Streaming Potential Coefficient Research Articles

Streaming Potential in Gas Phase Discrete Unsaturated Pore

The seismoelectric effect of porous media is the main basis for seismoelectric logging. At present, most of the studies on the seismoelectric effect in unsaturated porous media adopt the model of pores with continuous distribution of gas and liquid. There is a lack of theoretical research on the micro mechanism of the seismoelectric effect of unsaturated porous media with discrete gas phase, and the existing studies do not consider the effect of the electric double layer at the gas–liquid interface on the seismoelectric effect. Based on the capillary model, this work adopted the gas phase discrete model, combined the electric double layer theory and the seepage principle, considered the effect of electric double layer at the pore wall and the gas–liquid interface, and studied the micro principle of the seismoelectric effect of unsaturated porous media. Firstly, we studied the variation of gas–water two-phase flow pattern with saturation in unsaturated pores, then proposed the equivalent principle of series circuits, deduced the effective streaming current and conductance of a pore containing multiple bubbles, and then deduced the streaming potential coupling coefficient in the unsaturated pores. We also studied the effect of pore parameters such as saturation, pore size, bubble spacing, pore fluid viscosity, and salinity on the streaming potential coupling coefficient. The results show that the streaming potential coupling coefficient first increases and then decreases with the decrease in saturation, which is the same as the trend measured in Allègre’s experiment, and provide a theoretical explanation for the non-monotonic change in the coupling coefficient with saturation in unsaturated porous media.

A unified model for the permeability, electrical conductivity and streaming potential coupling coefficient in variably saturated fractured media

ABSTRACTWe present a new unified model for the permeability, electrical conductivity and streaming potential coupling coefficient in variably saturated fractured media. For those, we conceptualize the fractured medium as a partially saturated bundle of parallel capillary slits with varying sizes. We assume that the fracture size distribution of the corresponding medium follows a fractal scaling law, which allows us to establish a pressure head‐saturation relationship based on the Laplace equation. We first describe the flow rate, the conduction current and the electrokinetic streaming current within a single fracture. Then, we upscale these properties at the scale of an equivalent fractured media partially saturated in order to obtain the relative permeability, the electrical conductivity and the streaming potential coupling coefficient. The newly proposed model explicitly depends on pore water chemistry, interface properties, microstructural parameters of fractured media and water saturation. Model predictions are in good agreement with both experimental and simulated data and with another model from the literature. The results of this work constitute a useful framework to estimate hydraulic properties and monitor water flow in fractured media.

Electrokinetic coupling in unsteady pressure-driven flow through a porous transducer: Fractal capillary bundle model

• A fractal capillary bundle model is used to predict the frequency-dependent streaming potential coefficient of porous media. • The theoretical model of electrokinetic coupling in the unsteady pressure-driven flow is established to characterize the frequency response of the streaming potential coefficient. • An analytic solution of Poisson-Boltzmann equation is obtained for the capillary bundle of porous media with high zeta potential condition. • The proposed model shows satisfactory agreement with experiments and can optimize the parameter configuration of transducers. The liquid circular angular accelerometer (LCAA) is a novel fluid-mechanical device based on the principle of electrokinetic effect, which is featured as the streaming potential coefficient (SPC) to convert elastic wave energy into electromagnet energy within the porous transducer. To determine its dynamic response, we present an analytical expression for the frequency-dependent SPC of a porous transducer in the LCAA. The electrokinetic coupling process in unsteady pressure-driven flow is developed by using the fractal capillary bundle model. Electrical potential distribution in a torturous capillary of the arbitrary radius is modeled by a modified approximation for the Poisson-Boltzmann equation with high zeta potential condition. We present an apparatus for measuring the dynamic SPC of transducers that are sintered with glass beads and have high porosity. Measurements on natural and sintered porous samples validate the proposed model and demonstrate satisfactory agreement. According to the sensitivity analysis, changes in fluid and structural properties have a significant impact on the unsteady SPC response, particularly the solution concentration and mean particle size. The findings of this study can pave the way for the frequency-dependent SPC prediction of the porous transducer as well as a better understanding of the LCAA design optimization.

