Effective conductivity and permittivity of unsaturated porous materials in the frequency range 1 mHz-1GHz.

4.9 MB·English

by  Revil, A

WATERRESOURCESRESEARCH,VOL.49,306–327,doi:10.1029/2012WR012700,2013
Effective conductivity and permittivity of unsaturated porous
materials in the frequency range 1 mHz–1GHz
A.Revil1,2
Received21July2012;revised26November2012;accepted29November2012;published24January2013.
[ ] Amodelcombininglow-frequencycomplexconductivityandhigh-frequency
1
permittivityisdevelopedinthefrequencyrangefrom1mHzto1GHz.Thelow-frequency
conductivitydependsonporewaterandsurfaceconductivities.Surfaceconductivityis
controlledbytheelectricaldiffuselayer,theoutercomponentoftheelectricaldoublelayer
coatingthesurfaceoftheminerals.Thefrequencydependenceoftheeffectivequadrature
conductivityshowsthreedomains.Belowacriticalfrequencyf ,whichdependsonthe
p
dynamicporethroatsize(cid:2),thequadratureconductivityisfrequencydependent.Betweenf
p
andasecondcriticalfrequencyf ,thequadratureconductivityisgenerallywelldescribed
d
byaplateauwhenclaymineralsarepresentinthematerial.Clay-freeporousmaterialswith
anarrowgrainsizedistributionaredescribedbyaCole-Colemodel.Thecharacteristic
frequencyf controlsthetransitionbetweendoublelayerpolarizationandtheeffectofthe
d
high-frequencypermittivityofthematerial.TheMaxwell-Wagnerpolarizationisfoundto
berelativelynegligible.Forabroadrangeoffrequenciesbelow1MHz,theeffective
permittivityexhibitsastrongdependencewiththecationexchangecapacityandthespecific
surfacearea.Athighfrequency,abovethecriticalfrequencyf ,theeffectivepermittivity
d
reachesahigh-frequencyasymptoticlimitthatiscontrolledbythetwoArchie’sexponents
mandnlikethelow-frequencyelectricalconductivity.Theunifiedmodeliscomparedwith
variousdatasetsfromtheliteratureandisabletoexplainfairlywellabroadnumberof
observationswithaverysmallnumberoftexturalandelectrochemicalparameters.Itcould
bethereforeusedtointerpretinducedpolarization,induction-basedelectromagnetic
methods,andgroundpenetratingradardatatocharacterizethevadosezone.
Citation: Revil, A. (2013), Effective conductivity and permittivity of unsaturated porous materials in the frequency range 1 mHz–
1GHz, Water Resour.Res.,49, doi:10.1029/2012WR012700.
1. Introduction just to cite few examples [e.g., Linde et al., 2006]. That
said,thereisnotasinglemodelabletoexplainthecomplex
[2] Thereareanumberofelectricalandelectromagnetic
conductivityofunsaturatedmaterialsaswellastheireffec-
(EM) geophysical methods providing nonintrusively infor-
tive permittivity and their frequency dependences in the
mationrelativetothetexture,watercontent,andinterfacial
broad range of frequencies of interest in geophysics
electrochemistry in the shallow subsurface in an extremely
(1mHz–1GHz).
broad range of frequencies (1 mHz–1 GHz). The relevant
arsenal of nonintrusive techniques include self-potential, [3] In the low-frequency domain (<10 kHz), new mod-
els have been developed recently to describe the complex
direct current (DC) resistivity, frequency and time-domain
conductivity of saturated and unsaturated sands and clayey
inducedpolarization,frequencyandtime-domaininduction
materials [Revil and Florsch, 2010; Revil, 2012; Revil
EMmethods,groundpenetratingradar(GPR),andtheseis-
et al., 2012a, 2012b]. There is a vast literature regarding
moelectric/electroseismic methods. These nonintrusive
this topic in interfacial electrochemistry and colloidal
methods have many applications in hydrogeophysics to
chemistry [e.g., Schwarz 1962; Dukhin and Shilov, 2002;
assessgroundwaterresourcesandtomonitorgroundwater
Grosse,2002]aswellasingeophysics[TarasovandTitov,
quality or for vadose zone hydrogeology and agriculture
2007]. At very high frequencies, there are also predictive
models describing the high-frequency permittivity of po-
1DepartmentofGeophysics,ColoradoSchoolofMines,GreenCenter, rous rocks. These models are empirically based like the
Golden,Colorado,USA. Topp model forsoils [Topp andReynolds,1998], based on
2ISTerre,CNRS,UMRCNRS5275,Universit(cid:3)edeSavoie,LeBourget upscaling methods like the volume-averaging method
duLac,France.
[Pride,1994],orbasedontheeffectivedifferentialmedium
Correspondingauthor: A.Revil,DepartmentofGeophysics,Colorado approach or mixing laws (see discussion in Jones and Or
School of Mines, Green Center, 1500 Illinoisstreet, Golden, CO 80401, [2002], Cosenza et al. [2003], Robinson and Friedman
USA.([email protected])
[2003],andMiyamotoetal.[2005]).Notethatwiththevol-
ume-averaging approach, the different phases are consid-
©2012.AmericanGeophysicalUnion.AllRightsReserved.
0043-1397/13/2012WR012700 ered symmetrically while with the differential effective
306
REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
medium approach, one phase has to be chosen as the host
(for instance, grains immersed in a background fluid or
fluidinclusioninamatrix).
[4] The situation is much more confused in the interme-
diate range of frequencies (10 kHz–10 MHz). Typically, a
number of authors in geophysics have extended the high-
frequency behavior to lower frequencies by adding a con-
tribution related to the Maxwell-Wagner polarization
mechanism [Chen and Or, 2006]. Other approaches have
relied on using the so-called standard model of colloidal
chemistry (developed initially by Dukhin and Shilov and
based on the polarization of the diffuse layer, see Grosse
[2002])andapplyingthismodeltolow-frequencydatasets
[Garrouch and Sharma, 1994]. To fit the data, a stretch of
thetexturalparametersisusuallyrequiredandtheobtained
set of interfacial properties are not compatible with the tri-
ple layer complexation models commonly used in surface
electrochemistryofminerals.
