A physically-based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media
AA physically-based analytical model todescribe effective excess charge for streaming potential generation in watersaturated porous media
L. Guarracino , , + , ∗ and D. Jougnot , + May 1, 2019 (1) Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, Facultad de Ciencias As-tron´omicas y Geof´ısicas, Universidad Nacional de La Plata, La Plata, Argentina(2) Facultad de Ciencias Naturales y Museo, Universidad Nacional de La Plata, La Plata,Argentina(3) Sorbonne Universit´e, CNRS, EPHE, UMR 7619 Metis, F-75005, Paris, France(+)Both authors contributed equally to this publication(*)Corresponding author: [email protected] paper has been published in
Journal of Geophysical Research – Solid Earth , pleasecite as:
L. Guarracino, D. Jougnot (2018) A physically-based analytical model todescribe effective excess charge for streaming potential generation in wa-ter saturated porous media, Journal of Geophysical Research - Solid Earth,123(1), 52-65, doi:10.1002/2017JB014873. a r X i v : . [ phy s i c s . g e o - ph ] A p r eypoints • Derivation of a physically-based analytical model to determine the effective excesscharge • Mechanistic explanation for the empirical dependence between effective excess chargeand permeability • The new model reproduces experimental data for different media and a broad rangeof ionic concentrations
Abstract
Among the different contributions generating self-potential, the streaming po-tential is of particular interest in hydrogeology for its sensitivity to water flow. Es-timating water flux in porous media using streaming potential data relies on our ca-pacity to understand, model, and upscale the electrokinetic coupling at the mineral-solution interface. Different approaches have been proposed to predict streamingpotential generation in porous media. One of these approaches is the flux averag-ing which is based on determining the excess charge which is effectively dragged inthe medium by water flow. In this study, we develop a physically-based analyti-cal model to predict the effective excess charge in saturated porous media using aflux-averaging approach in a bundle of capillary tubes with a fractal pore-size distri-bution. The proposed model allows the determination of the effective excess chargeas a function of pore water ionic concentration and hydrogeological parameters likeporosity, permeability and tortuosity. The new model has been successfully testedagainst different set of experimental data from the literature. One of the mainfindings of this study is the mechanistic explanation to the empirical dependencebetween the effective excess charge and the permeability that has been found byseveral researchers. The proposed model also highlights the link to other lithologicalproperties and it is able to reproduce the evolution of effective excess charge withelectrolyte concentrations. Introduction
The self-potential (SP) method is a passive geophysical method based on the measure-ments of the electrical field which is naturally generated in the subsurface. The SP methodwas first proposed by Robert Fox in 1830 [
Fox , 1830] and it is considered as one of the old-est geophysical methods. Although SP data are relatively easy to measure, the extractionof useful information is a non-trivial task since the recorded signals are a superposition ofdifferent SP components. In natural porous media, signals are mainly generated by elec-trokinetic (water flux) and electrochemical (ionic fluxes or redox reactions) phenomena.For more details of this method and for an overview of all possible SP sources we refer to
Revil and Jardani [2013].In the present study, we focus on the electrokinetic (EK) contribution to the SP, thatis the part of the signal generated from the water flow in porous media (often referredto as streaming potential). The surface of the minerals that constitute porous mediais generally electrically charged, creating an electrical double layer (EDL) containing anexcess of charge that counterbalances the charge deficiency of the mineral surface [see
Hunter , 1981,
Leroy and Revil , 2004]. Figure 1a shows how the EDL is composed: a Sternlayer that contains only counterions coating the mineral with a very limited thickness, anda diffuse layer that contains both counterions and coions but with a net excess charge.The shear plane, that can be approximated as the limit between the Stern layer anddiffuse layer [e.g.
Leroy and Revil , 2004], separates the mobile and immobile part of thewater molecules when subjected to a pressure gradient. This plane is characterized by anelectrical potential called ζ -potential [see Hunter , 1981]. When the water flows throughthe pore it drags a fraction of the excess charge that give rise to a streaming current andan electrical potential field. E xc e ss c ha r ge , [ C m − ] P o r e v e l o c i t y , v [ m s − ] S u rf ace o f s ili ca m i n e r a l s O - O - O - OHOHOH S t e r n l a y e r Shear plane O - O - QQ F r ee e l ec t r o l y t e Capillary radius R OHO - a. Q v Q v ζ D i ff u s e G ouy - C h a p m a n l a y e r d λ D − − r [m]0 0.2 0.4 0.6 0.8 1x 10 − − r [m] C = 10 − mol/LC = 10 − C = 10 − b.c. mol/Lmol/L Figure 1: (a) Scheme of the electrical double layer for a given capillary of radius R .Distribution of the excess charge (b) and the pore vater velocity (c) from the shear plane( r = 0 m) to the center of the capillary ( r = R = 10 − m). The excess charge is calculatedusing (14) for different ionic concentrations of NaCl and the Jaafar et al. [2009] model forthe ζ -potential. Figure modified from Jougnot et al. [2015].The EK phenomenon has been studied experimentally and theoretically for more thana century due its relevance in many practical applications in various fields (e.g., microflu-3dic, chemical engineering, fluid mechanics). In hydrogeophysics, its interest relies on thefact that SP data is sensitive to groundwater flow. Indeed, groundwater flow monitoringcan otherwise be performed locally using rather intrusive methods (i.e. hydraulic headmeasurements between boreholes or in situ sensors, heat or chemical tracer evolution froma single well). However, given that SP only provides indirect measurements (electrical po-tential) of water flux, it is necessary to rely on a theoretical framework and petrophysicalrelationships that can predict the EK phenomenon at macroscopic scales.Two main approaches to simulate streaming current generation in fully saturatedporous media can be found in the literature. The classical approach relies on the use ofthe coupling coefficient, which is a rock-dependent property that relates the differenceof hydraulic pressure to the difference in electrical potential. This property has firstbeen described experimentally [e.g.