Application of critical path analysis to streaming potential coupling coefficient in porous media

Streaming potential is a passive hydrogeophysical method that can be used to monitor water flow in porous media. However, a quantitative use of this method requires a good understanding of the streaming potential generation in water saturated porous media. In this work, we used the Critical Path Analysis (CPA) method to propose a new model to estimate the streaming potential coupling coefficient (SPCC) in porous media. By comparing the proposed model with 109 experimental data points for both unconsolidated samples such as uniform glass bead and sand packs and consolidated samples such as sandstones, carbonates and with 36 simulated data points, we demonstrated that the CPA is applicable to model the SPCC for porous media with both narrow and broad pore size distributions. Additionally, from the obtained result for the uniform grain packings, we found the rc–d relationship between the critical pore radius rc and grain diameter d that was validated by using 70 experimental data points available in literature. The obtained rc–d relationship may be helpful for applications of CPA-based models for uniform grain packings. We believe that the CPA-based model is an useful alternative approach for modeling the streaming potential coupling coefficient in porous media along with two other models available in literature.

Time-resolved pressure-induced electric potential in nanoporous membranes: Measurement and mechanistic interpretation

Performance of charged nanoporous membranes in ion separations and electrokinetic energy conversion is controlled by their surface-charge density. This (or related such as zeta-potential) parameter has often been obtained from measurements of pressure-induced electric potential typically referred to as streaming potential. However, with nanoporous membranes in dilute electrolyte solutions, determination of genuine streaming potential is non-trivial and requires time-resolved measurements. A new approach to the interpretation of such measurements developed and tested experimentally in this study makes possible parallel determination of several membrane transport properties while previously only streaming-potential coefficient was determined. In nanoporous membranes, this coefficient is a non-monotone function of surface-charge density, which makes difficult unambiguous determination of the latter. On the contrary, the other properties determined in this study are monotone functions of surface-charge density. This approach has been validated via characterization of two nanoporous track-etched membranes (with the pore sizes of 25 nm and 35 nm) in KCl solutions and revealed an excellent applicability. The transport properties “extracted” from experimental data were used as input of full (numerical) version of space-charge model. This made possible determination of surface-charge density as well as some properties of hypothetical gel layers reportedly surrounding pores of nanoporous grades of track-etched membranes.

Understanding the Influence of Rock Content on Streaming Potential Phenomenon of Soil-Rock Mixture: An Experimental Study.

To applicate streaming potential phenomenon to study the seepage feature in the soil–rock mixture (SRM), research on the variation in the streaming potential phenomenon of SRM is the precondition. This paper deals, in assistance with the streaming potential test apparatus, with the streaming potential effect response of SRM subjected to different rock contents. The test results show that when the rock content increases from 10% to 30%, the streaming potential coupling coefficient increases with the increases in rock content at 85% compactness and 0.01 mol L−1 salinity. When the rock content is more than 30%, the streaming potential coupling coefficient decreases with the increases in rock content. As the rock content increases, the permeability coefficient has a negative correlation with the streaming potential coupling coefficient. The streaming potential increases first and then goes down with the increases in rock content, and the streaming potential decreases significantly when the rock content exceeds 50%. The findings indicate that the rock content is the key structural factor that restricts the streaming potential phenomenon of the SRM.

