· · · Institutional Repository

Macroscopic measurements of electrical resistivity require frequency-dependent effective models that honor the microscopic effects observable in macroscopic measurements. Effective models based on microscopic physics exist alongside with empirical models. We adopted an empirical model approach to modify an existing physical model. This provided a description of electrical resistivity as a function of not only frequency, but also water saturation. We performed two-electrode laboratory measurements of the complex resistivity on a number of fine and medium-grained unconsolidated sand packs saturated with water of three different salinities. For frequencies between 0.1 and 1 MHz, the data were fitted with the new model and compared to fits with Archie’s law. Our model described the relaxation times and DC resistivity values as negative exponential functions with increasing water saturation. All data could be accurately described as a function of frequency and water saturation with nine parameters.

We measured the electric parameters for four different configurations of unconsolidated homogeneous and layered sands as a function of frequency, water saturation, and salinity under fluid flow conditions. Our objective is to determine if the effect of heterogeneities at scales much smaller than the skin depth can be captured by introducing effective frequency‐dependent electrical values whose behavior can be described by simple functions. We employed the parallel plate capacitor technique to measure the complex impedance over a broad frequency range, from 100 kHz up to 3 MHz. We conducted main drainage and secondary imbibition cycles at atmospheric pressure and temperatures between 21°C and 22°C. The hysteretic effect in the real part of the effective complex permittivity at higher concentrations of NaCl is more pronounced for the homogeneous configurations than for the heterogeneous samples. Effective medium theory works well for dry and saturated layered sand, when the NaCl solution concentration is 1 mmol/l. It fails for fully saturated layered sands at salinities of 10 mmol/l or more. It also does not work for partially saturated sands, independent of salinity. A description of the electric properties of a layered sand at all saturation levels by means of an effective homogeneous medium will therefore require a dependence on frequency, saturation level, and salinity of the pore fluid. An extended version of the Cole‐Cole model fits the nonmonotonic behavior of the real part of permittivity versus saturation.

