Published online 26 May 2006
Published in Vadose Zone J 5:731-741 (2006)
DOI: 10.2136/vzj2005.0107
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Dependence of the Electrical Conductivity on Saturation in Real Porous Media
R. P. Ewinga,* and
A. G. Huntb
a Dep. of Agronomy, Iowa State Univ., Ames, IA 50011
b Dep. of Physics and Dep. of Geology, Wright State Univ., Dayton, OH 45435
* Corresponding author (ewing{at}iastate.edu)
Received 29 August 2005.
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ABSTRACT
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Archie's Law for the porosity and saturation dependence of electrical conductivity, 
(
) = ab
m(/)n (where is electrical conductivity, the subscript b denotes the brine or bulk solution, is porosity, is volume wetness, and a, m, and n are fitting parameters) was recently derived by applying continuum percolation theory to fractal porous media. We have recast Archie's Law in terms of saturation alone to obtain () = 0 ( – c)µ, where c is the critical volume fraction for percolation, and 0 = ab/(1 – c)µ. The value of the exponent, µ = 2.0 for three-dimensional systems and 1.28 for two, is consistent with theory and simulations. We examined the universality of the exponent's value, and the range of validity of our expression. Drawing on published data, we compared predicted and measured values of () across the full range of saturation, and found that the newly derived expression provides good predictions, is robust with respect to secondary effects such as residual salinity and contact resistance, and yields meaningful physical parameters.
Abbreviations: 2D, two-dimensional • 3D, three-dimensional • CPA, critical path analysis • PT, percolation theory • RS, truncated random fractal model of Rieu and Sposito
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INTRODUCTION
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A RECENT SERIES of publications has derived constitutive relationships of porous media using continuum percolation theory. These relationships have included the unsaturated hydraulic conductivity (Hunt and Gee, 2002a) and air permeability (Hunt, 2005a), solute and gas diffusivity as functions of saturation (Hunt and Ewing, 2003), pressure–saturation curves including hysteretic effects (Hunt, 2004c) and nonequilibrium conditions (Hunt and Skinner, 2005), and the electrical conductivity of saturated media (Hunt, 2004d).
Derivation of these constitutive relations necessitated adopting a model for porous media. Because of its known advantages in predicting water retention curves (Filgueira et al., 1999; Bird et al., 2000, Hunt and Gee, 2002b), the truncated random fractal model of Rieu and Sposito (1991; henceforth RS) was chosen. Because percolation theory (Stauffer and Aharony, 1994; henceforth PT) has been repeatedly shown to generate the most accurate predictions of upscaled transport coefficients in disordered media (Seager and Pike, 1974; Shah and Yortsos, 1996; Bernabé and Bruderer, 1998), it was chosen for calculations. Percolation theory deals with defining properties of random media close to the percolation threshold (the point at which continuous pathways are just barely connected), but it can also be applied to systems far from the threshold in the form of critical path analysis (Ambegaokar et al., 1971; Pollak, 1972; Katz and Thompson, 1986). The reason for the superiority of PT is that it quantifies both the otherwise qualitative notion of "finding the path of least resistance," and also the connectivity and tortuosity in the vicinity of the percolation threshold. Because it is impossible to make an explicit random fractal model that can be mapped to a regular lattice, we used continuum percolation (Berkowitz and Balberg, 1993; Stauffer and Aharony, 1994) rather than the more usual bond or site formulations. The following theoretical developments apply to both soils and rocks.
Almost all transport properties in soil and rock, including gas- and liquid-phase diffusivity (Hunt and Ewing, 2003) and air permeability (Hunt, 2005a), are well characterized by percolation scaling relationships; however, the unsaturated or relative hydraulic conductivity [K(), the ratio of the unsaturated to the saturated hydraulic conductivity] requires two separate techniques (Hunt, 2004a). Near the percolation threshold, unsaturated hydraulic conductivity is well described by the standard scaling analysis of PT (Stauffer and Aharony, 1994), which also gives the connectivity and tortuosity of the dominant pathways. Well above the percolation threshold, unsaturated hydraulic conductivity is better calculated using percolation theory's CPA (critical path analysis), which relates the bottleneck resistance value to the degree of saturation. Because electrical and hydraulic conductivity are somewhat similar, one objective of this study was to examine whether electrical conductivity requires both techniques, or only the scaling treatment. The second objective of this study was to examine experimental evidence regarding whether percolation scaling generally holds for the case of electrical conductivity in rocks and soils. In addressing this second objective, we necessarily encountered complications such as residual salinity, solid-phase conductance, and contact resistance.
