 0 and 1 will fit the data equally well. Catalogues
of curves used in some software to select a solution could lead to serious errors in estimating
the model parameters. Note that whatever the real value of m, inversion of the model
parameters in the time domain yields also an acceptable solution with 1m . In other words,
for any value of m, good fits of the time relaxation curve can be obtained for suitable values
of c and . Figure 6 shows for example, in connection with the set of model parameters (m =
0.2, c = 0.75, and =0.01 s, see curve a in Figure 7), the curve obtained with this set of value
and an alternative (and very different) solution with model parameters (m = 0.99, c = 0.75,
and = 0.0028 s, curve c on Figure 7). Figures 6 and 7 also show that two remote points in
the space parameter see case (a) and (c) on Figure 6, yields time responses that can not be
distinguished from each other. Moreover, as far as error bars are properly taken into account,
even case (b) can not be distinguished from cases (a) and (c). For comparison, the curve with
the fully non-compatible solution (m = 0.6, c = 0.2, and = 10 s) is also plotted (case (d) on
both figures).
4. Inversion in the Frequency Domain
Various inversion algorithms based on the non-linear, iterative, least-square method
have been proposed to invert spectral induced polarization (SIP) measurements using the
Cole-Cole model (see Pelton et al. 1978, 1984, Jaggar and Fell, 1988; Luo and Zhang 1998;
and Kemna et al., 2000). Xiang et al. (2001) proposed an alternative algorithm, called the
“direct inversion” that directly identifies the values of the Cole-Cole parameters. However,
their algorithm does not provide a full covariance analysis of the inverted values of the model
parameters. In this section, we apply the Bayesian algorithm to analyze the information that
can be retrieved from spectral induced polarization data. In addition, we compare the
uncertainties of the model parameters with those obtained in the time domain.
We first analyze the influence of the Cole-Cole parameters on the SIP response in the
frequency domain. Increasing the Cole-Cole exponent c increases both the steepness of the
phase peak and the slope of the amplitude curve response. The DC-resistivity 0 shifts the
amplitude curve vertically, and has no effect on the phase curve. The DC-resistivity is related
81
to the formation factor of the sample, the conductivity of the pore fluid, and the cation
exchange capacity of the medium (e.g., Revil et al., 1998).
In the present case and accounting for the additional parameter 0, the results consist
in the evaluation of 6 marginal pdfs, one for each couple of parameters. We use the same set
of synthetic data as proposed in the previous section and we investigate the frequency range
1.43 mHz to 12 kHz with a sampling rate given by NkHz 2/12 where N is number of
frequency used in the SIP FUCHS-II equipment (Radic Research). The a priori range of the
model parameters is the same as in the time domain case investigated in section 3 except for
the additional DC-resistivity, 0. For this parameter, we consider the a priori range (30-300
m) while the real value of this parameter is 100 m for all the synthetic cases analyzed
below. Since 0 is a Jeffrey’s parameter, we use the uniform probability density:
aa
)ln(')( 0
0
0 . (20)
In the time domain, we observed that the application of the inversion scheme requires
the computation of slowly convergent series. The inversion of the Cole-Cole parameters in the
frequency domain is hopefully simpler than in the time domain. Indeed, the Cole-Cole model
given by Eqs. (2) and (3) corresponds to an analytical complex function transfer, and is
consequently easier to invert than the slowly converging series arising in the time domain.
The cases are analyzed in Figures 8 to 11. Figure 8 presents inversion of synthetic data
in the spectral domain with ( 0 = 100 m, m = 0.8, c = 0.25, and = 0.01 s). Parameters are
well determined, only m have a small uncertainty of ± 0.1. Figure 9 shows results obtained
with parameters ( 0 = 100 m, m = 0.8, c = 0.25, and = 10 s). Large equivalence laws
appear between all the parameters; for example a strong correlation does exist between the
values of ’ and 0 for instance, 0=125 m (log 0=2.1), and s are also compatible
with the synthetic data. Figure 10 presents the results for the following values of the
parameters ( 0=100 m, m = 0.8, c = 0.75, and = 0.01 s). The distribution of the model
parameters c and m are wide while and 0 can be accurately determined. There is an ill-
posed correspondence between m and ))1/(log(' mmm that’s appears in m direction.
Results for the model parameters ( 0 = 100 m, m = 0.2, c = 0.25, and = 0.01 s) are
represented on Figure 11. In this case, all the model parameters can be accurately determined.
Note that the uncertainty of c and m are however correlated.
82
To finish this section, we perform a test using a set of real data. We use the data
published by Xiang et al. (2003). The results of the Bayesian analysis are shown on Figure 12.
This figure shows the marginal pdf when data are analyzed in the frequency interval (10 mHz
- 10 kHz). The space of the model parameters appears much simpler than in the time domain
(there is only one pdf maximum and the pdfs appear nearly Gaussian). Three model
parameters ( 0, m, and ) are well determined, but the Cole-Cole exponent c is poorly
determined. The maxima of the pdf obtained by the two-by-two marginal laws (the
coordinates of contour centres) are very close to the parameters derived from a classical least-
square method. So our approach validates the use of the classical least-square method in that
problem.
5. Use of Harmonics of Square Current Signals
An alternative way to perform field or laboratory measurements consists in injecting
square signal currents with different periods and to measure the associated voltage difference
using a pair of non-polarizing electrodes (e.g., Zonge Engineering and Research
Organization). In this case, the spectrum of the measured potential difference is the
superposition of the signal corresponding to each square signal. To analyze this case, we use
square current signals with the following periods 0.125, 1.0, and 8.0 seconds, respectively.
Then, we calculate the spectrum of frequencies 0.125, 1.0, and 8.0 Hz, including the 3rd, 5th,
7th, and 9th harmonics (Table 2) for each fundamental frequency that is presented in Figure
13a. The transfer function is:
)(
),()( 0
0nI
nVkn , (21)
where n is the harmonic number of each signal, 0 is the fundamental frequency, , V, and I
are the apparent resistivity, the voltage drop, and the injected current, respectively. This
function has uncertainties that increase with the number n of harmonics of the fundamental
frequency. These errors are incorporated in our Bayesian inversion approach (Figure 13b).
We assume that there is not error associated with the value of the injected current and that
there is only a constant deviation x associated with the measured potential drop. It follows
that the spectral deviation is the deviation X (Florsch et al., 1995):
83
N
xX , (22)
where N is the number of point of time sequence. The standard deviation of data can be
written:
)()( 0nX . (23)
We compute the a posteriori probability density function given by Eq. (5) using the
covariance matrix associated with the data given by Eq. (23). A synthetic example for the
inversion of the model parameters is presented on Figure 14. The values of the model
parameters are ( 0 = 100 m, m = 0.2, c = 0.75, and = 0.001 s). From the results plotted on
this figure, we conclude that when harmonics of a fundamental frequency are used to perform
induced polarization, only 0 is well-determined while the uncertainties associated with the
other parameters (m, c, are still very strong.
6. Conclusion
The Bayesian approach provides the information (or probability densities) associated
with the Cole-Cole parameters determined from induced polarization measurements in the
time or frequency domains. Because the Cole-Cole parameters can be used to determine the
hydraulic conductivity of aquifers, this approach has strong implications in hydrogeophysics
to determine the permeability using a Bayesian analysis. In this paper, the Bayesian approach
has been applied to the determination of the Cole-Cole parameters from measurements
simulated both in the time domain and in the frequency domain for which a wider range of
frequencies are investigated. While the use of the Cole-Cole model could show a limitation of
our approach, we point out that the Bayesian approach developed here could be used to
investigate the uncertainty associated with the inversion of alternative models like the Cole-
Davidson model.
The Bayesian procedure results in an enlightened explanation of why the classical time
domain approach cannot lead to a proper estimate of the Cole-Cole parameters. For
completeness, we investigated the frequency domain induced polarization by using the
harmonics of square current signals in a low-frequency range, theoretically reaching a broader
spectrum than just using the classical time domain relaxation associated with a single current
injection (corresponding to a single frequency). However, we showed that error bars increase
84
for the harmonics of each fundamental frequency. The Bayesian approach shows that, even in
this case, the correct estimation of the Cole-Cole parameters with the classical least-square
methods is difficult. Finally the Bayesian approach allows delineating “equivalent domains”
in the space of the model parameters accounting for the uncertainty in the measurements.
Acknowledgement. We are indebted to the INSU-ECCO program (project Polaris II, 2005-
2007) for support to this work. We thank the Referees and the Editor for their useful
comments.
85
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89
Appendix A. Determination of m’
To derive the non-informative structure of m, we consider an electrical equivalent
circuit where two resistors are involved (RA and RB). In this case, the chargeability m can be
expressed as 1
/1 AB RRm (Figure 1). Since RA and RB are both Jeffrey’s parameters,
their ratio AB RRS / is also a Jeffrey’s parameter. The homogeneous probability
distribution (HPD) of m is:
)1()(
mm
am , (A1)
where a is a constant and )1,0(m . Alternatively, the chargeability can be written as,
lim
lim 00
(A2)
0
1m . (A3)
However, is not a Jeffrey’s parameter since we have necessarily 0 . However, the
HPD of the normalized chargeability,
m
mm
1*
0
, (A4)
is,
*)1(**)(
mm
am , (A5)
where a is a constant. The HPD of m* has the same form than the HPD of m.
We look for an expression for the chargeability m’ satisfying )'(m is constant. If we
write )1/(1 Sm where S is a Jeffrey’s parameter, this yield,
m
mS SS
1)( , (A6)
and,
m
mS SS
1log)(log log (A7)
90
is a constant. If X is a Jeffrey’s parameter, 1 / X is also a Jeffrey’s parameter. It follows:
am
mm
1log' , (A8)
where a is a constant. Therefore the transform
m
mm
1log' , (A9)
is a suitable solution.
91
Captions
Figure 1. An equivalent circuit model corresponding to the Cole-Cole model.
Figure 2. Ill-posed correspondence between the chargeability m of the Cole-Cole
model and its normalized version ))1/(log(' mmm used for the Bayesian inversion.
Figure 3. Time domain inversion model with m = 0.8, c = 0.25, and = 0.01 s. (a) 3-D
space of a posteriori pdf. (b) a posteriori pdf on m interval parameter while fixed ( ’) and (c)
parameters.(c), (d) and (e) Marginal a posteriori pdf of two-by-two set of parameters.
Figure 4. Time domain inversion model with m = 0.8; c = 0.75, and = 10 s. (a) 3-D
space of a posteriori PDF. (b), (c) and (d) Marginal a posteriori pdf of two-by-two parameter.
Figure 5. Time domain inversion model with m = 0.2, c = 0.25, and = 0.01 s. (a) 3-D
space of a posteriori PDF. (b), (c) and (d) Marginal a posteriori pdf two-by-two of parameters.
Figure 6. Time domain inversion model with m = 0.2, c = 0.75, and = 0.01 s. (a) 3-D
space of a posteriori pdf. (b) Marginal a posteriori pdf law of parameter c alone. (c), (d) and
(e) Marginal a posteriori pdf two-by-two of parameters.
Figure 7. Partial Chargeability curve for a, b, c and d points represented on Figure 6.
Figure 8. The Marginal pdf of the Cole-Cole parameters ( 0 = 100 m, m = 0.8, c =
0.25, and = 0.01 s) for synthetic data.
Figure 9. The Marginal pdf of the Cole-Cole parameters ( 0 = 100 m, m =0.8, c =
0.25, and = 10 s) for synthetic data.
Figure 10. The Marginal pdf of the Cole-Cole parameters ( 0 = 100 m, m = 0.8, c =
0.75, and = 0.01 s) for synthetic data.
Figure 11. The Marginal pdf of the Cole-Cole parameters ( 0 = 100 m, m = 0.2, c =
0.25, and = 0.01 s) for synthetic data.
Figure 12. Marginal pdf of the Cole-Cole parameters ( 0=22 m, m = 0.13, c = 0.69,
and = 0.001 s) for the data of Xiang et al. (2003).
Figure 13. (a) Spectrum with fundamental frequency 0.125, 1.0, and 8.0 Hz, including
the 3rd, 5th, 7th, and 9th harmonics for each fundamental frequency. (b) Spectrum of
response divided by reference and error bars for each harmonic.
Figure 14. The Marginal pdf of the Cole-Cole parameters ( 0 = 100 m, m = 0.2, c =
0.75, and = 0.001 s) for synthetic data. Spectrum is created by 0.125, 1.0, and 8.0 Hz,
including the 3rd, 5th, 7th, and 9th harmonics for each fundamental fre
92
Tables
Table1. Number of chargeability slices and time interval sampling of decay curve in time
domain method in Cole mode of Syscal Pro® instrument. Vdly and Mdly are delay times (in
ms) from which the samples (sampling rate of 10 ms) are taken into account after injection
and after the current cut off, respectively. The period of current signal injection is 2 s.
Vdly Mdly 1 2-4 5-6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1260 20 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 180 200
Table 2. Fundamental frequencies and their harmonics used by the Zonge instrument in its
complex resistivity mode.
Harmonics Frequencies
(in Hz)
3 5 7 9
0.125 0.375 0.625 0.875 1.125
1 3 5 7 9
8 24 40 56 72
93
Figures
Figure 1
0 0.2 0.4 0.6 0.8 1m
-3
-2
-1
0
1
2
3
Log(m/(1-m))
Figure 2
(i x)-c
Ra
Rb
94
0 0.2 0.4 0.6 0.8 1m
0
0.2
0.4
0.6
0.8
1
PDF(m/c,log())
-4 -3 -2 -1 0 1 2 3 4log( )
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
-4 -3 -2 -1 0 1 2 3 4log( )
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
Figure 3
D2
D3
D1
m=0.8
c=0.25
=0.01 sec
(a) (b)
(c) (d) (e)
95
-4 -3 -2 -1 0 1 2 3 4log( )
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
-4 -3 -2 -1 0 1 2 3 4log( )
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
Figure 4
m=0.8
c=0.75
=10 sec
(a)
(b) (c) (d)
96
-4 -3 -2 -1 0 1 2 3 4log( )
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
-4 -3 -2 -1 0 1 2 3 4log( )
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
Figure 5
m=0.2
c=0.25
=0.01 sec
(a)
(b) (c) (d)
97
0 0.2 0.4 0.6 0.8 1c
0
0.2
0.4
0.6
0.8
1
PDF(c)
-4 -3 -2 -1 0 1 2 3 4log( )
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
-4 -3 -2 -1 0 1 2 3 4log( )
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
Figure 6
(a) (b)
m=0.2
c=0.75
=0.01 sec
(c) (d) (e)
ac
b
d a
c
b
d
a
c
b
d
98
0 0.4 0.8 1.2 1.6 2time (sec)
0
0.05
0.1
0.15
0.2
0.25P
art
ial C
harg
eabili
tya (m=0.2 c=0.75 =0.01sec)
b (m=0.04 c=0.86 =0.005 sec)
c (m=0.99 c=0.75 =0.0028 sec)
d (m=0.6 c=0.2 =10 sec)
ba
c
d
Figure 7
99
-4 -3 -2 -1 0 1 2 3 4log( )
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
log(
)
-4 -3 -2 -1 0 1 2 3 4log( )
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
log(
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
-4 -3 -2 -1 0 1 2 3 4log( )
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9m
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
log(
)
Figure 8
0=100 m
m=0.8
c=0.25
=0.01 sec
100
-4 -3 -2 -1 0 1 2 3 4log( )
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
-4 -3 -2 -1 0 1 2 3 4log( )
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1c
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
log
()
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9m
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
log(
)
-4 -3 -2 -1 0 1 2 3 4log( )
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
log
()
Figure 9
0=100 m
m=0.8
c=0.25
=10 sec
101
-4 -3 -2 -1 0 1 2 3 4
log( )
0
0.1
0.2
0.3
0.4
0.5
0.6
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Figure 10
0=100 m
m=0.8
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=0.01 sec
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-4 -3 -2 -1 0 1 2 3 4log( )
0
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)
Figure 11
0=100 m
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=0.01 sec
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Figure 12
0=22 m
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=0.001 sec
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0.1 1 10 100Frequency (Hz)
0
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mp
litu
de
(v)
0.1 1 10 100Frequency (Hz)
93
94
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98
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100
101
Resis
tivity (
m)
Figure 13
(a) (b)
105
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0
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()
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()
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()
Figure 14
0=100 m
m=0. 2
c=0.75
=0.001 sec
106
CR1Dinv: A Matlab program to invert 1D Spectral Induced Polarization data for Cole-Cole model include electromagnetic effects
A. Ghorbani*, C. Camerlynck*, N. Florsch*
* UMR 7619 « Sisyphe », Université Pierre et Marie Curie, Paris, France.
Abstract: An inversion code in Matlab is constructed to recover the parameters of Cole-Cole model from spectral induced polarization (SIP) data in a 1D earth. In a spectral induced polarization survey the impedances at various frequencies are recorded. Both induced polarization and electromagnetic coupling effects occur simultaneously in this frequency bandwidth, the latter being more and more dominate when frequency increases. We used CR1Dmode code published by Ingeman-Nielsen and Baumgartner (2006) as forward modeling. This code solves electromagnetic responses in the presence of complex resistivity effects in a 1D earth. A homotopy method is designed to overcome the local convergence of normal iterative methods. In addition, to further condition the inverse problem, we incorporate standard Gauss-Newton (or Quasi Newton) methods. The graphical user interfaces allows easy entering the data and the a priori model and also cable configuration. We present two synthetic examples to illustrate that the spectral parameters can be recovered from multifrequency complex resistivity data.
Keywords: Complex resistivity; Cole-Cole model; Spectral Induced Polarization; EM coupling, Homotopy inversion method.
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1. Introduction
Spectral Induced Polarization (SIP) is widely used in environmental and engineering
geophysical prospecting, as well as mineral exploration (e.g., Pelton et al., 1978, Luo and
Zhang, 1998), hydrogeophysics (e.g., Kemna et al., 1999; Kemna et al., 2004; Binley et al.,
2005), organic and non-organic contamination of soils and rocks (e.g., Vanhala, 1997, Abdel
Aal et al., 2006).
The mutual impedances of grounded wires (electromagnetic coupling between the
transmitter, the receiver and the ground) are of prime importance in SIP surveys. At low
frequencies, electromagnetic (EM) couplings and normal polarization effects of the
subsurface material have similar functional behavior with respect to the conductivity of the
earth and their combined effects are recorded in an SIP survey. EM coupling is a major
impediment in the interpretation of induced polarization (IP) data. EM coupling also increases
with the dipole length and dipole separation and with conductivity and frequency (Dey and
Morrison, 1973). For deep exploration, the dipoles and their separation must be large; hence,
the operational frequency is usually low to avoid EM coupling. Unfortunately, the natural EM
field spectrum rises steeply below 1.0 Hz enough (Gasperikova and Morrison, 2001). One
way to avoid EM coupling is measuring SIP data at frequencies low enough that any EM
coupling is either negligible or predictable (Katsube and Collett, 1973; Wynn and Zonge,
1977). Loke et al. (2006) use a regularized least-squares optimization method to recover the
SIP parameters in the 2D problem, but reduce the EM coupling phenomena by limit the
maximum frequency between 10 and 100 Hz. However, elimination of high frequency of IP
spectrum, it eliminate the important information.
Generally for Time Domain methods, one hopes and expects to avoid the problem of
EM coupling by use of the “delay time” (waiting a suitable length of time, a few tens of
milliseconds) after the transmitter is switched off before starting to acquire useful data.
During this delay it is expected that the EM coupling will varnish away to negligible levels.
However, it is not possible to avoid the problem this way in highly conductive environments.
In SIP literature, numerous approaches are to remove EM coupling effects in SIP data
(e.g., Coggon 1984; Song 1984; Pelton et al. 1978; Brown, 1985; Cao et al., 2005; Routh and
Oldenburg, 2001). However, all of these studies deal with Dipole-Dipole electrode array.
A forward modeling code has been developed that is capable of handling several
commonly used electrical and electromagnetic methods in a 1D environment by Ingeman-
Nielsen and Baumgartner (2006). This code calculates the mutual impedance in different
108
frequencies of 1D ground layers for Cole-Cole model and different grounded electrode arrays.
Also, it considers the influence caused by the placement of the wires.
Generally, the liberalized inversion method is used to solve of the nonlinear inversion
(e.g., Tarantola and Valette, 1982b). This method to solve of nonlinear equation like the
mutual impedance equation (Sunde, 1968) is justified when the initial solution is in the
neighborhood of the global minimum of the objective function. This is a reasonable
assumption where a good a priori knowledge of geology of field is available. Indeed, because
of numerous local minima in the objective function it is prevented the iterative optimization
techniques from working effectively.
Homotopy method is a powerful tool for solving nonlinear problems due to its widely
convergent properties (Watson, 1989). Homotopy was first used in geophysical applications
to solve the seismic ray-tracing problem (Keller and Perozzi, 1983). Vasco (1994 and 1998)
used homotopy to solve the inverse problem, illustrated its use for travel time tomography and
used to solve the regularized inverse problems in seismic problems. Everett (1996) applied the
method to solve the inverse electromagnetic problem based on a finite difference modeling.
Jegen et al. (2001) applied the classic Euler–Newton numerical continuation scheme to
inverse problem. Bao and Liu (2003) constructed a homotopy-regularization method to solve
inverse scattering problems with multi-experimental limited aperture data. Han et al. (2005)
used a homotopy method for the inversion of a two-dimensional acoustic wave equation.
The goal of this article is developing of the 1D inversion of IP and EM coupling
integral according to forward modeling code of Ingeman-Nielsen and Baumgartner (2006). A
homotopy method is applied to overcome the local convergence of Gauss-Newton and Quasi
Newton methods.
The plan of this paper is as follows. First, Background on EM coupling theory and forward
modeling and inversion process are reminded. Then, program structure is detailed. At last, the
results of inversion code for the synthetic data are presented.
2. Background on EM-coupling theory and forward modeling
In general, for two pairs of grounded electrodes at the earth’s surface, by definition, the
mutual impedance between the two electrode circuits is the ratio of the voltage in the
secondary to current in the primary.
)(/)()( 12 ωωω IEZ = (1)
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where I1 is the alternating current through the current electrodes and E2 is the electromotive
between the potential electrodes.
