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Geophysical Journal International Provided by RERO DOC Digital Library Geophys View metadata, citation and similar papers at core.ac.uk brought to you by CORE Geophysical Journal International provided by RERO DOC Digital Library Geophys. J. Int. (2010) 180, 225–237 doi: 10.1111/j.1365-246X.2009.04415.x A combination of the Hashin-Shtrikman bounds aimed at modelling electrical conductivity and permittivity of variably saturated porous media A. Brovelli1 and G. Cassiani2 1Laboratoire de technologie ecologique,´ Institut d’ingenierie´ l’environnement, Batimentˆ GR, Station No. 2, Ecole Polytechnique Fed´ erale´ de Lausanne, CH-1015 Lausanne, Switzerland. E-mail: alessandro.brovelli@epfl.ch 2Dipartimento di Geoscienze, Universita` di Padova, Via Giotto 1, 35127 Padova, Italy Accepted 2009 October 9. Received 2009 October 8; in original form 2009 March 4 SUMMARY In this paper, we propose a novel theoretical model for the dielectric response of variably satu- rated porous media. The model is first constructed for fully saturated systems as a combination of the well-established Hashin and Shtrikman bounds and Archie’s first law. One of the key advantages of the new constitutive model is that it explains both electrical conductivity—when surface conductivity is small and negligible—and permittivity using the same parametrization. The model for partially saturated media is derived as an extension of the fully saturated model, where the permittivity of the pore space is obtained as a combination of the permittivity of the aqueous and non-aqueous phases. Model parameters have a well-defined physical meaning, can be independently measured, and can be used to characterize the pore-scale geometrical features of the medium. Both the fully and the partially saturated models are successfully tested against measured values of relative permittivity for a wide range of porous media and saturat- ing fluids. The model is also compared against existing models using the same parametrization, showing better agreement with the data when all the parameters are independently estimated. An example is also presented to demonstrate how the model can be used to predict the relative permittivity when only electrical conductivity is measured, or vice versa. Key words: Electrical properties; Hydrogeophysics; Microstructures; Permeability and porosity. GJI Mineral physics, rheology, heat flow and volcanology as the ‘Lichteneker–Rother (LR) equation’ (Gueguen´ & Palciauskas 1 INTRODUCTION 1994): Many problems of environmental and geological interest are linked n to the presence and flow of water and other fluids in porous media. α α ε = φ ε , (1) The effective use of geophysical techniques to approach these prob- b i i i=1 lems is dependent on the availability of reliable constitutive laws. These models typically link the distribution of fluids in the pores where εb is the bulk relative permittivity of the porous medium, to the bulk physical properties of the medium as measured at the ϕi and εi are, respectively, the volume fraction and the (complex) field or laboratory scale. In recent years, electromagnetic methods, relative permittivity of the ith phase, n is the total number of phases such as ground penetrating radar (GPR), time domain reflectome- of which the medium is composed and the exponent α is a fitting try (TDR) and electrical resistivity tomography (ERT) have been parameter (–1 ≤ α ≤ 1). For α = 0.5, eq. (1) reduces to the well- widely utilized in hydrogeology, civil engineering, etc. (Vereecken known Complex Refractive Index Model (CRIM) (Birchak et al. et al. 2002, 2005, 2006; Butler 2005; Rubin & Hubbard 2005). In 1974; Wharton et al. 1980; Dobson et al. 1985; Heimovaara et al. these applications, the electromagnetic properties inferred at the 1994; Rubin & Hubbard 2005). The main advantages of the LR field-scale are translated, via constitutive models, into quantities of equation are its simplicity and the presence of fitting parameters practical interest, such as moisture content, solute concentration, (εs and α) that help to adequately match the experimental data. The petrophysical and geotechnical properties of the porous medium. LR mixing model (eq. 1) and particularly the CRIM are not purely Constitutive laws are often empirical equations recovered from fit- arbitrary fitting relationships, but have also some physical justifica- ting experimental data, for example, the well-known Topp et al. tions (Birchak et al. 1974; Zackri et al. 1998). Model parameters (1980) model. An alternative approach consists of using weighted are however not clearly connected to the petrophysical properties averages of the electromagnetic properties of the constituents, such of the medium, and their estimation might be difficult (Brovelli C 2009 The Authors 225 Journal compilation C 2009 RAS 226 