universal scaling of gas diffusion in porous media

RESEARCH ARTICLE10.1002/2013WR014790

Universal scaling of gas diffusion in porous media

Behzad Ghanbarian1 and Allen G. Hunt1,2

1Department of Earth and Environmental Sciences, Wright State University, Dayton, Ohio, USA, 2Department of Physics,Wright State University, Dayton, Ohio, USA

Abstract Gas diffusion modeling in percolation clusters provides a theoretical framework to address gastransport in porous materials and soils. Applying this methodology, above the percolation threshold the air-filled porosity dependence of the gas diffusion in porous media follows universal scaling, a power law in theair-filled porosity (less a threshold value) with an exponent of 2.0. We evaluated our hypothesis using 71experiments (632 data points) including repacked, undisturbed, and field measurements available in the lit-erature. For this purpose, we digitized Dp/D0 (where Dp and D0 are gas diffusion coefficients in porousmedium and free space, respectively) and e (air-filled porosity) values from graphs presented in seven pub-lished papers. We found that 66 experiments out of 71 followed universal scaling with the exponent 2, evi-dence that our percolation-based approach is robust. Integrating percolation and effective mediumtheories produced a numerical prefactor whose value depends on the air-filled porosity threshold and theair-filled porosity value above which the behavior of gas diffusion crosses over from percolation to effectivemedium.

1. Introduction

Diffusion, as a type of transport process in porous media, has been studied in many different communities,such as physics, hydrology, soil physics, chemical engineering, etc. The classic theory of diffusion proposedby Einstein [Einstein, 1905] in free space, e.g., air and water is not valid in porous media. In Einstein’s relation,mean-squared displacement hr2ðtÞi is linearly related to time t through a diffusion coefficient D0 [Havlinet al., 1983]

hr2 tð Þi5D0t (1)

where D0 is diffusion coefficient in free space (e.g., air).

In disordered, heterogeneous porous media, diffusion deviates from Einstein’s relation and becomes non-Fickian. In such media, the mean-squared displacement hr2ðtÞi is a nonlinear function of time [Havlin andBen-Avraham, 1987; Sahimi, 1993a; Ben-Avraham and Havlin, 2000], and can grow either more slowly (calledfractal transport) or more quickly (called superdiffusive transport) than linearly with time [Sahimi, 1993a]

hr2 tð Þi5Dpt2=dw (2)

where dw is random walk fractal dimension, and Dp is diffusion coefficient in porous medium. Note thatin equation (2), valid for both two and three dimension, Fickian diffusion corresponds to dw 5 2. In mostcases of interest, the mean-squared displacement does crossover to a linear dependence on time abovea certain length scale, allowing measurement of a diffusion constant. Nevertheless, the anomalousbehavior at smaller length scales has an important impact on the measured diffusion constant, as wewill discuss.

Experimental evidence [see, e.g., Currie, 1960, 1961, 1984; Currie and Rose, 1985; Moldrup et al., 2000a,2000b; Tuli and Hopmans, 2004; Moldrup et al., 2005a, 2005b, 2005c; Naveed et al., 2013] indicates that gasdiffusion in unsaturated (completely dry) porous media is a function of air-filled porosity, e (total porosity,/). Numerous empirical, semiphysical, and physical models were proposed to model gas diffusion [see, e.g.,Penman, 1940; Moldrup et al., 1997, 1999, 2000a, 2000b, 2004, 2005a]. Among them, Buckingham [1904]from his work on soils proposed (cited in Currie [1960])

Key Points:� Universal scaling of gas diffusion is

confirmed� Model parameters are physically

meaningful� The gas diffusion model presented is

robust and predictive

Correspondence to:B. Ghanbarian,[email protected]

Citation:Ghanbarian, B., and A. G. Hunt (2014),Universal scaling of gas diffusion inporous media, Water Resour. Res., 50,2242–2256, doi:10.1002/2013WR014790.

