Improving Petrophysical InterpretationWith Wide-Band

Electromagnetic MeasurementsEmmanuel Toumelin, SPE, and Carlos Torres-Verdín, SPE, The University of Texas at Austin, and Nicola Bona,

ENI E&P

SummaryBecause of their sensitivity to ionic content and surface texture,wide-band electromagnetic (WBEM) measurements of saturatedrocks exhibit frequency dispersions of electrical conductivity anddielectric constant that are influenced by a variety of petrophysicalproperties. Factors as diverse as fluid saturation, porosity, poremorphology, thin wetting films, and electrically charged clays af-fect the WBEM response of rocks. Traditional dielectric mixinglaws fail to quantitatively and practically integrate these factors toquantify petrophysical information from WBEM measurements.This paper advances a numerical proof of concept for useful petro-physical WBEM measurements. A comprehensive pore-scale nu-merical framework is introduced that incorporates explicit geo-metrical distributions of grains, fluids and clays constructed fromcore pictures, and that reproduces the WBEM saturated-rock re-sponse on the entire kHz-GHz frequency range. WBEM measure-ments are verified to be primarily sensitive (a) in the kHz range toclay amounts and wettability; (b) in the MHz range to pore mor-phology (i.e., connectivity and eccentricity), fluid distribution, sa-linity, and clay presence; and (c) in the GHz range to porosity, poremorphology and fluid saturation. Our simulations emphasize theneed to measure dielectric dispersion in the entire frequency spec-trum to capture the complexity of the different polarization effects.In particular, it is crucial to accurately quantify the phenomenaoccurring in the MHz range where pore connectivity effects areconfounded with clay polarization and pore/grain shape effectsusually considered in dielectric phenomena. These different sen-sitivities suggest a strong complementarity between WBEM andNMR measurements for improved assessments of pore-size distri-bution, hydraulic permeability, wettability, and fluid saturation.

IntroductionA number of experimental and theoretical studies suggest the mea-surable sensitivity of WBEM to various petrophysical factors, in-cluding porosity, brine salinity, fluid saturation and wettability,clay content, surface roughness, and even pore surface-to-volumeratio. Given the complexity of the different phenomena under con-sideration, practical models are designed to fit measured dielectricdispersions to ad-hoc models whose parameters are marginallysupported by quantitative petrophysical concepts.

Therefore, to assess whether accurate and reliable petrophysicalinterpretations are possible with WBEM measurements requires ananalysis that (a) incorporates pore structure, pore connectivity,multiphase saturation and electrochemical effects; and (b) quanti-fies the contributions of each factor in the measured WBEM dis-persions. However, extracting explicit petrophysical informationfrom WBEM responses is a difficult task. Myers (1991), for in-stance, illustrated the non-uniqueness of WBEM measurementswhen a decrease of water saturation, porosity, or brine salinityyielded similar responses. Recent advances in NMR logging andinterpretation (Freedman et al. 1990) can eliminate some of these

ambiguities with adequate experimental conditions, and if rockwettability is known. Conversely, WBEM measurements couldprovide independent wettability assessment in the cases whereNMR measurements alone reach their limits of sensitivity [forinstance, the impact of fluid saturation history on wettability de-termination was studied by Toumelin et al. (2006)]. Likewise, theinterpretation of NMR measurements can be biased by unac-counted rock morphology (Ramakrishnan et al. 1999) or by inter-nal magnetic fields in shaly or iron-rich sands (Zhang et al. 2003),whereas WBEM measurements provide independent informationon overall rock morphology. It is therefore timely to consider inte-grating both technologies for improving petrophysical analysis.

The objectives of this paper are twofold: (1) Review existingresults on the extraction of petrophysical information from rockWBEM measurements, and (2) establish a proof of concept for thenecessity to integrate electromagnetic measurements on the wide-frequency band from the kHz range to the GHz range, and studyhow WBEM techniques may yield petrophysical information un-available from other in-situ measurements. To reach the secondobjective, we introduce a generalized pore-scale simulation frame-work that allows incorporating arbitrary rock morphology andmultiphase fluid distribution.

Review of Wide-Band Electromagnetic Behaviorof Saturated RocksOrigins of Wide-Band Dispersion. As summarized by Sen andChew (1983), two main phenomena influence the dispersion ofconductivity and dielectric permittivity in saturated rocks at fre-quencies in the kHz-GHz range. The first effect is caused by theMaxwell-Wagner polarization in the MHz range, where variouspore-geometrical effects create sigmoid-type dispersions of con-ductivity and dielectric constant reviewed by Toumelin andTorres-Verdín (2007). The Maxwell-Wagner effect arises in thepresence of bimodal lossy and dielectric compounds (such asbrine/rock systems) where no surface zeta potentials (such as thoseencountered at clay surfaces) are present. Ions concentrate alongthe edges of elongated pores and create local capacitors in the porestructure whose effective capacitance and conductivity is fre-quency-dependent. Such a behavior solely depends on structuralaspect ratios regardless of size.

The second main electromagnetic phenomenon appearing inrocks consists of substantial enhancements of the dielectric con-stant as frequency decreases below the MHz range. In the early1980s, when WBEM studies for petrophysical applications were invogue, this low-frequency enhancement was regarded as measure-ment noise and spurious electrode-polarization effects. Subsequentquantification of electrode polarization exhibited strong negativepower laws of dielectric constant at kHz-range frequencies. Thispower-law effect has no apparent connection with Maxwell-Wagner polarization and is caused by the presence of static electriccharges at the interface between rocks (in particularly clay miner-als) and brine. The next sections of this paper review several modelsproposed to quantify such electrochemical effects, although none ofthem entirely captures the complexity of the phenomenon.

To understand the origin of low-frequency enhancement of themeasured dielectric constant, let us recall fundamental postulatesof electromagnetism: when an electric field E of radian frequency� illuminates a lossy material of conductivity �*(�) and dielectric

Copyright © 2008 Society of Petroleum Engineers

This paper (SPE 96258) was first presented at the 2005 SPE Annual Technical Conferenceand Exhibition, Dallas, 9–12 October, and revised for publication. Original manuscript re-ceived for review 13 July 2005. Revised manuscript received 12 September 2007. Paperpeer approved 7 October 2007.

