Journal of Colloid and Interface Science234,194–203 (2001)doi:10.1006/jcis.2000.7294, available online at http://www.idealibrary.com on

Frequency-Dependent Streaming Potentials

Philip M. Reppert,∗,1 Frank Dale Morgan, David P. Lesmes,† and Laurence Jouniaux‡∗Earth Resources Laboratory, Department of Earth Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, E34-356a, 42 Carleton Street,

Cambridge, Massachusetts 02142;†Department of Geology and Geophysics, Boston College, Chestnut Hill, Massachusetts 02167; and‡Laboratoirede Geologie, URA1316, Ecole Normale Superieure, 24 Rue Lhamond, 75231 Paris Cedex 05, France

Received June 23, 2000; accepted October 16, 2000

An experimental apparatus and data acquisition system was con-structed to measure the streaming potential coupling coefficients asa function of frequency. The purpose of the experiments was to mea-sure, for the first time, the real and imaginary portion of streamingpotentials. In addition, the measured frequency range was extendedbeyond any previous measurements. Frequency-dependent stream-ing potential experiments were conducted on one glass capillaryand two porous glass filters. The sample pore diameters rangedfrom 1 mm to 34 µm. Two frequency-dependent models (Packardand Pride) were compared to the data. Both Pride’s and Packard’smodels have a good fit to the experimental data in the low- andintermediate-frequency regime. In the high-frequency regime, thedata fit the theory after being corrected for capacitance effects ofthe experimental setup. Pride’s generalized model appears to havethe ability to more accurately estimate pore sizes in the porousmedium samples. Packard’s model has one unknown model param-eter while Pride’s model has four unknown model parameters, twoof which can be independently determined experimentally. Pride’sadditional parameters may allow for a determination of permeabil-ity. C© 2001 Academic Press

Key Words: electrokinetic; streaming potential; dynamic; fre-quency; ac.

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INTRODUCTION

The phenomenon of streaming potentials has been stufor many years. This has led to applications in such divefields as chemistry, biology, and geophysics (1–5). The owhelming majority of this work has been in the area ofstreaming potentials, while very little work has been donethe area of frequency-dependent streaming potentials. Pasoretical and experimental work dealing with the frequencysponse of streaming potentials has dealt with single-frequelow-frequency, and frequency response measurements. Sfrequency streaming potential measurements are made afrequency while varying the pressure in order to get pof streaming potentials versus pressure without using a flthrough apparatus (6). Low-frequency measurements are u

1 To whom correspondence should be addressed. Present address: Deparof Geological Sciences, 338 Brackett Hall, Clemson University, Clemson, S29634. Fax: (864) 656-1041. E-mail: reppert@clemson.edu.

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190021-9797/01 $35.00Copyright C© 2001 by Academic PressAll rights of reproduction in any form reserved.

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extrapolate the frequency-dependent streaming potential vto the dc limit. This is done for the purposes of determinthe zeta or surface potentials (7, 8). Low-frequency measments are also used to determine the effective pore sizehydraulic permeability (9). Frequency-response measuremexamine the frequency-dependent behavior of streaming potials (10, 11). Packard (10) proposed a frequency-depenstreaming potential theory for capillaries where the streampotential coupling coefficient remains constant at its dc vauntil the critical frequency is approached by the sinusoidal ding force. At frequencies higher than the critical frequency,streaming potential coupling coefficient decays with increasfrequency. Packard’s experiments were performed on a limnumber of capillary samples of large radii (2.083–0.589 mand with no changes in solution chemistry. Packard wasto achieve a maximum measuring frequency of 200 Hz. Co(11) attempted to duplicate Packard’s work, but with limitsuccess; he could not match his capillary data to the theorecurves of Packard. Cooke’s data for porous glass filters appto have the expected trend predicted by Packard’s theorycloser examination, however, Packard’s theory cannot be sfactorily fitted to Cooke’s data. No one to date has satisfactofitted frequency-dependent streaming potential data to theical curves for porous media of any pore diameter or for calaries with diameters less than 155µm.

In 1994, Pride (12) proposed a generalized theoryfrequency-dependent streaming potentials in porous media.theory relates the transport properties and pore-geometry paeters to the samples’ streaming potential frequency-respbehavior. No experimental work has been performed to valiPride’s theory.

The understanding of frequency-dependent streaming potials is vital to Earth scientists who study the electromagnsignals generated by seismic/acoustic waves propagating iearth. Earth scientists believe that these electromagneticnals are generated by oscillatory fluid flow in rocks relativethe mineral matrix (13). This relative flow induces a streaing current, which oscillates as an electric dipole at the sfrequency as the seismic/acoustic wave exciting the medThis frequency-dependent streaming current has a coucurrent that flows through the conductive part of the rock (bfluid) to develop a frequency-dependent streaming poten

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FREQUENCY-DEPENDENT

Frequency-dependent streaming potentials induced by amic wave are often referred to as the seismoelectric effect15). Some Earth scientists believe that the streaming potefrequency response may be useful in determiningin situperme-ability (16, 17).