Received 26 SEP 2013

Accepted 18 FEB 2014

Accepted article online 20 FEB 2014

Published online 12 MAR 2014

GHANBARIAN AND HUNT VC 2014. American Geophysical Union. All Rights Reserved. 2242

Water Resources Research

PUBLICATIONS

http://dx.doi.org/10.1002/2013WR014790

http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1944-7973/

http://publications.agu.org/

Dp

D05em (3)

where m was deduced to be 2 [Buckingham, 1904]. However, it was recently proposed that m should be notonly a function of e, but also soil tension head h or volumetric water content [see, e.g., Resurreccion et al.,2007; Chamindu Deepagoda et al., 2012]. In next section, we show how percolation theory provides a theo-retical framework for Buckingham’s results with the same exponent 2.

Penman [1940], a pioneer of the study of gas diffusion in soils, found a linear relation between Dp/D0 and eas follows

Dp

D05ae (4)

where a 5 0.66 given for a wide variety of porous media, but over a restricted range of air-filled porosity0< e< 0.6. However, van Bavel [1952] found a 5 0.58 for a more restricted range of dry granular materials(cited in Currie [1960]). Sallam et al. [1984] indicated that Penman’s model greatly overestimated Dp valuesat low air-filled porosities. The same results were also obtained by Moldrup et al. [2000b, 2003], Werner et al.[2004], and Moldrup et al. [2005c].

Millington and Quirk [1961] (MQ hereafter) presented a tortuosity model to predict unsaturated hydraulicconductivity

sg hð Þ5 /2

h10=3(5)

where sg(h) is saturation-dependent hydraulic tortuosity (>1), and h is water content (5/ 2 e). Althoughequation (5) was proposed for hydraulic conductivity, Millington and Quirk [1961] stated that it can be usedto estimate diffusion in unconsolidated media as Dp/D0 5 h10/3//2, followed widely in the literature [e.g.,Moldrup et al., 2000a, 2005a, 2005b, 2005c; Kawamoto et al., 2006; Resurreccion et al., 2007; Chamindu Deepa-goda et al., 2011a, 2011b] for soils. Note that one should substitute e for h to apply equation (5) to gas diffu-sion prediction [Moldrup et al., 2001]. Jin and Jury [1996] indicated that MQ model underestimates gasdiffusion. Their results are consistent with those of Moldrup et al. [2000b, 2003, 2005a, 2005b] from labmeasurements, and also those of Kawamoto et al. [2006] from field data. However, Kawamoto et al. [2006]found that the MQ model overestimated gas diffusion measured from lysimeter experiments.

In addition to the linear [Penman, 1940] and power-law [Buckingham, 1904] relationships, Troeh et al. [1982]proposed the empirical model

Dp

D05

e-u1-u

h iv(6)

where u and v are empirical constant coefficients. By directly fitting equation (6) to Buckingham’s gas diffu-sion data, Troeh et al. [1982] found u 5 0.1 and v 5 1.7. Using data from Buckingham [1904] and Currie[1960], they concluded that equation (6) fitted measured data better than the linear model of Penman[1940] and the power-law function of Buckingham [1904] with variable exponent m. Troeh et al. [1982] inter-preted correctly that the parameter u represents the dead-end air volume fraction whose value shoulddepend on the nature of the medium and the pore space. However, most models presented in the literatureignore this fraction. As we show in the following, the parameter u in equation (6) is equivalent to percola-tion threshold et in our universal scaling model for gas diffusion which has the similar form to equation (6)as was anticipated by Jin and Jury [1996]. However, Jin and Jury [1996] never applied percolation theory togas diffusion modeling. Note that using concepts from percolation theory, the threshold is a nonuniversalparameter depending on pore size distribution, pore structure, and connectivity.

