Frequency-DependentStreamingPotential ofPorousMedia—Part2 ...

Hindawi Publishing CorporationInternational Journal of GeophysicsVolume 2012, Article ID 728495, 17 pagesdoi:10.1155/2012/728495

Research Article

Frequency-Dependent Streaming Potentialof Porous Media—Part 2: Experimental Measurement ofUnconsolidated Materials

P. W. J. Glover,1 E. Walker,1 J. Ruel,2 and E. Tardif2

1 Department of Geology and Engineering Geology, Laval University, Quebec City, QC, Canada G1K 7P42 Department of Mechanical Engineering, Laval University, Quebec City, QC, Canada G1K 7P4

Correspondence should be addressed to P. W. J. Glover, [email protected]

Received 3 June 2011; Revised 1 November 2011; Accepted 12 December 2011

Academic Editor: Tsuneo Ishido

Copyright © 2012 P. W. J. Glover et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Frequency-dependent streaming potential coefficient measurements have been made upon Ottawa sand and glass bead packsusing a new apparatus that is based on an electromagnetic drive. The apparatus operates in the range 1 Hz to 1 kHz with samplesof 25.4 mm diameter up to 150 mm long. The results have been analysed using theoretical models that are either (i) based uponvibrational mechanics, (ii) treat the geological material as a bundle of capillary tubes, or (iii) treat the material as a porous medium.The best fit was provided by the Pride model and its simplification, which is satisfying as this model was conceived for porous mediarather than capillary tube bundles. Values for the transition frequency were derived from each of the models for each sample andwere found to be in good agreement with those expected from the independently measured effective pore radius of each material.The fit to the Pride model for all four samples was also found to be consistent with the independently measured steady-statepermeability, while the value of the streaming potential coefficient in the low-frequency limit was found to be in good agreementwith other steady-state streaming potential coefficient data.

1. Introduction

There have only been 10 measurements of the frequency-dependent streaming potential coefficient of porous geolog-ical and engineering materials. A review of the existing mea-surements was carried out by Glover et al. [1]. These previousmeasurements can be divided into two groups: (i) transientmeasurements with a percussive source and (ii) harmonicmeasurements with a vibrating source.

While the first of these approaches mimics many of thepossible applications more closely [2–4], it cannot providethe streaming potential coupling coefficient as a function offrequency without using the frequency domain filtering andFourier techniques. Such techniques can only be used in alinear system. Although the equations that describe thestreaming potential coefficient are linear below the transitionfrequency and there is no evidence that they become non-linear above that frequency, it has not yet been shown thatsuch an approach can be made to work for streaming po-

tential coupling coefficient measurements on rocks. Theprocessing of such data has, however, been discussed atlength in Reppert and Morgan [5]; they mention that inertialeffects may be seen if the transient signal has strong enoughhigh-frequency components.

The second approach is capable of providing the stream-ing potential coupling coefficient at each frequency directly.Its disadvantage is that a high-quality harmonic driving pres-sure is required to create the time-varying flow. Variousauthors have shown that measurements on a range of ma-terials are possible in the range 1 Hz to 600 Hz [6–10], butbefore the recent paper of Tardif et al. [11] only one meas-urement had been made on a geological material [10].

This paper reports research that uses the electromagneticdrive concept proposed by Glover et al. [1] to create anapparatus for measuring the frequency-dependent streamingpotential coupling coefficient of unconsolidated materialssuch as sands, gravels, and soils. Unconsolidated materialswere chosen because it is easier to arrange a sample holder

2 International Journal of Geophysics

with no need for a sleeve and a sleeve pressure. The experi-mental data have been analyzed using a number of theoreticalmodels. The first part of this paper is dedicated to describingthese models, followed by experimental measurements onsamples of sand and glass beads. The theoretical models havebeen compared with the measured data in order to obtain thetransition frequency, which has then been used to calculatethe effective pore radius of the sands and glass bead packsusing the theory in Glover and Walker [12].

2. Theoretical Models

The steady-state streaming potential coefficient (the stream-ing potential per driving fluid pressure difference) haslong been described by the Helmholtz-Smoluchowski (HS)equation, and is given in the form most convenient forapplication to rocks (e.g., [13]);

Cso = ΔV

ΔP= ε ζ

η f σ, where σ = σ f +

2Σs

Λ. (1)

In this equation ΔP (Pa) is the fluid pressure difference, ε(F/m) is the dielectric constant of the fluid, η f (Pa.s) is thedynamic viscosity of the fluid, ζ (V) is the zeta potential,ΔV (V) is the streaming potential, σ f (S/m) is the electricalconductivity of the bulk fluid, Σs (S) is the specific electricalconductance of the surface (i.e., that due to the double layer),σ (S/m) is the electrical conductivity of the mobile fluid,and Λ (m) is a characteristic length associated with themicrostructure of the pore network [14–18]. The steady-statestreaming potential is independent of the sample geometry.

The importance of considering the surface conductancewhen applying the HS equation to geological materials hasbeen discussed by a number of authors including [1, 14–17]. Recently several modified versions of the classical HSequation have been published that take into account thevariability of the streaming potential coupling coefficient asa function of grain size [17, 19–21], pore size [17], and porethroat size [17].

There are several theoretical models for the frequency-dependent streaming potential coupling coefficient. Themodels fall into three categories: (i) models based onlyon vibrational mechanics [22], (ii) models based on flowin capillary bundles [6], and (iii) those that have beendeveloped for porous media [23].