In this paper, the real and imaginary streaming potentialquency response for a capillary is developed based on Packmodel, which is then compared to Pride’s model. Then a discsion of the experimental system and methodology is providThis is followed by a presentation of the first data where the coplex frequency response for one glass capillary and two poglass filters is presented. These experiments cover a frequrange of 1–500 Hz.

DC STREAMING POTENTIALS

Streaming potentials are a subset of electrokinetic phenena, which includes electroosmosis, electrophoresis, andimentation potentials. Electrokinetic phenomena are a coquence of a mobile space charge region that exists at the intcial boundary of two different phases. This region is commoreferred to as the electrical double layer (EDL). The most simfied approximations of the EDL can be represented by a paraplate capacitor (Helmholtz model) or a charge distribution tdecays exponentially away from the surface, Gouy–Chapmodel (18). A more accurate model takes into account the fisize of ions by combining the Helmholtz and Gouy–Chapmmodels where a fixed layer exists at the surface (Stern laand a diffuse layer extends from the fixed layer into the bsolution (Gouy–Chapman diffuse layer). This model is oftenferred to as the Stern model of the electrical double layer. Mspecifically the interfacial region, shown schematically in Figconsists of the inner Helmholtz plane (IH) where ions aresorbed to the surface and the outer Helmholtz plane (OH) whions are rigidly held by electrostatic forces and cannot moThe closest plane to the surface at which fluid motion can tplace is called the slipping plane. The slipping plane has atential defined as the zeta potential (ζ ), which is a characteristicof the solid and liquid that constitute the interface. The diffulayer extends from the OH into the bulk of the liquid phaThe distance at which the diffuse layer potential (ψ0) has beenreduced toψ0/e is referred to as the Debye length. This is oftused as a measure of how far the diffuse layer extends intobulk fluid.

Streaming potentials occur when relative motion betweentwo phases displaces ions tangentially along the slipping pby viscous effects in the liquid. This displacement of ions gerates a convection current (Iconv) and has properties similar tan ideal current source. For a capillary,Iconv is defined by

Iconv(r ) =∫v(r )ρc(r ) dr, [1]

in whichv(r ) is the fluid particle velocity, dr is an infinitesimalpart of the cross section, andρc(r ) is the charge distribution

STREAMING POTENTIALS 195

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FIG. 1. (a) The Stern model of the electrical double layer. (b) One ofpossible potential distributions of the Stern model. This model assumes ththe Stern layer the potential varies linearly.

in the capillary. The solution to this integral can be found invariety of texts and is given by

Iconv(r ) = πεa2ζ1P

ηl, [2]

whereε is the permittivity of the fluid,a is the radius of thecapillary, l is the length of the capillary,1P is the pressureacross the sample,η is the viscosity of the fluid, andζ is the zetapotential.

In steady-state equilibrium the convection current mustbalanced by a conduction current (Icond) (18). Hence by ohmslaw,

Icond(r ) = πσa2

l1V, [3]

whereσ is the fluid conductivity and1V is the voltage mea-sured across the sample. Equating the convection and contion currents, which must be equal at equilibrium, leads toHelmholtz–Smoluchowski equation (18)

1V = εζ

ησ1P. [4]

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196 REPPER

It should be noted when viewing Eq. [4] that it does not inclugeometry terms for the specimen. The ratio1V/1P is referredto as the cross-coupling coefficient or simply the coupling coficient in the rest of the paper. When the capillary or pore-spthickness approach the dimensions of the diffuse layer (0.1µmfor 0.001 M solution), surface effects must be considered (2The radii of the samples used in our experiments are much lathan the Debye length; hence, these effects are not be addre

AC STREAMING POTENTIALS

The basic principles of ac streaming potentials are presefollowing the methodology of Packard (10). To better understthe physics of ac streaming potentials, appropriate compariare made to the frequency-dependent hydraulic problem pously solved by others (19).

