Frequency‐Dependent Electric Properties of Microscale Rock Models for Frequencies from One Millihertz to Ten Kilohertz

Abstract

In this study, spectral induced polariza on (SIP) spectra were generated numerically to be er understand how actual rock microstructure and electrolyte proper es in rock pores aff ect the spectral pa ern, i.e., the characteris c relaxa on me of polariza on as well as the polariza on strength of a rock pore system. The dynamics of charge carriers in threedimensional pore systems were simulated using a frequency-dependent formula on of the Nernst-Planck-Poisson (NPP) ion-transport equa ons. Basically, a pore-system model of alterna ng stacked cylinders of two diff erent sizes was studied considering the electrical double layer (EDL). A reduced ca onic mobility-resul ng from increased adsorp on of these ca ons at the rock-water interface-was assumed within the EDL. By solving the NPP equa ons using the fi nite element method, complex resis vity phase spectra were generated. Subsequently, the eff ect of pore structural proper es and electrolyte conduc vity on the magnitude and frequency posi on of the characteris c resis vity phase minimum of rocks was studied. The following results were found: First, regarding pore geometry, the characteris c frequency of the phase minimum f min decreases with increasing pore length of the large pore. Second, both small pores having a radius of a few Debye lengths combined with larger pores are needed to ensure detectable phase amplitudes. Third, with regard to electrolyte concentra on, the phase amplitude is inversely propor onal to the concentra on, whereas f min remains constant. Because the studied model does not provide a direct and exclusive link between the simulated electrical proper es and pore throat size, further research is needed here to specify a convincing SIP interpreta on method for improved permeability es ma on.Abbrevia ons: EDL, electrical double layer; SIP, spectral induced polariza on; SNP, short narrow pore.In geophysics, frequency-dependent complex resistivity measurements are a typical noninvasive method for fi eld-and laboratory-scale applications. Th is method is called spectral induced polarization (SIP) or the complex resistivity method, fi rst discovered by Schlumberger (1920). In other fi elds, it is known as impedance spectroscopy. Estimating permeability by such noninvasive electrical measurements has been extensively studied since the 1950s (e.g., Carman, 1956, Ch. 1) and is still a current fi eld of research (e.g., Binley et al., 2005;Titov et al., 2010). To achieve this estimation, one can take advantage of the dependence of both hydraulic and complex or frequency-dependent electric conductivity on microstructural rock properties. A simple model with respect to permeability k is given by the Kozeny-Carman (Carman, 1956, Ch. 1) and related equations (e.g., Katz and Th ompson, 1986). Th ese state a dependence of k on porosity φ and the surface area to volume ratio S por , which in turn can be linked to the electrical formation factor F and a characteristic hydraulic length scale l of the pore space. Th ere are several approaches to obtaining these ...

Geophysical Properties of the Near Surface Earth: Electrical Properties

“…These are not shown in Figure 16. A modeling study by Volkmann and Klitzsch (2010) also showed the presence of a low-frequency dispersion, but at slightly higher frequencies. Leroy et al (2008) measured a dispersion associated with particle size occurring at about 75 mHz for a random pack of spherical grains of modal grain size 187.5 mm and another dispersion that they associated with the characteristic size of asperities on the surface of their grains (i.e., surface roughness) at about 20 Hz, to which they attributed a characteristic scale of 12 mm.…”

“…The 'active' zones represent pore throat passages and are characterized by a reduced anion transport number. Volkmann and Klitzsch (2010) also noted that there is some evidence for a relationship with the conductivity of the pore fluid. For small excess electrolyte concentrations and electric fields, Marshall and Madden (1959) solved the dependence of the complex impedance as a function of the angular frequency and discovered that the relaxation time is related approximately to the square of the length of the active and passive pores.…”

“…By contrast, Volkmann and Klitzsch (2010) had focused on the microscopic effects that cause the polarization at the scale of single grains or pores. In 1962, Schwarz (1962) studied the dielectric dispersion of spherical particles in an electrolyte solution and found a Debye-type electrical dispersion with a relaxation time t that depends on the radius of the particle R:…”

Wideband impedance spectroscopy from 1mHz to 10MHz by combination of four- and two-electrode methods

“…In raw order of increasing frequency up to 10 MHz, we find Stern layer polarization (Schwarz, 1962;Revil and Florsch, 2010), possibly going together with membrane polarization (Marshall and Madden, 1959;Titov et al, 2002;Volkmann and Klitzsch, 2010;Bücker and Hördt, 2013) effects, followed by Maxwell-Wagner polarization and orientational polarization of water molecules (Maxwell, 1873;Wagner, 1914;Hanai, 1960;Chelidze and Guegen, 1999). Additionally, electrode polarization can occur in the presence of conductive minerals (Pelton et al, 1978).…”

Deep Geothermal Energy for Lower Saxony (North Germany) – Combined Investigations of Geothermal Reservoir Characteristics

“…We carried out calculations of an analytical membrane polarisation model (based on the model of Marshall and Madden [3]) with the aim to evaluate existing equations and refine them for geothermal application. The work is based on earlier numerical simulation efforts [4,5] and on existing laboratory investigations which include temperature as a variable parameter [6]. Fig.…”