Examination of a Theoretical Model of Streaming Potential Coupling Coefficient

Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.


Introduction
Electrokinetic phenomena are induced by the relative motion between a fluid and a solid surface and are directly related to the existence of an electric double layer between the fluid and the solid surface. Of electrokinetic phenomena, streaming potential and seismoelectric effects play an important role in geophysical applications. For example, streaming potential is used to map subsurface flow and detect subsurface flow patterns in oil reservoirs [1]. Streaming potential is also used to monitor subsurface flow in geothermal areas and volcanoes [2][3][4]. Monitoring of Streaming Potential anomalies has been proposed as a means of predicting earthquakes [5,6]. Seismoelectric and electroseismic effects have been developed in order to investigate oil and gas reservoirs [7], hydraulic reservoirs [8][9][10], and downhole seismoelectric imaging [11]. It should be noted that electrokinetic phenomena are not applicable for porous media totally saturated with oil. However, reservoir rocks are normally saturated with oil, salt water, and gas. The coupling coefficient of conversions between seismic waves and electromagnetic waves is a theoretically involved expression of frequency and Streaming Potential coupling coefficient [12].
The key parameter for electrokinetic phenomena is the Streaming Potential coupling coefficient and therefore the zeta potential [4]. Normally, the coupling coefficient is experimentally measured and the zeta potential is then deduced from that with knowledge of the electrical conductivity, viscosity, and electrical permittivity of the fluid. Glover et al. [13] developed a theoretical model to calculate the zeta potential and Streaming Potential coefficient of reservoir rocks and other porous media. By using reasonable values of input parameters that are supported by independent measurements or theory, the authors have obtained and compared the theoretical results to an experimental data set of 290 Streaming Potential coefficient measurements and 269 zeta 2 International Journal of Geophysics potential measurements using data from 29 publications. The comparison shows that the theoretical model can reproduce the main features of the experimental data.
However, the experimental data set was collected from different sources that may lead to a dissimilarity in fluid conductivity at the same pore fluid concentration (quality of deionized water used to make brine may be not the same), in pH of pore fluid (a big range of pH from 5.5 to 11), in temperature, and so forth. Some microstructure parameters that are needed for the model were also not given in the original publications. The zeta potential and surface conductance themselves may be also different from sample to sample [14]. Therefore, all of those dissimilarities could lead the experimental data to spread around the theoretically modeled graphs for a given set of material.
In this work we want to examine how well the model works for 20 consolidated samples with 7 different concentration NaCl solutions and 9 different temperatures. To do so, firstly the coupling coefficients and the zeta potentials were experimentally measured as a function of electrolyte concentration and temperature. Afterwards the model was used by taking fluid parameters, rock-fluid interface parameters already provided in [13], and rock microstructure parameters (porosity, permeability, and formation factor) measured by us for all the samples. The proton surface conductance was adjusted to fit the experimental data. The comparisons were carried out for each sample.
For electrolyte concentration dependence of the coupling coefficient, the comparison shows that the theoretical model is not in agreement with the experimental data, especially in low concentration range. However, if the zeta potential is considered to be constant over electrolyte concentration the model fits the experimental data very well in a whole range of concentrations. This leads to the suggestion that the formula of the zeta potential against electrolyte concentration in the model needs to be adjusted accordingly. Besides the model given in [13], we also use other empirical models from Pride and Morgan [15], Vinogradov et al. [16], Jaafar et al. [17], and Jouniaux and Ishido [18] to predict the Streaming Potential coupling coefficient as a function of electrolyte concentration. But none of them can fit experimental data at low electrolyte concentration. For temperature dependence of the coupling coefficient, we implemented a zeta potential offset to the model which improves the fit to experimental data. The comparison shows that the theoretical model does not fit the experimental data well but it can show the increase of coupling coefficient with increasing temperature as measured. The reason for that could be due to the variation of some input parameters describing the rock/fluid interface over temperature rather than being constant as assumed in the experiment and a possible process activated by temperature happening around 300 K.
This work is a continuation and extension of the earlier work [14] in which permeability dependence of Streaming Potential coupling coefficients for porous media with taking into account the difference in the zeta potential among samples and the change of surface conductance against electrolyte concentration was studied. This work includes five sections. Section 2 describes the theoretical background of the streaming potential including the calculation of Stern potential, zeta potential, surface conductance, and Streaming Potential coupling coefficient of porous media. Section 3 presents the experimental measurements. Section 4 contains the experimental results, discussion, and comparisons between the experimental data and the results of the theoretical model. Conclusions are provided in Section 5.