This mutual impedance involves a CR contribution due to the earth-return currents and an
induced coupling (IC) contribution due to induction between the wire loops above the ground
surface just as if the loops were not grounded (Brown, 1985). According to Sunde (1968), the
EM coupling between two grounded wires in an arbitrary configuration on the surface of the
earth can be calculated as an integration of mutual impedances of virtual dipoles along the
paths of the wires:
dsdSsS
rQrPZ
B
A
b
a∫ ∫
∂∂∂+= )(
cos)()(2
ξω
)()()()(cos)( BbQAbQBaQAaQdsdSrPB
A
b
a
+++= ∫ ∫ ξ (2)
where A, B and a, b are the end points of the transmitter and receiver wires (the grounding
points), dS and ds are infinitesimal elements of the two wires (the virtual dipoles), ξ is the
angle between the wire elements and r is the distance between them (Figure 1). Usually, when
dealing with EM coupling, the conductivity is considered constant and real and the
permittivity is most often neglected. With these assumptions on a homogeneous half-space,
the Q-function is real, constant (frequency independent), and only depending on the position
of the grounding points of the wires. It is therefore often referred to as the grounding function.
The P-function, although it contains a purely resistive term, is referred as the coupling
function.
One of the parts of forward modeling CR1Dmod (Ingeman-Nielsen and Baumgartner, 2006)
is dedicated to calculating EM response in the presence of CR effects. The software handles
frequency domain coupling for grounded wires. The wires may be divided into any number of
arbitrarily oriented linear segments, provided that the receiver and transmitter wires do not
intersect.
The usual practice is therefore to calculate a half-space response analytically based on the
properties of the first layer and add to this a correction term, which accounts for the summed
effect of the additional layering. Analytical solutions for the Hankel transforms in the P-
function of the grounded wire response are presented for non-magnetic homogeneous half-
spaces. Calculations can be performed using the full solutions, the new non-magnetic
approximation and the traditional quasistatic or low frequency approximation.
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3. Cole-Cole model
Many different models have been proposed for the description of the dispersive behavior of
the complex resistivity (Marshall and Madden, 1959; Van Voorhis et al., 1973; Vinegar and
Waxman, 1984), but the most widely used seem to be the empirical Cole-Cole model, which
was originally developed by Cole and Cole (1941) to describe dielectric dispersion. It was
shown by Pelton et al. (1978) to accurately describe the resistivity dispersion observed in field
data from areas with metallic mineral content. Recent works have shown that the Cole-Cole
parameters can be used to estimate the hydraulic conductivity of sediments (Binley et al.,
2005). The Cole-Cole model is given by:
( )
+−−= cj
mωτ
ρωρ1
111)( 0 (3)
where ρ0 is the resistivity in the DC limit, m is the chargeability, τ is the time constant and c is
the frequency dependence.
The Cole–Cole resistivity model is used in this forward modeling (Ingeman-Nielsen and
Baumgartner, 2006) that we used to inversion method.
4. Homotopy inversion theory
Homotopy is a method used to find solutions to general systems of nonlinear equations
L(P)=0.
A homotopy function H(P,λ) is constructed by adding to the “target function”L(P) a scalar
homotopy parameter λ and a second function g(P), so that
)()1()(),( PgPLPH λλλ −+= (4)
The “start equation” of nonlinear equations g(P)=0 can be chosen arbitrarily, the only
restriction is that it must possess at least one known solution, denoted by P=A.
For the initial value λ=0, the homotopy function H(P,λ) has a solution at A, since H(A,λ) =
g(A) = 0. With the aid of numerical continuation methods, starting from the known point A
and λ =0 a trajectory in (P,λ) space is mapped such that the homotopy function H(P,λ)
vanishes.
A solution P* of the target equation L(P)=0 occurs at any point where H(P,λ)=0, intersects
λ=1 since at these locations H(P*,1)=L(P*)=0. An application of homotopy methods to
geophysics is obvious if a geophysical inverse problem is formulated as a nonlinear equation
of L(P)=0, then all solutions P* satisfying L(P*)=0 that are found by the homotopy path-
tracking algorithm are solutions of the inverse problem (Jegen et al., 2001).
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5. Generalized nonlinear inverse problem
The knowledge of the probability law for each parameter, as shown by the Tarantola and
Valette (1982a):
( )( ))p(g
)p(g)p()p(
d
dpp µ
ρ⋅ρ=σ (5)
where:
(p)pσ is the a posteriori probability density for the vector of parameters p, (p)pρ the a priori
probability density for the data d, d = g(p), is the theoretical relationship may be considered
as exact, ( )g(p)dρ the probability density of the model and ( )g(p)dµ the null information (full
ignorance or homogeneous probability density HPD) on the parameter. This equation solves
the inverse problem for an exact non-linear theory with arbitrary a priori constraint on
parameters (pρ ) and an arbitrary probabilistic distribution of data ( dρ ).
For quasi-linear problems, if the relationship linking the observable data d to the model
parameters p, d = g(p), is approximately linear inside the domain of significant prior
probability, then the posterior distribution is just as simple as the prior distribution. For
instance, a gaussian a priori obviously leads to a gaussian a posteriori. In this case also, the
problem can be reduced to the computation of the mean and the covariance of the gaussian. It
is supposed hereafter, that the a priori information has a Gaussian form.
If the probability distributions are ‘bell-shaped’ (i.e., if they look like a Gaussian or like a
generalized Gaussian), then one may simplify the problem by calculating only the point
around which the probability is maximum, with an approximate estimation covariance matrix.
When one assumes that the probability densities are Gaussian:
( ) ( )
−−−=ρ −prior
1Tprior C
2
1const. ppppexp)p( pp (6)
( ) ( )
−−−=ρ −obs
1Tobs2
1const. ddCddexp)d( dd (7)
where Cp and Cd are covariance matrices parameters and data respectively. If we assume that
the nonlinearities are weak we obtain the followingleast square misfit function:
( ) ( ) ( ) ( )obs1T
obsprior1T
prior dpgCdpgppCppp dp −−+−−= −− )()()S( (8)
We apply the steepest decent algorithm (Gauss-Newton method) for the minimization of S(p)
function. If one considers the gradient αα ∂∂=γpS
, the Gauss-Newton algorithm is an iterative
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algorithm passing from point p to point 1k+p by the following function (Tarantola and
Valette, 2002):
( ) ( )( ) ( )[ ]priork1
obsk1T
k
11k
Tkkk1k C ppCdpgGCGCGpp
ppddppdd−+−×+−= −−−−−
+00000000
1ε (9)
where
priorp the initial estimation for the parameter vector p,
kp the estimated value at iteration step k,
00ppC the parameters covariance matrix (often diagonal by lack of knowledge);
00ddC the data covariance matrix (also diagonal for independent data),
kG and TkG are the derivative matrix of the data with respect to the parameters and its
transpose,
obsd the data vector,
kε is an ad-hoc (real, positive) value adjusted to force the algorithm to converge rapidly (if
kε is chosen too small the convergence may be too slow; it is chosen too large, the algorithm
may even diverge).
In the Gauss–Newton least-squares method, the Jacobian matrix is recalculated for all
iterations. The calculation of the Jacobian matrix can be the most time-consuming step of the
inversion process. In order to reduce the computing time, Loke and Dahlin (2002) used a
quasi-Newton method to estimate the Jacobian matrix values.
Jacobian matrix values can be calculated in first iteration. After each of iteration, the Jacobian
matrix is estimated by using the following updating equation:
Tkkk1k puGG ∆⋅+=+ (10)
Where kp∆ is the perturbation vector to the model parameters,
( ) kTkkkkk /. pppGgu ∆⋅∆∆−∆= , k1kk ggg −=∆ + , 1k+G is the approximate Jacobian matrix
for the (k+1)th iteration, kg is the model response for the kth iteration and kg∆ is the change
in the model.
In theory, the convergence rate of the quasi-Newton method is slower than the Gauss–Newton
method. While the quasi-Newton method might require more iteration to converge compared
to the Gauss–Newton method, the time taken per iteration can be much less. However, we
considered both methods in code.
Hereafter, the following hypotheses are considered:
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• Data, d, here a normally distributed Gaussian.
• Data are assumed to be independent from the parameters; it is simply stating that the
measurement action itself is without influence on the collected data.
• Parameters, p, for each horizontal layer is obtained by Cole-Cole model. Then invert
parameters are ρ0k, mk, ck, and τk respectively, static resistivity, chargeability,
frequency dependence, relaxation time and k is layer index.
• Homogeneous probability density (HPD) is taken constant; this term is not trivial and
Tarantola and Valette (1982a) propose to define HPD on a parameter from the
parameterization invariance with respect to a group transformation. In our case, the
parameters ρ0k and τk are positive and then its logarithmic character lead to state that
HPD for ρ0k is ( )[ ] .constlog k0 =ρµ and for τk is ( )[ ] constlog k =τµ . Here, ( )[ ]klog τµ
shows the null information or HPD for ( )kτlog . We considered
.constm1
mlog =
−µ and .const
c1
clog =
−µ (the change affecting m is
explained in details by Ghorbani et al., 2007).
• There is an explicit relationship d=g(p), where d is the data vector and p the parameter
vector to be sought; in our case, this is provided by equation 2. The relationship
d=g(p) is assumed to be exact.
6. Inversion process
In this section we explain the continuation (homotopy) inversion process considered here.
The input data consist of amplitude and phase at a number of frequencies that are converted to
real and imaginary resistivity values during the inversion processes. The measurements
should be performed for at least four frequencies spread over a suitably wide range since a
single simple Cole-Cole model has four parameters. The parameter vector p finally reduces to
[ ]Tiiii0i h,τ,c,m,ρ where )(ˆi0i0 ρ=ρ log , )
1log(ˆ
i
ii m
mm
−= , )
1log(ˆ
i
ii c
cc
−= , )log(ˆ
ii hh = and i is
layer index.
The first part is the determination of a 1D initial resistivity model from the DC apparent
resistivity values by an approximate inversion. The amplitudes of the apparent-impedance
values that are measured at the lowest frequency can be used as the DC resistivity values. To
invert the i0ρ and ih in initial model, mi is considered zero. Therefore in this case, forward
114
modeling is similar to DC resistivity model. The Gauss-Newton method (equation 9) is then
used to construct the initial resistivity and thickness model.
The second part consist the homotopy method that use Gauss-Newton or Quasi-Newton
methods to minimization of misfit function in each step.
The parameters obtained in the earlier part are used to compute the starting model of the
homotopy inversion. We use a linear combination function of g(P):
)()1(),( 00 PgdPh λ−+λ=λ (11)
Where:
dobs the vector of in-phase and out-of-phase parts of complex resistivity measurements;
g0 the vector of real and imaginary resistivity obtained from forward modeling by starting
parameter vector p0 ;
p the parameter vector and
λ is a real variable that changes between 0 and 1.
Here, it is assumed that the physics-based forward response capable of predicting the data
vector, d=g(p*). The first value of λ is 0 and increases in later steps. It is clear that g(P,λ=0) =
g0(P0) is the same starting model vector. When λ increases, the other model vectors are
estimated so the effect of data measurements vector increases and the effect of the initial
model vector decreases. In this work, path tracking is based on a set of predictor-corrector
steps. The predictor step consists of varying the model vector (P,λ) by ∆λ along the λ-
direction (Benavides and Everett, 2006). We used an isometric division for ∆λ or N
jλ j =
where N,1,j L= as it proves to give satisfactory results (Benavides and Everett, 2006; Han
et al., 2007). The corrector step is a Gauss-Newton (or Quasi-Newton) algorithm that is used
for minimization of the misfit function in each step of homotopy inversion. Once the corrector
step is terminated, the next predictor step is initiated. At the j th predictor-corrector step, the
predicted model vector is (pk, j.∆λ). Assuming that the solution pj of the j th equation has
already been obtained, the successive Gauss-Newton (or Quasi-Newton) method can be used
to solve the (j+1)th equation. Then the iteration formula for solve j th step of homotopy
equation 11 with considering Gauss-Newton equation 9 is following:
( ) ( ) ( )( ) ( )[ ]priork1
jkk1T
k
1
kTkk
jk
j1k ppCphpgCGCGCGpp ppddppdd −+−×+−= −−−−−
+ 00000000,11 λε (12)
where k is iteration index of the Gauss-Newton iterations and T,k,,k L10= .
Or:
115
( ) ( ) ( ) ( )
−+
+
−−×+−= −−−−−+ priork
1obsk
1Tk
11k
1Tkk
jk
j1k N
j
N
j1 ppCdpgpgCGCGCGpp ppddppdd 00000000 00ε (13)
The Jacobian-matrix values are obtained from numerical derivative of forward modelling.
Figure 2 shows the algorithm schema that is used during the inversion process.
7. Program structure
CR1Dinv consists of three main windows: CR1Dinv window, SondagePoint window and
Calculate window. CR1Dinv window controls the starting parameters and its standard
deviation and number of sounding data; SondagePoint window controls the configuration of
the sounding points; Calculate window controls the calculation specific parameters and
inversion routines. The left-hand side of CR1Dinv window features an interactive plot of the
half-space model (Figure 3). By mouse clicking into the plot, the user can insert layer
boundaries and drag them to the desired position. In the right-hand side of the same window,
the lower section gives input fields for the a priori layer parameters, including a priori Cole–
Cole model parameters and its standard deviation, relative permittivity (εr=ε/ε0) and magnetic
susceptibility (χ=µ/µ0-1), it also allows for adding and deleting layers in the model. The upper
right hand section of this window gives control of the measurement configurations. Dipole-
Dipole array and general surface array (GSA) are accessible. GSA allows for arbitrary
location of receiver and transmitter electrodes on the surface of the layered half-space.
Before entering the data, number of sounding points is determinate in this part. To enter the
new sounding data, one selects “new data” item from “File”. SondagePoint window (figure
4), has appeared and allows the user to enter the position of electrodes. At the bottom of this
window, there is a select field to enter the number of frequencies. We note that number of
frequencies is constant for all of sounding points. Therefore, before clicking to topview
button, it is necessary to select the number of measured frequencies. There is a column of
topview buttons on left hand of SondagePoint window. By left and then right clicking the
mouse on the topview buttons, a special window is called that allows the user to place the
electrodes, either by dragging them with the mouse, or by entering new coordinates in the
input fields, in left part of one. The topview windows also let the user add and move segments
of the receiver and transmitter wires. Thus, the influence caused by the placement of the wires
can be taken into account in the frequency domain responses. In the right hand of this window
is an Excel spread sheet that let the user enters the data spectrum (frequency in Hertz,
amplitude in ohm meter, negative phase in milliradian, Amplitude error in percent and phase
error in milliradian).
116
In Calculate window (Figure 5), parameters specific to the type of calculation can be adjusted
before calling the forward modeling routine (Ingeman-Nielsen and Baumgartner, 2006) and
also specific inversion routines. For configurations allowing a choice (array configuration),
the frequency domain calculation is selected as well as the ‘‘full’’ or ‘‘quasi-static’’ modes. In
full mode, CR1Dinv selects either the full solution or the non-magnetic first layer solution
depending on the magnetic susceptibility specified for the first layer. In quasi-static mode, the
program assumes both non-magnetic and quasi-static approximations regardless of the
specified susceptibility (Ingeman-Nielsen and Baumgartner, 2006). Forward modeling and
inversion routines that calculate frequency domain response of the grounded wire
configurations and inverse problem are solved for that, emgsafwd.m and Inversion.m
respectively.
Different parameters that are used in the inversion part of Calculate window are following:
- The homotopy coefficient determines the maximum division number along λ-
direction. λ is the homotopy parameter that varies between 0 and 1.
- The improvement in the RMS error is considered as criterion for termination at each
homotopy inversion step.
- Gauss-Newton and Quasi-Newton options allow using for minimization of the misfit
function in each step of homotopy inversion.
- Iteration adjustment is the same ε ad-hoc (real, positive) value that is explained in
section 5.
After the calculations have ended, results are saved into a binary Matlab file along with the
model, configuration and inversed parameters.
Finally, calculated responses are plotted as Argand diagrams onto the screen. For each layer
plots present convergence processes for each of parameter.
8. Results
The results from two synthetic examples with the inversion method are presented. Minimum
version is a Matlab environment version 7.0.4 (R14). We used 17 frequencies in the 0.183 Hz
to 12 kHz range with a logarithmic step NkHz 2/12 where N is number of frequency used in
the SIP FUCHS-II equipment (Radic Research).
The first example model consists of a half-space earth. Table 1 shows the real and inversion
values of parameters. The amplitude and phase values are calculated for 5 sounding points,
obtained from a Dipole-Dipole array. The distance array line AB=MN=50 m and n, Dipole-
117
Dipole separation, changes from 1 to 5. Transmitter and receiver cables are in same line. The
recovered parameters agree well with the true values. The number of parts in λ-direction is 10
(∆λ=0.1). Root mean square error (RMS) is obtained 0.3%.
We used inversion process for different starting parameter vectors and traced path tracking
curves (the curve that is obtained by parameters versus λ where 0),( =λPh in equation 11).
The results show that all the path tracking curves converges true value (Figure 6).
The second test model consists in a two horizontal layers mode with thickness of upper layer
5 m. Table 2 shows the model and inverted parameters. The amplitude and phase values are
calculated for 5 sounding points, obtained from Schlumberger array. The transmitter lines AB
change from 0.6 to 60 meter. Transmitter and receiver cables are in same line. The other
conditions that are supposed to inversion process are:
- The homotopy inversion consist 5 steps.
- The Quasi-Newton is considered for misfit minimization.
Table 2 shows the recovered electrical parameters agree well with the true values.
9. Conclusion
A homotopy inversion method for SIP data using the Cole-Cole model is proposed
based on electromagnetic formulation of a 1D earth. This method further widens the domain
of convergence of traditional methods. In this work, path tracing is based on a set of
predictor-corrector steps. The predictor step consists of varying the model vector (P,λ) by ∆λ
along the λ-direction. Locally inversion methods of Gauss-Newton and/or quasi Newton are
used as corrector step. Homotopy parameter λ is divided isometric.
An approximate inversion of the DC resistivity model is first carried out to construct the
initial resistivity and thickness models, and this is then used as the starting model in the final
inversion.
The inversion results show the recovered parameters agree well with the true values.
We test the inversion results with field data in a future paper.
Acknowledgement
We are indebted to the INSU-ECCO program (project Polaris II, 2005-2007) in France for
support to this work.
118
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122
Captions
Figure 1: Earth model and grounded wire in common array of two wires on the
surface. A, B and a, b are the end points of the transmitter and receiver wires, respectively. dS
and ds are infinitesimal elements of the two wires, ξ is the angle between the wire elements
and r is the distance between them. ρ*(z) indicates the complex resitivity in Z direction.
Figure 2: Simple schema of inversion algorithm.
Figure 3: CR1Dinv window: a graphical representation of the model, in which the user
can add or delete layers and move boundaries. Right-hand side of the window gives control of
the configuration, a priori layer parameters and a priori layer parameters standard deviation of
the model.
Figure 4: Sounding point window of CR1Dinv features a spread sheet that allows
entering the electrodes coordinate. Left and then right Click the mouse gives access to topwive
windows as edit the cable configuration of each sounding point.
Figure 5: Calculate window of CR1Dinv gives user control over calculation domain,
transform types, tolerances and spline interpolations.
Figure 6: Homotopy path tracking curves of Cole-Cole chargeability parameter on a
complex reistivity half space earth. Homotopy parameter lag ∆λ=0.1 is constant. The
chargeability values vary in starting parameters models.
123
Tables
Table 1: Inversed parameters are obtained during inversion algorithm for synthetic spectral induced polarization data. Earth is proposed half-space. Real values of parameters are presented.
Layers Parameters
ρ0(Ωm) m c τ(s)Standard deviation of
parameters
0ρ m c τRMS error
(%)
Real values 50 0.8 0.3 10-1
Inversion values 1
49.98 0.7998 0.300 0.0998 1 0.5 0.5 1 0.3
Table 2: Inversed parameters are obtained during inversion algorithm for synthetic spectral induced polarization data. Earth is proposed containing two layers. Real values of parameters are presented.
Layers
Parameters ρ0(Ωm) m c τ(s) h(m)
Standard deviation of parameters
0ρ m c τ h
RMS error (%)
Real values 50 0.30 0.30 1 5
Inversion values 1
49.9 0.30 0.30 0.97 4.96 1 0.5 0.5 1 0.1
Real values 300 0.05 0.80 0.01 -
Inversion values 2
299.7 0.05 0.77 0.01 - 1 0.5 0.5 1 -
2.6
124
Figures
Figure 1
125
Figure 2
Hom
otopy algorithm
A priori Entering: N: number of layers
Parameters: ρ0i,mi,ci,τi and hi
Standard deviation of parameters
[ ]TP iiii0i h,τ,c,m,ρ0 =)( 00 Pg
10 ≤λ<
)()1(),( 00 PgdPg λ−+λ=λ
Gauss-Newton or Quasi-Newton algorithms
Initial parameters vector: P Data vector: g(P,λ)
New P
λ=1
Yes
No Increase λ
Determinated P*
DC-resistivity inversion (Gauss-Newton method)
i0ρ andih
Search for starting point
126
Figure 3
Figure 4
127
Figure 5
0 0.2 0.4 0.6 0.8 1λ
0
0.2
0.4
0.6
0.8
1
m
Real parameters m*=0.8, ρ∗0=50 Ωm, c*=0.3, τ∗=0.1 s
initial parameters:m=0.01, ρ0=100 Ωm, c=0.1, τ=10 s
m=0.99, ρ0=100 Ωm, c=0.1, τ=10 s
m=0.49, ρ0=100 Ωm, c=0.1, τ=10 s
Figure 6
128
Non-invasive monitoring of water infiltration in a silty clay loam soil using
Spectral Induced Polarization
A. Ghorbani*, Ph. Cosenza*, S. Ruy#, C. Doussan
#, N. Florsch*
* UMR 7619 Sisyphe, Université Pierre et Marie Curie-Paris6, CNRS, Paris, France # INRA- UMR Climat, Sol Environnement, Avignon, France.
Abstract: An experimental investigation was undertaken to study the ability of Spectral
Induced Polarization (SIP) to monitor water infiltration in a silty clay loam soil. It was based
on the coupled acquisition of tensiometer data and Spectral Induced Polarization (SIP) spectra
(1.46 Hz to 12 kHz) during the infiltration event created by an artificial constant rainfall rate
of about 15 mm/hr.