The scaling of conductivity, whether electrical or hydraulic, is given near the percolation threshold by (Stauffer and Aharony, 1994)
 | [1] |
where p is a bond, site, or volume (in continuum percolation) fraction, and pc is its critical value for percolation. The exponent µ is a universal exponent from PT; for three-dimensional (3D) bond and site systems, it takes the value 2.0 (Gingold and Lobb, 1990; Clerc et al., 2000), and in two-dimensional (2D) systems, 1.28 (Derrida and Vannimenus, 1982). The 3D value was until recently thought to be 1.88, as given by the analysis of Skal and Shklovskii (1975). For either electrical or hydraulic conductivity, if the continuum formulation of PT is used, the volume wetness is used in place of p, and its critical value c in place of pc. Also for continuum percolation, the exponent µ may take a value greater than 2.0 in 3D; we will return to this issue of universality below.
In the case of electrical conductivity due to brine-filled pore space, Hunt (2004d, 2005b) argued that Eq. [1] applies to typical soils across the entire range of moisture contents following the substitutions p 
and pc c. The purpose of Hunt's (2004d) work was to provide a theoretical basis for Archie's Law (Archie, 1942), an empirical expression for the electrical conductivity, , of porous media. Archie's Law can be written
 | [2] |
with b being the electrical conductivity of the bulk brine solution, the porosity, and a, m, and n fitting parameters. The parameter a is usually close to unity, and some formulations (such as Archie's original presentation) omit it altogether. The exponent m is called the "cementation index" while n is referred to as the "saturation index"; both generally take values near 2.0 (although Hunt [2004d] based his analysis on Skal and Shklovskii's [1975] value of 1.88). In the case of a saturated medium, = and so Archie's Law reduces to
 | [3] |
The physical basis for Archie's Law has been the subject of much work during the last 20 yr (Thompson et al., 1987; Adler et al., 1992; Nigmatullin et al., 1992; Berkowitz and Balberg, 1993; Kuentz et al., 2000). In particular, both Thompson et al. (1987) and Berkowitz and Balberg (1993) considered the possibility that Archie's Law could follow from considerations of continuum percolation theory. Thompson et al. (1987) rejected the possibility while defending the universality of the exponent's value, then thought to be 1.88. Berkowitz and Balberg (1993) allowed it, but only under the condition that the critical volume fraction for percolation c approach zero. Hunt (2004d) argued that c proportional to would also suffice. This proportionality had already been suggested to hold in media with negligible clay content, with values of the proportionality constant typically in the range one-tenth to one-sixth (Hunt, 2004b).
Substituting for p and c for pc into Eq. [1], and inserting the prefactor ab, one finds
 | [4] |
which, adjusted for the limiting condition () = b at = = 1, becomes
 | [5] |
Where appropriate, we will simplify our notation by using 0 
ab/(1 – c)µ as the prefactor for percolation scaling.
The basic form of Eq. [5] is similar to (and probably more correct than) the equation proposed by Hunt and Ewing (2003) for liquid-phase diffusion in unsaturated soils. This is no coincidence: there is a strong similarity between liquid-phase electrical conductivity and diffusion in porous media. The parallel was first noted by Einstein (1905), and was further developed for soils by Schofield and Dakshinamurthi (1948) and Klinkenberg (1951). Electrical conductivity can even be used to estimate the diffusion coefficient (or vice versa), as per work by Snyder (2001) and Garrouch et al. (2001). The analogy is weaker for media with electrically conducting solids, high specific surface, or charged surfaces, but serves as a first-order approximation nonetheless. Investigations and analyses of the two phenomena have been largely independent from each other, so we know of no data examining whether identical exponents or threshold values are seen in practice.