[5] In this paper, following my previous works [Revil
and Florsch, 2010; Revil, 2012; Revil et al., 2012a,
2012b], I claim that the diffuse layer is not the main con-
tributortolow-frequencypolarizationofclayeymaterials.I Figure 1. The REV. (a) Sketch of the REV comprised
also claim that the Maxwell-Wagner polarization is not ei- between two reservoirs. (b) Sketch of the electrical double
ther the dominating mechanism of polarization at interme- layeratthesolidwaterinterface.Thedoublelayerincludes
diatefrequencies.Anextremelysimplemodelisdeveloped the Stern layer and the diffuse layer. The charges of these
based on the low-frequency polarization of the Stern layer two layers compensate the charge on the mineral surface
(the inner layer of the electrical double layer). This model (o plane). The d plane corresponds to the separation plane
is proposed for saturated and unsaturated clayey materials between the Stern and the diffuse layers. (c) Sketch of the
(water being the wetting phase). It explains for the first solid-waterandwater-airinterfaces.
time an important amount of literature data in the fre-
quencyrange1mHzto1GHzinaverysimpleway. @D
r(cid:2)H¼Jþ ; (1)
@t
2. Theory where J ¼ (cid:5)(cid:3) E denotes the conduction current density (in
A m(cid:4)2), (cid:5)(cid:3) ¼(cid:5)0þi(cid:5)00 represents the complex conductiv-
[6] I consider an unsaturated porous material as shown
ity, J ¼@D=@t corresponds to the displacement current
inFigure1awithanelectricaldoublelayercoatingthesur- d
density(Am(cid:4)2),D¼"Erepresentsthedielectricdisplace-
face of the grains (Figure 1b). In the classical literature in
ment (in C m(cid:4)2), H denotes the auxiliary magnetic field
geophysics,itiscustomarytoseeapolarizationsketchlike
(A m(cid:4)1), and " denotes the permittivity or dielectric con-
theoneshowninFigure2a[e.g.,Hasted,1961].Inthefre-
quency range 1 mHz–1 GHz, the apparent dielectric con- stant (in F m(cid:4)1) of the material. I consider anpffihffiffiaffiffirffiffimonic
external electrical field E¼E expð(cid:4)i!tÞ (i¼ (cid:4)1 is the
stant of a porous material is explained by two dissipation 0
pure imaginary number, f is the frequency in Hz, ! ¼ 2(cid:6)f
mechanisms only: the Maxwell Wagner polarization
the angular frequency (in rad s(cid:4)1), and E denotes a con-
(called the (cid:2) polarization in electrochemistry, see Grosse 0
stantamplitudeofthealternatingelectricalfield).Thetotal
[2002])andthepolarizationofthewatermoleculespastthe
current density J is the sum of J and J . This yields J ¼
gigahertz frequency (called the (cid:3) polarization). That said, t d t
ð(cid:5)(cid:3)(cid:4)i!"ÞE [Vinegar and Waxman, 1984]. This total cur-
we know that in induced polarization, the polarization of
rentdensitycanbewrittenas,
the electrical double layer plays an important role below
10kHz.Thisnondielectricdispersionmechanismisassoci- J ¼(cid:5)(cid:3) E; (2)
ated with the electromigration and accumulation/depletion t eff
of charge carriers at discontinuities in the migration path- where(cid:5)(cid:3) ¼(cid:5) (cid:4)i!" istheeffectivecomplexconduc-
eff eff eff
waysofthechargecarriers.Thismechanismiscalledthe(cid:4) tivity and (cid:5) and " are real positive frequency-depend-
eff eff
polarization in electrochemistry, and its role is such that it entscalarsdefinedby
may be the dominating mechanism of polarization for a
very broad frequency range as discussed in this paper (see (cid:5)eff ¼(cid:5)0; (3)
Figure2b).
" ¼"0(cid:4)(cid:5)00=!: (4)
eff
2.1. EffectiveConductivityandPermittivity:
Definitions Itisalsopossibletodefineaneffectivequadratureconduc-
tivity(inSm(cid:4)1)thatisdirectlyrelatedtotheeffectiveper-
[7] I first define the concepts of effective conductivity, mittivityas,
effective permittivity, and effective quadrature conductiv-
ity. Ampe`re law is given by [e.g., Vinegar and Waxman,
1984], (cid:5)0e0ff ¼!"eff ¼(cid:4)(cid:5)00þ!"0: (5)
307
REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
where e denotes the local electrical field (in V m(cid:4)1), N
w
denotethenumberofionicspecies,q thechargeofspecies
i
i,andC theirconcentration.Themobilitiesb (inNsm(cid:4)1)
i i
enteringequation(6)arerelatedtotheclassicalionicmobi-
lities (cid:2) (in m2 s(cid:4)1 V(cid:4)1) by b ¼(cid:2)=jqj and to the ionic
i i i i
self-diffusion coefficients D by D ¼k Tb, where k
i i b i b
denotes the Boltzmann constant (1.3807 (cid:2) 10(cid:4)23 J K(cid:4)1),
and T the absolute temperature (in K). There is also a cur-
rentdensityassociatedwiththegrains,j duetotheelectri-
s
caldoublelayercoatingthesurfaceoftheinsulatinggrains.
TheboundaryconditionsattheinterfaceS separating the
sw
solidphaseandtheporespaceare
(cid:3) (cid:4)
n (cid:7) e (cid:4)e ¼0; (7)
s s w
n (cid:7)ðj (cid:4)j Þ¼0; (8)
s s w
where n denotes the normal unit vector to S directed
s sw
fromtheporetothesolidphase(Figure1c).Theporespace
is filled with two immiscible fluids, the water and air
phases (water is assumed to be the wetting phase). The
boundaryconditionsattheinterfaceS separatingthenon-
aw
wettingphase(air)andwaterare
(cid:3) (cid:4)
n (cid:7) e (cid:4)e ¼0; (9)
a a w
(cid:3) (cid:4)
n (cid:7) j (cid:4)j ¼0; (10)
a a w
where n denotes the normal unit vector to S directed
Figure 2. Dielectric dispersion for polarization in a aw
from the water to the nonwetting phase (Figure 1c). In the
charged porous media. (a) Classical representation from
following,Iwillconsiderthatthesurfacechargedensityat
Poleyetal.[1978]andfoundinanumberofpapersingeo-
the interface between water and the nonwetting phase can
physics. (b) Representation of the three types of polariza-
be neglected. The reason to neglect this contribution is
tion discussed in the text. The (cid:4) polarization corresponds
because there is still a gap of knowledge in describing the
to the polarization of the electrical double layer, the (cid:2)
conductivity and polarization of this interface, which is,
polarization corresponds to the Maxwell-Wagner polariza-
however, described by an electrical double layer [Leroy
tion,andthe(cid:3) polarizationconcernsthepolarizationofthe
et al., 2012] and therefore likely characterized by surface
watermolecules.
conductionandpolarization.
