Duy Luong

Damien Thanh
email: thanhlud@tlu.edu.vn

Santiago Jougnot

Nguyen Solazzi

Phan Van Nghia

Van

Luong Duy

Damien Jougnot

Santiago G Solazzi

Nguyen Van Nghia

Phan Van Do

N V Nghia

Dynamic streaming potential coupling coefficient in porous media with different pore size distributions

Keywords: Numerical modelling, Fractals and multifractals, Permeability and porosity List of Symbols

D f

Fractal dimension for pore space no units at the interface between pore fluid and mineral surfaces. Minerals composing geological media generally acquire electrostatic charges at their surfaces when brought into contact with an electrolyte, such as water. This leads to a charge distribution known as the EDL prevailing at the vicinity of the water-mineral solid interfaces. The EDL consists of an excess of electrical charges in the pore water to compensate for the charged mineral surfaces. When a seismic wave propagates in fluid-filled porous media, it generates relative displacement of water with respect to the pore walls and, thus, relative movement of the charges in the EDL with respect to the charged pore surface. This process creates an electrical current and a resulting electrical field that can be measured at the earth surface or within a geological medium. Given the sensitivity of the seismoelectric signal to medium and fluid properties, the seismoelectric method attracts the interest of researchers in different areas, from oil and gas reservoir characterization to hydrogeophysics (e.g., [START_REF] Butler | Seismoelectric effects of electrokinetic origin[END_REF][START_REF] Mikhailov | Using borehole electroseismic measurements to detect and characterize fractured (permeable) zones[END_REF][START_REF] Garambois | Seismoelectric wave conversions in porous media: field measurements and transfer function analysis[END_REF][START_REF] Thompson | Electromagnetic-to-seismic conversion: a new direct hydrocarbon indicator[END_REF]). Additionally, it is indicated that the seismoelectric method is also feasible in well logging for oil and gas exploration (e.g., [START_REF] Dupuis | Anatomy of a seismoelectric conversion: Measurements and conceptual modeling in boreholes penetrating a sandy aquifer[END_REF][START_REF] Hu | Theoretical simulation of electroacoustic borehole logging in a fluid-saturated porous formation[END_REF]Wang et al. 2015a).

L
The seismoelectrical effect was studied by pioneering authors in the 1930 s (e.g., [START_REF] Thompson | A note on the seismic-electric phenomenon[END_REF][START_REF] Frenkel | On the theory of seismic and seismoelectric phenomena in a moist soil[END_REF]) and it remains a highly active research field (e.g., [START_REF] Pride | Electroseismic wave theory of frenkel and more recent developments[END_REF][START_REF] Revil | The Seismoelectric Method: Theory and Applications[END_REF][START_REF] Jouniaux | A review on electrokinetically induced seismo-electrics, electro-seismics, and seismo-magnetics for earth sciences[END_REF][START_REF] Jougnot | Predicting the frequency-dependent effective excess charge density: A new up-scaling approach for seismoelectric modelling[END_REF]). To model the seismoelectric effect, one normally uses the electrokinetic coupling coefficient C EK (ω), which is a frequency dependent parameter that relates the measured electrical potential difference with the fluid pressure difference driving the pore fluid flow. Several models have been proposed to describe C EK (ω) in the literature.

The two most used models for C EK (ω) were proposed by (i) [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] and (ii) [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF]. [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] proposed a model for C EK (ω) that is valid for capillary tubes. [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] obtained

C EK (ω) based on first principles and an upscaling approach based on volume averaging. Both of these models provide a good agreement with respect to measured data of the frequency dependent streaming potential (e.g., [START_REF] Reppert | Frequency-dependent streaming potentials[END_REF][START_REF] Tardif | Frequency-dependent streaming potential of ottawa sand[END_REF]Glover et al. 2012a;[START_REF] Jouniaux | Electrokinetics in earth sciences: A tutorial[END_REF]Wang et al. 2015b[START_REF] Wang | The evaluation of rock permeability with streaming current measurements[END_REF] and the seismoelectric conversion [START_REF] Zhu | Experimental studies of electrokinetic conversions in fluidsaturated borehole models[END_REF][START_REF] Zhu | Experimental measurements of the streaming potential and seismoelectric conversion in berea sandstone[END_REF].

Note that the dependence of C EK (ω) on thickness of the EDL was also presented by [START_REF] Shi | Dependence of dynamic electrokinetic-coupling-coefficient on the electric double layer thickness of fluid-filled porous formations[END_REF].

An alternative approach was proposed for the determination of the streaming potential coupling coefficient via the excess charge that is effectively dragged by the flow of pore water through the pore space (e.g., [START_REF] Kormiltsev | Do magnetic fields induced by the fluid flow movement in porous media[END_REF][START_REF] Revil | Constitutive equations for ionic transport in porous shales[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF][START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF][START_REF] Jougnot | Modeling Streaming Potential in Porous and Fractured Media, Description and Benefits of the Effective Excess Charge Density Approach[END_REF]. A similar approach was proposed for the seismoelectric effect using effective excess charge density (e.g., [START_REF] Revil | Stochastic inversion of permeability and dispersivities from time lapse selfpotential measurements: A controlled sandbox study[END_REF]. The effective excess charge density can be determined either by an empirical expression from permeability of porous media (e.g., [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF][START_REF] Revil | The Self-Potential Method: Theory and Applications in Environmental Geosciences[END_REF][START_REF] Cherubini | Streaming potential coupling coefficient and transport properties of unsaturated carbonate rocks[END_REF] or by a theoretical expression from macroscopic hydraulic parameters Dynamic streaming potential coupling coefficient in porous media with different pore size distributions 5 and electrokinetic parameters of porous media (e.g., [START_REF] Guarracino | A physically based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media[END_REF][START_REF] Soldi | An analytical effective excess charge density model to predict the streaming potential generated by unsaturated flow[END_REF]. [START_REF] Revil | Coupled hydromechanical and electromagnetic disturbances in unsaturated Dynamic streaming potential coupling coefficient in porous media with different pore size distributions 27 porous materials[END_REF] proposed an empirical expression for the dependence of the effective excess charge density on frequency by taking into account a relaxation time governing the transition between the lowfrequency and highfrequency regimes of the Navier-Stokes equations. Very recently, [START_REF] Jougnot | Predicting the frequency-dependent effective excess charge density: A new up-scaling approach for seismoelectric modelling[END_REF] presented a novel approach to up-scale the frequency-dependent effective excess charge density from the pore scale by conceptualizing porous media as a bundle of capillaries having a singular capillary size.