Quincke , 1859,
Dorn , 1880] and later quantified by
Helmholtz [1879] and von Smoluchowski [1903]. The so-called Helmholtz-Smoluchowskicoupling coefficient has been developed from a capillary-tube model and its final expressiondoes not depend on the geometrical properties of the porous medium. Therefore, it hasbeen used for any kind of medium under the assumption that the electrical conductivity ofthe mineral surface could be neglected. When it is not the case, alternative formula havebeen proposed by several researchers [e.g.,
Revil et al. , 1999,
Glover and D´ery , 2010].Various model using capillaries to predict the coupling coefficient under saturated andpartially saturated conditions can be found in the literature [e.g.
Rice and Whitehead ,1965,
Ishido and Mizutani , 1981,
Jackson , 2010,
Jackson and Leinov , 2012,
Thanh et al. ,2017]. The second approach to simulate streaming current generation is more recentand focuses on the excess charge that is dragged by the water flow. The first referenceto this approach in the English literature can be found in
Kormiltsev et al. [1998] andmore detailed theoretical frameworks can be found in
Revil and Leroy [2004],
Revil et al. [2007] and
Revil [2016a,b]. In this approach, the coupling parameter is the excess chargethat is effectively dragged by water in the porous media. Then, the streaming currentand streaming potential distribution can be generated by multiplying the effective excesscharge by the water velocity. Note that both approaches describe the same physics; themain difference between them relies on the parameter (coupling coefficient or excess chargedensity) used to describe the electrokinetic coupling between fluid flow and streamingpotential generation.In this work, we focus on the excess charge approach in the framework proposed by
Sill [1983] and modified by
Kormiltsev et al. [1998] and
Revil et al. [2007], where SPsignals can be directly related to the water velocity: ∇ · ( σ ∇ ϕ ) = ∇ · (cid:16) ˆ Q v v D (cid:17) (1)where, σ is the electrical conductivity (S m − ), ϕ the electrical potential (V) and ˆ Q v theexcess charge density (C m − ) effectively dragged by the water flow at a given Darcyvelocity v D (m s − ). From Eq. (1), it becomes clear that determining the effective excesscharge is crucial to predict streaming potential signal in hydrosystems and to infer waterflux from SP measurements.Experimental evidences has shown that the effective excess charge depends on theporous medium permeability [ Titov et al. , 2002,
Jardani et al. , 2007,
Bol`eve et al. , 2012].The first relationship to estimate the effective excess charge from permeability can be4ound in
Titov et al. [2002] (Fig. 2 of their paper). Some years later,
Jardani et al. [2007]proposed a more precise empirical relationship that has been proven to be very useful as itdecreases the number of variables to be estimated and provides good estimates of efectiveexcess charge for different types of porous media. This relationship has been used in manystudies involving electrokinetic phenomena, such as dam leakage [e.g.
Bol`eve et al. , 2009,
Ikard et al. , 2014], surface-groundwater interaction [e.g.
Linde et al. , 2011], seismoelectricstudies [e.g.
Mahardika et al. , 2012,
Jougnot et al. , 2013,
Revil et al. , 2015,
Monachesiet al. , 2015], hydraulically-active fracture identification [e.g.
Roubinet et al. , 2016] andpermeability field characterization [e.g.
Jardani and Revil , 2009,
Soueid Ahmed et al. ,2014]. However, Eq. (26) has been obtained regardless of pore water composition andother hydraulic parameters of porous media. Unfortunately, this relationship does nottake into account the dependence of the EDL on the ionic concentration of the electrolyte[see the discussion in
Jougnot et al. , 2015].The most rigorous approach to construct an accurate description of any phenomenaat macroscopic scales is to start with a pore-scale description and then apply an upscalingmethod to the microscopic equations [
Bernab´e and Maineult , 2015]. In this study, theporous media is conceptualized as a bundle of capillary tubes with a fractal pore-sizedistribution. The porosity and permeability of the porous media are estimated using anequivalent medium theory. This geometrical description of the porous media and up-scaling procedure have been successfully used to describe water flow in fractured media[
Guarracino , 2006,
Guarracino and Quintana , 2009], the evolution of multiphase flowproperties during mineral dissolution [
Guarracino et al. , 2014] and relations between hy-draulic parameters [
Guarracino , 2007]. On the other hand, the effective excess charge ina single tube can be calculated from the radial distributions of excess charge and watervelocity. Then the effective excess charge at the macroscopic scale is estimated usingthe flux-averaging technique proposed by
Jougnot et al. [2012]. In that paper, they showthat by combining these hydraulic and EK properties, a closed-form expression for theeffective excess charge can be obtained in terms of permeability, porosity, tortuosity, ionicconcentration, ζ -potential, and Debye length. The dependence of the developed modelon ionic concentration, permeability, porosity and also grain size is tested using differentset of experimental data. It is also shown that the proposed model allows to derive fromphysical concepts the empirical relation between effective excess charge and permeabilityfound by Jardani et al. [2007].
