Effective Excess Charge Density in Water Saturated Porous Media - Luong Duy Thanh

VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 9 Effective Excess Charge Density in Water Saturated Porous Media Luong Duy Thanh* Thuy Loi University, 175 Tay Son, Dong Da, Ha Noi, Vietnam Received 03 October 2018 Revised 28 OCtober 2018; Accepted 01 November 2018 Abstract: A model for the effective excess charge in a capillary as well as in porous media is developed for arbitrary pore scales. The prediction of the model is then compared with another published model that is limited for a thin electric double layer (EDL) assumption. The comparison shows that there is a deviation between two models depending on the ratio of capillary/pore radius and the Debye length. The reasons for the deviation between two models are not only due to the thin EDL assumption to get electrical potential and charge distribution in pores but also to some other approximations for integral evaluations. The results suggest that the model developed in this work can be used with arbitrary capillary/pore scale and thus is not restricted to the thin EDL assumption. Keywords: Zeta potential, porous media, electric double layer, effective excess charge. 1. Introduction The self-potential (SP) method is a passive geophysical method based on the measurements of the natural electrical potentials at the ground surface of the Earth. The SP survey has the advantage of being non-destructive, fast, inexpensive and very simple to perform in the field requiring only a high impedance millivoltmeter and at least two non-polarizable electrodes connected through a reel of wire. The SP method has been used for a variety of geophysical applications. For example, the SP measurements could be used to characterize active volcanic areas [1, 2], to detect and monitor ground- water flow [3, 4], contaminant plumes [5, 6] or to study landslides [7, 8]. In natural porous media, several main mechanisms proposed to explain SP signals are associated with streaming potential, electrochemical, thermoelectric, redox, or piezoelectric effects [9]. An overview of other possible mechanisms is available in [e.g., 10]. In this work, the SP signal that is often referred to as streaming potential due to the water flow in porous media is considered. The surface of the minerals that constitute porous media is generally ________ Tel.: 84-936946975. Email: luongduythanh2003@yahoo.com https//doi.org/ 10.25073/2588-1124/vnumap.4294 L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 10 electrically charged, creating an electrical double layer (EDL) containing an excess of charge that counterbalances the charge deficiency of the mineral surface [11, 12]. Fig. 1 shows structure of the EDL: a Stern layer that contains only counterions coating the mineral with a very limited thickness and a 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 and diffuse layer separates the mobile and immobile part of the water molecules when subjected to a fluid pressure difference. The electrical potential at the shear plane is called the zeta potential (ζ) [11]. In the bulk liquid, the number of cations and anions is equal so that it is electrically neutral. There are two main approaches to model streaming potential in saturated porous media. The first one is based on the use of the coupling coefficient which is a rock-dependent property that relates the hydraulic pressure difference to the electrical potential difference [e.g., 13, 14]. The second one is more recent and focuses on the excess charge that is dragged by the water flow [15-17, 18]. In this approach, the coupling parameter is the excess charge that is effectively dragged by water in the porous media. Recently, Guarracino and Jougnot (2018) have developed a theoretical model to predict the effective excess charge density in saturated porous media using a bundle of capillary tubes of different radius with a fractal pore-size distribution [19]. They used the thin EDL assumption (i.e., the thickness of the double layer is small compared to the pore size). Their proposed model shows the dependence of the effective excess charge on the electrolyte concentration and microstructure parameters of porous media like porosity, permeability and tortuosity. Their model has been successfully examined by different sets of experimental data. Figure 1. Schematic view of the EDL model. (a) Charge