The approach, which was applied both in the field and in a soil column, confirms the
existence of a significant phase drop in the high-frequency domain (typically greater than 1
kHz) during the first infiltration cycles. The interpretation of the tensiometer and SIP data
show that this phase drop is correlated with the water filling of pores in the [30-85] m
diameter range.
The phase drop is qualitatively and quantitatively interpreted as a Maxwell-Wagner effect
associated with the electrical heterogeneity of the soil. It could correspond to the transition
between two physical states. In the first state before the arrival of the wetting front, highly
polarized and wet aggregates are embedded in an electrically isolating phase, i.e. air. In the
second state after the arrival of the wetting front, structural pores between the aggregates are
filled with a connected and conducting phase i.e. water, leading macroscopically to a decrease
in bulk soil polarizability.
The experimental and theoretical results of this study suggest strongly that the SIP method
can be used to monitor the water filling of structural or draining pores in the field. This
original result requires validation in other sites.
Keywords: Spectral Induced Polarization, complex resistivity, infiltration.
Index terms: 0925 Exploration Geophysics: Magnetic and electrical methods (5109).1835 Hydrology:
Hydrogeophysics. 1838 Hydrology: Infiltration. 1875 Hydrology: Vadose zone
129
1. Introduction
Both dielectric and electrical methods have been used for decades in soil science in order
to quantify the changes in soil water content and soil salinity [e.g., Smith-Rose, 1933].
Dielectric methods, which measure the relative dielectric permittivity in the few dozen MHz
to 1 GHz frequency range, include (a) time domain reflectometry (TDR) probes [e.g., Davies
et al., 1977], capacitance sensors [e.g., Tran Ngoc et al., 1972] and ground-penetrating radar
[e.g., Chanzy et al., 1996]. The electrical methods, which are often called DC (Direct Current)
or low-frequency electrical techniques and measure the electrical conductivity (EC) in the few
Hz to 10 kHz frequency range, include (a) low-cost electric resistance methods (gypsum
block for matric potential measurements) [e.g., Bouyoucos and Mick, 1940] and (b) four-
electrode methods used both in the laboratory and the field [e.g., Tabbagh et al., 2000].
In electrical methods, the measured EC is an in-phase conductivity and is associated with
the in-phase conduction of the reference electric current. However, the measured electrical
impedance is generally a complex quantity with a corresponding in-phase and quadrature
component [e.g., Keller and Frischknecht, 1982; Ward, 1990]. In the frequency domain, this
quadrature conduction is related to a phase shift between the measured voltage and the
applied alternative current. The quadrature conduction at low frequencies (typically from 10
mHz up to 10 kHz) is referred to in geophysics as induced polarization (IP), complex
electrical conductivity or complex resistivity. The method used to measure the spectra of
these complex quantities is called Spectral Induced Polarization (SIP).
Initially, the IP method was developed for detecting small concentrations of disseminated
mineralization in base metal exploration [e.g., Marshall and Madden, 1959; Van Voorhis et
al., 1973]. However, recently, attention has focused on IP in rocks and soils as a means of
determining surface properties indirectly (cation exchange capacity and specific surface area)
[Vinegar and Waxman, 1984; Olhoeft, 1985; Boerner and Schön, 1991; Slater and Glaser,
2003; Lesmes and Morgan, 2001; Slater et al., 2005] and hydraulic conductivity [Boerner et
al., 1996; Lima and Niwas, 2000; Comas and Slater, 2004; Binley et al., 2005; Tong et al.,
2006]. The experimental investigations of Titov et al. (2002) and Scott and Barker (2003)
showed that pore-throat diameters and pore geometry in water-saturated geomaterials
contribute significantly to both in-phase and out-phase conduction at low frequencies. IP
measurements performed on unsaturated sands submitted to various infiltration-drainage
cycles have shown that IP parameters exhibit a complex saturation dependence and are a
function of saturation range and saturation history [Ulrich and Slater, 2004]. Titov et al.
[2004] confirmed the previous results and proposed an IP model for unsaturated sands. This
130
model is able to detect the critical water saturation, which corresponds to the change in pore-
water geometry from bulk water to an adsorbed film existing on the quartz grains. However,
not all of the aforementioned experimental investigations examined natural soils with an
associated broad distribution of particles, pores and aggregate sizes.
In summary, although the IP method has become increasingly popular in environmental
and groundwater investigations [e.g., Binley and Kemna, 2005; Boerner, 2006], its full
potential has yet to be realized in hydrological sciences, especially in topics related to flow
and transport in partially saturated soils.
Indeed, among these topics, the non-invasive characterization of water infiltration in
unsaturated soils is still an active field of research [e.g., Dingman, 2002]. Although the DC
electrical method has been reported to be able to monitor flows in the vadose zone [e.g.,
Hagrey and Michaelsen, 1999; Zou et al., 2001], the IP method has never been used for this
purpose. The aim of this work was to address experimentally the capabilities of the SIP
method for characterizing vertical flows in the vadose zone during an infiltration test.
Our approach is based on both field and laboratory experiments that couple SIP
measurements and hydraulic measurements as performed in a natural agricultural soil.
First, the IP (and hence SIP) method and the underlying physical mechanisms are briefly
reviewed and the various IP parameters are discussed. The field and laboratory experiments
are presented in the second and the third part of the paper, respectively. Finally, the results are
discussed in the fourth part.
2. Induced Polarization: theoretical and phenomenological aspects
2.1 Complex electrical properties
The concepts of complex conductivity and complex resistivity are directly related to
Maxwell’s postulate: the total electric current density Jt is the sum of the conduction current
density Jc and the displacement current density Jd, which are both defined by the following
constitutive equations for time harmonic fields [e.g., Guéguen and Palciauskas 1994]:
EEJ c1
(1)
ED
J d 0it
(2)
where is electrical conductivity, is electrical resistivity, E is applied electric field, D is
the electric displacement, t is time, i is 1 , 0 is the permittivity of the vacuum and is
the complex relative permittivity. Equation (1) is referred to as Ohm’s law.
131
Consequently, by using equations (1) and (2), the total current density can be expressed as
EEEJJJ dct*
1*)i( 0 (3)
where * and * are the complex conductivity and the complex resistivity, respectively.
By comparison with (2), equation (3) can be rewritten in order to introduce an effective
complex relative permittivity eff*:
EJ efft 0i * (4)
The complex electrical properties, *, * and eff* are linked with each other by the
following relationships:
**
* eff0i1
(5)
Generally, these complex properties measured in earth materials are frequency dependent,
i.e. they show an electrical dispersion. They can be split into two parts, i.e. a real component
and an imaginary (or in-phase and quadrature) component. They can also be expressed in
polar form as a magnitude and a phase. For instance, complex resistivity can be written as:
)exp(*'''* ii (6)
where ’, ’’ and * are the real part, the imaginary and the resistivity amplitude of
respectively, and is the phase of that can be also written as follows:
)mradsiftypically('
''
'
''tan 1001 (7)
or from equation (5):
)mradsiftypically('
''
'
''tan
''
'tan
eff
eff10011 (8)
where ’eff and ’’eff are the real and imaginary parts of the complex relative permittivity
respectively. Parameters ’ and ’’ are the real and imaginary parts of the complex
conductivity respectively.
2.2 Polarization mechanisms and models in the low frequency range
The purpose of this section is not to establish an exhaustive summary of the experimental
and theoretical works devoted to polarization mechanisms but to give the main physical
concepts widely used. For a comprehensive review of such works, the reader is referred to
132
Chelidze and Gueguen [1999], Lesmes and Morgan [2001], Santamarina [2001] and
references therein.
In general, any mechanism, which restrains the relative displacement of charges, can be
considered as a polarization mechanism (charge displacement without restriction renders
conductivity) [e.g., Santamarina, 2001]. This leads to an accumulation of negative charges,
(i.e. an electrical pole) on one side of the medium and positive charges (i.e. another pole) on
the opposite charges. In effect, the medium becomes polarized.
In the low frequency ranges (typically from 10 mHz up to 10 kHz), the restrained relative
displacement of charges in geomaterials may have two origins.
The first origin is electrochemical and it is related to the existence of an Electrical Double
Layer (EDL) at the interface between solid and liquid phases. The EDL results from usually
negative electrical surface charges existing in the solid particles and appears at the solid-liquid
interface, as an oppositely charged ion atmosphere. In the EDL, the average concentration of
cations (in the case of negatively charged surfaces) is larger than that of anions. There are two
clearly distinguishable groups of electrochemical mechanisms and models. In the first group,
models have been introduced to account for the dielectric response of dilute suspensions of
spherical particles from Colloid Science [e.g, Lyklema, 1995] (Figure 1a). In the second, the
Marshall-Madden model has been especially proposed for rocks with a dispersed clay fraction
[e.g., Marshall and Madden, 1959]. In this model, rock is considered as a serial connection of
active (ion-selective) and passive (non-selective) zones. Clay minerals coating the quartz
grains or located in pore throats [Ward, 1990] and pore throats themselves associated with an
EDL [Scott and Barker, 2003; Titov et al., 2002] are selective zones (Figure 1b and 1c). The
first case in which clay minerals play a significant role is often called “membrane
polarization” (Figure 1b). These active and passive zones have different transport Hittorf
numbers, which produces local concentration gradients under applied external voltage. These
concentration gradients induce local solute flows and hence an additional electrical current
leading to the phase shift and the frequency dependency of the resistivity observed in IP
measurements [Titov et al., 2002].
The second main origin of low-frequency polarization in earth materials is geometrical
and interfacial (GI). The GI or spatial polarization mechanism results from differences in
conductivity and polarizability among components in a mixture, producing charge
accumulation at the interface. The first calculation related to GI polarization was made by
Maxwell [1891], who considered layered materials (Figure 1d and Figure 1e). As Wagner
[1924] solved the complex permittivity of a dilute suspension of conductive spheres (Figure
133
1f), interfacial polarization is also known as the Maxwell-Wagner effect. This mechanism,
which can be seen as a bulk effect, has a pure macroscopic definition and does not require any
comprehensive understanding of the physical process of charge accumulation at the molecular
level.
The most popular formulation for modeling the Maxwell-Wagner effect in Earth Sciences
is the Maxwell-Wagner-Hanai-Bruggeman (MWHB) equation [e.g., Chelidze and Guéguen,
1999; Lesme and Morgan, 2001; Cosenza et al., 2003], which corresponds to a Differential
Effective Medium (DEM) theory. Consider a mixture of two components: spheroidal
inclusions with an effective complex permittivity i are embedded in a matrix characterized
by an effective complex permittivity m. The effective complex permittivity of the mixture
mix is given by the MWHB equation:
i
m/
mix
m
mi
mixi d1
1
(9)
where di is the volume fraction of spheroidal inclusions, m is a particle shape factor
related to the eccentricity of the spheroidal inclusions and is also called the “cementation
exponent”. Obviously, from equations (5), the MWHB equation (9) can also be expressed in
terms of complex conductivity or complex resistivity. Thus, Samstag and Morgan [1990]
used a similar equation expressed in terms of complex conductivity to model the IP of
saturated shaly sands.
The GI and electrochemical models are not incompatible since they operate at different
scales. For instance, Lesmes and Morgan [2001] proposed a granular model for the electrical
properties of saturated sedimentary rocks that combines both approaches. On the basis of their
model, the MWHB equation was considered as a mixture formula in which the
electromagnetic properties of components, especially the clay fraction, are governed by
microscopic and physico-chemical laws.
3. Field experiments
3.1 Soil, experimental set-up and procedure
Field experiments were undertaken on a silty clay loam soil of an agricultural INRA field
site located near Avignon in Southern France. The A horizon (0-60cm) is made of a tilled
layer (0-30 cm) overlapping an undisturbed layer with centimetric (0-15 cm layer) to
decimetric (15-30 cm layer) polyhedric clods. The main characteristics are given in Table 1.
134
A square plot of 0.99 x 0.99 m was established (Figure 2). A trench about 30 cm deep was
dug around the perimeter of the plot. A PVC wall was placed in the trench against the inside
wall of the plot, except on one side so that possible runoff water could drain away (Figure 2).
Twenty-four automatic pressure transducers connected to micro-tensiometers (2.2 mm in
diameter, SDEC France) were placed in the plot 20 cm and 40 cm apart at depths of 8 cm and
13 cm (Figure 2). The 24 tensiometers were grouped into 4 subgroups by 4 switching boxes
connected to a datalogger (CR10X, Campbell Sci).
SIP measurements were carried out with the SIP-FUCHS II device (Radic Research)
which measures complex resistivity over 7 decades of frequency (1.4 mHz up to 12 kHz) with
4 electrodes. It consists of two remote units that record current I injected by two electrodes A
and B and voltage U signals measured between two electrodes M and N (Figure 2). The
apparent complex resistivity * is given by:
)(ie)(*I
Ua)(* 2 (10)
where (i2=-1); a is the electrode separation; )(* and )( are the resistivity
amplitude and the phase respectively. They are a priori a function of the angular frequency .
Equation (10) gives an “apparent” value of the resistivity, which is the resistivity of a
homogeneous ground that will give the same impedance for the same electrode array.
To measure the whole spectrum of both parameters, )(* and )( , the SIP FUCHS-
II apparatus starts with the highest frequency, 12 kHz, and the N other decreasing frequencies
are obtained by the following division: 12 kHz/2N. Optical fibers are used in order to
minimize electromagnetic cross-couplings between the transmitter and the receiver and to
ensure a system-wide synchronization. The measured data are transferred to the base unit,
where the impedance amplitude and the phase shift are determined. The SIP FUCHS-II is
connected to a computer in order to record the data and to display the results in real time.
To minimize the so-called “electrode polarization” associated with electrochemical
reactions occurring at the electrode-sample boundary [e.g., Chelidze et al., 1999], the A and B
current electrodes were Cu/CuSO4 electrodes whereas the M and N electrodes were Pb/PbCl2
electrodes [e.g., Petiau, 2000].
With this experimental set-up and within the impedance range of 1-10 kOhm, the absolute
impedance error is less than 0.1 %. Within the same impedance range and for frequencies less
than 1 kHz, the phase error was estimated to be less than 1 mrad. When the frequency is in the
range of 1 kHz to 12 kHz, the phase error increases to 1 mrad.
135
The electrodes were vertically installed in the plot. A Wenner array was used with the
electrodes 20 cm apart. In this array, the four electrodes are collinear and the intervals
between adjacent electrodes are equal (Figure 2). If a homogeneous soil is being studied, the
depth of investigation of such an array, for which the sensitivity to vertical changes in
electrical properties is maximum, is about 10.4 cm, i.e. ABx0.519 [e.g., Edwards, 1977],
which is between the two depths of the tensiometers. Nevertheless, it should be borne in mind
that: (a) the value of 10.4 cm is an order of magnitude used for planning the field experiment
since the studied system (water front displacement in a dry soil) is intrinsically
heterogeneous; (b) the measured amplitude and phase are complex averages of these
parameters over the depth of investigation, so they are not local measurements, as is the case
for tensiometer measurements.
Water infiltration in the soil was achieved by a rainfall simulator (Institut de Recherche
pour le Développement-IRD model) which provided a constant rainfall rate in the range of
10-100 mm.hr-1
. Fresh water taken from the water table located at the site was used as the
wetting fluid for all experiments. Its conductivity was equal to 930 S/cm at 20 °C.
The field experiment was conducted in two main phases (Figure 3). First, an infiltration
test (Test 1) with a rainfall rate of 15.4 mm/h was performed for about two hours. Before,
during and after this rainfall event, an electrical resistivity spectrum between 1.46 Hz to 12
kHz was measured every two minutes. As the soil was initially in a dry state (initial
volumetric water content of 0.240 m3.m
-3, which corresponds to a matric potential lower than
– 10 m), tensiometers were inserted into the soil only 30 min before the beginning of the
rainfall simulation in order to avoid their desaturation. It follows that the first pressure
measurements are not representative of the soil water matric potential as the tensiometers
were not in equilibrium with the soil water. Representative measurements are obtained only
after the arrival of the wetting front at the depth of insertion. In the second phase (Test 2 in
Figure 3) one day later, a new infiltration test with a higher rainfall rate of 32.4 mm/h was
carried out for about one hour. The soil was wet (matric potential about -0.35 m which
corresponds to a water content of 0.350 m3.m
-3). Electrical resistivity spectra were measured
again with the same procedure as in Test 1, except for the frequency range: 0.732 Hz to 12
kHz. During both phases, the soil water matric potentials were recorded with a 10-s sampling
interval.
136
3.2 Results
During Test 1, no surface ponding occurred at any time. The rainfall rate was equal to the
infiltration rate into the soil. The matric potential measurements located at the same depth, as
indicated in Figure 4 for six tensiometers, showed significant differences in terms of
amplitude and temporal dynamics. This spatial variability is also illustrated in Figure 5 where
the times corresponding to the arrival of a wetting front (i.e. fast increase of the matric
potential, becoming less negative) are plotted. Both figures confirm an infiltration process
associated with a significant spatial heterogeneity of the structured soil studied.
To compare these results with geophysical measurements in order to obtain average
values of SIP parameters over about 10 cm, tensiometer values were averaged for the two
groups of 12 tensiometers located at the two depths, 8 cm and 13 cm (Figure 6). The average
value at 13 cm clearly shows that, during water infiltration, the initial and rapid increase of
the matric potential (at 22 min) was stopped at about 50 min, after which the matric potential
evolved more slowly. The time of 22 min (respectively 50 min hereafter referred to as td) is
interpreted as the average time for the wetting front to reach the depth of 13 cm (resp. as the
average time to reach the steady state).
During Test 1, both components (phase and resistivity amplitude) of the complex
resistivity spectrum vs. time are given in Figures 7 and 8. The resistivity amplitude decreased
during water infiltration and increased slowly when rainfall was stopped (Figure 7). The good
sensitivity of resistivity amplitude to soil water content changes is already known: resistivity
is a decreasing function of soil water content [e.g., Rhoades et al., 1976]. Moreover,
resistivity amplitude measurements show a very weak frequency dependence: the observed
resistivity spectra were flat. This classical feature justifies the use of low-frequency electrical
techniques for measuring soil Direct Current (DC) resistivity [e.g., Tabbagh et al., 2000].
In comparison with amplitude, phase displayed a more complex behavior (Figure 8). Early
during water infiltration, the phase angle increased slowly but significantly as the water
content increased in the first centimeters of the soil. However, at time td (see arrow in Figure
8), the phase measured in the high frequency domain had dropped significantly with a higher
level of noise. This rapid drop, which was not observed in the lower part of the frequency
range, is related to the arrival of the wetting front in a zone where the sensitivity of the
electrical parameters is maximal. Since this drop mainly concerns the phase measurements at
time td and is enhanced in the higher part of the frequency range, a GI polarization mechanism
was suspected to occur during the wetting front diffusion. This assumption will be discussed
137
on the basis of results provided by a laboratory investigation and a subsequent model, both of
which are presented in the last section.
During infiltration, the phase continued to decrease slowly with a significant level of
noise, likely related to the heterogeneity of the soil and the associated water movement. When
the rainfall stopped, the phase returned almost to the initial level existing before the phase
drop, and thereafter decreased slowly during the drainage phase. It should be noted that this
phase reversibility was not associated with a hydraulic reversible behavior, as indicated in
Figure 6.
Concerning Test 2, the matric potential prior to infiltration (Figure 9) was higher
compared to Test 1 before the application of the rainfall rate. One-day drainage, redistribution
and evaporation did not restore the initial soil-water state that existed before Test 1. A small
amount of ponding was observed during Test 2. The resistivity spectra shown in Figure 10
were still sensitive to water content changes induced by water infiltration. The phase values
were lower than those obtained during Test 1 (Figure 11). Moreover, contrary to Test 1, the
phase spectra did not show any significant correlations with the transient matric potential
variation given in Figure 9. Indeed, the phase drop observed during Test 1 was not detected
during Test 2. This physical phenomenon is consequently related to a dry soil and possibly to
a characteristic size of air-filled pores, above which the phase drop would not exist. This
aspect will be discussed hereafter.
In summary, the results of this field experiment confirm the practical interest in using
resistivity measurements to monitor soil water content but also demonstrate a surprising phase
drop occurring during water infiltration in a dry soil. This peculiar pattern associated with
phase measurements has not observed in the literature to date and requires further
experimental and theoretical work.
4. Column experiments
4.1 Objective, experimental set-up and procedure
The objectives of this laboratory study were: (a) experimental confirmation of the phase
drop measured in the field; (b) characterization in controlled conditions in order to
substantiate a physical mechanism.
The laboratory PVC column (Figure 12) was 280 mm high, had a 150 mm internal
diameter and was filled with undisturbed core soil from the field experiment. Soil sampling
was carried out carefully in order to obtain an undisturbed and representative soil sample in a
dry state. The first 15 cm layer of the column was cloddy whereas the bottom part of the soil
138
core was more massive. The column contained 10 micro-tensiometer ports located at depths
given in Figure 12.
The infiltration events were simulated by a rainfall simulator which was calibrated to
provide a rainfall rate similar to that applied during the field experiments (i.e. 15 mm/hr). The
simulator area was close to that of the column in order to constrain the wetting profile to one
dimension. The base of the column was not sealed in order to maintain a drained condition.
The drainage water was weighed and its electrical conductivity was measured. The electrical
conductivity of the infiltration water used for the experiment was 660 µS cm-1
.
A laboratory complex resistivity system (1253 Gain-Phase analyzer Solartron
Schlumberger) was used for electrical data acquisition. Four Cu/CuSO4 electrodes were
positioned in a horizontal plane located at 7 cm depth (Figure 12) where the soil structure was
the most similar to that in the field. This electrode configuration offers two advantages. First,
it makes it possible to test a very different configuration compared to that used in the field, i.e.
the sensitivity of the geometric electrode array compared to the main direction of the water
infiltration can be estimated. Second, its maximum sensitivity is in the electrode plane, i.e. the
influence of the physical processes located at the top and the bottom of the column is
minimized. Three high frequencies were considered: 93.5 Hz, 187 Hz and 1.5 kHz in order to
speed up the acquisition process with a high rate of sampling. Note that all three frequencies
were also used by the SIP FUCHS-II field device. Moreover, in this laboratory configuration,
the apparent complex resistivity is not given by equation (10): a new geometric factor, i.e. the
2 a term in equation (10), had to be calculated. This factor related to the electrode
configuration was quantified numerically by solving the Laplace equation with a finite-
difference scheme.