The proposed model is based on the macroscopic description of effective excess chargein porous media from the upscaling of pore-size flow and electrokinetic phenomena. Theporous medium is conceptualized as an equivalent bundle of water-saturated capillarytubes with a fractal law distribution of pore sizes. First, we derive expressions for themost important macroscopic hydraulic parameters (porosity and permeability). Then,we obtain an approximate expression for the excess charge in a single tube and, fromthis result, we estimate the effective excess in the porous media in terms of the hydraulicparameters.To derive both hydraulic and electrokinetic properties we consider as representative5lementary volume (REV) a cylinder of radius R REV (m) and lenght L (m). The poresare assumed to be circular capillary tubes with radii varing from a minimum pore radius R min (m) to a maximum pore radius R max (m).The cumulative size-distribution of pores whose radii are greater than or equal to R (m) is assumed to obey the following fractal law [ Tyler and Wheatcraft , 1990,
Yu et al. ,2003,
Guarracino et al. , 2014]: N ( R ) = (cid:18) R REV R (cid:19) D , (2)where D is the fractal dimension of pore and 0 < R min ≤ R ≤ R max < R REV . Byusing the Sierpinski carpet (a classical fractal object)
Tyler and Wheatcraft [1990] showthat the fractal dimension of (2) ranges from 1 to 2, with the highest values associatedwith the finest textured soils. Capillary tube models with this type of fractal pore-sizedistributions have been successfully used to describe macroscopic process and hydraulicproperties in different porous media [e.g.
Yu et al. , 2003,
Guarracino et al. , 2014,
Soldiet al. , 2017].Note that the cumulative number of pores given by (2) decreases with the increase ofpore radius R , then differentiating (2) with respect to − R we can obtain the number ofpores whose radii are in the infinitesimal range R to R + dR : dN = DR DREV R − D − dR. (3) The porosity φ of the REV defined above can be straightforward computed from itsdefinition: φ = Volume of poresVolume of REV = (cid:82) R max R min V p ( R ) dNπR REV
L , (4)where V p ( R ) = πR l (m ) is the volume of a single tube of radius R and length l (m).Substituting (3) into (4) and solving the definite integral we obtain φ = τ DR − DREV (2 − D ) (cid:0) R − Dmax − R − Dmin (cid:1) , (5)where τ = l/L is the dimensionless hydraulic tortuosity of the capillary tubes [ Scheidegger ,1958]. Note that the model assumes a single value of τ for all capillary tubes, this valuemust be considered as a mean tortuosity value of all tube sizes.In order to obtain both the permeability and the excess charge of the REV we need todescribe the water flow in a single tube. For a laminar flow rate, the velocity distributioninside the tube can be described by the Poiseuille model [ Bear , 1988]: v ( R, r ) = ρ w g ητ (cid:2) R − ( R − r ) (cid:3) ∆ hL , (6)where r (m) is the distance from the pore wall ( r = 0) to the center of the tube ( r = R ), ρ w the water density (kg/m ), g the gravitational acceleration (m/s ), η the dynamic viscosity6Pa s), and ∆ h the pressure head drop across the REV (m). The average velocity v (m/s)in the capillary tube has the following expression: v ( R ) = ρ w g ητ R ∆ hL . (7)The total volumetric flow through the REV V Q (m /s) is the sum of the volumetricflow rates of all individual tubes. According to (7) and (3) V Q can be computed as follows: V Q = (cid:90) R max R min v ( R ) πR dN = ρ w g ητ πDR DREV − D (cid:0) R − Dmax − R − Dmin (cid:1) ∆ hL . (8)On the basis of Darcy’s law (macroscopic scale), V Q can also be expressed as V Q = πR REV ρ w gη k ∆ hL , (9) k being the intrinsic permeability of the porous media (m ).Then, combining (8) and (9) we obtain the following expression for permeability k interms of the geometrical parameters of the porous media: k = D τ (4 − D ) R − DREV (cid:0) R − Dmax − R − Dmin (cid:1) . (10)It is important to remark that a similar equations for φ and k have been recentlyderived by Soldi et al. [2017] assuming constrictive capillary tubes. In the limit case ofstraight tubes both expressions for φ and k are identical (see equations (11) and (15) oftheir paper).For most porous media it can be assumed that R min << R max [ Yu and Li , 2001].Then, equations (5) and (10) can be reduced to φ = τ D (2 − D ) R − DREV R − Dmax , (11) k = D τ (4 − D ) R − DREV R − Dmax . (12)Finally, combining (11) and (12) we obtain a simple relationship to estimate perme-abilty from porosity k = γφ − D − D , (13)where γ = R REV τ (4 − D ) (cid:0) − DτD (cid:1) − D − D . Note that for D = 1 the exponent of the porosity is 3 and(13) is equivalent to Kozeny’s equation [ Kozeny , 1927].
