We address the problem of predicting saturation-dependent electrical conductivity {\sigma} in packings of spheres during drainage and imbibition. The effective-medium approximation (EMA) and the universal power law of percolation for {\sigma} are used, respectively, at higher and low water saturations to predict the conductivity, with the crossover between the two occurring at some intermediate saturation Swx. The main input to the theory is a single parameter that we estimate using the capillary pressure data. The predictions are compared with experimental, as well as numerical data for three distinct types of packings. The results for drainage in all the packings indicate that the universal power law of percolation is valid over the entire range of Sw. For imbibition, however, the universal power law crosses over to the EMA at Swx = 0.5. We also find that the effect of the pore-size distribution on the {\sigma}-Sw relation is minimal during both drainage and imbibition.
Deep Dive into Hysteretic behavior of electrical conductivity in packings of particles.
We address the problem of predicting saturation-dependent electrical conductivity {\sigma} in packings of spheres during drainage and imbibition. The effective-medium approximation (EMA) and the universal power law of percolation for {\sigma} are used, respectively, at higher and low water saturations to predict the conductivity, with the crossover between the two occurring at some intermediate saturation Swx. The main input to the theory is a single parameter that we estimate using the capillary pressure data. The predictions are compared with experimental, as well as numerical data for three distinct types of packings. The results for drainage in all the packings indicate that the universal power law of percolation is valid over the entire range of Sw. For imbibition, however, the universal power law crosses over to the EMA at Swx = 0.5. We also find that the effect of the pore-size distribution on the {\sigma}-Sw relation is minimal during both drainage and imbibition.
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Hysteretic behavior of electrical conductivity in packings of particles
Behzad Ghanbarian1* and Muhammad Sahimi2
1 Bureau of Economic Geology, Jackson School of Geosciences, University of Texas at
Austin, Austin, Texas 78713, USA
2 Mork Family Department of Chemical Engineering and Materials Science, University
of Southern California, Los Angeles, California 90089-1211, USA
Abstract We address the problem of predicting saturation-dependent electrical
conductivity σ in packings of spheres during drainage and imbibition. The effective-
medium approximation (EMA) and the universal power law of percolation for σ are used,
respectively, at higher and low water saturations to predict the conductivity, with the
crossover between the two occurring at some intermediate saturation Swx. The main input
to the theory is a single parameter that we estimate using the capillary pressure data. The
predictions are compared with experimental, as well as numerical data for three distinct
types of packings. The results for drainage in all the packings indicate that the universal
power law of percolation is valid over the entire range of Sw. For imbibition, however,
the universal power law crosses over to the EMA at Swx = 0.5. We also find that the effect
of the pore-size distribution on the σ-Sw relation is minimal during both drainage and
imbibition.
Key words: Drainage . Imbibition . Packings of spheres . Electrical conductivity .
Partially saturated
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1 Introduction
The relationship between the effective permeability k and electrical conductivity σ of
porous media has been studied for decades (see, for example, Johnson et al. 1986; Katz
and Thompson 1986,1987; Bernabe and Bruderer 1998; for a comprehensive review, see
Sahimi 2011). Since k has the units of a length squared, its square root is a characteristic
length scale of porous media, representing a static property. On the other hand, σ is a
dynamic property that characterizes flow of electrons in a pore space saturated by a
conducting fluid, such as brine. Thus, in principle, there is no exact relation between k
and σ. Despite this, σ is commonly used in geophysics and hydrology to predict k and the
relative permeabilities (see, for example, Revil and Cathles 1999; Doussan and Ruy
2009; Niu et al. 2015; Mawer et al. 2015). For this reason accurate prediction of σ has
been a problem of great interest for a long time (see e.g., Knight 1991; Friedma, 2005;
Zhan et al. 2010; Revil 2016), particularly under partially-saturated conditions, for which
various models have been proposed to predict σ. Among them is the empirical Archie’s
law (Archie 1942), which is valid only for water-wet rocks deprived of microporosity or
substantial clay-exchange cations (Toumelin and Torres-Verdín 2008), and has been
extensively used to determine σ. A recent review of the theoretical models of σ is given
by Ghanbarian et al. (2015a).
Mualem and Friedman (1991) represented the pore space by a bundle of series-
parallel capillary tubes (Mualem 1976) in order to predict σ. By including the concept of
residual volumetric water content, they proposed power laws for σ under fully- and
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partially-saturated conditions. The exponents that characterized their power laws were 1.5
and 2.5 [see their Eqs. (30) and (31)] for, respectively, the fully- and partially-saturated
media. The effect of hysteresis, pore shape, pore-size distribution (PSD), viscosity, and
wettability on saturation-dependent σ has been well studied using pore-network models
(PNMs) and/or solving the Laplace equation in digitized images (see Schwartz and
Kimminau 1987; Tsakiroglou and Fleury 1999a,b; Man and Jing, 2000; Bekri et al.,
2003). Bryant and Pallatt (1996) argued that σ is dependent upon the geometry of the
pore space, and proposed a PNM for computing it for a packing of spheres. The
predictions were in good agreement with experimental data. Suman and Knight (1997)
studied the effect of the pore structure and wettability on saturation-dependence of σ
during drainage and imbibition by means of a PNM. The effect of the wettability was
investigated, and the influence of the pore structure was studied by varying the breadth of
the PSD and the pore-size correlations. Suman and Knight (1997) reported that the effect
of hysteresis is more significant in oil-wet porous media than in the water-wet ones.
Hysteresis in both types of porous media decreased, however, as the extent of the
correlation between the pores increased. Li et al. (2015) computed σ during drainage and
imbibition, modeled by invasion percolation (Chandler et al. 1982; Wilkinson and
Willemsen 1983, Knackstedt et al. 2000) in the PNMs under completely water-wet
conditions. The computed σ did not generally follow Archie’s law (consistent with the
experimental data of Longeron et al. 1989), and exhibited either downward or upward
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