It is well known that porous media usually have complicated and disordered pore structure with variation of pore sizes. Therefore, conceptualizing a porous medium as a bundle of capillaries with a unique pore size, as performed in [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] and [START_REF] Jougnot | Predicting the frequency-dependent effective excess charge density: A new up-scaling approach for seismoelectric modelling[END_REF], does not reflect the highly complex microstructure of porous media. Additionally, to the best of our knowledge, the surface electrical conductivity, a parameter that plays an important role in electrokinetics for low fluid electrical conductivity, has not yet been considered in the available models for C EK (ω). Note that the surface electrical conductivity is typically taken into account in the streaming potential coupling coefficient, that is, the quasistatic coefficient via the modified Helmholtz-Smoluchowski equation (e.g., [START_REF] Morgan | Streaming potential properties of westerly granite with applications[END_REF][START_REF] Revil | Streaming potential in porous media 1. theory of the zeta potential[END_REF]Glover et al. 2012b).

Capillary tube models assume that flow channels are generated within the pore space. The characteristics of these channels are effectively represented employing the capillary tube geometry, using different tortuosities and pore radii distributions. Models based on this approach have, despite their simplicity, proven to be a highly effective tool for the realistic description of, for example, the water content (e.g., [START_REF] Tyler | Fractal processes in soil water retention[END_REF], the permeability (e.g., [START_REF] Yu | A fractal permeability model for bi-dispersed porous media[END_REF][START_REF] Nghia | Predicting water flow in fully and partially saturated porous media: a new fractal-based permeability model[END_REF], the electrical conductivity (e.g., [START_REF] Niu | The use of electrical conductivity measurements in the prediction of hydraulic conductivity of unsaturated soils[END_REF][START_REF] Thanh | A physically based model for the electrical conductivity of water-saturated porous media[END_REF][START_REF] Rembert | A fractal model for the electrical conductivity of watersaturated porous media during mineral precipitation-dissolution processes[END_REF], the thermal conductivity (e.g., [START_REF] Chu | Enhanced fractal capillary bundle model for effective thermal conductivity of composite-porous geomaterials[END_REF], the electrokinetic coupling (e.g., [START_REF] Jackson | Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model[END_REF][START_REF] Soldi | An analytical effective excess charge density model to predict the streaming potential generated by unsaturated flow[END_REF][START_REF] Vinogradov | Influence of pore size distribution on the electrokinetic coupling L.D.Thanh[END_REF]) and the water flow in frozen soils (e.g., [START_REF] Watanabe | Capillary bundle model of hydraulic conductivity for frozen soil[END_REF].

This work proposes a physically-based model for the C EK (ω) by conceptualizing porous media as a bundle of tortuous capillaries. We consider both lognormal and fractal pore size distributions. We also analyze the effects of the surface electrical conductivity in the proposed model. Parameters influencing C EK (ω) are investigated and explained. The proposed model is then compared with published data and previous models.

THEORETICAL DEVELOPMENT

Pore scale

Let us consider a capillary of radius r (m) and length L τ (m) that is saturated by water with viscosity η w = 10 -3 (Pa s) and density ρ w = 1000 (kg/m 3 ) at temperature 20 o C. This capillary is submitted to an oscillatory frequency-dependent pressure difference ∆P (Pa) of angular frequency ω (rad/s) which is given by

∆P (t) = ∆P * e -iωt , (1)
where t (s) is the time, ∆P * (Pa) is the amplitude of pressure difference across the capillary tube and i is the imaginary unit. The superscript * denotes the amplitude of the oscillatory pressure difference and of other oscillatory variables from now on. Following previous publications (e.g., [START_REF] Reppert | Frequency-dependent streaming potentials[END_REF][START_REF] Solazzi | Dynamic permeability functions for partially saturated porous media[END_REF][START_REF] Jougnot | Predicting the frequency-dependent effective excess charge density: A new up-scaling approach for seismoelectric modelling[END_REF], we drop the harmonic term e -iωt to simplify the notation.

From the Navier-Stokes equations, it is possible to obtain the water velocity profile in the capillary as a function of the distance from the axis of the capillary x and the angular frequency v * (x, ω) (m/s) (e.g., [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF][START_REF] Reppert | Frequency-dependent streaming potentials[END_REF]. Additionally, the distribution of the excess charges ρ e (x) (C/m 3 ) in the pore water can be attained by solving the Poisson-Boltzmann equation (e.g., [START_REF] Rice | Electrokinetic flow in a narrow cylindrical capillary[END_REF]. From v * (x, ω) and ρ e (x), one can find the streaming current in the capillary using the approach that has been already presented by [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] or [START_REF] Reppert | Frequency-dependent streaming potentials[END_REF]. Using the assumptions of Debye-Hückel and a thin EDL (i.e., the thickness of the double layer is small compared to the pore radius), the frequency dependent streaming current i c (r, ω) in the capillary of radius r (m)

under the pressure difference ∆P * is given by (e.g., [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF][START_REF] Reppert | Frequency-dependent streaming potentials[END_REF])

i * c (r, ω) = 2π rζ∆P * (ω) η w L τ κ J 1 (κr) J 0 (κr) , (2)
where = r 0 (F/m) is the permittivity of the pore water with the permittivity of vacuum 0 = 8.86 × 10 -12 F/m and the relative permittivity of water r =80.1 at temperature 20 o C, ζ (V) is the zeta potential, J 0 and J 1 are the first kind Bessel functions of the zeroth and first order, respectively, and κ (m -1 ) is defined as

κ 2 = - iωρ w η w . (3)
In this work, we consider the thin EDL assumption, which is satisfied in a wide variety of natural systems (e.g., seawater) (see the discussion section in [START_REF] Jougnot | Exploring the effect of the pore size distribution on the streaming potential generation in saturated porous media, insight from pore network simulations[END_REF]). However, it is important to remark that that this assumption may fail when: (i) the pore fluid has a low salinity, such as porous sediments containing fresh water; (ii) the EDL thickness becomes comparable to the characteristic pore size, such is the case of clay rocks. In this context, the thick EDL assumption should be used Dynamic streaming potential coupling coefficient in porous media with different pore size distributions 7 instead. For further details regarding the thin/thick EDL assumptions, we refer the readers to the work of [START_REF] Jackson | On the validity of the thin and thick double-layer assumptions when calculating streaming currents in porous media[END_REF].