Since electrokinetic phenomenon is caused by the coupling of fluid flow and charge dis-tribution at pore scale, the magnitude of this phenomenom will be mainly determined bythe macroscopic hydraulic and electrical properties of the porous medium. Based on the7revious description of hydraulic properties we will compute the effective excess chargedensity ˆ Q REVv carried by the water flow in the REV (C/m ). The effective excess chargedensity, also called dynamic excess charge depending on the authors and symbolized byˆ Q v or ¯ Q effv , has to be distinguished from the other excess charge densities contained inthe pore space: Q v the total excess charge density (C/m ), which includes all the chargesfrom the Stern and diffuse layers of the EDL, and ¯ Q v the excess charge located only inthe diffuse layer (C/m ) (Fig. 1a) [see the discussion in Revil , 2017]. It is important toremark that in this study we do not consider the charges that are located in the Sternlayer as they are fixed on the pore wall and do not contribute to the streaming current.We start this derivation by defining the excess charge distribution ¯ Q v in a capillarytube saturated by a binary symetric 1:1 electrolyte (e.g., NaCl) as follows¯ Q v ( r ) = N A e C (cid:20) e − e ψ ( r ) kBT − e e ψ ( r ) kBT (cid:21) , (14)where N A is the Avogadro’s number (mol − ), e the elementary charge (C), C the ionicconcentration far from the mineral surface (mol/m ), ψ the local electrical potential inthe pore water (V), k B the Boltzman constant (J/K), and T is the absolute temperature(K). For the thin double layer assumption (i.e., the thickness of the double layer is smallcompared to the pore size) the local electrical potential can be expressed [ Hunter , 1981]: ψ ( r ) = ζe − rlD , (15) l D = (cid:115) (cid:15)k B T N A C e , (16)where ζ (V) is the ζ -potential on the shear plane (Fig. 1a), l D the Debye lenght (m), and (cid:15) the water dielectric permittivity (F/m).As proposed by Jougnot et al. [2012], the effective excess charge density ˆ Q Rv carriedby the water flow in a single tube of radius R is defined byˆ Q Rv = 1 v ( R ) πR (cid:90) A Q v ( r ) v ( R, r ) dA, (17)being A the cross sectional area of the tube (m ). Using a polar coordinate system withthe pole located in the center of the tube, equation (17) becomesˆ Q Rv = 2 v ( R ) R (cid:90) R Q v ( r ) v ( R, r )( R − r ) dr. (18)In order to obtain a closed-form analytical expression of ˆ Q Rv we approximate the ex-ponential terms of (14) by a four-term Taylor serie: e ± e ψ ( r ) kBT = 1 ± e ψ ( r ) k B T + 12 (cid:18) e ψ ( r ) k B T (cid:19) ± (cid:18) e ψ ( r ) k B T (cid:19) . (19)8ubstituting (19) in (14) and solving (18) we obtainˆ Q Rv = − N A e C ζk B T ( R/l D ) (cid:26) − e − RlD (cid:20)(cid:16) Rl D (cid:17) + 3 (cid:16) Rl D (cid:17) + 6 (cid:16) Rl D (cid:17) + 6 (cid:21)(cid:27) + N A e C ζk B T ( R/l D ) (cid:26) − e − RlD (cid:20)(cid:16) Rl D (cid:17) + 2 (cid:16) Rl D (cid:17) + 2 (cid:21)(cid:27) − N A e C ζk B T ( R/l D ) (cid:110) − e − RlD (cid:104)(cid:16) Rl D (cid:17) + 1 (cid:105)(cid:111) − N A e C ζ k B T ) (3 R/l D ) (cid:26) − e − RlD (cid:20)(cid:16) Rl D (cid:17) + 3 (cid:16) Rl D (cid:17) + 6 (cid:16) Rl D (cid:17) + 6 (cid:21)(cid:27) + N A e C ζ ( k B T ) (3 R/l D ) (cid:26) − e − RlD (cid:20)(cid:16) Rl D (cid:17) + 2 (cid:16) Rl D (cid:17) + 2 (cid:21)(cid:27) − N A e C ζ k B T ) (3 R/l D ) (cid:110) − e − RlD (cid:104)(cid:16) Rl D (cid:17) + 1 (cid:105)(cid:111) . (20)For the thin double layer assumption (i.e., the thickness of the double layer is smallcompared to the pore size) we consider l D << R and (20) can be reduced toˆ Q Rv = 8 N A e C ( R/l D ) (cid:34) − e ζk B T − (cid:18) e ζ k B T (cid:19) (cid:35) . (21)Figure 2 (a) shows the effective excess charge ˆ Q Rv predicted by (20) and (21) for aionic concentration C = 1 mol/m . Note that even though the number of terms ofequation (21) is drastically reduced, both equations predict similar values of ˆ Q Rv . In orderto test the general validity of (21)under the thin double layer assumption, we compareapproximate values of ˆ Q Rv with exact values obtained by the numerical solution of (18)assuming pore-sizes R greater that 5 Debye lenghts. Figure 2 (b) presents the goodnessof the fit for different values of ionic concentration C . From the analysis of Figure 2,we conclude that equation (21) predicts fairly well the effective excess charge in capillarytubes for a wide range of radius and ionic concentration values.The effective excess charge ˆ Q REVv carried by the water flow in the REV is defined byˆ Q REVv = 1 v D πR REV (cid:90) R max R min ˆ Q Rv ( R ) v ( R ) πR dN, (22)where v D = ρ w gη k ∆ hL is the Darcy’s velocity (m/s) (macroscopic scale). Substituting (21),(7) and (3) in (22) and assuming R min << R max yieldsˆ Q REVv = 8 N A e C (cid:34) − e ζk B T − (cid:18) e ζ k B T (cid:19) (cid:35) − D − D (cid:18) l D R max (cid:19) . (23)Finally, combining (11), (12) and (23) we obtain the following expression for ˆ Q REVv :ˆ Q REVv = N A e C l D (cid:34) − e ζk B T − (cid:18) e ζ k B T (cid:19) (cid:35) τ φk . (24)The above equation constitutes the main result of this paper. Note that (24) predictsthe effective excess charge density in terms of both macroscopic hydraulic parameters9 −6 −4 −2 −8 −7 −6 −5 −4 −3 −2 Q ^ v R ( C / m ) R (m) a) Analytical estimates of Q^ vR Eq. (19) Eq. (20) 10 −3 −2 −1 −8 −7 −6 −5 −4 Q ^ v R ( C / m ) R (m) b) Numerical and analytical estimates of Q^ vR C =0.1 mol/m C =1 mol/m C =10 mol/m C =100 mol/m Figure 2: (a) Analytical estimates of ˆ Q Rv using equations (20) and (21) for a ionicconcentration C = 1 mol/m ; (b) numerical (points from Eq. 17) and analytical (linesfrom Eq. 21) estimates of ˆ Q Rv for different concentration values ( R > l D ). The ζ -potentialvalues are computed using (25).(porosity, tortuosity and permeability) and electrokinetic parameters (ionic concentration, ζ -potential and Debye lenght). This equation gives insight into the role of macroscopichydraulic parameters and it can be considered a starting point for designing non-invasivemethods to monitoring groundwater flow using self-potential measurements. a) log (Q^ vREV ) for C =100 mol/m φ [-] 10 -20 -18 -16 -14 -12 -10 -8 P e r m eab ili t y k [ m ] -6-4-2 0 2 4 6 8 10 b) log (Q^ vREV ) for k=10 -14 m φ [-] 10 -1 I on i c c on c en t r a t i on C [ m o l / m ] (Q^ vREV ) for φ =0.2 10 -20 -16 -12 -8 Permeability k [m ] 10 -1 I on i c c on c en t r a t i on C [ C / m ] -6-4-2 0 2 4 6 8 10 Figure 3: Parametric analysis of effective excess charge ˆ Q REVv : (a) sensitivity to porosityand permeability for a fixed value of ionic concentration, (b) sensitivity to porosity andionic concentration for a fixed value of permeability, (c) sensitivity to permeability andionic concentration for a fixed value of porosity.In order to study the role of porosity φ , permeability k , and ionic concentration C on effective excess charge, we perform a parametric analysis of equation (24). The ionicconcentration dependence of ζ -potential is assumed to obey the relation proposed by Prideand Morgan [1991]: ζ ( C ) = a + b log ( C ) , (25)where a and b are fitting parameters. For this study we use the parameter values obtainedby Jaafar et al. [2009] on silicate-based materials for NaCl brine a =-6.43 mV and b =20.85mV. 10igure 3 summarizes the parametric analysis of log (cid:16) ˆ Q REVv (cid:17) for the following rangesof variability of ionic concentration C , permeability k and porosity φ : 10 − mol/m ≤ C ≤ mol/m , 10 − m ≤ k ≤ − m and 0 . ≤ φ ≤ .
5. Figure 3a showsthe effect of porosity and permeability on ˆ Q REVv for a fixed value of ionic concentration( C = 100 mol/m ). It can be observed that ˆ Q REVv is strongly determined by perme-ability while porosity only produces a slightly increase of ˆ Q REVv values. Figure 3b showsthe effect of porosity and ionic concentration on ˆ Q REVv . As shown in this panel, theseparameters can change ˆ Q REVv values in two orders of magnitude for a fixed value of per-meability ( k = 10 − m ). An increase of ˆ Q REVv can be observed when ionic concentrationdecreases and porosity increases. Finally, Figure 3c shows the role of permeability andionic concentration on ˆ Q REVv for a fixed value of porosity ( φ = 0 . Q REVv is much more significant than ionic concentration.From this parametric analysis we can conclude that effective excess charge ˆ Q REVv ishighly sensitive to permeability values. However, porosity and ionic concentration canmodify ˆ Q REVv values in two orders of magnitude for a given value of permeability [seediscussions in
Jougnot et al. , 2012, 2015]. The increase of porosity or the decrease of ionicconcentration produce an increase of ˆ Q REVv values.