Comparison of Eq. [3] with experimental data from 50 different rocks with a wide range of permeabilities gave µ = 1.82 ± 0.16 in 3D (Thompson et al., 1987). This reasonable correspondence with theoretical values suggests that rocks, and possibly soils, follow the predictions of continuum percolation theory with universal scaling. We will examine experimental evidence of that possibility.
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BACKGROUND
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Solid and Fluid Contributions to Relative Electrical Conductivity
In both petroleum engineering and soil science, it is frequently assumed that the electrical conductivity has contributions from both the solid and liquid phases (Patnode and Wyllie, 1950; Cremers and Laudelout, 1965; Rhoades et al., 1976). A widely used relationship in soil science is
 | [6] |
where the first term, s, is referred to as a "surface" or "solid" term (Cremers and Laudelout, 1965; Rhoades et al., 1976). Interpreted as the contribution of hydrated clay minerals, this term is considered independent of moisture content except under extremely dry conditions. The second term, attributed to conducting fluid in the pore space, is the product of the conductivity of the soil solution, the water content, and a transmission coefficient that is itself a linear function of the water content (Rhoades et al., 1976). Across a limited range of values, (a + b) can look like ( – c)m for m = 2.0 (Fig. 1
), so this traditional phenomenology may mask a universal dependence compatible with Archie's Law.
Because the solid phase is always well above the percolation threshold, a "surface" or "solid" conductivity term should be independent of saturation. In practical terms, it is more important at low water contents (Letey and Klute, 1960; Cremers et al., 1966): as the soil drains, approaching the percolation threshold from above, its relative contribution increases. So while surface conductivity is typically neglected, it may dominate the system conductivity if the brine electrical conductivity is low, the medium has low porosity or a low degree of saturation, or the medium has a high specific surface (Klein and Santamarina, 2003).
If the contrast between the solid and fluid conductivities is small, individual conducting pathways will tend to go through both phases. There are, therefore, three approximations in order of increasing complexity:
- The solid phase is assumed to have zero conductivity, so all current flows through the liquid phase only (Eq. [5]).
- The solid and liquid phases are assumed to conduct strictly in parallel, and so a constant solid-phase conductivity s is added to Eq. [5]:
 | [7] |
This approach is quite common, and yet it is most appropriate for s << b (the usual case) or s >> b, cases in which only a negligible quantity of current flows from one phase to another.
- If the solid and fluid conductivities do not have a large contrast, then an optimized path of conduction will sometimes go through the solid phase to bypass a more tortuous path through the liquid, and sometimes through liquid to bypass a higher resistance solid path. In other words, the two phases will not conduct strictly in parallel, and in fact the degree of interaction in the conducting pathways will vary with the relative conductivities of the two phases, as well as with the water content. There is no universally agreed-upon mathematical formulation for this interaction, which has been an active area of research for decades. We conjecture that such a phenomenon would reduce the value of µ by 0.12, the tortuosity contribution to the conductivity exponent (Stauffer, 1979). This would give a conductivity exponent µ = 1.88 when there is little contrast between solid and liquid conductivities.
Nonuniversality
Both models and data exist (Kogut and Straley, 1979; Feng et al., 1987; Balberg, 1987) supporting a nonuniversal scaling (i.e., µ taking different values in different systems) of electrical or hydraulic conductivity for certain specific cases in continuum percolation. Feng et al. (1987) obtained the then "universal" value µ = 1.88 for a conducting matrix from which equal-sized, overlapping spherical voids are removed. This so-called "random void" or "Swiss cheese" model is functionally equivalent to a conducting fluid with insulating spherical inclusions, e.g., a brine-saturated sandstone, and its low percolation threshold and universal exponent support a percolation basis for Archie's Law. In their model, the width of the conducting necks remaining between voids was denoted 
, and its values were argued to be uniformly distributed. For the inverse case, that of overlapping spherical conductors in an insulating matrix (the inverted random void model), Feng et al. (1987) obtained the nonuniversal value µ* = µ + 0.5 = 2.38 (where the asterisk denotes a nonuniversal value). This latter case is the negative of a typical particulate geological medium composed of round insulating particles and a conducting fluid, so its nonuniversal exponent need not imply nonuniversal behavior in rock and soil.