Theoretical Background of
Streaming Potential 2.1. Streaming Potential Coupling Coefficient. The streaming current is created by the relative motion of the diffuse layer with respect to the solid surface induced by a fluid pressure drop over the channel and is directly related to the existence of an electric double layer (EDL) between the fluid and the solid surface (for more details, see [24]). This streaming current is then balanced by a conduction current, leading to the streaming potential. In a porous medium, the electric current density and the fluid flux are coupled, so fluids moving through porous media generate a streaming potential [25]. The streaming potential increases linearly with the fluid pressure difference that drives the fluid flow, provided that the flow remains laminar [26]. The steady state Streaming Potential coupling coefficient is defined when the total current density is zero as where Δ is the streaming potential, Δ is the fluid pressure difference, is the relative permittivity of the fluid, 0 is the dielectric permittivity in vacuum (8.854 × 10 −12 F/m), is the dynamic viscosity of the fluid, is the zeta potential, and eff is the effective conductivity. The zeta potential is the electric potential on the shear plane when a part of the diffuse layer is transported by fluid flow (for more details, see [11,27]). The effective conductivity includes the intrinsic fluid conductivity and the surface conductivity. If the fluid conductivity is much higher than the surface conductivity (the fluid conductivity is larger than 0.13 S/m or 0.01 M NaCl solution for our measurement as stated in [14]), the effective conductivity is approximately equal to the fluid conductivity, eff = . Therefore, the coupling coefficient becomes the well-known Helmholtz-Smoluchowski equation: According to [14], (1) can be written as where is the fluid conductivity, Σ is the surface conductance, is the formation factor, and is the permeability of the porous medium. Equation (3) shows that, besides the zeta potential, fluid conductivity, and surface conductance, International Journal of Geophysics 3 the coupling coefficient also depends on the microstructure of porous media such as permeability, porosity, and tortuosity expressed via the formation factor.
To calculate the coupling coefficient, one needs to know (1) the zeta potential, (2) the fluid relative electric permittivity, (3) the fluid viscosity, (4) the fluid conductivity, and (5) the surface conduction as a function of electrolyte concentration and temperature as well as the rock microstructure parameters such as permeability and formation factor. All of those are theoretically presented in [13]. In (3), we just rewrite the key expressions for the zeta potential and the surface conduction.

Zeta
Potential. The electrical potential distribution in the EDL has, approximately, an exponential distribution given in [13,28,29] = exp (− ) , where is the Stern potential (V) given by and is the Debye length (m) given by and is the distance from the mineral surface. Equation (4) is, in fact, the solution of the linearized Poisson-Boltzmann equation describing the electrostatic potential distribution in the electrical double layer and is called the Debye-Hückel approximation [12]. The zeta potential can then be calculated as where = 2.4 × 10 −10 m is the shear plane distance (the distance from the mineral surface to the shear plane).
In (5) and (6), is Boltzmann's constant, is temperature (in K), 0 is the dielectric permittivity in vacuum, is the relative permittivity, is the elementary charge, is Avogadro's number, pH is the fluid pH, is the binding constant for sodium adsorption, (−) is the disassociation constant for dehydrogenization of silanol surface sites, Γ is the surface site density, is the disassociation constant of water, and is the electrolyte concentration.

