VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 treaming Potential in Unconsolidated amples aturated with Monovalent Electrolytes Luong Duy Thanh 1,*, Rudol prik 2 1 Thuy Loi University, 175 Tay on, Dong Da, Hanoi, Vietnam 2 Van der Waals-Zeeman Institute, University o Amsterdam, The Netherlands Received 30 July 2017 Revised 22 eptember 2017; Accepted 16 October 2017 Abstract: The streaming potential coeicient o liquid-rock systems is theoretically a very complicated unction depending on many parameters including temperature, luid concentration, luid ph, as well as rock parameters such as porosity, grain size, pore size, and ormation actor etc. At a given porous media, the most inluencing parameter is the luid conductivity or electrolyte concentration. Thereore, it is useul to have an empirical relation between the streaming potential coeicient and electrolyte concentration. In this work, the measurements o the streaming potential or our unconsolidated samples (sandpacks) saturated with our monovalent electrolytes at six dierent electrolyte concentrations have been perormed. From the measured streaming potential coeicient, the empirical expression between the streaming potential coeicient and electrolyte concentration is obtained. The obtained expression is in good agreement with those available in literature. Additionally, it is seen that the streaming potential coeicient depends on types o cation in electrolytes and on samples. The dependence o the streaming potential coeicient on types o cation is qualitatively explained by the dierence in the binding constant or cation adsorption on the silica suraces. The dependence o the streaming potential coeicient on samples is due to the variation o eective conductivity and the zeta potential between samples. Keywords: treaming potential coeicient, zeta potential, porous media, sands. 1. Introduction The streaming potential is induced by the relative motion between the luid and the solid surace. In porous media such as rocks, sands or soils, the electric current density, linked to the ions within the luid, is coupled to the luid low. treaming potential plays an important role in geophysical applications. For example, the streaming potential is used to map subsurace low and detect orresponding author. Tel.: 84-936946975. Email: luongduythanh2003@yahoo.com https//doi.org/ 10.25073/2588-1124/vnumap.4213 14
L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 15 subsurace low patterns in oil reservoirs [e.g., 1]. treaming potential is also used to monitor subsurace low in geothermal areas and volcanoes [e.g., 2, 3]. Monitoring o streaming potential anomalies has been proposed as a means o predicting earthquakes [e.g., 4, 5] and detecting o seepage through water retention structures such as dams, dikes, reservoir loors, and canals [6]. The streaming potential coeicient (P) is a very important parameter, since this parameter controls the amount o coupling between the luid low and the electric current low in porous media. The P o liquid-rock systems is theoretically a very complicated unction depending on 18 undamental parameters including temperature, pore luid concentration, luid ph, as well as rock parameters such as porosity, grain size, pore size, and ormation actor etc [e.g., 7, 8]. At a given porous media, the most inluencing parameter is the luid conductivity. Thereore, it is useul to have an empirical relation between the P and luid conductivity or electrolyte concentration. For example, Jouniaux and Ishido [8] obtain an empirical relation to predict the P rom luid conductivity based on numerous measurements o the streaming potential on sand saturated by Nal which have been published. By itting experimental data collected or sandstone, sand, silica nanochannels, tainton, and Fontainebleau with electrolytes o Nal and Kl, Jaaar et al. [9] obtain another empirical expression between the P and electrolyte concentration. However, experimental data sets they used or itting are rom dierent sources with dissimilar luid conductivity, luid ph, temperature, mineral composition o porous media. All those dissimilarities may cause the empirical expressions less accurate. To critically seek empirical expressions to estimate the P