The basic electrokinetic principles of dc and ac streampotentials are the same. The difference is in the hydrodynapart of the solution where the constant pressure of the dc careplaced by a sinusoidal pressure in the ac case. Thereforestarting point for the derivation must go back to the hydronamic problem where the Navier–Stokes equation,

ρ∂v(r, ω)

∂texp(−iωt)

= −∇P(ω) exp(−iωt)+ η∇2v(r, ω) exp(−iωt), [5]

is shown with a sinusoidal driving pressure applied acrosssample. Equation [5] can be rearranged into the form

[∂2

∂r 2+ 1

r

∂

∂r+ k2

]v(r, ω) = 1P(ω)

ηl, [6]

where the gradient of the pressure has been replaced by1P(ω)/ land

k2 = −iωρ

η. [7]

It is apparent that the solution can be expressed using Befunctions, with the general solution to Eq. [6] expressed as

v(r, ω) = −1P(ω)

k2lη+ C1J0(kr)+ C2Y0(kr). [8]

The applied boundary conditions arev(r, ω) = 0, whenr = a;v(r, ω)= finite, whenr = 0, and also notingY0(0)= ∞, whichthen requires thatC2 = 0. This leads to

C1 = −1P(ω)

ηlk2

1

J0(ka). [9]

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Substituting Eq. [9] into Eq. [8] gives

v(r, ω) = 1P(ω)

ηlk2

[J0(kr)

J0(ka)− 1

][10]

as the frequency-dependent velocity inside a capillary. Integing Eq. [10] over the cross-sectional area of the capillary giv

v(ω) = 1P(ω)

ηlk2

[2

ka

J1(ka)

J0(ka)− 1

][11]

as the average fluid velocity inside a capillary, which is tfrequency-dependent hydraulic solution for a capillary.

Now that the frequency-dependent fluid flow inside a capillahas been presented, Eq. [10] can be substituted into Eq. [1give

Iconv(ω) =∫

2πrρc(r )v(r, ω) dr, [12]

which is the frequency-dependent convection current insidcapillary, where the charge density in the capillary in cylindriccoordinates is given by

ρc(r ) = −ε∇2ψ0(r ) = −εr

∂

∂r

(r∂ψ0(r )

∂r

). [13]

Integrating Eq. [12] from the center of the capillary to the sliping plane under the assumption that the capillary radius is lacompared to the Debye length of the EDL (10, 12) gives

Iconv(ω) = −2πεaζ1P(ω)

ηlk

J1(ka)

J0(ka)[14]

as the frequency-dependent convection current for a capillaAs in the DC case, at equilibrium, the convection current m

be balanced by a conduction current. The conduction currendetermined using Ohm’s law, where the conduction currengiven by

Icond(ω) = 1V(ω)πa2σ

l. [15]

Setting the convection current equal to the conduction currgives

C(ω) = 1V(ω)

1P(ω)=[εζ

ση

]−2

ka

J1(ka)

J0(ka). [16]

This is the ac Helmholtz–Smoluchowski equation in a forsimilar to that presented by Packard (10), whereC(ω) isthe frequency-dependent cross-coupling coefficient. TheHelmhotz–Smoluchowski equation reduces to the dc form inlimit asω goes to zero, which can be demonstrated using Be

function recursive relations or by taking the low-frequency ap-proximation of Eq. [16]. Figure 2 shows the real and imaginary

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FREQUENCY-DEPENDEN

FIG. 2. Theoretical comparison of the frequency-dependent hydraulic stion, Eq. [11] and the frequency-dependent streaming potential solution. Thand imaginary portions of both solutions are shown. The coupling coefficare normalized.

portions of the ac coupling coefficient, Eq. [16], and a plotthe frequency-dependent hydraulic equation, Eq. [11]. Figushows the phase part of Eqs. [11] and [16]. It can be seethese two figures that the general behavior of the two equais similar. In particular, the low-frequency behavior is identibut the high-frequency behavior diverges as one goes to hifrequencies.

At first examination of frequency-dependent streaming potials and frequency-dependent hydraulics, it might be expethat the two phenomena would have identical behavior, sincefrequency-dependent streaming potential behavior is goveby the frequency-dependent fluid flow. The different responof the ac Helmholtz–Smoluchowski equation, Eq. [16], andfrequency-dependent hydraulic equation, Eq. [11], are easiunderstand by looking at the series and asymptotic approxtions for the low- and high-frequency cases, respectively.lowing the methodology of Crandall (20) (which was developfor the acoustic case), the low-frequency and high-frequeapproximations are made. The two separate equations arecombined to form a single equation.

FIG. 3. Phase comparison of the frequency-dependent coupling coeffic

and frequency-dependent hydraulics, where the phase is 0◦ for both curves at0 Hz.