205June 2008 SPE Journal

permittivity �*(�), it generates a current density J��*E and anelectric displacement field D��*E through that material. In gen-eral, both �* and �* are complex numbers and can be written interms of their real and imaginary parts as �*���−i�� and�*���+i��, where i2�−1 (the minus sign in �* is consistent withthe expressions that follow). As in the case of any causal physicalmechanism, �� and �� are coupled through the Kramers-Kronigrelationship (Landau and Lifschitz 1960):

����� = 1 +2

� �0

��0����0�

�02 − �2

d�0. . . . . . . . . . . . . . . . . . . . . . . . . . (1)

The latter relationship is generally used to verify the compat-ibility of experimental measurements with a complex expressionof the dielectric constant (Milton et al. 1997). A similar relationholds between �� and −��. With the e−i�t convention for the time-harmonic electric field of radian frequency �, the total densitycurrent J enforced through the material is given by

J = J +�D

�t= ��* − i��*� E = �*E, . . . . . . . . . . . . . . . . . . . . . (2)

where �* is the total effective complex conductivity of the mate-rial. The previously described total current density can also beexpressed in terms of the real effective conductivity �eff and per-mittivity �eff:

J = ��eff − i��eff�E, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

so that

�eff = Re��* − i��*� = �� + ���, . . . . . . . . . . . . . . . . . . . . . . (4a)

�eff = Re��* + i�*��� = �� +��

�, . . . . . . . . . . . . . . . . . . . . . . (4b)

and the measured dielectric constant (or relative dielectric permit-tivity) �eff, is equal to �eff divided by the vacuum permittivity�0�8.854×10−12 Farad/m. When static charges are negligible atthe rock/brine interface, the current J and displacement D are inphase with E, and both �* and �* remain real. However, if thesurface of an obstacle to the propagation of E is electricallycharged, then an ionic double layer develops at that interface.Complex values of �* and �* ensue, depending on the size of theobstacle, which explains positive out-of-phase conductivity �� andfrom Eq. 4b implies an enhancement of �eff at low frequencies.

Power-Laws, Double-Layers, and Fractal Geometries. If ��varies smoothly at low values of frequency, a power-law of expo-nent (–1) follows from Eq. 4b between dielectric constant andfrequency. The power law becomes a linear relationship between�eff and frequency on a bilogarithmic scale. This very intuitiveapproach may justify the systematic power-law behavior experi-mentally observed in clay-bearing rocks (Knight and Nur 1987;Bona et al. 1998), illustrated in Fig. 1, although to date there is noconclusive physical explanation that justifies it. Different modelsand approaches have been used to quantify such low-frequencybehavior of the dielectric constant. All these models are consideredas multiscale (or, to some extent, fractal) systems.

The basic element of such multiscale constructions is the elec-trochemical double layer (EDL) that develops outside a chargedrock surface. Hydrated sodium cations in brine solution concen-trate at the surface of negatively charged rock surfaces to enforceelectrical neutrality of the rock/fluid system. According to theStern model, a fixed layer of charged cations ensues that is ad-sorbed to the rock surface. A second, diffusive layer of electriccharge extends into the pore brine with a negative exponentialprofile of cation concentration. The ions in this EDL migrate underthe excitation of time-varying electric field, and the EDL as awhole behaves in the form of a dipole with complex electricalproperties that depend on frequency, ion mobility, pore brine sa-linity, surface charge, and surface geometry (Fixman 1980; Limaand Sharma 1992; Lesmes and Frye 2001). Computation of theEDL potentials proceeds from the combined solution of the elec-

trical and diffusion potentials with boundary conditions specific tothe EDL (DeLacey and White 1981; Chew and Sen 1982; Caoet al. 1994).

Thevanayagam (1997) proposed an approach that posits thefractal nature of the observed power-laws between �eff and �.Such a model is intended to capture the effective electrical prop-erties at each rock scale. Brine chemistry and electrical propertiesare assumed to vary at each scale (bound fluid, clay-bound doublelayer, free fluid). Starting from the smallest scale, effective elec-trical properties are computed at each scale from (a) the brineproperties at that scale and (b) the effective properties of the nextsmaller scale through nested mixing-laws. Thevanayagam recog-nized the necessity of incorporating EDL effects at the smallestscale; however, he used arbitrary values of brine conductivity anddielectric constant at each scale. In essence, Thevanayagam’s(1987) approach complements Sen et al.’s (1981) self-similar mod-els using stepwise iterations of compounds whose electrical prop-erties vary from scale to scale, instead of infinitesimal dilution ofcompounds that remain self-similar at all scales.

Rather than using arbitrary values of brine conductivities ateach fractal level, Lima and Sharma (1992) incorporated funda-mental electrochemical double-layer calculations around sphericalclay grains based on Fixman’s (1980) model to quantify the ef-fective conductivity and dielectric constant for clay zones. Thesevalues were then upscaled with Sen et al.’s (1981) self-similar,infinite-dilution mixing law by regarding brine as the host of themixture, rock as spherical grain inclusions, and clay as eitherspherical pellets or as shells coating the rock grains. In Thevanaya-gam’s model, the latter approach constitutes a two-step fractalprocess with a different mixing law. Lima and Sharma (1992),however, made no mention of power-law effects in their work. TheEDL low-frequency dielectric enhancement takes the form of asigmoid which converges at frequencies lower than 1 kHz to highvalues of dielectric constant, depending on clay geometry andsurface charge. It can be argued that, if the sigmoid extends alongseveral decades of frequency and if it is truncated before reachingits low-frequency asymptote, then that dielectric constant trendbehaves in similar fashion to a power law. This assumption is veryplausible, considering the measurements of Lesmes and Frye(2001) of saturated Berea samples: as � decreased from 1 kHz and0.1 Hz, they measured an increase of �eff that exactly followed apower law with exponent (–1), while below 0.01 Hz, �eff con-verged to static values in a sigmoid fashion. Such a behaviorsuggests that observed power laws may be formed by truncatedhigh-amplitude sigmoids which converge outside the measurementrange. This result supports the assumption of an EDL origin for theso-called dielectric low-frequency power law.

Fig. 1—Wide-band dispersions of dielectric constant measuredon four 22%-porosity Berea samples treated to exhibit differentwettabilities and saturated with different brine salinities. OW:oil-wet samples; WW: water-wet samples. The conductivity ofdeionized water is 1.57×10−3 S/m at 100 Hz, and that of salinebrine (75.8 g/l NaCl) is 9.5 S/m. After Bona et al. (1998); datacourtesy of Nicola Bona.