rom electrolyte concentration, we have carried out streaming potential measurements or a set o our sandpacks saturated by our monovalent electrolytes (Nal, NaI, Kl and KI) at six dierent electrolyte concentrations (10 4 M, 5.0 10 4 M, 10 3 M, 2.5 10 3 M, 5.0 10 3 M, and 10 2 M). From the measured P, we obtain the empirical expression between the P and electrolyte concentration. The obtained expression is in good agreement with those reported in [8, 9] in which the P in magnitude is inversely proportional to electrolyte concentration. Additionally, it is seen that the streaming potential coeicient depends on types o cation in electrolytes and on samples. The dependence o the streaming potential coeicient on types o cation is qualitatively explained by the dierence in the binding constant or cation adsorption on the silica suraces. The dependence o the streaming potential coeicient on samples is due to the variation o eective conductivity and the zeta potential between samples. This paper includes ive sections. ection 2 describes the theoretical background o streaming potential. ection 3 presents the experimental measurement. ection 4 contains the experimental results and discussion. onclusions are provided in the inal section. 2. Theoretical background o streaming potential 2.1. Electrical double layer A porous medium is ormed by mineral solid grains such as silicates, oxides, carbonates etc. When a solid grain surace is in contact with a liquid, it acquires a surace electric charge [12]. The surace charge repels ions in the electrolyte whose charges have the same sign as the surace charge (called the coions ) and attracts ions whose charges have the opposite sign (called the counterions and normally cations) in the vicinity o the electrolyte silica interace. This leads to the charge distribution known as the electric double layer (EDL) as shown in Fig. 1. The EDL is made up o the tern layer, where cations are adsorbed on the surace and are immobile due to the strong electrostatic attraction, and the diuse layer, where the ions are mobile. The distribution o ions and the electric potential
16 L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 within the diuse layer is governed by the Poisson Boltzman (PB) equation which accounts or the balance between electrostatic and thermal diusion orces [12]. The solution to the linear PB equation in one dimension perpendicular to a broad planar interace is well-known and produces an electric potential proile that decays approximately exponentially with distance as shown in Fig. 1. In the bulk liquid, the number o cations and anions is equal so that it is electrically neutral. The closest plane to the solid surace in the diuse layer at which low occurs is termed the shear plane or the slipping plane, and the electrical potential at this plane is called the zeta potential (ζ). The zeta potential plays an important role in determining the degree o coupling between the electric low and the luid low in porous media. Most reservoir rocks have a negative surace charge and a negative zeta potential when in contact with ground water [13, 14]. The characteristic length over which the EDL exponentially decays is known as the Debye length and is on the order o a ew nanometers in most rock-water systems. The Debye length is a measure o the diuse layer thickness; its value depends solely on the properties o the luid and not on the properties o the solid surace [15] and is given by (or a symmetric, monovalent electrolyte such as Nal) o rkt b d, (1) 2 2000Ne where k b is the Boltzmann s constant, ε 0 is the dielectric permittivity in vacuum, ε r is the relative permittivity o the luid, T is temperature (in K), e is the elementary charge, N is the Avogadro s number and is the electrolyte concentration (mol/l). Figure 1. tern model [10, 11] or the charge and electric potential distribution in the electric double layer at a solid-liquid interace. In this igure, the solid surace is negatively charged and the mobile counter-ions in the diuse layer are positively charged (in most rock-water systems). Figure 2. Development o streaming potential when an electrolyte is pumped through a capillary (a porous medium is made o an array o parallel capillaries).
