# Frequency-Dependent Streaming Potentials: A Review

Authors: L. Jouniaux; C. Bordes
Journal: International Journal of Geophysics
 (2012)

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## Abstract

The interpretation of seismoelectric observations involves the dynamic electrokinetic coupling, which is related to the streaming potential coefficient. We describe the different models of the frequency-dependent streaming potential, mainly Packard's and Pride's model. We compare the transition frequency separating low-frequency viscous flow and high-frequency inertial flow, for dynamic permeability and dynamic streaming potential. We show that the transition frequency, on a various collection of samples for
which both formation factor and permeability are measured, is predicted to depend on the permeability as inversely proportional to the permeability. We review the experimental setups built to be able to perform dynamic measurements. And we present some measurements and calculations of the dynamic streaming potential.


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## Body

## 1. Introduction

Electrokinetics arise from the interaction between the rock matrix and the pore water. Therefore electrokinetic phenomena are often observed in aquifers, volcanoes, and hydrocarbon or hydrothermal reservoirs. Observations show that seismoelectromagnetic signals associated to earthquakes can be induced by electromagnetic induction [1, 2] or by electrokinetic effect [3, 4]. The electrokinetic phenomena are due to pore pressure gradients leading to fluid flow in the porous media or fractures and inducing electrical fields. These electrokinetic effects are associated to the electrical double layer which was originally described by Stern. The electrokinetic signals can be induced by global displacements of the reservoir fluids (streaming potential) or by the propagation of seismic waves (seismoelectromagnetic effect). As soon as these pressure gradients have a transient signature, the dynamic part of the electrokinetic coupling has to be taken into account by introducing the dependence on fluid transport properties.It is generally admitted that two kinds of seismoelectromagnetic effects can be observed. The dominant contribution, commonly called “coseismic”, is generated close to the receivers during the passage of seismic waves. The second kind, so-called “interfacial conversion” [5], is very similar to dipole radiation and is generated at physicochemical interfaces due to strong electrokinetic coupling discontinuities. This interface conversion is often perceived to have the potential to detect fine fluids transitions with higher resolution than seismic investigations, but in practice, signals are often masked by electromagnetic disturbances, especially when generated at great depth.Nevertheless recent field studies have focused on the seismoelectric conversions linked to electrokinetics in order to investigate oil and gas reservoirs [6] or hydraulic reservoirs [5, 7–13]. It has been shown using these investigations that not only the depth of the reservoir can be deduced, but also the geometry of the reservoir can be imaged using the amplitudes of the electroseismic signals [14]. Moreover fractured zones can be detected and permeability can be measured using seismoelectrics in borehole [15–18]. This method is especially appealing to hydrogeophysics for the detection of subsurface interfaces induced by contrasts in permeability, in porosity, or in electrical properties (salinity and water content) [19–21].The analytical interpretation of the seismoelectromagnetic phenomenon has been described by Pride [22], by connecting the theory of Biot [23] for the seismic wave propagation in a two-phase medium with Maxwell’s equations, using dynamic electrokinetic couplings. The seismoelectromagnetic conversions have been modeled in homogeneous or layered saturated media [12, 21, 24–26] with applications to reservoir geophysics [27].Theoretical developments showed that the electrical field induced by theP-waves propagation is related to the acceleration [12]. The electrokinetic coupling is created at the interface between grains and water, when there is a relative motion of electrolyte ions with respect to the mineral surface. Thus, seismic wave propagation in fluid-filled porous media generates conversions from seismic to electromagnetic energy which can be observed at the macroscopic scale, due to this electrokinetic coupling at the pore scale. The seismoelectric coupling is directly dependent on the fluid conductivity, the fluid density, and the electric double layer (the electrical interface between the grains and the water) (see [28], in this special issue “Electrokinetics in Earth Sciences” for more details). For more details on the surface complexation reactions see Davis et al. [29] or Guichet et al. [30]. It can be accurately quantified in the broad band by a dynamic coupling [22] which can be linked in the low-frequency limit to the steady-state streaming potential coefficient largely studied in porous media [30–44].Laboratory experiments have also been investigated for a better understanding of the seismoelectric conversions [45–56]. These papers describe the laboratory studies performed to investigate this dynamic coupling. An oscillating pore pressure must be applied to a rock sample, and because of the relative motion between the rock and the fluid, an induced streaming potential can be measured. Depending on the oscillating frequency of the fluid, the fluid makes a transition from viscous dominated flow to inertial dominated flow. As the frequency increases, the motion of the fluid within the rock is delayed and larger pressure is needed. In order to know the dynamic coupling, both real and imaginary parts of the streaming potential must be measured.