With this experimental set-up and within the impedance range of 1-10 kOhm, the absolute
impedance error and the phase error were estimated to be less than 0.1 % and 2 mrad,
respectively.
The laboratory experiment was conducted in two cycles (Figure 13). Each cycle consisted
in an infiltration step by application of rainfall and a subsequent free drainage step with no
applied rainfall. During both infiltration and drainage cycles, the matric potentials and the SIP
parameters were recorded with a 2-minute sampling interval.
139
4.2 Results
Figures 14 and 15 show the matric potential variation during the first infiltration-drainage
cycle. The results confirm that the water flow is vertical, one-dimensional and homogeneous
at least until a depth of about 10 cm (Figure 14). On the other hand, the matric potential
readings in the lower part of the column (Figure 15) strongly suggest that the infiltration
became complex and preferential: the tensiometer C5 located at a depth of 20 cm reacted
before the tensiometer D5 located at a depth of 17 cm. Moreover, the response time lag
between tensiometers D1 and C6 located at the same depth (23 cm) was greater than 10
minutes (Figure 15).
Figure 16 shows the evolution of the resistivity amplitude during the first infiltration-
drainage cycle for the three frequencies, 93.7 Hz, 187 Hz and 1.5 kHz. The results again
validate the good sensitivity of resistivity to soil water content changes, i.e. the resistivity
evolution reacted immediately when the rainfall rate was applied or stopped. They were also
weakly frequency-dependent, as were the field measurements. Nevertheless, the results show
two features that were not observed in the field: (a) a resistivity peak was recorded at time 20
min; (b) the resistivity began to increase at time 66 min during water infiltration, whereas the
soil water content continued to increase.
The first feature, i.e. the resistivity peak, is clearly associated with the arrival of the water
front at the maximum sensitivity plane of the electrode array (see the C2 tensiometer
evolution located at the same depth of 7cm). It can be interpreted as a classical “à-coup-de-
prise” which is usually observed in electrical investigations when a current electrode (A or B)
is placed in contact with a highly resistive or conductive heterogeneity. This phenomenon is
enhanced when high and shallow contrasts of resistivity exist at the soil surface. Moreover, it
is interesting to note that a similar peak was observed in a rather different context during the
migration of a hydrocarbon liquid through a porous medium within a laboratory column
[Chambers et al., 2004].
The second feature (the increase of resistivity) is clearly related to the first arrival of water
at the base of the column (Figure 16). Since resistivity is primarily a function of salinity and
water content, the resistivity increase is here interpreted as a decrease in the salinity of the soil
solution. As the infiltration process evolves, the percolating fresh water interacts
geochemically with the solid and the initial water of the micropores, which is usually more
concentrated [e.g., Blackmore, 1978]. As the seepage began at the base of the column and as
fresh water was provided continually into the column, the pore water became more and more
140
diluted by the fresh water. The EC of the retrieved water during the first 2 hours of drainage
was greater (1713 µS/cm) than the irrigation water (660 µS/cm).
The phase evolution during the first infiltration-drainage cycle is shown for the three
frequencies in Figure 17. The values of the tensiometer C2 located at the same depth (i.e.,
7cm) are also given in the same figure. In comparison with the field measurements, the
laboratory results confirmed the rapid phase drop during the water infiltration. This phase
drop, which is clearly associated with the water front past the plane of electrodes, is also
frequency-dependent and is higher for the highest frequency (~ 15 mrad at 1.5 kHz). As for
the resistivity measurements, an inverse phase peak was also detected, which might have the
same cause, i.e. an “à-coup-de-prise”.
However, contrary to the field experiment, an additional feature was present, i.e. the
transient phase evolution was less noisy and erratic than that measured in the field. In our
opinion, this feature seems to confirm the relationship between the electrical noise recorded
for the phase measurements and the level of heterogeneity of the studied system. The good
signal-to-noise ratio obtained in the laboratory is likely due to the location of the zone of the
electrical maximum sensitivity in a homogenous part of the soil column, whereas the poor
signal-to-noise ratio obtained in the field was mainly the consequence of preferential flow
water in a heterogeneous soil at the scale of the electrical set-up (see Figures 5a and 5b).
Moreover, contrary to the field experiment, the phase did not increase when the rainfall
stopped. This point can be explained by the relatively good homogeneity of the soil column in
the plane of the measurement electrodes. Contrary to the field experiment where macropores
with different sizes played a significant role, the soil involved in the column experiment was
likely associated with much smaller draining pores, which were still saturated when the
rainfall stopped. On the other hand, in the field experiment, the numerous larger pores drained
off rapidly as the rainfall ended. Consequently, as discussed hereafter, this comparison
suggests that draining pore size is a key parameter in understanding the evolution of the
measured polarizability.
The resistivity amplitude and the phase measured during the second infiltration-drainage
experiment are plotted in Figures 18 and Figure 19, respectively. As was the case during the
first cycle, the evolution in resistivity amplitude exhibited the same transient signature with
the same salinity effect during the water seepage at the base of the column. However, no
resistivity peak was observed. This can be explained by a much lower contrast between the
conductivity of the percolating water and that of the background.
141
Similarly to the field experiment during Test 2, no significant phase drop was measured
during the second infiltration-drainage cycle: the phase measurements were almost
independent of the infiltration history of this cycle.
In summary, the laboratory measurements confirmed that the field observations: the SIP
parameters, i.e. the resistivity amplitude and the phase, are saturation-dependent [e.g., Ulrich
and Slater, 2003]. Moreover, the results validate the phase drop evidenced during water
infiltration in the field in a dry soil. The existence of this phase drop does not depend on the
electrode arrangement relative to the major direction of the water flow. At this stage, in order
to use the SIP approach for characterizing vertical flows in soils, it is necessary to provide
some physical hypotheses to explain this new and original result. This is the main objective of
the following section.
5. Discussion
Our purpose in this section is not to provide a quantitative model of the field and
laboratory experiments. There are at least two reasons for this. Firstly, from a hydrological
point of view, the results show that the vertical flows were complex and likely preferential in
both cases. Moreover, many of the hydraulic parameters (hydraulic conductivity, distribution
of the initial porosity and water content etc.) are unknown. Secondly, from a geophysical
point of view, complex apparent resistivities were measured that are not intrinsic soil
parameters: these apparent parameters depend on the electrode arrangement so a
multielectrode system with a specific inversion model would be necessary in order to obtain a
spatial distribution of the intrinsic parameters.
The ensuing discussion focuses on the phase drop measured in both experimental studies,
which showed the following features:
The phase drop exists whatever the electrode arrangement (at least for the two
configurations used in this study). It seems to be enhanced when the plane of the
electrodes is perpendicular to the main direction of the water front.
It occurs only if the soil is initially in a dry state, typically here with a matric
potential lower than a few meters (Figures 6, 14 and 15). Indeed, both in the field
and the laboratory, the following infiltration–drainage cycle did not exhibit any
measurable phase drops.
The second point suggests that, for this particular soil, a characteristic size of water-filled
pores might exist, above which the phase drop vanishes. This characteristic size, rc, can be
142
bounded by two others, rmin and rmax, which are respectively the smaller pore radius value
estimated from the first infiltration-drainage cycle (exhibiting a phase drop) and the higher
pore radius value estimated from the second infiltration-drainage cycles (with no phase drop):
rmin < rc < rmax, (11)
Both bounds can be quantified from the Jurin equation:
iw hg
cosTr
2 (12)
where T is surface tension (=0.075 N/m for an air-water interface), w is density of water,
g is the gravitational acceleration, is the angle contact (taken equal to 0, in our case) and hi is
the matric potential measured prior to the application of the rainfall during the first
infiltration-drainage cycle (to calculate rmin) or during the second infiltration-drainage cycle
(to calculate rmax). To obtain the narrowest interval in the inequality (11), the tensiometer
values from the field experiment were preferred since they provided the lowest initial matrix
potential (the smaller pore radius) in the first infiltration-drainage cycle and the highest initial
matrix potential (the highest pore radius) in the second infiltration-drainage cycle.
By using equation (12) and the field results (figured 6 and 9), the parameters rmin and rmax
are respectively 14 m (hi =-1.1 m) and 43 m (hi =-0.35 m), respectively. Consequently, the
corresponding pore diameter range is about [30-85 m]
An interesting point arises from this estimation: this estimated range can be referred to as
the pore size of mesopores [30-75 m], in accordance with the Soil Science Society of
America’s terminology (Table 2). Although the class limits given in Table 2 are necessarily
somewhat arbitrary, this estimated range of [30-85 m] also corresponds to the normally
draining pores (or transmission pores) [Greenland, 1981; Kay and Angers, 2000] or to
structural pores [Stengel, 1979]. Consequently, this statement strongly suggests that the
measured phase drop is related to the transition of the water filling of mesopores or structural
pores. In the following, the term “structural pores” is used since it covers a broader class of
transmission draining pores (see Table 2).
Nevertheless, at this stage, an important question remains: what is the physical origin of
this phase drop that occurs as the water fills the structural pores? This question can also be
reformulated by two others: Is the observed phase drop associated with a polarization
mechanism? If so, among the different polarization mechanisms described previously, what is
the relevant process?
143
To answer these questions, it may be relevant to express our results in terms of complex
permittivity or complex conductivity *. As previously shown in experimental works
[Chelidze et al., 1999; Slater and Glaser, 2003; Ulrich and Slater, 2004], polarization
magnitude is primarily included in the imaginary part ’’ (or in the real part ’eff) whereas the
complex path of charge carriers in porous materials, i.e., charge transfer phenomena, is related
to the real part ’ (or in the imaginary part ’’eff). In other words, the complex resistivity *
and especially the phase , are not direct measurements of polarization. In fact, as shown in
equation (8), the phase almost defines the polarization magnitude relative to the conduction
magnitude.
Figures 20 and 21 show the evolution of the imaginary part ’’ and the real part ’ of the
complex conductivity measured in the field and the laboratory for the 1.5 kHz frequency.
Both figures confirm that the phase drop is clearly due to a decrease in the imaginary
component ’’ i.e., a decrease in soil polarizability as the infiltrated water filled the structural
pores.
Since the phase drop is not related to the filling of micropores and is enhanced at high
frequencies, it is safe to assume that the underlying polarization process associated with the
phase drop is not purely electrochemical in relation with the EDL but rather a GI polarization.
As far as a GI mechanism is concerned, the simplest depolarization effect related to the phase
drop is a two-step process (Figure 22):
1. Before the arrival of the wetting front (stage 1 in Figure 22), water is mainly in the
microporosity. The wet aggregates associated with the fine fraction of the soil
generate a significant polarization magnitude whose physical origin is likely
complex (i.e., combination of GI and EDL effect) and will not be discussed in
detail here. Nevertheless, it is easy to imagine that during the application of an
external electrical field, the interfaces between the wet aggregates and the
interaggregate pores filled with air would constitute electrical barriers in which a
significant amount of electrical charge carriers would be blocked, leading to the
polarization of the wet aggregates.
2. The fresh water fills the structural pores bypassing the wet polarized aggregates,
leading to a decrease in the bulk polarization amplitude (stage 2 in Figure 22).
This process can be enhanced by the release of a large part of the charge carriers
blocked in the aggregates by diffusion in the water-filled structural pores.
144
This kind of Maxwell-Wagner effect (decrease in bulk polarizability due to filling of
structural pores) can be illustrated by a simple calculation based on a MWHB model. In this
approach, the polarized wet aggregates composed of the soil fine fraction and the micropores
(with possibly smaller pores) are modeled as oblate inclusions alternatively embedded in air
(stage 1) and in fresh water (stage 2). The complex conductivity of the soil * corresponding
to both stages is calculated from equation (9) (MWHB model) expressed in complex
conductivities:
ag
m/*
w,a
*
w,a
*
ag
*
agd
*
*1
1
(13)
where dag is the volumetric fraction of aggregates; *
w,a is alternatively the complex
conductivity of air (stage 1) or water (stage 2); *
ag is the complex conductivity of the wet
polarized aggregates that is considered as a fitting parameter since the polarization
mechanisms related to the sole aggregates will not be discussed. As in Samstag and Morgan
[1990], the imaginary part ,,
ag of the aggregates was estimated from the literature by
assuming that the dielectric behavior of the aggregates with microporosity is similar in order
of magnitude to that of a clay-rich rock with a very small structural porosity (i.e., argillite).
The range of values of parameters *
w,a and *
ag is given in Table 3. The results of a
sensitivity analysis (not shown) demonstrated that a change in +/- 50 % of the ,,
ag value
given in Table 2 had a low effect on the soil bulk imaginary conductivity. The particle shape
factor (often called “cementation exponent”) for all inclusions m in equation (13) is taken to
be equal to 2 [e.g., Rhoades et al., 1976; Revil et al., 1998].
A numerical application of this modeling approach is given in Figure 23 for two high
frequencies, 1500 Hz and 150 Hz. The real conductivity of the aggregates has been chosen to
be two-fold higher than the real conductivity of water. The volumetric fraction of wet
aggregates has been chosen in the range of 60-90 % since the structural porosity is typically in
the range 10-40 % in clay loamy soils [e.g., Stengel, 1979].
The results in Figure 23 clearly show a drastic decrease in the soil imaginary conductivity
between the two stages (before and after structural pore filling) for a frequency of 1.5 kHz.
Compared to the experimental decrease, this calculated decrease is possibly overestimated by
the underlying simplifications used in the model: in reality, from stage 1 to stage 2, charge
carriers can also be transferred efficiently by diffusion in the water-filled structural pores and
between aggregates, which may contribute to lowering the bulk polarizability. These aspects
145
are not rigorously taken into account in the modeling process. Nevertheless, despite the
uncertainties included in the calculations, this model is in agreement with the experimental
trend; the drastic modeled decrease could explain quantitatively the phase drop observed in
the field and in the laboratory.
Moreover, one may wonder whether the MWHB equation is able to model the frequency
dependence of the measured electrical properties. In other words, can this equation show that
the observed phase drop is enhanced in the high part of the used frequency range? To answer
this question, one has to introduce into the model a relationship between the electrical
properties of aggregates and the frequency: indeed, there is no reason to suppose that the
electrical properties of aggregates do not depend on frequency. Unfortunately, this
relationship is unknown and is a priori difficult to obtain. However, a value of the relative
dielectric permittivity of aggregates equal to 108 at 150 Hz (Table 3), which is not an unusual
value for dense clayey rocks at low frequencies [e.g., Comparon, 2005; Cosenza et al., 2007],
leads to obtaining a lower phase drop compared to that calculated at 1500 Hz (Figure 23).
Although this calculation is not a validation of the proposed model, it confirms the
experimental trend i.e., the decrease in the observed phase drops with frequency. At this
stage, we are aware that further experimental and theoretical investigations are required to
fully validate this approach with regard to the frequency-dependent phenomena observed. In
particular, we need to understand and to model the low frequency electrical spectra of highly
clayey materials.
6. Conclusion
In situ and laboratory experiments were undertaken to estimate the capabilities of the SIP
method to characterize vertical flows in the vadose zone during water infiltration. These
experimental investigations had two original features: (a) they were performed in the same
silty clay loamy soil and (b) they coupled SIP and tensiometer measurements.
During both experiments, the evolution in phase variation showed a significant drop at
high frequencies (typically greater that 1 kHz) during the first infiltration cycles. These phase
drops were correlated to the water filling of pores whose equivalent diameters were estimated
to be in the range of [30-85 m]. This range might be related to structural porosity in this
fine-grained soil.
146
These phase drops were qualitatively and quantitatively interpreted as a GI mechanism,
i.e. the decrease in polarization amplitude is mainly due to the filling of a conducting liquid in
the structural pores between high polarized and wet aggregates.
This work strongly suggests the need for further experimental and theoretical
investigations in two directions. Firstly, from a practical point of view, the SIP method seems
to be able to monitor the filling of drainage pores and possibly in the field, thus providing
indications about soil structural features. However, further field work is needed in different
sites and other hydrological situations to validate this capability. Secondly, from a
fundamental point of view, there is still no comprehensive understanding of the polarization
mechanisms involved at the different pore scales (from micropores to macropores) so the SIP
method cannot yet be used efficiently in the field. This is why theoretical investigations on the
fundamental polarization mechanism that occurs in earth materials must remain an active field
of research.
Acknowledgements. This research was supported by the Agence Nationale de la
Recherche (ANR) –ECCO programs (POLARIS II Project: “Polarisation Provoquée
Spectrale” – Spectral Induced Polarization and «Flux d’infiltration, recharge et hétérogénéité:
une approche multi-échelle de la colonne à la parcelle» project n° 161 ACI ECCO 2003).
147
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151
Captions
Table 1 Main mean (n=2) characteristics of the soil for the 30 cm upper layer. Corg =
content of organic carbon.
Table 2. Pore size classification and corresponding functional descriptions (Soil
Science Society of America, 1997, from Kay and Angers, 2000; modified from Greenland,
1981).
Table 3. Typical range of values for the complex electrical conductivity of the three
phases (air, water and aggregates). The parameters ’ag and ’’ag are estimated from the
literature by assuming that the dielectric behavior of the aggregates is similar to that of a clay-
rich rock.
Figure 1. Electrical polarization mechanisms. (a) Electrical Double Layer (EDL)
polarization. The figure illustrates excess and deficiency in ion concentration around a
polarized particle. Dotted line indicates the local diffusion flows of both cations and anions;
dashed line gives the local diffusion flows near the solid surface. (b) Membrane polarization.
Clay particles at pore throats constitute ion-selective zones. The EDL is not drawn for clarity.
(c) Polarization associated with pore throats. Pore throats with corresponding EDL constitute
ion-selective zones. The EDL is not drawn for clarity. (d) No interfacial polarization: the
layered medium is parallel to the electric field; (e) Maxwell (interfacial) polarization: the
layered medium is perpendicular to the electric field; (f) Wagner (interfacial) polarization.
Dielectric host with conductive inclusions.
Figure 2. Schematic view of the field experimental set-up. A. Cross-section B. Top
view.
Figure 3. Schematic view of the experimental procedure.
Figure 4. Matric potentials measured in 6 tensiometers (T1 to T6) vs. time during Test
1. Time 0 corresponds to the beginning of the rainfall event of Test 1. The arrow indicates the
arrival time of the wetting front at the tensiometer.
Figure 5. Arrival time for the wetting front observed for the 12 tensiometers located 8
cm deep (6a) and 13 cm deep (6b).
Figure 6. Mean matric potential at two depths (8 cm and 13 cm) vs. time (Test 1). The
beginning and the end of rainfall application are also shown. The arrow indicates a
152
characteristic event which is compared to SIP measurements (see Figures 7 and 8). Typical
errors bars are also given.
Figure 7. Resistivity amplitude spectra vs. time (Test 1). The beginning and the end of
rainfall application are indicated by dashed lines. Note that at about time 145 minutes, a larger
spectrum (from 91 mHz up to 12 kHz) was measured.
Figure 8. Phase spectra vs. time (Test 1). The beginning and the end of rainfall
application are also given. For clarity, the spectra below 187.5 Hz are not numbered. Typical
errors bars are also given. The phase drop indicated by an arrow is related to the average time
for the wetting front to reach a steady state (see Figure 6).
Figure 9. Mean matric potential at two depths (8 cm and 13 cm) vs. time (Test 2). The
beginning and the end of rainfall application are also given.
Figure 10. Resistivity amplitude spectra vs. time (Test 2). The beginning and the end
of rainfall application are also indicated by dashed lines. For comparison, the same scale of
amplitude used in Figure 7 has been used.
Figure 11. Phase spectra vs. time (Test 2). The beginning and the end of rainfall
application are also given. For clarity, not all the spectra are numbered. For comparison, the
same scale of phase used in Figure 8 has been used. Typical error bars are also given.
Figure 12. Schematic diagram of laboratory column in (a) section and (b) plan view
(not to scale). Distance of tensiometer from soil surface is given in brackets.
Figure 13. Schematic view of laboratory experimental procedure.
Figure 14.Tensiometer values at three depths (4 cm, 7 cm and 10 cm) vs. time (Cycle
1). At time 0, the rainfall rate was applied. The end of rainfall application is also indicated by
a dashed line.
Figure 15.Tensiometer values at three depths (17 cm, 20 cm and 23 cm) vs. time
(Cycle 1). At time 0, the rainfall rate was applied. The end of rainfall application is also
indicated by a dashed line.
Figure 16. Resistivity amplitude spectra vs. time (1st infiltration-drainage cycle). The
beginning and the end of rainfall application are indicated by dashed lines. The time related to
the seepage at the bottom of the column is indicated by an arrow.
153
Figure 17. Phase spectra vs. time (1st infiltration-drainage cycle). The values of the
tensiometer located at the same depth (7cm) are also given. C2 The beginning and the end of
rainfall application are indicated by dashed lines. The time related to the seepage at the
bottom of the column is indicated by an arrow. Typical error bars as a function of frequency
are also given.
Figure 18. Resistivity amplitude spectra vs. time (2nd infiltration-drainage cycle). The
beginning and the end of rainfall application are indicated by dashed lines. The time related to
the seepage at the bottom of the column is indicated by an arrow. For comparison, the same
scale of amplitude used in figure 17 has been used.
Figure 19. Phase spectra vs. time (2nd
infiltration-drainage cycle). The beginning and
the end of rainfall application are indicated by dashed lines. The time related to the seepage at
the bottom of the column is indicated by an arrow. Typical error bars as a function of
frequency are also given.
Figure 20. Imaginary conductivity vs. time for field and laboratory experiments
measured for a high frequency (1.5 kHz) (1st infiltration-drainage cycle). The times
corresponding to the observed phase drop are indicated by arrows.
Figure 21. Real conductivity vs. time for field and laboratory experiments measured
for a high frequency (1.5 kHz) (1st infiltration-drainage cycle). The times corresponding to
the observed phase drop are indicated by arrows.