It was early argued (Feng et al., 1987) that the exponent's taking a universal or nonuniversal value hinged on whether the conducting phase was positively curved like spheres, or negatively curved like the solid part of Swiss cheese. Recent work (Balberg, 1998) has given a more powerful explanation. When the random void model is generalized to a power-law size distribution for the insulating spheres, it no longer has a uniform distribution of overlaps as in the Feng et al. (1987) Swiss cheese model; rather, a power-law distribution of overlaps is obtained. In such a case, the universality of the conductivity exponent depends solely on the exponent governing the power-law distribution, with no dependence on curvature. In applied terms, broader distributions of pore size (and thereby conductivity) are more likely to show nonuniversal behavior. Rock and soil with broad pore size distributions also tend to have higher fractal dimensions; this tendency is relevant to our discussion below on the range of validity of percolation scaling for electrical conductivity.
Other plausible systems can also generate nonuniversal exponents of percolation. Consider a medium composed of conducting, nonoverlapping spherical particles. The electrical conductivity at zero saturation is effectively zero, because the diameter of the conducting path at the contact points is theoretically zero. Now add a small amount of a wetting conducting liquid (e.g., brine) to the medium. The liquid will form pendular rings (capillary bridges) at the contact points, with the resulting geometric structure resembling two spheres connected by a cylinder of diameter . As saturation increases, and therefore () also increase. While the particle sizes need not continue to zero radius, the radii of the pendular rings do approach zero in the limit of zero saturation. Furthermore, the critical volume fraction for percolation would be zero, satisfying Balberg's (1987) criterion, because the solid portion of the medium would only just conduct at zero saturation; increasing the liquid content would merely increase the radii of the connecting "bridges."
This model of spheres connected by pendular rings, however, is only superficially similar to Feng et al.'s (1987) inverted random void model, which predicted a nonuniversal power µ* = 2.38 for electrical conductivity. At low saturation, such that the radius of meniscus curvature is less than the radius of the spheres, the liquid "contact area" connecting the spheres—the cross-sectional area of the liquid cylinder joining the spheres—scales as the square root of the ring's volume (Rose, 1958). In the special case that the particles and brine have equal conductivities, electrical conductivity would scale as the 1/4 power of the brine content. A smaller exponent would be obtained for a lower brine/solid conductivity ratio; for example, data from an analogous experiment by Géminard and Gayvallet (2001) for the case b = s/2 imply a power of 1/6 (but only up to 
25% of saturation, above which point the growing pendular rings are no longer the dominant contributor to conductivity, and so the system changes behavior). In any case, and consistent with our conjecture above, a conducting solid may result in a scaling exponent whose value is less than the universal value across a limited range of saturation.
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RANGE OF VALIDITY OF PERCOLATION SCALING
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Critical path analysis (Ambegaokar et al., 1971; Pollak, 1972) applies PT to find the rate-limiting conductance of systems with wide ranges of local transport coefficients. In the following discussion, we assume that the liquid is electrically conducting but the solid is not. Applying CPA to an electrically conducting viscous fluid in a network model with a wide range of pore sizes, and assuming Poiseuille flow, the same bottleneck pore radius rc controls both K (hydraulic conductivity) and (Friedman and Seaton, 1998; Hunt, 2001). For network models that use a rigid lattice, the pore radii r take on a range of values while the lengths l are constant, with the result that both the rate-limiting electrical conductance, c, and hydraulic conductance, Kc, depend on the same rc (Friedman and Seaton, 1998):
 | [8] |
 | [9] |
identical dependencies except for the different exponents on rc.