## 2. From Dynamic Streaming Potential to Seismoelectromagnetic Coupling

The steady-state streaming potential coefficient is defined as the ratio of the streaming potential to the driving pore pressure:(1)Cs0=ΔVΔP=εζησf,
which is called the Helmholtz-Smoluchowski equation, where σf, ε, and η are the fluid conductivity, the dielectric constant of the fluid, and the fluid dynamic viscosity respectively (see [28]). In this formula the surface electrical conductivity is neglected compared to the fluid electrical conductivity. The potential ζ is the electrical potential within the double layer on the slipping plane. Although the zeta potential can hardly be modeled for a rock and although it cannot be directly measured within a rock, the steady-state streaming potential coefficient can be measured in laboratory, by applying a fluid pressure difference (ΔP) and by measuring the induced streaming electric potential (ΔV) [30, 38, 39, 44, 57]. The electrical potential ζ itself depends on fluid composition and pH and the water conductivity [29–31, 38, 40, 42, 44, 58].


### 2.1. Packard’s Model

Packard [59] proposed a model for the frequency-dependent streaming potential coefficient for capillary tubes, assuming that the Debye length is negligible compared to the capillary radius, based on the Navier-Stokes equation:(2)Cs0(ω)=ΔV(ω)ΔP(ω)=(εζησf)(2aiωρf/ηJ1(aiωρf/η)J0(aiωρf/η)e-iωt),
where ω is the angular frequency, a is the capillary radius, J1 and J0 are the Bessel functions of the first order and the zeroth order, respectively, and ρf is the fluid density.The transition angular frequency for a capillary is(3)ωc=ηρfa2.
More recently Reppert et al. [60] used the low- and high-frequency approximations of the Bessel functions to propose the following formula, which corresponds to their equation 26 corrected with the right exponents −2 and -1/2:(4)Cs0(ω)=(εζησf)[1+(-2aηωρf(12-12i))-2]-1/2
with the transition angular frequency(5)ωc=8ηρfa2
and showed that this model was not very different from the model proposed by Packard [59].The complete development relating Biot’s theory and Maxwell’s equations has been published by Pride in 1994 [22].


### 2.2. Pride’s Model

Pride [22] derived the equations governing the coupling between seismic and electromagnetic wave propagation in a fluid-saturated porous medium from first principles for porous media. The following transport equations express the coupling between the mechanical and electromagnetic wavefields ([22] (174), (176), and (177)):(6)J=σ(ω)E+L(ω)(-∇p+iω2ρfus),-iωw=L(ω)E+k(ω)η(-∇p+iω2ρfus).In the first equation, the macroscopic electrical current densityJ is the sum of the average conduction and streaming current densities. The filtration velocity w of the second equation is separated into electrically and mechanically induced contributions. The electrical fields and mechanical forces that create the current density J and filtration velocity w are, respectively, E and (-∇p+iω2ρfus), where p is the pore-fluid pressure, us is the solid displacement, and E is the electric field. The complex and frequency-dependent electrokinetic coupling L(ω), which describes the coupling between the seismic and electromagnetic fields [22, 60], is the most important parameter in these equations. The other two coefficients, σ(ω) and k(ω), are the electric conductivity and dynamic permeability of the porous material, respectively.The seismoelectric coupling that describes the coupling between the seismic and electromagnetic fields is complex and frequency-dependent Pride [22]:(7)L(ω)=L0[1-iωωcm4(1-2dΛ)2(1-i3/2dωρfη)2]-1/2,
where L0 is the low-frequency electrokinetic coupling, d is related to the Debye-length, Λ is a porous-material geometry term [65], and m is a dimensionless number (detailed in Pride [22]).The transition angular frequencyωc separating low-frequency viscous flow and high-frequency inertial flow is defined as(8)ωc=ϕηα∞k0ρf,
where ϕ is the porosity, k0 is the intrinsic permeability, and α∞ is the tortuosity.


### 2.3. Further Considerations

The low-frequency electrokinetic couplingL0 is related to the steady-state streaming potential coefficient Cs0 by(9)L0=-Cs0σr,
where σr is the rock conductivity. The electrokinetic coupling L(ω) can be estimated by considering that steady-state models of Cs0 can be applied to the calculation of L0. When writting σr=σf/F with surface conductivity neglected, the steady-state electrokinetic coupling can be written as(10)L0=-εζηF.
We can see that the steady-state electrokinetic coupling is inversely proportional to the formation factor.The transition angular frequency separating viscous and inertial flows in porous medium can be rewritten by insertingα∞=ϕF with F, as follows:(11)ωc=1Fηk0ρf,
where F is the formation factor that can be deduced from resistivity measurements using Archie’s law.Since the permeability and the formation factor are not independent but can be related byk0=CR2/F [66] with C being a geometrical constant usually in the range 0.3–0.5 and R being the hydraulic radius, the transition angular frequency can be written as(12)ωc=ηρfCR2.
Equation (12) shows that the transition angular frequency in porous medium is inversely proportional to the square of the hydraulic radius.Recently Walker and Glover [74] proposed a simplified equation of Pride’s development assuming that the Debye length is negligible compared to the characteristic pore size, and assuming the following parameter:(13)m=8(Λreff)2
leading to (14)L(ω)=L0[1-2iωωc(Λreff)2]-1/2
with reff being the effective pore radius, and a transition angular frequency being(15)ωc=8ηρfreff2.Garambois and Dietrich [12] studied the low-frequency assumption valid at seismic frequencies, meaning at frequencies lower than Biot’s frequency separating viscous and inertial flows and gave the coseismic transfer function for low-frequency longitudinal plane waves. In this case, and assuming Biot’s moduli C≪H, they showed that the seismoelectric field E is proportional to the grain acceleration:(16)E≃-L0σrρfü=ϵζησfρfü.Equations (16), (9), and (1) show that transient seismo-electric magnitudes will be affected by the bulk density of the fluid, and the streaming potential coefficient which is inversely proportional to the water conductivity and proportional to the zeta potential (which depends on the water pH).