Note that in the conventions used above, (cid:5)00 is positively [9] The microscopic equations given above are now
eff averagedatthescaleofanREV.TheREVcorrespondsto
defined while (cid:5)00 is negatively defined. Two phase angles
the rock volume located in between two large-parallel
canbedefined.Theformerisdeterminedfromthecomplex
circular disks of area A separated by the distance L. This
conductivity tan(cid:7)¼(cid:5)00=(cid:5)0 and the second from the effec-
is typically the case of a jacketed cylindrical sample in
tive parameters tan(cid:8)¼(cid:5)00 =(cid:5) ¼!" =(cid:5) (note that
eff eff eff eff the laboratory. We assume that there is a macroscopic
(cid:7)(cid:5)0while(cid:8) (cid:6)0).
electrical potential difference (cid:4) . When this potential
2.2. ConductivityandPermittivity: difference is divided by L, one obtains the equivalent
AVolume-AveragingApproach macroscopic field perpendicular to the end faces of the
[8] Pride [1994] developed a volume-averaging REV. We note z the unit vector normal to the end faces
approachusedtoupscaletheNernst-Planckequationtothe (Figure 1a). For example, for the electrical field at the
scale of a representative elementary volume (REV) of a pore or grain scale, we have r(cid:2)e(cid:9) ¼0, so the electrical
porous rock. He was however concerned only by the satu- fields e(cid:9) can be derived from scalar electrical potentials
ratedcaseandmotivatedbythemodelingoftheseismoelec- e(cid:9) ¼(cid:4)r (cid:9) where the index (cid:9) defined the phase (w for
tric effect. We use this approach to upscale the Nernst- water, s for solid, a for air). The macroscopic electrical
Planckequationtoaporousmaterialsaturatedbytwoimmis- fieldiswrittenas
cible fluid phases (water and air). The local current density
j (in A m(cid:4)2) occurring in the water phase is determined (cid:4)
w z(cid:7)E¼(cid:4) ; (11)
from the Nernst-Planck equation, which is written, in ab- L
senceofmacroscopicconcentrationgradients,as
where(cid:4) ¼ ðLÞ(cid:4) ð0Þdenotesthedifferenceofelec-
(cid:9) (cid:9)
trical potential between the two reservoirs (Figure 1a). In
N
j ¼X(cid:3)q2bC(cid:4)e : (6) the pore water phase, the fundamental Laplace problem is
w i i i w
i¼1 definedby
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REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
r2G ¼0; inV ; (12) wheref denotesthevolumetricfractionofphase(cid:9)(f ¼(cid:7),
w w (cid:9) f
(cid:7)denotes theconnected porosity,and f ¼1(cid:4)(cid:7)is the so-
s
n (cid:7)rG ¼0;onS ; (13)
s w sw lidity, f ¼s (cid:7)¼(cid:10) is the volume fraction of water, also
w w
na(cid:7)rGw ¼0;onSaw; (14) called the water content in hydrogeology, and fa ¼
ð1(cid:4)s Þ(cid:7) denotes the volume fraction of the nonwetting
w
(cid:5)
L;onz¼L phase).Therefore,wehave,
G ¼ (15)
w 0; onz¼0;
A¼ð1(cid:4)(cid:7)Þa þs (cid:7)a þð1(cid:4)s Þ(cid:7)a : (26)
s w w w a
where S and S denote the solid-water and air-water
sw aw
interfaces, respectively (Figure 1c), and Vw denotes the [11] We first define the electrical conductivity of the
water phase volume. The electrical potential in the water porewater,whichisgivenby,
phase can be written as ¼G (cid:4) =L and the local elec-
w w
tricalfieldbye ¼(cid:4)rG (cid:4) =L.Asimilarboundary-value N N
w w X X
problem of the normalized effective potential Ga can be (cid:5)w ¼ q2ibiCi¼ jqij(cid:2)iCi: (27)
definedinthenonwettingphase(V denotestheairphase), i¼1 i¼1
a
[12] From equation (26), the average conduction current
r2G ¼0; inV ; (16) densityisgivenby
a a
na(cid:7)rGa¼0;in Saw; (17) 1 Z 1 Z
J¼ (cid:5) edV þ (cid:5) e dV; (28)
(cid:5)L; onz¼L V s s V w w
Ga¼ 0; onz¼0: (18) Vs Vw
J¼ð1(cid:4)(cid:7)Þ(cid:5) e þs (cid:7)(cid:5) e : (29)
s s w w w
[10] The volume average of a quantity a(cid:9) (a(cid:9) is a scalar,
avector,orpossiblyatensor)is [13] Thetortuosityofthewaterphase(cid:4)wisdefinedby
1Z 1
ha(cid:9)i¼V a(cid:9)dV; (19) ew ¼(cid:4) E; (30)
w
V(cid:9)
1 1 Z 1 Z
(cid:8)1þ z(cid:7)nG dSþ z(cid:7)n G dS; (31)
whereV(cid:9) isthevolumeofthe(cid:9)phasewithintheREV.Slat- (cid:4)w Vw s w Vw a w
tery’sclassicaltheoremstates[Slattery,1967], Ssw Saw
where E ¼(cid:4)z(cid:4) =H denotes the macroscopic electrical
1Z
hra i¼rha iþ n a dS; (20) field. From equations (25) and (26), the macroscopic elec-
(cid:9) (cid:9) V (cid:9)(cid:5) (cid:9)
tricalfieldisalsogivenby
S(cid:9)(cid:5)
(no summation on the indices) where n(cid:9)(cid:5) represents the E ¼hesiþhewiþheai; (32)
unit outwardly directed normal vector for the (cid:9) phase and
E¼ð1(cid:4)(cid:7)Þe þ(cid:7)s e þ(cid:7)ð1(cid:4)s Þe : (33)
S represents the interfacial area contained within the s w w w a
(cid:9)(cid:5)
averagingvolume.Applyingthistothewater,solid,andair
The phase average of the electrical field e can be also
phases and using the unit vectors defined in Figure 1c, I a
related tothemacroscopicelectrical fieldvia thetortuosity
obtain,
ofthenonwettingphase(cid:4) ,
a
1Z 1Z
hra i¼rha iþ na dSþ n a dS; (21) 1
w w V s w V a w ea¼(cid:4) E; (34)
Ss Sw a
1Z 1 1 Z
hrasi¼rhasi(cid:4)V nsasdS; (22) (cid:4)a (cid:8)1(cid:4)Va z(cid:7)naGadS: (35)
Ss Ssa
1Z I need now to find a relationship between the tortuosity of
hra i¼rha i(cid:4) n a dS; (23)