Surface Conduction.
The surface conductance is calculated using the techniques described in [13,28,29] in which the surface conductance is given by where Σ EDL , Σ Stern , and Σ Prot are the contributions to the surface conductivity from ionic conduction in the EDL, in the Stern layer, and associated with proton transfer, respectively. Even the above formula is developed for 1 : 1 electrolyte of the aqueous NaCl solution and quartz; it still works for other porous materials such as Fontainebleau, Stainton, St. Bees sandstones, or Carbonates (see [13] for more details), where in which = √(2 × 10 3 0 )/( + 10 −pH ) and = √8 × 10 3 0 ( + 10 −pH + 10 pH− ). In (9)-(11) is ionic Stern-plane mobility, Na + , H + , Cl − , OH − are ionic mobility of Na + , ionic mobility of H+, ionic mobility of Cl − , and ionic mobility of OH − in the solution, respectively, and Prot is proton surface conductance.

Disassociation Constant.
This parameter defines the fluid pH which is required to calculate the Stern potential and surface conductance. The value of varies with temperature and is approximately in the range 0 ∘ C to 100 ∘ C as follows [30]: where is in ∘ C.
Indeed, the fluid pH is also defined by the reaction of water with carbon dioxide from the air which generates bicarbonate (HCO 3 ) and hydrogen ions (H + ) [13]. This leads to the fact that water that has been exposed to air is slightly acidic. In the model, one needs to know the pH to calculate the concentrations of hydrogen ions (10 −pH ) and hydroxyl ions (10 pH− ) in the fluid. The pH values required for the model are directly measured by a pH meter.

Experiments
Streaming Potential measurements have been performed on 20 consolidated samples (55 mm in length and 25 mm in diameter) including both natural and artificial samples from different sources (see Table 1). The natural samples numbered from 1 to 7 were obtained from Shell company [31]. Artificial samples numbered from 8 to 13 were obtained from HP Technical Ceramics Company in England. The natural samples numbered from 14 to 19 were obtained from Berea Sandstone Petroleum Cores Company in the USA. The last one numbered 20 was obtained from Delft University. The mineral compositions of all samples are also shown in Table 1. The experimental setup for the Streaming Potential measurement as a function of concentration (at room temperature −22 ∘ C) as well as the method to measure the porosity, solid density, permeability, and formation factor is well described in [14]. For measuring the streaming potential as a function of temperature, we immersed the core holder and the bottle containing the solution into the bath of a thermally stabilized water bath (Thermo Fisher Scientific SC150 and A25) as shown in Figure 1.
The solutions used in our measurements were NaCl solutions with 7 different electrolyte concentrations (4 2.0 × 10 −2 M, and 5.0 × 10 −2 M). Temperatures used in our measurements are 6 ∘ C, 11 ∘ C, 16 ∘ C, 16 ∘ C, 21 ∘ C, 26 ∘ C, 31 ∘ C, 36 ∘ C, 41 ∘ C, and 46 ∘ C, respectively. It should be noted that the model presented in Section 2 is valid in the range 0 ∘ C-100 ∘ C [13]. The solution was circulated through the samples until the electrical conductivity and pH of the solution reached a stable value. The pH values of equilibrium solutions are in the range 6.0 to 7.5 and the conductivities are saved for the zeta potential calculations. To avoid CO 2 uptake from the air that leads to change of conductivity and pH of the solutions, the solution container was tightly covered during experiment.