### 2.4. The Electrokinetic Transition Frequency Compared to Hydraulic’s One

The theory of dynamic permeability in porous media has been studied by many authors [61, 65, 75–77].The frequency behavior of the permeability is given by Pride (1994) [22](17)k(ω)k0=[(1-iωωc4m)1/2-iωωc]-1.The transition angular frequency for a porous medium is the same as (8). Charlaix et al. [62] measured the behavior of permeability with frequency on capillary tube, glass beads, and crushed glass. The dynamic permeability is constant up to the transition frequency above which it decreases, and the more permeable the sample is, the lower the transition frequency is. Other measurements have been performed on glass beads and sand grains [61]. The transition frequency (fc=ωc/2π) varies from 4.8 Hz to 149 Hz for samples having permeability in the range 10-8 to 10-10 m2 (see Table 1), which are extremely high permeabilities.Table 1
Measured or predicted transition frequency for dynamic streaming potential and permeability, for samples of porosityϕ, formation factor F, permeability k0, and half of the mean particle size r, from (SED) Smeulders et al. [61], (CKS) Charlaix et al. [62], (SG) Sears and Groves [63], (P) Packard [59], (TGR) Tardif et al. [64], and (RMLJ) Reppert et al. [60]. *indicates predicted transition frequency from (3) and **Indicates the transition frequency computed by the authors.
SampleParticle sizeμmϕ [%]Fk0 [m2]fc [Hz]SourceCapillary254 (radius)10-810–2.5* HzCKSCapillary508 (radius)1.3–0.62* HzSGCapillary G4720 (radius)0.31*–0.28** HzPCapillary G2826 (radius)0.23*–0.21** HzPCapillary 1800–1100 (radius)7.1 HzRMLJGlass beads1.25–1.75327.84.2×10-94.8 HzSEDGlass beads850 (r)502.810-86.2 HzCKSGlass beads580–700318.79×10-1020 HzSEDGlass beads450 (r)503.22×10-925 HzCKSGlass beads250 (r)5035×10-10108 HzCKSGlass beads200–2703191.4×10-10126 HzSEDCrushed glass440 (r)50310-944 HzCKSCrushed glass265 (r)503.22×10-1045–103 HzCKSPorous filter A72.5–87269 HzRMLJPorous filter B35–50710 HzRMLJSand grains1000–200031926×10-106.7 HzSEDSand grains150–3002910.710-10149 HzSEDOttawa sand200–250 (r)314.71.2×10-10230–273 HzTGRThe transition frequency indicates the beginning of the transition for both the permeability and the electrokinetic coupling. However the transition behavior and the cuttoff frequency are different between permeability and electrokinetic coupling ((7) and (17)), both depending on the pore-space geometry term m but in different manner.We calculated the predicted transition frequencyfc=ωc/2π with ωc from (11) with η=10-3 Pa.s and ρf=103 kg/m3. The other parameters F and k0 are measured from different authors cited in Bernabé[78] (see Table 2). We also calculated the parameters for four Fontainebleau sandstone samples. It has been shown for these samples that F=ϕ-2.01 (from Ruffet et al. [79]) and that k0=aϕn with different values for n according to the porosity. The following laws were chosen: k0=1.66×10-4ϕ8 for ϕ<6% and k0=2.5×10-10ϕ3 for ϕ ranging between 8 and 25% [80]. We can see that the transition frequencies are of the order of kHz and MHz and no more from 0.2 to 150 Hz as measured or calculated on glass beads, sand grains, crushed glass, or capillaries. We plotted the results of the transition frequency as a function of the permeability on these various samples in Figure 1. Although the formation factor is not constant with the permeability, it is clear that the transition frequency is inversely proportional to the permeability as(18)log10(fc)=-0.78log10(k)-5.5,
and varies from about 100 MHz for 10-17 m2 to about 10 Hz for 10-8 m2, so by seven orders of magnitude for nine orders of magnitude in permeability.Table 2
Predicted transition frequency (from (11)), for dynamic streaming potential, for samples of porosity ϕ, formation factor F, and permeability k0, from (1) calculated in the present study, and measured by (2) Taherian et al. [67], (3) Morgan et al. [68], (4) Fatt [69], (5) Wyble [70], (6) Dobrynin [71], (7) Chierici et al. [72], and (8) Yale [73].
Sampleϕ [%]Fk0 [m2]fc [Hz]Fontainebleau sandstone120252×10-123.2 kHzFontainebleau sandstone115458×10-134.4 kHzFontainebleau sandstone1101022.5×10-136.2 kHzSandstone-S22231.262.7×10-129.7 kHzSandstone-S4722014.48.5×10-1313 kHzBoise826129×10-1314.7 kHzBerea sandstone500820204.9×10-1316.2 kHzSandstone-S42219.714.76.7×10-1316.2 kHzSandstone-S4522111.77.2×10-1318.8 kHzFahler 162832942.7×10-1420 kHzSandstone-S43221.2135.1×10-1323.5 kHzPliocene 41721144.94.2×10-1426.1 kHzPliocene 35720156.23.7×10-1427.5 kHzBerea sandstoneC2H319.815.13.8×10-1327.7 kHzSandstone-S50218.317.23.1×10-1330 kHzTriassic3872112.64×10-1231.4 kHzTriassic3472013.93.5×10-1332.7 kHzBerea sandstoneB2320.315.22.64×10-1339.7 kHzSandstone-S5226.48.74.1×10-1345 kHzSandstone-S35218.7517.42×10-1346.5 kHzMassillon DH81623.81.3×10-1351.4 kHzCambrian 16714312.59.5×10-1553.6 kHzFontainebleau sandstone154126.5×10-1559.4 kHzBerea sandstoneD1318.518.41.3×10-1366.5 kHzTensleep141518.91.2×10-1370.3 kHzTertiary 80782214.91.5×10-1371.1 kHzCambrian 678.190.92.3×10-1476.1 kHzTorpedo62041.74.5×10-1484.9 kHzMiocene 778.3384.64.4×10-1594 kHzCambrian 1471152.63.2×10-1494.5 kHzSandstone Triassic27718207.2×10-14110.5 kHzSandstone-S9220.9121×10-13126.2 kHzTriassic2671817.26.8×10-14135.7 kHzSandstone-S6222.810.68.3×10-14180.7 kHzBerea 100H81717.24.9×10-14188.4 kHzSandstone S15221.813.94.5×10-14256.7 kHzKirkwood515401.2×10-14331.6 kHzIndiana DV827123×10-14440.3 kHzIsland Rust A1314.652.55.2×10-15579 kHzBradford511902.5×10-15700.3 kHzAustin chalk323.622.79.7×10-15763 kHzMassillon DV81927.86.9×10-15830.4 kHzSandstone-S34221.3513.71.1×10-141.06 MHzSandstone S44215.724.54.2×10-151.5 MHzIndiana L. SA131829.21.9×10-152.9 MHzTennessee A135.5180.32.3×10-163.8 MHzAZPink (Coconino)310.362.46.3×10-164.04 MHzLeuders L.SA1315.241.57.1×10-165.3 MHzSandstone-S40210.91301.9×10-166.4 MHzSandstone-S23218.840.74.8×10-168.1 MHzFahler 18981.9714.32×10-1711.1 MHzPenn blue A133.92196.2×10-1711.7 MHzAZChoclate239.5159.35.8×10-1717.2 MHzFahler 16182.3416.71×10-1738.2 MHzFahler 14287.61642×10-1748.5 MHzSandstone S21212.1653×10-1781.7 MHzFahler 15484.6263.17×10-1886.4 MHzFahler 19284.4128.29×10-18137.9 MHzThe transition frequencyfc=ωc/2π (in Hz) predicted in the present study with ωc from (11) with η=10-3 Pa.s and ρf=103 kg/m3 as a function of the permeability (in m2). The transition frequency varies as log10(fc)=-0.78log10(k)-5.5. The parameters of the samples, F and k0, are measured from different authors on various samples cited in Tables 1 and 2.
(a)(b)