The dependence of ζ (V) on the ionic concentration C w (mol/L) of pore water is given by (e.g., [START_REF] Pride | Electrokinetic dissipation induced by seismic waves[END_REF]:

ζ = {a + b log 10 (C w )} × 10 -3 , (4)
where a and b are fitting parameters. In this work, we use a = -6.43 mV and b = 20.85 mV reported by [START_REF] Jaafar | Measurement of streaming potential coupling coefficient in sandstones saturated with high salinity nacl brine[END_REF] for silica-based rocks for modeling.

As a consequence of the streaming current in the capillary, an oscillatory electrical potential difference called the frequency dependent streaming potential ∆V * (ω) is built up between the ends of the capillary. This streaming potential causes a frequency dependent electric conduction current in the capillary, which, in turn, can be determined using Ohm s law. By considering both bulk and surface electrical conductions in the water saturated capillary, the conduction current is given by [START_REF] Birdi | Handbook of Surface and Colloid Chemistry[END_REF][START_REF] Thanh | A fractal model for streaming potential coefficient in porous media[END_REF]Thanh et al. , 2020a)

i * c (r, ω) = ∆V * (ω)σ w πr 2 L τ + ∆V * (ω)Σ s 2πr L τ = π∆V * (ω) L τ σ w r 2 + 2Σ s r , (5)
where σ w is the electrical conductivity of the fluid and Σ s is the specific surface conductance at the interface between fluid and the solid. Note that, following previous publications (e.g., [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF][START_REF] Reppert | Frequency-dependent streaming potentials[END_REF][START_REF] Jougnot | Seismoelectric effects due to mesoscopic heterogeneities[END_REF][START_REF] Jougnot | Predicting the frequency-dependent effective excess charge density: A new up-scaling approach for seismoelectric modelling[END_REF], we do not take into account the frequency dependence of the electrical conductivity.

As performed in the DC case [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF][START_REF] Reppert | Frequency-dependent streaming potentials[END_REF][START_REF] Thanh | A fractal model for streaming potential coefficient in porous media[END_REF]Thanh et al. , 2020a)), at equilibrium, the streaming current is balanced by the conduction current in the capillary. Setting Eq.

(2) equal to Eq. ( 5) and using the definition of C * EK (r, ω) as

C * EK (r, ω) = ∆V * (ω) ∆P * (ω) , (6)
the following expression is obtained for a single capillary

C * EK (r, ω) = ∆V * (ω) ∆P * (ω) = ζ η w (σ w + 2Σs r ) 2 κr J 1 (κr) J 0 (κr) . ( 7
)
If the surface conductivity is negligible (Σ s =0), Eq. ( 7) reduces to the model proposed by [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] or [START_REF] Reppert | Frequency-dependent streaming potentials[END_REF].

C * EK (r, ω) = ∆V * (ω) ∆P * (ω) = ζ η w σ w 2 κr J 1 (κr) J 0 (κr) . ( 8
)

REV scale

In order to obtain the upscaled electrical streaming current and conduction, we consider a cubic representative elementary volume (REV) of the porous media of side-length L o and cross-section area A REV , the latter being perpendicular to the flow direction. In the presence of a fluid pressure gradient, flow channels are generated within the pore space. The characteristics of these channels are usually modelled employing the capillary tube analogy. In this context, the REV is conceptualized as composed by an equivalent bundle of capillary tubes with radii varying from a minimum pore radius r min to a maximum pore radius r max . The pore size distribution f (r) in the REV is such that the number of capillary tubes with radius in the range from r to r + dr is given by f (r)dr. Note that the pore size distributions f (r) can be obtained from the hydrodynamic characteristic curves of a porous medium, using either the capillary pressure-saturation or relative permeability-saturation relationships (e.g., [START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF].

Following the approach used in [START_REF] Thanh | A fractal model for streaming potential coefficient in porous media[END_REF] for the DC case, the frequency dependent streaming current I * s (ω) and conduction current I * c (ω) through the REV are given by

I * s (ω) = rmax r min i * s (r, ω)f (r)dr = rmax r min 2π rζ∆P * (ω) η w L τ κ J 1 (κr) J 0 (κr) f (r)dr, (9)
and

I * c (ω) = rmax r min i * c (r, ω)f (r)dr = rmax r min π∆V * (ω) L τ σ w r 2 + 2Σ s r f (r)dr. ( 10
)
We apply the similar procedure to get the C * EK (r, ω) at pore scale, that is, we set Eq. ( 9) equal to Eq. ( 10) and, then, we obtain the following expression for C * EK (ω) at the macro-scale:

C * EK (ω) = rmax r min 2 ζ ηwκ J 1 (κr) J 0 (κr) rf (r)dr rmax r min [σ w r 2 + 2Σ s r] f (r)dr . ( 11
)
Recall that the transition from viscous to inertia dominated flow occurs at the so-called transition angular frequency ω t above which the amplitude of fluid flow decreases with frequency, which is given by (e.g., [START_REF] Solazzi | Dynamic permeability functions for partially saturated porous media[END_REF])

ω t = 2η w ρ w r 2 , ( 12
)
where r is a characteristic radius representative of the saturated pores.

In this work, we analyze two different PSDs: (i) lognormal and (ii) fractal distributions, for modeling C * EK (ω) as presented below. However, it is important to remark that Eq. ( 11) has been built up for any pore size distribution f (r).