Results and Discussion
4.1. Porosity, Solid Density, Permeability, and Formation Factor. Porosity, density, and permeability of the samples are shown in Table 1 with an error of 3%, 5%, and 6%, respectively. Values of the formation factor and corresponding tortuosity are also reported in Table 1 with an error of 6% and 9%, respectively.
International Journal of Geophysics 5

Streaming Potential as a Function of Concentration.
The way used to obtain the streaming coupling coefficients is the same as described in [14]. The streaming coupling coefficients at different solutions for all samples are shown in Table 2. It should be noted that experimental data for coupling coefficients of the samples numbered from 8 to 19 have been already published in [14]. There are no coupling coefficients for two samples DP43 and DP172 at concentrations of 2 × 10 −2 M and 5 × 10 −2 M because these samples are so permeable that they need a very large flow rate to generate measurable electric potentials at high concentration solutions. The maximum error of the coupling coefficients is 15%. By using the input parameters presented by Glover et al. [13] in Table 3 except proton surface conductance Prot which is adjusted to fit data as described in [14], we have calculated the change of the zeta potential, the fluid viscosity, the relative permittivity, and the surface conduction against the electrolyte concentration.
According to Glover et al. [13], one can improve the fit of the model to the experimental data by introducing a constant zeta potential offset . It means that (7) is replaced by = exp(− / ) + . The zeta potential offset enables the model to reproduce the Streaming Potential coupling coefficient and zeta potential at high salinity. The physical meaning behind that is still unclear even though a possible explanation was partially given in [13]. Therefore, more research needs to be carried out to understand that parameter.
The relationship between the pore electrolyte concentration and fluid conductivity is now needed for the model. This relationship was found by fitting experimental data as shown, for example, in Figure 2 for the sample IND01.  shows that the relationship follows = 9.5 + 0.0085 (S/m) for the sample IND01 and it is in good agreement with the model of Sen and Goode [32] for fluid conductivity of a NaCl solution as a function of electrolyte concentration in the ranges 10 −6 < < 1 M and 15 < < 25 ∘ C ( = 10 ). Similarly, the relationships between the pore electrolyte concentration and fluid conductivity were obtained for other samples.
All data is now sufficient to model the coupling coefficient as a function of electrolyte concentration with input 6 International Journal of Geophysics  parameters given by [13] in Table 3. Figure 3 shows the experimental and modeled results of Streaming Potential coupling coefficient for sample IND01. The modeled result was implemented for three values of pH (pH = 6, 7, and 8) with the zeta potential offset = 0.045 V and the proton surface conductance Prot = 3×10 −28 S m 2 /site. Figure 3 shows that the model of coupling coefficient against the electrolyte concentration is very sensitive to the pH of the fluid as expected in [13]. The model is not in good agreement with the experimental result especially at a low concentration, where the deviation of coupling coefficient from experimental data becomes bigger.
The deviation between the modeled and experimental results could arise from the formula proposed to calculate the zeta potential in (5)- (7). According to the model, the zeta potential in magnitude would decrease drastically with increasing concentration as shown in Figure 4 (the solid line). However, based on the three measured coupling coefficients for all samples in Table 2 at high concentrations of 10 −2 M, 2 × 10 −2 M, and 5 × 10 −2 M, the zeta potentials at those corresponding concentrations were calculated by using (2) as shown, for example, in Figure 4 (the dashed line) for sample US5. It is worthwhile noting that the surface conductance is neglected when the electrolyte concentration is higher than 10 −2 M as mentioned in [14]. It is obvious that the International Journal of Geophysics measured zeta potential slightly decreases with increasing concentration and can be roughly considered constant over the studied range of the electrolyte concentration. Similarly, the zeta potentials were experimentally obtained for other samples.
We therefore assume the zeta potential to be constant over the electrolyte concentration and that is also inferred from the collected experimental data presented in [18]. The dependence of the coupling coefficient on the electrolyte concentration for all samples with corresponding proton surface conductance is experimentally and theoretically shown in All theoretical results were carried out for pH of 6.7 because this value is the average of measured values (between 6 and 7.5). For reasons of clarity, the coupling coefficient is plotted against the reverse of electrolyte concentration rather than electrolyte concentration as shown in Figure 5. This is because we want to avoid the experimental data locating just on the bottom left corner of the graphs at low electrolyte concentrations. Figure 5 shows that when the zeta potential is assumed to be constant over the electrolyte concentration, the model is well capable of producing the main features of the experimental data. It is also seen that the coupling coefficient is linearly proportional to the inverse of electrolyte concentration as expected at high concentrations when the surface conductance is ignored. This observation leads to the suggestion that the formula for the calculation of the zeta potential (see (4) to (7)) needs to be revised especially at a low concentration.
Due to the limitation of the model developed by Glover et al. [13] at low concentration, we have looked at other models to predict the streaming potential as a function of electrolyte concentration. For example, by fitting experimental data for quartz and NaCI or KCI at pH = 7 and = 25 ∘ C, Pride and Morgan [15] obtain the relation = 8 + 26 log 10 ( ) , (13) where is in mV and is the electrolyte concentration. Similarly, Vinogradov et al. [16] obtain the relation between the zeta potential and electrolyte concentration based on published zeta potential data for quartz, silica, and glass in NaCl at pH = 6-8 as = −9.67 + 19.02 log 10 ( ) .
By fitting experimental data collected for sandstone, sand, silica nanochannels, Stainton, and Fontainebleau with electrolytes of NaCl and KCl at pH = 6-8, Jaafar et al. [17] obtain an empirical expression between the coupling coefficient and electrolyte concentration: where is expressed in mV MPa −1 . Jouniaux and Ishido [18] obtains the other relations to predict the coupling coefficient from fluid conductivity based on numerous measurements of the streaming potential on sand with NaCl at pH = 7-8 which have been published: where is the fluid conductivity and is in V Pa −1 . Putting (13) and (14) (16), the fluid conductivity as a function of electrolyte concentration obtained by fitting experimental data as shown in Figure 2 is used. The prediction of the coupling coefficient as a function of electrolyte concentration from all four empirical models is shown in Figure 6.
Because all empirical models are mostly obtained for the silica based samples, experimental data for Bentheim sandstone mainly made up by silica (see Table 1) are used for comparison as shown in Figure 6. The prediction from the model of Glover et al. with the adjustment of constant zeta potential for Bentheim sandstone shown in Figure 5(c) is also repeated in Figure 6. The comparison shows that all models converge and agree with the experimental data at high electrolyte concentration. However, at low concentration  Figure 6). Therefore, all four empirical models fail to predict the Streaming    Potential coupling coefficients as a function of the electrolyte concentration even for the most used materials such as silica and NaCl especially at low electrolyte concentrations. Figure 6 shows that the best approach to predict the coupling coefficient as a function of electrolyte concentration is to use the modified model of Glover et al. Besides four above models, there are also other ones available in literature, for example, from [26,33,34] that are not shown in Figure 6 due to the confusion.