L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 17 2.2. treaming potential The dierent lows (luid low, electrical current low, heat low etc.) are coupled by the general equation J i = n Lij X j, (2) j 1 which links the orces X j to the macroscopic luxes J i through transport coupling coeicients L ij [16]. onsidering the coupling between the hydraulic low and the electric current low, assuming a constant temperature and no concentration gradients, the electric current density J e (A/m 2 ) and the low o luid J (m/s) can be written as the ollowing coupled equation [e.g., 8]: J e = - 0 V Lek P. (3) k0 J = - Lek V P, (4) where P is the pressure that drives the luid low (Pa), V is the electrical potential (V), 0 is the bulk electrical conductivity, k 0 is the bulk permeability (m 2 ), is the dynamic viscosity o the luid (Pa.s), and L ek is the electrokinetic coupling (A.Pa -1.m -1 ). The irst term in Eq. (3) is the Ohm s law, and the second term in Eq. (4) is the Darcy s law. The coupling coeicient L ek is the same in Eq. (3) and Eq. (4) because the coupling coeicients must satisy the Onsager s reciprocal relation in the steady state. From these equations, it is possible to notice that even i no potential dierence is applied ( V = 0), then simply the presence o a pressure dierence can produce an electric current. On the other hand, i no pressure dierence is applied ( P = 0), the presence o an electric potential dierence can still generate a luid low by electroosmosis. The P (V/Pa) is deined when the electric current density J e is zero, leading to V L (5) ek. P 0 This coeicient can be measured by applying a pressure dierence P to a porous medium and by detecting the induced electric potential dierence V (see Fig. 2). The driving pressure induces a streaming current (second term in Eq. (3)) which is balanced by the conduction current (irst term in Eq. (3)) which leads to the electric potential dierence that can be measured. In the case o a unidirectional low through a cylindrical saturated porous rock, this coeicient can be expressed as [e.g., 1, 4, 8] (6) r o, e where σ e is the eective conductivity, and ζ is the zeta potential. The eective conductivity includes the luid conductivity and the surace conductivity. To characterize the relative contribution o the surace conductivity, the dimensionless quantity called the Dukhin number has been introduced [17]. The P can also be written as [e.g., 8, 18] (7) r o, F r
18 L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 where σ r is the electrical conductivity o the sample saturated by a luid with a conductivity o σ and F is the ormation actor. The electrical conductivity o the sample can possibly include surace conductivity. I the luid conductivity is much higher than the surace conductivity, the eective conductivity is approximately equal to the luid conductivity, σ e = Fσ r = σ and the P becomes the well-known Helmholtz-moluchowski equation: (8) r o. 3. Experiment treaming potential measurements have been perormed on our unconsolidated samples o sand particles with dierent diameters (ee Table 1). The samples are made up o blasting sand particles obtained rom Unicorn I BV ompany. Four monovalent electrolytes (Nal, NaI, Kl and KI) are used with 6 dierent concentrations (10 4 M, 5.0 10 4 M, 10 3 M, 2.5 10 3 M, 5.0 10 3 M, and 10 2 M). All measurements are carried out at room temperature (22 ±1 o ). 3.1. ample assembly amples are constructed by illing polycarbonate plastic tubes (1 cm in inner diameter and 7.5 cm in length) successively with 2 cm thick layers o particles that are gently tamped down, and they are then shaken by a shaker (TIRA-model TV52110). Filter paper is used in both ends o the tube to retain the particles and is permeable enough to let the luid pass through. The samples are lushed with distilled water to remove any powder or dust. 3.2. Porosity, permeability, and ormation actor measurements The porosity is measured by a simple method mentioned in [19] and reerence therein. The sample is irst dried in oven or 24 hours, then cooled to room temperature, and inally ully saturated with deionized water under vacuum. The sample is weighed beore (m dry ) and ater ull saturation (m wet ) by a vacuum desiccator and the porosity is determined as ( mwet m dry ) /, (9) AL where ρ is density o the water, A and L are the cross sectional area and the physical length o the samples, respectively. The measured porosity o the samples is 0.39 independently o the size o sand particles with an error o 5%. This value is in good agreement with [20] or a random packing o spherical particles. Table 1. ample ID, diameter range, permeability k o, porosity, ormation actor F and tortuosity α. ample ID ize (μm) k o (m 2 ) F α 1 1 300-400 13.2 10-12 4.2 1.64 2 2 200-300 7.5 10-12 4.0 1.56 3 3 90-150 3.8 10-12 4.2 1.64 4 4 0-50 0.7 10-12 4.3 1.68