The lognormal distribution of capillary tubes is frequently applied to porous media (e.g., [START_REF] Kosugi | Three-parameter lognormal distribution model for soil water retention[END_REF][START_REF] Jougnot | Exploring the effect of the pore size distribution on the streaming potential generation in saturated porous media, insight from pore network simulations[END_REF][START_REF] Ghanbarian | Applications of critical path analysis to uniform grain packings with narrow conductance distributions: Ii. water relative permeability[END_REF] and is given by

f (r) = A √ 2πs r exp - ln( r rm ) √ 2s 2 , r min ≤ r ≤ r max ( 13
)
where A is a normalizing prefactor, r m is the geometric mean pore radius, and s is the log-normal standard deviation. Note that the total number of capillaries N t following the lognormal distribution in the REV is given by

N t = rmax r min f (r)dr = A 2 erf ln( rmax rm ) √ 2s -erf ln( r min rm ) √ 2s (14)
Similarly, substituting Eq. ( 13) into Eq. ( 11), we obtain the C * EK (ω) with the lognormal distribution as:

C log, * EK (ω) = ζ η rmax r min 2 κr J 1 (κr) J 0 (κr) r exp - ln( r rm ) √ 2s 2 dr σ w rmax r min r exp - ln( r rm ) √ 2s 2 dr + 2Σ s rmax r min exp - ln( r rm ) √ 2s 2 dr . (15)
For sufficiently low frequencies, Eq. ( 15) reduces to

C log,0 EK = ζ η rmax r min r exp - ln( r rm ) √ 2s 2 dr σ w rmax r min r exp - ln( r rm ) √ 2s 2 dr + 2Σ s rmax r min exp - ln( r rm ) √ 2s 2 dr , (16)
where

rmax r min rexp - ln( r rm ) √ 2s 2 dr = - √ π √ 2 sr 2 m e 2s 2 erf 2s 2 -ln( rmax rm ) √ 2s -erf 2s 2 -ln( r min rm ) √ 2s , (17)
and

rmax r min exp - ln( r rm ) √ 2s 2 dr = - √ π √ 2 sr m e s 2 2 erf s 2 -ln( rmax rm ) √ 2s -erf s 2 -ln( r min rm ) √ 2s . ( 18
)
Substituting Eq. ( 17) and Eq. ( 18) into Eq. ( 16), the following is obtained

C log,0 EK = ζ η 1   σ w + 2Σse -3s 2 2 rm erf s 2 -ln( rmax rm ) √ 2s -erf s 2 -ln( r min rm ) √ 2s erf 2s 2 -ln( rmax rm ) √ 2s -erf 2s 2 -ln( r min rm ) √ 2s   (19)
Following [START_REF] Jougnot | Predicting the frequency-dependent effective excess charge density: A new up-scaling approach for seismoelectric modelling[END_REF], we can write Eq. ( 15) as

C log, * EK (ω) = C log,0 EK C rel,log, * EK (ω), (20)
where C rel,log, * EK (ω) is the frequency dependent relative streaming potential coefficient for the lognormal distribution with respect to the value at 0 Hz and is given by

C rel,log, * EK (ω) = rmax r min 2 κr J 1 (κr) J 0 (κr) rexp - ln( r rm ) √ 2s 2 dr rmax r min rexp - ln( r rm ) √ 2s 2 . (21)

Fractal Distribution

Fractal distribution has been already applied for porous media in many publications (e.g., [START_REF] Yu | A fractal permeability model for bi-dispersed porous media[END_REF][START_REF] Guarracino | A physically based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media[END_REF][START_REF] Soldi | An analytical effective excess charge density model to predict the streaming potential generated by unsaturated flow[END_REF]Thanh et al. 2020b,c). The fractal distribution of capillary tubes is given by (e.g., [START_REF] Tyler | Fractal processes in soil water retention[END_REF][START_REF] Yu | A fractal permeability model for bi-dispersed porous media[END_REF]Thanh et al. 2020b)

f (r) = D f r D f max r -D f -1 , ( 22
)
where D f is the fractal dimension for pore space that is between 1 and 2 in two-dimensional spaces and between 2 and 3 in three dimensional spaces. Note that the total number of capillaries N t following the fractal distribution in the REV is given by

N t = rmax r min f (r)dr = r max r min D f -1 ≈ r max r min D f . ( 23
)
Here, we consider that r max >> r min , that is, rmax r min D f is much larger than 1, which is normally valid in porous media (e.g., [START_REF] Yu | A fractal permeability model for bi-dispersed porous media[END_REF]Thanh et al. 2020b).

Substituting Eq. ( 22) into Eq. ( 11), we obtain the C * EK (ω) with the fractal distribution as:

C f ra, * EK (ω) = ζ η σ w + 2Σs rmax 2-D f 1-D f 1-α 1-D f 1-α 2-D f 2 -D f (1 -α 2-D f )r 2-D f max rmax r min 2 κr J 1 (κr) J 0 (κr) r 1-D f dr, (24)
where α = r min /r max .

Eq. ( 15) and Eq. ( 24) indicate that the dynamic streaming potential coupling coefficient is a complex function depending on the properties of water (η, σ w , ), mineral-water interfaces (ζ, Σ s ), microstructural parameters of porous media (D f , r min , r max for the fractal distribution and r m , s, r min , r max for the lognormal distribution) and frequency (ω).

When ω → 0, the pressure difference tends to a steady-state condition across the REV and, thus, the parameter κ approaches zero as indicated by Eq. ( 3). The limit of 2 κr J 1 (κr) J 0 (κr)

becomes (e.g., [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF][START_REF] Reppert | Frequency-dependent streaming potentials[END_REF])

lim κ→0 2 κr J 1 (κr) J 0 (κr) = 1. ( 25
)
Dynamic streaming potential coupling coefficient in porous media with different pore size distributions 11

Therefore, the integral term in Eq. ( 24) reduces to

lim κ→0 rmax r min 2 κr J 1 (κr) J 0 (κr) r 1-D f dr = r 2-D f max 2 -D f (1 -α 2-D f ), (26)
and Eq. ( 24) becomes

C f ra,0 EK = ζ η σ w + 2Σs rmax 2-D f 1-D f 1-α 1-D f 1-α 2-D f . ( 27
)
Note that, in Eq. ( 27), the proposed model reduces to that of [START_REF] Thanh | A fractal model for streaming potential coefficient in porous media[END_REF], which was proposed and validated for DC conditions.