Streaming Potential as a Function of Temperature.
Measurement of the coupling coefficient as a function of temperature normally takes about 8 h for each sample. During that time, the coupling coefficient may change. To see how the coupling coefficient varies over time, we measured the coupling coefficient as a function of time at room temperature as shown in Figure 7(a) by continuously pumping the fluid through the sample and measuring the coupling coefficient after every fixed time interval. Figure 7 shows that firstly the coupling coefficient decreases drastically over time which could be due to the change of conductivity and pH (CO 2 uptake or electrode polarization drift or mineral decay from samples) and after about 40 h the coupling coefficient gets relatively stable. Therefore, the coupling coefficients as a function of temperature would be picked after 40 h.
To check how long it takes to get thermal equilibrium for the sample inside the core holder, we fixed a sensor probe tip of the digital temperature meter (Omega DP460) on the sample outlet surface and found that the time period to get thermal equilibrium is around 45 mins. Therefore, we take 45 mins as a time sampling (time between two consecutive measurements).
We also measured viscosity of the liquid as a function of temperature as shown in Figure 7(b) by comparing the slopes of flow rate-pressure difference straight lines (see [24]) at different temperatures to the reference slope at room temperature at which the fluid viscosity is known to be 0.96 × 10 −3 Pa⋅s [35]. Figure 7(b) shows that the measured viscosities are in very good agreement with the model suggested by [35]. Therefore, the sample completely gets thermal equilibrium after 45 mins. To minimize the change of fluid conductivity over time, we used a solution of the relatively high concentration 5 × 10 −3 M.
To model the coupling coefficient as a function of temperature, the same argument as described in Section 4.2 was used. The relative permittivity, viscosity, surface conductance, disassociation constant, and fluid conductivity at the concentration of 5 × 10 −3 M as a function of temperature are theoretically given in [13]. The zeta-potential offset was found by fitting the experimental data of the coupling coefficient for each sample. Figure 8 shows the experimental results and modeled results of coupling coefficient for sample IND01 in which three pH values (6, 7, and 8), three corresponding zeta-potential offsets ( = 0.045 V, 0.083 V, and 0.12 V), and proton surface conductance Prot = 3 × 10 −28 S m 2 /site were used to model. It was seen that the coupling coefficient versus temperature is also very sensitive to fluid pH and pH = 6 gives better fit than the others. Therefore, pH = 6 was used to model the coupling coefficient versus temperature for representative samples of EST, BereaUS5, DP217, and BER502 as shown in Figure 9. Both the experimental and theoretical results show that the coupling coefficient increases with increasing temperature. However, Figure 9 shows an essential deviation between experimental data and the model. The origin of that deviation is still unclear. One of the reasons may be due to a temperature activated transport process happening around 300 K, which obviously cannot be explained by the model and needs to be explored.