2.1. Vibrational Mechanics Models. If we apply the amplitudeof the critically damped second-order vibrational behaviour[22] to the frequency-dependent streaming potential cou-pling coefficient, we get

Cs(ω)Cso

=(

1 +(ω

ωt

)2)−1/2

, (2)

where Cs(ω) (in V/MPa) is the streaming potential couplingcoefficient at a frequency ω (in Hz), Cso (in V/MPa) is thesteady-state streaming potential coupling coefficient, and ωt

(in Hz) is the transition frequency. This model only exists inthe real domain and is a special case (for ξ = 1) of the general

second-order vibrational behaviour with variable dampingthat is given by

Cs(ω)Cso

=⎛⎝(

1−(ω

ωt

)2)2

+(

2ξ(ω

ωt

))2⎞⎠−1/2

, (3)

for the frequency-dependent streaming potential couplingcoefficient. In this equation ξ is the damping factor of thesystem. This equation provides the modulus or magnitudeof the vibration as is usually measured experimentally eitherby a peak-to-peak or an RMS measurement.

Both equations can be fitted to experimental data whereCs(ω)/Cso is plotted as a function of frequency. For (2) thetransition frequency is the only fitting variable, while (3)has two fitting variables: the transition frequency and thedamping factor.

These vibrational mechanics models are purely formaland contain no underlying physics. They are interesting inthat they can show that a system is behaving in a certainmanner, but no inference can be made, for example, aboutwhat controls the damping coefficient. This lack of specificityoften allows such models to apparently fit the data better thanother models which include more of the underlying physics.

2.2. Capillary Tube Models. The capillary tube model wasintroduced by Packard [6] together with a small number ofexperimental measurements. It is given by

Cs(ω)Cso

=(−2kr

J1(kr)J0(kr)

), (4)

where

k2 = −iωρ f

η f, (5)

where i = √−1, ρ f (kg/m3) is the density of the bulk fluid, ω(rad/s) is the angular frequency, r (m) is the radius of thecapillary in which the flow takes place, and J0 and J1 areBessel functions of the zeroth and first order, respectively.Note that k has dimensions of inverse length; hence, kris dimensionless, and the real part of the Bessel functionexpression�[(2J1(kr)/krJ0(kr))] varies between unity at lowfrequencies and zero at high frequencies, and its imaginarypart �[(2J1(kr)/krJ0(kr))] is zero at both low and highfrequencies, but attains a peak value at a frequency knownas the transition frequency.

Reppert et al. [9] have provided a simplification of (4):

Cs(ω)Cso

=(

1− 2r

√η f

ωρ f

{1√2− 1√

2i})

, (6)

(their Equations 26 and 38). However, a recent studyshowed that the simplification is incorrect [11]. The correctsimplification, which is consistent with the simplified modelsolution shown in their Figure 4 [9] (T. Ishido, pers. comm.,2011), is

Cs(ω)Cso

=⎛⎝1 +

[−2r

√η f

ωρ f

{1√2− 1√

2i}]−2

⎞⎠−1/2

. (7)

International Journal of Geophysics 3

When (4) or (7) is fitted to experimental data where theviscosity and density of the process fluid is known, the onlyfitting variable is the effective capillary radius r.

2.3. Porous Media Models. An extremely important studyby Pride [23] has provided a model of streaming potentialcoupling coefficient for porous media. It takes the form

Cs(ω) =(

εζ

η f σ

{1− 2

δ

Λ

})

×⎛⎝1− i

ω

ωt

m∗

4

{1− 2

δ

Λ

}2{

1− i3/2δ

√ωρf

η f

}2⎞⎠−1/2

,

(8)

where

ωt ≡ φ

τeκDC

η f

ρ f, (9)

and

m∗ ≡ φΛ2

τeκDC. (10)

Equation (8) can be rewritten in the same form as theprevious models as

Cs(ω)Cso

=⎛⎝1− i

ω

ωt

m∗

4

{1− 2

δ

Λ

}2{

1− i3/2δ

√ωρf

η f

}2⎞⎠−1/2

,

(11)

noting that the steady-state term in this model includes anadditional factor Cso = εζ/(η f σ){1− 2(δ/Λ)}.

In (8)–(11), τe (unitless) is the electrical tortuosityof the pore network, φ (unitless) is the porosity of thesample, κDC (m2) is the steady-state fluid permeability, andδ (m) is the Debye length. The parameter ωt (rad/s) is thetransition frequency, at which the quadrature component ofthe dispersive system is greatest. While this equation mayseem complex, it should be noted that for most geologicalmedia the pore fluid is sufficiently saline for the Debye lengthto be much smaller than the characteristic length scale (i.e.,δ� Λ), which allows significant simplifications to be made.

Recently, such a simplification of (11) has been proposed[19]:

Cs(ω)Cso

=(

1− im∗

4ω

ωt

)−1/2

, (12)

where

ωt = φ

τeκDC

η f

ρ f= 8

r2eff

η f

ρ f, (13)

where reff (m) is the effective pore radius of the rock.Equation (12) makes the assumption that δ � Λ, which isvalid for the majority of porous rocks that are saturated with

saline fluids with a concentration of 10−3 mol/dm3 or more.If we take m∗ = 8/3 as suggested by [19], we get

Cs(ω)Cso

=(

1− i23ω

ωt

)−1/2

, (14)

which is dependent solely on the transition frequency.If either the full Pride model (11) or its simplifications

(12) and (14) are fitted to experimental data where theviscosity and density of the pore fluid and the microstruc-tural parameters (δ,Λ,φ, τe, κDC) of the porous medium areknown, the only fitting variable is the transition frequency.

2.4. Physical Interpretation of Theoretical Models. Untilrecently only the Packard model [6] and its simplification[9] had been tested against data from a few measurementson capillary tubes and filter material [9, 10]. Recently [11]published some data for Ottawa sand which suggested thatthe vibrational mechanics models [22] fitted their data best.They reported that the best fit of all of the models wasa damped second-order vibration mechanics model (3).However, there were errors in their data processing. A correctanalysis of their data shows that their data follow the Pridemodel best, which agrees very well with the results of thisstudy.

