Journal of Applied Geophysics 67 (2009) 109–113

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Journal of Applied Geophysics

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Modelling of Maxwell–Wagner induced polarisation amplitude for clayey materials

Alain Tabbagh ⁎, Philippe Cosenza, Ahmad Ghorbani, Roger Guérin, Nicolas FlorschUMR 7619 Sisyphe, UPMC/CNRS, case 105, 4 place Jussieu, 75252 Paris cedex 05, France

⁎ Corresponding author. Fax: +33 386 6947 33.E-mail address: alain.tabbagh@upmc.fr (A. Tabbagh)

0926-9851/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.jappgeo.2008.10.002

a b s t r a c t

a r t i c l e i n f o

Article history:

Induced polarisation exists Received 19 September 2007Accepted 7 October 2008

Keywords:Numerical modellingElectrical methodsInduced polarisationElectrical properties of clay

for a wide variety of heterogeneous media in which conductive particles areembedded in a resistive coating. This phenomenon can be explained either by reversible chemical “reactive”processes or by the Maxwell–Wagner (M–W) interfacial polarisation effect. Modelling of the amplitude of thelater effect for both isotropic and platelet like polarisable cells, shows that it can adequately explain theresults obtained in clayey materials.In order to investigate the role of platelet flatness, we firstly consider the isolated spherical water dropmodel, then the platelet model, and finally take into account the influence of coupling between cells. It isfound that the platelet flatness coefficient has a greater influence on M–W relative permittivity than on itselectrical conductivity, which suggests that effective permittivity can be a useful indicator of texture.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Induced polarisation (IP), which differs from the phenomenon ofelectromagnetic induction, exists for a wide range of undergroundmedia, and is the underlying phenomenon on which the mostcommonly used surface mineral prospection techniques are based. Itcorresponds either to the generation, in the time domain, of a transientsignal after the turn-off of an injected current, or to the presence, in thefrequency domain, of a quadrature component and a frequencydependent variation of electrical resistivity (or conductivity). Botheffects can be described by considering an abnormally high dielectricpermittivity.

The IP phenomenon appears in heterogeneous media whereconductive particles are embedded in amore resistive coating (Vacquieret al., 1957; Marshall and Madden, 1959; Olhoeft, 1985; Lesmes andMorgan, 2001). It is very widespread, and is particularly significant inmaterials which contain disseminated mineralisations such as semi-conductive sulphide or graphite particles. It is also observed in media,with electrical resistivities lying typically between 50 and 100 Ωm, themost common values for soils (Michot et al., 2003) which contain asignificant but non-dominant clay phase. Its frequency (or time)dependence has been modelled (Luo and Zhang, 1998), using Cole–Cole (1941) model which is more sophisticated representation of theDebye model (1929), and which makes use of two parameters:polarisability and relaxation time constant.

Two types of physical process can explain IP (Lesmes and Morgan,2001): (i) the Maxwell–Wagner (M–W) effect (Maxwell Garnett, 1904;Wagner, 1914), an interfacial polarisation process where ion displace-ments are limited to the scale of pores or particles, and (ii) an

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electrochemical process corresponding to a wide variety of reversiblephysico-chemical phenomena at grain surfaces (Schurr, 1964; Fixman,1980). Thefirst type refers toboth freedisplacement inporewaterand totangential displacement at grain surface, the second corresponds toelectron exchanges at surfaces. These two types of process are notexclusive and may coexist simultaneously. Nevertheless, whenever IPphenomena are observed, attention tends to be focused on the potentialpresence of the M–W effect, as it is the simplest process and can bemodelled without the need to assume any reversible electro-chemicalprocesses. With such an approach, when platelet-like volumes areinvolved, the sample's polarisability will depend on the locations andnumbers of electric charges.

Although frequency dependence (or relaxation time range) is avery important aspect of IP, in the present paper we consider only theamplitude of the IP phenomenon, in an effort to evaluate themaximumrange of values reached as a consequence of the M–W effect alone. Ifthis range is found to be significantly smaller than that observedexperimentally, the additional presence of electrochemical processeswould need to be considered.

Sen (1981) has already studied this question on the basis ofeffective medium theory. He showed that high aspect ratio “particles”(e.g., plate-shaped grains) can generate large values of rock permit-tivity. In the present paper, a numerical approach is used which allowsthe inherent approximation associated with mixing rules to beavoided, such that the local electric field and hence the effectivepermittivity are determined numerically rather than analytically.Calculations are achieved under the static case hypothesis, withoutconsidering phase lags. The same modelling scheme is applied toelectrical conductivity, in order to compare the influence of particleshapes on both properties.