Jardani et al. [2007]
The empirical relationship to estimate the effective excess charge ˆ Q REVv from permeability k proposed by Jardani et al. [2007] reads as follows:log ( ˆ Q REVv ) = A + A log ( k ) , (26)where A = − . A = − . φ = ( k/γ ) − D − D .Replacing this expression in (24) and taking the logarithm on both sides of the resultingequation, we obtain exactly (26) but with the following constants: A = log (cid:26) N A e C γ − D − D (cid:20) − e ζk B T − (cid:16) e ζ k B T (cid:17) (cid:21) (cid:0) l D τ (cid:1) (cid:27) ,A = − − D . (27)According to our model, the log ( ˆ Q REVv )-intercept A mainly depends on chemical andinterface parameters while the slope A only depends on the fractal dimension of the poresize distribution (1 < D < − < A < − .
666 and that the value obtained by
Jardani et al. [2007] A = − . D = 1 . Q REVv data determined by severalauthors [
Ahmad , 1964,
Cassagrande , 1983,
Friborg , 1996,
Pengra et al. , 1999,
Revil et al. ,2005, 2007, 2012,
Boleve et al. , 2007,
Jardani et al. , 2007,
Glover and D´ery , 2010,
Zhuand Toks¨oz , 2012,
Jougnot and Linde , 2013]. The best fit of our model is obtained forfractal dimension D = 1 . D can fit a wide range of soiltextures, then it can be considered a mean value of all pore size distributions.It is worth mentioning that ˆ Q REVv data displayed on Fig. 4 are obtained from elec-trokinetic coupling coefficient C EK values (V/Pa) using the following equation ( Revil andLeroy [2004],
Revil et al. [2007]): ˆ Q REVv = − C EK σηk , (28)where σ (S/m) is the electrical conductivity, η (Pa s) the dynamic viscosity, and k (m ) thepermeability. The values of σ and k of each sample have been measured independently. Permeability, k [m ] -20 -18 -16 -14 -12 -10 -8 -6 E ff e c t i v e e xc e ss c ha r ge ,[ C m - ] -4 -2 Clayed soils (Cassagrande, 1983)Glacial tills (Friborg, 1996)Sand (Ahmad, 1964)Berea sandstone (Zhu & Toksoz, 2012)Saprolites (Revil et al., 2012)Glass beads (Boleve et al., 2007; Pengra et al., 1999)Limestones (Revil et al., 2007; Pengra et al., 1999)Clayrocks (Revil et al., 2005)Alluvium (Jardani et al., 2007)Sandstone (Pengra et al., 1999)Sand (Jougnot & Linde, 2013)Glass beads (Glover & Dery, 2010)Jardani et al. (2007)This model with D = 1.571
Figure 4: Comparison between the effective excess charge ˆ Q REVv as a function of the per-meability k . Symbols represent experimental data from literature for different lithologies.Solid lines are the fit of the proposed model (with D = 1 . Jardani et al. [2007] empirical relationship (26).If D = 1 .
527 allows us to retrieve the exact same trend as
Jardani et al. [2007], weare left with a cloud of values that spread around the proposed model. To test our modelwith more accuracy, we focus on a selection of experimental data where all the parametershave been measured in the following section.
In order to analyze the effect of salinity on ˆ Q REVv we test the proposed model (24) withlaboratory data obtained by
Pengra et al. [1999]. These authors performed an exhaustive12etrophysical characterization of a collection of rock and glass bead samples and theymeasured the electrokinetic coupling coefficient C EK for different NaCl brine concentra-tions. For this test ˆ Q REVv values are obtained from C EK using (28). The only parameterof(24) which is not measured by Pengra et al. [1999] is the hydraulic tortuosity τ . Thus,we fit this parameter using a least square algorithm. The ionic concentration dependenceof ζ -potential is assumed to obey (25).Figure 5 shows the fit of (24) to experimental values of effective excess charge ˆ Q REVv measured at different ionic concentrations C for 3 sandstones samples of different per-meabilities and 1 fused glass bead sample. The fitted values of tortuosity and measuredvalues of porosity and permeability of each sample are listed in the figure caption. Exper-imental data show that ˆ Q REVv decreases with the increase of ionic concentration, and thisbehaviour can be adequately described by the proposed model. The decrease of ˆ Q REVv with the increase of C was also predicted and discussed by Jougnot et al. [2015]. Q ^ v R EV ( C / m ) C (mol/m ) a) Fontainebleau-A model data 0.1 1 10 100 1000 Q ^ v R EV ( C / m ) C (mol/m ) b) Fontainebleau-B model data 1 10 100 100 1000 Q ^ v R EV ( C / m ) C (mol/m ) c) Fontainebleau-C model data 0.01 0.1 1 100 1000 Q ^ v R EV ( C / m ) C (mol/m ) d) Glass beads 200 micron model data Figure 5: Comparison between measured values of ˆ Q REVv and predicted values usingthe proposed model (Eq. 24) for different ionic concentrations. Experimental data areobtained from
Pengra et al. [1999]. Hydraulic parameters of soil samples: a) φ = 0 . k = 2 . × − m , τ = 1 .