Figure 22. Schematic describing the two physical conditions (stages) related to the
observed phase drop. The first stage corresponds to the condition before the arrival of the
wetting front - in other words - before the water filling of the mesopores (pores typically
greater than 30 m). At this stage, water is located in the microporosity inside the polarized
aggregates. Charge carriers are blocked in the aggregates since the air is electrically an
isolator. The second stage represents the physical condition after the arrival of the wetting
front: the mesopores are now filled with connected fresh water, leading to a decrease in the
polarizability of the aggregates.
Figure 23. Calculated imaginary soil conductivity as a function of volumetric
aggregate content. Both stages 1 and 2 (before and after the arrival of the wetting front) are
considered.
154
Tables
Table 1
% Clay
(g g-1)
% Silt
(g g-1)
CEC
(cmol kg-1)
Ca++
(cmol kg-1)
Na+
(cmol kg-1)
Mg++
(cmol kg-1)
K+
(cmol kg-1)
Corg
(g kg-1)
34.5 54.0 11.9 39.5 0.72 1.52 0.5 14.4
Table 2
Class Class limit (equivalent
pore diameter) ( m)
Functional description
Macropores > 75 Structural
Pores Mesopores 30-75
Normally draining
pores
Transmission
pores
Micropores 5-30 Slowly draining pores
Ultramicropores 0.1-5 Useful water retention
capacity
Storage pores
Cryptopores < 0.1 Non useful water
content
Residual pores
155
Table 3
Complex electrical
conductivity
Range (S/m) References and comments
Air
’a= ’’a= 0
Water
’w 9.3 10-2
Measured at 20 °C
’’w 7. 10-6
(f=1500 Hz)
3.5 10-6
(f=750 Hz)
The imaginary part ’’w is calculated
from the relative permittivity of
water i.e. 80: 802 0fw
'' where f
is the frequency; 0 is the free space
permittivity equal to 8.85 10-12
F/m.
Aggregates
'’ag 8.5 10-4
( ’ag =104)
(f=1500 Hz)
8.5 10-1
( ’ag =108)
(f=150 Hz)
The corresponding real relative
dielectric permittivity is also given
’ag
Comparon (2005), Cosenza et al.
(2007)
156
Figures
(a) (b)
(c) (d)
(e) (f)
Figure 1
157
A M N B
Rainfall simulator
Cover against wind 12 tensiometers at depth 8 cm
Switching boxto tensiometers
12 tensiometers at depth 13 cm
Electrodes
Transmitter
ReceiverControlUnit PC
Optical fibercable
Switching box
A
runoff
Z
Y
X
Datalogger
L=0.99 m
Tensiometer z=0.08 m
Tensiometer, z=0.13 m
A
M
N
B
Electrode
T12
T11
T10
T09
T08
T07
T06
T05
T04
T03
T02
T01
T18
T17
T16
T15
T14
T13
T24
T23
T22
T21
T20
T19
PVC jacket
runoff
Z
Y
X
Datalogger
L=0.99 m
Tensiometer z=0.08 m
Tensiometer, z=0.13 m
A
M
N
B
Electrode
T12
T11
T10
T09
T08
T07
T06
T05
T04
T03
T02
T01
T18
T17
T16
T15
T14
T13
T24
T23
T22
T21
T20
T19
PVC jacket
Figure 2
158
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Time (hours)
Tensiometers
SIP measurements
Rainfall (15.4 mm/h) Rainfall (32.4 mm/h)
SIP measurements
Test 1 Test 2
Figure 3
-25 0 25 50 75 100 125 150 175 200Time (min)
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
Ma
tric
pote
ntial (m
)
T1 (8 cm)
T2 (13 cm)
T3 (8 cm)
T4 (13 cm)
T5 (8 cm)
T6 (13 cm)
Figure 4
159
Figure 5
160
-20 0 20 40 60 80 100120140160180200Time (min)
-1.2
-0.8
-0.4
0M
atr
ic p
ote
ntia
l (m
)Applied Rainfall rate
StoppedRainfall rate
8 cm
13 cm
Figure 6
161
-20 0 20 40 60 80 100 120 140 160 180 200Time (min)
16
20
24
28
32A
mp
litu
de
(O
hm
.m)
Applied rainfall rate
Stopped rainfall rate
fmin= 91 mHz
fmin= 1.46 Hz
fmax= 12 kHz
Test 1
Figure 7
162
-20 0 20 40 60 80 100 120 140 160 180 200
Time (min)
0
5
10
15
20
25
-Pha
se (
mra
d)
Appliedrainfall rate
Stoppedrainfall rate
12 kHz
6 kHz
3 kHz
1.5 kHz
750 Hz375 Hz
187.5 Hz
Test 1
1.46 Hz
Figure 8
163
-20 -10 0 10 20 30 40 50 60 70 80
Time (min)
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
Ma
tric
po
ten
tia
l (m
)StoppedRainfall rate
Applied Rainfall rate
Test 2
13 cm
8 cm
Figure 9
164
-20 -10 0 10 20 30 40 50 60 70 80
Time (min)
16
20
24
28
32
Am
plit
ude (
Ohm
.m)
StoppedRainfall rate
0.732 HzTest 2
12 kHz
Applied Rainfall rate
Figure 10
-20 -10 0 10 20 30 40 50 60 70 80
Time (min)
0
5
10
15
20
25
-Pha
se
(m
rad)
Applied Rainfall rate
StoppedRainfall rate
12 kHz
6 kHz
3 kHz
Test 20.732 Hz
Figure 11
165
Figure 12
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Time (hours)
Tensiometers
SIP measurements
Rainfall (15 mm/h) Rainfall (15 mm/h)
SIP measurements
Figure 13
166
0 100 200 300 400 50050 150 250 350 450
Time (min)
-5
-4
-3
-2
-1
0M
atr
ic p
ote
ntia
l (m
)
Cycle 1 -Upper part
C1 (4 cm)
D4 (4 cm)
C2 (7cm)
C3 (10cm)
D3 (10cm)
0 20 40 60
-5
-4
-3
-2
-1
0
Stoppedrainfall rate
Figure 14
167
0 100 200 300 400 50050 150 250 350 450
Time (min)
-6
-4
-2
0
-5
-3
-1
Ma
tric
po
ten
tia
l (m
)Cycle 1 -Lower part
C4 (17cm)
D2 (17cm)
C5 (20cm)
C6 (23cm)
D1 (23cm)
0 20 40 60 80
-6
-4
-2
0
Stopped RainfallRate
Figure 15
168
0 40 80 120 160 200 240 280
Time (min)
32
36
40
44
48
52A
mp
litu
de (
Oh
m.m
)
Cycle 1
93.5Hz
187Hz
1.5kHz
Applied Rainfall rate
Stopped Rainfall rate
Seepage
Figure 16
169
0 40 80 120 160 200 240 280
Time (min)
0
5
10
15
20
25
-Ph
ase (
mra
d)
-6
-4
-2
0
Ma
tricp
ote
ntia
l(m
)Cycle 1
93.5 Hz
187 Hz
1.5 kHz
Tensiometer C2
Stopped Rainfall rate
Applied Rainfall rate
Seepage
Figure 17
170
0 40 80 120 160
Time (min)
32
36
40
44
48
52
Am
plit
ud
e (
Oh
m.m
)
Cycle 2
93.5 Hz
187 Hz
1.5 kHz
Applied Rainfall rate
Stopped Rainfall rate
Seepage
Figure 18
0 40 80 120 160
Time (min)
0
5
10
15
-Ph
ase (
mra
d)
Cycle 2
93.5 Hz
187 Hz
1.5 kHz
Applied Rainfall rate
Stopped Rainfall rate
Seepage
Figure 19
171
0 50 100 150 200
Time (min)
0
0.001
0.002
0.003
0.004
0.005
0.006
Ima
gin
ary
co
nd
uctivity
'' (d
S/m
)
Field experiment
Laboratory experiment
Phase drop
Figure 20
0 50 100 150 200
Time (min)
0.2
0.3
0.4
0.5
0.6
Rea
l conductivity
' (d
S/m
)
Field experiment
Laboratory experiment
Phase drop
Fig. 21
172
Figure 22
173
0.6 0.7 0.8 0.9
Volumetric aggregates content
0.0001
0.001
0.01
0.1
1Im
ag
ina
ry s
oil
co
nd
uctivity (
dS
/m)
MWHB model
Stage 1-before wetting front
Stage 2- after wetting front (1500 Hz)
Stage 2 - after wetting front (150 Hz)
Phase drop
1500 Hz
150 Hz
Figure 23
174
Effects of drying on the low-frequency electrical properties of Tournemire
argillites
Philippe COSENZA*, Ahmad GHORBANI*, Nicolas FLORSCH*, André REVIL#
* UMR 7619 Sisyphe, Université Pierre et Marie Curie-Paris6, Paris, France #
UMR 6635, CNRS-CEREGE, Université d’Aix-Marseille III, Aix-en-Provence, France.
Abstract: Nearly water-saturated argillite samples (initial water content near 3.4 wt%)
were cored from an undisturbed area of an underground facility of the French Institute for
Radioprotection and Nuclear Safety (IRSN), located at Tournemire (Aveyron, France). These
samples were subjected to the following desiccation path: (a) a desaturation phase during
which the samples were dried at ambient temperature conditions, relative humidity equal to
43 % in average and (b) a heating phase during which the same samples were heated at four
temperature levels from 70 °C up to 105 °C. During both phases, the low-frequency complex
resistivity (0.18Hz- 12 kHz) was recorded by a four-electrodes device.
The amplitude of the complex resistivity was extremely sensitive to water content change. At
the end of the isotherm desaturation phase, it has been multiplied by a factor of 3 to 5. During
the heating phase, the resistivity increased by more than two orders of magnitude compared to
the initial state. The percentage of Frequency Effect shows a low sensitivity to water content
changes during the desaturation stage while it increased by two orders of magnitude during
the heating phase. This results confirms that low-frequency spectral signature is extremely
sensitive to textural changes (i.e., thermal-induced microcracking in this case) that occurred
during heating. Moreover, the complex resistivity of the samples show a strong anisotropy (a
ratio of 10 between both amplitudes measured in the perpendicular directions). The classical
Cole-Cole model can not be used to fit the experimental data obtained in the heating phase. A
generalized formulation of this model is required and was successfully applied to represent
the complex resistivity data.
Keywords: Spectral Induced Polarization, complex resistivity, argillite, drying, microcracks.
Abbreviated title: Drying and SIP of Tournemire argillites.
175
1. Introduction
In many industrial countries (such as Belgium, Germany, France, Japan, Spain,
Switzerland) deep argillaceous formations are considered as potential host media for high
level radioactive wastes. This is because clayey geomaterials have the ability to adsorb large
amount of ions and they possess the low permeability required to slow down the percolations
of fluids (e.g. Mitchell, 1993). In France, in order to improve the knowledge of the
containment properties of such formations, two geological sites are currently under study: the
Callovo-Oxfordian argillites of the Paris Basin (e.g., ANDRA, 2005a) and the Toarcian
argillites of Tournemire, in the Southern France (e.g., Cabrera et al., 2001).
For safety assessment of long-term radioactive waste disposals, a critical issue is the
impact of the thermo-hydro-mechanical loading induced by the excavation of deep galleries
and by the exothermic canisters (e.g., Kaluzny, 1990; ANDRA, 2005b). In this framework, it
is desirable to have non-invasive tools in order to determine in situ petrophysical parameters
(like the porosity, the permeability, the water content) and the textural changes that can be
induced by thermal and mechanical induced cracking during the construction and the life of
the underground repository.
Complex resistivity method also named Spectral Induced Polarization (SIP) consists in
measuring both the low-frequency resistivity and the phase shift between the voltage and the
applied current. This is a very promising geophysical method to monitor both water content
and fracturing of argillaceous rocks. Indeed, low-frequency and DC resistivity are often used
to determine the water content of porous materials (e.g., Dannowski and Yaramanci, 1999)
and electrical spectroscopy can be used as an efficient tool to detect fractures (Glover et al.,
1997; Glover et al., 2000; Nover et al., 2000; Heikamp and Nover, 2003). Moreover, recent in
situ investigations in the Mont Terri underground laboratory (Kruschwitz and Yaramanci,
2004) showed that SIP can be qualitatively used to characterize the so-called “excavation
damaged zone” (EDZ) located around the galleries. The EDZ is generated by the excavation
because of the stress and temperature changes and the desiccation of argillites.
Nevertheless, to our knowledge, no extensive experimental SIP investigations has been
performed on argillaceous rocks with a high clay content (typically greater than 30 wt%) in
the low-frequency range, typically 10 mHz to 10 kHz. To our opinion, this is mainly due to
two reasons. On the one hand, the oil and mineral companies were not interested to
investigate the physical properties of shale for a long tine. In the other hand, SIP
measurements in the low-frequency range (typically less than 10 Hz), are difficult to conduct
since they require to remove the spurious electrode polarizations occurring at the boundary
176
between the sample and the measuring electrodes (e.g., Vinegar and Waxman, 1984,
Garrouch and Sharma, 1994; Levitskaya and Stenberg, 1996; Carrier and Soga, 1999;
Chelidze et al., 1999). The substitution technique based on electrolytic cells between the
sample and the current electrodes (e.g. Vinegar and Waxman, 1984; Olhoeft, 1985) can not be
used for argillaceous rock as their texture is extremely sensitive to water content and salinity
changes. Moreover, the use of rigid and “nonpolarizing” electrodes with a platinum coating
requires an extremely good sample to electrode contact in order to minimize the
corresponding contact resistance. This contact may be difficult to get and it is well-known
that black platinum electrodes are brittle and easily damaged when clamped against rock
samples (see Levitskaya and Steinberg, 1996).
This is why the first goal of this study is to establish a simple and reliable procedure in
order to measure low-frequency complex electrical resistivity of an argillaceous rock. The
second goal of our study in relation to the monitoring of underground facilities, is to assess
the ability of SIP to measure both water content changes and fracturing. In this purpose, the
desiccation path is preferred, starting with samples close to saturation, cored in the
undisturbed zone of the site. The third objective of our paper concerns the quantitative
modelling of the experimental data. We examine below the simplest models describing the
effects of desiccation on the measured low-frequency electrical properties of the argillites.
2. Geological overview and sampling strategy
The site of Tournemire was chosen as a test site by the French Institute for
Radioprotection and Nuclear Safety (IRSN) to study the confining properties of argillites for
research purpose. This site is located in southern France, in the western border of the Causses
Basin, a Mesozoic sedimentary basin. The general stratigraphy of the Tournemire massif is
sub-horizontal (Figure 1A.). Three major formations are identified. The lower formations
(Hettangian, Sinemurian and Carixian) and the upper formations (Upper Aalenian, Bajocian
and Bathonian) consist in limestones and dolomites. The intermediate formations (Domerian
and Toarcian) correspond to marls and argillaceous rocks. The main geological target of the
studies conducted by IRSN is the upper Toarcian formation corresponding to a layer 160 m
thick of argillites.
The Toarcian formations is intercepted by a set of boreholes and two galleries drilled
from an old railway tunnel crossing the Upper Toarcien (Cabrera et al., 2001) (Figure 1A.).
The mineralogical compositions of samples taken from boreholes shows that clay minerals
177
(kaolonite, illite, and interstratified illite-smectite) represent about 40 wt.% of the bulk-rock
composition (De Windt et al. 1998; Charpentier, et al., 2004). The coarse fraction contains
quartz (~20 wt.%), K-feldspars, bioclasts, and pyrite (~2 wt.%). Carbonates (calcite with
minor dolomite and siderite) constitute 15 wt.% of the bulk-rock composition. Geochemical
studies (e.g., Bonin, 1998, De Windt et al. 1998) and mechanical investigations (e.g. Niandou
et al., 1997; Cosenza et al., 2002) confirmed that Tournemire argillite consists in an
transverse isotropic material. Some physico-chemical properties of Tournemire argillites are
given in Table 1.
A set of four samples was taken from the boreholes TM90 and TM180 (Figure 1B.),
which were drilled in 2005 perpendicularly to the wall of the main tunnel. The TM90
borehole is subhorizontal and parallel to the bedding in the Toarcian formation and TM180
was drilled vertically in the same formation perpendicularly to the tunnel axis. This
configuration allow to investigate the degree of anisotropy of this material. The drilling was
carried out with air to avoid any contact of the formation with any kind of aqueous solutions.
Samples were immediately taken after the completion of drilling and put in Al-coated plastic
bag under confined N2 atmosphere. The samples were located between 12 and 14 meters from
the gallery wall in a preserved zone outside the so-called Excavation Disturbed Zone (EDZ)
and outside the influence of the ambient air. The samples have been cored far from the wall of
the tunnel wall in order to study near-saturated and undamaged argillite samples. This state
corresponds therefore to an initial reference state (IRS). In a first stage, in order to understand
separately the impact of water content changes and fracturing, it was easier to work with this
IRS and not to mix the effects of various (hydraulical and mechanical) processes existing in a
material located in the EDZ.
The geometrical features of the samples and their water content are given in Table 2.
The low values of water content (3.33 wt.% in average) are typical of this argillite in a natural
state (Bonin, 1998). From the variation ranges of the porosity and the grain density given in
Table 2, we checked that this average corresponds to a near-saturated state, between 90 % and
100% of saturation degree.
3. Methodology
3.1. Experimental set-up
The SIP FUCHS-II measure the complex resistivity over 7 decades of frequency (1.4
mHz up to 12 kHz). It consists in two remote units that record the current I and voltage U
signals (Figure 2).
178
The complex impedance Z* is defined by (e.g., Pelton et al., 1983):
)()(*)(* ieZI
UZ (1)
where (i2=-1); )(*Z and )( are the impedance amplitude and the phase
respectively. They are a priori function of the angular frequency . The existence of the
phase )( results from polarization processes occurring in the sample, i.e. the so-called
Induced Polarization (IP) effect.
To measure the whole spectrum, the SIP FUCHS-II apparatus starts with the highest
frequency, 12 kHz, and the N other decreasing frequencies are obtained by the following
division: 12 kHz/2N. Optical fibres are used in order to minimize electromagnetic cross-
couplings between the transmitter and the receiver. The measured data are transferred to the
base unit, where the apparent resistivity and the phase shift are determined. The SIP FUCHS-
II is connected to a computer, to record the data and to display the results in real-time.
As mentioned previously, the main difficulty at low frequency is to avoid two spurious
effects that interact each other: (a) the disturbing impedance of the electrode-sample contact
and (b) the so called “electrode polarization” associated with electrochemical reactions
occurring at the electrode-sample boundary (e.g., Vanhala and Soininen, 1995; Levitskaya,
and Stenberg, 1996; Chelidze et al., 1999).
To minimize these effects in a simple way, measurements have been performed with a
four-electrode device using low-cost medical electrodes. The potential (or measuring)
electrodes and the current (or source) electrodes are electrocardiogram (ECG) Ag/AgCl
electrodes (Asept Co.) and thin carbon films (Valutrode® electrodes from Axelgaard
Manufacturing Co.), respectively. Silver-silver chloride (Ag/AgCl) electrodes which are
known to be stable almost non-polarizable in comparison with metal electrodes are widely
used in laboratory SIP measurements (e.g. Vinegar and Waxman, 1984; Vanhala and
Soininen, 1995). In our study, ECG Ag/AgCl electrodes consists of a small metal round piece
(10 mm diameter) galvanized by silver and covered with a soft sponge imbibed of a AgCl gel.
The carbon films (50 mm diameter, 1 mm thick), that are used in electrotherapy are circular
and covered with a conductive adhesive gel which provides a good electrical contact between
the electrodes and the rock sample (i.e., no air gaps in between).
Our four-electrode device was validated using porous and electrically inert samples. We
chose “clean” limestone samples. Indeed, “clean” (i.e. with no clay fraction) limestone
saturated with a highly-conductive brine will not generate spurious polarization (e.g., Van
179
Voorhis et al., 1973; Vinegar and Waxman, 1984). Figure 3 shows the resistivity and the
phase spectrum in the range 0.18 Hz -12 kHz of two limestone samples from Lavoux and
Vilhonneur. They were characterized by low clay fractions typically less than 1 wt%, (Bemer
et al., 2004) saturated with a 0.1 .m brine (NaCl) conductivity. No significant polarization
has been measured in the frequency range 0.18 Hz to 12 kHz except for the highest frequency
value of 12 kHz for which a small but significant phase shift is observed and likely associated
with an electromagnetic coupling/noise effect. The existence of this phase shift implies a
small error of about +/- 1 mrad on the measurements for frequencies greater than 6 kHz. For
lower frequencies, the phase shift is less than +/- 0.5 mrad that is a typical error value for the
SIP-Fuchs (e.g. Weller et al., 1996; Binley et al., 2005).
In fact, following the experimental configuration given in Figure 2, and in order to
compare the data from the different samples (with different geometries), a geometrical factor
K (expressed in meters) has to be calculated in order to convert the measured complex
impedance Z* as complex resistivity *:
*( )= K Z*( ) (2)
The parameter K has been calculated numerically by solving the Laplace equation with a
finite-difference scheme. Its value for each sample is given in Table 1. It should be noted that
)(* will be called hereafter the amplitude of the complex resistivity.
3.2 Experimental procedure
The samples were monitored during two phases. In the first phase, called below the
“desaturation phase”; the samples were dried at ambient temperature conditions. In the second
phase, called below the “heating phase”, the same samples were heated at four temperature
levels (70, 80, 90 and 105 °C).
The desaturation phase provides a desiccation path with samples that are initially nearly
water-saturated. It can also mimic the effect of ventilation in an underground gallery. During
this phase, different parameters were recorded over time (up to one hour to four days) as a
function of the dehydration rate. These parameters are the loss of weight induced by
desiccation, the complex resistivity spectrum (phase and amplitude) in the frequency range
0.18 Hz-12 kHz, the relative humidity (RH%), the room temperature and the temperature of
the surface of the sample by a non-contact thermometer. The desaturation phase was
considered as finished when the sample weights did not evolve anymore; it lasted from 19 to
22 days.
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The heating phase was started immediately after the end of the desaturation phase. The
same set of samples was subjected to four background temperature levels by an oven: 70 °C,
80 °C, 90 °C, and 105 °C. At each level, temperature was maintained constant during 24
hours. The last temperature level (105 °C) allowed to measure the initial water content on the
basis of the loss of weight of the samples. Before and after each temperature level, the loss of
weight and the complex resistivity spectrum were measured. It should be noted that the
dielectric spectrum was not recorded immediately after removal from the oven but at least 5
hours afterwards, in order to obtain a thermal equilibrium, i.e., to equilibrate sample
temperature and room temperature.