distribution. (b) Electric potential distribution. Figure is modified from [13]. Such a thin EDL assumption, however, is not always satisfied, especially for natural or artificial porous media with low permeability, such as mudstone or shale saturated by low salinity water [20]. Therefore, the model proposed by [19] is not valid for those systems of porous media and water. In this work, a model for the effective excess charge is developed for arbitrary pore scales based on a simpler picture of porous media. Namely, a representative elementary volume (REV) of a porous medium (see L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 11 Fig. 2) with the physical length L and cross-sectional area A can be approximated as an array of N parallel capillaries of length L with the same radius r [13]. The prediction from the obtained model is then compared with [19]. The comparison shows that there is a deviation between two models depending on the ratio of capillary/pore radius and the Debye length. The reasons for the deviation between two models are not only due to the thin EDL assumption but also to some other approximations for integral evaluations. The results suggest that the model developed in this work can be used with arbitrary capillary/pore scale and thus is not restricted to the thin EDL assumption. Figure 2. The REV of a porous medium composed of a large number of tortuous capillaries with same radius. 2. Theoretical development The distribution of the excess charges in the diffuse layer of a capillary is governed by the Poisson- Boltzmann equation [21]: 0 )()(1   r r dr rd r dr d r       (1) where ψ(r) and ρ(r) is the electric potential (in V) and the volumetric charge density (in C m-3) in the liquid at the distance r from the axis of the capillary, respectively; εr is the relative permittivity of the fluid (78.5 at 25oC for water) and εo is the dielectric permittivity in vacuum (8.854×10−12 C2 J−1 m−1). For symmetric electrolytes such as NaCl or CaSO4 in the liquid, ρ(r) is given by [22] ) )( sinh(2)( Tk reZ eZCNr b fA    (2) where fC is the electrolyte concentration in the bulk fluid representing the number of ions (anion or cation) (mol m−3), e is the elementary charge (e = 1.6×10−19 C), Z is the valence of the ions under consideration (dimensionless); kb is the Boltzmann’s constant (1.38×10-23 J/K), T is the Kelvin temperature (in K) and NA is the Avogadro’s number (6.022 ×1023 /mol). Putting Eq. (2) into Eq. (1), one obtains ) )( sinh( 2)(1 0 Tk reZeZCN dr rd r dr d r br bA          (3) L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 12 The boundary conditions to be satisfied for the cylindrical capillary surface are: (i) the potential at the surface r = R (R is the radius of the capillary),  )(R ; (ii) the potential at the center of the capillary r = 0, 0 )( 0  rdr rd [19, 22] By solving Eq. (3) with the linear approximation ( Tk reZ Tk reZ bb )( ) )( sinh(   for small values of Tk reZ b )( [23]), the analytical solution of ψ(r) and ρ(r) are obtained as [22] )( )( )(   R I r I r o o  (4) )( )( )( 2 0      R I r I r o o r (5) where Io is the zero-order modified Bessel function of the first kind and  is the Debye length characterizing EDL thickness given by fA bro CeZN Tk 222    (6) It should be noted that the electrical potential ψ(r) for a thin EDL assumption is given by [e.g., 19, 23]  rR er    .)( (7) By solving the Navier-Stokes equation, the velocity profile of the fluid in a cylindrical capillary with pressure driven is given by [24]  22 4 )( rR L P rv     (8) where ΔP is the pressure difference across the capillary, η is the dynamic viscosity of the fluid and L is the real length of the tortuous capillary that is related to the length of the porous media by  .LL  ( is the tortuosity of the capillary). The effective excess charge density R vQˆ carried by the water flow in a single tube of radius R is defined by [e.g., 19]  R av R v rdrrvr Rv Q 0 2 2).().( . 1ˆ   (9) L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 13 where avv is the average velocity in the capillary and that is given by [25] L RP vav 8 . 2  (10) Putting Eq. (5), Eq. (8) and Eq. (10) into Eq. (9), one obtains:         R o R o o rR v drr r Irdr r IR R IR Q 0 3 0 2 24 0 )()( )(... 4ˆ     (11) After evaluating the integral and making arrangements, Eq. (11) is written as              1 )( )( . 2 . 