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

SUMMARY Seismoelectric signals are generated by electrokinetic coupling from seismic wave propagation in fluid-filled porous media. This process is directly related to the existence of an electrical double layer at the interface between the pore fluid and minerals composing the pore walls. The seismoelectric method attracts the interest of researchers in different areas, from oil and gas reservoir characterization to hydrogeophysics, due to the sensitivity of the seismoelectric signals to medium and fluid properties. In this work, we propose a physically based model for the dynamic streaming potential coupling coefficient (SPCC) by conceptualizing a porous medium as a bundle of tortuous capillaries characterized by presenting different pore size distributions (PSD). The results show that the dynamic streaming potential coupling coefficient is a complex function depending on the properties of pore fluid, mineral–pore fluid interfaces, microstructural parameters of porous media and frequency. Parameters influencing the dynamic SPCC are investigated and explained. In particular, we show that the PSD affects the transition frequency as well as the shape of the SPCC response as a function of frequency. The proposed model is then compared with published data and previous models. It is found that the approach using the lognormal distribution is in very good agreement with experimental data as well as with previous models. Conversely, the approach that uses the fractal distribution provides a good match with published data for sandstone samples but not for sand samples. This result implies that the fractal PSD may not be pertinent for the considered sand samples, which exhibit a relatively narrow distribution of pore sizes. Our proposed approach can work for any PSD, for example, including complex ones such as double porosity or inferred from direct measurements. This makes the proposed models more versatile than models available in literature.

Coupled inversion of hydraulic and self‐potential data from transient outflow experiments to estimate soil petrophysical properties

AbstractHydraulicproperties of soils could play an important role in affecting the partitioning of precipitation in the critical zone. In addition to traditional approaches, in the last two decades, many geophysical methods have been used to aid the hydrologic characterization and measurement of geological materials. In particular, the self‐potential (SP) method shows great potential in these hydrogeophysical applications. The objective of this study is to evaluate whether the addition of SP data can improve the estimation of hydraulic properties of soils in an outflow experiment. A stochastic, coupled hydrogeophysical inversion was developed, in which the governing equations were solved using the finite volume method and the parameter estimation was conducted using a Bayesian approach associated with the Markov chain Monte Carlo technique. The results show that the addition of SP data in the inversion could reduce the uncertainty related to the estimated hydraulic parameters of soils and the length of the associated 95% confidence interval can be shortened by ∼1/3. It is also shown that the electrical properties of soils at saturated and unsaturated conditions may also be estimated from the outflow experiment when SP data are available. Compared with hydraulic parameters, the accuracy of the estimated electrical properties is slightly lower. Among them, the saturated streaming potential coupling coefficient Csat has the highest accuracy and lowest uncertainty since Csat directly influences the magnitude of SP signals. The accuracy of other electrical parameters is lower than that of Csat (and hydraulic parameters), and the associated uncertainty can be one order of magnitude larger.

Experimental Study on the Streaming Potential Phenomenon Response to Compactness and Salinity in Soil–Rock Mixture

The study on the effect of material structure and solution properties on the streaming potential of the soil–rock mixture (SRM) will be beneficial for improving the reliability of the measurement results for self-potential monitoring in embankment dams. We design two experimental groups and investigate the changes of potential and pressure during seepage of SRM (slightly clay materials) with different compactness and different concentration. The effects of the compaction degree and solution concentration on the streaming potential coupling coefficient and streaming potential were analyzed. The test results demonstrate that when the clay content in soil matrix is slight, the potential has a linear relationship with the hydraulic head difference, and seepage obeys Darcy’s law. The surface conductivity is negligible at 0.01 M (1 M corresponds to a concentration of 58.4 g L−1) salinity, the compactness of the SRM decreases (the permeability coefficient increases), the apparent streaming potential coupling coefficient and pressure difference decrease is the reason streaming potential decreases. The permeability coefficient of the SRM is not affected by the change in salinity (0.0001–1 M) at 85% compactness, and its seepage characteristics are related to the mineral composition, morphology and the thickness of the bound water layer (electric double layer). This study lays a foundation for further research on the self-potential method to monitor the structure of embankment dams.