A. Brovelli and G. Cassiani & Cassiani 2008). More theoretically sound are methods based are the prediction of the electrical conductivity of partially molten for example on effective medium or mean-field approximations earth mantle (e.g. Waff 1974; Park & Ducea 2003; Park 2004), (e.g. Wagner 1924; Bergman 1978; Sihvola & Kong 1988, 1989; and the mechanical properties of soils and granular mixtures (e.g. Friedman 1997; Cosenza et al. 2003) and the volume averaging ap- Watt & Peselnick 1980). In this context, it has been shown that a proach (e.g. Pride 1994). For example, Friedman (1998) proposed combination of the upper and lower bounds can be used to estimate a constitutive relationship for unsaturated soils as a linear combina- the expected bulk properties of the mixture, for example, the elas- tion of two effective medium models. The constitutive equation of tic constants (Hill 1952; Thomsen 1972; Watt & Peselnick 1980; Friedman (1998) produces a reasonably good data fit for a number Berryman 2005). In this work, we follow a similar strategy but, of cases (e.g. Miyamoto et al. 2005; Blonquist et al. 2006; Chen rather than taking the arithmetic or geometric mean of the upper and & Or 2006). More recently, the volume averaging approach was lower bounds—as done in some of the works mentioned above—we used by Linde et al. (2006) to further extend the expression of Pride compute the bulk response of the mixture as a weighted average of (1994) for the dielectric response of soils, by accounting for the ef- the upper and lower bound. The weighting factor explicitly incor- fect of variable water saturation. Although a convincing validation porates some information about the geometry and topology of the of the model of Linde et al. (2006) against experimental data is pore structure, and consequently we expect an improved estimate still missing, the proposed relationship is particularly attractive in of the bulk transport coefficient. that it models electrical conductivity and dielectric properties using the same parametrization, namely the electrical formation factor F, the cementation factor m and the saturation exponent n.These 2.1 Relationship between bulk electrical conductivity parameters depend on the geometrical properties of the pore-space and permittivity of a porous medium geometry (e.g. Revil & Glover 1998; Revil & Cathles 1999). It can therefore be expected that models based on these parameters Due to the formal equivalence of the governing macroscale equa- might provide additional information regarding the structure and tions, transport processes in porous media can be modelled using topology of the porous medium. The impact of the pore-scale geo- a unified treatment (Berryman 1992, 2005; Pride 1994; Revil & λ metrical properties on both the electrical and dielectrical response Linde 2006). For example, the thermal conductivity , the electrical ε of the porous medium has been investigated (Madden & Williams permittivity (under quasi-static conditions) and the low frequency σ 1993; Jones & Friedman 2000; Robinson & Friedman 2001; electrical conductivity are described by similar boundary-value Friedman & Robinson 2002). All these studies concluded that problems (Gueguen´ & Palciauskas 1994; Pride 1994): changes in particle distribution and size modify the tortuosity and ∇·(λ∇T ) = 0(2) connectivity of the pore-space and therefore the bulk response of the medium. It is also observed that these changes cannot be accounted for by porosity variations only. ∇·(ε∇V ) = 0(3) The aim of this paper is to develop a new constitutive law for the electromagnetic response of saturated and unsaturated porous ∇·(σ ∇V ) = 0, (4) media, including the effect of the pore-scale geometrical features of the composite. The model is based on a linear interpolation where V is the electric potential and T the temperature. Indeed, of analytical and exact upper and lower bounds. The model in- while the boundary-value problem is identical in the case of ther- volves only parameters that can be measured independently and are mal conductivity and permittivity (Revil 2000; Berryman 2005), related to the description of other physical properties of a multi- the additional contribution to the total conductivity of the excess phase porous medium such as electrical conductivity. The proposed charge at the interface between water and grain minerals further relationship is subsequently tested using experimental data taken complicates the problem for the effective electrical conductivity from the literature, and model predictions in terms of the micro- (Brovelli et al. 2005). The surface conductivity is mainly due to geometrical parameters (Archie’s cementation factor and saturation impurities (clays and oxides) lining the pores and to the presence of exponent) are compared with those obtained from
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