Philippe Cosenza

Complex conductivity of water-saturated packs of glass beads

The low-frequency conductivity response of water-saturated packs of glass beads reflects a combination of two processes. One process corresponds to the polarization of the mineral/water interface coating the surface of the grains. The other process corresponds to the Maxwell-Wagner polarization associated with accumulation of the electrical charges in the pore space of the composite medium. A model of low-frequency conductivity dispersion is proposed. This model is connected to a triple-layer model of electrochemical processes occurring at the surface of silica. This model accounts for the partition of the counterions between the Stern and the diffuse layers. The polarization of the mineral/water interface is modeled by the electrochemical polarization model of Schurr for a spherical grain. We take into account also the DC surface conductivity contribution of protons of the sorbed water and the contribution of the diffuse layer. At the scale of a macroscopic representative elementary volume of the porous material, the electrochemical polarization of a single grain is convoluted with the grain size distribution of the porous material. Finally, the Maxwell-Wagner polarization is modeled using the complex conductivity of a granular porous medium obtained from the differential effective medium theory. The predictions of this model agree well with experimental data of spectral induced polarization. Two peaks are observed at low frequencies in the spectrum of the phase. The first peak corresponds to the distribution of the size of the beads and the second peak is due to the roughness of the grains.

Effective medium theories for modelling the relationships between electromagnetic properties and hydrological variables in geomaterials: a review

The paper reviews the effective medium theories used for modelling the relationships between electromagnetic properties (low-frequency conductivity and high-frequency permittivity) and hydrological variables (water content, salinity, suction, permeability) in soils and rocks. It aims a) to provide a simple presentation of these theoretical approaches, b) to present their theoretical and practical limitations and c) to establish some connections with empirical equations usually used in hydrogeophysics (i.e., Archie relationships, Topp equation and complex refractive index model). This review demonstrates that two groups of effective medium theories can be clearly identified. The first group constituted by the Maxwell-Wagner and Symmetric-Bruggeman rules is characterized by easy-to-use models. When the volumetric water content and the texture are known, they allow to obtain fast estimates of effective electromagnetic properties. In the second group, the differential effective medium schemes that are more complex from a mathematical point of view are preferred when frequency-dependent properties are studied. In particular, differential effective medium schemes are used for providing some insights into the physical basis of spectral induced polarization measurements in hydrogeophysical applications.

Differential effective medium schemes for investigating the relationship between high‐frequency relative dielectric permittivity and water content of soils

Received 15 October 2002; revised 27 January 2003; accepted 4 April 2003; published 3 September 2003. [1] Differential effective medium (DEM) theory is presented and used to calculate the effective HF dielectric permittivity k of unsaturated soils considered as mixtures of solid and two fluid phases. In the case of coarse-grained soils for which dielectric losses are negligible, a good agreement with the empirical equation of Topp et al. [1980] is obtained using only a spherical grain shape. Moreover, results of DEM schemes have shown that ohmic losses induced by salinity modify significantly the relationship between the apparent dielectric permittivity at 120 MHz and the volumetric water content q, for NaCl concentration greater than 0.045 mol/L. In the case of clayey soils, simulations based on the same theoretical approach show that the available data can not be modeled by considering only the geometrical effect associated with the ‘‘platy’’ units and the ohmic losses associated with the surface conductivity. INDEX TERMS: 0699 Electromagnetics: General or miscellaneous; 1866 Hydrology: Soil moisture; 5112 Physical Properties of Rocks: Microstructure;

A new method for quantitative petrography based on image processing of chemical element maps: Part II. Semi-quantitative porosity maps superimposed on mineral maps

Abstract Visualizing and quantifying the spatial heterogeneity of rock textures (i.e., mineral and porosity distributions) is of great interest in petrology or for understanding petrophysical properties. Spatial heterogeneities are not accurately revealed by usual techniques based on microscopy or bulk physical measurements. Detailed mineral mapping is already available from processing of chemical element maps acquired using an electron probe microanalyzer (Part I). The present paper is devoted to developing a new, coupled method for obtaining porosity maps from the same initial data. According to the difference between measured and theoretical sums of oxide weight percentages, a mean porosity is semi-quantitatively estimated for each pixel of the map (i.e., not fully absolute or accurate). All pores, including nanometer-size ones, are taken into account, whereas a sample area of several square millimeters is analyzed (spatial resolution of a few micrometers). The textural heterogeneities are thus visualized from the complementary maps of solids and voids. By superimposing these two maps, both the mean porosity and a porosity histogram associated with each rock-forming mineral are obtained. Such porosity measurements integrate the pore amounts within mono-crystals larger than the X-ray emission volume or between nanometer-size crystals of a matrix. When porosity changes are associated with a given mineral (various crystal arrangements, dissolution, etc.), pluri-modal distributions appear on porosity histograms. Thresholding each histogram mode then allows these processes to be localized. We used the MX80 bentonite to test this methodology, which represents a useful tool to study the local deformations and alteration of each rock-forming mineral, as well as to model transport properties.