Predicting Electrokinetic Coupling and Electrical Conductivity in Fractured Media Using a Fractal Distribution of Tortuous Capillary Fractures

Electrokinetics methods have attracted increasing interest to characterize hydrogeological processes in geological media, especially in complex hydrosystems such as fractured formations. In this work, we conceptualize fractured media as a bunch of parallel capillary fractures following the fractal size distribution. This conceptualization permits to obtain analytical models for both the electrical conductivity and the electrokinetic coupling in water saturated fractured media. We explore two different approaches to express the electrokinetic coupling. First, we express the streaming potential coupling coefficient as a function of the zeta potential and then we obtain the effective charge density in terms of macroscopic hydraulic and electrokinetic parameters of porous media. We show that when the surface electrical conductivity is negligible, the proposed models reduces to the previously proposed one based on a bundle of cylindrical capillaries. This model opens up a wide range of applications to monitor the water flow in fractured media.

Electro-Osmotic Behavior of Polymeric Cation-Exchange Membranes in Ethanol-Water Solutions.

The aim of this work is to apply linear non-equilibrium thermodynamics to study the electrokinetic properties of three cation-exchange membranes of different structures in ethanol-water electrolyte solutions. To this end, liquid uptake and electro-osmotic permeability were estimated with potassium chloride ethanol-water solutions with different ethanol proportions as solvent. Current–voltage curves were also measured for each membrane system to estimate the energy dissipation due to the Joule effect. Considering the Onsager reciprocity relations, the streaming potential coefficient was discussed in terms of ethanol content of the solutions and the membrane structure. The results showed that more porous heterogeneous membrane presented lower values of liquid uptake and streaming potential coefficient with increasing ethanol content. Denser homogeneous membrane showed higher values for both, solvent uptake and streaming coefficient for intermediate content of ethanol.

A Physically Based Model for the Streaming Potential Coupling Coefficient in Partially Saturated Porous Media

The electrokinetics methods have great potential to characterize hydrogeological processes in porous media, especially in complex partially saturated hydrosystems (e.g., the vadose zone). The dependence of the streaming coupling coefficient on water saturation remains highly debated in both theoretical and experimental works. In this work, we propose a physically based model for the streaming potential coupling coefficient in porous media during the flow of water and air under partially saturated conditions. The proposed model is linked to fluid electrical conductivity, water saturation, irreducible water saturation, and microstructural parameters of porous materials. In particular, the surface conductivity of porous media has been taken into account in the model. In addition, we also obtain an expression for the characteristic length scale at full saturation in this work. The proposed model is successfully validated using experimental data from literature. A relationship between the streaming potential coupling coefficient and the effective excess charge density is also obtained in this work and the result is the same as those proposed in literature using different approaches. The model proposes a simple and efficient way to model the streaming potential generation for partially saturated porous media and can be useful for hydrogeophysical studies in the critical zone.

Experimental Measurement of Frequency-Dependent Permeability and Streaming Potential of Sandstones

Hydraulic flow, electrical flow and the passage of elastic waves through porous media are all linked by electrokinetic processes. In its simplest form, the passage of elastic waves through the porous medium causes fluid to flow through that medium and that flow gives rise to an electrical streaming potential and electrical counter-current. These processes are frequency-dependent and governed by coupling coefficients which are themselves frequency-dependent. The link between fluid pressure and fluid flow is described by dynamic permeability, which is characterised by the hydraulic coupling coefficient (Chp). The link between fluid pressure and electrical streaming potential is characterised by the streaming potential coefficient (Csp). While the steady-state values of such coefficients are well studied and understood, their frequency dependence is not. Previous work has been confined to unconsolidated and disaggregated materials such as sands, gravels and soils. In this work, we present an apparatus for measuring the hydraulic and streaming potential coefficients of high porosity, high permeability consolidated porous media as a function of frequency. The apparatus operates in the range 1 Hz to 2 kHz with a sample of 10 mm diameter and 5–30 mm in length. The full design and validation of the apparatus are described together with the experimental protocol it uses. Initial data are presented for three samples of Boise sandstone, which present as dispersive media with the critical transition frequency of 918.3 ± 99.4 Hz. The in-phase and in-quadrature components of the measured hydraulic and streaming potential coefficients have been compared to the Debye-type dispersion model as well as theoretical models based on bundles of capillary tubes and porous media. Initial results indicate that the dynamic permeability data present an extremely good fit to the capillary bundle and Debye-type dispersion models, while the streaming potential coefficient presents an extremely good fit to all of the models up to the critical transition frequency, but diverges at higher frequencies. The streaming potential coefficient data are best fitted by the Pride model and its Walker and Glover simplification. Characteristic pore size values calculated from the measured critical transition frequency fell within 1.73% of independent measures of this parameter, while the values calculated directly from the Packard model showed an underestimation by about 12%.