95, b) φ = 0 . k = 9 . × − m , τ = 1 .
83, c) φ = 0 . k = 5 . × − m , τ = 3 .
24, d) φ = 0 . k = 5 . × − m , τ = 1 .
90. Tortuosityvalues are obtained by fitting (24) to experimental data.
To test the effect of porosity on ˆ Q REVv estimates we fit the proposed model to experimentalvalues of ˆ Q REVv in terms of both k and k/φ obtained by Pengra et al. [1999] for C = 0 . k . The fitting parameter is the ratio φ/τ and the RMS of the fit is 41.9713/m . Figure 6b shows the fit of the model to the same experimental values of ˆ Q REVv but in terms of k/φ . In this case the fitting parameter is τ and the RMS is reduced to19.64 C/m . Even though more experimental evaluations of the model are needed, thistest shows that the inclusion of porosity improves the estimate of ˆ Q REVv . -2 -1 -15 -14 -13 -12 -11 -10 Q ^ v R EV ( C / m ) k/ φ (m ) a) Fitting parameter τ =2.18, RMS=19.64 C/m model sandstones carbonates glass beads 10 -2 -1 -15 -14 -13 -12 -11 Q ^ v R EV ( C / m ) k (m ) b) Fitting parameter φ / τ =0.0427, RMS=41.97 C/m model sandstones carbonates glass beads Figure 6: Comparison between measured values of ˆ Q REVv vs k/φ and predicted valuesusing (24) for C = 0 . Among the available data in the SP literature,
Glover and D´ery [2010] studied the effectof grain size on the electrokinetic coupling coefficient C EK . These authors measured C EK on 12 packs of glass bead having different grain diameters d ( d ∈ [1 , × − m) attwo pore water salinities: C = 2 × − and 2 × − mol/L.To test the performance of our model for different grain sizes, we used the measure-ments of Glover and D´ery [2010] to predict the effective excess charge. For each grainsize, we used the permeability and mercury porosity listed in their Table 1. For ionicconcentrations C = 2 × − and 2 × − mol/L we use, respectively, the mean values ζ = -71.62 and -24.89 mV of ζ -potentials calculated for all the samples (given in theirTable 2). Note that the glass bead diameter should not influence the interface propertiesof the quartz mineral, this is an artifact due to surface conductivity effects [see also Leroyet al. , 2012,
Li et al. , 2016].In order to compare our model prediction with the measurements, we compute thecoupling coefficient C EK using (28) and the electrical conductivity model proposed by Glover and D´ery [2010]: σ = 1 F (cid:18) σ w + 4 mF Σ S d (cid:19) , (29)where σ w is the electrical conductivity of pore water (calculated from C , see [ Sen andGoode , 1992]); Σ S , the glass beads surface conductance; m , the cementation index; and F = φ − m , the formation factor of the bead samples. The only parameter which is notprovided in the study is the hydraulic tortuosity τ .14igure 7a shows the predicted effective excess charge for the two ionic concentrationsas a function of the glass bead diameter and τ = 1 . Jardani et al. [2007] empirical relationship which only depends onpermeability k . Figure 7b illustrates the pretty good fit between the coupling coefficientmeasurements and predicted values of our model for both pore water salinities. Notethat τ = 1 .
20 is the only fitting parameter of the model and this parameter allows todescribe all measured data (different glass bead sizes and two ionic concentration values).It is worth mentioning that we could improve the fit between the proposed model andexperimental data by adjusting τ for each glass bead diameter (especially for d (cid:28) × − m), but we prefer keeping the number of fitting parameters as small as possible. Glass bead diameter, d [m] -6 -5 -4 -3 E ff e c t i v e e xc e ss c ha r ge , Q v e ff [ C m - ] -4 -2 Model (2 x -4 mol/L)Model (2 x -3 mol/L)Jardani et al. (2007) Glass bead diameter, d [m] -6 -5 -4 -3 C oup li ng c oe ff i c i en t, C E K [ V P a - ] -8 -7 -6 -5 -4 Data (2 x -4 mol/L)Data (2 x -3 mol/L)Model (2 x -4 mol/L)Model (2 x -3 mol/L)Jardani et al. (2007)Jardani et al. (2007) a.b. Figure 7: (a) Model prediction of ˆ Q REVv for glass beads pack with different diameters attwo different ionic concentrations: C = 2 × − and 2 × − mol/L in blue and red,respectively. (b) Comparison between the measured and predicted coupling coefficientswhen using the proposed model with τ = 1 .
20. Note that the black dashed lines are the
Jardani et al. [2007] empirical relationship prediction (Eq. 26) for the effective excesscharge in (a) and the coupling coefficient in (b).