Assuming that the pore size distribution mainly accounts for gas diffusion in porous media, Moldrup et al.[1999] presented the soil water retention curve-dependent model called Buckingham-Burdine-Campbell(BBC hereafter) [Buckingham, 1904; Burdine, 1953; Campbell, 1974]

Water Resources Research 10.1002/2013WR014790


Dp

D05/2 e

/

� �213=b

(7)

where /2 term represents the Buckingham’s gas diffusivity at e 5 /, and b is the Campbell’s model expo-nent controlling the pore size distribution broadness. Equation (7) was successfully evaluated for six undis-turbed soil samples with b values ranging from 4.6 to 14 [Moldrup et al., 1999]. However, equation (7) waslater modified by Moldrup et al. [2000a] by replacing Buckingham’s expression with the empirical term 2e3

100

10:04e100 and / with e100 (where e100 is the air-filled porosity at tension head h 5 100 cm). Moldrup et al.[2000a] concluded that their modification predicted gas diffusion more accurately than the Penman andMQ models.

In 2004, Moldrup et al. modified the BBC model to avoid the Campbell b parameter whose calculationrequires soil water retention curve measurements, and presented a three-porosity model (TPM hereafter).Iiyama and Hasegawa [2005] who compared the TPM with the MQ model to predict gas diffusion in undis-turbed peat soils found the MQ model more accurate than the TPM. They also mentioned that Dp/D0 valuesat low air-filled porosity e< 0.1 were very small, near zero, and suggested the presence of noneffective air-filled pores which do not account for gas pathways. This result is consistent with the concept of the thresh-old value, the critical air-filled porosity below which the pore space loses its macroscopic connectivity, frompercolation theory.

Hunt and Ewing [2003] used simulations of Ewing and Horton [2003] to interpret a phenomenological repre-sentation for the diffusion constant given in Moldrup et al. [2001] in terms of percolation theory. The resultsof the simulation and the reported experiments of Moldrup could be represented in the form, Dp/D0 5 1.1 h(h – ht) where ht is the critical water content, which for small values of the threshold is very nearly the perco-lation scaling result, but is, strictly speaking, incorrect. A slightly different theoretical result was generatedfor gas diffusion, which was based on the observation that solutes can diffuse through water films belowthe percolation threshold, an alternate pathway unavailable in gas diffusion.

Kristensen et al. [2010] and Hamamoto et al. [2011] developed two-regime (dual-porosity) models for gas dif-fusion. In particular, Kristensen et al. [2010] considered a linear function like Penman’s equation and a gen-eral power-law function for the first and second regimes, respectively. However, empirical constantcoefficients included in their model have to be determined experimentally, which restricts their model’sapplication to arbitrary porous media.

Literature on gas diffusion is rather confusing due to numerous empirical and semiphysical models pre-sented. Moldrup et al. [1999, 2007] among others assumed that gas diffusion coefficient is mainly related topore size distribution. In contrast, we recently demonstrated [Ghanbarian-Alavijeh and Hunt, 2012] that thesaturation dependence of the air permeability in porous media, as expected [Hunt, 2005], follows universalscaling with a universal exponent of 2 in 3-D systems from percolation theory. In this article, we will demon-strate that the saturation dependence of the gas diffusion coefficient does so as well. A result that universalscaling describes the behavior of a property means that effects from the pore size distribution are negligi-ble. Its effect, however, can be seen in the percolation threshold parameter.

The reason for investigating the saturation dependence of gas diffusion in the context of percolation theorywas contextual. One of the present authors has already shown that the electrical conductivity as a functionof saturation follows universal scaling, at least in a large majority of cases [Ewing and Hunt, 2006], as doesthe solute diffusion constant [Hunt and Ewing, 2009] and, more recently, the air permeability [Ghanbarian-Alavijeh and Hunt, 2012]. In the case of the hydraulic conductivity, however, universal scaling is relevant atmost in a relatively narrow range of moisture contents near the threshold moisture for percolation (alsoknown as the irreducible moisture content), while in the thermal conductivity, universal scaling is relevantmostly to the density dependence [Hunt and Ewing, 2009]. Thus, among the relatively simple properties ofporous media, gas diffusion remained to be investigated [Hunt and Ewing, 2009]. It is important to developa general understanding of which properties have such a simple behavior, and which are strongly affectedby the details of the medium. Therefore, the main objectives of this study are (1) to present theoreticalframework and universal scaling laws based on concepts from percolation theory, and (2) to evaluate ourpercolation theoretical approach with a large database previously used in the literature to test othermodels.