206 June 2008 SPE Journal

WBEM for Quantifying Petrophysical Properties. Previousstudies examined the influence of specific petrophysical propertiesof interest on WBEM measurements, including clay presence,multiphase saturation, rock wettability, and pore size; however,such studies failed to quantify the influence of the same petro-physical properties on WBEM measurements. This section sum-marizes results from several previous studies and emphasizes theirpractical limitations.

Because electrochemical and geometrical phenomena affect thedielectric response of saturated rocks, Myers (1991, 1996) consid-ered the dispersion of dielectric constant measured in the 20–200MHz range as the sum of (1) a clay term, where clay cationexchange capacity appears, (2) a geometric term, and (3) a vuggyporosity term. For that frequency range, Myers showed that terms(1) and (2) are sufficient to fit the dielectric dispersion of sand-stones saturated with two fluid phases, while terms (2) and (3) areappropriate to fit the dielectric measurements of single-phase het-erogeneous carbonates. The fitting parameters, however, have noexplicit petrophysical significance and are yet to be quantitativelyrelated to rock morphology and macroscopic petrophysical prop-erties, such as hydraulic permeability (Myers 1996).

Bona et al. (1998) used a similar superposition concept to fitdielectric rock measurements performed under partial-saturationconditions with (1) a power-law term added to (2) a Maxwell-Wagner sigmoid term, but on a broader 100 Hz–100 MHz fre-quency range. As shown in Fig. 1, Bona et al.’s (1998) measure-ments of water-wet (WW) Berea samples clearly exhibit power-law behavior below 1 MHz. By contrast, measurements performedon samples treated to be oil-wet (OW) at similar values of watersaturation reach sigmoid-like asymptotic behavior at values of di-electric constant much smaller than for the case of water-wetsamples. Although Bona et al. (1998) did not advance quantitativemethods to calculate fluid saturation under different wettabilityconditions, they did emphasize the potential of WBEM measure-ments in the kHz–MHz range to characterize rock wettability withno need for prohibitively long Amott wettability tests.

The complex refractive index method (CRIM) is known toquantify partial fluid saturations reasonably well in the GHz range;uncertainties associated with this method are lower than those oflow-frequency Archie-based conductivity calculations. However,CRIM and similar models do not account for heterogeneous car-bonate pore structure and are inaccurate to quantify fluid saturationin these conditions (Myers 1996). CRIM also serves as a startingpoint for Seleznev et al. (2006) to derive a mixing law that correctsfor the same geometrical parameters and that fits dielectric mea-surements of geometrically challenging carbonate rocks, eventhough their range of application was limited to 100 MHz–1 GHz.Such ongoing efforts suggest that more quantitative morphologicdetails can be incorporated in the interpretation of WBEM mea-surements to improve the estimation of petrophysical properties.

Beyond their observation of power-law behaviors in the sub-MHz range, Knight and Nur (1987) also observed a correlationbetween measured dielectric constant and rock surface-to-volumeratio for partially saturated rocks in the range of 60 kHz–4 MHz.However, their results lend themselves to ambiguous interpreta-tion. The largest dispersions of dielectric constant (correlated withthe highest surface-to-volume ratios) also happened to be mea-sured in tight samples with less than 1 mD permeability, less than8% porosity, and substantial clay content. Furthermore, these mea-surements were made in deionized water, which displaces the fre-quency range of pore geometrical effects below 1 MHz. Eventu-ally, the exact origin of the correlation between dielectric constantand surface-to-volume ratio is difficult to determine.

Finally, Olhoeft’s (1985) comprehensive review on mHz–MHzwide-band electrical phenomena in rocks cites oxidation-reductionreactions as a common brine/rock interface phenomenon. Whensurface reactions take place in a diffusion-limited (by opposition tokinetics-limited) fashion, the electrical response of a compoundgenerally depends on the size of metallic bodies in the mixture. Inrocks, these bodies can consist of pores coated with metallic de-posits, or of metallic inclusions embedded in the rock matrix. Toillustrate this argument, Olhoeft noted that suspensions of 1-mm

metallic particles exhibit a 10-Hz relaxation mode in their dielec-tric spectrum, while suspensions of 10-�m particles exhibit a 100-kHz mode. An adaptation of double-layer models would providesimilar results, although a macroscopic model that incorporatesboth kinetics- and diffusion-limited reactions in saturated rocks isyet to be proposed. The possibility of inferring pore-size informa-tion from WBEM measurements becomes very attractive to vali-date NMR pore-size estimates.

All these approaches therefore attempt to extract petrophysicalinformation from WBEM measurements, but remain qualitative atbest for not individualizing the contributions of the processes si-multaneously occurring in rocks. In order to study the interactionsbetween the different phenomena, we develop a pore-scale modelgeneralized from that of Toumelin and Torres-Verdín (2007) forquantifying Maxwell-Wagner geometrical polarization effects.

Development of a Generalized 2D WBEMNumerical FrameworkGeometrical Pore-Scale Framework. Toumelin and Torres-Verdín (2007) developed a numerical procedure to study the limitsof dielectric mixing laws to characterize pore structures in theabsence of clays and saturated exclusively with brine, in agreementwith both Maxwell’s equations and Kramers-Kronig relations.That model incorporates interfacial WBEM effects arising fromthe accumulation of charges at the boundaries between brine andgrains and is based on the following procedure:

(1) Build disc-shape binary pore maps where pixels representeither grain or brine in a binary fashion.

(2) Assign each such pixel with DC conductivity and dielectricconstant of the modeled medium (grain or brine).

(3) For each given frequency between 10 kHz and 1 GHz,numerically solve Maxwell’s equations with the method of mo-ments to compute the internal electric field and electric currentwithin each pixel of the pore map when subject to transverse-magnetic excitation. Fig. 2 illustrates the distribution of internal cur-rents obtained for different angles of incidence of the exciting wave.

(4) Use the close-form analytical solution of electromagneticscattering within a disc to determine the homogeneous DC con-ductivity and dielectric constant of a disc that minimize the dis-crepancy between the electric fields and the induced currents cal-culated within the pore map and within the disc.

Instead of assigning arbitrary pore shapes to the model, weextract the pore maps from rock digital pictures at different rates ofmagnification. Fig. 3 shows an example of a digitized picture of a24%-porosity sample, which is binarized and truncated into anappropriate disc-shape pore map (white and blue pixels). This newrock model exhibits explicit distributions of pores and throatslarger than the pixel resolution of the rock image. Extending themodel in three dimensions on the binary skeleton of high-resolution rock tomography (Arns et al. 2005) should shed addi-tional insight to geometrical polarization effects in complexrock structures.