L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 19 Figure 3. The low rate against pressure dierence. Two runs are shown or the sample 3. Permeability is determined by a constant low-rate experiment (see [19] and reerence therein or more details). A high pressure pump (LabHut, eries III- Pump) ensures a constant low through the sample, a high precision dierential pressure transducer (Endress and Hauser Deltabar PMD75) is used to measure the pressure drop. For low velocities Darcy s law holds ka o P Q, L (10) where Q is the luid volume low rate, k o is the permeability, P is the dierential pressure imposed across the sample, η is the viscosity o the luid. The permeability is then determined rom the slope o the low rate - pressure graph (see Figure 3 or the sample 3, or example) with the knowledge o L, A and η (10-3 Pa.s). The graph shows that there is a linear relationship between low rate and pressure dierence and Darcy s law is obeyed. imilarly, the permeability o other samples is obtained and reported in Table 1 with an error o 10 %. Figure 4. chematic o the experimental setup to measure the saturated sample resistance. Method o determining the ormation actor and tortuosity is introduced in [19] and reerence therein. The ormation actor F is deined as: F, (11) r
20 L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 where α is the tortuosity, ϕ is the porosity o the sample, σ is the electrical conductivity o the luid directly measured by a conductivity meter (onsort 861) and σ r is the electrical conductivity o the ully saturated sample. Eq. (11) is only valid when surace conductivity eects become negligible (σ is higher than 0.60 /m as stated in [21] or silica-based samples). chematic o the experimental setup to measure resistances o saturated samples is shown in Figure 4. The electrodes or the resistance measurements are silver meshes. The electrodes are placed on both sides against the sample that is ully saturated successively with a set o Nal solutions with high conductivities. The sample resistance is measured by an impedance analyzer (Hioki IM3570) to calculate σ r with the knowledge o the geometry o the sample (the length and the diameter). From the measured values o σ and σ r, the ormation actor and corresponding tortuosity are determined with an error o 6 % and 9 %, respectively (see Table 1). The measured ormation actors o the samples are the range rom 4.0 to 4.3. According to Archie s law, F m in which is the porosity o the sample and m is the so called cementation exponent. For unconsolidated samples made o spherical particles, the exponent is around 1.5 [22]. Thereore, the measured ormation actors are in good agreement with Archie s law. Figure 5. Experimental setup or streaming potential measurements. Figure 6. treaming potential as a unction o pressure dierence or the sample 3 and electrolyte Kl at a concentration o 2.5 10-3 M. Three representative measurements are shown.
L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 21 3.3. treaming potential measurement The experimental setup or the streaming potential measurement is shown in Fig. 5. The solution is circulated through the samples until the electrical conductivity and ph o the solution reach a stable value measured by a multimeter (onsort 861). The ph values o equilibrium solutions are in the range 6.0 to 7.5. Electrical potential dierences across the samples are measured by a high input impedance multimeter (Keithley Model 2700) connected to a computer and controlled by a Labview program (National Instruments). Pressure dierences across a sample are measured by a highprecision dierential pressure transducer (Endress and Hauser Deltabar PMD75). Table 2. The streaming potential coeicient (in mv/bar) or dierent electrolyte concentrations. ample ID Electrolyte 10 4 M 5.0 10 4 M 10 3 M 2.5 10 3 M 5.0 10 3 M 10 2 M Nal -1250-202 -120-41 -20 NaI -1275-206 -125-44 -22 1 Kl -1033-183 -91-30 -11 KI -1233-200 -118-38 -19 Nal -1950-380 -179-70 -32 NaI -2067-403 -185-87 -35 2 Kl -1230-263 -127-58 -26 KI -1700-350 -170-60 -27 Nal -2100-625 -357-145 -63-31 NaI -2466-763 -400-160 -77-34 3 Kl -1567-570 -313-120 -65-25 KI -1665-573 -330-123 -58-26 Nal -4021-842 -429-149 -69-33 NaI -4067-850 -435-151 -81-36 4 Kl -2333-576 -290-106 -71-29 KI -3933-836 -430-147 -75-31 The way used to collect the P is already described in [23] and reerence therein. treaming potential across the sample (ΔV) is measured as a unction o applied pressure dierence (ΔP). The P is then obtained as the slope o the straight line (see Fig. 6). Three measurements are perormed to ind the average value o the P. The P or all samples at dierent electrolyte concentrations is shown in Table 2 except or two samples 1 and 2 at electrolyte concentration o 10-2 M. Because these samples are very permeable, they need a very large low rate to generate measurable electric potentials at high electrolyte concentration. The maximum error o the P is 10 %. It is ound that the P is negative or all samples and all