Surface Probing with Streaming Potential: The Polymer Flooding Scenario

The first experimental evidence of polymer adsorption in natural carbonates obtained from streaming potential measurements is presented. The surface probing of the Estaillades core was realized in a highly saline environment with I ~ 3 M similar to formation water naturally found in some oil reservoirs, which are typical target for enhanced oil recovery (EOR) by Smart Water Injection Methods (SWIM) and Hybrid SWIM. A highly sensitive setup with an exceptional signal-to-noise ratio even at high salinity, allowing to discriminate voltage fluctuations below 0.1 mV, was used to probe the streaming potential coupling coefficient. The injection of partially hydrolyzed polyacrylamide Flopaam 3130S was followed by chase flooding with formation water for several pore volumes, to assure that any changes measured after the polymer flooding are due to the retained polymer. We show that the streaming potential coupling coefficient changes sign after polymer adsorption. This result not only is an unambiguous evidence for polymer adsorption but has implications on chemical EOR scenarios that need to consider wettability changes due to inversion of surface charge.

Streaming Potential Measurements on the Binary Mixture Triethylamine-Water Near the Demixing Phase Transition

Large density fluctuations developing near the phase transition point of the binary mixture affect physical parameters directly related to the electrokinetic coupling coefficient. Here the first electrokinetic measurements for a porous rock sample are carried out with a critical binary mixture of triethylamine-water, especially around the phase transition point. From the measured streaming potential coefficient, the zeta potential is obtained for the critical composition. The results show that there is no anomaly in the streaming potential coefficient as the temperature approaches the demixing temperature. It is also seen that the streaming potential coefficient and the zeta potential in magnitude decreases with increasing temperature. This observation is opposite to what has been observed in literature. It means that the properties of the electric double layer for the mixtures are different from those for aqueous electrolytes. Additionally, the zeta potential for the critical composition is predicted to fluctuate around the critical point.

Examination of the fractal model for streaming potential coefficient in porous media

In this work, the fractal model for the streaming potential coefficient in porous media recently published has been examined by calculating the zeta potential from the measured streaming potential coefficient. Obtained values of the zeta potential are then compared with experimental data. Additionally, the variation of the streaming potential coefficient with fluid electrical conductivity is predicted from the model. The results show that the model predictions are in good agreement with the experimental data available in literature. The comparison between the proposed model and the Helmholtz-Smoluchowski (HS) equation is also carried out. It is seen that that the prediction from the proposed model is quite close to what is expected from the HS equation, in particularly at the high fluid conductivity or large grain diameters. Therefore, the model can be an alternative approach to obtain the zeta potential from the streaming potential measurements.

