Reservoir characterization

Stress Dependence of Sandstone Electrical Properties and Deviations From Archie’s Law

The resistivity index (RI) of Fontainebleau and Bentheimer sandstones was investigated at ambient and reservoir pressures down to low water saturations. The RI measurements show that both sandstones display Archie behavior at elevated pressure.


The resistivity index (RI) of Fontainebleau and Bentheimer sandstones was investigated at ambient and reservoir pressures down to low water saturations. The RI measurements show that both sandstones display Archie behavior at elevated pressure. However, at ambient pressure, the RI for Fontainebleau sandstone deviates from Archie behavior at low water saturations. The pore-space images suggest that the deviation from Archie behavior is attributable to the presence of conductive percolating grain-contact regions.


Deviations from Archie’s law are known to occur, particularly at low water saturations, even for clean sandstones. It is unclear whether the deviations from Archie behavior observed at low pressures are also displayed at elevated pressures. In this paper, the authors present laboratory measurements of RI for two strongly water-wet sandstones at ambient and elevated pressures. The measurements are supplemented with high-resolution microcomputed-tomography (CT) imaging in dry and wet states at ambient pressure to determine an accurate description of the open resolved pore space and to attribute a finite porosity to fluid-saturated grain contacts at elevated pressure. Assuming that the main elements responding to effective stress are the grain contacts, grain-contact conductivities are estimated at elevated confining pressures with actual formation-­factor measurements for saturated samples at the same confining pressures. These are compared with computations on the ­micro-CT images. For Bentheimer, which contains image-resolvable clay regions, the clay regions are considered as additional conductive pathways with different stress dependence.


Experimental Procedure. Rock. The Fontainebleau sandstone samples used exhibit relatively low porosity in the range of 4 to 5%, with sample permeabilities of approximately 0.03 to 2.2 md at confining pressures of 500 to 6,000 psi. The Bentheimer sandstone samples used have a porosity of approximately 23.7 to 25.9%, with permeability of the intact samples at approximately 2,000 md over a range of 500 to 6,000 psi in confining pressure.

Fluid. RI and drainage capillary pressure experiments were performed with brine (2% by weight sodium chloride) and air as the wetting and the nonwetting phases, respectively. The density and viscosity of brine were measured to be 1.0166 g/cm3 and 1.0 cp, while 0.00129 g/cm3 and 0.0185 cp were used for the ­density and viscosity of air, respectively.

Core preparation, as well as image analysis and acquisition, is described in detail in the complete paper. In Fig. 1, the authors illustrate the final phase assignments for Fontainebleau sandstone, including grain/grain boundary labels for a few different saturations and boundary assignments.

Fig. 1—Cross sections (top row is 840×840 voxels, bottom row is 800×800 voxels, alternate grain partition and distinctive grain-contact labeling on the basis of a more-conservative segmentation) through phase distributions of Fontainebleau sandstone forming the basis for resistivity and RI calculations at different water saturations. In 1a through 1c, red indicates nonwetting fluid, blue is water, and grain contacts are green. Water saturations: (a) 8%, (b) 32%, (c) 61% for the case of all grain contacts conducting, and for different contact labeling. Water saturations for all grain contacts conducting: (d) 12%, (e) 18%, (f) 37%.


Laboratory Apparatus. The electrical-resistivity measurements were performed with equipment designed for both drainage capillary pressure and different confining pressures up to 10,000 psi. ­Resistances were measured in 6- to 12-hr increments, and values were recorded. Samples were flushed with 3 to 5 pore volumes of the saturating brine between resistance measurements to ensure consistent salinity over the experiment. The resistance reading was considered stable when three subsequent measurements were within ±3% of each other. Resistance measurements from the samples were then converted to a resistivity value.

Calculation of Electrical Properties. Electrical conductivity and RI are derived directly on the tomographic images by use of the voxel discretization naturally provided by the tomograms. Each voxel is assigned a distinct resistivity according to its phase label, and the Laplace problem of electrical conductivity is solved with a finite-element method implemented in parallel. For the samples used, the authors selected the z-­direction of the samples and applied a fixed potential of zero and unity to inlet and outlet, respectively. Effective resistivities result from a volume average over the resulting current field according to Ohm’s law. RIs are calculated by solving twice, for the fully saturated system and partially saturated system, and deriving the conductivity or resistivity ratio. For Fontainebleau, it is known that the percolation behavior of grain contacts at ambient conditions plays a critical role. Accordingly, the authors considered two different grain/grain partitions and different grain-­contact-conductivity scenarios, as follows:

  • The aggressive segmentation in Figs. 1a through 1c results in some resolved porosity in grain contacts. There are two compliant porosity elements: grain/grain contact voxels exhibiting nonresolved porosity and ones at which porosity is resolved but a different label is given. The authors differentiate grain-contact conductivities for “grain contact” and “resolved porosity” accordingly. The resulting dependence of porosity on confining pressure matches the experimental data.
  • A variation of the previous scenario with significantly smaller conductivities scaled to model the experimental data while maintaining a nonzero grain-contact conductivity for all grain-contact voxels.
  • A best-guess segmentation and grain partition with fewer grains (Fig. 1d through 1f), setting grain contact everywhere to a corresponding grain-contact thickness of 170 nm or doing so for only a selection of the grain contacts in a random manner.