If the surface conductivity can be neglected (Σ s = 0), Eq. ( 19) and Eq. ( 27) reduce to the widely used Helmholtz-Smoluchowski (HS) equation which does not include information about the medium geometrical properties and has been proven to be useful in a large range of natural geological media [START_REF] Smoluchowski | Contribution à la theorie de l'endosmose electrique et de quelques phenomènes correlatifs[END_REF]. Namely, the HS equation is given by

C HS EK = ζ ησ w . ( 28
)
It is also shown that the PSD does not have effect on the quasi-static streaming potential coupling coefficient as long as the surface conductivity is negligible, which is in agreement with the result reported in [START_REF] Jougnot | Exploring the effect of the pore size distribution on the streaming potential generation in saturated porous media, insight from pore network simulations[END_REF].

Similarly, the frequency dependent relative streaming potential coupling coefficient for the fractal PSD with respect to the zero frequency is given by

C rel,f ra, * EK (ω) = 2 -D f (1 -α 2-D f )r 2-D f max rmax r min 2 κr J 1 (κr) J 0 (κr) r 1-D f dr. (29)
Eq. ( 29) can be solved numerically once the parameters D f , r min and r max of porous media are defined.

RESULTS AND DISCUSSION

3.1 Sensitivity of the model

Dynamic streaming potential coupling coefficient in a single capillary

Figure 1 shows the dynamic relative streaming potential coupling coefficient C * EK (r, ω) as a function of frequency for a single capillary predicted Eq. ( 7): (a) real component and (b) imaginary component of C * EK (r, ω). In Eq. ( 7), we use representative values of r = 10 µm, C w = 10 -3 mol/L (from Eq. ( 4), ζ is obtained) and three representative values Σ s (0, 5×10 -9 and 10×10 -9 S). Note that we consider C w = 10 -3 mol/L as it has been proven to be a pertinent value in the context of groundwater studies (e.g., Jackson et al. 2012). The range of Σ s between 0 and 10×10 -9 S is commonly reported for (1953). It is seen that the surface conductivity has an influence on real and imaginary part of C * EK (r, ω). When ignoring the surface conductivity, one may overestimate the C * EK (r, ω), especially for small capillaries and low ionic concentration.

Quasi-static streaming potential coupling coefficient in porous media

As previously mentioned, we employ fractal and lognormal PSDs (see Figure 2) for pore sizes ranging from r min = 1 µm to r max = 100 µm. In particular, we consider the following characteristics for the (f) lognormal with r m = r max /3 and s = 0.4. Note that D f , r m , r min , r max and s can be selected with different values.

Figure 3 shows the variation of the quasi-static streaming potential coupling coefficient C 0 EK with ionic concentration. Fig. 3 (a) is predicted from Eq. ( 27) for the fractal distribution with three values of D f (1.4, 1.6, 1.8). Fig. 3 (b), (c) are predicted Eq. ( 19) for the lognormal distribution with three values of r m (r max /3, r max /10, r max /20 while fixing s = 0.1) and three values of s (0.1, 0.25, 04 while fixing r m = r max /10), respectively. The surface conductance Σ s is taken as 5 × 10 -9 S. Fig. 3 (a) shows that the C 0 EK from the fractal distribution is sensitive to D f and decreases with an increase of D f at low ionic concentration C w . When C w is larger than a certain value, that means, the surface conductivity is negligible, C 0 EK approaches C HS EK (see black solid line) irrespective of the geometrical properties (D f ) of the medium. The reason for the decrease of C 0 EK with increasing D f is that, when D f increases, the number of capillaries characterized by relatively small radii increases, as shown by the shift from Fig. 2 (a) to (d). Consequently, the surface conductivity of the REV increases and, thus, C 0 EK decreases. Note that the surface conductivity of porous media is dominated by the contribution from the smaller capillaries for the same surface conductance and electrical conductivity of water. Fig.

(b) shows that the C 0

EK predicted from the lognormal distribution is sensitive to r m and decreases with a decrease of r m at low ionic concentration C w . When C w is larger than a certain value, C 0 EK approaches C HS EK irrespective of the r m values, as discussed above (see black solid line). The decrease of C 0 EK with decreasing r m is explained in the similar way to that of Fig. 3 (a). Namely, when r m decreases, the number of smaller radius capillaries increases as shown by the shift from Fig. 2 (b) to (e). Consequently, the surface conductivity of the REV increases and C 0 EK decreases. Fig. 3 (c) shows that the C 0 EK is much less sensitive to the lognormal standard deviation s at low ionic concentration C w . When C w is larger than a certain value, C 0 EK also reduces to C HS EK (see black solid line). These results are consistent with previously published studies (e.g., [START_REF] Jougnot | Exploring the effect of the pore size distribution on the streaming potential generation in saturated porous media, insight from pore network simulations[END_REF]). Recently, [START_REF] Vinogradov | Influence of pore size distribution on the electrokinetic coupling L.D.Thanh[END_REF] proposed to use a rock-specific PSD directly extracted from the petrophysical characterization of a sample. This PSD is non-monotonic and follows a form given by three intervals:

f (r) =            B 1 r-r min rmax-r min m 1 , for r min ≤ r ≤ r 1 constant, for r 1 < r < r 2 B 2 r-rmax r min -rmax m 2 , for r 2 < r ≤ r max (30)
where B 1 and B 2 represent the normalisation factors of the first and third interval, respectively; 0 < m 1 , m 2 < ∞ (constant, unitless) are the respective skewing constants and r 1 and r 2 are limits of the second interval. The limits of each interval are determined by experimental data for the PSD. For example, for a sample of Berea sandstone reported in literature, [START_REF] Vinogradov | Influence of pore size distribution on the electrokinetic coupling L.D.Thanh[END_REF] 4 (a), we can predict the variation of C 0 EK with ionic concentration using the same approach as previously mentioned (see Fig. 4 (b) with Σ s = 5×10 -9 S). Note that we obtained C 0 EK numerically for a new PSD proposed by [START_REF] Vinogradov | Influence of pore size distribution on the electrokinetic coupling L.D.Thanh[END_REF]. The HS equation is also used for comparison in Fig. 4 (b). It is seen that the non-monotonic PSD proposed by [START_REF] Vinogradov | Influence of pore size distribution on the electrokinetic coupling L.D.Thanh[END_REF] is quite relevant to the lognormal PSD that is later shown to be pertinent for both consolidated samples (e.g., rocks) or unconsolidated ones (e.g., sand packs). Therefore, the non-monotonic PSD proposed by [START_REF] Vinogradov | Influence of pore size distribution on the electrokinetic coupling L.D.Thanh[END_REF] can provide realistic description of porous rocks and can be applied to study transport phenomena in porous media.