## 2.1. Packard’s Model

Packard [59] proposed a model for the frequency-dependent streaming potential coefficient for capillary tubes, assuming that the Debye length is negligible compared to the capillary radius, based on the Navier-Stokes equation:(2)Cs0(ω)=ΔV(ω)ΔP(ω)=(εζησf)(2aiωρf/ηJ1(aiωρf/η)J0(aiωρf/η)e-iωt),
where ω is the angular frequency, a is the capillary radius, J1 and J0 are the Bessel functions of the first order and the zeroth order, respectively, and ρf is the fluid density.The transition angular frequency for a capillary is(3)ωc=ηρfa2.
More recently Reppert et al. [60] used the low- and high-frequency approximations of the Bessel functions to propose the following formula, which corresponds to their equation 26 corrected with the right exponents −2 and -1/2:(4)Cs0(ω)=(εζησf)[1+(-2aηωρf(12-12i))-2]-1/2
with the transition angular frequency(5)ωc=8ηρfa2
and showed that this model was not very different from the model proposed by Packard [59].The complete development relating Biot’s theory and Maxwell’s equations has been published by Pride in 1994 [22].


## 2.2. Pride’s Model

Pride [22] derived the equations governing the coupling between seismic and electromagnetic wave propagation in a fluid-saturated porous medium from first principles for porous media. The following transport equations express the coupling between the mechanical and electromagnetic wavefields ([22] (174), (176), and (177)):(6)J=σ(ω)E+L(ω)(-∇p+iω2ρfus),-iωw=L(ω)E+k(ω)η(-∇p+iω2ρfus).In the first equation, the macroscopic electrical current densityJ is the sum of the average conduction and streaming current densities. The filtration velocity w of the second equation is separated into electrically and mechanically induced contributions. The electrical fields and mechanical forces that create the current density J and filtration velocity w are, respectively, E and (-∇p+iω2ρfus), where p is the pore-fluid pressure, us is the solid displacement, and E is the electric field. The complex and frequency-dependent electrokinetic coupling L(ω), which describes the coupling between the seismic and electromagnetic fields [22, 60], is the most important parameter in these equations. The other two coefficients, σ(ω) and k(ω), are the electric conductivity and dynamic permeability of the porous material, respectively.The seismoelectric coupling that describes the coupling between the seismic and electromagnetic fields is complex and frequency-dependent Pride [22]:(7)L(ω)=L0[1-iωωcm4(1-2dΛ)2(1-i3/2dωρfη)2]-1/2,
where L0 is the low-frequency electrokinetic coupling, d is related to the Debye-length, Λ is a porous-material geometry term [65], and m is a dimensionless number (detailed in Pride [22]).The transition angular frequencyωc separating low-frequency viscous flow and high-frequency inertial flow is defined as(8)ωc=ϕηα∞k0ρf,
where ϕ is the porosity, k0 is the intrinsic permeability, and α∞ is the tortuosity.


## 2.3. Further Considerations

The low-frequency electrokinetic couplingL0 is related to the steady-state streaming potential coefficient Cs0 by(9)L0=-Cs0σr,
where σr is the rock conductivity. The electrokinetic coupling L(ω) can be estimated by considering that steady-state models of Cs0 can be applied to the calculation of L0. When writting σr=σf/F with surface conductivity neglected, the steady-state electrokinetic coupling can be written as(10)L0=-εζηF.
We can see that the steady-state electrokinetic coupling is inversely proportional to the formation factor.The transition angular frequency separating viscous and inertial flows in porous medium can be rewritten by insertingα∞=ϕF with F, as follows:(11)ωc=1Fηk0ρf,
where F is the formation factor that can be deduced from resistivity measurements using Archie’s law.Since the permeability and the formation factor are not independent but can be related byk0=CR2/F [66] with C being a geometrical constant usually in the range 0.3–0.5 and R being the hydraulic radius, the transition angular frequency can be written as(12)ωc=ηρfCR2.
Equation (12) shows that the transition angular frequency in porous medium is inversely proportional to the square of the hydraulic radius.Recently Walker and Glover [74] proposed a simplified equation of Pride’s development assuming that the Debye length is negligible compared to the characteristic pore size, and assuming the following parameter:(13)m=8(Λreff)2
leading to (14)L(ω)=L0[1-2iωωc(Λreff)2]-1/2
with reff being the effective pore radius, and a transition angular frequency being(15)ωc=8ηρfreff2.Garambois and Dietrich [12] studied the low-frequency assumption valid at seismic frequencies, meaning at frequencies lower than Biot’s frequency separating viscous and inertial flows and gave the coseismic transfer function for low-frequency longitudinal plane waves. In this case, and assuming Biot’s moduli C≪H, they showed that the seismoelectric field E is proportional to the grain acceleration:(16)E≃-L0σrρfü=ϵζησfρfü.Equations (16), (9), and (1) show that transient seismo-electric magnitudes will be affected by the bulk density of the fluid, and the streaming potential coefficient which is inversely proportional to the water conductivity and proportional to the zeta potential (which depends on the water pH).