8ˆ 1 2 0   R I R I RR Q o rR v (12) where I1 is the first-order modified Bessel function of the first kind. The effective excess charge density REV vQˆ carried by the water flow in the REV consisting N capillaries of the same radius R is defined in the same way as mentioned in [19] Av vRNQ Q D av R vREV v . ...ˆˆ 2  (13) where A is the cross sectional area of the REV (see Fig. 2) and L P vD   .   is the Darcy velocity (m/s) ( is the permeability of the porous medium). Eq. (13) is now rewritten as   L LR A RNQ Q R vREV v 8 . . ..ˆˆ 22  (14) The porosity  of the REV of the porous media is defined as T V V V  (15) where VV is the volume of void-space (  LRNVV .. 2 ) and VT is the total volume of the REV ( LAVT . ). Consequently, the porosity is given by LA LRN V V T V . .. 2   (16) Combining Eq. (12), Eq. (14) and Eq. (16) yields           2 1 02 2 2 2 1 .1 )( )( . 2 8 . .ˆ. 8 . .ˆˆ                      R I R I R R Q L LR QQ o r R v R v REV v (17) L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 14 For the case where porous media made up of the capillaries with the same radius, the permeability of the porous media is given by [e.g., 14] 2 2 8   R  (18) Combining Eq. (6), Eq. (17) and Eq. (18) yields     2 1 0 1 .1 )( )( . 2ˆ        XI XI X Q o r REV v (19) where Tk CeZNR X bro fA    . 16 222  . Eq. (14) and Eq. (19) show the dependence of the effective excess charge density on microstructure parameters of porous media (capillary radius, porosity, tortuosity and permeability) and electrokinetic parameters (ionic concentration, valence of ions and the zeta potential). 3. Results and discussion In this work, a system of 1:1 symmetric electrolytes such as NaCl, KNO3 (Z = 1) and silica-based surfaces are considered at room temperature (T=295 K) for the modeling because of the availability of input parameters. For silica-based rocks saturated by 1:1 symmetric electrolytes, the Cf - relation is found to follow [26]: ζ = a + blog10(Cf) (20) where a = -9.67 mV, b = 19.02 mV (ζ in mV). (a) (b) Figure 3. The variation of the electric potential )(r with respect to distance r using Eq. (4) and Eq. (7) at Cf = 1 mol/m3 (a) and Cf = 50 mol/m3 (b). L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 15 The variation of the electric potential )(r with respect to distance r from the center of the capillary predicted from Eq. (4) and Eq. (7) at two different electrolyte concentrations of Cf = 1 mol/m3 and Cf = 50 mol/m3 is shown in Fig. 3 for the radius R = 10-8 m. To get Fig.3, the values of  and  calculated from Eq. (6) and Eq. (20) at Cf = 1 mol/m3 and Cf = 50 mol/m3 are 0.96×10-8 m, -69 mV and 0.13×10-8 m, -34 mV, respectively. It is shown that when the radius R is approximately the same as the Debye length λ at Cf = 1 mol/m3, the electric potential predicted from Eq. (4) is much different from Eq. (7). However, the predictions from Eq. (4) and Eq. (7) are in good agreement with each other as expected when a thin EDL assumption is valid at Cf = 50 mol/m3 (λ =0.13×10-8 m and R/λ 8). From Eq. (12), the dependence of the effective excess charge density R vQˆ on the capillary radius R at two different electrolyte concentrations of Cf = 1 mol/m3 (a) and Cf = 50 mol/m3 (b) is shown in Fig. 4 (the dash lines). It is seen that the effective excess charge density decreases with increasing capillary radius. Additionally, Guarracino and Jougnot (2018) obtained the effective excess charge density in a capillary R vQˆ and in a porous medium REV vQˆ after some approximations and under a thin EDL assumption as below [19]:                  3 2 3 2 )/( 8 ˆ Tk e Tk e R eCN Q bb fAR v   (21) and      2 3 2 1 3 2ˆ                  Tk e Tk e eCNQ bb fA REV v (22) (a) (b) L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 16 Figure 4. Dependence of the excess charge density R vQˆ on the capillary radius R predicted from this work (the dashed lines) and from Guarracino and Jougnot (2018) (the solid lines) at Cf = 1 mol/m3 (a) and Cf = 50 mol/m3 (b). The variation of the effective excess charge density R vQˆ with the capillary radius R predicted from Eq. (21) is also shown in Fig. 4 at the same electrolyte concentrations (the solid lines). It is found that the model of [19] always overestimates the effective excess charge density for any value of the capillary radius at both electrolyte concentrations. Similarly, the result shows that when the radius R is approximately the same as the Debye length λ at Cf = 1 mol/m3, the effective