Conclusion
We have measured Streaming Potential coupling coefficients for 20 consolidated samples with 7 different concentration NaCl solutions and 9 different temperatures under wellcontrolled conditions (pH, temperature, and quality of distilled water). Afterwards the model has been used by taking fluid parameters, rock-fluid interface parameters already provided in [13], and rock microstructure parameters (porosity, permeability, and formation factor) measured by us for all the samples. The proton surface conductance was adjusted to fit the experimental data. The comparisons were carried out for each sample.
For concentration dependence of the coupling coefficient, the comparison shows that the theoretical model is not in agreement with the experimental data, especially in the low concentration range. However, if the zeta potential is considered to be constant over a small range of electrolyte concentrations the model fits the experimental data very well.  This leads to the suggestion that the formula of the zeta potential against electrolyte concentration presented in the model needs to be recalculated and adjusted accordingly. Besides the model given in [13], we also use other models to predict the streaming potential as a function of electrolyte concentration. But none of them can fit the experimental data at low electrolyte concentration. Therefore, further revisions of those models need to be carried out for a better fit at low electrolyte concentration. Of all the models used for the comparison, the modified model of Glover et al. with assuming of constant zeta potential that is experimentally measured is the best approach to predict the coupling coefficient as a function of electrolyte concentration.
For temperature dependence of the coupling coefficient, we implemented a zeta-potential offset to the model which improves the fit to experimental data. The comparison shows that the theoretical model does not fit the experimental data well, but it can show the increase of coupling coefficient with increasing temperature as measured. The reason for that could be due to the variation of the input parameters over temperature because the Streaming Potential coefficient is sensitive to changes in the input parameters that describe the rock-fluid interface, such as the surface site density (Γ ), the binding constant ( ), and the disassociation constant ( (−) ) [13]. Therefore, more research about the change of the input parameters versus temperature needs to be conducted for a better explanation. A possible explanation of the deviation at around 300 K could be a temperature activated transport process. The results also show that the model is very sensitive to the pH of the pore fluid, especially for the Streaming Potential coefficient as a function of temperature. So the fluid pH needs to be carefully taken into account during measurements. Ellermeijer for his assistance in building up the experimental setups.