## 2.4. The Electrokinetic Transition Frequency Compared to Hydraulic’s One

The theory of dynamic permeability in porous media has been studied by many authors [61, 65, 75–77].The frequency behavior of the permeability is given by Pride (1994) [22](17)k(ω)k0=[(1-iωωc4m)1/2-iωωc]-1.The transition angular frequency for a porous medium is the same as (8). Charlaix et al. [62] measured the behavior of permeability with frequency on capillary tube, glass beads, and crushed glass. The dynamic permeability is constant up to the transition frequency above which it decreases, and the more permeable the sample is, the lower the transition frequency is. Other measurements have been performed on glass beads and sand grains [61]. The transition frequency (fc=ωc/2π) varies from 4.8 Hz to 149 Hz for samples having permeability in the range 10-8 to 10-10 m2 (see Table 1), which are extremely high permeabilities.Table 1
Measured or predicted transition frequency for dynamic streaming potential and permeability, for samples of porosityϕ, formation factor F, permeability k0, and half of the mean particle size r, from (SED) Smeulders et al. [61], (CKS) Charlaix et al. [62], (SG) Sears and Groves [63], (P) Packard [59], (TGR) Tardif et al. [64], and (RMLJ) Reppert et al. [60]. *indicates predicted transition frequency from (3) and **Indicates the transition frequency computed by the authors.
SampleParticle sizeμmϕ [%]Fk0 [m2]fc [Hz]SourceCapillary254 (radius)10-810–2.5* HzCKSCapillary508 (radius)1.3–0.62* HzSGCapillary G4720 (radius)0.31*–0.28** HzPCapillary G2826 (radius)0.23*–0.21** HzPCapillary 1800–1100 (radius)7.1 HzRMLJGlass beads1.25–1.75327.84.2×10-94.8 HzSEDGlass beads850 (r)502.810-86.2 HzCKSGlass beads580–700318.79×10-1020 HzSEDGlass beads450 (r)503.22×10-925 HzCKSGlass beads250 (r)5035×10-10108 HzCKSGlass beads200–2703191.4×10-10126 HzSEDCrushed glass440 (r)50310-944 HzCKSCrushed glass265 (r)503.22×10-1045–103 HzCKSPorous filter A72.5–87269 HzRMLJPorous filter B35–50710 HzRMLJSand grains1000–200031926×10-106.7 HzSEDSand grains150–3002910.710-10149 HzSEDOttawa sand200–250 (r)314.71.2×10-10230–273 HzTGRThe transition frequency indicates the beginning of the transition for both the permeability and the electrokinetic coupling. However the transition behavior and the cuttoff frequency are different between permeability and electrokinetic coupling ((7) and (17)), both depending on the pore-space geometry term m but in different manner.We calculated the predicted transition frequencyfc=ωc/2π with ωc from (11) with η=10-3 Pa.s and ρf=103 kg/m3. The other parameters F and k0 are measured from different authors cited in Bernabé[78] (see Table 2). We also calculated the parameters for four Fontainebleau sandstone samples. It has been shown for these samples that F=ϕ-2.01 (from Ruffet et al. [79]) and that k0=aϕn with different values for n according to the porosity. The following laws were chosen: k0=1.66×10-4ϕ8 for ϕ<6% and k0=2.5×10-10ϕ3 for ϕ ranging between 8 and 25% [80]. We can see that the transition frequencies are of the order of kHz and MHz and no more from 0.2 to 150 Hz as measured or calculated on glass beads, sand grains, crushed glass, or capillaries. We plotted the results of the transition frequency as a function of the permeability on these various samples in Figure 1. Although the formation factor is not constant with the permeability, it is clear that the transition frequency is inversely proportional to the permeability as(18)log10(fc)=-0.78log10(k)-5.5,
