Sketch of the electrical double layer coating the surface of the grains. The symbol represents the metal cations (e.g., or ) and the anions (e.g., ). The charge density (in ) denotes the surface charge density of the surface of the mineral (located on the -plane), whereas (in ) is the surface charge density of the Stern layer (compact layer located on the -plane), and (in ) denotes the equivalent surface charge density of the diffuse layer (the sum of the three charge densities is equal to zero). The d-plane separates the outer part of the Stern layer from the diffuse layer. The -plane represents the true mineral surface whereas the -plane represents the mean plane for the sorption of the counterions in the Stern layer.
CT scans for the Fontainebleau core samples F2, M13, Z07Z, and Z14Y. The scale, given as a white line, is 2000 μm (note that the grains have a diameter close to 250 μm). The porosity of these core samples is given as void percentage in the figures, and their remaining properties are reported in Table 1.
Electrical conductivity of the porous material versus pore fluid electrical conductivity using NaCl solutions. (a) The log-log plot shows the electrical conductivity data of four representative samples (at four distinct porosities from 0.05 to 0.21) and for seven pore water conductivities (NaCl solutions). The lines correspond to the fit with the linear model discussed in the main text and is used to determine the (intrinsic) formation factor and the surface conductivity according to equation 4. The salinity dependence of surface conductivity (e.g., Leroy and Revil, 2004; Weller et al., 2013) is neglected following the same (b) for the sample F3 of Börner (1992).
Electrical formation factor versus porosity for the complete set of samples. A modified Archie’s relationship is used to determine the log–log relationship between electrical formation factor and porosity introducing a percolation threshold. The percolation porosity is in the range from 0.015 to 0.023 (best fit: 0.019) and the cementation exponent is in the range 1.4–1.5 (best fit: 1.46). The data points from Börner (1992) concern the core samples F1 (, ) and F3 (, ).
Cementation exponent versus connected porosity. When corrected with a percolation threshold, the cementation exponent has a value close to the theoretical value of 1.5 corresponding to spherical grains and obtained from the differential effective medium theory (Sen et al., 1981). Note the increase of the scatter in the data at low porosities, i.e., in the vicinity of the percolation threshold, probably because the size of the representative elementary volume becomes comparable (or is larger) to the size of the core samples. The data points from Börner (1992) concern the core samples F1 and F3.
Bulk tortuosity of the pore space versus connected porosity . Note the substantial increase in the bulk tortuosity of the pore space in the vicinity of the percolation threshold. The curve corresponds to the equation with (porosity exponent) and (percolation porosity). The data points from Börner (1992) concern samples F1 and F3 of Fontainebleau sandstones.
Surface conductivity versus bulk tortuosity of the pore space. The straight plain line corresponds to the best fit of the model equation with a single grain diameter. The data point from Börner (1992) concerns the sample F3 of Fontainebleau sandstone with a porosity of 0.068, a formation factor of (tortuosity 8.00), a quadrature conductivity of , a permeability of , and a surface conductivity of .
Sketch illustrating the potential effect of quartz cementation upon the effective grain diameter. (a) At high porosities, the effective grain diameter is equal to the size of the grains. (b) Some pore channels between grains are closed to fluids when the quartz sand undergoes cementation. There is an increase in the clustering of grains with cementation. However, this increase may not influence the local mean radius of curvature in the pores that remain open.
Permeability versus electrical formation factor. The formula derived in the main text is compared to experimental data measured using the full data set of the 69 Fontainebleau samples. The observed exponent for the dependence of permeability with the formation factor of is close to the theoretical value .
Measured permeability versus predicted porosity using the porosity, the percolation porosity, and the porosity exponent. The experimental data measured on 69 Fontainebleau samples were fitted with equation 12. Additional data are from Kieffer et al. (1999) for air permeability and total porosity. There is a good agreement between the model and the data. Note the strong decrease in permeability in the vicinity of percolation threshold.
Permeability and formation factor as a function of the reduced porosity (connected porosity minus the porosity at the percolation threshold ). We observe that once corrected for the percolation threshold, permeability and formation factor scale linearly, in a log–log plot, with reduced porosity. In both cases, we used . (a) Modified Revil and Cathles’ (1999) equation (, , and kept fixed). (b) Modified Archie’s equation (Archie, 1942, ). Data source: this work.