a a V a a the pore space (cid:4) (related to the definition of the electrical
Sa formation factor F at saturation by F ¼(cid:4)=(cid:7)) and the tor-
tuosities of the wetting and nonwetting phases. If we note
where dS is an infinitesimal surface volume element. The
e the local electrical field of the fluid phase (comprising
volumetric phase average a and the total volumetric aver- f
(cid:9) the wetting plus nonwetting fluid phases), the phase aver-
ageA arererelatedtoeachotherby,
age of e is related to the phase average of the electrical
f
field in the wetting phase e and to the phase average of
w
a(cid:9) ¼ha(cid:9)i=f(cid:9); (24) theelectricalfieldinthenonwettingphasee by
a
X X
A¼ ha i¼ f a ; (25)
(cid:9) (cid:9) (cid:9)
(cid:9) (cid:9) ef ¼swewþð1(cid:4)swÞea: (36)
309
REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
Theelectricfielde isrelatedtothemacroscopicfieldEby, expression of the electrical conductivity is introduced by
f
usingthefollowingchangeofvariables,
1
e ¼ E: (37)
f (cid:4) (cid:7)sw () 1sn: (47)
(cid:4) F w
w
Combining equations (30), (34), (A36), and (37), I obtain
after few algebraic manipulations the following relation- From equation (39), Icanalsouse thefollowingchangeof
ship between the tortuosity of the pore space, the tortuosiy variables,
ofthewaterphase,andthetortuosityoftheairphase,
1 ¼ sw þ1(cid:4)sw: (38) (cid:7)ð1(cid:4)(cid:4)aswÞ () F1(cid:3)1(cid:4)snw(cid:4): (48)
(cid:4) (cid:4) (cid:4)
w a
Usingequations(47)and(48)intoequation(45),Iobtained
Therefore, the tortuosity of the pore space is equal to the the following expression for the electrical conductivity of
harmonicaverageofthetortuositiesofeachphaseweighted theporousmedium,
by their relative saturation. If we multiply equation(38) by
theporosity,weobtainarelationshipbetweenthetortuosity
of the air phase, the formation factor, and the tortuosity of h(cid:5)i¼(cid:5)sþF1snwð(cid:5)w(cid:4)(cid:5)sÞ(cid:4)F1(cid:3)1(cid:4)snw(cid:4)(cid:5)s; (49)
thewaterphase,
1 1
h(cid:5)i¼ sn(cid:5) þ(cid:5) (cid:4) (cid:5) ; (50)
(cid:7)ð1(cid:4)s Þ 1 (cid:7)s F w w s F s
w ¼ (cid:4) w: (39)
(cid:4)a F (cid:4)w h(cid:5)i¼ 1(cid:6)sn(cid:5) þðF(cid:4)1Þ(cid:5) (cid:7): (51)
F w w s
Theelectricalformationfactorcaninturnberelatedtothe
porositybyapower-lawfunction[Archie,1942], In the following, I remove the bar on the pore water and
surfaceconductivitytermsanddroptheaveragesymbolfor
F¼(cid:7)(cid:4)m; (40) the electrical conductivity to keep the notations as simple
as possible. The same chain of algebraic manipulations
calledthefirstArchie’slaw,wheremiscalledthecementa- could be applied also to complex conductivity and there-
tion exponent (m (cid:6) 1 is a strict bound and (1 (cid:5) m (cid:5) 3 for fore I can write the following expression of the complex
sandstones and sands according to experimental data, see conductivityoftheclayeymaterial,(cid:5)(cid:3),asafunctionofthe
Revil et al. [1998, Figure 5]). The conduction current den- conductivityoftheporewater(cid:5)w andthecomplexconduc-
sitycanbewrittenas tivity of the solid phase (cid:5)(cid:3) ¼(cid:5)0 þi(cid:5)00 (complex because
S S S
of the polarization the electrical double-layer coating the
(cid:6) (cid:7) grains),
J¼(cid:5) E(cid:4)(cid:7)s e (cid:4)(cid:7)ð1(cid:4)s Þe þs (cid:7)(cid:5) e ; (41)
s w w w a w w w
J¼(cid:5)sEþ(cid:7)swð(cid:5)w(cid:4)(cid:5)sÞew(cid:4)ð1(cid:4)swÞ(cid:7)(cid:5)sea; (42) (cid:5)(cid:3)¼ 1(cid:6)sn(cid:5) þðF(cid:4)1Þ(cid:5)(cid:3)(cid:7): (52)
F w w S
(cid:8) (cid:7)s ð1(cid:4)s Þ(cid:7) (cid:9)
J¼ (cid:5) þ wð(cid:5) (cid:4)(cid:5) Þ(cid:4) w (cid:5) E: (43)
s (cid:4)w w s (cid:4)a s [14] Notethattheporefluidconductionandsurfacecon-
ductionaddinparallelinthismodel,andthereforetheelec-
Theeffectiveelectrical conductivityofthematerialh(cid:5)i(in trical conductivity of the porous material is a linear
Sm(cid:4)1)isdefinedbythemacroscopicOhm’slaw, function of the conductivity of the pore fluid. This is
expectedasthevolume-averagingapproachtreatsthesolid
and fluidphases symmetrically,and they are bothassumed
J ¼h(cid:5)iE: (44)
c
tobecontinuousthroughtheREV.AsdiscussedinAppen-
dixA,thesituationisdifferentwhenadifferentialeffective
Combiningequations(43)and(44)yields
mediumapproachisused.Whenwearedealingwithgran-
ular media, the solid phase is discontinuous and grains are
h(cid:5)i¼(cid:5) þ(cid:7)swð(cid:5) (cid:4)(cid:5) Þ(cid:4)ð1(cid:4)swÞ(cid:7)(cid:5) : (45) immersed in the background pore water. In this case, the
s (cid:4)w w s (cid:4)a s differential effective medium theory yields a nonlinear
relationship between the electrical conductivity of the po-
Note that in the case where there is no surface conductiv- rousmaterialandtheconductivityoftheporefluid(seefor
ity, we can connect equation (45) to the second Archie’s instance, Revil et al. [1998] and Revil [1999]). Sands and
law by sandstones are somewhere in between these two types of
modelsasthediffuselayeriscontinuous(overlappingfrom
grain-to-grain) while the Stern layer is discontinuous. In
(cid:7)s 1
limh(cid:5)i¼ w(cid:5) ¼ sn(cid:5) ; (46) addition, it follows that the diffuse layer does not polarize
(cid:5)s!0 (cid:4)w w F w w whiletheSternlayerpolarizesandthattheSternlayercan-
notcontributetotheDCelectricalconductivity.