Inclusion of Clays. Next, we incorporate clays in the rock skel-eton by arbitrarily distributing clay pixels throughout the poremap, as illustrated in Fig. 3 (magenta pixels) where clay amountsto 3% of the rock volume. At each considered frequency, insteadof grain or brine properties, clay pixels are assigned effectiveelectrical properties following Lima and Sharma’s (1992) treat-ment of an ionic double-layer around a spheroid clay grain (seeAppendix A). If the grains are larger than the pixel resolution, thenseveral pixels form one clay-coated grain and each pixel is as-signed the same electrical properties. On the other hand, if grainsare smaller than pixel resolution, then one pixel represents a groupof clay-coated grains that span a surface equal to that pixel. Thisnumerical framework therefore upscales the response of the clay-bearing rock at the pixel level regardless of mixing laws.

Inclusion of Immiscible Fluids and Wettability. The 2D poremap is assimilated to a cross section of 3D rock geometry, wheremultiphase saturations can also be distributed in the pore map asthey would appear on a 2D cross-section of a 3D saturated rock.

207June 2008 SPE Journal

Fluid distributions described in the pore maps that follow are de-signed to resemble a cross-section through a rock where fluiddisplacement takes place in three dimensions. Fig. 4 illustrates thisconcept and emphasizes the pixel-based structure of such a 2Dmodel for a water-wet rock. In this model, a layer of one to twowater pixels is preserved at the rock surface to enforce a strongelectrical contrast at rock/water and oil/water interfaces, while oilis assumed to invade the pore structure starting from the outside ofthe disc.

Simulating wettability effects on WBEM measurements is akey objective to this numerical exercise. As shown in Fig. 1, largeenhancements of the dielectric constant are measured at low fre-quencies in WW rocks with low values of water saturation. Con-tinuity between thin brine films in the WW pores prevents energystorage within the films; therefore, the strong enhancement ofdielectric constant at low values of frequency in WW rocks isprimarily caused by EDL effects. In strongly OW conditions, theseEDLs are inhibited because no cations are in contact with thecharged clay surface. The activation of EDLs in the magenta pixelsof Figs. 3 and 4 characterizes the water wettability of the rockmodel depicted in Fig. 3. These principles are used to constructWW and OW multiphase pore models.

Assessing Rock Morphology From Single-PhaseWBEM MeasurementsWe now distribute the wide-band effective clay properties calcu-lated in Appendix A into the clay pixels of Fig. 3 pore maps. Pixels

labeled as “clay” in that pore map are successively assigned theelectrical properties of (a) grain, (b) 0.1 S/m brine, and (c) 100-nmclay in 0.1 S/m brine background. Fig. 5 shows the correspondingdielectric dispersions for the three cases and emphasizes the in-fluence of rock texture in the MHz range. In that figure, conductivepaths created by brine in Case B decrease the ability of the rockmodel to store energy compared to Case A. The amplitude ofthe dielectric dispersion at low values of frequency is lower inCase B than in Case A. In Case C, where the inclusion of 3%dispersed clay is simulated over the entire frequency range, thedielectric dispersion is similar to that simulated for Case B at 10MHz, but diverges exponentially below this frequency. All CasesA through C are associated with similar dielectric behavior above10 MHz.

If we vary brine salinity in the clay-free configuration (Case A),then the simulated dielectric response (Fig. 6) exhibits the ex-pected scaling with brine conductivity (Kenyon 1984; Toumelinand Torres-Verdín 2007). On the other hand, if brine salinity isvaried in the 3%-clay rock model, then the scaling of dielectricconstant with conductivity disappears and the dispersion shown inFig. 7 ensues. We also note that high brine salinity preserves rocktexture information in the MHz range while double-layer phenom-ena are prominent at lower values of frequency. By contrast, lowbrine salinities induce a dielectric response where both effectsoverlap in an undistinguishable manner. Such a numerical ap-proach is therefore necessary to quantify—and ultimately invert for—rock morphological effects confounded with clay polarization.

Fig. 2—Amplitude distribution of internal current density J (A/m2) induced in a pore map exhibiting elliptic inclusions, under twoperpendicular angles of incidence at 100 MHz, and using different scales. Arrows indicate the directions of plane-wave incidence.

Fig. 3—Sandstone micrograph (left panel; gray: grain; black: pore space) and corresponding digital pore map (right panel; gray:grain; white: pore space) scattered with arbitrary clay inclusions (red pixels). The scale on the pore map axes is given in pixels,with the pixel resolution equal to 5 µm.

208 June 2008 SPE Journal

This observation appears particularly useful when other petro-physical methods fail to properly quantify pore structure and con-nectivity. For example, the low surface relaxation of carbonaterocks prevents the NMR characterization of certain bimodal po-rosities. Appendix B describes the construction of a model for suchrocks, where the simulated WBEM response departs greatly fromthe usual sigmoidal response of sedimentary rocks (Fig. 1) forexhibiting flat dispersions of both conductivity and dielectric con-stant across the entire kHz–GHz frequency range. Toumelin andTorres-Verdín (2007) also demonstrated that connectivity betweenpores of equal eccentricity affect both electrical dispersions, asrecapitulated in Figs. 8 and 9. The different inflection behavior ofthe dielectric spectra simulated for pore maps D1 and D2 alsosuggests a possible dependence of dispersion-inflection frequen-cies upon the degree of pore isolation, as the internal electriccurrents distribute differently in the pore space when percolation isreached across the entire pore space. It is therefore important toquantify the contribution of pore geometry to the WBEM responsein that frequency range.

Assessing Wettability and Fluid SaturationPrevious simulation work (Toumelin et al. 2006) emphasized theconceptual ambiguity of wettability interpretation using 2D NMRtechniques for given conditions of oil viscosity and saturation his-tory. We now conceptually confirm that WBEM techniques can

circumvent these ambiguities in rocks with minimal amounts ofclay. As mentioned earlier, EDL effects fully develop in water-wetcases, possibly enhanced by fractal surface roughness, while theyare inhibited in OW cases. Following the method developed above,we populate the pore map of Figs. 3 and 4 with wetting andnonwetting fluid pixels to describe the progressive drainage of aWW rock by oil (Fig. 10). By contrast, Fig. 11 models the pro-gressive drainage of an OW rock by water.