excess charge density R vQˆ predicted from this work is far different from [19] as shown in Fig. 4a. However, values of R vQˆ predicted from this work and from [19] are almost the same as shown in Fig. 4b when a thin EDL assumption is valid at Cf = 50 mol/m3 (R/ λ  8). The ratio of R/ λ  8 is in good agreement with the value of R/ λ  5 when the thin EDL assumption is valid as reported in [19]. The predictions of the excess charge density REV vQˆ in the porous medium as a function of electrolyte concentration Cf and the permeability  from this work and from [19] are shown in Fig. 5 and Fig. 6, respectively. Input parameters used for modeling in Fig. 5 are 2.0 , 95.1 ,  = 10- 14 m2 and in Fig. 6 are 2.0 , 95.1 , Cf = 1 mol/m3. The reason to use the above parameters is that they have been used for modeling in [19]. Fig. 5 shows that the effective excess charge density REV vQˆ decreases with increasing electrolyte concentration. This is explained by the decrease of the EDL thickness when the electrolyte concentration increases [e.g., 11, 13, 14]. Consequently, the excess charge in the EDL and the effective excess charge density decrease. Fig. 6 shows that the REV vQˆ decreases with increasing permeability of porous media. This behavior is suitable with what is predicted in Fig. 4 because there is a link between the pore radius and permeability of porous media as shown in Eq. (18). Additionally, the results in Fig. 5 and Fig. 6 show that the model proposed by [19] overestimates the effective excess charge density at a given electrolyte concentration as well as permeability. The deviation between this work and [19] depends on the ratio of capillary/pore radius and the Debye length. L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 17 Figure 5. The variation the effective excess charge REV vQˆ with electrolyte concentration Cf predicted from this work (the dashed line) and from Guarracino and Jougnot (2018) (the solid line). (a) (b) Figure 6. The variation of the effective excess charge REV vQˆ with permeability  predicted from this work (the dashed lines) and from Guarracino and Jougnot (2018) (the solid lines) at Cf = 1 mol/m3 (a) and Cf = 50 mol/m3 (b). The reasons for a difference between this work and [19] are not only due to the thin EDL assumption to get ψ(r) and ρ(r) but also to approximations to evaluate integrals as performed by [19]. Namely, Guarracino and Jougnot (2018) applied Taylor series expansions of exponential functions and only used the first four terms [19]. They then used a thin EDL assumption again to simplify integrals. Therefore, those approximations lead to another deviation contributing to the difference between this work and L.D. Thanh / VNU Journal of Science: Mathematics – Physics, Vol. 34, No. 4 (2018) 9-18 18 [19]. However, when the thin EDL assumption is valid (R/ λ  8), the model proposed by [19] is in good agreement with the model obtained in this work for arbitrary pore scales. 4. Conclusion In this work, a model for the effective excess charge in a capillary R vQˆ as well as in porous media REV vQˆ is developed for arbitrary pore scales. The results of R vQˆ and REV vQˆ as a function of the capillary radius, electrolyte concentration and permeability predicted from the obtained model are then compared with those from [19]. The comparisons show that the effective excess charge densities predicted from the model of [19] are always larger than those predicted from this work. The deviation between them depends on the ratio of capillary/pore radius and the Debye length R/λ. When the pore radius is approximately the same as the Debye length, there is a big difference between two models. However, when the ratio of R/λ is larger than 8, two models are in very good agreement with other. The possible reasons for a difference between two models are not only due to the thin EDL assumption to get ψ(r) and ρ(r) but also to some approximations for integral evaluations used in [19]. The results suggest that the obtained model is more general and can be used with arbitrary capillary/pore scale and thus is not restricted to the thin EDL assumption. Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.99-2016.29. References [1] M. Aubert, Q.Y. Atangana, Groundwater, 34 (1996) 1010–1016. [2] A. Finizola, J.-F. Lénat, O. Macedo, D. Ramos, J.-C. Thouret, F. Sortino, J. Volcanol. 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