Predicted versus measured permeabilities for the 69 samples. The predicted permeability is determined from the formation factor, the porosity, and the surface conductivity. We used the following values of the parameters: and . Note that there are no fudge factors in the model used here.
Ri of Fontainebleau sandstone versus water saturation at high salinity (negligible surface conductivity). The data exhibit an Archie-type behavior with the second Archie’s exponent down to a residual water saturation approximately given by the ratio of the porosity at percolation (0.019) and the measured porosity (0.159). Data are from Durand (2003), Knackstedt et al. (2007), Han et al. (2009) (sample porosity 0.22), and Yanici et al. (2013).
Experimental setup. (a) Position of the electrodes on the core sample. (b) Position of the current ( and ) and voltage ( and ) electrodes.and ZEL-SIP04-V02 impedance meter built by Egon Zimmermann. The data acquisition system operates in the frequency range from 1 to 45 kHz with a phase accuracy close to 0.1 mrad below 1 kHz.
Conductivity spectra for sample Z16X. (a) In-phase conductivity showing nearly no dependence with frequency. (b) Quadrature conductivity showing a relaxation peak at 0.8 Hz. The high-frequency polarization is probably due to Maxwell-Wagner polarization.
Relationship between the Cole–Cole relaxation time and two different pore sizes reported in Revil et al. (2014) submitted for a collection of clayey sandstone samples. (a) Correlation between the Cole–Cole relaxation time and the pore size determined from the formation factor and permeability (2.2 μm for sample Z16X and 14.9 μm for sample Z7Y). (b) Correlation between the Cole–Cole relaxation time (or the characteristic time associated with the peak frequency for the Fontainebleau sandstones) and the peak of the pore size distribution determined from mercury intrusion measurements. (c) For the Fontainebleau sandstone, we use the mercury porosimetry data of Bourbié and Zinsner (1985) based on the porosity value of the core sample (7 μm for sample Z16X and 20 μm for sample Z7Y). The data for the sample at 13% porosity are from David et al. (2013).
Quadrature conductivity versus bulk tortuosity of the pore space. The straight line corresponds to the best fit of the model equation. The data point from Börner (1992) concerns the sample F3 of Fontainebleau sandstone with a porosity of 0.068, a formation factor of (tortuosity 7.84), a quadrature conductivity of , and a surface conductivity of .
Comparison of the relationship determined by Weller et al. (2013) (only the clayey materials) and additional data including oil/gas shales with low porosities and samples of Fontainebleau sandstones (seven samples from this work and sample F3 from Börner, 1992).
Determination of the partition coefficient though a triple layer model for silica (Leroy et al., 2008) for different values of the pH and salinity of NaCl solutions. The filled circle represents the conditions used for this work in terms of salinity and pH (5.6).
Relationship between apparent formation factors (defined as the ratio of the resistivity by the resistivity of the pore water) and intrinsic formation factors accounting for the effect of the surface conductivity in the relationship between the conductivities of the sample and the pore water. The horizontal line identifies the value of the intrinsic formation factor , whereas the curved line identifies a fit with equation 4 in terms of an apparent formation factor. The gray areas correspond to typical conductivity values of freshwater aquifers (80–).
Relationship between apparent formation factors (defined as the ratio of the resistivity by the resistivity of the pore water, horizontal line) and intrinsic formation factors accounting for the effect of the surface conductivity in the relationship between conductivity of the sample and the pore water conductivity (curved line). We use samples F3 from Börner (1992) and Z14Y (this work). The data for these two samples characterized by the same porosity and formation factor are consistent. The surface conductivity is estimated to be . The gray area corresponds to the typical conductivity values of freshwater aquifers (80–).
Microstructure of the grains in the Fontainebleau sandstone. The detrital grains with minor amounts of quartz overgrowth (the Fontainebleau sandstone has undergone a minor degree of early quartz cementation) are coated by a layer of amorphous silica. In turn, this layer is coated with a microcrystalline quartz layer.
Main experimental results obtained from 69 core samples of the Fontainebleau sandstone: and represent twice the standard deviations of formation factor and surface conductivity, respectively. In this table, represents the correlation coefficient describing the quality of the fit of the electrical conductivity data using equation 4.