STREAMING POTENTIALS 197

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Starting with the ac Helmholtz–Smoluchowski expressiEq. [16], and substituting the low-frequency approximationsthe Bessel functions (20)J0 andJ1, with

J0(x) =∞∑

n=0

(−1/4x2)k

k!0(n+ 1)[17]

and

J1(x) = x

2

∞∑n=0

(−1/4x2)k

k!0(n+ 2), [18]

wherex represents ka and ka< 1. Substituting these two approximations into Eq. [16] and taking the Limn→∞ gives

1V(ω)

1P(ω)= lim

n→∞

[εζ

ση

] 2

ka

x2

∑∞n=0

(−1/4x2)k

k!0(n+2)∑∞n=0

(−1/4x2)k

k!0(n+1)

=[εζ

ση

][2

ka

x

2

]. [19]

Sincex= ka, the low-frequency approximation reduces to

Cka<1

(ω) = 1V(ω)

1P[ω]=[εζ

ση

]= 1V

1P. [20]

The high-frequency approximation used for the Bessel functiJ0 andJ1 are given by

J1(x√−i )

J0(x√−i )

= −i, [21]

which can be found in Crandall (20) or which can be easproven using the asymptotic approximations of Abramowitz aStegun (21). In Eq. [16],

ka= a

√− iρω

η, [22]

from which the following substitution is made in Eq. [22]:

x√−i = a

√− iρω

η= a

√ρω

η

√−i . [23]

Then substituting Eq. [21] and Eq. [23] into Eq. [16] gives

C(ω) =[εζ][−2i

]=[εζ] −2i√

. [24]

ka>10 ση ka ση a −iρω

η

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t

a

2

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198 REPPER

FIG. 4. Normalized comparison of the Bessel function solution, Eq. [1to the approximated Bessel function solution, Eq. [26].

Equation [24] can then be modified to

C(ω)ka>10

=[εζ

ση

][−2

a

√η

ωρ

(1√2− 1√

2i

)]. [25]

Combining the low-frequency and high-frequency approximtions gives

CA(ω) =[εζ

ση

][1− 2

a

√η

ωρ

(1√2− 1√

2i

)], [26]

where CA(ω) represents the approximated cross-coupling cficient. In Fig. 4, the solution using Bessel function approximtions, Eq. [26], is plotted against the Bessel function soluof Eq. [16]. It can be seen in Fig. 4 that the two solutionsnearly identical. There is a slight divergence in the intermedifrequency range, 1> ka> 10, which was not accounted forthe approximation. However, the error is smaller than measments can detect.

It can be seen in Eq. [26] and Figs. 2, 3, and 4 that asfrequency is increased inertial effects start to retard the moof the fluid within the pore space. This occurs as the fluid maa transition from viscous dominated flow to inertial dominaflow. At the higher frequencies the flow becomes inefficierequiring more energy to move the same amount of liquidsame distance. Therefore at frequencies higher than the trtion from viscous to inertial flow, more pressure is requiredshear the same quantity of ions from the diffuse zone than wstrictly in the viscous flow regime. It can also be seen in Eq. [and Fig. 2 that at higher frequencies the real and imaginary pof the coupling coefficient solution are decreasing at the srate. This explains the 45◦ phase angle found at high frequencin Fig. 3.

One might ask why the frequency-dependent hydraulicsfrequency-dependent streaming potentials do not havesame behavior. By comparing the form of the frequendependent streaming potential (FSP) approximate solutiothe frequency-dependent hydraulic (FDH) solution, valuable

sight into the physics that causes the difference in behaviortween FDH and FSP solutions can be obtained. This compar

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is made using Eqs. [20], [25], and [26] of the FSP approximtion and Eq. [11] of the (FDH) solution. The low-frequenapproximation to Eq. [11] is (20)

H (ω)ka<1=(

8

a2+ 4

3iρω

)−1

, [27]

where H (ω) represents the frequency-dependent hydrasolution. When ka< 1, the real term in Eq. [27] dominates ovthe imaginary term. This implies that at low frequencies the flis real (no vorticity present) and as the frequency is increaan inertial component starts to develop in the flow (vorticpresent). However, when ka< 1 the inertial term remainsinsignificant and the flow is essentially viscous. Therefoboth low-frequency FDH and FSP solution approximatioapproach the dc limit at low frequencies.

The high-frequency approximation of the FDH equationgiven by (20)

H (ω)ka>10

=(

iρω + 1

r2η√ρω

2η(1+ i )

)−1

, [28]

where (2η/ρω)1/2 is the viscous skin depth. When examininEq. [28], it can be seen that the bulk of the fluid is governedthe imaginary terms and thus inertial flow exists. However, whthe viscous skin depth is sufficiently small, a second-order efstarts to dominate and the imaginary term starts to decreathe same rate as the real term, giving rise to a diminished pagation velocity. It becomes apparent from the approximaanalysis that the high-frequency solution causes the differebetween the frequency-dependent hydraulics and the frequedependent streaming potential behavior. When the integraEq. [12] is evaluated, most of the contribution to the integoccurs along the wall of the capillary where most of the chadistribution is located. This region near the wall is also whthe second-order effect of the hydraulic solution starts to dinate. Consequently, the frequency-dependent streaming ptial exhibits high-frequency behavior that follows the formthe second-order effect of the hydraulic solution.