and varies from about 100 MHz for 10-17 m2 to about 10 Hz for 10-8 m2, so by seven orders of magnitude for nine orders of magnitude in permeability.Table 2
Predicted transition frequency (from (11)), for dynamic streaming potential, for samples of porosity ϕ, formation factor F, and permeability k0, from (1) calculated in the present study, and measured by (2) Taherian et al. [67], (3) Morgan et al. [68], (4) Fatt [69], (5) Wyble [70], (6) Dobrynin [71], (7) Chierici et al. [72], and (8) Yale [73].
Sampleϕ [%]Fk0 [m2]fc [Hz]Fontainebleau sandstone120252×10-123.2 kHzFontainebleau sandstone115458×10-134.4 kHzFontainebleau sandstone1101022.5×10-136.2 kHzSandstone-S22231.262.7×10-129.7 kHzSandstone-S4722014.48.5×10-1313 kHzBoise826129×10-1314.7 kHzBerea sandstone500820204.9×10-1316.2 kHzSandstone-S42219.714.76.7×10-1316.2 kHzSandstone-S4522111.77.2×10-1318.8 kHzFahler 162832942.7×10-1420 kHzSandstone-S43221.2135.1×10-1323.5 kHzPliocene 41721144.94.2×10-1426.1 kHzPliocene 35720156.23.7×10-1427.5 kHzBerea sandstoneC2H319.815.13.8×10-1327.7 kHzSandstone-S50218.317.23.1×10-1330 kHzTriassic3872112.64×10-1231.4 kHzTriassic3472013.93.5×10-1332.7 kHzBerea sandstoneB2320.315.22.64×10-1339.7 kHzSandstone-S5226.48.74.1×10-1345 kHzSandstone-S35218.7517.42×10-1346.5 kHzMassillon DH81623.81.3×10-1351.4 kHzCambrian 16714312.59.5×10-1553.6 kHzFontainebleau sandstone154126.5×10-1559.4 kHzBerea sandstoneD1318.518.41.3×10-1366.5 kHzTensleep141518.91.2×10-1370.3 kHzTertiary 80782214.91.5×10-1371.1 kHzCambrian 678.190.92.3×10-1476.1 kHzTorpedo62041.74.5×10-1484.9 kHzMiocene 778.3384.64.4×10-1594 kHzCambrian 1471152.63.2×10-1494.5 kHzSandstone Triassic27718207.2×10-14110.5 kHzSandstone-S9220.9121×10-13126.2 kHzTriassic2671817.26.8×10-14135.7 kHzSandstone-S6222.810.68.3×10-14180.7 kHzBerea 100H81717.24.9×10-14188.4 kHzSandstone S15221.813.94.5×10-14256.7 kHzKirkwood515401.2×10-14331.6 kHzIndiana DV827123×10-14440.3 kHzIsland Rust A1314.652.55.2×10-15579 kHzBradford511902.5×10-15700.3 kHzAustin chalk323.622.79.7×10-15763 kHzMassillon DV81927.86.9×10-15830.4 kHzSandstone-S34221.3513.71.1×10-141.06 MHzSandstone S44215.724.54.2×10-151.5 MHzIndiana L. SA131829.21.9×10-152.9 MHzTennessee A135.5180.32.3×10-163.8 MHzAZPink (Coconino)310.362.46.3×10-164.04 MHzLeuders L.SA1315.241.57.1×10-165.3 MHzSandstone-S40210.91301.9×10-166.4 MHzSandstone-S23218.840.74.8×10-168.1 MHzFahler 18981.9714.32×10-1711.1 MHzPenn blue A133.92196.2×10-1711.7 MHzAZChoclate239.5159.35.8×10-1717.2 MHzFahler 16182.3416.71×10-1738.2 MHzFahler 14287.61642×10-1748.5 MHzSandstone S21212.1653×10-1781.7 MHzFahler 15484.6263.17×10-1886.4 MHzFahler 19284.4128.29×10-18137.9 MHzThe transition frequencyfc=ωc/2π (in Hz) predicted in the present study with ωc from (11) with η=10-3 Pa.s and ρf=103 kg/m3 as a function of the permeability (in m2). The transition frequency varies as log10(fc)=-0.78log10(k)-5.5. The parameters of the samples, F and k0, are measured from different authors on various samples cited in Tables 1 and 2.
(a)(b)