whereniscalledthesecondArchie’sexponentinthelitera- [15] Using the displacement current density rather than
ture.Inthefollowing,thissecondArchie’sexponentinthe the electromigration current density, an equation similar to
310
REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
equation (52) can be obtained for the high-frequency dielec- where equation (55) has been used. Therefore, the real (in
tricconstantusingthesamechainofalgebraicmanipulations, phase) low-frequency component of the solid phase con-
ductivityisgivenby,
"0¼ 1(cid:6)sn" þ(cid:3)1(cid:4)sn(cid:4)" þðF(cid:4)1Þ" (cid:7): (53)
F w w w a S (cid:5)0 ¼ð1=F(cid:7)Þsn(cid:4)1(cid:2) (cid:11) ð1(cid:4)f ÞCEC; (59)
S w ðþÞ S M
[16] Now coming back to equation (52), I need to
where I used the following change of variables
express the low-frequency conductivity of the solid phase
ð1=(cid:4) Þ()ð1=F(cid:7)Þsn(cid:4)1 resulting from equation (47). I
correspondingtoaninsulatingsolidphasecoatedbyacon- w w
replacebelowtheexponent(n(cid:4)1)intheexpressionofthe
ductiveandpolarizableelectricaldoublelayerasafunction
complex surface conductivity by the exponent p because
of the surface charge density, the mobility of the counter-
thetruetortuositythatshouldbeusedistheoneassociated
ions, and the tortuosity of the electromigration pathways
with pathways along the grains, not strictly speaking, the
along the surface of the grains. I assume that at low fre-
tortosity of the water phase adjacent to the grains (I will
quency, only the conduction of the electrical diffuse layer
showhoweverinsection7thattheapproximationp(cid:9)n(cid:4)
participates to the surface conductivity. Indeed, the grains
1 is a good one). This approach is consistent with the one
are in contact to each other (a very important distinction
derived by Vinegar and Waxman [1984] and Slater and
with colloidal suspensions) and the diffuse layer is always
Glaser [2003]. The low-frequency (in phase) conductivity
above a percolation threshold. This assumption has been
isgivenby
checkedexperimentallybyVaudeletetal.[2011a,2011b].
[17] The electrical conductivity of the grains is given by
the mobility of the counterions (corrected by the tortuosity (cid:5) ¼ 1(cid:8)sn(cid:5) þsp(cid:10)F(cid:4)1(cid:11)(cid:2) (cid:11) ð1(cid:4)f ÞCEC(cid:9): (60)
around the grains) times the concentration of the charge 0 F w w w F(cid:7) ðþÞ S M
carrierstimestheircharge.Thisyields,
[18] Asimilarapproachcanbefollowedforthehigh-fre-
quency electrical conductivity, for which now both the
(cid:2) S
(cid:5)0S ¼ (cid:4)ðþÞQdV ; (54) Stern and diffuse layers contribute to the overall surface
w S conductivity [Revil, 2012; Revil et al., 2012a]. Following
where Sdenotesthe surface areaof the grains (inm2),Q is the same approach as used to derive equation (60), this
the surface charge density of the diffuse layer (in C m(cid:4)d2), yields
andV isthevolumeofthegrains(inm3).Notethatthemo-
bilitySof the charge carriers is corrected for the tortuosity of (cid:5) ¼ 1(cid:5)sn(cid:5) þsp(cid:10)F(cid:4)1(cid:11)(cid:11) h(cid:2) ð1(cid:4)f Þþ(cid:2)S f iCEC(cid:12):
the electromigration of the counterions along the surface of 1 F w w w F(cid:7) S ðþÞ M ðþÞ M
thegrainsinthewaterphase(thecounterionsarenotallowed (61)
topassthroughthegrainsthatareinsulating).Thefractionof
the volumetric charge density associated with the diffuse where (cid:2)S denotes the mobility of the counterions in the
ðþÞ
layeriswrittenas Stern layer (as expected (cid:2)S (cid:10) (cid:2) , see Revil [2012] for
ðþÞ ðþÞ
claysandReviletal.[2012a]arguedfor(cid:2)S ¼(cid:2) insilica).
ðþÞ ðþÞ
(cid:10)1(cid:4)(cid:7)(cid:11) [19] An important parameter in the description of fre-
ð1(cid:4)fMÞQV ¼(cid:11)S s (cid:7) ð1(cid:4)fMÞCEC; (55) quency-domain inducedpolarization,but moreimportantly
w
in the description of time domain induced polarization, is
where f (dimensionless) denotes the fraction of counter- the normalized chargeability. In a time-domain induced
M
ions in the Stern layer (the counterions compensating the polarization experiment, the chargeability M characterizes
negative charge of the mineral surface are distributed howthedecayofthesecondary voltageswhentheprimary
between the diffuse layer and the Stern later, see Revil current is shut down. This chargeability can be normalized
[2012] and references therein). Consequently, ð1(cid:4)fMÞ to obtain the normalized chargeability Mn ¼M(cid:5)1 ¼(cid:5)1
denotesthefractionofcounterionsinthediffuselayer.The (cid:4)(cid:5)0, and therefore from equations (60) and (61), the nor-