In a fractal porous medium, self-similarity requires that l 
r (Hunt, 2001). Equations [8] and [9] therefore reduce to
 | [10] |
 | [11] |
In fractal media, continuum percolation is preferable to site and bond percolation treatments (Hunt, 2001). In continuum percolation, the relevant variable is a volume fraction, which for both hydraulic and electrical conductivity is the water content, . The random fractal model applied was chosen as continuous, but constrained to be compatible with the RS discrete random fractal model. There,
 | [12] |
with D the fractal dimensionality of the pore space, and r0 (rm) the minimum (maximum) radius over which the fractal description holds. For the RS model, rc at saturation is (Hunt and Gee, 2002a)
 | [13] |
which as a function of becomes (Hunt and Gee, 2002a)
 | [14] |
Using Eq. [11], the dependence of K on via rc() is thus (Hunt and Gee, 2002a)
 | [15] |
where Ks is the value of K at saturation, i.e., at = . Notice that Eq. [13]
through [15] diverge as D 3.0, i.e., the range of pore sizes is infinite. In the analogous equation for electrical conductivity, the exponent in Eq. [15] becomes 1/(3 – D):
 | [16] |
due to the different dependencies of electrical and hydraulic conductivity (Eq. [10] and [11]).
Percolation theory gives a scaling relationship for unsaturated hydraulic conductivity,
 | [17] |
similar to Eq. [4] and [5] for electrical conductivity, with K0 Ks. Equation [17] indicates that the conductivity at the percolation threshold K(c) = 0, but Eq. [15] yields rc(c) = r0, which is incompatible with K(c) = 0 except for the special case r0 = 0. When c, however, the dependence of K on the bottleneck radius (as given by CPA) is secondary to its dependence on the density of water-carrying pathways (as given by scaling arguments of PT; see Hunt, 2004a). In other words, a change from CPA to PT occurs because the separation of these paths cannot be less than the correlation length from PT, 
. It is known from PT that in the neighborhood c, ( – c)–
with = 0.88 in 3D and 1.33 in 2D (Stauffer and Aharony, 1994). In such a case, the percolation scaling shown in Eq. [1] and [17] applies.
The crossover point from CPA (Eq. [15]) to PT (Eq. [17]) was found (Hunt, 2004a) by forcing both K and dK/d to be continuous at some crossover water content x, which is then given by
 | [18] |
Repeating the procedure for (), using the analogous Eq. [16], yields
 | [19] |
Note that if the formulation for K() in Eq. [13] and [14], and for () in Eq. [15], are recast using the Balberg (1987) assumptions, their exponents will change to D/(3 – D) and (D – 2)/(3 – D), respectively. The same substitutions will then apply to the denominators of Eq. [18] and [19]. We raise this issue because the Balberg formulation appears to slightly overestimate the conductivities, while that given above underestimates them; they thus serve to constrain but not solve the issue.
We can now consider the theoretical crossover water contents in comparison with soils. Typical values for soils are = 0.4 and D = 2.81 (Bittelli et al., 1999). Taking a typical value of the critical volume for percolation, c = /10 = 0.04, we have x = 0.134 for hydraulic conductivity and x = 0.570 for electrical conductivity. In other words, for this "typical" soil, K() is defined by percolation scaling only for < 0.134, while () is determined by percolation scaling right up to saturation. The precise value of the crossover water contents will vary depending on specifics of the system, but the order 0 
c < x (hydraulic) < x (electrical) will hold. This explains why at saturation can be a power law in , but K cannot, except in the case of quite ordered (low D) systems (Hunt, 2005b). When slightly exceeds x, the pore size distribution starts to influence the electrical conductivity; for >> x, Archie's Law becomes completely invalid. The observed spread in values of m (Thompson et al., 1987) is roughly in accord with predicted increase due to wide pore size distributions (Hunt, 2005b).
As a condition for the validity of the percolation scaling relationship for electrical conductivity (Eq. [5]), let us require x 
0.9 in Eq. [19] (noting that, at least for soils, 100% saturation is seldom reached anyway). This condition divides the (D,) plane into one region where x 0.9 and so Eq. [5] is valid, and another where x < 0.9 and so Eq. [5] is invalid. A similar demarcation can be defined for hydraulic conductivity. Figure 2
shows the demarcation lines according to both this discussion and that of Balberg (1987) for () and K(), along with known (D,) data for some 150 geological media. Regions in which scaling is valid are below and to the left of the demarcation lines. While most sandstones (Thompson et al., 1987) and some Hanford site soils (Hunt and Gee, 2002b) fall within the region where Eq. [5] would be invalid, a regression of these two groups of media falls very close to the demarcation line. Most of the 50+ Bemidji North Pool soils (W. Herkelrath, unpublished data, 1992) fall within the region in which Eq. [5] is valid. Because of this, we suppose, given no information to the contrary, that Eq. [5] is valid for any geological porous medium throughout its full range of saturation.