Streaming Potential in Unconsolidated Samples Saturated with Monovalent Electrolytes

The streaming potential coefficient of liquid-rock systems is theoretically a very complicated function depending on many parameters including temperature, fluid concentration, fluid pH, as well as rock parameters such as porosity, grain size, pore size, and formation factor etc. At a given porous media, the most influencing parameter is the fluid conductivity or electrolyte concentration. Therefore, it is useful to have an empirical relation between the streaming potential coefficient and electrolyte concentration. In this work, the measurements of the streaming potential for four unconsolidated samples (sandpacks) saturated with four monovalent electrolytes at six different electrolyte concentrations have been performed. From the measured streaming potential coefficient, the empirical expression between the streaming potential coefficient 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 coefficient depends on types of cation in electrolytes and on samples. The dependence of the streaming potential coefficient on types of cation is qualitatively explained by the difference in the binding constant for cation adsorption on the silica surfaces. The dependence of the streaming potential coefficient on samples is due to the variation of effective conductivity and the zeta potential between samples.

A study on the variation of zeta potential with mineral composition of rocks and types of electrolyte

Streaming potential in rocks is the electrical potential developing when an ionic fluid flows through the pores of rocks. The zeta potential is a key parameter of streaming potential and it depends on many parameters such as the mineral composition of rocks, fluid properties, temperature etc. Therefore, the zeta potential is different for various rocks and liquids. In this work, streaming potential measurements are performed for five rock samples saturated with six different monovalent electrolytes. From streaming potential coefficients, the zeta potential is deduced. The experimental results are then explained by a theoretical model. From the model, the surface site density for different rocks and the binding constant for different cations are found and they are in good agreement with those reported in literature. The result also shows that (1) the surface site density of Bentheim sandstone mostly composed of silica is the largest of five rock samples; (2) the binding constant is almost the same for a given cation but it increases in the order KMe(Na+) < KMe(K+) < KMe(Cs+) for a given rock.References Corwin R. F., Hoovert D.B., 1979. The self-potential method in geothermal exploration. Geophysics 44, 226-245. Dove P.M., Rimstidt J.D., 1994. Silica-Water Interactions. Reviews in Mineralogy and Geochemistry 29, 259-308. Glover P.W.J., Walker E., Jackson M., 2012. Streaming-potential coefficient of reservoir rock: A theoretical model. Geophysics, 77, D17-D43. Ishido T. and Mizutani H., 1981. Experimental and theoretical basis of electrokinetic phenomena in rock-water systems and its applications to geophysics. Journal of Geophysical Research, 86, 1763-1775. Jackson M., Butler A., Vinogradov J., 2012. Measurements of spontaneous potential in chalk with application to aquifer characterization in the southern UK: Quarterly Journal of Engineering Geology & Hydrogeology, 45, 457-471. Jouniaux L. and T. Ishido, 2012. International Journal of Geophysics. Article ID 286107, 16p. Doi:10.1155/2012/286107. Kim S.S., Kim H.S., Kim S.G., Kim W.S., 2004. Effect of electrolyte additives on sol-precipitated nano silica particles. Ceramics International 30, 171-175. Kirby B.J. and Hasselbrink E.F., 2004. Zeta potential of microfluidic substrates: 1. Theory, experimental techniques, and effects on separations. Electrophoresis, 25, 187-202. Kosmulski M., and Dahlsten D., 2006. High ionic strength electrokinetics of clay minerals. Colloids and Surfaces, A: Physicocemical and Engineering Aspects, 291, 212-218. Lide D.R., 2009, Handbook of chemistry and physics, 90th edition: CRC Press. Luong Duy Thanh, 2014. Electrokinetics in porous media, Ph.D. Thesis, University of Amsterdam, the Netherlands. Luong Duy Thanh and Sprik R., 2016a. Zeta potential in porous rocks in contact with monovalent and divalent electrolyte aqueous solutions, Geophysics, 81, D303-D314. Luong Duy Thanh and Sprik R., 2016b. Permeability dependence of streaming potential coefficient in porous media. Geophysical Prospecting, 64, 714-725. Luong Duy Thanh and Sprik R., 2016c. Laboratory Measurement of Microstructure Parameters of Porous Rocks. VNU Journal of Science: Mathematics-Physics 32, 22-33. Mizutani H., Ishido T., Yokokura T., Ohnishi S., 1976. Electrokinetic phenomena associated with earthquakes. Geophysical Research Letters, 3, 365-368. Ogilvy A.A., Ayed M.A., Bogoslovsky V.A., 1969. Geophysical studies of water leakage from reservoirs. Geophysical Prospecting, 17, 36-62. Onsager L., 1931. Reciprocal relations in irreversible processes. I. Physical Review, 37, 405-426. Revil A. and Glover P.W.J., 1997. Theory of ionic-surface electrical conduction in porous media. Physical Review B, 55, 1757-1773. Scales P.J., 1990. Electrokinetics of the muscovite mica-aqueous solution interface. Langmuir, 6, 582-589. Behrens S.H. and Grier D.G., 2001. The charge of glass and silica surfaces. The Journal of Chemical Physics, 115, 6716-6721. Stern O., 1924. Zurtheorieder electrolytischendoppelschist. Z. Elektrochem, 30, 508-516. Tchistiakov A.A., 2000. Physico-chemical aspects of clay migration and injectivity decrease of geothermal clastic reservoirs: Proceedings World Geothermal Congress, 3087-3095. Wurmstich B., Morgan F.D., 1994. Modeling of streaming potential responses caused by oil well pumping. Geophysics, 59, 46-56.