volumetric charge density Q (in C m(cid:4)3) denotes the total malizedchargeabilityisgivenby
V
charge density of the double-layer per unit volume of the
porewater(includingtheSternanddiffuselayerstakento- (cid:10)F(cid:4)1(cid:11)(cid:10) 1 (cid:11)
M ¼sp (cid:11) (cid:2)S f CEC: (62)
gether). The relationship between QV and its value at satu- n w F F(cid:7) S ðþÞ M
ration(QS ats ¼1)isQ ¼QS=s .Wealsohave
V w V V w
We can express also the conductivity terms in term of the
S
ð1(cid:4)fMÞQV ¼QdV : (56) volumetricchargedensityQV,whichyields,
p
1
Therefore, the surface charge density can be expressed in (cid:5)0 ¼Fsnw(cid:5)wþ(cid:5)0s; (63)
termsoftheCECby
1
(cid:5) ¼ sn(cid:5) þ(cid:5)1: (64)
Q S¼ð1(cid:4)f ÞQ V ; (57) 1 F w w s
d M V w
S
Q ¼(cid:11) ð1(cid:4)f ÞCEC; (58) in which we have entered the term ðF(cid:4)1Þ=F (cid:9)1 in
dVS S M the expressions of the surface conductivities. In these
311
REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
in the literature. The expression for Að(cid:7);mÞ contrasts with
the highsalinity conductivity model developed recently by
Revil[2012],whichyields,
2(cid:10) (cid:7) (cid:11)
Að(cid:7);mÞ¼mðF(cid:4)1Þ : (68)
3 1(cid:4)(cid:7)
For spherical grains, F(cid:7)(cid:9)3=2(more precisely (cid:6)/2 for the
ionstomovearoundaperfectsphericalgrain)andtherefore
thepresentlyderivedexpressionofAð(cid:7);mÞis
2 (cid:10) (cid:7) (cid:11)
Að(cid:7);mÞ(cid:9) ðF(cid:4)1Þ : (69)
3 1(cid:4)(cid:7)
Thedifferencebetweenequations(67)and(68)istherefore
really in the factor m in the expression derived from the
effective medium theory. Note for spherical grain
(F(cid:7)(cid:9)3=2) and at high porosity, the formation factor is
givenby
3(cid:10)1(cid:4)(cid:7)(cid:11)
F ¼(cid:7)(cid:4)3=2(cid:9)1þ : (70)
2 (cid:7)
Equations(69)and(70)yields
Að(cid:7);mÞ(cid:9)1: (71)
With this approximation and using p ¼ n (cid:4) 1, the high-
frequencyandlow-frequencyconductivitiesaregivenexactly
bytheWaxmanandSmits[1968]equation(seealsoWaxman
andThomas[1974]forthequadratureconductivity):
1 (cid:8) QS(cid:9)
Figure 3. Effective quadrature conductivity versus fre- (cid:5) ¼ sn (cid:5) þ(cid:2) ð1(cid:4)f Þ V ; (72)
0 F w w ðþÞ M s
quency. (a) Type A. Berea sandstone (data from Lesmes w
and Frye [2001], pH 8, 10(cid:4)1 M NaCl solution). The data 1 (cid:5) h iQS(cid:12)
arecharacterizedbythreecomponents: aminimumcritical (cid:5)1 ¼Fsnw (cid:5)wþ (cid:2)ðþÞð1(cid:4)fMÞþ(cid:2)SðþÞfM sV : (73)
w
frequencyfromwhichthemagnitudeofthequadraturecon-
ductivity is going down when the frequency decreases, a
The present analysis therefore provides a theoretical foun-
plateau (over at least four decades in frequency), and a
dation to the Waxman and Smits [1968] model in which
high-frequencydomaininwhichthereisanincrease ofthe
magnitude of the apparent quadrature conductivity. Filled
circles: measurements with four electrodes, filled squares,
Table1. PropertiesoftheBereaSandstone(ClayType:Kaolinite
measurements with two electrodes. (b) Type B. shaly con-
glomerate(thisstudy,pH7,10(cid:4)1MNaClsolution). andIllite)
Property Value
Archieexponentm(–) 1.78,a1.69,b1.8560.05e
Saturationexponentn(–) 1.83,a1.9860.23,e2.11–2.17i
relationships, the low-frequency and high-frequency sur- SpecificsurfaceareaS (m2g(cid:4)1) 0.74c–0.93d
sp
faceconductivitytermsareexpressedas Porosity(cid:7)(–) 0.2060.03c
FormationfactorF(–) 18to26b
Graindensity(cid:11) (kgm(cid:4)3) 2660i
(cid:5)0¼sp Að(cid:7);mÞ(cid:2) ð1(cid:4)f ÞQS; (65) CEC(Ckg(cid:4)3) S 256Ckg(cid:4)1h
s w F ðþÞ M V ExcesschargedensityQ (Cm(cid:4)3) 2.7(cid:2)106j–5.8(cid:2)106i
v
Að(cid:7);mÞh i aJun-ZhiandLile[1990].
(cid:5)1s ¼spw F (cid:2)ðþÞð1(cid:4)fÞþ(cid:2)SðþÞfM QSV; (66) bFrom the raw data of Lesmes and Frye [2001], corrected for surface
conductivity.
cLesmesandFrye[2001](onesample).
andtheparameterAð(cid:7);mÞisgivenby dZhanetal.[2010].
eAttia[2005](eightsamples).
1 (cid:10) (cid:7) (cid:11) fAggouretal.[1994].
Að(cid:7);mÞ¼ ðF(cid:4)1Þ : (67) gZhuandToksöz[2012]
F(cid:7) 1(cid:4)(cid:7)
hUsingCEC¼Q S withQ ¼0.32Cm(cid:4)2andS ¼0.8m2g(cid:4)1.
s sp s sp
iVinegar and Waxman [1984, Tables 7 and 10] (porosity 0.19, four
Note that the ratio of the surface conductivity to the pore samples).
water conductivity is often called the Dukhin number (Du) jUsingQV¼(cid:11)S[(1(cid:4)(cid:7))/(cid:7)]CEC.
312
REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
Table2. PropertiesoftheBereaSandstone (cid:2)2
(cid:12) ¼ s2; (75)
p 2DS w
Permeability PoreThroat ðþÞ
Porosity¼(cid:7)(–) k (mD) Sizea(cid:2)(inmm)(1)
0
!