When D increases slightly past the limit of validity of Eq. [5], electrical conductivity is underpredicted by percolation scaling (Hunt, 2005b); with further increases in D, percolation scaling is no longer a useful framework for analysis. Note that for some combinations of and D, the difference is subtle (Fig. 3
), and data may present as percolation scaling (Eq. [5]) with an exponent µ* > µ. The above analysis therefore offers a new explanation of some experimentally observed nonuniversal scaling exponents: they may simply result from percolation scaling being applied outside of its range of validity.
When data are within the range of validity of percolation scaling, and yet indicate a nonuniversal value µ*, some caution is still advisable. Incorrect estimates of c can produce apparently nonuniversal values for µ*; hence simultaneous fitting for both c and µ has built-in pitfalls. Underestimation of c can result in apparent values of µ* > µ, while overestimation of c can produce values of µ* < µ (Fig. 4
). As a practical matter, it is best to start with the assumption that universality is observed, and only resort to nonuniversality when other, more mundane explanations have been exhausted.
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COMPARISON WITH EXPERIMENT
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We applied the percolation scaling framework to the analysis of experimental data. To the best of our knowledge, all of these data are from water-wet media, and hysteresis (if present) is ignored. The data sets examined here (Table 1) represent both coarse and fine soils, and both igneous and clastic sedimentary rock. We proceed from simpler to more complex cases, with the complex cases illustrating how a sound theoretical footing can help handle potentially confounding issues.
The data of Rhoades et al. (1976) (Fig. 1) were originally plotted in a form that subtracted out any surface or solid conductivity, so analysis simply involved fitting values for a and c. Our Eq. [5] yielded the same R2 as that of Rhoades et al. (1976, Eq. [6]), but where their a and b are clearly meaningless fitting parameters, our parameters a and c have physical significance: a gives the medium's tortuosity at saturation, while c is the critical volume for percolation. For the Indio soil represented here, we had a tortuosity at saturation of 1.232, and a critical volume fraction c = 0.073. Figure 1b, the same data plotted in logarithmic coordinates, highlights the percolation scaling formulation's clear superiority at low water contents.
Archie's (1942) seminal paper presented electrical resistivity data for a number of saturated consolidated Gulf Coast sandstones, and for samples of saturated unconsolidated Nacatoch sand. Fitting Eq. [5] to his sandstone and sand data (with – c substituted for – c), we obtained correlation slopes of almost precisely one and intercepts near zero, in contrast to his slopes of 0.66 (sandstone) and 1.55 (sand) (Table 1). As expected, tortuosity was lower in sand than in sandstone. The critical volume for percolation in the sand was just 1% of porosity; that in the sandstone (2% of porosity) would probably be higher if the sandstones were strongly cemented.
Binley et al. (2001) used their data to make inferences regarding moisture content and were content with a simple calibration to Archie's Law. A second set of data from the same sandstone ( = 9.3%) was published in 2002. Cassiani et al. (2005) tested their own model using the data from Binley et al. (2001), and found a constant solid contribution to the electrical conductivity of 0.00143 S m–1 added to the 0.0156 S m–1 electrical conductivity of the fully saturated pore space; however, the Cassiani et al. (2005) analysis implies a relatively weak -dependent contribution, as seen in Fig. 5
. Using the numerical values from Cassiani et al. (2005) and the common assumption that at typical experimental frequencies the two conduction mechanisms operate in parallel, we have
 | [20] |
as a specific instance of Eq. [7]. We had no independent basis on which to choose a value of c in these sandstones. We could have assumed that its value is 0.1, but it is more likely that, in a medium with significant solid conductivity, the critical volume for percolation is zero. As in the example of conducting spheres with pendular bridges, any water at all should increase conductivity. Using the numerical values from Cassiani et al. (2005), and assuming c = 0, gave a no-parameter fit that is clearly superior to the fit of Cassiani et al. (2005) (Fig. 5). Fitting gave values slightly different from those of Cassiani et al. (2005), and yielded a slope of 1.00 and an intercept of –0.0004 for regressing predicted against observed values. Notice that if the solid contribution (s in Eq. [20]) had been underestimated, the prefactor 0 = 0.0156 would be overestimated. That is, if we had optimized for c alone, our value would be dependent on the accuracy of the estimated s.