Most of the theoretical models have a real and imaginarypart. In this paper we have analysed these two contributionsseparately, comparing the measured data with the overallmagnitude and each of the complex components of eachmodel. We have taken an RMS measurement approach whichprovides the magnitude of the variation and maximises theprecision with which it can be measured.

It is interesting to consider the physical meaning ofthe real and imaginary contributions to the streamingpotential coupling coefficient. Currently we do not havesufficient information to answer this question with author-ity. However, there are some indications. The streamingpotential coupling coefficient is defined as the ratio of thestreaming potential to the pressure drop across the sample.However, it is the fluid velocity that separates the charge andcauses the streaming potential. This implies that the fre-quency dependence of the streaming potential coupling coef-ficient depends on the frequency dependence of the dynamicfluid permeability. The dynamic fluid permeability at lowfrequencies is controlled by viscous flow that is representedby the real part of the dynamic permeability. However, whena critical frequency is reached, the inertial acceleration of thefluid begins to control the flow (e.g., [9, 24]). The inertialacceleration is represented by the imaginary part of thedynamic permeability. Hence, we might expect the real andimaginary parts of the streaming potential coupling coeffi-cient to be influenced by the same transition from viscous-dominated to inertial-dominated fluid flow. In this scenario,the transition frequency is the same as the critical frequencyat which viscous-dominated fluid flow becomes inertiallydominated. It is becoming clear, however, that the frequencydependence of fluid flow and of the streaming potentialcoupling coefficient are subtly different. The normalised


dynamic permeability of a bundle of capillary tubes approx-imately follows a Debye model for all frequencies, while thenormalised streaming potential coupling coefficient followsthe same Debye model up to the transition frequency thendeviates from it considerably [25].

3. Experimental Tests

3.1. Sample Material and Basic Characteristics. Experimentaltests were carried out on samples of Ottawa sand and onpacks of glass beads using the apparatus described in theas- sociated paper [1]. Their main properties are shown inTable 1.

Ottawa Sand. Ottawa sand was obtained from Fisher Sci-entific and washed repeatedly in distilled water in order toremove any rock powder before being dried in a vacuumoven prior to use. The pore throat and grain size distribu-tions of the sample material were measured using mercuryinjection porosimetry and are shown as Figure 1 in [11]together with the grain size distribution obtained by laserdiffraction measurements using a Malvern Mastersizer 2000.The porosity of sand samples was also measured using ahelium pycnometer. The complex electrical properties of asaturated sample of the sand were measured at 60 frequen-cies between 1 Hz and 1 MHz using a Solartron 1260A Im-pedance Analyzer. The absolute value of the complex electri-cal conductivity at the frequency for which the out-of-phaseconductivity was minimum has been used together with themodal grain size from the laser diffraction measurementsand the helium porosity in order to calculate the mean poresize of the sand samples using the method of [12].

The quasi-steady-state permeability was obtained bycalculating the volume of fluid flowing through the sampleper second at 10 Hz using the measured piston displacementand also measuring the pressure required to move this fluid.The permeability at 10 Hz was κ10 = 1.19× 10−10 m2. We havetaken this value to represent the steady-state permeability inthe absence of steady-state permeability on the sample.

The measurements shown in this paper are the sameas those reported in [11]. However, it should be notedthat there were major errors in the analysis of the data in[11] which are corrected in this work. Consequently, thiswork represents the correct treatment of the Ottawa sanddata, while the previous paper should be referred to for adetailed petrophysical description of the material as well asmeasurements of the petrophysical properties of the Ottawasand that have been made by other authors.

Glass Beads. Three sizes of soda lime glass bead (nominally0.5, 1, and 2 mm in diameter) were obtained from EndecottsLtd.. Samples of the beads were washed repeatedly in distilledwater before being dried in a vacuum oven prior to use.No mercury porosimetry was carried out on the beadsbecause they are too expensive to be disposed of after onlyone use. The grain size distribution was obtained by laserdiffraction measurements and using the detailed calibrationinformation provided by Endecotts Ltd., which is based on

a sieve analysis. The porosity of the glass bead samples wasmeasured using a helium pycnometer. The complex electricalproperties of a saturated sample of the sand were measuredat 60 frequencies between 1 Hz and 1 MHz using a Solartron1260A Impedance Analyzer, and the results were used tocalculate the mean pore size of each bead pack using theGlover and Walker method [12] in the same way as for theOttawa sand.

The steady-state permeability of each bead pack wasmeasured using a gravitational pressure head. The results areshown in Table 1. These values are in very good agreementwith the value predicted from the grain size and electricalmeasurements using the RGPZ method [26] (their Equation(10)).

Fluids. The fluid used in the experiments was 10−3 mol/LNaCl with a measured density of 997 kg/m3, which agreeswell with the equation of state of NaCl solutions (e.g., [27])and a viscosity of 8.94 × 10−4 Pa.s calculated using themodel of [28]. The electrical conductivity of the fluid wasmeasured at σ f = [1.23 ± 0.05]×10−2 S/m at 25◦C beforeuse, which is in good agreement with the model of Sen andGoode [29] at [1.23± 0.05]×10−2 S/m. For the Ottawa sand,the conductivity of the fluid emerging from the apparatusduring the measurement was extremely close to the originalconductivity of the fluid ([1.21±0.05]×10−2 S/m at 25◦C). Inthe case of the glass beads, the conductivity of the fluid slowlyincreased to 1.48 × 10−2 S/m, 1.38 × 10−2 S/m, and 1.33 ×10−2 S/m for 0.5, 1, and 2 mm diameter beads, respectively,while being circulated through the sample for 24 hoursbefore the electrical and electrokinetic measurements weremade. This amounts to an increase of concentration from1 × 10−3 mol/L to 1.23 × 10−3 mol/L, 1.15 × 10−3 mol/L,and 1.10 × 10−3 mol/L, respectively, which we associate withdissolution of the grains during the attainment of physico-chemical equilibrium between the grains and the fluid. Forthe Ottawa sand experiment, the pH of the fluid during themeasurement was pH 6. In the case of the glass beads, theinitial pH of the fluid was pH 6.9, which reduced during therecirculation of the fluids. The pH of the fluid was measuredon samples of fluid emerging from the apparatus duringthe electrokinetic experiment. The stable values were pH6.4, 6.6, and 6.7 for the 0.5, 1, and 2 mm diameter beads,respectively. We note again that the changes in the fluid con-ductivity and pH are not as great as some authors have expe-rienced (e.g., Leroy et al. [30]), and we associate this withthe repeated washings that we subjected the material toinitially. However, we note later that modelling of the steady-state streaming potential coupling coefficient is extremelysensitive to the fluid conductivity and the pH. Hence it isextremely important for these parameters to be measured onthe fluid emerging from the experimental apparatus duringthe electrokinetic measurement.