## 3. Experimental Apparatus and Procedure

Several experimental setups were proposed to provide the sinusoidal pressure variations.The first experimental apparatus proposed a sinusoidal motion delivered by a sylphon bellows which was driven by a geophone-type push-pull driver (Figure2 from Packard [59]). The low-frequency oscillator (0.01 Hz to 1 kHz) was used for operation of the push-pull geophone driver. Similar setups were proposed by Thurston [81] (Figure 3) and Cooke [82], so that frequency of this kind of source was 1–400 Hz [82], 20–200 Hz [59], and 10–700 Hz [81]. The induced pressure was up to 2 kPa. More recently Schoemaker et al. [83] used a so-called Dynamic Darcy Cell (DCC) with a mechanical shaker connected to a rubber membrane leading to a frequency range for the oscillating pressure 5 to 200 Hz. The sinusoidal fluid flow was also applied by a displacement piston pump directly connected to the electrodes chambers (Figure 4 from [63, 88]). The piston was mounted on a Scotch Yoke drive attached to a controllable speed AC motor [84]. The frequency range of this source was then 0.4 Hz to 21 Hz and the pressure up to 15 kPa. Pengra et al. [85] used a piston rod attached to a loudspeaker driven by an audio power amplifier (Figure 5). They performed measurements up to 100 Hz, with an applied pressure of 5 kPa RMS. More recently it was proposed by Reppert et al. [60] to use an electromechanical transducer (Figure 6), and these authors covered a frequency range 1–500 Hz. The vibrating exciter proposed by Schoemaker et al. [86] was used from 5 Hz to 200 Hz. Recently Tardif et al. [64] used an electromagnetic shaker operating in the range 1 Hz to 1 kHz and provided measurements up to 200 Hz. Higher frequencies have been investigated [49, 50, 52, 54, 55] for the detection of the interfacial conversions.Figure 2
The sylphon bellows is driven by a geophone-type push-pull driver to apply a sinusoidal motion to the sample (modified from [59]).Figure 3
Experimental setup used by Thurston [87] (modified from [87]).Figure 4
Experimental setup used by Groves and Sears (modified from [88]).Figure 5
Experimental setup used by Pengra et al. [85] for streaming potential and electro-osmosis measurements (modified from [85]).Figure 6
Experimental setup used by Reppert et al. [60] (modified from [60]).The electromagnetic noise radiating from such equipment must be suppressed by shielding the setup and wires (shielded twisted cable pairs) [64, 86]. Moreover it is essential to have a rigid framework. A mechanical resonance can occur in the cell/transducer system (at 70 Hz in [85]), and the noise associated with mechanical vibration can be suppressed puting an additional mass to the frame [64].Once the oscillatory pressure is applied, the pressure must be measured. Most of the setups include piezoelectric transducers to measure the pressure difference over the capillary or the porous sample. Reppert et al. [60] proposed to use hydrophones that have a flat response from 1 to 20 kHz. Tardif et al. [64] proposed to use dynamic transducers with a low-frequency limit 0.08 Hz and a maximum frequency of 170 kHz.The electrodes are usually Ag/AgCl or platinum electrodes. The electrodes used by Schoemaker et al. [86] were sintered plates of Monel (composed of nickel and copper). The electrical signal must be measured using preamplifiers or a high-input impedance acquisition system. Since the impedance of the sample depends on the frequency, one must correct the measurements from this varying-impedance to be able to have a correct streaming potential coefficient [60]. Moreover the electrodes at top and bottom of the sample can behave as a capacitor, requiring a correction using impedance measurements too [86].The sample is usually saturated and it is emphasized that the sample should be left until equilibrium with water. This equilibrium can be obtained by leaving the sample in contact with water for some time, and by flowing the water within the sample several times by checking thepH and the water conductivity until an equilibrium is reached [39]. The procedure including water flow is better because the properties of the water can be measured. When the properties of the water are measured only before saturating the sample, the resulting water once in contact with the sample is not known. Usually the water is more conductive when in contact with the sample, and the pH can change. Recalling that the streaming potential is proportional to the zeta potential (which depends on pH) and inversely proportional to the water conductivity (1), it is essential to know properly the pH and the water conductivity.