Dynamic streaming potential coupling coefficient in porous media

Figure 5 shows the dynamic relative streaming potential coupling coefficient as a function of frequency predicted from the lognormal distribution, given by Eq. ( 21) taking r min = 1 µm, r max = 100 µm, and considering three representative values of r m (r max /3, r max /10, r max /20) and s = 0.1. We illustrate the (a) real component, (b) imaginary component and (c) phase of C rel, * EK (ω). For comparison, the black line represents [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] prediction given by Eq. ( 8) for a single capillary of radius r = 10 µm. It is observed that the general behavior of C rel, * EK (ω) predicted from Eq. ( 21) for porous media and that predicted from Eq. ( 8) for a single capillary are similar. The noticeable characteristic of the curves is that the transition frequency shifts to higher frequency when r m decreases. This relationship is explained based on Eq. ( 12) for the transition frequency ω t . Namely, when r m decreases, the number of capillaries with smaller radii increases, as mentioned earlier. Consequently, the characteristic radius r decreases and the transition frequency ω t increases with decreasing r m . It is seen that the magnitude of the C rel, * EK (ω) is stable at low frequencies and decrease for frequencies greater than the transition frequency. The reason is that when the frequency increases, the inertia regime starts to prevail and the fluid movement starts to be increasingly out of phase with the applied oscillatory pressure, thus reducing the fluid velocity within the pore (e.g., [START_REF] Zhou | First-principles calculations of dynamic permeability in porous media[END_REF]. Therefore, the C rel, * EK (ω) decreases. Fig. 5 (a) and (b) shows that the real and imaginary parts of C rel, * EK (ω) follow the similar behaviors for different values of r m and decrease at the same rate at high frequencies. This explains the 45 o phase angle found at high frequencies as shown in Fig. 5 (c) and that is in agreement with [START_REF] Reppert | Frequency-dependent streaming potentials[END_REF].

Figure 6 shows the dynamic relative streaming potential coupling coefficient as a function of frequency predicted from the lognormal distribution with r min = 1 µm, r max = 100 µm for three different values of s (0.1, 0.25, 0.4) and a representative of r m = r max /10: (a) real component and (b) imaginary component of C rel, * EK (ω). Again, the black solid line is predicted from [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] given by Eq. ( 8) for a single capillary of radius r = r max /10 = 10 µm for comparison. We note that the observed behaviors in Fig. 6 are similar to Fig. 5. The coefficient C rel, * EK (ω) is sensitive to s and ω t decreases with an increase of s for a given value of r m . The reason is that when s increases, r increases (see shift from Fig. 2 (c) to (f)). Consequently, ω t decreases with increasing s.

Figure 7 shows the dynamic relative streaming potential coupling coefficient as a function of frequency predicted from the fractal distribution, given by Eq. ( 29 EK (ω). The black solid line is predicted from [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] given by Eq. ( 8) for a single capillary of radius r = 10 µm for comparison. It is observed that, regardless of D f , the real component of the C rel, * EK (ω) is rather stable at low frequencies and decrease for frequencies greater than a given threshold value. It is seen that Eq. ( 29) can reproduce the main trend predicted by the model of [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] that is valid for a single capillary at low frequency. However, the behavior diverges for higher frequencies. We also observe that the transition frequency ω t for porous media Dynamic streaming potential coupling coefficient in porous media with different pore size distributions 17 EK (ω). The black line is predicted from [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] given by Eq. ( 8) for a single capillary of radius r = 10 µm for comparison. moves to higher frequencies with an increase of D f . The reason for this is that when D f increases, the number of capillaries with small radius increases. Consequently, the characteristic radius r decreases and the transition frequency ω t increases with increasing D f . [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] given by Eq. ( 8) for a single capillary of radius r = r max /10 for comparison.

Comparison with published data

0.15×10 -6 , 0.065×10 -6 , 0.035×10 -6 and 0.024×10 -6 V/Pa for 0.012, 0.048, 0.095, 0.18 and 0.32 S/m, respectively. From Eq. ( 21) and Eq. ( 29), we can obtain C rel, * EK (ω) with fitting parameters and therefore C * EK (ω). Note that, due to numerical constraints associated with the integrations indicated by Eq. ( 21) and Eq. ( 29), we do not intent here to perform an exhaustive inversion of the parameters of the proposed model from experimental datasets. Our intention is to show that the proposed approach is capable of reproducing experimental results and, thus, we empirically search for the parameters that provide a relatively good fit. For the lognormal PSD, we found that the parameters r m = 5.8 µm, s = 0.1, r min = 0.13 µm and r max = 27 µm provide a good fit. For the fractal PSD, a good fit is obtained for D f = 1.5, r min = 0.13 µm and r max = 27 µm. It is seen that the proposed model using both distributions predicts very well the experimental data. The root mean square deviation (RMSD) for the lognormal and fractal distributions are 1.15×10 -9 and 2.23×10 -9 V/Pa. Therefore, the lognormal distribution provides a slightly better result than the fractal distribution. EK (ω). The black solid line is predicted from [START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] given by Eq. ( 8) for a single capillary of radius r = r max /10 for comparison.

fractal PSD, we take D f = 1.6, r min = 24 nm and r max = 4.7 µm. We observe that the proposed model using the lognormal and fractal PSDs can reproduce experimental data reported by [START_REF] Peng | Seismo-electric conversion in shale: experiment and analytical modelling[END_REF].