There were several goals that we pursue by doing the measurements during the heating
phase. They are (a) to provide some physical insights concerning the effect of temperature
induced by exothermic nuclear wastes on the dielectric behaviour of argillite, (b) to remove a
significant part of the “bound” water in the microporosity, (c) to generate microcracks in the
samples. These microcracks are formed in response to two processes. First, heating is
responsible for local thermal stresses in the different components (the different minerals and
the interstitial liquid). In turns, these thermal stresses induces local thermal strain
incompatibilities and hence microcracks. In addition, the cooling of the sample after their
removal from the oven generates a high tensile stress (e.g., Bérest and Weber, 1988), which
may go beyond the low tensile strength of the argillaceous rock, typically in the range of 2-10
MPa (e.g., Rousset, 1988).
4. Results and qualitative interpretation
4.1 Loss of weight under desiccation
Figures 4a and 4b show the evolution of the water content during both phases
(desaturation and heating) for each sample. As mentioned previously, the duration of the
desaturation phase was ranging from 19 days (TM90-1260) up to 27 days (TM180-1288). The
samples with the same size and from the same borehole (i.e., TM90-1260 and TM90-1280)
show similar dehydration rates during the desaturation phase (Figure 4a). During this phase,
the relative humidity was equal to 43 % in average. This corresponds to a rather dry
atmosphere. Despite of this low value of relative humidity, the desaturation induced a limited
loss of weight for the whole set of samples. Indeed, they kept the half of their initial water
content (between 48 % and 54 %, see Figures 4a and 4b) at the end of this phase.
Consequently, it is suspected that this desaturation phase involved mainly the water removal
from the macro and mesoporosity (typically for pore sizes the range 0.8 nm up -50 nm, see for
181
instance Rouquerol et al., 1994) and hence moderate textural changes. Indeed, if an adsorbed
water layer of thickness dw surrounding platy particles is considered, the corresponding water
content w is proportional to the specific surface Sa, as follows (e.g., Mitchell, 1993):
wwa dSw (3)
where w is water density. By using the approximate equation (3), a mean value of Sa equal to
26 m2/g, measured by the BET/N2 method on Tournemire argillite (Bonin, 1998; Devivier et
al., 2004) and a value of dw equal to 0.8 nm (about two diameters of H2O molecules) give a
value of water content associated with microporosity equal to 2.1 %. This order of magnitude
is confirmed by the results of others experimental studies performed on the same material
(Valès et al., 2004). Furthermore, it should be emphasized that the water content calculated on
the basis in a drying oven at 105°C leads to underestimate the real water content in argillite.
Indeed, a drying at 105°C during 24 h does not remove all the water, specially “bound” water,
in smectite clays (e.g., Tessier, 1978).
During the heating phase, the small “rebounds” observed in the figures 4a and 4b are related
to the removal of the samples from the oven in order to return to a thermal equilibrium with
the room.
4.2 Effect of drying on amplitude spectra
Amplitudes obtained during both experimental stages (desaturation and heating phase) at two
extreme frequencies (0.18 Hz and 12 kHz) are presented in Figures 5a and 5b as a function of
water content.
Considering the desaturation phase, comparatively to the initial near-saturated state,
amplitudes of samples TM 90 and TM 180 boreholes were multiplied by a factor 3 and 5,
respectively. This result demonstrates that DC or low-frequency electrical methods are very
promising to monitor the evolution of water content. In addition, the anisotropic texture of
argillite influences significantly the electrical properties. Indeed, figure 5a and figure 5b show
that there exists about one order of magnitude between the amplitude parallel to the
stratification and that perpendicular to the stratification. Nevertheless, we are aware that this
strong contrast may be lower in the field since electrical properties, as for many other physical
properties are stress dependent (e.g., Glover et al., 1997). Indeed, the core decompression
during the drilling may induced a small but significant opening of some natural bedding
planes in the samples (e.g., Niandou et al., 1997).
182
Moreover, considering the desaturation phase, results show that the relationship
between amplitude * and the water content w is slightly frequency-dependent. This result
is also illustrated in Figure 6 where the amplitude spectrum of sample TM 90-1280 is plotted
at different values of water content. On the basis of the discussion given in the previous
section, this suggests that the loss of free water from macroporosity does not generate
significant distortion of the low-frequency amplitude spectrum. Furthermore, an amplitude
spectrum measurement will not provide any significant additional insights in regard to in situ
monitoring of water content.
During the heating phase, the amplitude increases drastically by 2 to 3 orders of
magnitude when the background temperature increases. Consequently, the thermal-induced
microcracking and the loss of a significant part of “bound” water in the microporosity affect
significantly the amplitude spectrum of the complex resistivity (Figure 6). This can be shown
by using the percentage of frequency effect, pFE, (e.g., Van Voohris et al., 1973)
)12(*
)12(*)18.0(*(%)
kHz
kHzHzpFE , (4)
where )18.0(* Hz and )12(* kHz are the amplitudes measured at the two extreme
frequencies (0.18 Hz and 12 kHz). The parameter pFE is plotted in Figure 7 as a function of
water content. The results illustrate clearly the previous observations given in Figures 5a, 5b,
and 6, the pFE remains low and almost constant during the desaturation phase (i.e,. no change
in amplitude spectrum) while it increases drastically during the heating phase. Consequently,
this parameter could be used to discriminate the evolution associated with free water and that
with microcracking.
At this stage, it is difficult to identify clearly the polarization processes related to the
values of pFE measured during the desaturation phase. When clay minerals are present in a
sedimentary rock, several researchers have consider that SIP is explained by the “membrane
polarization” model, for which the sediment is described by two zones: selective and non-
selective domains, which are connected in series (Marshall and Madden, 1959; Klein and Sill,
1982; Vinegar and Waxman, 1984; Titov et al., 2002). Following this model, the zones are
associated with different effective mobilities (or Hittorf transport numbers) of cations and
anions which, under the applied external electrical field, produce local charge accumulations
and charge density gradients that are responsible for the frequency dependence of the
resistivity. However, others polarization processes can be also invoked: the Maxwell-Wagner
effect related to the heterogeneity of the material and macroscopic space charge distributions
183
(Olhoeft, 1985) and the polarization of the electrical double layer (see Dukhin and Shilov,
1974; Garrouch and Sharma, 1994; Vanhala, 1997; Revil et al., 1998).
Nevertheless, considering the heating phase associated with much higher values of pFE,
the most appealing explanation is to consider the role of the thermal-induced microcracks as
micro-capacitors contributing to create displacement currents and hence to induce
macroscopically frequency-dependent resistivity measurements. This aspect will be discussed
further.
4.3 Effect of drying on phase spectrum
The phase measured during both experimental stages (desaturation and heating phase) at
two extreme frequencies (0.18 Hz and 12 kHz) are presented in Figure 8a and figure 8b as a
function of the water content.
At the opposite of amplitude measurements, a significant difference in the measured
phases between both samples extracted from the same borehole TM180 (called hereafter
TM180 samples) can be observed. Since this is not the case for both samples taken from
borehole TM90 (horizontally drilled), this difference may be attributed to a small vertical
variation in the mineralogy and/or in the texture. Note that both TM180 samples have
different initial water content (Table 1) and are located at 1424 mm and 1288 mm from the
gallery wall, respectively.
In comparison with amplitude spectra (Figures 5a. and 5b), Figures 8a and 8b indicate
that an evolution of the phase over two decades in the whole range of water content values.
The phase is less sensitive than amplitude to change in the water content. It should be
mentioned that the erratic phase variations for high water content for TM90 samples at 0.18
Hz (Figure 8b) are due to the measurements of low phase values: these values have the same
order of magnitude than the instrumental error.
However, the comparison between the amplitude and phase spectra evidences also some
similar features. Indeed, both parameters show a continuous evolution with water content for
both experimental stages (desaturation and heating) for TM180 samples. At the opposite of
the TM90 samples, no significant discontinuity in the amplitude and phase can be observed at
the desaturation-heating transition. We suggest that this remarkable feature is associated with
the anisotropy of the Tournemire argillite. Indeed, the texture of an argillite is characterized
by two components (e.g., Sammartino et al., 2003) (Figure 9A.): a coarse and non-conducting
phase consisting of grains (mainly tectosilicates and carbonates, in the range of a few dozens
up to few hundreds of micrometers), often aligned sub-horizontally, i.e. perpendicular to the
184
compaction and in the other hand, a fine and electrical-conducting fraction, i.e. the clay
matrix in which clay particles (individual and/or aggregates of 2:1 illite, smectite or 1:1
kaolinite units) are also oriented perpendicularly to the compaction (e.g. Djéran-Maigre et al.,
1998). Moreover, scanning electron microscope (SEM) images reveal that a major fraction of
the macroporosity in argillite is located between the coarse phase and the clay matrix
(Sammartino et al., 2003). This explains why voids and microcracks appear primarily around
carbonates and quartz grains when a saturated argillite begins to dry (e.g., Gasc-Barbier et al.,
2000). As the desaturation process evolve, subhorizontal fractures parallel to the stratification
are generated as a result of the coalescence of the initial voids and microcracks in the
anisotropic clay matrix (Figure 9B.). Finally, the heating favours the occurrence of additional
cracks mainly associated with the clay matrix (Figure 9C.).
Consequently, following these simple ideas, the complex resistivity measured
perpendicularly to the stratification (TM180 samples) and thus perpendicularly to the
desiccation-induced fractures are mainly affected by the non-conductive barriers consisting in
these microcracks. The current flow has to face directly the fractures. As a result, the
microfractures which appears in the clay matrix during the heating phase have a more little
influence. At the oposite, the complex resistivity measured parallel to the stratification (TM90
samples) is less influenced by the desiccation-induced microfractures generated during the
desaturation phase since the latter are sub-parallel to the direction of the current flow.
Measurements for TM90 samples will be more sensitive to microcracks in the clay matrix.
This explains why a clear discontinuity in the evolution of SIP parameters is observed in
figures 5a, 5b, 8a, and 8b at the desaturation/heating transition.
5. Representation of Polarization Spectra
5.1 Conventional empirical models
In this section, the conventional empirical models widely used in IP studies are
presented. Our objective is twofold: (1) to invert the data with these conventional models and
(2) to provide a practical model (i.e., with the smallest number of independent parameters) to
characterize the SIP spectra of clayey rocks undergoing drying.
The most popular empirical model is probably the Cole-Cole (CC) model (e.g., Cole and
Cole, 1941; Pelton et al., 1983; Vanhala, 1997):
cim
)(1
111)(* 0 , (5)
185
where 0 is the direct-current resistivity ( m) (i.e. 0 when 0 ), m is the
chargeability, c is the Cole-Cole exponent, is a time constant (s); is the angular frequency
(rad/s) and i2=-1. The chargeability m describes the magnitude of the polarization effect while
is related to the position of the phase peak, fc, at which the phase reaches a maximum
(Pelton et al., 1983). This critical frequency is given by:
ccm
f2/1)1(2
1. (6)
In order to represent simply their data, Cole and Cole (1941) chose to represent the real
part Re( *) or ’ and the imaginary part Im( *) or ’’ of the measured complex resistivity *
at each function, by points in the complex plane. This synthetic way to display dielectric data
is now extensively used and some examples are given in Figure 10. With this method of
representation called hereafter Argand representation, the CC model appears as a circular arc
with its center either on or below the real axis of the complex plane (Figure 10).
However, experimental investigations on rock samples have shown in several cases
(porphyry copper deposits, some sedimentary rocks) that complex resistivity data could
appear as a straight line in the Argand plot (Van Voohris et al., 1973; Weller et al., 1996): the
real and imaginary parts obey identical power laws of frequency. Such data are represented
the so-called Drake model, also called the Constant-Phase-Angle (CPA) model. It can be
expressed by the equation:
ai )1(
1)(* 0 , (7)
where only three parameters 0 , and a are unknown. At high frequencies ( >> 1), the
amplitude is given by:
a)()(* 0 , (8)
and the phase is given by:
a2
. (9)
In order to model asymmetrical circular arc in the Cole-Cole representation, the Cole-
Davidson (CD) model has been proposed by Davidson and Cole (Davidson and Cole, 1950)
(Figure 10):
aim
)1(
111)(* 0 (10)
186
The CC, CPA, and DC models can be also generalized into a single formulation; the so-
called the generalized, GCC model, (Pelton et al., 1983)
acim
))(1(
111)(* 0 . (10)
When a=1 or c=1, one obtains the CC model or the CD model, respectively. When c=1 and
m=1, the CPA model is obtained. Klein and Sill (1982) showed that the GCC model is the
best model to fit the experimental data obtained from artificial clay-bearing sandstones.
5.2 Inversions of the dielectric spectra
The dielectric spectra have been inverted using a non-linear iterative least-square
method (Tarantola and Valette, 1982) on the basis of the three representation models given
above.
With respect to the desaturation phase, the inversion of the spectra of the TM90 samples
shows that the CC formulation provides the best agreement when the chargeability is fixed
equal to 1 (Figure 11). The inversions of TM180 spectra were a priori more complex: as it is
shown for the example given in Figure 11 (TM180-1424, w=3.49 %), these spectra indicate a
linear element at the lowest part of the frequency range (0.18 Hz-5.86 Hz) i.e., with a
frequency behaviour corresponding to the power–law given by equation (8) ( a ).
Consequently, the most suitable model for the TM180 samples is likely a combination of the
CC model (for the high part of the frequency range) and the CPA model (for the low part of
the frequency range).
Figure 12 illustrates the inversions corresponding to the data of the heating phase. The
best agreement with the dielectric spectra was obtained with the GCC model which can
reproduce the specific asymmetrical arcs observed in the Argand diagram of the complex
resistivity (Figure 12). For both desiccation phases, the evolutions of CC and GCC parameters
as a function of the water content are given in Figures 13a, 13b, 13c 13d and 13e. In these
figures, the TM180 samples are not considered in the desaturation phase since their spectra
could not be modelled by a single CC model.
Figures 13a and 13d show that the parameters 0 and are very sensitive to changes in
the water content. The water content dependence of 0 is not surprising since DC-resistivity is
known to be a good indirect predictor of water content (e.g., Dannowski, and Yaramanci,
1999). However, the water content dependence of is more difficult to explain. This
parameter is often linked to the size of polarizable grains (e.g., Vanhala, 1997). Nevertheless,
187
when clay-bearing sedimentary rocks are considered, various formulations (e.g., Madden and
Marshall,1959; Vinegar and Waxman, 1984; Titov et al., 2002; Titov et al., 2004) agree with
the fact that the relaxation time depends on a characteristic length l between two selective
zones following:
2l . (11)
In shaly sands, a selective zone, which corresponds to an accumulation of electrical
charges (i.e., electrical poles) is typically an aggregate of clay minerals or the contact between
two adjacent quartz grains coated by clay particles. This relationship (l) explains also why
Klein and Sill (1982) found a good correlation between values inverted by a CGG model
and the size of isolating beads in their artificial clay-glass beads mixtures.
We suggest that our values confirm these previous theoretical and experimental
studies. As the heating occurs, the average distance between selective zones consisting in
interfaces between shrunk clay matrix and some microcraks, increases due to the growth of
the microcracks and the generation of new microcracks in the medium (Figure 14). The
corresponding electrical tortuosity increases. As a result, the parameter evolves drastically
over five decades following a power-law as given in (11). This process is likely associated
with the generation of new selective zones (i.e. new electrical poles). This is confirmed by the
increase of the chargeability, m with the heating (Figure 13b).
6. Conclusions
Low-frequency dielectric spectra measurements have been conducted on a set of
argillite samples taken from the Tournemire test site, which is under study by the IRSN. The
dielectric spectra have been recorded with samples submitted to the following desiccation
path: (a) a desaturation phase under ambient air and then (b) a heating phase corresponding to
four temperature levels (70 °C, 80 °C, 90 °C, and 105 °C).
This experimental investigation demonstrates (a) the interest of using the SIP method to
remotely monitor both the water content and the thermally-induced microcracks and (b) that
the best models to invert the dielectric spectra is the CC model and the GCC model for the
desaturation phase and the heating phase, respectively. However, the data show that the
development of the anisotropy can induced different spectral signatures and polarization
processes. This may lead to difficulty for the use of a comprehensive model. In this situation,
a complex resistivity tensor should be introduced depending on the fabric tensor of the
medium.
188
The next steps of this work would be (1) to improve our knowledge of the underlying
physical process involved in the IP effect of clayey rocks (2) to apply the SIP method in the
field, i.e., in the gallery of a deep repository. Following the first task, as mentioned
previously, the involved physical and chemical processes are complex and numerous: the
membrane polarization effect, the Maxwell-Wagner effect, the polarization of the Electrical
Double Layer. To our opinion, the best way to reach successfully this objective, would be to
use upscaling approaches associated with well-characterized physical processes previously
identified by relevant microscopic observations (SEM etc..).
Acknowledgements. This research was supported by the Agence Nationale de la
Recherche (ANR) –ECCO program (POLARIS Project: “Polarisation provoquée spectrale” –
Spectral Induced Polarization). We thank also Elizabeth Bemer and Jean-François Nauroy of
French Petroleum Institute (IFP) for providing the limestone samples.
189
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193
Captions
Figure 1. A: Geological cross-section of the Tournemire tunnel (modified from
Cabrera et al., 2001). The tunnel is drawn in bold line; the transversal galleries and all the
previous boreholes are not shown. B: Locations of the TM90 borehole and TM180 borehole
(not to scale).
Figure 2. Experimental set-up, RU-0: Remote Unit 0 (or Transmitter); RU-1: Remote
Unit 1 (or Receiver). Note that both remote units are mobile since SIP-Fuchs II equipment is
used for field measurements. ECG P1 and ECG P2: electrocardiogram Ag/AgCl electrodes as
potential electrodes; CF C1 and CF C2: Carbon films as current electrodes.
Figure 3. a. Amplitude spectra of limestone samples. b. Phase spectra of limestone
samples.
Figure 4. a. Loss of weight of samples TM90-1260 and TM90-1280 as a function of
time. b. Loss of weight of samples TM180-1424 and TM180-1288 as a function of time.
Figure 5. a. Amplitude at 12 kHz as a function of water content. b. Amplitude at 0.18
Hz as a function of the water content. The direction of the current flow I comparatively to the
stratification is also shown.
Figure 6. Amplitude spectra of sample TM90-1280 during the desaturation and the
heating phase.
Figure 7. Percentage of Frequency Effect (pFE) as a function of the water content.
Figure 8. a. Phase at 12 kHz as a function of water content. b. Phase at 0.18 Hz as a
function of water content. The direction of the current flow I comparatively to the
stratification is also given.
Figure 9.A. Simplified texture of argillite in the saturated case (modified from
Sammartino et al. 2003). B. Textural evolution during the desaturation phase. In a first step,
microcracks are generated from macropores mainly located at the clay-grain contacts, due to
the shrinkage of the clay fraction. In a second step, these microcracks propagated
subhorizontally into the anisotropic clay matrix. C. Textural evolution during the heating
phase (adapted from Sammartino et al., 2003).
Figure 10. Examples of conventional empirical models displayed in an Argand
representation: Cole-Cole model (with 0=30 .m, =10-2
s, m=0.7, c=0.4); CPA model (with
0=35.5 .m, =105 s, c=0.008), Davidson-Cole model 0=30 .m, =10 s, m=0.8, c=0.13).
194
Figure 11. Examples of experimental and modelled dielectric spectra obtained during
the desaturation phase in an Argand diagram.
Figure 12. Experimental and modelled dielectric spectra of TM90-1280 sample
obtained during the heating phase in an Argand diagram. The real and imaginary parts are
normalized by the greatest value taken in each spectrum.
Figure 13. Inverted parameters of the Cole-Cole model and the Generalized Cole-Cole
model as a function of water content. (a) The DC-resistivity 0. (b) The chargeability m. (c)
The Cole-Cole exponent c; (d) The time constant . (e) The exponent a of the Generalized
Cole-Cole model.
Figure 14. Schematic representation of the polarization process during the heating
195
Tables
Table 1. Physico-chemical properties of Tournemire (from Bonin, 1998; Cabrera et al., 2001)
Density 2.5-2.6 103 kg.m
-3
Grain density 2.7-2.8 103 kg.m
-3
Pore size centred around 2.5 nm
Porosity 6 –9 %
Gravimetric water content 3.5 – 4 %
Specific surface area 23-29 m2/g
Cation exchange capacity 9.5-10.8 meq/100 g
Hydraulic permeability 10-14
-10-15
m/s (laboratory)
10-11
-10-14
m/s (in situ)
Table 2. Geometrical features and water content of the samples.