Fig. 12 shows the wide-band dispersions of dielectric constantsimulated for all the pore maps described in Figs. 10 and 11, fortwo values of brine conductivity, �w�0.1 and 1 S/m. Each panelof Fig. 12 groups the simulation results by brine conductivity orwettability type. Several remarks stem from this comparison. First,the scaling of the dielectric constant with conductivity remainspresent in the EDL-free OW cases regardless of fluid saturation.This scaling disappears in the WW cases, which makes saturationestimates nontrivial. Second, each panel of Fig. 12 shows that thehierarchy of the dielectric dispersion curves with respect to watersaturation or brine conductivity is enforced in the GHz range re-gardless of the competition between geometrical and electrochemi-cal effects at lower values of frequency. The values of dielectricconstant simulated for low values of water saturation are smallerthan those simulated for high values of water saturation, whichhonors CRIM results. Likewise, the dielectric dispersion sigmoidsof the bottom right panel of Fig. 12 exhibit a shift to the left (lowerfrequencies for same values of dielectric constant) when conduc-

Fig. 4—Graphical example of two-phase fluid distribution within the pore map of Fig. 3. Colors are coded as follows: gray: grain;red: clay cements; green: non-wetting hydrocarbons; white: water-filled pore space. The local enlargement shown on the rightpanel displays the pixel-based construction of the 2D model.

Fig. 5—Frequency dispersion of dielectric constant simulatedfor the rock model shown in Fig. 3 assigning three differenttextures to its red pixels. Brine conductivity is taken equal to0.1 S/m.

Fig. 6—Frequency dispersion of dielectric constant simulatedfor the rock model of Fig. 3 (without clay double-layers) and forthree different values of brine salinity.

209June 2008 SPE Journal

tivity decreases. Such a behavior is consistent with the notion ofconductivity scaling introduced by Kenyon (1984). However,when frequency decreases in the MHz range where EDL effectsbecome important, this curve hierarchy disappears as dispersioncurves intersect and change order.

Simulation results described in Fig. 12 exhibit a low-frequencypower-law behavior for the dielectric constant of WW rock mod-els. This power-law behavior is remarkable given the small amount

of clay pixels (as low as 3% of the total number of rock pixels) andthe small value of surface charge (�11 �C/cm2) considered inthe model. Simulation results agree very well with the measuredtrends of dielectric constant of WW rocks (Fig. 1). Dielectric dis-persions obtained from simulations and experiments exhibit (a)low-frequency convergence in OW cases, (b) low-frequency ex-ponential behavior in WW cases, and (c) dual inflection in theMHz-range in some WW cases. Quantitatively, the absolute valuesof simulated and measured dielectric constants differ by a factorapproximately equal to 20. It is likely that larger values of claysurface charge, clay amount, or a more accurate double-layermodel, would reduce this discrepancy. However, the relative wide-band behavior of WW dielectric dispersions is identical for bothsimulations and measurements. In the 1-MHz range, between thetwo inflection points of the dielectric dispersion curves, both simu-lations and measurements yield values of dielectric constant equalto 10 times the value of their respective high-frequency asymp-totes. In the 10-kHz range, this enhancement factor becomes ap-proximately equal to 30 for both simulations and measurements.

It is also important to note that electrical measurements ingeneral, and water-wet dielectric low-frequency enhancements inparticular, are insensitive to oil viscosity. The only assumptionformulated on the saturating oil was to assign it a dielectric con-stant equal to 2 over the entire kHz–GHz frequency spectrum.

ConclusionsFractal and double-layer concepts were reconciled to explain thelow-frequency dielectric behavior of saturated rocks. A general-ized 2D pore-scale electromagnetic model ensued, which incorpo-rates the simultaneous dielectric effects of pore geometry, chargedclays and minerals, wettability, and fluid saturations. Despitequantitative limitations owing to the spatial resolution of the poremap and the double-layer model, we obtained very encouragingresults when modeling explicit rock WBEM measurements inde-pendently from mixing rules or fitting parameters:

First, WBEM measurements are theoretically proven to be sen-sitive to both pore shape and pore connectivity. The flat dielectricdispersion simulated in the diffusively-coupled microporous rockmodel is an example where WBEM becomes a viable technique tocharacterize complex rocks where classic NMR interpretation failsto correctly predict petrophysical properties. One can envision ageneralization of WBEM measurements to diagnose the presenceof complex pore inhomogeneity, including micritized, oolithic, orvuggy carbonate rocks. However, WBEM measurements have notheoretical sensitivity to pore size except when EDL develops aspore-surface coating.

Second, simulations reproduced WBEM measurements fairlywell for partially saturated rocks. They characterized wettabilitytrends in clay-bearing sandstones, even for clay concentrations aslow as 3%, and regardless of oil grade. Wide-band EM techniques

Fig. 7—Frequency dispersion of dielectric constant simulatedfor the rock model of Fig. 3 (with clay double-layers character-ized in Fig. A-1) and for three different values of brine salinity.

Fig. 8—Pore maps modified after map D and exhibiting increas-ing connectivity between brine pores. Brine inclusions (blackpixels) are embedded in the rock host (gray pixels).

Fig. 9—Frequency dispersions of effective dielectric constant(top panel) and electrical conductivity (bottom panel) for poremaps D00 to D3. Simulation results are identified with markersfor �host=2. The two continuous sigmoids identify the bestmatches to the simulated results obtained using two analyticalapproches: Effective-Medium-Approximation in continuouscurve (Kenyon 1984) and CRIM-like power-law in dotted curve.

210 June 2008 SPE Journal

can therefore help to determine in-situ sandstone wettability andcomplement NMR wettability estimates which are ambiguouswhen oil viscosity exceeds a few cp.

A comprehensive WBEM model needs to be reliable in thepresence of charged clay to perform quantitative petrophysicalassessments. Clays usually form dia- or para-magnetic impuritiescausing internal magnetic fields which make NMR interpretationunreliable for petrophysical estimates. Consequently, WBEMtechniques can improve the reliability of NMR techniques in shalyor iron-rich sands.