As geophysicists we are interested in the capabilities of electromagnetic methods for exploration, characterization, and monitoring of the subsurface. For frequencies below 1 MHz, the electromagnetic field is primarily diffusive, rendering the method very little resolving power. For this reason the resistivity values of layered structures cannot be resolved separately, but some average resistivity representing a stack of layers can be found from measured data. Diffusive electromagnetic methods find their application in discriminating large contrasts in electric conductivity values. These contrasts can occur between soil/rock and metallic inclusions or between the soil/rock mass and its pore fluid. In this thesis we are interested in the electromagnetic effects of the possibly strong contrast between the rocks or soils and the pore fluid. The electric response of fluid filled rocks and soils depends mainly on saturation level, porosity, the electric properties of the constituents, the pore fluid concentration, and pore geometry. In this thesis we try to improve our understanding of the effective electromagnetic response behavior of un-compacted sand, clay, and their layered mixtures. Measurements were conducted at frequencies from 100 kHz to 3 MHz during continuous inflow and outflow of water as a function of saturation level and pore water salinity using NaCl with concentrations of 1, 10, or 100 mmol/l. The effective electric conductivity is measured for homogeneous and layered sand and clay samples and layered combinations of sand and clay. The experimentally obtained macroscopic, frequency-dependent, and therefore complex-valued, electric conductivity is analyzed as a function of the various parameters. Initially, we used a much wider frequency range from 210 Hz to 3 MHz as offered by the measuring device, but due to strong electrode polarization in the two-electrode setup it was finally decided to restrict the analysis to the frequency band from 100 kHz to 3 MHz where we are sure no electrode polarization effect is present in the data. The procedures used to account for the electrode polarization effects are described in detail in Chapter 1. The electrode polarization cannot be calibrated for, but can be incorporated in the model increasing the number of unknown parameters. This is a viable option in principle, but leads to the uncertainty that errors in the electrode polarization model leads to unknown errors in the obtained effective conductivity values that cannot be identified as such. For this reason and to be more accurate it was decided to limit the frequency range from 100 kHz to 3 MHz. The objective of Chapter 2 is to determine whether the effect of heterogeneities at scales much smaller than the skin depth on a macroscopic electric measurement can be captured by introducing an effective frequency-dependent electrical resistivity that can be described by simple functions of macroscopic properties, such as porosity, water saturation, and salinity. For the experimental part of our study, we employed the two-electrode cell technique to measure the complex impedance over a broad frequency range, from 100 kHz up to 3 MHz. We conducted main drainage and secondary imbibition cycles at atmospheric pressure and temperatures between 21 ◦C and 22 ◦C. We found that the hysteretic effect, being the difference in the measured electric impedance between first drainage and second imbibition, is more pronounced for the homogeneous configurations than for the heterogeneous samples in the real part of the effective complex permittivity at higher concentrations of NaCl. Known effective medium theory for layered media works well for dry and fully saturated layered sand, when the NaCl solution concentration is 1 mmol/l. It fails for fully saturated layered sands at salinities of 10 mmol/l or more. Known effective medium models that are based on first principles are static limits or linearizations of non-linear behavior. When these linearizations are adequate descriptions of the effective behavior the models work well. At the salinity levels above 1 mmol/l, there appear to be measurable effects of effective non-linear behavior of the layered sands. To describe such behavior, new models need to be developed. The static effective medium theory does not work for partially saturated sands, independent of salinity, indicating that improved effective medium models should accommodate the water saturation dependence as well as frequency and salinity dependence. We introduce a special case of an existing five parameter model, which is a reduced form of the double Cole-Cole model, to describe the measured effective electric resistivity. This model very well fits the frequency-dependent complex electric resistivity data of homogeneous and layered unconsolidated sands during the drainage and the imbibition cycles under continuous flow conditions and the three different salinities used in these experiments. In Chapter 3, we investigate the effects of pore liquid salt concentration on the complex electric response of two different unconsolidated samples, layered sand and sand-clay, in the frequency range of 100 kHz to 3 MHz to study the effect of grain size on the effective response. The electric parameters that describe the electric response of the samples are obtained with the same two-electrode measurement cell as described in Chapter 2. Under continuous fluid flow conditions, first drainage and secondary imbibition cycles were conducted for the two different three-layered samples saturated with saline-water in three different NaCl solution concentrations at atmospheric pressure and temperatures between 21 ◦C and 22 ◦C. Analysis revealed that for a fixed frequency but varying water saturation the real part of electric permittivity increases with increasing salt concentration. For the highest NaCl solution concentration, the hysteretic effect becomes more pronounced and remains present at higher frequencies. For the two samples, plots of the conductivity amplitude and phase as a function of frequency show large variation with water saturation and NaCl solution concentrations, suggesting that this sensitivity is useful for environmental and geo-engineering characterization in vadose zone. Determining the five parameters in our model is numerically challenging, because it is a strongly non-linear and ill-posed inverse problem. The inversion has to be carried out for each water saturation level and each salt concentration level. It would be very beneficial to have a model that captures not only frequency dependence, but also water saturation dependence. Archie’s law is the best known candidate to do that. However, we found that Archie’s law cannot capture the variations in the complex resistivity with saturation, especially for samples with substantial fine-grain fractions. Archie’s law is valid for the real part of the complex resistivity and for samples with coarse-grain fractions. As an alternative, we propose a relaxation model. Its parameters vary exponentially with saturation. This model is inspired by the experimentally observed dependence on water saturation on all samples used in our tests. The relaxation model provides a good fit to the saturation dependence of our^ complex resistivity data and improves significantly on Archie’s classic law and is described in detail in Chapter 4. Measured electric responses of samples provide strong indications of changes in fluid saturation, salt concentration, and heterogeneity (here only layering). This makes it a promising technique for real time in-situ measurements. One application is in Enhanced Oil Recovery (EOR), where real-time monitoring of production-related changes is useful. Moreover, the results of our investigations could be applied to borehole logging data, because the frequency range and physical scale that we use are similar to those in borehole logging.