## 4. Measurements and Calculations of the Dynamic Electrokinetic Coefficient

The absolute magnitude of the streaming potential coefficient normalized by the steady-state value was calculated by Packard [59] as(19)f(Ya)=(-2YaiiJ1(iYa)J0(iYa)e-iωt),
which is equal to (2), but expressed as a function of the parameter Ya=a(ωρf)/η, the transition frequency being obtained for Ya=1 (Figure 7). The streaming potential coefficient is constant up to the transition angular frequency and then decreases with increasing frequency.Figure 7
The absolute magnitude of the normalized streaming potential coefficient calculated by Packard [59] using (19) where Ya=aωρf/η, equivalent to (2) (modified from [59]).Sears and Groves [63] measured the streaming potential coefficient on a capillary of radius 508 μm which was coated with clay-Adams Siliclad and then incubated with 1% bovine serum albumin and filled with 0.02 M Tris-HCl at pH 7.32. They reported the streaming potential and the pressure difference as a function of frequency in the range 0–20 Hz. We calculated the resulting streaming potential coefficient (see Figure 8) which decreases from about 1.3×10-7 to 4×10-8 V/Pa. These authors computed the zeta potential and concluded that the zeta potential is independent of the frequency with an average value of 28.8 mV. Moreover they concluded that the zeta potential is also independent of the capillary radius and capillary length.Figure 8
The streaming potential coefficient measured as a function of frequency by Sears and Groves [63] on a capillary coated with clay, incubated with BSA in 0.02 M Tris-HCl.The value of the streaming potential coefficient on Ottawa sand measured at 5 Hz by Tardif et al. [64] was -5.2×10-7 V/Pa using a 0.001 mol/L NaCl solution to saturate the sample. Values between 1 and 2×10-8 V/Pa were measured on samples saturated by 0.1 M/L NaCl brine [85]. A compilation of numerous streaming potential coefficients measured on sands and sandstones at various salinities in DC domain [44] showed that Cs0=-1.2×10-8σf-1, where Cs0 is in V/Pa and σf in S/m. A zeta potential of −17 mV can be inferred from these collected data, assuming the other parameters (see (1)) independent of water conductivity. These assumptions are not exact, but the value of zeta is needed for numerous modellings which usually assume the other parameters independent of the fluid conductivity. Therefore an average value of −17 mV for such modellings can be rather exact, at least for medium with no clay nor calcite.Reppert et al. [60] calculated the real part and the imaginary part of the theoretical Packard’s streaming potential coefficient (2) for different capillary radii (see Figure 9). It can be seen that the larger the radius is, the lower the transition frequency is, as shown previously by the different theories. Recent developments by the group of Glover have been performed to build a new setup and to make further measurements on porous samples: two papers detail these studies in this special issue on Electrokinetics in Earth Sciences.Figure 9
The real and imaginary part of Packard's model (2) calculated by Reppert et al. [60] for three capillary radii: 100 μm (continuous line), 50 μm (dashed line), and 10 μm (point line) (modified from [60]).


## 5. Conclusion

Since the theory of Pride in 1994 [22], the dynamic behavior of the streaming potential is known for porous media. However few experimental results are available, because of the difficulty to perform correct measurements at high frequency. Up to now, measurements of the frequency-dependence of the streaming potential have been performed up to 200 Hz on high-permeable samples. The main difficulty arises from electrical noise induced by mechanical vibration. Moreover it has been emphasized that the measurements must be corrected by impedance measurements as a function of frequency too because the impedance of the sample depends on frequency. Further theoretical developments performed by Garambois and Dietrich [12] studied the low-frequency assumption valid at frequencies lower than the transition frequency. We show that this transition frequency, on a various collection of samples for which both formation factor and permeability are measured, is predicted to depend on the permeability as inversely proportional to the permeability.



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*Source: 648781-2012-04-18.xml*