Table 1 summarizes the different sensitivities that are requiredto perform accurate petrophysical assessments of a general rockmodel. Pore size and connectivity are vital to assess hydraulicpermeability in both clean and clayey sands. NMR measurementsusually quantify pore-size distribution but not pore connectivity,unlike WBEM measurements. It is also necessary to identify avariety of fluids to perform adequate fluid typing. NMR measure-ments detect medium- and light-oil grades quite well, but areriddled by acquisition and inversion problems in the presence ofgas and heavy oils. Oil grades whose relaxation times and diffu-sivities are close to those of water within confining rocks alsocause NMR interpretation problems. WBEM measurements, onthe other hand, are independent of hydrocarbon type; therefore,they allow wettability assessment but not fluid typing. Table 1emphasizes the sensitivities of NMR and WBEM techniques forthese and other essential criteria. Once combined, the two mea-surement techniques would permit a full coverage of the sensitivitydomain required to perform accurate and reliable petrophysicalassessments.

AcknowledgmentsWe acknowledge stimulating discussions with M. Sharma toimplement the Fixman EDL model. Our sincere gratitude goes to

two anonymous reviewers for their constructive technical and edi-torial feedback. Funding for the work reported in this paper wasprovided by (a) the American Chemical Society through grant no.PRF 37519-AC9, (b) the U.S. Department of Energy through con-tract no. DE-FC26-04NT15518, and (c) UT Austin’s ResearchConsortium on Formation Evaluation, jointly sponsored by Ana-darko, Aramco, Baker Atlas, BHP Billition, BP, British Gas,ConocoPhillips, Chevron, ENI E&P, ExxonMobil, Halliburton En-ergy Services, Hydro, Marathon, the Mexican Institute for Petro-leum, Occidental Oil and Gas Corporation, Petrobras, Schlum-berger, Shell International E&P, Statoil, Total, and Weatherford.

ReferencesArns, C.H. et al. 2005. Pore-Scale Characterization of Carbonates Using

X-Ray Microtomography. SPEJ 10 (4): 475–484. SPE-90368-PA.DOI: 10.2118/90368-PA.

Bona, N. et al. 1998. Characterization of rock wettability through dielectricmeasurements. Revue de l’Institut Français du Pétrole 53 (6): 771–783.

Cao, Q.-Z., Wong, P.-Z., and Schwartz, L.M. 1994. Numerical studies ofthe impedance of blocking electrodes with fractal surfaces. PhysicalReview B 50 (8): 5771–5774. DOI: 10.1103/PhysRevB.50.5771.

Chew, W.C. and Sen, P.N. 1982. Dielectric enhancement due to electro-chemical double layer: thin double layer approximation. J. of ChemicalPhysics 77 (9): 4683–4693. DOI: 10.1063/1.444369.

DeLacey, E.H.B. and White, L.R. 1981. Dielectric response and conduc-tivity of dilute suspensions of colloidal particles. J. of the ChemicalSociety, Faraday Transactions 2 (77): 2007–2039. DOI: 10.1039/f29817702007.

Fixman, M. 1980. Charged macromolecule in external fields. I. The sphere.J. of Chemical Physics 72 (9): 5177–5186. DOI: 10.1063/1.439753.

Freedman, R. et al. 1990. Wettability, Saturation, and Viscosity FromNMR Measurements. SPEJ 8 (4): 317–327. SPE-87340-PA. DOI:10.2118/87340-PA.

Fig. 11—Fluid distribution in oil-wet rock models for three values of water saturation Sw. Gray pixels represent grain matrix; whitepixels, water-saturated porosity; and green pixels, oil-saturated porosity.

Fig. 10—Fluid distribution in water-wet rock models with clays and three values of water saturation Sw. Gray pixels represent grainmatrix; red pixels, clay; white pixels, brine-saturated porosity; and green pixels, oil-saturated porosity.

211June 2008 SPE Journal

Fig. 12—Frequency dispersions of dielectric constant simulated for the pore maps of Figs. 10 (water-wet cases, WW) and 11 (oil-wetcases, OW), for two values of brine conductivity. Top left panel: all simulation results for �w=1 S/m; top right panel: for �w=0.1 S/m;bottom left panel: all water-wet geometries; bottom right panel: all oil-wet geometries.

212 June 2008 SPE Journal

Glover, P.W.J., Meredith, P.G., Sammonds, P.R., and Murrell, S.A.F.1994. Ionic surface electrical conductivity in sandstones. J. of Geo-physical Research 99 (B11): 21635–21650. DOI: 10.1029/94JB01474.

Kanket, W., Suddhiprakarn, A., Kheoruenromne, I., and Gilkes, R.J. 2005.Chemical and crystallographic properties of Kaolin from ultisols inThailand. Clays and Clay Minerals 53 (5): 478–489. DOI: 10.1346/CCMN.2005.0530505.

Kenyon, W.E. 1984. Texture effects on megahertz dielectric properties ofcalcite rock samples. J. of Applied Physics 55 (8): 3153–3159. DOI:10.1063/1.333315.

Knight, R.J. and Nur, A. 1987. The dielectric constant of sandstones, 60kHz to 4 MHz. Geophysics 52 (5): 644–654. DOI: 10.1190/1.1442332.

Landau, L.D. and Lifschitz, E.M. 1960. Electrodynamics of continuousmedia. Oxford, UK: Pergamon Press.

Lesmes, D.P. and Frye, K.M. 2001. Influence of pore fluid chemistry onthe complex conductivity and induced polarization responses of Bereasandstone. J. of Geophysical Research 106 (B3): 4079–4090. DOI:10.1029/2000JB900392.

Lima, O.A. and Sharma, M.M. 1992. A generalized Maxwell-Wagnertheory for membrane polarization in shaly sands. Geophysics 57 (3):431–440. DOI: 10.1190/1.1443257.

Milton, G.W., Eyre, D.J., and Mantese, J.V. 1997. Finite Frequency RangeKramers-Kronig Relations: Bounds on the Dispersion. Physical ReviewLetters 79 (16): 3062–3065. DOI: 10.1103/PhysRevLett.79.3062.

Myers, M.T. 1991. A saturation interpretation model for the dielectricconstant of shaly sands. Paper 9118. SCA International Symposium,San Antonio, Texas, 21–22 August.

Myers, M.T. 1996. A pore geometry dispersion model for the dispersion ofthe dielectric constant. Paper 9626. SCA International Symposium,Montpellier, France, 8–10 September.

Olhoeft, G.R. 1985. Low-frequency electrical properties. Geophysics 50(12): 2492–2503. DOI: 10.1190/1.1441880.