3.2. Experimental Methodology. The cell was loaded witheither a sample of sand or beads in layers of 1 cm with lighttamping between the layers in the case of the sand, andwith agitation after each layer in the case of the glass beads.


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4

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16

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2

4

6

8

10

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id p

ress

ure

dif

fere

nce

(kP

a)

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amin

g po

ten

tial

(m

V)

Time (s)

Streaming potentialFluid pressure differencePiston displacement

−10

−8

−6

−2

−4

20 Hz

0.5 0.55 0.6 0.65−20

−16

−12

−8

−4

(a)

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10 100

Piston displacement, peak to peak (mm)

Amplitude decreasingAmplitude increasing

-Cs

(V/M

Pa)

(b)

Figure 1: (a) The measured waveforms (streaming potential, fluid pressure and piston position (LVDT)) at 20 Hz. (b) The calculatedstreaming potential coupling coefficient as a function of piston amplitude at 10 Hz for Ottawa sand saturated with 10−3 mol/L NaCl solutionat 24◦C (φ = 0.325, κ10 = 1.19 × 10−10 m2). The errors in the frequency were calculated from the analysis of a train of approximately 500cycles, while the errors in the streaming potential coupling coefficient were calculated from the errors in the RMS streaming potential andthe measured RMS pressure difference (500 cycles).

The system was fully saturated with the process fluid, usingback-pressure where necessary to remove all air bubbles.Once saturated, the steady-state permeability of the glassbeads was measured using gravity-driven flow. The processfluid was then recycled through the sample for 24 hours toensure full physicochemical equilibrium. During this timethe permeability of the Ottawa sand was measured at a fre-quency of 10 Hz using the pressure transducers and calculat-ing the flow by measuring the piston displacement with theLVDT.

Figure 1 shows the typical measured waveforms for 20Hz, noting that there is a tendency that higher frequenciesprovide better-quality waveforms. High-quality measure-ments were possible between 5 and 200 Hz for the Ottawasand and between 10 Hz and 500 Hz for the glass bead packs.Both the streaming potential and the dynamic pressure canbe measured with acceptable levels of noise. Figure 1(a)shows that the fluid pressure and piston displacement are inphase with each other and in antiphase with the streamingpotential, as theory requires. The compressive part of eachcycle is sinusoidal as expected, whereas the backstroke isslightly distorted due to the inflow for new fluid throughthe check valve. This slight asymmetry might be correctedusing digital filtering of the measured data or by imposinga background DC fluid flow and a back-pressure instead ofusing check valves. This latter approach would also removeany tendency for the fluid to cavitate.

In the case of Ottawa sand, tests were made up to 600 Hz,when the sample tube failed. It was observed that the sealbetween the piston and the tube let in air at frequencieshigher than 200 Hz. Although the data for frequencies greaterthan 200 Hz seem to behave well, we have not reported thembecause the presence of air bubbles may make the meas-urements unreliable. We corrected the air leakage for the

glass bead pack measurements simply by lubricating the pis-ton seal.

This paper contains results for three diameters of glassbead (0.5, 1, and 2 mm). We also attempted to make mea-surements on glass bead packs with a 0.25 mm and 3.35 mmnominal diameter. Unfortunately we could not generate suf-ficient pressure to produce a streaming potential of sufficientsize to measure the 3.35 mm beads with accuracy, and theexperiment with the 0.25 mm beads did not provide data ofsufficient quality to report.

The frequency-dependent streaming potential coeffi-cients were calculated using the methods described inReppert et al. [9] and Reppert and Morgan [5], the mostimportant step being the renormalisation of the data takinginto account of the frequency-dependent impedance of thesample and measuring circuit, which is shown for eachsample at each of the measurement frequencies in Figure 2.This procedure ensures that the streaming potential coef-ficient is calculated with the correct sample conductivity,that is, that which relates to the frequency of the data.Without such a step, the data seem to fit better the vibrationalmechanics models, as was erroneously reported by us in[11]. Inclusion of the correction results in the data fittingthe Pride model [23] better than the other models. This is asatisfying result as the Pride model was specifically conceivedfor porous media.