Sample Length (mm) Diameter (mm) Water content
(wt. %)
Geometrical
factor (m)
TM90-1280 193 79.5 3.42 0.0848
TM90-1260 205 79.5 3.33 0.0848
TM180-1424 52.5 79.5 3.49 0.1751
TM180-1288 88 79.5 3.09 0.1731
196
Figures
N Sm
m
2
3
4
5
6
7
8
TunnelTunnel
Cernon Fault
TM180
TM90
Tunnel
S
N
A
B
1. Hettangian: limestone and dolomite2. Sinemurian: limestone and dolomite3. Carixian: limestone4. Domerian: marls5. Toarcian: argilites and marls6. Aalenian: limestone7. Bajocian: limestone and dolomite8. Bathonian: limestone and dolomite
TM Boreholes
1 m
Tournemire site
0 1000 2000-500 500 1500
200
400
600
800
1000
200
400
600
800
1000
1
Figure 1
197
Figure 2
198
10-1
100
101
102
103
104
105
Frequency (Hz)
10-1
100
101
Am
plit
ude
(m
)
Brine (0.1 m)
Vilhonneur limestone
Lavoux limestone
10-1
100
101
102
103
104
105
Frequency (Hz)
-10
-8
-6
-4
-2
0
2
4
6
8
10
-pha
se
(m
rad
)
Brine (0.1 m)
Vilhonneur limestone
Lavoux limestone
Figure 3
0 10 20 30Time (days)
0
1
2
3
4
Wa
ter
co
nte
nt
(%)
TM90-1280
TM90-1260
HeatingDesaturation
70 °C
80 °C
90 °C
105 °C
0 10 20 30 40Time (days)
0
1
2
3
4
Wa
ter
co
nte
nt
(%)
TM180-1288
TM180-1424
TM90-1260:
HeatingDesaturation
70 °C
80 °C
90 °C
TM180-1288:
105 °C
Figure 4
(a) (b)
(a) (b)
199
0 1 2 3 4Water content (%)
100
101
102
103
104
105
106
Am
plit
ude (
m)
Frequency: 12 kHzTM90-1280
TM90-1260
TM180-1424
TM180-1288
Desaturation
Heating
current
current
0 1 2 3 4Water content (%)
100
101
102
103
104
105
106
Am
plit
ude (
m)
Frequency: 0.18 HzTM90-1280
TM90-1260
TM180-1424
TM180-1288
Desaturation
Heating
current
current
Figure 5
(a) (b)
200
10-1
100
101
102
103
104
105
Frequency (Hz)
101
102
103
104
105
Am
plit
ude (
m)
w=0.11%
w=0.4%
w=1.14%
w=2.07%
w=2.27%w=2.74%
w=3.13%
TM 90-1280Heating phase:
Desaturation phase:
Figure 6
0 1 2 3 4Water content (%)
100
101
102
103
104
PF
E (
%)
TM90-1280
TM90-1260
TM180-1424
TM180-1288
Desaturation
Heating
Figure 7
201
0 1 2 3 4
Water content (%)
101
102
103
104
-Ph
ase
(m
rad
)Frequency: 12 kHz
TM90-1280
TM90-1260
TM180-1424
TM180-1288
Desaturation
Heating
0 1 2 3 4
Water content (%)
10-1
100
101
102
103
-Ph
ase
(m
rad
)
Frequency: 0.18 HzTM90-1280
TM90-1260
TM180-1424
TM180-1288
Desaturation
Heating
Figure 8
(a) (b)
202
Figure 9
203
10 15 20 25 30 35
Real part ( m)
0
1
2
3
4
5-I
mag
inary
part
(m
)
Cole-Cole model
CPA model
Cole-Davidson model
Figure 10
204
20 30 40 50 60 70 80 90 100 110 120 130
Real part of Resistivity ( m)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-Im
ag
ina
ry p
art
of
Re
sis
tivity (
m)
Desaturation phaseData: TM90-1280
Model
Data: TM180-1424
w=
1.6
5 %
w=
2.0
7 %
w=
2.1
2 %
w=
2.2
7 %
w=
2.4
5 %
w=
2.7
4 %
w=
3.0
2 %
w=
3.1
0 %
w=
3.1
3 %
w=
3.4
9 %
0.183 Hz
1.465 Hz
23.44 Hz
375 Hz
12000 Hz
3000 Hz
Figure 11
0 0.2 0.4 0.6 0.8 1 1.2
Real part of normalized Resistivity ( m)
0
0.1
0.2
0.3
-Im
agin
ary
part
of
norm
aliz
ed
Re
sis
tivity (
m) TM90-1280
Heating phase
Data
Model
w=
0.1
1%
w=
1.1
4%
w=
0.7
0%
0.183 Hz
1.465 Hz
23.44 Hz
375 Hz
12000 Hz
3000 Hz
Figure 12
205
0 1 2 3 4Water content (%)
101
102
103
104
105
106
107
TM90-1280
TM90-1260
TM180-1424
TM180-1288
Heating
Desaturation
0 1 2 3 4Water content (%)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Charg
eabili
ty (
m)
TM90-1280
TM90-1260
TM180-1424
TM180-1288
Heating
Desaturation
0 1 2 3 4Water content (%)
0
0.2
0.4
0.6
0.8
1
Co
le-C
ole
exp
one
nt
(c)
TM90-1280
TM90-1260
TM180-1424
TM180-1288Heating
Desaturation
0 1 2 3 4Water content (%)
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
se
c)
TM90-1280
TM90-1260
TM180-1424
TM180-1288
Heating
Desaturation
0 0.4 0.8 1.2 1.6Water content (%)
0
1
2
3
Co
le-C
ole
exp
on
en
t (a
)
Heating phaseTM90-1280
TM90-1260
TM180-1424
TM180-1288
Figure 13
(e)
(a) (b)
(c) (d)
206
Figure 14
207
208
Compaction of quartz sands by pressure solution
using a Cole-Cole distribution of relaxation times
A. Revil,1 P. Leroy,1 A. Ghorbani,2 N. Florsch,2 and A. R. Niemeijer3
Received 7 November 2005; revised 6 May 2006; accepted 5 June 2006; published 19 September 2006.
[1] Stressed water-infiltrated silica rocks may deform by pervasive pressure solutiontransfer (PPST), which involves dissolution of the grain-to-grain contacts, transport bydiffusion of the solute, and precipitation on the free surfaces of the grains. A fundamentalquestion regarding this process is how to model rheological behavior at stresses andtemperatures typical of the crust of the Earth. A Voigt-type poroviscoplastic model ismodified by using a Cole-Cole distribution of relaxation times rather than a Diracdistribution used previously. The motivation of this choice is to account forthe distribution of the grain size in the compaction of the porous aggregate assuming thatthis distribution obeys approximately a log normal distribution. This grain size distributiondepends upon the initial grain size distribution and cataclasis in the early stage ofcompaction. We compared this modified viscoplastic model with the full set ofexperimental data obtained in various conditions of mean grain size, effective stress, andtemperature by Niemeijer et al. (2002). These data provide tests of all aspects of themodel, which can be considered to have no free parameters. We show the experiments ofNiemeijer et al. (2002) on PPST are primarily diffusion-limited. The grain sizedistributions observed for three samples imply that the distribution of the relaxation timecovers 5 orders of magnitude in grain size.
Citation: Revil, A., P. Leroy, A. Ghorbani, N. Florsch, and A. R. Niemeijer (2006), Compaction of quartz sands by pressure solution
using a Cole-Cole distribution of relaxation times, J. Geophys. Res., 111, B09205, doi:10.1029/2005JB004151.
1. Introduction
[2] Pervasive pressure solution transfer (PPST) describesthe irreversible compactional process of mass transfer inrocks in response to stress and temperature fluctuations[e.g., Rutter, 1976, 1983; Niemeijer and Spiers, 2002;Skvortsova, 2004, and references therein]. Other mechanismof deformation are possible like those associated with micro-cracking [Karner et al., 2003]. PPST is associated with stressconcentration at grain-to-grain contacts increasing solubilityof the solid in the pore fluid, diffusion of the solute alonggrain-to-grain contacts, and precipitation on free faces of thegrains. The understanding of PPST can lead to the under-standing of locking/unlocking processes that affect granulargouge of active faults during the tectonic cycle (see recentpapers by Montesi [2004] and Yasuhara et al. [2005]) andcompaction of quartz sands in sedimentary basins [e.g.,Wahab, 1998]. In addition, PPST could explain soft creeprheology observed in the brittle-ductile transition zone of thecrust and within the seismogenic crust itself [Ivins, 1996].
[3] For quartz sands, several research scientists havemodeled pressure solution using a Newtonian viscous law[e.g., Rutter, 1976, 1983; Dewers and Hajash, 1995;Renard et al., 1997; Yang, 2000; He et al., 2002, 2003].In contrast, Stephenson et al. [1992] described the occur-rence of PPST of quartz sands in sedimentary basins, overlong periods of time, as being a purely plastic compaction.Revil [1999] proposed a unified model, which takes theform of a poroviscoplastic (Voigt-type) linear model with asingle relaxation time associated with the mean grain size ofthe porous medium (Figure 1). The microscopic reasons forthis behavior were explored by Revil [2001], who proposedthe existence of a stress threshold at the grain-to-grain contactbelow which PPST stops. Additional evidences in favor ofthis model were recently presented by Yasuhara et al. [2003,2004, 2005] and Alcantar et al. [2003]. Implications of thismodel for travelling solitary waves were explored by Yang[2002]. However, there is the need to test further the validityof the model for a wide range of mean grain sizes, temper-ature, and effective stresses and to incorporate in this model awider grain size distribution than just using a single value forthe grain size.[4] In section 2, the model of Revil [1999] is modified to
account for the distribution of grain size in the rheologicalmodel. To reach this goal, we use a Cole-Cole distribution ofrelaxation times (rather than a single value) in the viscoplasticresponse of the porous aggregate. This is equivalent toassuming that the grain size distribution obeys a log normaldistribution. The model of Revil [1999] is also modified to
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, B09205, doi:10.1029/2005JB004151, 2006ClickHere
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FullArticle
1Department of Hydrogeophysics and Porous Media, CNRS-CEREGE,University Paul Cezanne-Aix-Marseille III, Aix-en-Provence, France.
2Department of Applied Geophysics, University Paris VI, Paris, France.3High Pressure and Temperature Laboratory, Faculty of Geosciences,
Utrecht University, Utrecht, Netherlands.
Copyright 2006 by the American Geophysical Union.0148-0227/06/2005JB004151$09.00
B09205 1 of 11
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account for the nonlinear behavior of the deformation/effective stress constitutive law observed at high effectivestresses (section 3). Despite the fact that this modificationchanges the results little at effective stress conditions pre-vailing in the upper crust of the Earth, it seems to us importantto extend our model to a wide range of T-P conditions. Insection 4, a nonlinear optimization scheme based on theSimplex algorithm is used to compare the compactionalmodel to all the experimental data of Niemeijer et al.[2002]. Niemeijer et al. [2002] presented a unique series ofexperimental data in which they investigated the influence oftemperature (in the range 400–600C), effective stress (in therange 50–150 MPa), and mean grain size (10–86 m) uponthe compactional response of several quartz aggregates. Inaddition, the grain size distributions resulting from thecompaction process and cataclasis was measured byNiemeijer et al. [2002] for three samples, spanning over2 orders of magnitude in grain size. Consequently, thesedata offer a unique opportunity to test our compaction model.[5] Five predictions made by our model are tested for the
first time in this paper: (1) The plastic limit of the compac-tional response of the porous aggregate is independent ofthe mean grain size of the aggregate. (2) The temperaturedependence of the (long-term or plastic) compaction coef-ficient can be determined using an Arrhenius law. (3) Themean relaxation time of the viscoplastic model can be
described (within a factor 2) using the mean grain diameterof the aggregate and temperature. (4) The relaxation timedistribution (RTD) can be directly related to the particle sizedistribution (PSD) assuming a log normal distribution for thelatter; the model should be flexible enough to incorporateother PSDs. (5) The same compactional model can be used toexplain the full range of data obtained by Niemeijer et al.[2002] including the short-term viscous behavior and thelong-term plastic limit.
2. Theoretical Background
2.1. Debye Distribution of Relaxation Times
[6] We make the following assumptions: (1) the rock isan isotropic granular material, (2) the mineral of the granularaggregate is silica, and (3) in this section the grain size
Figure 1. Sketch of the compactional model. (a) The deformation of a representative elementary volumeof quartz sand follows a linear poroviscoplastic (Voigt-type) rheological behavior. The springs in parallelwith the dashpot represent the plastic (thermostatic) equilibrium state, whereas the dashpots represent thekinetics of PPST at the grain-to-grain contacts (the dashpots ‘‘p’’ and ‘‘d’’ correspond to dissolution/precipitation chemistry and diffusion-limited processes, respectively) (modified from Revil [2001]). Anadditional spring models the poroelastic response of the medium. (b) Analogy between a Voigt-typeviscoplastic model (a dashpot in parallel with an anelastic spring) and an electrical circuit in which a resistor(R is the resistance) is in parallel with a capacitor (C is the capacitance). Such an electrical model isclassically used to model the induced polarization response of water-saturated porous rocks.
Table 1. Temperature Dependence of Model Parametersa
Parameter Temperature Dependence = 0 exp(E/RT)
Solubility C0 = 67.6 kg m3 (1) EC = 21.7 kJ mol1 (1)Diffusivity D0 = 5.2 108 m2 s1 (2) ED = 13.5 kJ mol1 (2)Rate constant k+
0 = 31.3 mol m2 s1 (3) Ek = 71.3 kJ mol1 (3)Compressibility 0 = 2.6 108 Pa1 (4) E = 17 kJ mol1(4)
aNotes: 1, Iler [1979]; 2, Revil [2001]; note for comparison, Dewers andOrtoleva [1990] used D0 1 108 m2 s1 and ED = 40 kJ mol1 whileNakashima [1995] used ED = 15 kJ mol1; 3, Dove and Crerar [1990]; 4this work.
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distribution obeys a delta (Dirac) distribution with d0 themean grain size. Under these assumptions and neglectingmechanical deformation (grain rearrangement [e.g., Revil etal., 2002]), the compactional response of the porous aggre-gate is described by the following linear constitutive law[Revil, 1999]:
~ tð Þ ¼ tð Þ deff tð Þ
dtþ eff tð Þ
d tð Þ
dt; ð1Þ
~ tð Þ 0 tð Þð Þ=0; ð2Þ
tð Þ 1 exp t=0ð Þ½ ; ð3Þ
where circled cross stands for the Stieltjes convolutionproduct. In equations (1)–(3), t is time, (t) is the time-dependent compressibility of the porous aggregate, is itslong-term plastic compressibility (compaction coefficient),0 is the relaxation time, eff = p is the effective stress, is the confining pressure, p the pore fluid pressure, (t) isporosity at time t, and 0 is the initial porosity at thebeginning of the PPST process. The origin of time coincideswith the instant at which the material, in a relaxed referencestate, is subjected to the application of effective stress. The
porosity 0 is the porosity following the application of thestress. So that the initial poroelastic compaction representsthe starting reference state; initial poroelastic compaction isinvestigated in Appendix A. This elastic compaction processfollows instantaneously the stress variations and is reversible.[7] The rock behavior is modeled to have memory of the
highest effective stress experienced by the grain-to-graincontact during its history because of the increase of the grain-to-grain contiguity associated with the compaction. Theprevious compactional response holds as long as
deff =dt 0: ð4Þ
When the effective stress decreases, the deformationresponse follows essentially a poroelastic behavior corre-sponding to the relaxation of the elastic energy stored in thesample.[8] The relaxation time in equation (3) is given by
(Figure 1a),
0 ¼ =Q; ð5Þ
Q1 ¼ Q1d þQ1
p ; ð6Þ
Qd ¼32W
kbT
C!D
gd30; ð7Þ
Qp ¼3W2
kbTN
kþ
d0; ð8Þ
where T is the temperature (in K), C (in kg m3) is thesolubility of the grain surface in the pore water solution inequilibrium with quartz at fluid pressure and temperature, !is the effective thickness of the diffusion pathways at thegrain-to-grain contacts (2 nm, see discussions by Revil[2001]), D (in m2 s1) is the diffusivity of silica at the grain-to-grain contacts, d0 (in m) is the grain diameter, W is themolecular volume of silica (3.77 1029 m3), k+ is thedissolution rate constant, kb is the Boltzmann constant(1.381 1023 J K1), g is the density of the grains, and Nis Avogadro’s number (6.02 1023 mol1). The numericalconstants entering equations (7) and (8) and the temperaturedependence of the various parameters involved in theseequations are reported in Table 1.[9] Revil [2001] showed that diffusion of the solute at the
grain-to-grain contacts is one or 2 orders of magnitudesmaller than that in the bulk pore water. This was recentlyconfirmed by laboratory experiments [e.g., Alcantar et al.,2003; Yasuhara et al., 2003, 2004]. In low-porosity shalesand in compacted bentonites (both having small pore sizes),Revil et al. [2005] and Leroy et al. [2006] reached similarconclusions based on modeling of coupling phenomena thataffects diffusion of ionic species in charged media.[10] The model of Revil [1999] corresponds to a Voigt-
type linear viscoplastic model represented by a dashpot inparallel with an anelastic spring. From the perspective of thedifferential equation governing the system, the behavior isanalogous to an electrical circuit formed by a resistor in
Figure 2. Distribution of relaxation times P(s) where s ln( /0) for various values of the Cole-Cole exponent c. Notethat smaller values of the exponent corresponds to broaderdistributions of relaxation times. The case c = 1 yields a Diracdistribution. The case 0.5 c 1 is very similar to aGaussian distribution. For c < 0.5, the Cole-Cole distributionhas a longer tail than the Gaussian distribution.
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parallel with a capacitor (Figure 1b). In this case, therelaxation time is given by a Dirac (delta) distribution andtakes the discrete value 0. In the study of the inducedpolarization of porous materials, this is known as a Debyerelaxation or distribution.
2.2. Cole-Cole Distribution of the Relaxation Times
[11] When the grain size distribution cannot be describedby a unique value, the relaxation time distribution (RTD)exhibited by the compactional response of a porous aggregatecannot be described by a single value of the relaxation time,hence by a Dirac distribution. If the effective stress variationfollows a Heaviside (step) distribution, grain crushing asso-ciated with cataclasis in the first stage of deformation,increases the distribution of the grain size [e.g., Gratier etal., 1999]. This distribution needs to be accounted for, at leastin a simple fashion, in the compactional law. Indeed, becauserelaxation times entering the compaction law are directlyassociated with the grain size distribution (equations (5) to(8)), a wide distribution of grain sizes implies a widedistribution of relaxation times. The inclusion of a realistic
RTD is especially important if the process is diffusion-limitedbecause of the power law relationship relating the grain sizesand the relaxation times in that case (equations (5) and (7)).[12] In the study of the electrical properties of saturated
porous rocks, it is customary to represent broad distributionsof relaxation times in the induced polarization of saturatedrocks and soils with the so-called ‘‘Cole-Cole’’ distribution[see Cole and Cole, 1941; Taherian et al., 1990]. The Cole-Cole distribution is characterized by a single coefficient, c,that accounts for the broadness of the distribution of relax-ation times and therefore of the broadness of the distributionof the grain sizes [see Chelidze et al., 1977; Pelton et al.,1978; Chelidze and Gueguen, 1999; Kemna, 2000].[13] Under the assumption that the distribution of relax-
ation times is represented by a Cole-Cole distribution, thecompressibility of our rheological model becomes
tð Þ F t=0; cð Þ; ð9Þ
F t=0; cð Þ 1X
1
n¼0
1ð Þn t0
nc
G 1þ ncð Þ; ð10Þ
F t=0; cð Þ X
1
n¼1
1ð Þnþ1 t0
nc
G 1þ ncð Þ; ð11Þ
where G( ) is the gamma function defined by
G xð Þ ¼
Z
1
0
ux1eudu ð12Þ
(x > 0). Note that the series development involved inequation (11) converges very slowly for t/0 > 10 and c < 1;thus a significant number of terms is required for a goodconvergence of the series. In the case c = 1, we recover theDebye distribution
F t=0; 1ð Þ X
1
n¼1
1ð Þnþ1 t0
n
G 1þ nð Þ; ð13Þ
F t=0; 1ð Þ 1X
1
n¼0
1ð Þn t0
n
G 1þ nð Þ; ð14Þ
Ft
0; 1
1X
1
n¼0
t0
n
n!¼ 1 exp
t
0
; ð15Þ
as required for the internal consistency of the model. TheRTD is given with the probability [Cole and Cole, 1941]
P sð Þ ¼1
2
sin 1 cð Þ½
cosh csð Þ cos 1 cð Þ½ ; ð16Þ
Figure 3. Long-term compaction coefficient is deter-mined using the plastic equilibrium compactional responseof the aggregates and optimization of the model parameters.The solid circles correspond to the data from Niemeijer etal. [2002] (all runs). (a) The temperature dependence of thecompaction coefficient can be described using an Arrhenius’slaw. (b) The compaction coefficient does not depend on themean grain size. The thickness of the grey band representsthe uncertainty in the determination of the compactioncoefficient.
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where s ln( /0), describes the distribution of relaxationtimes and 0 describes the peak of relaxation times given insection 2.1. The peak of the RTD is associated with the peakd0 of the grain size distribution and c characterizes thebroadness of the distribution. The probability distributionP(s) has the property
Z
þ1
1
P sð Þds ¼ 1: ð17Þ
The Cole-Cole equation produces a very broad distribution ofrelaxation times when c is small (see Figure 2). For examplefor c = 0.25, 72 percent of the relaxation times are included inthe range 103 < /0 < 10
3. When c is in the range 0.5–1, theCole-Cole distribution is quite similar to a log normaldistribution. However, the Cole-Cole distribution has alonger tail than the log normal distribution when c is less than0.5. The parameter c introduced in our model is not just acurve fitting parameter. It corresponds directly to thestandard deviation of the logarithm of the grain sizedistribution assuming a log normal distribution. Accordingto the dependence between relaxation times and the grain size(equations (5), (7), and (8)), the RTD can be compareddirectly with the particle size distribution (PSD).[14] Some granular aggregates may exhibit RTDs that are
not described by the Cole-Cole distribution. In these cases,other types of distribution can be considered in our model.For example, the Cole-Davidson distribution [Davidson andCole, 1951] can account for nonsymmetrical PSD distribu-tions of sediments or fault gouge. Using the properties ofthe convolution product, any distribution can be consideredcorresponding to a given grain size distribution. However,the mathematics are likely to become complex, and we willtreat this elsewhere. In the present model, we show that thelog normal PSD is a good approximation and thereforejustifies the use of the Cole-Cole distribution.
3. Deviation From the Linear Model
[15] We now adapt the model of Revil [1999] to a widerange of effective stresses. At high effective stresses (typi-cally 100 MPa), the experimental data by Niemeijer et al.[2002] show a clear departure from the linear model proposedin section 2. In the case of these experimental data, theeffective stress history is simply
eff tð Þ ¼ effH tð Þ; ð18Þ
where H (t) is the Heaviside or step function (H (t) = 0 fort < 0 and H (t) = 1 for t 0) and eff is the imposed(constant) effective stress. Under this situation, the convolu-tion integral in equation (1) yields the following linearcompactional response
tð Þ ¼ 0 1 tð ÞeffH tð Þ
ð19Þ
tð Þ ¼ 0 1 F t=0; cð ÞeffH tð Þ
: ð20Þ
Figure 4. Comparison of the model predictions andlaboratory experiments for the long-term compactionalresponse 1/0 of porous aggregates. (a) Test of the linearrelationship is shown. Note the discrepancy between the dataand the model at high effective stresses. (b) Test of theexponential relationship is shown. The experiments cover abroad range of temperatures (400 to 700C) and effectivestresses (50 to 344 MPa) (solid circles, Niemeijer et al.[2002]; solid squares, Lockner and Evans [1995]). The finalporosity is independent of the mean grain size. Indeed, thesamples cover a broad mean grain size spectrum (from 5 to85 m). The gray bands represent uncertainty in determina-tion of the porosity ratio.