AC STREAMING POTENTIALS IN POROUS MEDIA

A model for ac streaming potentials in a porous mediumrived from first principles was developed by Pride (12). Pridversion of the ac Helmholtz–Smoluchowski equation is givby

CP(ω) = 1V(ω)

1P(ω)

=[εζ

ησ

][1− i

ω

ωt

m

4

(1−2

d

3

)2(1− i

32 d

√ωρ

η

)2]− 1

2

,

ison [29]

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a

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i

i

ncyhemowely,fects

th is

thesel-is

are

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FREQUENCY-DEPENDENT

where CP(ω) represents the AC coupling coefficient for poromedia using Pride’s model,d is the Debye length, and3 is atypical pore radius representing a weighted volume-to-surfratio (19). The transition frequency

ωt ≡ φ

α∞k0

η

ρ, [30]

as defined by Pride (12), separates the low-frequency visflow regime from the high-frequency inertial flow regime.Fig. 2 the transition frequency occurs where the hydraulic iminary curve intercepts the hydraulic real curve. Porosity is giby φ tortuosity byα∞, and the dc permeability byk0. The di-mensionless numberm is defined as

m≡ φ

α∞k032, [31]

which is also a function of the pore microgeometry, whichduces to 8 for a capillary. Closer examination of the ac pormedia coupling coefficient, Eq. [29], reveals that to a first-orapproximation the response is determined by the transitionquencyωt and the dc coupling coefficient.

Figure 5 shows the normalized real and imaginary parts oftheoretical ac coupling for three capillaries of different radusing Eq. [16]. The obvious characteristic of the curves is thattransition frequency shifts with changing radius of the capillaThis relationship is easily evident when looking at the equatfor the transition frequency, Eq. [30], and realizing that thepermeability for a capillary is

k0 = a2

8. [32]

The transition frequency for capillary then becomes

ωt(cap)= 8

a2

η

ρ. [33]

FIG. 5. Normalized coupling coefficient of the real and imaginary compnents for three capillaries of different radii.

STREAMING POTENTIALS 199

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COMPARISON OF PACKARD’S AND PRIDE’S MODELS

Pride (12) generated his model by creating low-frequeand high-frequency models separately and then combining tinto a single combined model. Equations [34] and [35] shPride’s low-frequency and high-frequency models, respectivwhen capillary geometry terms are used and second-order efare neglected. Capillary geometry terms imply thatm= 8,φ =1, α∞ = 1, 3 = a, andk0 = (a2)/8, wherea is the capillaryradius. Second-order effects occur when the Debye lenglarge compared to the capillary radius:

CPLAC = εζ

ησ[34]

CPHAC = −2

[εζ

ησ

]i

12 δ

3. [35]

Equation [35] can be rewritten as

CPHAC =[εζ

ησ

][−2

a

√η

ρω

√i

], [36]

whereδ is the viscous skin depth and is given by (2η/ρω)1/2.Equation [36] can also be rewritten as

CPHAC =[εζ

ησ

][−2

a

√η

ρω

(1√2− 1√

2i

)], [37]

which clearly shows that the real and imaginary parts ofhigh-frequency solution are identical to Eq. [25] in the Besfunction approximation. One possibility of combining the lowfrequency and high-frequency solutions into a single model

CP(ω) =[εζ

ησ

][1− 2

a

√η

ωρ

(1√2− 1√

2i

)], [38]

where capillary geometry is used and second order-effectsneglected.

The simplified version of Pride’s model, Eq. [38], as oposed to the full equation, is compared to Packard’s equafor two reasons. First, the second-order effects do not affecmodel within the range of frequency or concentrations for whPride’s model is defined. Second, the simplified model of PrEq. [38], is identical to the approximated form the Packarmodel, Eq. [26]. This implies, when using capillary geometerms and neglecting second-order effects, that Pride’s mis identical to Packard’s model when the series and asympapproximations are used.

A visual comparison of Packard’s and Pride’s models canmade by looking at Fig. 4, where Eqs. [16] and [38] are copared. A slight discrepancy between the two curves becoapparent, which is addressed earlier in this paper. To comple

o-approximate the Bessel function, the range where ka is greaterthan one but less than 10 must be addressed. Packard addresses

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200 REPPER

this range because the Bessel function form inherently cothe whole range. It appears from the comparison of PackaBessel function form to Pride’s model that Pride’s model donot completely address this region. When Fig. 4 is compato actual measurements, however, it is difficult to resolvedifference between the two curves when experimental erroaccounted for in the data. Consequently, Pride’s model is aquate in this region.

