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

Abstract

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.

Laboratory Measurements of Zeta Potential in Fractured Lewisian Gneiss: Implications for the Characterization of Flow in Fractured Crystalline Bedrock

“…Multiplying Equation ( 3) by the cross-section area to the flow, 2wH, and integrating across the fracture aperture (between −w and +w) yields the well-known cubic law [64] for the total volumetric flow rate, Q: Equation ( 5) is consistent with a recently published study [65] in the limit of a single fracture, and it implies that for a decrease of approximately 16% in sample's permeability (from 51 mD to 43 mD), the corresponding expected decrease in fracture aperture (2w) will be around 8% (from 0.782 mm at 4 MPa to 0.718 mm at 7 MPa). Decreasing fracture aperture implies a smaller distance of separation between the fracture walls, and as such, a higher confining pressure can potentially have an impact on the electrical properties of hydraulically active fractures via an increased impact of the surface electrical conductivity and the zeta potential.…”

A Fractal Model for Effective Excess Charge Density in Variably Saturated Fractured Rocks

“…By combining this result and the hydraulic properties derived in Guarracino (2006) and Monachesi and Guarracino (2011) for the same fracture network, we obtain a closed‐form expression for $${\hat{Q}}_{v}$$ in terms of permeability, porosity, saturation and electrokinetic parameters. This modeling strategy has been successfully used for estimating $${\hat{Q}}_{v}$$ in classical porous media described by capillaries (e.g., Guarracino & Jougnot, 2018; Jougnot et al., 2019; Rembert et al., 2020; Soldi et al., 2020, 2018; Thanh et al., 2021).…”

“…3 of 11 we obtain a closed-form expression for 𝐴𝐴 Q𝑄𝑣𝑣 in terms of permeability, porosity, saturation and electrokinetic parameters. This modeling strategy has been successfully used for estimating 𝐴𝐴 Q𝑄𝑣𝑣 in classical porous media described by capillaries (e.g., Jougnot et al, 2019;Rembert et al, 2020;Soldi et al, 2020Soldi et al, , 2018Thanh et al, 2021).…”

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

“…However, to the best of our knowledge, there are only few analytical models for fractured rocks in the literature. For example, Thanh et al (2021) proposed a model for the electrical conductivity and streaming potential coupling coefficient in fractured media under saturated conditions using a capillary bundle model following the fractal scaling law. Guarracino and Jougnot (2022) presented a model to predict the effective excess charge density for fully and partially water saturated fractured media that are described by the fractal Sierpinski carpet.…”

“…The aim of this study is to develop a unified model for the permeability, electrical conductivity, and streaming potential coupling coefficient in fractured media by extending the work proposed by Thanh et al (2021) to partially saturated conditions. For this purpose, we conceptualize a fractured medium as a partially saturated bundle of parallel capillary slits following the fractal scaling law.…”