Ramakrishnan, T.S., Schwartz, L.M., Fordham, E.J., Kenyon, W.E., andWilkinson, D.J. 1999. Forward models for nuclear magnetic resonancein carbonate rocks. The Log Analyst 40 (4): 260–270.

Seleznev, N., Habashy, T., Boyd, A., and Hizem, M. 2006. Formationproperties derived from a multi-frequency dielectric measurement. Pa-per VVV. SPWLA Annual Logging Symposium, Veracruz, Mexico,4–7 June.

Sen, P.N. and Chew, W.C. 1983. The frequency dependent dielectric andconductivity response of sedimentary rocks. J. of Microwave Power 18(1): 95–104.

Sen, P.N., Scala, C., and Cohen, M.H. 1981. A self-similar model forsedimentary rocks with application to the dielectric of fused glassbeads. Geophysics 46 (5): 781–795. DOI: 10.1190/1.1441215.

Sonon, L.S. and Thompson, M.L. 2005. Sorption of a nonionic polyoxy-ethylene lauryl ether surfactant by 2:1 layer silicates. Clays and ClayMinerals 53 (1): 45–54. DOI: 10.1346/CCMN.2005.0530106.

Stroud, D., Milton, G.W., and De, B.R. 1986. Analytical model for thedielectric response of brine-saturated rocks. Physical Review B 34 (8):5145–5153. DOI: 10.1103/PhysRevB.34.5145.

Thevanayagam, S. 1997. Dielectric dispersion of porous media as a fractalphenomenon. J. of Applied Physics 82 (5): 2538–2547. DOI: 10.1063/1.366065.

Toumelin, E. and Torres-Verdín, C. 2007. 2D pore-scale simulation ofwide-band electromagnetic dispersion of saturated rocks. Geophysics72 (3): F97–F110. DOI: 10.1190/1.2561301.

Toumelin, E., Torres-Verdín, C., Chen, S., and Fisher, D.M. 2003. Rec-onciling NMR Measurements and Numerical Simulations: Assessmentof Temperature and Diffusive Coupling Effects on Two-Phase Carbon-ate Samples. Petrophysics 44 (2): 91–107.

Toumelin, E., Torres-Verdín, C., Sun, B., and Dunn, K.-J. 2006. Limits of2D NMR Interpretation Techniques To Quantify Pore Size, Wettabil-ity, and Fluid Type: A Numerical Sensitivity Study. SPEJ 11 (3):354–363. SPE-90539-PA. DOI: 10.2118/90539-PA.

Zhang, G.Q., Hirasaki, G.J., and House, W.V. 2003. Internal field gradientsin porous media. Petrophysics 44 (6): 422–434.

Appendix A—Quantitative Incorporation ofElectrical Double LayersFollowing de Lima and Sharma (1992), we embed the effectivevalues of conductivity and dielectric constant for the ionic double

layers of wet clay from Fixman’s (1980) model. Clay particles arerepresented as spheres of radius a and consist of resistive core andcounterion surface charge density (in units of 1/m2). Ions in thedouble-layer of diffusivity D displace around the insulating sphereand build up a stationary charge. The corresponding diffusion time is given by the ratio

=a2

2D, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)

while the double-layer efficiency � is defined by the dimension-less ratio

� =

aC, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

where C is the density of cations in the bulk water (in units of1/m3). The complex conductivity associated with this double-layer�DL* can be written as a function of the pore water conductivity,�w*, as follows:

�*DL =��*w

1 − �Y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

In this equation, Y stands for the frequency-dependent conductivitybuildup caused by the time oscillation of charges around the in-sulating grain, and is defined by the equation

Y = −1 + �1 − i���

2 + 2�1 − i��� − 2i�. . . . . . . . . . . . . . . . . . . . . . (A-4)

From these expressions, it follows that the values of real effectiveconductivity and dielectric constant for the clay particle and itsEDL can be approximated by the equations

�eff = �� Re� �

1 − �Y� + � �0�� Im� �

1 − �Y�, . . . . . . . . (A-5a)

and

�eff = �� Re� �

1 − �Y� −��

� �0Im� �

1 − �Y� + �grain. . . . . (A-5b)

The practical use of Eqs. A-5 requires quantification of the con-stants and C included in Eq. A-2. As noted by Glover et al.(1994), depends on ion surface adsorption and brine pH, as wellas electrolyte concentration. However, it is reasonable to assume aconstant clay surface charge density in the order of 10–20 �C/cm2,or 6–12×1017 ions/m2, as measured by Sonon and Thompson(2005) in smectites. In this paper, we assume a conservative valueof �11 �C/cm2 or 6.9×1017 ions/m2 It must be pointed out thatvalues as high as 100 �C/cm2 have been reported for kaolinites(Kanket et al. 2005), which would make the impact of clays on theelectrical response of rocks much more significant than that shownin this paper.

The brine cationic density C is linked to the electrical proper-ties of the brine. Stroud et al. (1986) recapitulate existing corre-lations between �w (in S/m), �w, temperature T (in °C), and salinityX (in kppm) through the equations

�w =T + 21.67

45.56 �0.0123 +3647.5

�1000X�0.995�−1

, . . . . . . . . . . . (A-6a)

and

�w = � 1

87.69 − 0.3921T + 0.000217T2+

2.4372X

58.443�1000 − X��−1

. . . . . . . . . . . . . . . . . . . . . . . (A-6b)

By dividing X by the molecular atomic mass of the dissolved saltand multiplying the result by the elementary charge, one obtainsthe expression for C.

Using Eqs. A-1 through A-6 for different clay sizes a�100 and1000 nm, and for brines with different conductivities of 0.1, 1, and10 S/m, we generated the wide-band dielectric dispersions shownin Fig. A-1. These results exhibit expected sigmoidal shapes

213June 2008 SPE Journal

whose amplitudes vary primarily with salinity. For 100-nm clayparticles, if the frequency range is limited to frequencies above 10kHz, only brine conductivities lower than 10 S/m will affect therock dielectric response (�eff≈�grain); when clay size increases, thiseffect becomes more important. As salinity decreases, �eff≈�w for100-nm clay particles, but �eff >>�w for larger clay size at lowfrequency. If the frequency range is truncated in the kHz range, theascending part of the sigmoids locally fits a power-law expressionbetween frequency and dielectric constant. Such a behavior is inagreement with discussions included in previous sections ofthis paper.