3.3. Displacement Tests. One of the characteristics of anelectromagnetic shaker is that the piston amplitude decreaseswith frequency for any set driving current [1]. However, thedifferential fluid pressure generated by the piston increaseswith frequency [1]. If the driving current is kept constantthroughout a suite of tests at different frequencies, thecombination of these two effects is to generate smaller


0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000

Nor

mal

ized

ele

ctri

cal i

mpe

dan

ce (

V/M

Pa)

Frequency (Hz)

(a) Ottawa sand

01 10 100 1000

Nor

mal

ized

ele

ctri

cal i

mpe

dan

ce (

V/M

Pa)

Frequency (Hz)

0.2

0.4

0.6

0.8

1

1.2

(b) Glass beads, d = 0.5 mm

1 10 100 1000

Nor

mal

ized

ele

ctri

cal i

mpe

dan

ce (

V/M

Pa)

Frequency (Hz)

Measurements

0

0.2

0.4

0.6

0.8

1

1.2

(c) Glass beads, d = 1 mm

1 10 100 1000

Nor

mal

ized

ele

ctri

cal i

mpe

dan

ce (

V/M

Pa)

Frequency (Hz)

Measurements

0

0.2

0.4

0.6

0.8

1

1.2

(d) Glass beads, d = 2 mm

Figure 2: Normalised electrical impedance data for (a) Ottawa sand, and glass beads with (b) d = 0.5 mm, (c) d = 1 mm, and (d) d = 2 mm.

differential fluid pressures at high frequencies, lowering thesignal-to-noise ratio. One solution to this problem is to varythe drive current in order to generate a differential fluidpressure that can be measured with good signal-to-noiseratio and to keep that differential pressure constant for asgreat a range of frequencies as possible. Such a procedureimplies that the piston amplitude will be different for eachfrequency. Current understanding of frequency-dependentstreaming potentials does not indicate that there is a pistonamplitude below which the coupling is not fully developedor any other reason why the streaming potential couplingcoefficient should vary with piston amplitude. However, itis important experimentally to know if we will introduce anysystematic errors by allowing the piston amplitude to varyand also to know if there is a piston amplitude below whichmeasurements become unreliable due to the signal-to-noiseratio.

We have carried out tests to examine the measuredstreaming potential as a function of the piston amplitudeusing a sample of the Ottawa sand and an arbitrary frequencyof 10 Hz. Dynamic fluid pressure, dynamic streaming poten-tial, and instantaneous piston position measurements weremade while decreasing the shaker driving current in incre-ments (and hence the piston amplitude) until the measuredvalues were below the noise threshold. Measurements werethen made while incrementally increasing the driving currentuntil the maximum displacement was reached.

The results are shown in Figure 1(b). The initial peak-to-peak displacement was just over 20 mm, and this decreasedincrementally until it was 0.5 mm, then increased againto a maximum about 21.7 mm. In the range about 2 mmto 21.7 mm, the measured streaming potential and thecalculated streaming potential coupling coefficient remainedstable with Cs = 0.5374 ± 0.0029 V/MPa, which represents a


1 10 100 1000

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)

Critically damped second order [22]Second order, damping = 1.5 [22]Pride [23] and Walker and Glover [19]Packard [6] and Reppert et al. [9]Measurements

MagnitudeOttawa sand

0

0.2

0.4

0.6

0.8

1

1.2

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)

Pride [23] and Walker and Glover [19]Packard [6] and Reppert et al. [9]Real part of measurements

RealOttawa sand

(b)

1 10 100 1000

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)

Pride [23] and Walker and Glover [19]Packard [6] and Reppert et al. [9]Imaginary part of measurements

ImaginaryOttawa sand

0

0.1

0.2

0.3

0.4

0.5

0.6

(c)

Figure 3: The calculated streaming potential coupling coefficient (normalised to the value at 5 Hz, which was 0.518 V/MPa) as a functionof frequency for Ottawa sand saturated with 10−3 mol/L NaCl solution at 24◦C (φ = 0.325, κ10 = 1.19 × 10−10 m2). (a) Magnitude withsix models shown fitting the data [6, 9, 19, 22, 23], (b) real component for models [6, 9, 19, 23], and (c) imaginary component for models[6, 9, 19, 23].

variability of about 0.5%. The values during reduction andaugmentation of piston displacement were Cs = 0.5369 ±0.0026 V/MPa and 0.5379 ± 0.0032 V/MPa, respectively, sothere is no sensitivity to whether the piston amplitude isincreasing or decreasing. We can remark, therefore, that inthe range 2 mm to 21.7 mm the streaming potential and thecalculated streaming potential coupling coefficient measuredby this instrument are independent of piston amplitudeand direction of piston amplitude change. Hence the piston

amplitude may be varied to optimize the measurementconditions.

The measured streaming potential was affected at pistonamplitude less than about 2 mm. Under these conditionsthe measured pressure difference is very small for our highpermeability sample, and it is difficult to distinguish the mea-surements from the background noise. We believe thatthe observed increase in the streaming potential couplingcoefficient for displacements less than 2 mm is due to the


1 10 100 1000

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)

Critically damped second order [22]Second order, damping = 0.8555Pride [23] and Walker and Glover [19]Packard [6] and Reppert et al. [9]Measurements

Magnitude

0

0.2

0.4

0.6

0.8

1

1.2

Glass beads, d = 0.5 mm

(a)

Frequency (Hz)


Real

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000


Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

(b)

0

0.2

0.3

0.1

0.4

0.5

0.6

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)


Imaginary

1 10 100 1000


(c)

Figure 4: The calculated streaming potential coupling coefficient (normalised to the value at 2 Hz, which was 1.37 V/MPa) as a functionof frequency for a pack of nominally 0.5 mm diameter glass beads saturated with 10−3 mol/L NaCl solution at 24◦C (φ = 0.383, κDC =1.62× 10−10 m2). (a) Magnitude with six models shown fitting the data [6, 9, 19, 22, 23], (b) real component for models [6, 9, 19, 23], and(c) imaginary component for models [6, 9, 19, 23].

difficulty in measuring these small pressures. There was noevidence for turbulent fluid flow at large piston amplitudes.