Table 2. Conditions of the Experiments of Niemeijer et al. [2002]a
Run T, C d, m eff, MPa ib f
c
Cpf3 500 47.1 100 0.2731 0.1271Cpf4 500 42.9 150 0.2350 0.1051Cpf5 500 40.5 50 0.2612 0.1660Cpf6 400 40.9 100 0.2514 0.1536Cpf7 600 43.2 100 0.3138 0.0970Cpf8 500 11.7 100 0.3108 0.1465Cpf9 500 85.9 100 0.2451 0.1032
aThese experiments cover a wide range of effective stresses, temper-atures, and mean grain size.
bPorosity at the beginning of the compaction process.cPorosity at the end of the compaction process.
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We show now that the previous equations can be modifiedto account for the observed nonlinear behavior of thecompactional response at high effective stresses.
3.1. Long-Term Plastic Limit
[16] The long-term limit of the compactional response ofthe porous aggregate corresponds to timescales for whichthe duration of the experiment is much larger than the peakof the distribution of the relaxation times. When t 0, thecompactional response reaches a linear ‘‘plastic’’ limit[Stephenson et al., 1992]
1=0 ¼ 1 eff : ð21Þ
The term ‘‘plastic’’ means that the compactional response isirreversible and that time does not appear explicitly in theconstitutive rheological law. Equation (21) is valid under theassumption that eff 1 [Revil, 2001] and a linearization ofthe compaction law can be performed. However, experimentsperformed at high effective stresses show unambiguously that
there is a deviation from linearity [see, e.g., Niemeijer et al.,2002, Figure 2b]. We propose to correct this nonlinearity byreplacing equation (21) with the plastic limit
1 ¼ 0 exp eff
; ð22Þ
which admits equation (21) as a limit when eff 1, so forlow effective stress levels (100 MPa). Our choice of anexponential law is related to the exponential relationshipbetween the solubility and the effective stress at themicroscopic level.[17] The first prediction of our model (prediction 1) is
that the poroplastic limit is independent of the grain sizedistribution of the quartz sand. This prediction is in agree-ment with the experimental data of Niemeijer et al. [2002,Figure 2d]. Indeed for experiments performed with a widerange of mean grain sizes, in the range 12–86 m (not to bemisled with the grain size distribution of each sample), thecompactional response 1/0 was observed for the narrowrange 0.40–0.48. Figure 3b shows that the compactioncoefficient is independent on the mean grain size of theporous aggregates.[18] The dependence of the compaction coefficient on
temperature is determined using an Arrhenius’s law for (Table 1) (prediction 2). Note that is inversely proportionalto the critical stress c, which defines the limiting stressthat grain-to-grain contacts can support without creeping[Stephenson et al., 1992; Revil, 2001]. A similar behaviorcould exist for tectonic faults. We expect that the criticalstress sc is temperature-dependent. Revil [1999] used a lineardependence between this critical stress and the temperature.The dependence of the compaction coefficient on temper-ature can be also fitted with an Arrhenius law. This Arrhe-nius law is calibrated against the data of Niemeijer et al.[2002]. The final result is shown in Figure 3a and is reportedin Table 1.[19] A comparison between the long-term linear compac-
tion law (equation (21)) and the exponential law(equation (22)) (Figure 4) shows that the exponential lawprovides a better description of the experimental data in theplastic limit of compaction. As these experimental data covera broad range of temperatures (400 to 700C) and effectivestresses (50 to 344 MPa), the exponential model is likely tobe more appropriate than the linear model.
3.2. Short-Term Compaction
[20] As for the long-term limit of compaction, we have toaccount for the nonlinear behavior discussed above for the
Figure 5. (a) Compaction curve of a quartz aggregate inthe initial stage of deformation. Data are fromNiemeijer et al.[2002] (run Cpf8, effective pressure: 100 MPa, mean graindiameter: 12 m, effective stress 100 MPa, temperature500C, i = 0.311). (b) Comparison between the predictedporosity and the measured porosity is shown. (c) Comparisonbetween the measured grain size distribution and thatpredicted using the Cole-Cole distribution (plain line) isshown.
Table 3. Model Prediction of Experiments of Niemeijer et al.
[2002]
Run Qp,a Pa s1 Qd,
a Pa s1 Qd,a Pa s1
LimitedProcessb
0,c
hours
Cpf3 6.70 1014 1.20 1014 1.02 1014 DL 145Cpf4 7.36 1014 1.59 1014 1.31 1014 DL 113Cpf5 7.79 1014 1.89 1014 1.52 1014 DL 97Cpf6 17.0 1014 9.32 1015 6.02 1015 DL 174Cpf7 2.31 1013 2.58 1014 2.32 1014 DL 83Cpf8 2.70 1013 7.83 1013 2.01 1013 DPL 7.4Cpf9 3.67 1014 1.98 1015 1.88 1015 DL 788
aDetermined from equations (6), (7), and (8).bDL, diffusion limited; DPL, dissolution/precipitation kinetics limited.cDetermined from equation (5) and Table 1.
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full compactional response following the application of thestress. Indeed, very high values for the effective stresses wereused in the experiments described by Lockner and Evans[1995] and Niemeijer et al. [2002]. Therefore we replaceequation (19) by
tð Þ ¼ 0 exp tð Þeff
H tð Þ; ð23Þ
for t > 0. equation (23) admits equation (19) in the limiteff 1.
4. Comparison With Experimental Results
[21] We use all the experimental data by Niemeijer et al.[2002] to test the compaction model developed above.Seven experiments were reported by Niemeijer et al. [2002]at various conditions of temperature, mean grain size, andeffective pressure typical of the crust of the Earth. Theseconditions are reported in Table 2. Niemeijer et al. [2002]point out that an abrupt change in strain rate occurred, in allruns, after a few hours to 12 hours. They argue that thiscould be due to the fact that dissolved copper strongly hindersthe dissolution rate of quartz and infiltration of dissolvedcopper from the cell into the samples during these experi-ments may have slow down compaction. However, suchcontamination is difficult to explain because the flux ofwater is outward from the samples during their compac-tion. Diffusion is a too slow process at this timescale
Table 4. Results of Inversion for Experimental Data of Niemeijer
et al. [2002]a
Run 0 c , 108 Pa1
Cpf3 0.276 ± 0.002 0.35 ± 0.01 1.80 ± 0.02Cpf4 0.245 ± 0.002 0.31 ± 0.01 2.45 ± 0.01Cpf5 0.266 ± 0.002 0.34 ± 0.01 1.95 ± 0.03Cpf6 0.262 ± 0.002 0.28 ± 0.01 1.17 ± 0.03Cpf7 0.313 ± 0.002 0.31 ± 0.01 2.45 ± 0.13Cpf8 0.308 ± 0.004 0.29 ± 0.01 1.07 ± 0.04Cpf9 0.256 ± 0.002 0.30 ± 0.01 2.53 ± 0.04
aThe reported values are the mean and twice the standard deviation of thea posteriori distributions of the model parameters.
Figure 6. (a) Compaction curve of a quartz aggregate inthe initial stage of deformation. Data are fromNiemeijer et al.[2002] (run Cpf3, effective pressure 100 MPa, mean graindiameter 47 m, effective stress 100 MPa, temperature500C, i = 0.273). (b) Comparison between the predictedand measured porosities is shown. (c) Comparison betweenthe measured grain size distribution and that predicted usingthe Cole-Cole distribution (plain line) is shown.
Figure 7. (a) Compaction curve of a quartz aggregate inthe initial stage of deformation. Data are fromNiemeijer et al.[2002] (run Cpf9, effective pressure 100 MPa, mean graindiameter 86 m, effective stress 100 MPa, temperature500C, i = 0.24). (b) Comparison between the predicted andmeasured porosities is shown. (c) Comparison between themeasured grain size distribution and that predicted using theCole-Cole distribution (plain line) is shown.
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over the length of the sample to allow copper to infiltratethrough its pore network. As shown below, there is noneed of such an hypothesis to explain the shape of thecompaction curves.[22] There are also some controversies regarding the
relative roles of pressure solution versus cracking at grain-to-grain contacts. Cataclastic compaction by cracking that isfluid-assisted through reaction at the crack tip (and thereforethermally activated) can also be an important process ofcompaction, and the samples of Niemeijer et al. [2002] showabundant evidence of cracking. However, the high temper-atures of the Niemeijer et al. [2002] experiments would tendto favor pressure solution as the dominant mechanism ofdeformation.
[23] For t > 0, the nonlinear compactional law is writtenexplicitly as
tð Þ ¼ 0 exp effX
1
n¼1
1ð Þnþ1 t0
nc
G 1þ ncð Þ
2
4
3
5: ð24Þ
[24] We first compare the prediction of this equation withthe experimental data in the early stage of compaction (seeFigures 5 to 8) using the fitting algorithm described inAppendix B. In Figure 9, we plot the mean relaxation times0 determined by fitting equation (24) to the experimentaldata as a function of the relaxation times derived indepen-dently, in Table 3, using equations (6), (7), and (8). There is
Figure 8. Compaction curves of quartz aggregates in the initial stage of deformation. Data are fromNiemeijer et al. [2002]. The parameters given are fitted parameters, but their values are very close toindependent evaluations from the equations developed in section 2.
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good agreement between the two estimates, so it followsthat prediction 3 of the model is fine (the mean relaxationtime of the viscoplastic model can be described (within afactor 2) using the mean grain diameter of the aggregate andtemperature). Therefore, to diminish the number of fittingparameters when we consider all the experimental data, weuse the value of the relaxation time 0 directly inferred fromequations (6), (7), and (8).[25] From the considerations given above, the parameters
to invert are 0, , c. We use the inversion algorithmdescribed in Appendix B. The results of the inversion, forthe full set of the experimental data, are reported in Table 4.The fitted porosity 0 is very close to the initial porosity.This is expected from the small contribution of the poroe-lastic contribution (see demonstration in Appendix A). Thecompaction coefficient increases with temperature accord-ing to the Arrhenius law (Figure 3a) and the range of valuesof the compaction coefficient is consistent with that reportedby Revil [2001]. The value of c implies a broad relaxationtime distribution in agreement with the observed PSD(Figures 5, 6, 7, and 8) and the fact that the process isdiffusion-limited. Indeed, as shown in Table 3, the kineticsof the compaction experiments are limited by the diffusionof the solute species at the grain-to-grain contacts. Thisimplies that the relaxation time is an function of the grainsize with a power law exponent equal to three, in agreementwith available data (see Figures 5c, 6c, and 7c). Soprediction 4 of the model, stated at the end of section 1,is also checked.[26] All the parameters involved in our compaction model
can be obtained independently. It follows that the only
factor that must be adjusted in our model is the Cole-Coleexponent. From the experiments made by Niemeijer et al.[2002], the Cole-Cole exponent falls in a narrow rangec = 0.30 ± 0.05. It follows that this range of values isrecommended to model pervasive pressure solution transferin such materials. Additional experiments would be useful totest whether the parameter value c = 0.30 is universal. If so,our model has no free parameters. The compaction modelwith c = 0.30 and experimental data are compared in Figure 9and Figure 10. The agreement between the model and theexperimental data provides a test of prediction 5: the samecompactional model can be used to explain the full range ofdata obtained by Niemeijer et al. [2002] including the short-term viscous behavior and the long-term plastic limit.[27] If the grain size distribution evolves during compac-
tion, this implies that the value of c (that reflects the standarddeviation of the lognormal distribution of the PSD) changesduring the compaction. For example, if the grain size distri-bution appears more restricted at the end of the experimentsthan at the start of a run, this implies that the value of cincreases during the experiment. Inversion for c, at differenttime steps, during compaction may be used to monitor theevolution of particle size distribution with time. This could beimportant for permeability modeling, for example, duringfault zone healing (T. Dewers, personal communication,2005).
5. Concluding Statements
[28] The model of Revil [1999] is modified using a Cole-Cole distribution of relaxation times and nonlinear com-pactional law at high effective stresses. The Cole-Cole
Figure 9. Comparison between the mean time 0 constantobtained by fitting the data to equation (24) (values reportedin Figures 5 to 8) and the time constant determinedindependently from equations (5) to (8) and the values ofparameters in the last column of Table 3. Cpfi corresponds tothe experimental run ‘‘i’’. The grey band represents theuncertainty in determination of the mean relaxation time.
Figure 10. Test of the model for the effective stress. Thecompaction model is compared to compaction curves of threequartz aggregates, at the same temperature, but at differenteffective stresses (c = 0.30 and all the other parameters aredetermined from the equations developed in themain text andTable 1). Experimental data are from Niemeijer et al. [2002].
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distribution is characterized by a single exponent c, whichaccounts for the distribution of the relaxation times. The casec = 1 corresponds to a Dirac distribution, which is in turnassociated with a very sharp distribution of grain size (e.g., aGaussian distribution with a very small variance). The use ofa Cole-Cole response is intended to account for large grainsize distributions in the compactional response of the porousaggregates. This modified model is compared to the exper-imental data obtained byNiemeijer et al. [2002] for which thegrain size distributions cover more than 2 orders of magni-tude, resulting in part from cataclasis in the early stage ofdeformation. We find a good agreement between the predic-tion of the model and the experimental data with c = 0.30 ±0.05. Because the kinetics of compaction are limited bydiffusion (see Table 3), we expect that the RTD covers 5–6 orders of magnitude. This is in agreement with theprevious value for the Cole-Cole exponent. Broad distribu-tions of grain size likely exist during natural compaction anddeformation of fault gouge.[29] This new model has important implications for the
dynamics of faults, the compactional response of sandstonesin sedimentary basins, and subsidence related to changes ofthe effective stress in clastic oil/gas reservoirs. The modelyields different predictions for the compactional response ofporous aggregates from those using other rheologies (e.g.,viscous or poroelastic). This model has also some applica-tions to the study of solitary waves in the crust of the Earth[see Revil and Cathles, 2002; Revil et al., 2003].
Appendix A: Poroelastic Contribution
[30] The total porosity change of the porous aggregate iswritten as
@ tð Þ
@t¼
@
@t
e
þ@
@t
i
; ðA1Þ
where the first term of the right-hand side of equation (A1)(subscript e) corresponds to the poroelastic contribution andthe second term of the right-hand side of equation (A1)(subscript i) corresponds to the (irreversible) compactionalresponse associated with pressure solution. In Biot’s theory,the poroelastic contribution to deformation is given by
@ tð Þ
@t
e
¼ e@
@t
@p
@t
; ðA2Þ
where is the porosity, is the confining pressure, and p isthe pore fluid pressure. The elastic porosity compressibilitye and the porosity effective stress coefficient are definedby
e ¼1
B
1
K
1
Ku
KðA3Þ
¼=Kf þ 1=Bð Þ 1=K 1=Kuð Þ=B
=K 1=K 1=Kuð Þ=B; ðA4Þ
where B is the Skempton’s coefficient, K is the drained bulkmodulus, Ku is the undrained bulk modulus, and Kf is the
bulk modulus of the pore water. A simple analysis of theporoelastic compactional response indicates that it is muchsmaller than the PPST response in the experiments byNiemeijer et al. [2002]. Typically, (/)e < 5%.
Appendix B: Fitting Procedure
[31] Equation (24) can be expressed as a nonlinearfunctional relationship of the form d = G(m) between thevector of model parameters m and the vector of porositydata d:
d ¼ 1; 2; . . . ; 3½ T ðB1Þ
m ¼ 0; ; c½ T ; ðB2Þ
where T signifies transpose. We use an a priori densityprobability corresponding to the null information in therange of possible values of the model parameters, which are0 2 [0.20; 0.35], b 2 [0.5 108/Pa; 3.0 108/Pa], andc 2 [0; 1]. The Simplex algorithm [Caceci and Cacheris,1984] is used to minimize the least squares objectivefunction
minR ¼k G mð Þ d0 k2; ðB3Þ
where kvk = (vTv)1/2 denotes the Euclidian (L2) norm and dois the vector of the observed porosity data.[32] We use an a priori model chosen randomly in the a
priori set of values of the model parameters given above.Then, we determine the mean and the standard deviation ofthe optimized model parameters resulting from the use ofthe optimization process. Five hundred terms were consid-ered in the series development involved in equation (24).The results are reported in Table 4.
[33] Acknowledgments. A.R. is grateful to L. M. Cathles for fruitfuldiscussions when he was at Cornell University and the American RockMechanic Association (ARMA) for the 2004 Rock Mechanics ResearchAward he received and which has stimulated the present work. ANDRA isthanked for a grant given to P. Leroy during his Ph.D. thesis. This paperis the GDR-FORPRO Contribution 2005/11A.D and is also a contributionfrom the project POLARIS (ANR-ECCO-PNRH). D. Coelho, S. Altmann,and J. Lancelot are thanked for their supports. We thank J. Weertman,Andreas Kronenberg, E. H. Rutter, T. Dewers, and two anonymous refereesfor their constructive comments regarding this work.
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A. Ghorbani and N. Florsch, University Paris VI, UMR 7619 ‘‘Sisyphe’’Universite Pierre et Marie Curie, Paris 6, Tour 56, 4 place Jussieu, F-75252Paris Cedex 05, France.P. Leroy and A. Revil, CNRS-CEREGE, BP 80, F-13545 Aix-
en-Provence, Cedex 4, France. ([email protected]; [email protected])A. R. Niemeijer, HPT Laboratory, Faculty of Geosciences, Utrech
University, P.O. Box 80.021, NL-3508 TAUtrech, Netherlands. ([email protected])
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General conclusion and perspectives
This thesis has been dealt with the evaluation and applications of SIP method in
environmental exploration is summarized in five papers. Two papers are presented in the
context of the inversion inference and inversion methods. The results of these papers are
following:
Bayesian approach was developed to invert of the Cole-Cole model parameters in the
time or in the frequency domains. The Bayesian approach consists in propagating the
information provided by the measurements through the model, and to combine this
information with an a priori knowledge of the data.
We have shown that the Bayesian approach provides a better analysis of the uncertainty
associated with the parameters of the Cole-Cole model by comparison with more
conventional methods based on the minimization of a cost function using the least-square
criteria. The Bayesian approach allows delineating “equivalent domains” in the space of the
model parameters accounting for the uncertainty in the measurements.
The Bayesian procedure results demonstrated that the classical time domain approach cannot
lead to a proper estimate of the Cole-Cole parameters. Morover, the frequency domain
induced polarization signals that use the harmonics of square current signals in a low
frequency range cannot lead to estimate the Cole-Cole parameters.
On the opposite, the Cole-Cole parameters can be more correctly inverted in the frequency
domain. These results are also valid for other models describing the SIP response like the
Cole-Davidson or power law models.
We developed the 1D inversion of IP and EM coupling integral according to forward
modeling code taken from Ingeman-Nielsen and Baumgartner (2006). We used the general
Homotopy method, as path tracking is based on a set of predictor–corrector steps. The
corrector steps are Gauss–Newton or Quasi Newton sequence of iterations. The inversion
results show the recovered parameters agree well with the true values.
Two other papers are presented in the context of the experimental applications of SIP
method. The results of these papers that are following:
Spectral induced polarization is used to monitoring of the water infiltration in a clay loamy
soil. Both in situ and soil column experiments were based on the coupled acquisition of
tensiometer data and SIP spectra (1.46 Hz to 12 kHz) during a water infiltration achieved by a
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artificial constant rainfall rate of about 15 mm/hr. This approach confirmed the existence of
significant phase drops in the high-frequency domain (typically greater that 1 kHz) during the
first infiltration cycles. The crossed interpretation of tensiometer data and SIP data has shown
that these phase drops were correlated with the water filling of pores whose equivalent
diameter were estimated in the range of [30-85 m]. This class of diameters correspond to the
transmission or structural pores.
These phase drops were qualitatively and quantitatively interpreted as a Maxwell-Wagner
effect: during the water infiltration, the decrease of the polarization amplitude is due to a
release of charge carriers initially blocked in the microporosity, leading macroscopically to a
decrease of the bulk soil polarizability.
Consequently, the experimental and theoretical results of this study suggest strongly that the
SIP method would be able to control in the field the water filling the structural pores.
Spectral induced polarization was used to monitor the water content and the thermally-
induced microcracks for four nearly water-saturated argillite samples of test site. The SIP
measurements (0.1 Hz to 12 kHz) recorded by a four-electrodes device, during two following
desiccation path: (a) a desaturation phase under ambient air and then (b) a heating phase
corresponding to four temperature levels (70 °C, 80 °C, 90 °C, and 105 °C).
The amplitude of the complex resistivity was extremely sensitive to water content change. At
the end of the isotherm desaturation phase, it has been multiplied by a factor of 3 to 5. During
the heating phase, the resistivity increased by more than two orders of magnitude compared to
the initial state. The percentage of Frequency Effect shows a low sensitivity to water content
changes during the desaturation stage while it increased by two orders of magnitude during
the heating phase. This result confirms that low-frequency spectral signature is extremely
sensitive to textural changes (i.e., thermal-induced microcracking in this case) that occurred
during heating. Moreover, the complex resistivity of the samples shows a strong anisotropy.
The best models to invert the dielectric spectra is the Cole-Cole model and the general Cole-
Cole model for the desaturation phase and the heating phase, respectively.
However, the data show that the development of the anisotropy can induced different spectral
signatures and polarization processes.
This work strongly suggests further experimental and theoretical investigations in three
directions.
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1 It is important to confirm in others sites that SIP method is able to provide
structural geometrical properties (types of pores and induced-microcracks
etc.).
2 Until now the application of the results of small-scale laboratory
experiments to large-scale field investigations has been very problematical,
especially in environmental geophysics. We have to develop relevant
conceptual tools to achieve the upscaling of the SIP parameters at different
scales.
3 The need to expand the frequency range towards both lower and higher
frequencies is evident and a one-decade increase of up to 104 Hz seems to
be realistic. Until now, the limitations are usually due to inductive
coupling. Electronic improvements have to be pursuing to enhance the
development of new applications in environmental geophysics.
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