For all cases and both sandstones, the authors introduced a dependency of grain-contact porosity on confining pressure and for Bentheimer a clay-­region porosity dependence on confining pressure. It is assumed that the effect of change in grain-contact conductivity on effective conductivity of the samples is a first-order effect and that, to a first-order approximation, the geometry of the pore space stays and the water distribution remains unchanged with increasing pressure. The dependence of porosity on confining pressure by contact porosity alone was possible only for the first previously described scenario and not possible at all for Bentheimer sandstone because of the higher porosity and associated smaller grain-contact area.

Experimental Results

Porosity. Both Bentheimer and Fontainebleau samples display decreasing porosity with increasing confining pressure. The change of the porosity values over the confining-pressure range was approximately 1.2 times for Fontainebleau and approximately 1.05 times for Bentheimer. The difference is likely a result of the higher porosity of the Bentheimer samples. In relative terms, the significance of confining pressure for Fontainebleau sandstone is much higher than for Bentheimer. In absolute terms, a reduction in porosity of approximately 1 porosity unit is seen for Bentheimer; a reduction of 0.3–0.8 porosity units is seen for Fontainebleau.

For both rocks in the first stage of confining pressure up to 2,000 psi, the porosity response is nonlinear, and then it varies almost linearly to 6,000 psi. The response of the Bentheimer and Fontainebleau sandstones, therefore, is almost elastic, with a nonlinear behavior in the early stage of the compression.

Electrical Properties. Electrical resistivity was measured on six core plugs of 1.5‑in. diameter and four core plugs of 1-in. diameter by use of the porous-plate-desaturation technique. Electrode spacing was 15.5 and 12 mm for the 1.5- and 1-in. plugs, respectively. All data were taken at 1.0 kHz.

Formation Factor. For the Bentheimer samples, the increase in formation factor was approximately 10%, and it was up to 100% for Fontainebleau (up to 200% for individual samples). A reference point of 500 psi was used as confining pressure for both rocks. The associated cementation exponent increased by approximately 9% for the Fontainebleau samples and by less than 1% for the Bentheimer samples across the whole pressure range. The samples with lower porosity display a higher formation factor for both samples. The effect of the confining pressure was higher on samples of lower porosity.

RI. The RI data sets measured in this study are in good agreement with previously reported measurements at ambient conditions for Fontainebleau and Bentheimer sandstone over the pressure and saturation ranges for which RI was measured.

At 500-psi confining pressure, Bentheimer behaves as Archie rock. Fontainebleau sandstone exhibits clear non-Archie behavior at water saturations of less than 20%. The present finding of non-Archie behavior at ambient conditions for Fontainebleau sandstone is consistent with the literature.

At 6,000-psi confining pressure, Bentheimer sandstone remains an Archie rock. Fontainebleau sandstone displays Archie behavior to much lower water saturations (less than 10%) than at low confining pressure, where deviations start at water saturations less than 15%. Deviations from Archie behavior at low water saturations expressed by a change of slope are much weaker at high confining pressure than at ambient conditions.

This study supports the finding of non-Archie behavior at ambient conditions for Fontainebleau sandstone reported in previous studies. The authors conclude that the most likely explanation of the disappearance or weakening of non-Archie behavior of Fontainebleau at high confining pressures is the closure or ­partial closure of grain/grain contacts at elevated confining pressures.

Numerical results for both rocks are described in detail in the complete paper.


  • RI measurements at different confining pressures of clean, strongly water-wet Bentheimer rock samples display a typical Archie behavior for water saturations down to 10%.
  • Clean strongly water-wet Fontainebleau low-porosity sandstone displays a negative deviation of RI from Archie’s law at ambient conditions because of the presence of conductive percolating grain-contact regions, which are mainly closed at 6,000‑psi confining pressure, returning the rock to Archie behavior by reducing the saturation point at which deviations from Archie’s law appear.
  • Numerical simulations, including those on very-low-porosity Fontainebleau sandstone, can depict the main behavior of the experimental results by considering the percolation behavior of the grain-contact network for Fontainebleau. Keeping all grain contacts conducting in the numerical approach does not capture the qualitative behavior seen in the experiment.

It is confirmed that the effect of ­reservoir-stress conditions on grain contacts leads to the observed waning of non-Archie behavior toward high confining pressure. The observed behavior requires a distribution of grain/grain-contact conductivities, with some grain contacts becoming nonconductive. Such variations in grain-contact conductivity would be a natural consequence of local stress conditions, where highly stressed contact regions provide little to no contribution to electrical conduction.
This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 181484, “Experimental and Numerical Investigations on Stress Dependence of Sandstone Electrical Properties and Deviations From Archie’s Law,” by M.F. Farid, J.-Y. Arns, W.V. Pinczewski, and C.H. Arns, University of New South Wales, prepared for the 2016 SPE Annual Technical Conference and Exhibition, Dubai, 26–28 September. The paper has not been peer reviewed.