For unsaturated, sandy clay soils of varying water content, therange of experimental values of effective relative permittivity is

Fig. 1. Experimental effective relative permittivity of a sandy clay soil (30% clay) as afunction of frequency, for three different values of aqueous volumetric content (0.03,0.148, 0.43) (Tabbagh, 1994).

110 A. Tabbagh et al. / Journal of Applied Geophysics 67 (2009) 109–113

illustrated in Fig. 1, for frequencies ranging between 1 kHz and 1 MHz(Tabbagh, 1994). When they are dry, the samples exhibit a lowdielectric permittivity (around 3). For a low level of water content(0.148, which is lower than the wilting point), it exceeds 100 at 1 kHz,and in the case of high water content it reaches 1400, which is morethan twenty times greater than the value expected from molecularpermittivity considerations (Weast, 1983). Other authors, for example(Levitskaya and Stenberg, 1996; Talalov and Daev, 1996; Chen and Or,2006), have published results confirming the possible strongamplitude of the effective permittivity at low frequencies.

2. Spherical isolated water drop model

The simplest model which can be used, in a first approach tomodelling of theM–Weffect, is that of the spherical isolatedwater drop.In the absence of an external electric field, the cations and anions arerandomly distributed inside the sphere, and the barycentres of bothanion and cation clouds coincide. The application of an external electricfield generates a small shift between these two centres, such that themaximumamplitude of the drop's polarisation (volumetric bulk electricmoment) cancels the electric field inside the drop. This form ofpolarisation is usually represented by fictitious equivalent electriccharges at the drop's surface.

To calculate the maximum polarisation of the drop one first adoptsthe hypothesis, which will be justified later, that the number of ions itcontains is sufficient to enable the external field to be compensated.

If q cosθ is the superficial charge density equivalent to thepolarisation, this leads to:

E0 = ∫0

π qcosθ4πe0er

2πcosθdθ;

where E0 is the applied external field, ε0 the vacuum permittivity and εris the relative (molecular) permittivity inside the drop. It follows that:q = 4

π E0e0er; i.e. the total surface charge (positive or negative) equalsQ=8a2E0ε0εr, and that the bulk electricmoment of the sphere is given byM=a3E04πε0εr (a being the radius of the sphere).

For water with a resistivity of 20 Ω m and a concentration of0.005mole l−1 (of monovalent equivalent ions), the total mobile electriccharge would correspond to 2.10−6 C for a sphere of 1 mm radius,

whereas the charge Q needed to balance an applied field of 1 V m−1 isaround 6.10−15 C. The above (M–Weffect) hypothesis is thus likely.

One can also consider a sphere of conductivity σ located inside anisolating medium (of relative permittivity εr) exposed to a uniformfield E0: the secondary potential outside the sphere is that of a dipoleof moment M=a3E04πε0εr.

The electric moment of the sphere thus depends on its volume, onthe primary field intensity and on the molecular permittivity εr. Theeffective permittivity of anM–Wpolarisable medium comprising a setof cells analogous to the sphere will thus be independent of theelectrical conductivity of the cells. The effective permittivity of theisolated sphere can be determined by associating the above expres-sion for the dipole moment with that of a sphere of internalpermittivity εeff, located inside an external medium of permittivityε1, such that M = 4πe0e1a3E0

eeff −e1eeff + e1

. This requires that eeff−e1eeff + e1

= 1, i.e.εeff→∞, which is in total agreement with the hypothesis of completecompensation of the external field inside the sphere.

However, if a fully representative model is required, one mustaccount for the fact that isotropic polarisable cells are not adequate tomodel clay particles, and that they are numerous and scattered.Electrical coupling between cells must also be taken into account.Consequently, as for the determination of molecular permittivity(Tabbagh et al., 2000), both the shape and cross coupling character-istics of polarisable cells must be considered.

3. Isolated platelet model

For clay platelets the relevantmodel is that of an electric double layerwithanexternal cation cloudsurrounding theanions. Thepresenceof anelectric field induces a small displacement in the barycentres of theoppositely charged ion clouds, thus leading to polarisation of theplatelet.

The electric field generated by self polarisation (assumed to beuniform) inside a flat rectangular cell of dimensions δx, δy and δz(δV=δx.δy.δz) must be such that the total electric field cancels out tozero at the centre of the cell. As the electrostatic potential generatedby a dipole of M moment can be expressed by U = 1

4πe0erYM Yr

r3, thecorresponding electric field can be found by integrating the electricfield over the volume of the cell:

−E0 =1

4πe0eru−grad

YMδV

Yrr3

!dxdydz:

This integral is calculated by assuming thatM is constant inside thecell. It contains a singularity (r=0) that can be avoided by analyticallyintegrating over two variables, before performing numerical integrationof the remaining variable. The resulting moment can be expressed by:

M=AδV.E04πε0εr, where the integration result, A =u A

As

Yrr3

� �dv,

(Ys being the unit vector parallel to YM) is a dimensionless geometricalfactor related to the shape of the cell.