3.4. Initial Frequency-Dependent Streaming Potential Cou-pling Coefficient Results. Figures 3, 4, 5 and 6 show the nor-malised measured streaming potential coupling coefficientfor Ottawa sand and the three grades of glass bead as a func-tion of frequency. The normalised streaming potential coup-

ling coefficient was calculated by dividing the measuredstreaming potential coupling coefficient at a given frequencyby that measured at the lowest frequency available (5 Hz forOttawa sand and 2 Hz for the glass beads). If one can assumethat the streaming potential coupling coefficient at this lowfrequency approximates to that during steady-state flow, it ispossible to say that the data shown in Figures 3–6 representthe right-hand side of (2), (3), (4), (11) and (14), that is, thefrequency-dependent terms.


Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)


Magnitude

1 10 100 10000

0.2

0.4

0.6

0.8

1

1.2

Glass beads, d = 1 mm

(a)

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)


Real

1 10 100 10000

0.2

0.4

0.6

0.8

1

1.2


(b)

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)


Imaginary

0

0.1

0.2

0.3

0.4

0.5

0.6

1 10 100 1000


(c)

Figure 5: The calculated streaming potential coupling coefficient (normalised to the value at 2 Hz, which was 1.61 V/MPa) as a function offrequency for a pack of nominally 1 mm diameter glass beads saturated with 10−3 mol/L NaCl solution at 24◦C (φ = 0.38, κDC = 5.95 ×10−10 m2). (a) Magnitude with six models shown fitting the data [6, 9, 19, 22, 23], (b) real component for models [6, 9, 19, 23], and (c)imaginary component for models [6, 9, 19, 23].

It should be noted in these figures that the error barsbecome larger at the higher frequencies. This is due to thedifficulty in measuring small streaming potentials at frequen-cies greater than the transition frequency.

4. Analysis and Modelling ofthe Experimental Results

4.1. Steady-State Streaming Potential Coupling Coefficient.The majority of the data analysis will concentrate on the fre-

quency-dependent part of the streaming potential couplingcoefficient. However, we should say a few words aboutthe steady state streaming potential coupling coefficient.Although this was not measured in our apparatus, we canperhaps use the streaming potential coupling coefficientat the lowest frequency as a reasonable indication of thatunder true steady-state conditions considering that Figures3–6 show that these values tend towards the steadystatevalue. The lowest frequency for the Ottawa sand was 5 Hzwhile those for the glass beads was 2 Hz. We will call the


Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)


Magnitude

1 10 100 10000

0.2

0.4

0.6

0.8

1

1.2


(a)

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)

Real part of measurements

Real

1 10 100 10000

0.2

0.4

0.6

0.8

1

1.2

Pride [23] and Walker and Glover [19]Packard [6] and Reppert et al. [9]


(b)

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)

Imaginary part of measurements

Imaginary

0

0.2

0.3

0.1

0.4

0.5

0.6

1 10 100 1000

Pride [23] and Walker and Glover [19]Packard [6] and Reppert et al. [9]


(c)

Figure 6: The calculated streaming potential coupling coefficient (normalised to the value at 2 Hz, which was 1.80 V/MPa) as a functionof frequency for a pack of nominally 2 mm diameter glass beads saturated with 10−3 mol/L NaCl solution at 24◦C (φ = 0.382, κDC =27.2× 10−10 m2). (a) Magnitude with six models shown fitting the data [6, 9, 19, 22, 23], (b) real component for models [6, 9, 19, 23], and(c) imaginary component for models [6, 9, 19, 23].

streaming potential coupling coefficients at these frequenciesthe quasi-steady-state values. The quasi-steady-state values ofthe streaming potential coupling coefficient for each of thesamples are shown in Table 2. This table also shows someelectrokinetic modelling that we have carried out using thesame approach as Glover and Dery [17] and Glover et al.[16]. In this modelling we kept the following parametersconstant: Γo = 5 sites/nm2, pKme = 7.5, pK− = 8, whilethe formation factor, porosity, cementation exponent, grain

diameter fluid concentration, and pH were set to the valuesrelated to each sample (Table 1). It can be seen from Table 2that the modelled values of the steady-state streaming poten-tial coupling coefficient slightly overestimate the measuredvalues. We need to put the measured and modelled valuesin the context of the measurements made by others and theexperimental errors. Figure 7 shows the measured values andthe model curves in the context of a database of other steady-state streaming potential coupling coefficient measurements


-Str

eam

ing

pote

nti

al c

oupl

ing

coef

fici

ent

(V/P

a)

Pore fluid concentration (mol/L)

1.E−11

1.E−10

1.E−09

1.E−05 1.E−04 1.E−03 1.E−02

1.E−08

1.E−07

1.E−06

1.E−05

1.E−04

1.E−03

1.E−02

1.E+00 1.E+011.E−01

(a)

-Str

eam

ing

pote

nti

al c

oeffi

cien

t (V

/Pa)

Pore fluid concentration (mol/L)

1.E−05

1.E−07

1.E−06

1.E−04 1.E−03 1.E−02

database by Jaafar [31-32]Jaafar [31-32]Model Ottawa sand, pH 6

Measurement, Ottawa sand, pH 6

Model glass beads, d = 0.5 mm, pH 6.4Model glass beads, d = 1 mm, pH 6.6Model glass beads, d = 2 mm, pH 6.7

Measurement, glass beads, d = 0.5 mm, pH 6.4Measurement, glass beads, d = 1 mm, pH 6.6Measurement, glass beads, d = 1 mm, pH 6.7

(b)

Figure 7: (a) The steady-state streaming potential coupling coef-ficient measured in this work shown with a compilation of silica-based earth materials measured by (open symbols) or compiled by(solid symbols) Jaafar [31, 32]. The lines represent the theoreticallymodelled streaming potential coupling coefficient using the methodof Glover et al. [16] and Glover and Dery [17] and with theparameters Γo = 5 sites/nm2, pKme = 7.5, pK− = 8, andT = 24.0◦C, with the porosity, cementation exponent, grain size,and pH for the individual samples (given in Table 1). (b) The samediagram as (a) on an expanded scale.

