Electrical Conductivity of Clayey Rocks and Soils: A Non‐Linear Model

Abstract

Electrical conductivity of different rocks and soils lays the base for electrical logging and electrical and electromagnetic (E&EM) methods that have been extensively used for near-surface, environmental, and hydrogeological investigations (e.g., Glover, 2015). It is the pioneering work of Archie (1942) that makes the quantitative assessment of porosity available and paves the way for other petrophysical parameters. Several years later, Patnode and Wyllie (1950) found that Archie's law is only suitable to "clean" samples as clayey minerals in the "dirty" ones present themselves as "conductive solids". Winsauer and McCardell (1953) ascribed the clay conductivity to its electrical double layer (EDL) for the first time. Thus the additional component Patnode and Wyllie (1950) added to Archie's law shall be called the surface conductivity. Soon after, Wyllie and Southwick (1954) discovered that the surface conductivity is not constant but differs at low and intermediate salinities, and they presented a three-resistor (TR) model to explain this non-linearity. It shall be clarified here that "cleanness" and "dirtiness" are relative terminologies that are essentially a question of the ratio between surface and pore water conductivity (i.e., the Dukhin number. cf, Shilov et al., 2001). Indeed, one can find surface conductivity even in clean glass beads (e.g., Bolève et al., 2007).Although research on the electrical conductivity of clayey rocks and soils progressed significantly in the 1950s, the following years are not always the case. Despite the fact that many models have been presented to account for this non-linear characteristic, they are argued not to comply with electrochemical theories or not coincide with measurements, and most are too complex to use (Bussian, 1983). For example, the TR model (Wyllie & Southwick, 1954) can explain measurements well, yet it is exclusive of EDL, clay minerals being treated as classic conductors. The Waxman-Smits (WS) model (Waxman & Smits, 1968), probably the most popular model, postulates that the ionic mobility decreases with the diminishing ionic concertation which is opposite to the electrochemical theory (e.g., Bockris & Reddy, 1998). The Padé approximant (PA) model (Schwartz et al., 1989) first calculates the low-salinity conductivity with surface tortuosity and the high-salinity conductivity with bulk

Insights on the Interplay of Rifting, Transcrustal Magmatism and Formation of Geothermal Resources in the Central Segment of the Ethiopian Rift Revealed by 3‐D Magnetotelluric Imaging

“…Considering reported groundwater electrical conductivities with a mean of 0.3 S/m in the area (Burnside et al., 2021), the most widely distributed observed bulk conductivities within C1 ( σ = 0.1–0.2 S/m) would require an unreasonably large fluid fraction within C1 (see Text S6.2 in Supporting Information S1). It is thus likely that enhanced conductivities in C1 are attributed to a superposition of pore fluid conduction and conduction through a conductive soil component such as clays (e.g., Pride, 1994; Qi & Wu, 2022). Clays also form through rock weathering processes and are commonly found in soils around the study area (Fritzsche et al., 2007).…”

Geophysical–geotechnical methodology for assessing the spatial distribution of glacio‐lacustrine sediments: The case history of Lake Seracchi

“…According to the TDR measurements, in most parts of the lake the sediment electrical conductivity is about 0.003–0.004 S/m—higher than that of water (0.0016 S/m). This higher conductivity is probably due to the presence of ions on the surface of clay‐size grains (Qi & Wu, 2022; Ward, 1990). In two small areas (a few square meters) in the very proximity of the inflows, the electrical conductivity of the sediment is lower than the average 0.003–0.004 S/m (Figure 4d).…”

Induced Polarization of Clayey Rocks and Soils: Non‐Linear Complex Conductivity Models

“…Data used in this study can be found in the previously published literature of Börner (1992), , Slater and Glaser (2003), Mendieta et al (2021), and Waxman and Smits (1968). Materials for the models and results presented in this paper are available in Qi and Wu (2023).…”

“…BHS model (Bruggeman, 1935; Hanai, 1960; Sen et al., 1981) $$${\sigma }^{\ast }={\sigma }_{\mathrm{w}}{\phi }^{m}{\left(\frac{1-{\sigma }_{\mathrm{c}}^{\ast }/{\sigma }_{\mathrm{w}}}{1-{\sigma }_{\mathrm{c}}^{\ast }/{\sigma }^{\ast }}\right)}^{m}$$$ is an exceptional precedent that can both explain real and quadrature nonlinearities concurrently, and it downscales to Archie's law when in the one‐phase situation. Compared to such an eminent model, as discussed in Qi and Wu (2022), Equation 9 bears the advantages of mathematical simplicity and parametric consistency with Archie's law. The former is because BHS is an implicit function with σ * on both sides that needs complicated methods to solve (Glover et al., 2010; Lesmes & Morgan, 2001), while the latter results from its parameter m links to both bulk and surface continuity (de Lima et al., 2005; Niwas et al., 2006, 2007).…”

“…Furthermore, if we recall the three‐resistor model for the real conductivity (Lévy et al., 2018; Wyllie & Southwick, 1954) $$${\sigma }^{\prime }=\frac{{\sigma }_{\mathrm{w}}}{F}+\frac{(y+z){\sigma }_{\mathrm{c}}{\sigma }_{\mathrm{w}}+x{\sigma }_{\mathrm{c}}^{2}}{xz{\sigma }_{\mathrm{c}}+yz{\sigma }_{\mathrm{w}}},$$$ where x , y , and z are empirical factors. It can be easily tested that the second term of three‐resistor model equals Weller‐Salter model, which are the real (Qi & Wu, 2022) and quadrature parts of Equation 9, respectively. It is not surprising that three‐resistor and Weller‐Slater models work so well for SIP since our model reconciles these two models with the fact that surface and quadrature conductivities follow the same EDL‐and‐water pattern as in Equation 9, such similarity being phenomenologically discovered by Börner (1992) and confirmed by Weller et al.…”