Our predicted values compared well with the observations of Binley et al. (2001 and 2002) (Fig. 6
). For both data sets (2001 and 2002), individually as well as combined, Eq. [7] with c = 0 matched the data with slope near one, intercept near zero, and a high correlation coefficient (Table 1). In support of our conjecture about the exponent taking a value µ* = 1.88 for conducting solids, we found that equally good fits were given for the two values of the exponent.
A solid-phase conductivity is shown more dramatically in silica sand data (Fig. 7
) supplied by Ren (personal communication, 2003; Ren et al., 1999). Here it is clear that the solid phase makes a constant contribution to the overall conductivity—again suggesting the assumption c = 0—with the remaining conductivity varying with the conductivity and volume fraction of the solution. When we subtracted the solid contribution, estimated as the mean conductivity for the b = 0 solution, the data fell on lines of µ = 2 in logarithmic space (Fig. 8
). Because each datum is from a separate packed core, the data as a whole are somewhat noisy, but the figure shows reasonable prediction of total conductivity from only the known values b and , the assumed c = 0, and the readily estimated value of s. A slight improvement was given by optimizing for a and c. We note in passing that the lower slopes at low water contents, as suggested by the data, are consistent with our conjecture that the nonuniversal value µ* = 1.88 may be more correct for media with similar solid- and liquid-phase conductivities.
Roberts and Lin (1997) examined electrical conductivity of 9.3% porosity tuff as a function of water content, temperature, and solution concentration. Two solutions were used: deionized water (DW), and a standard "J-13" solution. We examined only the draining leg of their data, because it was done at a single temperature (25°C) and the wetting leg was much noisier. Examination of the data indicated that solid-phase conductivity was negligible, but the nearly identical behaviors of the DW- and J-13-saturated samples suggested the presence of residual salinity in the medium. Making the assumption that any residual salinity dissolves completely at any non-zero water content, we modified Eq. [7] to account for residual salinity:
 | [21] |
where r is the residual salinity's contribution to electrical conductivity. The factor (b + r)/ accounts for both solution and residual salinity contributions. We fit both the DW and J-13 data in a single optimization, using the known conductivity value for J-13 (b = 0.0256 S m–1), and assuming that b = 0.0 S m–1 for DW and that all samples had identical critical volumes. The resulting fit (Fig. 9
) had an R2 of 0.96 for DW and 0.92 for J-13; the log axes accentuate the errors at low water content. Optimization gave a value of c = 0.002 and a residual salt equivalent conductivity of 0.004 S m–1, approximately 15% of the J-13 conductivity.
The ability to account for conductivity caused by residual salt or exchangeable cations is particularly useful in soils, where such salts may contribute a significant fraction of the liquid-phase conductivity. For example, Rinaldi and Cuestas (2002) packed loess soils with known volume fractions of NaCl solution, and measured electrical conductivity at different water contents (their Fig. 12). In the zero-electrolyte treatment, the increase in electrical conductivity with water content can reasonably be attributed to residual salts in the soil. Fitting Eq. [21] to their data provided an excellent fit (Fig. 10
) with an R2 of 0.98. The optimized value for the residual salt equivalent conductivity was 0.085 S m–1. If the relative concentration of the various cations were known, their absolute concentrations could be determined.
The data of Abu-Hassanein et al. (1996) provide another example of the importance of the contribution of residual salt. They presented data on four soils differing in texture and clay mineralogy; each soil was tested at three different degrees of compaction. Tap water (b = 9.5 x 10–3 S m–1) was used throughout. Because we didn't know a priori whether any given soil would have residual salt or solid-phase conductivity, we added a solid-phase conductivity to Eq. [21]. As it turned out, the solid contribution was precisely zero for all but Soil D, which had a negligible value of s = 6.6 x 10–5 S m–1, so it was dropped from the analysis. Fitting each soil in turn, we obtained R2 between 0.82 and 0.97 (Fig. 11
). Residual salt accounted for 92 to 99% of the saturated conductivity, with higher percentages in soils with higher clay contents. The critical volume for percolation was also higher in higher clay soils, as expected.