EXPERIMENTAL APPROACH

The approach used to collect the streaming potential datato hold the capillary or porous filter stationary and oscillatefluid back and forth through the sample using a sinusoidal ding pressure at one end while having the other end open toatmosphere. Silver silver–chloride electrodes were placedeither side of the sample in fluid ports to keep them out offluid flow path (22). The frequency response of the electrowas measured using a four-electrode method and found to bin the region used in this experiment. The electrode measments were also verified through the use of a capacitive couantenna placed around the outside of the sample. The prewas monitored on the high-pressure and atmospheric-pressides of the sample by miniature hydrophones that have afrequency response from 1 to 20 kHz. The acrylic apparawhich holds the sample, electrodes, and hydrophones, is shschematically in Fig. 6. The enclosure around the samplepreamplifiers is constructed of Mu-metal. Mu-metal is usedcause of its superior electromagnetic shielding properties.acrylic housing that holds the sample is additionally encloin an aluminum housing, and the electromechanical shakerproduces the driving pressure is enclosed in a separate steeThis large quantity of shielding is required for two reasons. Tfirst is that the laboratory is an electrically noisy environmeThe second is that the pressure source is being generatedelectromechanical transducer, which is driven at the samequency as the measured streaming potential. The driving trducer and associated leads emit enough electromagnetic s(EMF) at the same frequency being measured to sometimefect the measured results. Depending on the pore or capidiameters, the resistance of the sample could be several hun

FIG. 6. A simplified schematic representation of the test cell and pres

source. The apparatus is constructed of Plexiglas and bolted to a heavy mplate.

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FIG. 7. A simplified representation of the streaming potential data acqution circuit is shown. A Labview 12-bit AD board was used to acquire the din the computer.

megaohms, which causes the sample to act as an antennthe EMF fields produced by the electromechanical transduThe acrylic housing has 50 kg of lead placed on it to reducesample vibration.

The waveforms were sampled using two instrument preplifiers and a 12-bit analog-to-digital board with 24.8-µV res-olution. Electrometers or FET preamplifiers were requiredinput buffers due to the high impedance of the samples (.5Äto 1 GÄ). A schematic representation of the data acquisitsystem is shown in Fig. 7. The data were analyzed using stral analysis with a Hanning window applied to the data prto spectral analysis. In addition to spectral analysis, the tidomain voltage and pressure were monitored at each drifrequency. The raw data were saved for possible future procing. Cross-correlation analysis of the signals was also perforas a verification of the amplitude spectrum measurements.

DATA/RESULTS

To test Packard’s and Pride’s models experimentally, the pdiameters of the samples must be known. The pore diametused as an adjustable parameter in the models to fit the thto the data. Actual pore diameters are then compared toexperimentally determined pore diameters. The pore diameof the glass samples were provided by the manufacturer.

The glass samples consist of a capillary and two porousters. Capillary 1 has an inner diameter that ranges from 0.1.1 mm, and a length of 60 cm. Porous Filter A has manufactudetermined pore diameters ranging from 145 to 175µm, anoverall diameter of 1 cm, and a thickness of 2 mm. Porous FB has manufacturer-determined pore diameters ranging fromto 100µm, an overall diameter of 1.9 cm, and a thickness1 cm.

The experiments were carried out using 10−3 and 10−3.7 MKCl with a pH of 5.5. Assuming a concentration of 10−4 M KCl,the Debye length was found to be much less than the radiuthe pore space. Therefore, second-order effects can be negin Pride’s model.

etalFigure 8 shows the measurements of the coupling coefficientversus frequency for Capillary 1 along with a theoretical curve

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FIG. 8. Cross-coupling real and imaginary uncorrected data for Capil1 plotted along with the theory for a 0.8-mm capillary.

for Eq. [26]. It can be seen that the data do not have a satisfafit to the theory. In addition to the streaming potential dataFig. 8, the frequency-dependent impedance of the samplemeasuring circuit was also determined and is shown in FigThe impedance of the sample and measuring circuit wastermined using a four-electrode method. The RC of the cirwas found to have a resistance of 1 GÄ and a capacitance o25 pF. The resistance was determined to be on the order oresistance of a 0.8-mm capillary filled with 10−3.7 M KCl. Thecapacitance was found to be on the order of the input captance of the amplifier and the capacitance of the wire leads.data in Fig. 8 were then corrected for this impedance by normizing the real and imaginary portions of the impedance atparticular frequencies of the streaming potential measuremThis normalized impedance was then divided into the data.corrected data are shown in Fig. 10, where a good correlabetween the theory and the data is evident. At the higherquencies, signal-to-noise problems caused the fit to have amore scatter. The transition frequency determined from the fithe theoretical curve is 7.1 Hz, which gives a capillary diame

FIG. 9. Capillary 1 impedance data plotted with the best RC circuit

sponse. The resistance and capacitance of the RC circuit are 1 GÄ and 25 pF,respectively.