For a cube, as for a sphere, A = 34π = 0:239. For a cell of constant

volume (δV), a value of A=0.1826 is found when δx=2δy=2δz, i.e. whenthe cell is elongated in the x direction, parallel to the external field. For aflat cell with δx=0.1δy=0.1δz, aligned perpendicularly to the externalfield, A=1.775, and for an even flatter cell with δx=0.01δy=0.01δz, thevalue of A is 17.68. The shape of the platelet thus has a strong influenceon its electric moment.

There are at least two ways to define the effective relativepermittivity of a given volume. The electric field or electric potentialgenerated outside the volume can be used to compute its effectivepermittivity, by defining it to be the relative permittivity of an identicalvolume of homogeneous material, generating the same external fieldor potential (Tabbagh et al., 2002). The effective relative permittivitycan also be written as the ratio of the internal resultant electricalinduction vector to the internal resultant electric field: eeff =

YDe0YE. The

first approach requires the arbitrary choice of a reference body,

Table 1Effective relative permittivity variations in M–W polarisable cells, as a function ofvolumetric content and flatness coefficient

Polarisable cell

Volumetric content

10% 5% 2,5%

c/e=1 18 16,5 15,8Flatness c/e=10 37 26 20,5Coefficient c/e=20 36,6 28

c/e=40 37,5

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whereas the second can be used whatever the shape of therepresentative elementary volume (REV) is under consideration. Fora single platelet, the second approach leads to: εeff =1+4πAεr. As anexample, for εr=15 and δx=0.1δy=0.1δz, this leads to εeff =336 for alldirections in the plane of the platelet, and εeff =17 in the orthogonaldirection.

4. Effective permittivity of a set of isotropic coupled cells

In order to take into account the effects of coupling betweenelementary cells, one can consider that every cell ‘sees’ a total field Et,given by the sum of the primary uniform external field (E0) and that ofthe fields generated by itself and all of the other cells, and that thepolarisation is such that the total field is cancelled at its centre. Inorder to simplify the calculation, we consider all cells to have the sameshape and volume. Such an approach is similar to that of the momentmethod used to numerically model soil and rock conductivities(Tabbagh et al., 2002).

At each point, the secondary field can be written as:

Et−E0 =1

4πe0e1∑cells

−grad YMi

Yrir3i

!;

where themoments of each cell,Mi, are expressed byMi=AδV.Eti 4πε0εr inaccordance with the hypothesis of total field cancellation at the centre ofeach cell. The values ofEti at the centreof each cell are obtainedbywriting:

Eti−E0 = ∑j≠i−AδVgrad Etj

rr3

� �+ u

i−grad Eti

rr3

� �dxdydz;

A linear system of equations can be deduced from this expression.The resulting values of effective permittivity are shown to

increase regularly with the content of M–W polarisable cells. InFig. 2, for a mediumwith a relative molecular bulk permittivity of 15,the effective permittivity exceeds 28 when the volumetric content,Ct, reaches 0.48. Such an increase is significant, but does not changethe order of magnitude of the effective permittivity. This is in perfectagreement with experimental data for both clean rocks and sands,when they are saturated with a saline solution (Guéguen andPalciauskas, 1994).

5. Effective permittivity of a set of coupled platelets

The calculation scheme is the same as in the preceding case butwith the proper value of the geometrical factor A.

Fig. 2. Numerically modelled results, showing variations in effective static relativepermittivity as a function of content in isotropic M–W polarisable cells.

In order to compare these results with experimental data obtainedin the presence of clay, one can consider anisotropic cells correspond-ing to square platelets of side c and thickness e, with a resulting‘flatness coefficient’ defined by c/e. To simplify the calculations, all ofthe platelets are assumed to have the same shape and volume, butrandom orientations.