0.206b 498b 7.9 (cid:12) ¼ 4kF s2; (76)
0.205b 496b 7.9 p DS w
0.207b 477b 7.7 ðþÞ
0.180c 228c 6.0
0.230d 450d 6.8 whereDS denotesthediffusioncoefficientofthecounterions
ðþÞ
aFrom(cid:2)¼pffi8ffiffiFffiffiffikffiffi0ffiffi¼pffi8ffiffi(cid:7)ffiffiffi(cid:4)ffiffimffiffiffikffiffi0ffiffiwithm¼1.74. (in m2 s(cid:4)1), k denotes the permeability (in m2), and (cid:2) is a
bFromAggouretal.[1994]. characteristicporethroatsizethatcanbederivedfromtheper-
cFromLesmesandFrye[2001]. meability and the formation factor (see for instance, Table 2
dFromZhuandToksöz[2012]. for the Berea sandstone and a definition of this parameter in
AppendixB).Thediffusioncoefficientisrelatedtothemobil-
their mobility B is related to the mobility (cid:2) by B¼ ity of the counterions in the Stern layer, (cid:2)S , by the Nernst-
ðþÞ ðþÞ
(cid:2)ðþÞð1(cid:4)fMÞ. Einstein relationship DS ¼k T(cid:2)S =jq j, where k is the
[20] In the high-porosity limit discussed previously, the ðþÞ b ðþÞ ðþÞ b
normalizedchargeabilityisgivenas
(cid:10)2(cid:11)
M ¼sn(cid:4)1 (cid:11) (cid:2)S f CEC; (74)
n w 3 S ðþÞ M
whichisexactlyequation(43)ofRevil[2012]forthequad-
rature conductivity obtained through the differential effec-
tive medium approach. Up to this point, I have not
discussed the frequency dependence of the electrical con-
ductivity.Thiswillbedoneinthenextsection.Also,rather
than writing the surface conductivity and the normalized
chargeabilityasafunctionoftheCEC,wecanusetherela-
tionship developed by Revil [2012] to express the parame-
ters as a function of the specific surface area. The
relationship between the CEC (in C kg(cid:4)1) and the specific
surface area S (in m2 kg(cid:4)1) is CEC ¼Q S , where Q
Sp S Sp S
denotes the mean charge density for clay minerals (typi-
cally, two elementary charges per nm2, i.e., Q ¼ 0.32 C
S
m(cid:4)2, see Revil [2012]). Note that Q and Q are related to
S d
eachotherbyð1(cid:4)f ÞQ ¼Q .
M S d
2.3. FrequencyDependenceoftheEffective
ConductivityandPermittivity
[21] Figure3showstheeffectivequadratureconductivity
spectrum for a Berea sandstone (data from Lesmes and
Frye[2001]),whichisaclayeysandstonewithaverysmall
amountofclayminerals.ThepropertiesoftheBereasand-
stone are summarized in Tables 1 and 2. I call Type A,
clayeymaterialsshowingthetypeofquadratureconductivity
spectrashowninFigure3a.Thistypeofquadratureconduc-
tivitydistributionexhibitsthreedistinctregions: (i)Domain
Iisdefinedbyfrequenciessuchasf <f ((cid:9)0.03Hzforthe
p
BereasandstonedatashowninFigure3a).Inthisregion,the
effective quadrature conductivity decreases with the fre-
quency(with(cid:5)00ð!¼0Þ¼0).DomainIIisdefinedbyf (cid:5)
p
f (cid:5)f ((cid:9)0.3MHzinFigure3a).Inthisdomain,thequadra- Figure 4. Relationship between the normalized charge-
d
tureconductivityreachesaplateau.DomainIIIisdefinedby ability and the quadrature conductivity. (a) Eight bentonite
f >f .Inthisdomain,theeffectivequadratureconductivity sand mixtures (R ¼ 0.97). Data from Slater and Lesmes
d
increaseswiththefrequencyandiscontrolledbythepermit- [2002].(b)Onesaprolitecoresample(S16,seeReviletal.
tivityofthematerial.Ifirstprovidesomeexpressionsforthe [2012b]). The normalized chargeability is determined from
two critical frequencies and then I will provide expressions thehighandlowasymptoticvaluesofthe(in-phase)electri-
foreachdomain. calconductivity.Thediscrepancyatlowquadratureconduc-
[22] In Appendix B, I show that a characteristic relaxa- tivity may be dueto the higher uncertainty in extrapolating
tiontimeofaclayeymaterialcanbedeterminedbythefol- thehigh-frequencyandlow-frequencylimitsfortheelectri-
lowingequations calconductivity.
313
REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
Table3. PropertiesoftheSandstonesFromtheVinegarandWaxman[1984]DatabaseAnalyzedWiththeLinearConductivityModel
(ShalySands,NaCl)
Sample Mineralogya F(–)b (cid:5) c(10(cid:4)4Sm(cid:4)1) (cid:7)((cid:4))d CECd(Ckg(cid:4)1) logQ e(Cm(cid:4)3) ff(–)
S V
3477 K,C 15.1 30.6 0.201 237.74 6.40 0.85
3336A K 23.0 23.6 0.205 393.65 6.61 0.93
3478 K,C 18.9 21.9 0.181 417.70 6.70 0.94
101 K,I 16.4 43.0 0.210 531.40 6.72 0.91
102 K,I 16.2 13.1 0.205 599.84 6.79 0.97
CZ10 C 17.8 72.6 0.233 772.91 6.83 0.89
3833A C 13.9 60.7 0.224 1154.1 7.03 0.94
3126B I,S 11.8 96.2 0.249 1446.1 7.06 0.92
3847A I,C 47.7 29.6 0.126 754.56 7.14 0.96
3283A K,C,I 19.4 44.0 0.186 1245.8 7.16 0.96
3885B I,C 28.9 50.4 0.204 1676.7 7.24 0.97
3972E S,M,I 21.7 44.8 0.187 1546.7 7.25 0.97
3258A I 49.9 86.0 0.128 2022.1 7.56 0.95
3891A C,I 39.3 222.4 0.192 3325.2 7.57 0.93
3308A S,C 12.9 381. 0.270 5498.4 7.59 0.91
3323F S,C 14.4 1065 0.288 10145 7.82 0.89
3324A S,C 29.9 932 0.220 6622.6 7.79 0.84
3323E S,C 15.1 1124 0.281 9801.6 7.82 0.87
3324B S,C 29.6 907 0.204 7843.3 7.91 0.87
3306F S,C 47.8 853 0.171 7122.6 7.96 0.87
aFromVinegarandWaxman[1984,Table1],K:kaolinite,C:chlorite,I:illite,andS:smectite.
bFromthefitofthein-phaseconductivitydatausingthelinearconductivitymodel.
cFromthefitofthein-phaseconductivitydatausingthelinearconductivitymodel.
dFromVinegarandWaxman[1984].
eFromQ andporosityassumingamassdensityforthegrainsof2650kgm(cid:4)3.
V
fUsingf¼1(cid:4)Q^ 1/Q .
V V
Boltzmannconstant(1.3807(cid:2)10(cid:4)23JK(cid:4)1),Tistheabsolute surface of clay minerals is in average Q ¼ 0.32 C m(cid:4)2
S
temperature (in K), and jq j is the absolute value of the (see Revil [2012]). So the average distance between the
ðþÞ
charge of the counterions in the Stern layer. Using the data- charged sites is 1.8 nm. Using this distance, equation (78)
base of Vinegar and Waxman [1984], Revil [2012] found yieldsacriticalfrequencyof0.3MHz.