STREAMING POTENTIALS 201

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FIG. 10. Corrected cross-coupling real and imaginary data for 60-mm-l0.8-mm-diameter capillary plotted along with the theory for a 0.8-mm capilError bars are not shown because they fall within the size of the data poin

of 0.8 mm. This is in agreement with the manufacturer-providiameter of 0.8–1.1 mm for this type of capillary. LookingFig. 10, it can be seen that for the higher frequencies, ineflow essentially dominates in Capillary 1.

The corrected data for Porous Filter A, shown in Fig. 11,plotted against Eq. [26]. As can be seen, the theory fits thewell. Fitting the theoretical curve to the data for Porous FiA gives a pore radius of 65µm and a transition frequency o269 Hz. The manufacture-provided pore radii for the PorFilter A are 72.5–87 mm.

Figure 12 shows the corrected data for Porous Filter B awith the best-fit theoretical curve. Porous filter B has its besto Pride’s theory, Eq. [26], using a 40-µm pore radius and a transition frequency of 710 Hz. Recall that the pore radii providedthe manufacturer for Porous Filter B are 35 to 50µm. The datafor Porous Filter B fits the theory very well in the low-frequenand the intermediate-frequency regions. At the higher freqcies, there is a little more scatter than at the low frequenciesto poor signal-to-noise ratio.

FIG. 11. Porous Filter A corrected cross-coupling real and imaginaryplotted with the theoretical response. The transition frequency was determto be 269 Hz, which gives a pore radius of 65µm. The manufacturer provide

pore radius for porous filter B ranges from 72.5 to 87µm. Error bars are notshown because they fall within the size of the data points.

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FIG. 12. Porous Filter B corrected cross-coupling frequency responseplotted along with the theoretical response. The transition frequency wastermined to be 710 Hz, which gives a pore radius of 40µm. The manufacturerprovided pore radius for porous filter B ranges from 35 to 50µm. Error bars arenot shown because they fall within the size of the data points.

It appears from the data that the experimentally determipore radii tend toward the low range of radii provided by tmanufacturers. This may imply that the frequency-dependstreaming potential response is controlled by the smallest pgeometry in a fluid path. Further research is required to saycertain why the frequency response tends toward the smpore sizes. At the present time it appears to be related onthe hydrodynamic portion of the problem.

DISCUSSION AND APPLICATIONS

The emphasis of this study has been to examine, for thetime, complex streaming potentials and to extend the frequerange covered by this type of experiment. As a consequencthis study Packard’s and Pride’s theories have been verifieda range of pore sizes. Both Packard’s and Pride’s theories fidata nearly identically. The transition frequency is the imptant parameter to be determined uniquely, for both capillaand porous filters, from frequency-dependent streaming potial curves. From the transition frequency, the pore size candetermined. The transition frequency can be determined graically from the data or by curve fitting the theory to the daand then determining the transition frequency from the theoPackard’s theory for a capillary is related to the capillary radand the viscosity and density of the fluid. In Pride’s model oadditional parameter (m) is included if the second-order paramters are neglected and3 is taken as the pore radius. The seconorder effect parameters are important only where the pore sare very small and/or the electrolyte concentration is very lo

The data collected on glass filters with pore diameters mlarger than the diffuse zone indicate that the curves generusing Pride’s and Packard’s theories cannot be distinguisfrom each other. This result confirms the analysis done elier in the paper. What is different between the theories is t

Pride’s model includes second-order effects, which are elimnated in the approximation made during the solution of Eq. [1

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Packard’s model. In this respect, Pride’s model appears tmore complete and general, but the accuracy in these reghas not been confirmed. From the data thus far collecteporous filters, it can be concluded that the curves generby Packard’s and Pride’s models can accurately fit the dTherefore, both Packard’s and Pride’s models give identicasults when used with realistic frequency ranges and soluchemistries as shown from the theoretical analysis and theperimental data.