The present study is focused on the estimate of the effective excess charge ˆ Q REVv infully saturated porous media, a key parameter to understand and model the streaming15otential generation. Based on physical and geometrical concepts we derive a closed-formanalytical expression to estimate ˆ Q REVv from electrokinetic and hydraulic parameters. Themathematical development involves up-scaling procedures at pore and REV scales, similarto the numerical up-scaling proposed by
Jougnot et al. [2012] for partially saturated soils.The extension of the present model to unsaturated conditions is not straightforward as arelationship between permeability and saturation needs to be derived for the fractal poresize distribution.The first step of this study consists in the up-scaling of EK properties for a binarysymetric 1:1 electrolyte from the EDL scale to the capillary tube or pore scale (i.e., fromfew nm to possibly several mm). The approximate expression (21) relates effective excesscharge ˆ Q Rv to the radius of the tube R and chemical parameters (ionic concentration C , ζ -potential and Debye length l D ). The accuracy of this equation has been tested tobe correct as long as the electrical double layer from the walls of the capillary do notoverlap ( R > l D ), i.e. under the thin double layer assumption (Fig. 2). An extension ofthis model for EDL overlapping could be obtained by truncation of the diffuse layer [e.g. Gon¸calv`es et al. , 2007].The second step consists in the up-scaling of both hydraulic and EK properties from thepore scale (capillary tube) to the REV scale. The REV of porous media is conceptualizedas a bundle of capillary tubes with a fractal pore-size distribution. Fractal models havebeen proven very useful to obtain different hydraulic properties of rocks and soils [
Tylerand Wheatcraft , 1990,
Yu et al. , 2003,
Guarracino , 2007,
Guarracino et al. , 2014,
Soldiet al. , 2017]. In this study, the fractal distribution approach allows us to obtain simpleexpressions for porosity φ (11) permeability k (12) and effective excess charge ˆ Q REVv (23) in terms of the fractal dimension D , the maximum pore radius R max and the REVradius R REV . Finally, combining these macroscopic properties we obtain the closed-formexpression (24) to estimate ˆ Q REVv from hydraulic ( φ , k , τ ) and pore water chemistry ( C , l D , ζ ) parameters.The proposed model is derived from physical and geometrical concepts and providesa mechanistic framework for understanding the role of hydraulic parameters on ˆ Q REVv . Inparticular, the model corroborates the empirical relationship (26) of
Jardani et al. [2007]which relies on an increasing number of data sets with different lithologies and it is used bymany researchers. It is important to remark that the link between effective excess chargeand permeability is the subject of debates in the scientific community [e.g.
Jouniaux andZyserman , 2016,
Zyserman et al. , 2017,
Revil , 2017] and our model provides a theoreticaljustification to this link. However, in this study we show that it would be better to linkˆ Q REVv to the ratio of permeability and porosity ( k/φ ) instead of k (see Fig. 6).Another limitation of the relationship proposed by Jardani et al. [2007] is that it doesnot explicitly depend on the pore-water chemistry, i.e. on the ζ -potential, Debye lengthand ionic concentration. Indeed, the newly proposed model takes into account the ionicconcentration and its ζ -potential dependence and is therefore able to reproduce laboratorydata from Pengra et al. [1999] where all the parameters but one (tortuosity) have beenestimated independently.The proposed model shows that ˆ Q REVv contains information on the lithology as pre-dicted by
Revil and Jardani [2013] (p. 64). Indeed, three petrophysical parameters areexplicitly identified in the model: the permeability, k , the porosity, φ , and the hydraulic16able 1: Comparison between fitted and predicted hydraulic tortuositiesMaterial τ from fit τ e from Eq. (30) Glass beads packs from Glover and D´ery [2010]Mean value over 12 packs 1.20 1.24
Samples from Pengra et al. [1999]Fontainebleau A 1.95 1.59Fontainebleau B 1.83 1.82Fontainebleau C 3.24 3.03Glass beads (200 µ m) 1.90 1.64tortuosity, τ . It is worth to emphazise here that the relation between hydraulic and elec-trical tortuosity is not straightforward [e.g. Clennell , 1997]. In the present work, we testthe simple model of
Winsauer et al. [1952] to determine the hydraulic tortuosity used inEq. (24) based on parameters that can be measured electrically [see
Clennell , 1997, fora discussion]: ( τ e ) = F φ, (30)where τ e is the hydraulic tortuosity determined electrically from the formation factor andthe porosity. Table 1 show the comparison between the fitted tortuosity τ to the onepredicted by (30). One can see that predicted and fitted tortuosities fall fairly close oneto an other, which is a promising preliminary result. However, further work is needed toestablish a better relation between these electrical and hydraulic properties, therefore tolink the effective excess charge ˆ Q REVv to electrical conductivity.The proposed model represents a major step forward in understanding the links be-tween hydraulic and electrokinetic parameters in the framework of the excess chargeapproach. The model provides a theoretical basis for the empirical relationship withthe medium permeability and describes the dependence of the excess charge on otherpetrophysical parameters. The simplicity of the model and its excellent fit to differentset of experimental data opens-up new possibilities for a broader use of the SP methodin hydrogeophysics studies based on the increasingly popular excess charge approach.The analytical development of a model for partially saturated porous media using thepresented approach will be the natural next step to pursue this work.
Acknowledgments
The data used to test the proposed model is listed in the references. This researchis partially supported by Universidad Nacional de La Plata and Consejo Nacional deInvestigaciones Cient´ıficas y T´ecnicas (Argentina). The authors strongly thank the editorand the three reviewer for their nice and really constructive comments.17 eferences
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