Numerical modeling of the role of water and clay content in soils' and rocks' bulk electrical conductivity

Received 23 October 2000revised 1 May 2002;accepted 24June 2002;published25November2002. [1] The influence of clay and water content in the electrical conductivity of rocks and soils has been experimentally established and is expressed by simple empirical laws: the Archies law and the addition law between volume water conductivity and surface shale conductivity. Two independent numerical modeling techniques, the moment method and the finite difference method, are presented here and are used, first, to verify the agreement between Maxwells equation based theoretical approaches and the empirical laws and, second, to begin to investigate for a possible effect of the microscopic geometry over macroscopic conductivity. A good agreement between simulation results and Archies law is obtained when both randomly distributed isotropic and elongated microvolumes of conducting water are considered and a slight difference appears between these two microstructures. For low clay contents in clay-dispersed media, the clay-associated conductivity is shown to be proportional to a specific clay area, which is in good agreement with the addition empirical law.

Numerical modeling for investigating the physical meaning of the relationship between relative dielectric permittivity and water content of soils

The relative permittivity of soils is related to the volumetric liquid water content by a monotonous and general relationship. Among others, Topp et al. [1980] have proposed a simple empirical function to allow a direct determination of the water content. The moment method is applied here to calculate the apparent permittivity of a given volume of soil resulting from the electrical coupling between its different constituents and to attempt to reproduce the empirical function. This is not possible when considering randomly isotropically distributed elementary volumes of water, solid, and air. To reproduce the empirical function, it is necessary to consider that as the water content increases, the elementary volumes of water become more elongated since the liquid phase is continuous.

A physical model of the low‐frequency electrical polarization of clay rocks

Low-frequency (0.18 Hz to 1.5 kHz) effective dielectric spectra have been measured on a set of near-saturated samples of argillite. The measured spectra of the real part of the complex apparent permittivity did not show significant correlation with cation exchange capacity (CEC) per unit mass of rock values. They satisfied a power law relationship with the frequency, at least for samples with CEC values lower than 10 cmol/kg. The Maxwell-Wagner-Hanai-Bruggeman formulation used for a two-phase mixture has been modified to account for mutual polarization between the pockets of water located in the micropores and those located in the macropores. The results of the modeling calculations illustrate (1) the ability of this new formulation to reproduce the power law relationships of the measured spectra of the real and imaginary components of the complex permittivity and (2) the strong impact of the pore electrical conductivity.

Comment on “Generalized effective-medium theory of induced polarization” (

Zhdanov (2008) developed recently a self-consistent effective-medium approach to model induced polarization of porous media with the goal to “provide a unified mathematical model of heterogeneity, multiphase structure, and the polarizability of rocks” (Zhdanov, 2008, p. F197). In this discussion paper, we claim that whereas the work of Zhdanov is mathematically correct, it does not incorporate most of the fundamental mechanisms involved in induced polarization. In short, his model accounts for a special form of polarization mechanism (e.g., associated with the discontinuity of the electrical current displacement at the interface between different phases or electron transfer at the electrolyte/metallic particle interface in the linear asymptotic limit of the Butler-Volmer equation; see Bockris and Reddy, 1970).

Effect of the local clay distribution on the effective electrical conductivity of clay rocks

The “local porosity theory” proposed by Hilfer was revisited to develop a “local clay theory” (LCT) that establishes a quantitative relationship between the effective electrical conductivity and clay distribution in clay rocks. This theory is primarily based on a “local simplicity” assumption; under this assumption, the complexity of spatial clay distribution can be captured by two local functions, namely, the local clay distribution and the local percolation probability, which are calculated from a partitioning of a mineral map. The local clay distribution provides information about spatial clay fluctuations, and the local percolation probability describes the spatial fluctuations in the clay connectivity. This LCT was applied to (a) a mineral map made from a Callovo-Oxfordian mudstone sample and (b) (macroscopic) electrical conductivity measurements performed on the same sample. The direct and inverse modeling shows two results. First, the textural and classical model assuming that the electrical anisotropy of clay rock is mainly controlled by the anisotropy of the sole clay matrix provides inconsistent inverted values. Another textural effect, the anisotropy induced by elongated and oriented nonclayey grains, should be considered. Second, the effective conductivity values depend primarily on the choice of the inclusion-based models used in the LCT. The impact of local fluctuations of clay content and connectivity on the calculated effective conductivity is lower.

Chapter 8 - Semipermeable Membrane Properties and Chemomechanical Coupling in Clay Barriers

Abstract In this chapter, the semipermeable behavior of clay-rocks is mainly exemplified on fluid flow. For such media and besides pressure gradients, other driving forces not accounted for in the classical form of Darcys law describes fluid flow. In the context of coupled transport processes, the additional driving forces are chemical, electrical potential and thermal gradients. Consequently, the resulting so-called osmotic fluxes must be considered in clay media for hydrodynamic and transport calculations. All available osmotic conductivity data have been collected here, including the most recent values acquired in the course of ongoing research on nuclear waste confinement in clay-rock formations. By means of adjusted curves, these data can be directly used as abacuses. The data are also used to support some theoretical calculations, some of which are presented and discussed here. These predictive models (theoretical expressions or abacuses) for osmotic parameters make it possible to perform some fluid and transport calculations within clay-rock formations. Therefore, they can be conveniently used for pore pressure profiles interpretation within clay stratigraphic layer, calculation of characteristic times of solute migration through Peclet number characterization, or even steady or transient state direct transport calculations. For transient state hydrodynamics, the identification of the required chemomechanical coupling coefficient is analyzed in the light of the most recent theoretical work.