electrolytes used in this work. 4. Results and discussion From Table 2, the dependence o the P on types o electrolyte is shown in Fig. 7 or a representative sample (or example, sample 3). Fig. 7 shows that the magnitude o the P decreases with increasing electrolyte concentration or all samples and all electrolytes as reported in literature [e.g., 7, 8]. The experimental result also shows that the P mostly depend on types o cation in electrolytes. The dependence o the P on types o cation can be qualitatively explained by the dierence in the binding constant o cations. For example, the binding constant o K + is larger than Na + [7]. Thereore, at the same ionic strength more cations o K + are absorbed on the negative solid surace than cations o Na +. This makes the electric potential on the shear plane (the zeta potential)
22 L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 smaller in the electrolyte containing cations o K + than that in the electrolyte containing cations o Na +. Thereore, cations have eect on the zeta potential and on the P. Namely, the P in magnitude is larger in the electrolyte containing cations o Na + than that in the electrolyte containing cations o K +. The experimental results also show that anions have only a small eect on the P as observed in [24]. Figure 7. The magnitude o the P as a unction o electrolyte concentration or dierent electrolytes and or the sample 3. Fig. 8 shows the variation o the P with samples or electrolyte NaI, or example. It is seen that at a given electrolyte concentration, the P depends on the samples. Namely, the magnitude o the P increases in the order rom 1, 2, 3 and 4 respectively. This may be due to the variation o eective conductivity with particle size o the samples and the dierence in the zeta potential between the samples. From Table 2, the magnitude o the P as a unction o electrolyte concentration is plotted or all samples saturated by dierent electrolytes (see Fig. 9). By itting the experimental data shown by the solid line in Fig. 9, the empirical relation between the P in magnitude and electrolyte concentration is obtained as 9 2.0 10 (V/Pa) (12) where is electrolyte concentration. Figure 8. The magnitude o the P as a unction o electrolyte concentration or dierent samples saturated by electrolyte NaI.
L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 23 Expression in Eq. (12) has the similar orm as the empirical expression 1.36 10 / 9 0.9123 obtained by Jaaar et al. [9] by itting experimental data collected or sandstone, sand, silica nanochannels, tainton, and Fontainebleau with electrolytes o Nal and Kl at ph = 6-8. Additionally, by itting experimental data on sand saturated by Nal at ph = 7-8 which are available 8 in literature, Jouniaux and Ishido [8] obtain the expression 1.2 10 / ( is the luid conductivity). The relation between luid conductivity o a Nal solution and electrolyte concentration in the range 10-6 M < < 1 M (15 o < temperature < 25 o ) is given as 10 [25]. Thereore, we 9 get the expression 1.2 10 / based on [8] and that is also similar to Eq. (12). The prediction o P as a unction o electrolyte concentration rom the empirical expressions o [8, 9] is also shown in Fig. 9 (see the dashed lines). It is seen that the predictions rom [8, 9] have the same behavior as that obtained in this work but give smaller values o the P at a given electrolyte concentration. The reason or the deviation between the empirical expressions may be due to dissimilarities o luid conductivity, luid ph, mineral composition o porous media, temperature etc. Figure 9. P in magnitude as a unction o electrolyte concentration or dierent samples (1, 2, 3 and 4) and dierent electrolytes (Nal, NaI, Kl and KI). ymbols are experimental data. olid line is the itting line and two other dashed lines are predicted rom [8] and [9]. 5. onclusions The measurements o the streaming potential o our unconsolidated samples saturated with our monovalent electrolytes at dierent electrolyte concentrations have been perormed. From the measured P, the empirical expression between the P and electrolyte concentration is obtained. The empirical expression is in good agreement with those reported in literature in which the P in magnitude is inversely proportional to electrolyte concentration. This work has added an additional empirical expression to the existing ones that allow us to predict the P rom electrolyte concentration or silica-based samples saturated by monovalent electrolytes. Additionally, it is seen that the streaming potential coeicient depends on types o cation in electrolytes and on samples. The dependence o the streaming potential coeicient on types o cation is qualitatively explained by the dierence in the binding constant or cation adsorption on the silica suraces. The dependence o the streaming potential coeicient on samples is due to the variation o eective conductivity and the zeta potential between samples.