(from [31, 32]). It should be noted that the experimentalvalues are plotted here at the fluid concentrations calculatedfrom the fluid electrical conductivity that was measuredon the fluid leaving the apparatus during the experimentrather than that of the original stock fluid, which was0.001 mol/L. The error bars are approximate and represent aconservative assessment of the measurement errors (30% forthe streaming potential coupling coefficient and 10% for thepore fluid concentration). It can be seen that the measuredvalues are in fact in extremely good agreement with theexisting data and the model. The model is extremely sensitiveto the values of fluid concentration and pH used, and henceit is extremely important to have an accurate measurementof the pH value and the electrical conductivity of the fluidthat emerges from the experimental apparatus during theexperiment in order that the measurements can be comparedwith the model at an acceptable level of precision.

4.2. Frequency-Dependent Streaming Potential Coupling Coef-ficient. Figures 3–6 show the experimental data togetherwith theoretical curves that (i) are based on standard equa-tions in vibration mechanics, for example, [22], (ii) havebeen developed for capillary tubes [6, 9], and (iii) have beenderived for porous media [19, 23]. In each figure there arethree parts. The first shows the behaviour of the magnitude(absolute value, modulus) of the complex variable, while theother two parts show the real and imaginary componentsof the complex variable. The vibrational mechanics models[22], the Pride model [23], and its simplification [19] mayall be fitted to the data to obtain the transition frequencyωt, from which the characteristic pore radius of the sandcan be calculated using (13), while the Packard model [6]and its simplification [9] may be used to obtain the effectivecapillary radius of the sand directly. A single curve is includedfor the Packard capillary tube model [6] and the Reppert etal. simplification [9] because they are indistinguishable at thescale of the figures and which indicates that the correctedReppert et al. simplification (i.e., (8)) performs extremelywell. The Pride model [23] and its simplification by Walkerand Glover [19] are also represented by a single curve for thesame reason.

Figures 3–6 show that all of the models describe thedata fairly well. However, the following discussion showsthat some of these fits do not use parameters that areconsistent with other physical properties of the samples. Thebest fit for all the samples is provided by the Pride model[23] and its Walker and Glover simplification [19]. Thesemodels are specifically designed for porous media but donot implicitly take into account the surface conduction thatoccurs naturally in geological porous media. In fitting thismodel, we used the independently measured quasi-steady-state permeability kDC , the electrical tortuosity τe that wascalculated from the electrical impedance measurements, andthe porosity by helium pycnometry φ, all of which aregiven in Table 1. The fitting variables were the transitionfrequency and the characteristic length scale of the pore spaceΛ. The values of the fitting variables are given in Table 2.The transition frequency can be used to derive an effective

International Journal of Geophysics 13T

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the

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pore size for the sample reff using (13), which is also givenin Table 2. When this is done, it is clear that both Λ andreff agree well with the independently obtained pore size ofthe sample rp (Tables 1 and 2). We conclude that the Pridemodel and its simplification perform extremely well whencompared to experimental data.

While not as effective as the Pride model and itssimplification, the Packard model [6] and its Reppert et al.simplification [9] also provide a fairly good fit to the data.However, the discrepancies between these models and thedata clearly show that a model which is based on a bundleof capillary tubes is not as effective as the Pride model indescribing a porous medium when the fluid flow and electri-cal flow have a tortuosity which is significantly different fromunity such as in our samples where the electrical tortuosity isapproximately 1.5 (see Table 1). Many rocks have tortuositiesmuch higher than this. Hence, one would expect the Packardmodel and Reppert et al. models to perform worse for theserocks, overestimating the effective capillary radius and hencethe predicted permeability of the sample. The advantage withthese models is that when used with a single pore fluid at aconstant temperature and pressure, they use a single variable:the effective capillary radius, which is shown in Table 2 forcomparison with the independently measured value rp.

The critically damped second-order vibrational mechan-ics model also provides a reasonable fit to the data, givingtransition frequencies and effective pore radii that areconsistent with the independently obtained measurements(Table 2). However, the variably damped second-order vibra-tional mechanics model has difficulty fitting the data. Herethere are two fitting parameters, the transition frequency andthe damping coefficient. A large number of different combi-nations of these parameters provide curves that seem to fitthe experimental data approximately. Figures 3–6 show oneparticular combination where the damping coefficient ξ =1.5 and the transition frequencies are given in Table 2. Thetransition frequencies are clearly much too large and predicteffective pore radii that are badly underestimated. Increasingthe damping coefficient further allows the frequency roll-off to approach the experimental data, but only at theexpense of even higher predicted transition frequencies. Weconclude, therefore, that this model is of no practical usewhen describing frequency-dependent streaming potentialcoefficients of porous media.

4.3. Sensitivity of the Pride Model to Steady-State Permeability.The full Pride model calculates the transition frequency fromthe sample porosity, electrical tortuosity, and permeabilityas well as the density and viscosity of the pore fluid (9).It also calculates a parameter which we have called m∗

(10) from the porosity, characteristic length scale of thepores Λ, the electrical tortuosity, and the permeability ofthe sample. It became clear in our modelling that the fit tothe experimental data depends strongly on the value of m∗,which in turn depends upon the steady-state permeabilityof the rock kDC , its electrical tortuosity τe, and its porosityφ as well as the characteristic length scale of its pores Λ.While τe, φ, and Λ vary from sample to sample, they can

Nor

mal

ized

str

eam

ing

pote

nti

al c

oeffi

cien

t

Frequency (Hz)