(cid:2)S (Naþ, 25(cid:11)C) ¼ 1.5 (cid:2) 10(cid:4)10 m2s(cid:4)1V(cid:4)1. This yields the [24] Thesecondcharacteristicfrequencycomesfromthe
ðþÞ
transition between the low-frequency polarization of the
diffusion coefficient of the counterions DS (Naþ, 25(cid:11)C) ¼
ðþÞ electrical double layer and the high-frequency dielectric
3.8(cid:2)10(cid:4)12m2s(cid:4)1forclays.Forfullywater-saturatedmateri- effectintheoverallpolarizationofthematerial.Itisthere-
als,thelowercriticalfrequencyatwhichtheeffectivequadra-
foredefinedby
tureconductivitybecomesconstantisgivenby
(cid:12) ¼(cid:4)"0=(cid:5)00; (80)
(cid:2)2 d
(cid:12) ¼ : (77)
p 2DS where (cid:5)00 and "0 denotes the quadrature conductivity of the
ðþÞ
plateau and the high-frequency dielectric constant. From
Taking (cid:2)¼6mmandDS (Naþ,25(cid:11)C)¼3.8(cid:2)10(cid:4)12m2 equation (53), the high-frequency dielectric constant is
ðþÞ
s(cid:4)1, the critical frequency associated with the previous given(forsaturatedconditions)by
relaxationtimeis
1
f ¼ 1 ¼DSðþÞ; (78) "0¼F½"wþðF(cid:4)1Þ"s(cid:12): (81)
p 2(cid:6)(cid:12) (cid:6)(cid:2)2
p
Taking " ¼ 80" ," ¼ 5.9 " ,and F¼ 18 for the Berea
w 0 s 0
DS sandstone, we get "0 ¼ 10.0 " . The second critical fre-
f ¼ ðþÞ ; (79) 0
p 8(cid:6)Fk quencyisthereforegivenby
0
and I obtain f ¼ 0.034 Hertz in excellent agreement with 1 (cid:5)00
p f ¼ ¼(cid:4) : (82)
the data shown in Figure 3a. Note the measurement of this d 2(cid:6)(cid:12) 2(cid:6)"0
d
relaxation frequency can be used to estimate therefore the
permeability of the material using equation (79) (this idea Using the value of the quadrature conductivity for the
willbeexploredbelowinsection6). Berea sandstone, (cid:5)00 ¼ (cid:4)1.5(cid:2)10(cid:4)4 S m(cid:4)1 (see Figure 3),
[23] I follow here closely Vinegar and Waxman [1984] and "0 ¼ 10.0 "0 yields fd ¼ 0.3 MHz (see Figure 3a).
inestimatingthehighestfrequencyatwhichtheSternlayer Above this frequency, the polarization response of the ma-
polarization can play a role. The shortest possible spatial terialisdominatedbythedielectricpolarizationofthema-
variationbetweenchargedandunchargedzonesistheaver- terial (see Figure 3a). Note also if the quadrature
agedistancebetweenthesurfacesitesontheclayminerals. conductivityisontheorderof(cid:4)1(cid:2)10(cid:4)5Sm(cid:4)1,thecriti-
I have mentioned that the surface charge density on the calfrequencyf canbeaslowas10kHz.
d
314
REVIL:ELECTRICALPROPERTIESOFPOROUSMATERIALS
(f < f ), the quadrature conductivity decreases as the
p
inverse of the root mean square of the frequency. For type
Amaterialsexhibitingaplateauforthequadratureconduc-
tivity (see Figure 3a), the normalized chargeability is also
equaltothequadratureconductivity(M ¼(cid:4)(cid:5)00).Notethat
n
Lesmes and Frye [2001] and Slater and Lesmes [2002]
attemptedasemitheoreticalexplanationfortheequivalence
between normalized chargeability and imaginary conduc-
tivity. This equivalence is very well illustrated by the data
of Slater and Lesmes [2002] and Revil et al. [2012b] (see
Figure4).
[27] Wethereforeproposethefollowingmodelinwhich
thefrequencydependenceisexplicit:
(cid:10)f(cid:11)1=2
(cid:5)0¼(cid:5) þM ; f <f ; (83)
0 n f p
p
(cid:5)0 ¼(cid:5) ; f (cid:6)f ; (84)
1 p
(cid:10)f(cid:11)1=2
(cid:5)00¼(cid:4)M ; f <f (85)
n f p
p
(cid:5)00 ¼(cid:4)M ; f (cid:6)f ; (86)
n p
Figure 5. High-frequency volumetric charge density ver-
susthetotalchargedensity.Thehigh-frequencyvolumetric
charge density is determined from electrical conductivity
measurements at various salinities (from the surface con-
ductivity and the formation factor), while the total charge
density is determined from the porosity and the cation
exchangecapacity.
[25] Inowdescribethequadratureconductivityresponse
belowthecriticalfrequencyf forTypeAmaterials,which
p
are exhibiting a clear plateau in their polarization spectra.
This plateau is likely resulting froma broaddistributionof
polarization length scales and therefore from the self-simi-
larity of the pore space for a broad range of scales (this
idea was expressed by Vinegar and Waxman [1984] and is
consistent with the modeling performed by Cosenza et al.
[2008]andobservationsbyHyslipandVallejo[1997]).
[26] For clay minerals, the kinetics of sorption/desorp-
tionofcationsintheSternlayerisprettyfast(fewseconds
according to Pohlmeier and Ilic [1998]). Therefore at low
frequencies (typically < 1 Hertz), the kinetics of sorption/
desorption may influence the polarization of the electrical
double layer. Wong [1979] provided a general model
describing the polarization of metallic particles (grains)
exchanging charge carriers with the surrounding pore Figure 6. Pore water conductivity dependence of the con-
water. In his case, the charge carriers are redoxactive spe- ductivity of shaly sands (data from Vinegar and Waxman
cies (electron donors and electron acceptors) and the parti- [1984]). I use here a collection of five samples with more or
cle conductive to electrons. His model is however general lessthesameformationfactorbutdifferentCECandtherefore
to work also with insulating grains coated by an electrical surface conductivities. According to the model developed in
double layer. Wong [1979] showed that his rather complex the main text, the conductivity data should follow the same
model can be represented with a Warburg impedance trend,whichisthecase.TheparameterDudenotestheDukhin
model.TheWarburgmodelimpliesthatatlowfrequencies number,i.e.,theratioofsurfacetoporewaterconductivity.
315