An analytical effective excess charge density model to predict the streaming potential generated by unsaturated flow

Abstract

The self-potential (SP) method is a passive geophysical method that relies on the measurement of naturally occurring electrical field. One of the contributions to the SP signal is the streaming potential, which is of particular interest in hydrogeophysics as it is directly related to both the water flow and porous medium properties. The streaming current is generated by the relative displacement of an excess of electrical charges located in the electrical double layer surrounding the minerals of the porous media. In this study, we develop a physically based analytical model to estimate the effective excess charge density dragged by the water flow under partially saturated conditions. The proposed model is based on the assumption that the porous media can be represented by a bundle of tortuous capillary tubes with a fractal pore size distribution. The excess charge that is effectively dragged by the water flow is estimated using a flux averaging approach. Under these hypotheses, this new model describes the effective excess charge density as a function of saturation and relative permeability while also depending on the chemical and interface properties, and on petrophysical parameters of the media. The expression of the model has an analytical single closed-form which is consistent with a previous model developed from a different approach. The performance of the proposed model is then tested against previous models and different sets of laboratory and field data from the literature. The predictions of the proposed model fits fairly well the experimental data and shows improvements to estimate the magnitude of the effective excess charge density over the previous models. A relationship between the effective excess charge density and permeability can also be derived from the proposed model, representing a generalization to unsaturated conditions of a widely used empirical relationship. This new model proposes a simple and efficient way to model the streaming current generation for partially saturated porous media.

Exploring the Effect of the Pore Size Distribution on the Streaming Potential Generation in Saturated Porous Media, Insight From Pore Network Simulations

“…In order to propose an analytical solution for equation (16), Guarracino and Jougnot (2018) consider the Debye-Hückel approximation, an usual way to derive analytically the distribution of the local electrical potential (e.g., Guarracino & Jougnot, 2018;Jougnot et al, 2012Jougnot et al, , 2015Soldi et al, 2019). This approximation is an accurate solution of the Poisson-Boltzmann equation (equation (12)) for low local electrical potentials, that is, | | ≪ (k B T)∕|q i | ≃ 25.7 mV (for T = 298 K) and monovalent ions.…”

“…Guarracino and Jougnot's () model performs very well with different SP data sets from laboratory measurements (Glover & Déry, ; Pengra et al, ). Note that Soldi et al () propose an extension of this model to partially saturated conditions.…”

An effective excess charge model to describe hysteresis effects on streaming potential

“…Whereas all of these components depend on the water saturation, the behaviour of each one is different. The behaviour of the first two components under unsaturated conditions has been studied for decades (e.g., Archie et al, 1942;Waxman and Smits, 1968;Mualem, 1986;Lenhard and Parker, 1987), nevertheless, how the effective excess charge density varies with saturation is a current theme of study that requires more development (e.g., Jougnot et al, 2012;Revil, 2017;Thanh et al, 2018;Soldi et al, 2019).…”

Transpiration‐ and precipitation‐induced subsurface water flow observed using the self‐potential method

“…where R [m] is the capillary size, f D [−] is the equivalent pore-size distribution function inferred from hydrodynamic parameters (i.e., van Genuchten parameters and permeability k), and v R [m/s] is the mean pore-water velocity for a given capillary radius. This approach is more complex than Equation (10) but better describes SP amplitudes in natural media over a large range of saturations in which b Q v can increase multiple orders of magnitude as saturation decreases (Jougnot et al, 2012(Jougnot et al, , 2015Soldi, Jougnot, & Guarracino, 2019;Zhang, Vinogradov, Leinov, & Jackson, 2017). Given the large variation in saturation observed during the period of data collection in this work, we use Once b Q v S w ð Þ is defined, the streaming-current term of water movement in variable saturated conditions can be written:…”

“…The soil‐specific effective excess charge is then calculated by integrating the effective excess charge of saturated capillaries for given pore‐size distribution: where R [m] is the capillary size, f D [−] is the equivalent pore‐size distribution function inferred from hydrodynamic parameters (i.e., van Genuchten parameters and permeability k ), and v R [m/s] is the mean pore‐water velocity for a given capillary radius. This approach is more complex than Equation but better describes SP amplitudes in natural media over a large range of saturations in which can increase multiple orders of magnitude as saturation decreases (Jougnot et al, , ; Soldi, Jougnot, & Guarracino, ; Zhang, Vinogradov, Leinov, & Jackson, ). Given the large variation in saturation observed during the period of data collection in this work, we use the Jougnot et al () relative permeability flux‐averaging approach in this work (Figure ).…”