SUMMARY/CONCLUSION

Real and imaginary parts of frequency-dependent streing potentials were measured for the first time. A compariof streaming potential data to proposed models has beensented for various pore sizes. Although there is a slight discancy between the capillary model and the generalized mooverall both Packard’s and Pride’s models fit the frequendependent streaming potential capillary data. For porous mboth Packard’s capillary model and Pride’s porous media mfit the data, which allows the pore size to be estimated. The dience between the two models comes from the model paramused to fit the curve. When using Pride’s model, if eithermor3are know a priori, it allows determination of the other parameFrom the results thus far presented, the frequency streamintential response may be governed by the smallest pore geomin the fluid path.

Currently plans are underway to extend the frequency raof the experiment to 20–30 kHz. This will allow study of a widrange of samples, including rocks. Studying a broader sampof rocks will help us to understand the relationship betwfrequency-dependent streaming potentials and permeabilialluded to by others (16, 17). A set of samples with well-defingeometries will better help to confirm how well the models cpredict pore size and whether the response is governed bpore-throat size. Last, performing frequency-dependent stuat higher temperatures and pressures will help us to ascewhat is happening to the solution chemistry and electrical dble layer atin situ temperatures and pressures for rocks. Tunderstanding has importance to the oil industry as well aEarth scientists for the study of earthquake nucleation (23)

ACKNOWLEDGMENTS

We thank the United States Air Force for their grant in support of this wWe also thank Dereck Hirst and the MIT Laboratory of Nuclear Science MacShop personnel for their invaluable help in constructing the test cell andrequired equipment. This work was partially supported by DOE GrantFG02-00ER15041, a Lavoisier grant from the French Foreign office, andCompagine G´eneral de Geophysique (France).

REFERENCES

i-6],

1. Kurtz, R. J., Findl, E., Kurtz, A. B., and Stormo, L. C.,J. Colloid InterfaceSci.57,28–39 (1976).

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d,

nsof

.er-

FREQUENCY-DEPENDEN

2. Levine, S., Marriott, J. R., Neale, C., and Epstein, N.,J. Colloid InterfaceSci.52,136–149 (1975).

3. Rice, C. L., and Whitehead, R.,J. Phys. Chem.69(11) (1965).4. Morgan, F. D., Williams, E. R., and Madden, T. R.,J. Geophys. Res.94,

12,449–12,461 (1989).5. Ishido, T., and Mizutani, H.,J. Geophy. Res.86,1763–1775 (1981).6. Sears, A. R., and Groves, J. N.,J. Colloid Interface Sci.65,479 (1977).7. Pengra, D. P., Shi, L., Li, S. X., and Wong, P., “MRS Symposium Proce

ings, Boston, December 1995.”8. Wong, P., U.S. Patent 5,417,104 (1995).9. Li, S. X., Pengra, D. B., and Wong, P.,Phy. Rev. E51, 5748–5751

(1995).10. Packard, R. G.,J. Chem. Phys.21,303–307 (1953).11. Cooke, C. E.,J. Chem. Phys.23,2299–2303 (1955).12. Pride, S. R.,Phys. Rev. B.50,15,678–15,696 (1994).13. Thompson, A. H., and Gist, G. A.,The Leading Edge, December,1169

(1993).

14. Haartsen, M. W., and Pride, S. R.,in “64th Ann. Internat. Mtg. Soc. Expl.

Geophys., Expanded Abstracts,” pp. 1155–1158, 1994.

STREAMING POTENTIALS 203

ed-

15. Zhu, Z., Cheng, C., and Toks¨oz, M. N., in “64th Ann. Internat. Mtg. Soc.Expl. Geophys., Expanded Abstracts,” pp. 26–29, 1994.

16. Mikhailov, O. V., Haartsen M. W., and Toks¨oz, M. N.,Geophysics62,1–9(1997).

17. Reppert, P. M., and Morgan, F. D.,in “Transactions of the 99th AnnualMeeting of the American Geophysists Union,” T32H-04, 1998.

18. Overbeek, J. Th.,in “Irreversible Systems” (H. R. Kruyt, Eds.), Vol. 1.Elsevier, New York, 1952.

19. Johnson, D. L., Koplik, J., and Schwartz, L. M.,Phys. Rev. Lett.57,2564–2567 (1986).

20. Crandall, I. B., “Theory of Vibrating Systems and Sound.” Van NorstranNew York, 1926.

21. Abramowitz, M., and Stegun, I. A., “Handbook of Mathematical FunctioWith Formula, Graphs, and Mathematical Tables.” U.S. National BureauStandards, 1964.

22. Morgan, F. D.,in “Lecture Notes in Earth Sciences” (S. Bhattacharji, G. MFriedman, H. J. Neugebauer, and A Seilacher, Eds.), Vol. 27, Spring

Verlag, Berlin/New York (1989).

23. Morgan, F. D.,Eos Trans.AGU, F173 (1995).