It seems that, in this case, the fractal PSD provides a slightly better prediction than the lognormal PSD (RMSD values for the lognormal and fractal PSDs are 0.022 and 0.003). At high frequency, as seen in Fig. 9 and later figure, the magnitude of C rel, * EK for the lognormal PSD decreases with increasing frequency at a higher rate than that for the fractal PSD.

Note that r max of porous media can be predicted from permeability using an expression suggested by [START_REF] Cai | Prediction of maximum pore size of porous media based on fractal geometry[END_REF]

r max = 1 2 32τ k (4 -D f )(1 -φ) (2 -D f )φ , ( 31
)
where τ is the tortuousity of porous media which can be estimated using τ = 1 -2.02ln(φ) given by [START_REF] Peng | Seismo-electric conversion in shale: experiment and analytical modelling[END_REF]. From Eq. ( 31), r max is estimated to be 15 µm and 5.7 µm for the samples reported by [START_REF] Zhu | Experimental measurements of the streaming potential and seismoelectric conversion in berea sandstone[END_REF] and [START_REF] Peng | Seismo-electric conversion in shale: experiment and analytical modelling[END_REF], respectively. Those values are reasonably good agreement with ones we obtained when fitting the data with our models (27 µm and 4.7 µm, respectively).

Figure 10 shows the variation of the magnitude of C rel, * EK (ω) as a function of frequency measured by Wang et al. (2015b) for a sandstone sample (φ = 0.302 and k = 1435 mD) with an error bar of approximately ±10% that is deduced from Fig. 4 of Wang et al. (2015b). The proposed model with the lognormal PSD is used to reproduce experimental data by taking fitting parameters of r m = 127 µm, s = 0.1, r min = 5 µm and r max = 700 µm. It is seen that the proposed model using the lognormal PSD can reproduce the main trend of experimental data. For simplicity, we do not show the prediction from the fractal PSD, as it can not match the behavior of measured data. Note that r m = 127 µm is comparable to the effective pore radius of 358 µm of the sample as reported by Wang et al. (2015b). Glover et al. (2012a) for the Ottawa sand. It is seen that the proposed model with the lognormal PSD provides a very good match with experimental data. However, the proposed model with the fractal PSD is only in good agreement with data at low frequencies. It indicates that the fractal PSD may not be not pertinent for the Ottawa sand, which exhibits a narrow PSD.

Previous works in the literature provide with models for the dynamic streaming potential coupling coefficient in porous media. For example, in its pioneering work, [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] provided a model for the dynamic streaming potential coupling coefficient in porous media as where

C rel, * EK (ω) = 1 -i m * 4 ω ω t 1 - λ d Λ 2 1 -i 3/2 λ d ωρ w η w 2 -1/2 (32)
ω t = φη w τ kρ w (33)
and

m * = φΛ 2 τ k . ( 34
)
From Eq. (32) to Eq. ( 34), λ d (m) is the Debye length and Λ is the characteristic length scale.

Dynamic streaming potential coupling coefficient in porous media with different pore size distributions 23

The parameter ω t (rad/s) is the transition frequency. In most geological media, the condition that λ d is much smaller than Λ is normally satisfied [START_REF] Jougnot | Exploring the effect of the pore size distribution on the streaming potential generation in saturated porous media, insight from pore network simulations[END_REF]. Therefore, a significant simplification can be made on Eq. ( 32) as performed by [START_REF] Walker | Permeability models of porous media: Characteristic length scales, scaling constants and time-dependent electrokinetic coupling[END_REF]:

C rel, * EK (ω) = 1 -i m * 4 ω ω t -1/2 . ( 35
)
As reported by Glover et al. (2012a) for the Ottawa sand, we have τ = 1.52, φ = 0.32, k = 1.19×10 -10 m 2 . The Debye length is calculated to be λ d = 9.66 nm for 10 -3 mol/L NaCl electrolyte.

The characteristic length scale Λ is found to be 62 µm by fitting. It is seen that the models given by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] and [START_REF] Walker | Permeability models of porous media: Characteristic length scales, scaling constants and time-dependent electrokinetic coupling[END_REF] also provide a very good match with experimental data with the suitable fitting parameter Λ. However, it is important to remark that our proposed approach is designed to allow for virtually any PSD. We only analyze two cases (fractal and lognormal PSDs) in this work. However, our approach permits to compute the response for the double-lognormal PSD, for example, a case that can not be explored using the models given by [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] or [START_REF] Walker | Permeability models of porous media: Characteristic length scales, scaling constants and time-dependent electrokinetic coupling[END_REF], which are restricted to only one characteristic length scale Λ.

CONCLUSIONS

We have proposed a physically-based model for the frequency dependence of the streaming potential coupling coefficient C EK (ω) by conceptualizing a porous medium as a bundle of tortuous capillaries and using the lognormal and fractal PSDs. The surface electrical conductivity was also taken into account in the proposed approach. It is seen that C EK (ω) is a complex function depending on the properties of water (η, σ w , ), mineral-water interfaces (ζ, Σ s ), microstructural parameters of porous media (D f , r min , r max for the fractal PSD and r m , s, r min , r max for the lognormal PSD) and frequency. The results also show that the PSD does not have effect on the quasi-static streaming potential coupling coefficient as reported in the literature when the surface conductivity is negligible. Parameters influencing C EK (ω) are investigated and explained based on the PSD and the transition angular frequency. The proposed model is then compared with published data and other published models. We found that the proposed model using the lognormal PSD is in very good agreement with the experimental data and previous models in the literature. The proposed model using the fractal PSD provide a good match with published data for sandstone samples but not for the sand samples. The reason may be that the fractal PSD is not pertinent for samples with narrow PSDs. Our results suggest that the PSD of porous media plays an crucial role in the dynamic behaviour of C EK (ω). Finally, we remark that the proposed approach works for virtually any PSD, including ones that can be directly measured from rock characterization. This particular feature makes this model more versatile than previous models available in literature. A model for C EK (ω) under unsaturated conditions using the proposed technique will be performed in our future work.

o

  Length of the representative elementary volume (REV) m signals are generated by electrokinetic coupling from seismic wave propagation in fluid-filled porous media and are directly related to the existence of an electrical double layer (EDL)

Figure 1 .