The data sets from Mori et al. (2003) and Tuli and Hopmans (2004) are somewhat similar, so we present them together (Fig. 12
). We obtained R2 = 0.98 fitting Eq. [21] to the Mori et al. (2003) data, slightly lower than their 0.99 using the Rhoades equation (Eq. [6]), but we learned that c = 0.0 and a = 1.417. Tuli and Hopmans (2004) gave s = 0.0725 dS m–1 for Oso Flaco sand; this is the only medium we encountered that combined nonnegligible solid conductivity with a nonzero critical volume for percolation. Our fit yielded a mean absolute residual of only 0.02, compared with their value of 0.08.
We also examined an unexpected complication in the data published by Kechavarzi and Soga (2002). They presented triplicate calibration curves for their miniature resistivity probes, and reported that fitting Archie's Law to the data gave R2 = 0.91, a somewhat disappointing value for a calibration curve. A plot of the raw data (Fig. 13
) shows a marked decrease in the slope of the () curve; combined with the unknown characteristics of the miniature probe, this raised the possibility that there was some contact resistance in their experimental setup. The washed sand was unlikely to have residual salinity, and solid conductivity would curve the slope up rather than down. We accordingly allowed for contact resistance 
c in the () relationship through a modification of Eq. [7], giving
 | [22] |
where the constant c corrects for geometric factors specific to their experimental setup. This new equation fit the data quite well (Fig. 13), with R2 = 0.97. Allowing a different value of the product cc in each replicate increased the R2 to 0.98.
Our examination of the data sets discussed above found critical volume fractions for percolation ranging from 0.0 to 0.073 (Table 1), reasonably in line with the observed range of Hunt (2004b). Values of a, which we interpret as the electrical tortuosity at saturation, ranged from 1.07 to 2.73. In two cases, brine conductivity b was not given, forcing us to lump ab into a single parameter; when this is done, the value of the lumped parameter cannot yield useful information about its component parts. Conductivity attributable to residual salinity was encountered in tuff and several of the soils, but in only one sand and none of the sandstones. As predicted, the data fall on the line indicating an exponent µ = 2.0 (Fig. 14
), with deviations below the line indicating contact resistance (e.g., Kechavarzi and Soga, 2002), and deviations above the line indicating the effect of residual salinity (e.g., Roberts and Lin, 1997; Abu-Hassanein et al., 1996). Deviations attributable to residual salinity are most pronounced for low-conductivity solutions and low water contents (e.g., Ren et al., 1999).
Data sets for which multiple measurements were conducted on a single core tended to be much less noisy than data sets for which individually packed cores contributed a single point each, though of course some investigations preclude this approach. As our discussion indicates, analysis should not be attempted blind: information about the medium and methodology is frequently helpful in deciding, e.g., whether it is reasonable to suppose that the medium will have residual salinity. Equations with too many fitting parameters, even physically based ones, run the risk of overfitting to their data, so secondary parameters (s, r, and c) should be invoked only if there is a physical basis for doing so.
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CONCLUSIONS
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Our examination of predictions of the saturation dependence of electrical conductivity, as derived from percolation scaling, does not constitute proof of the validity of our approach; however, the predictions we have presented are generally at least as good as those based on other formulations, with porosity, contact resistance, and residual salinity contributing most of the uncertainty. More importantly, the parameters in our equations have physical meaning, and it is straightforward to adjust the basic equation to account for these secondary effects. Fits without optimization, presented for data by Ren (unpublished data, 2003) and Binley et al. (2001 and 2002), provided better than first-order approximations to the measured values. Optimization provided improved fits, and yielded reasonable values for parameters with physical significance. We conclude that the saturation dependence of electrical conductivity in porous media is consistent with Archie's Law as derived from continuum percolation theory.
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ABSTRACT
INTRODUCTION