Appendix B—Construction of a WBEM Model toQuantify WBEM Microporosity EffectsPresence of diffusion coupling in microporous carbonate rockswith low surface relaxivity prevents NMR measurements fromcorrectly quantifying pore-size distributions, bound fluid volumes,and hydraulic permeability (Ramakrishnan et al. 1999; Toumelinet al. 2003). In this paper, we show that WBEM measurementshave the potential to reduce ambiguities on such complex rockmorphologies. A WBEM geometric model is built from rock im-ages and two-step elecromagnetic upscaling in the absence ofdouble-layer effects. We synthesized a carbonate rock model from

core data, core pictures, and NMR measurements of a micritizedrock sample exhibiting diffusion coupling (Sample A fromToumelin et al. 2003). This sample includes a complex porositymade of dissolution vugs (amounting to 10 of the 24% porosity ofthe sample, with pores sizes in the 15-�m range), primary-porositymacropores (4% porosity, 5–15 �m range), and abundant micri-tized intragranular porosity (10% porosity, submicron range).

Following the dual-porosity structure shown by the high-resolution images of Fig. B-1, Fig. B-2 shows an equivalent two-scale 2D rock model amenable to the simulation of WBEM mea-surements. This figure comprises two disks with two differentspatial resolutions that constitute the two steps of the upscalingmethod. The left disk of Fig. B-2 forms a 501×501-pixel map of alow-resolution rock model (1 pixel�0.5 �m) and captures thegeneral dual porous structure of the micritized carbonate sample(10% intergranular and 15% intragranular porosities). However, itslow spatial resolution neglects the submicron micropores and mi-crothroats existing within the microporous grains. This is why wealso use a higher-resolution micromodel (right disk of Fig. B-2)that explicitly enforces the presence of 100-nm brine layers be-tween micrite grains, as shown in the right disk of 501×501-pixelresolution (1 pixel�50 nm). Once the dispersions of isotropicconductivity and dielectric constant are computed for the high-

Fig. B-1—High-resolution scanning electron microscope (SEM) images of a carbonate sample exhibiting diffusion pore couplingat 140X (left panel) and 1400X (right panel) magnifications (Toumelin et al. 2003a)

Fig. A-1—Frequency dispersions of pure clay dielectric constant simulated for different values of clay size and brine conductivitythrough Eq. 9b. Open markers: a=100 nm, closed markers: a=1 µm. Squares: �w=10 S/m; triangles: �w=1 S/m; circles: �w=0.1 S/m.Dotted lines identify the bulk dielectric constant of quartz (4) and water (80). The locally-straight lines obtained between 1 kHz and10 kHz for a=100 nm and between 100 kHz and 500 kHz for a=1 µm appear as power-laws of dielectric constant behavior withrespect to frequency.

214 June 2008 SPE Journal

resolution micromodel, they are assigned to the grain pixels in thelow-resolution rock model. The effective conductivity and dielec-tric constant obtained for the low-resolution model constitute theeffective properties of the coupled micritized rock.

Fig. B-3 illustrates the distribution of electric-current strengthcalculated at frequencies below 1 MHz within the low-resolutionrock model after two-step upscaling. Incorporation of the micro-scale model in the two-step upscaling process plays a fundamentalrole in the effective response of the coupled rock model. Fig. B-4compares the wide-band dielectric dispersions calculated for thecoupled model, with and without two-step upscaling, with thefrequency dispersions calculated for the generic model of Fig. 3. Inthe absence of upscaling, the dielectric response of the coupledmacromodel (left disc of Fig. B-2 only) is similar to that shown inFig. 5 for the generic rock, except for the amplitudes of the di-electric constant. That amplitude is larger in the generic modelthan in the coupled macromodel because eccentricities and dead-ends are more prominent in the generic model. Once the micro-model (right disc of Fig. B-2) is incorporated as the first step of the

upscaling, the dielectric dispersion of the coupled model becomesremarkably flat. Presence of a water layer between all micritegrains effectively connects all the pores of the macromodel andeliminates the geometric dielectric enhancement of the rock. Ex-istence of dead ends where electrical storage occurs in discon-nected water zones is therefore the key to dielectric enhancement.Given these simulation results, it is likely that a rock exhibitingstrong diffusion coupling will also exhibit high electrical conduc-tivity and abnormally constant electromagnetic dispersion. There-fore, special emphasis can be placed on WBEM measurements tocharacterize abnormal heterogeneous rock morphologies and re-fine the estimation of petrophysical properties performed withNMR measurements.

Emmanuel Toumelin has been with Chevron North AmericaE&P as a Petrophysicist for Midcontinent and Alaska since2006. Previously, he was a research assistant at the University ofTexas at Austin from 2000 to 2005, where he conducted re-search on pore-scale modeling and combined interpretationof NMR and dielectric rock properties. He is an associate editorfor Petrophysics (SPWLA) and a technical editor for SPE Journaland SPE Reservoir Evaluation and Engineering. He holds anengineering degree from the Ecole Centrale de Lille and MSand PhD degrees in petroleum engineering from the Universityof Texas at Austin. Carlos Torres-Verdín is an Associate Professorin the Department of Petroleum and Geosystems Engineeringof the University of Texas at Austin, where he conducts re-search on formation evaluation, petrophysics, well logging,borehole geophysics, and integrated reservoir characteriza-tion. Previously, he worked as Research Scientist for Schlum-berger-Doll Research and as Reservoir Specialist and ProductChampion for YPF S.A. (Buenos Aires, Argentina). In the past,he has been distinguished by the SPWLA with the TechnicalAchievement Award and with the 2003, 2004, 2006, and 2007Best Paper Awards by the journal Petrophysics. He directs theJoint Industry Research Consortium on Formation Evaluation.He holds a PhD degree in engineering geoscience from theUniversity of California at Berkeley. He is an associate editor forPetrophysics (SPWLA) and a technical editor for the SPE Journal.

Fig. B-3—Distribution of internal current-density amplitudes (A/m2) within the low-resolution microporous rock model below 1MHz. The arrow shows the direction of polarization of the inci-dent electric field.

Fig. B-4—Comparison of dielectric dispersions obtained for 1S/m brine with the generic model of Fig. 3 and with the coupledmodel of Fig. B-2.

Fig. B-2—Representation of the two-step 2D model of a rockdesigned to replicate the rock structure shown in Fig. B-1.

215June 2008 SPE Journal