 1 Figure 1. Dynamic relative streaming potential coupling coefficient C * EK (r, ω) as a function of frequency for a single capillary predicted Eq. (7) for representative values of r = 10 µm, C w = 10 -3 mol/L (from Eq. (4), ζ is obtained) and three values Σ s (0, 5×10 -9 and 10×10 -9 S): (a) real component and (b) imaginary component of C * EK (r, ω).

  PSD: (a) fractal with D f = 1.4; (d) fractal with D f = 1.8; (b) lognormal with r m = r max /3 and s = 0.1; (e) lognormal with r m = r max /20 and s = 0.1; (c) lognormal with r m = r max /3 and s = 0.1;

Figure 2 .

 2 Figure 2. PSDs used in this work. We consider r min = 1 µm and r max = 100 µm, and illustrate: (a) fractal PSD with D f = 1.4; (d) fractal PSD with D f = 1.8; (b) lognormal PSD with r m = r max /3 and s = 0.1; (e) lognormal PSD with r m = r max /20 and s = 0.1; (c) lognormal PSD with r m = r max /3, s = 0.1; (f) lognormal PSD with r m = r max /3 and s = 0.4.

Figure 3 .

 3 Figure 3. Variation of the quasi-static streaming potential coupling coefficient C 0 EK with ionic concentration: (a) is predicted from Eq. (27) for the fractal distribution with three values of D f (1.4, 1.6, 1.8); (b), (c) are predicted from Eq. (19) for the lognormal distribution with three values of r m (r max /3, r max /10, r max /20 while fixing s = 0.1) and three values of s (0.1, 0.25, 04 while fixing r m = r max /10), respectively.

  obtained r min Dynamic streaming potential coupling coefficient in porous media with different pore size distributions 15

Figure 4 .

 4 Figure 4. (a) A new PSD proposed by Vinogradov et al. (2021), fractal PSD (D f = 1.5) and lognormal PSD (r m = 11 µm and s = 0.15) in the same range of radii from r min = 5 µm to r max = 100 µm; (b) Variation of the quasi-static streaming potential coupling coefficient C 0 EK with ionic concentration for different PSDs (Σ s = 5 × 10 -9 S).

  ) taking r min = 1 µm, r max = 100 µm for three representative values of D f (1.4, 1.6, 1.8): (a) real component and (b) imaginary component of C rel, *

Figure 5 .

 5 Figure 5. The dynamic relative streaming potential coupling coefficient as a function of frequency predicted from the lognormal distribution -Eq. (21) with r min = 1 µm, r max = 100 µm for three representative values of r m (r max /3, r max /10, r max /20) and a representative of s = 0.1: (a) real component, (b) imaginary component and (c) phase (in degree) of C rel, *EK (ω). The black line is predicted from[START_REF] Packard | Streaming potentials across glass capillaries for sinusoidal pressure[END_REF] given by Eq. (8) for a single capillary of radius r = 10 µm for comparison.

Figure 8 Figure 6 .

 86 Figure 8 shows the variation of the magnitude of C * EK (ω) as a function of frequency measured by Zhu & Toksoz (2013) for a Berea sandstone sample (permeability k = 450×10 -15 m 2 and porosity φ = 0.23) saturated by different pore water conductivities. The proposed model with lognormal and fractal distributions is used to explain the experimental data (see solid lines in Fig. 8 (a) and (b), respectively). As reported by Zhu & Toksoz (2013), we consider the values of C 0 EK are 0.3×10 -6 ,

Figure 9 Figure 7 .

 97 Figure9shows the variation of the magnitude of C rel, * EK (ω) as a function of frequency measured by[START_REF] Peng | Seismo-electric conversion in shale: experiment and analytical modelling[END_REF] for a sandstone sample with φ = 0.0390 and k = 10.1 mD. The proposed model with the lognormal and fractal distributions is also used to reproduce experimental data. For the lognormal PSD, the data is fitted by taking r m = 1.4 µm, s = 0.1, r min = 24 nm and r max = 4.7 µm. For the

Figure 8 .

 8 Figure 8. Comparison between the amplitude of C * EK as a function of frequency (symbols) measured by Zhu & Toksoz (2013) and the model predictions (solid lines): (a) lognormal PSD with the best parameters of r m = 5.8 µm, s = 0.1, r min = 0.13 µm and r max = 27 µm; (b) fractal PSD with the fitting parameters of D f = 1.5, r min = 0.13 µm and r max = 27 µm.

Figure 9 .

 9 Figure 9. Comparison between the amplitude of C rel, * EK as a function of frequency for a sandstone sample measured by Peng et al. (2020) and the model predictions: (a) lognormal PSD with fitting parameters of r m = 1.4 µm, s = 0.1, r min = 24 nm and r max = 4.7 µm; (b) fractal PSD with fitting parameters of D f = 1.6, r min =

Figure 11

 11 Figure 11 shows the variation of C rel, * EK (ω) as a function of the frequency ω measured by Glover et al. (2012a) for Ottawa sand with modal grain radius of 235 µm saturated by 10 -3 mol/L NaCl electrolyte as shown by symbols: (a) magnitude, (b) real part, and (c) imaginary part of C rel, * EK (ω). The proposed model with the lognormal and fractal distributions is used to reproduce the experimental data. For the lognormal distribution, the fitting parameters are r m = 55 µm, s = 0.1, r min = 1.05 µm and r max = 105 µm. For the fractal distribution, the fitting parameters are D f = 1.1, r min = 1.05 µm and r max = 105 µm. Note that r m = 55 µm is quite close to the effective pore radius r p = 67 µm reported by Glover et al. (2012a) for the Ottawa sand. It is seen that the proposed model with

Figure 11 .

 11 Figure 11. The dynamic streaming potential coefficient as a function of frequency. Experimental data is obtained from Glover et al. (2012a) for Ottawa sand (φ = 0.32, k = 1.19×10 -10 m 2 ) and the predictions are from the proposed model, the models by Pride (1994) and Walker & Glover (2010): (a) Magnitude of the C rel EK (ω), (b) real component C rel EK (ω), and (c) imaginary component of C rel EK (ω).

ACKNOWLEDGMENTS

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103. 99-2019.316.

Data Availability Statement

The data underlying this article will be shared on reasonable request to the corresponding author.