24 L.D. Thanh, Rudol. / VNU Journal o cience: Mathematics Physics, Vol. 34, No. 1 (2018) 14-24 Acknowledgments The irst author would like to thank Vietnam National Foundation or cience and Technology Development (NAFOTED) under grant number 103.99-2016.29 or the inancial support. Reerences [1] B. Wurmstich, F. D. Morgan, Geophysics 59 (1994) 46 56. [2] R. F. orwin, D. B. Hoovert, Geophysics 44 (1979) 226 245. [3] F. D. Morgan, E. R. Williams, T. R. Madden, Journal o Geophysical Research 94 (1989) 12.449 12.461. [4] H. Mizutani, T. Ishido, T. Yokokura,. Ohnishi, Geophys. Res. Lett. 3 (1976). [5] M. Trique, P. Richon, F. Perrier, J. P. Avouac, J.. abroux, Nature (1999) 137 141. [6] A. A. Ogilvy, M. A. Ayed, V. A. Bogoslovsky, Geophysical Prospecting 17 (1969) 36 62. [7] P. W. J. Glover, E. Walker, and M. D. Jackson, Geophysics 77 (2012) D17 D43. [8] L. Jouniaux and T. Ishido, International Journal o Geophysics, vol. 2012, Article ID 286107, 16 pages, 2012. doi:10.1155/2012/286107. [9] Vinogradov, J., M. Z. Jaaar, and M. D. Jackson, Journal o Geophysical Research Atmospheres 115 (2010) B12204. [10] O. tern, Z. Elektrochem 30 (1924) 508 516. [11] T. Ishido, H. Mizutani, Journal o Geophysical Research 86 (1981) 1763 1775. [12] H. M. Jacob, B. ubirm, Electrokinetic and olloid Transport Phenomena, Wiley-Interscience, 2006. [13] K. E. Butler, eismoelectric eects o electrokinetic origin, PhD thesis, University o British olumbia, 1996. [14] H. Hase, T. Ishido,. Takakura, T. Hashimoto, K. ato, Y. Tanaka, Geophysical Research Letters 30 (2003) 3197 3200. [15] R. J. Hunter, Zeta Potential in olloid cience, Academic, New York, 1981. [16] L. Onsager, Physical Review 37 (1931) 405-426. [17]. Dukhin, V. hilov, Dielectric Phenomena and the Double Layer in Disperse ystems and Polyelectrolytes, John Wiley and ons, New York, 1974. [18] L. Jouniaux, M. L. Bernard, M. Zamora, J. P. Pozzi, Journal o Geophysical Research B 105 (2000) 8391 8401. [19] Luong Duy Thanh, Rudol prik, VNU Journal o cience: Mathematics-Physics 32 (2016) 22-33. [20] Paul W. Glover and Nicholas Déry. Geophysics 75(2010), F225-F241. [21]. F. Alkaee and A. F. Alajmi, olloids and uraces A 289 (2006) 141 148. [22] P. N. en,. cala, and M. H. ohen, Geophysics 46 (1981) 781 795. [23] Lưong Duy Thanh, Rudol prik, VNU Journal o cience: Mathematics and Physics 31 (2015) 56-65. [24] Luong Duy Thanh and Rudol prik (2016). Zeta potential in porous rocks in contact with monovalent and divalent electrolyte aqueous solutions geophysics, 81(4), D303-D314. [25] P. N. en and P. A. Goode, Geophysics 57 (1992) 89-96.