Streaming Potential Coupling Coefficient and Transport Properties of Unsaturated Carbonate Rocks

Core Ideas Streaming potential coupling coefficient is dependent on saturation in carbonate rocks. Salinity influences the streaming potential coupling coefficient. Water relative permeability can be estimated from streaming potential measurements. Transport properties of carbonate rocks can be modeled using the van Genuchten approach. We measured the streaming potential coupling coefficient of natural saturated and unsaturated carbonate rocks. Saturation was achieved with NaCl brines with salinities ranging from 2 × 10−3 to 2.0 mol L−1. The magnitude of the coupling coefficient increased with decreasing salinity, similarly to the trend observed for sandstones. The permeability had a low impact on the values of the streaming potential coupling coefficient at high and low salinity. The zeta potential was calculated at full saturation using a modified version of the Helmholtz–Smoluchowski equation that accounts for surface electrical conductivity. Under atmospheric conditions, the magnitude of the zeta potential decreased with the increase in salinity. We also explored the relationships between the streaming potential coupling coefficient and water saturation in three partially saturated limestones using a steady‐state flow experiment. We found good agreement between the van Genuchten approach and experimental data, and fitted both the relative permeability and capillary pressure curves with the same value of the van Genuchten exponent mv. We validated the predictive water relative permeability model described by Revil in water‐wet rocks when the second fluid phase is non‐polar.

A fractal model for streaming potential coefficient in porous media

ABSTRACTStreaming potential is the result of coupling between a fluid flow and an electric current in porous rocks. The modified Helmholtz–Smoluchowski equation derived for capillary tubes is mostly used to determine the streaming potential coefficient of porous media. However, to the best of our knowledge, the fractal geometry theory is not yet applied to analyse the streaming potential in porous media. In this article, a fractal model for the streaming potential coefficient in porous media is developed based on the fractal theory of porous media and on the streaming potential in a capillary. The proposed model is expressed in terms of the zeta potential at the solid−liquid interface, the minimum and maximum pore/capillary radii, the fractal dimension, and the porosity of porous media. The model is also examined by using another capillary size distribution available in published articles. The results obtained from the model using two different capillary size distributions are in good agreement with each other. The model predictions are then compared with experimental data in the literature and those based on the modified Helmholtz–Smoluchowski equation. It is shown that the predictions from the proposed fractal model are in good agreement with experimental data. In addition, the proposed model is able to reproduce the same result as the Helmholtz–Smoluchowski equation, particularly for high fluid conductivity or large grain diameters. Other factors influencing the streaming potential coefficient in porous media are also analysed.