Measurements

k = 2.72e − 9 m2

k = 1e − 9 m2

k = 2e − 9 m2

k = 3e − 9 m2

k = 4e − 9 m2

k = 5e − 9 m2

1 10 100 10000

0.2

0.4

0.6

0.8

1

1.2Glass beads, d = 2 mm

Figure 8: The calculated streaming potential coupling coefficient(normalised to the value at 2 Hz, which was 1.80 V/MPa) as afunction of frequency using the Pride model [23] for a pack ofnominally 2 mm diameter glass beads saturated with 10−3 mol/LNaCl solution at 24◦C (φ = 0.382, κDC = 27.2 × 10−10 m2). Eachsolid curve shows the results of the model for a different value ofpermeability. The dashed line is for the permeability of the samplethat was measured independently.

be measured independently. This is also true of the steady-state permeability of the rock. However, the steady-statepermeability of the rock can vary over many orders ofmagnitude which makes the Pride model extremely sensitiveto this parameter. Figure 8 shows the Pride model for a rangeof different permeabilities from 1 × 10−9 m2 to 5 × 10−9 m2

and the other parameters matching those for our 2 mmglass bead dataset. It is worthwhile noting that this is avery small permeability range considering (i) the precisionthat is usual when measuring permeability in the laboratoryand (ii) the natural variability of permeability from sampleto sample even in isotropic, homogeneous clean reservoirrocks. While this sensitivity might be considered to be aproblem when forward modelling, it is a huge advantagewhen backward modelling as it should allow the predictedpermeability to have a very high precision. Hence, althoughthese measurements are currently difficult to make, they havethe potential of providing an extremely precise method ofobtaining the permeability of porous media.

4.4. Transition Frequencies and Pore Sizes. The transitionfrequencies and capillary radii calculated from each of the 5models are given in Table 2. All of the transition frequencies,except those from the second order vibrational mechanicsmodel with variable damping, are in broad agreement andvary according to the pore size of the sample. Figure 9 showsall the existing data as compiled by Tardif et al. [11] togetherwith the Ottawa sand data of [11] and the new glass beaddata presented in this paper. The dashed lines represent the


1

10

100

1000

10000

Tran

siti

on f

requ

ency

(H

z)

1/(effective pore radius)2, (m−2)

1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10

0◦C 25◦C50◦C100◦C

(a)

10

100

1000

Tran

siti

on f

requ

ency

(H

z)

1/(effective pore radius)2, (m−2)

0◦C 25◦C

50◦C

100◦C

1.E+091.E+081.E+07

(b)

Figure 9: The electrokinetic transition frequency as a function of the inverse square characteristic pore size (a) in full and (b) at expandedscale. Black symbols, previous data for capillary tubes, filters frits, and rocks, Figure 6 of [19]; grey symbols, Ottawa sand measured at24.0◦C ± 0.8◦C; open symbols, glass bead packs. The dashed lines represent the theoretical result [19] at four different temperatures.

result of (13) [19] using the fluid viscosities and densities forthe pore fluid at various different temperatures. It is clear thatthe new data are compatible with the independently obtainedeffective pore radius measurements.

The transition frequency can be used to predict theeffective pore radius of each sample using (13). Table 2 showsthat all the predictions are in good agreement with the valueof effective pore radius derived from the measured graindiameter using the method of Glover and Walker [12] andwith the equivalent capillary radius derived from the Packardmodel [6].

The transition frequency can be used to predict thesteady-state permeability of the sample using (13). Table 2shows that all the predictions are in good agreement withthe measured values of permeability and the permeabilitypredicted using the RGPZ method [26]. It should be notedthat the predictions for the porous media models [19, 23]are exact because the permeability is an input to thesemodels. Hence, these models can only be used to predictpermeability if they are fitted to the experimental data withthe permeability as a variable parameter.

5. Conclusions

We have used the electromagnetic drive approach to create anexperimental apparatus to measure the dynamic streamingpotential coupling coefficient of disaggregated porous mediabetween 1 Hz and 1 kHz. The apparatus has been used tomeasure samples of Ottawa sand and glass bead packs. Meas-urements were made on Ottawa sand between 5 Hz and200 Hz, and on glass bead packs between 2 Hz and 500 Hz. Inmost cases the full variation either side of the transition fre-quency was captured. Measurements were possible up to

1 kHz, but in practice the streaming potential values becameso small at high frequencies that they were unreliable; onlythose with reasonably small errors have been included in thispaper.

Analysis of the steady-state part of the measured data,shows that the measured steady-state streaming potentialcoupling coefficient is compatible with the latest theoreticalmodels of electrokinetics.

The dynamic experimental data, in the form of nor-malised streaming potential coupling coefficient, have beenfitted with five theoretical models that were derived (i) fromvibrational mechanics theory, (ii) for bundles of capillarytubes, and (iii) for porous media. The Pride model andits simplification, which were developed for porous media,fitted the data best and provided transition frequencies,characteristic length scales, and effective pore radii thatwere consistent with independently measured values for thesamples. The Packard model and its simplification, whichwere developed for capillary tubes, also performed well.The second order vibrational mechanics model with variabledamping only fitted the data when unreasonable transitionfrequencies were used, but the critically damped secondorder vibrational model performed reasonably well.

We found that the Pride model and its simplificationmodels are extremely sensitive to the steady-state permeabil-ity which may cause difficulties in forward modelling giventhat this parameter is rarely known precisely and that there isoften a large range of permeabilities even in isotropic, homo-geneous, clean reservoir rock. However, the sensitivity is anadvantage in reverse modelling as it should allow precisepermeability determinations to be made by fitting this modelto experimental dynamic streaming potential coupling coef-ficient data.


Acknowledgments

This work has been made possible thanks to funding bythe Natural Sciences and Engineering Research Council ofCanada (NSERC) Discovery Grant Programme. The authorswould also like to thank Guillaume Lalande and the membersof the mechanical engineering workshop for their help.

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