A very significant dependence is observed between effectivepermittivity and flatness. As an example, for a relative bulk molecularpermittivity of 15 and a volumetric content of Ct=10% in M–Wpolarisable cells, an effective permittivity of 18 is computed forisotropic cells, as opposed to 37 for cells with a flatness coefficient of10. If the bulk molecular permittivity increases to 25, εeff starts at avalue of 26.3 for c/e=1 and reaches 62.4 for c/e=40, when Ct=2.5%. InTable 1, it can be seen that an increase in flatness more or lesscompensates for a decrease in platelets content. For smaller values offlatness the dependence is quasi-linear, as shown in Fig. 3, whereas forhigher values of flatness it exhibits saturation like behaviour (Fig. 4).Wherever the dependence is linear, it is reasonable to assume that εeffis proportional to the relative molecular permittivity, to the M–Wpolarisable cell content, and to the flatness coefficient c/e, leading tothe following approximate regression law:

eeff = 1 + 0:015Ctc=eð Þ

From this expression, it can be seen that the maximum effectiverelative permittivity (1400) found in the data of Fig.1 could be producedbyhighwater content soilwith amolecular relative permittivity near30,a flatness coefficient of 100 and anM–Wpolarisable cell content of 30%.In the case of very flat platelets, the same maximum value can bereached for a volumetric content of 15% (in fact the clay contentwas 20%but the flatness was unknown).

It is thus valid to consider, for clayey soils, that the amplitude ofexperimentally observed effective permittivities can be explained bythe M–Weffect in the presence of randomly orientated clay platelets.

6. Effect of flatness on conductivity of a set of coupled platelets

The application of the same modelling scheme to conductivityvariations may open new paths towards improved characterisation ofplatelets on the basis of their electrical properties. Here, we use themomentmethod (Tabbagh et al., 2002; Tabbagh andCosenza, 2006) andconsider a clay–water mixture with a water conductivity of 0.04 Sm−1

and a surface platelet conductivity of 0.2 Sm−1. These values correspondto continental surface clay formations and soils in which the water'sconductivity is considerably lower than that of clay particles (Gupta andHanks, 1972; Michot et al., 2003).

In Fig. 5, variations in apparent conductivity are shown as afunction of flatness coefficient, for a constant 10% volumetric contentof randomly orientated platelets. The increase in relative conductivityis significant but lower than the relative variation in M–Wpermittivity. For higher flatness coefficients and a 5% volumetriccontent (Fig. 6), the same tendency is confirmed: the increase inconductivity is significant, but clearly lower than that of thepermittivity. This numerical result is in agreement with other

Fig. 5. Numerically modelled results, showing variations in apparent conductivity as afunction of the flatness coefficient, for a molecular relative permittivity of 15 and avolumetric content of 10%, in randomly orientated M–W polarisable non-isotropic cells(to be compared with Fig. 3).

Fig. 3. Numerically modelled results, showing variations in effective static relativepermittivity as a function of the flatness coefficient (for flatnesses lower than 12), for abulk molecular relative permittivity of 15 and a volumetric content of 10%, in randomlyorientated M–W polarisable, non-isotropic cells.

112 A. Tabbagh et al. / Journal of Applied Geophysics 67 (2009) 109–113

theoretical studies (e.g., Sen, 1981; Cosenza and Tabbagh, 2004;Tabbagh and Cosenza, 2006), and demonstrates that bulk permittivityis more sensitive to the geometry and shape of heterogeneities (i.e.platelets) than bulk conductivity.

7. Conclusion

A non-electrochemical process, the M–Weffect in clay platelets, issufficient to explain strong experimentally observed effective permit-tivities in clayey materials. As permittivity is far more sensitive toplatelet flatness than conductivity, this characteristic can be used tofacilitate the identification of the type of clay present in a givenformation. In practical terms, however, for the effective permittivity tobe correctly measured in the laboratory or in situ, its valuemust not be

Fig. 4. Numerically modelled results, showing variations in static effective relativepermittivity as a function of the flatness coefficient (for high values of flatness), for amolecular relative permittivity of 30 and a volumetric content of 5%, in randomlyorientated M–W polarisable platelets.

negligible when compared to that of the electrical conductivity. If aclayey soil is considered with a conductivity of σ=0.03 Sm and aneffective permittivity of εeff =1000 at 1 kHz, the ratio j e0eeffωσ j (ω beingthe angular frequency) is equal to 0.002, such that the effectivepermittivity could be difficult tomeasure. On the other hand, the sameratio would reach a value of 0.2 at 100 kHz, such that the effectivepermittivity would be straightforward to measure using typical fieldinstrumentation.

Acknowledgements

The research presented in this paper was supported by thePOLARIS II project programme: “Polarisation Provoquée Spectrale”(Spectral Induced Polarization) of the French National Research

Fig. 6. Numerically modelled results, showing variations in apparent conductivity as afunction of the flatness coefficient, for a molecular relative permittivity of 30 and avolumetric content of 5%, in randomly orientated M–W polarisable platelets (to becompared with Fig. 4).

113A. Tabbagh et al. / Journal of Applied Geophysics 67 (2009) 109–113

Agency (ANR)— ECCO/PNRH (« ECosphère Continentale : processus etmodélisation », action thématique Hydrologie : cycle de l'eau et fluxassociés (matières, énergie)).

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