2. Theory

2.1. Diffusion in Percolation ClustersIn order to study fluid flow and solute transport in disordered porous media, percolation theory provides apromising framework. One of the main characteristics in percolation theory is presence of a threshold pc

below which a medium loses its connectivity. At this point it is important to distinguish between an applica-tion of percolation theory to a medium and its application to the fluids within it. Regardless of the optimalmodel (e.g., random pore network), one might choose to represent a porous medium, as the concentrationof any particular fluid within the medium is reduced, the connections of pores filled by that fluid diminishin accord with the predictions of percolation theory.

Below the percolation threshold pc (p< pc where p is the fraction of bonds or sites that are occupied orpresent, which physically means the bonds or sites through which a physical phenomenon such as fluidflow occurs) all clusters are finite in size, and the largest clusters have a typical size of the order of the finitecorrelation length (v) [Kirkpatrick, 1973; Ben-Avraham and Havlin, 2000; Hunt and Ewing, 2003]. In general,there is no spanning (‘‘infinite’’) cluster when p< pc [Feder, 1988]. At pc, an incipient infinite cluster (a ran-dom fractal percolation cluster [Feder, 1988]) occurs along with other finite clusters which are not differentfrom those which form below pc [Kirkpatrick, 1973; Ben-Avraham and Havlin, 2000]. In this case, the mass ofthe spanning cluster increases with the size, L, of the lattice as a power-law Ldf where df is the mass fractaldimension of the fractal cluster [Feder, 1988]. Above the percolation threshold (p> pc), there are still incipi-ent infinite cluster and finite clusters [Kirkpatrick, 1973]. At or below the percolation threshold, the typicalsize of the largest finite clusters is on the order of the finite correlation length. However, the incipient infi-nite cluster is different than the one at pc [Ben-Avraham and Havlin, 2000]. When the system size L is lessthan the correlation length (L< v), the system is heterogeneous, statistically self-similar fractal, and thus dif-fusion is non-Fickian [Gefen et al., 1983; Sahimi, 1993b]. For L> v, the system is macroscopically homogene-ous, the geometry is Euclidean, and diffusion is Fickian [Feder, 1988; Sahimi, 1993b].

Another remarkable property of percolation processes is that transport near the threshold pc is independentof the structure and geometry of the pore network, a feature called universality. This behavior follows apower-law scaling (e.g., equation (10)) whose critical exponent is universal and only depends on the Euclid-ean dimension of the system [Sahimi, 2011]. However, the percolation threshold pc for each network is sen-sitive to the microstructure of the system and thus nonuniversal.

The fluids in porous media can thus be regarded as random fractals where random motions of moleculesmay be described as diffusion in percolation clusters. In such a case, diffusion can be interpreted in two dif-ferent ways: (1) diffusion is restricted to the incipient infinite cluster at p 5 pc and (2) all (both the incipientinfinite and finite) clusters are involved in diffusion process. Ben-Avraham and Havlin [2000] suggested thelatter being relevant and applicable to many practical purposes. Sahimi [1993b, 2012], however, considersthe first case (diffusion only in the incipient infinite cluster) for solute diffusion near percolation threshold inporous media. Sahimi [1993b] states that ‘‘although a particle can diffuse on all clusters, only diffusion onthe sample-spanning cluster contributes significantly to Dp.’’ Following Ben-Avraham and Havlin [2000], herewe assume that gas diffusion occurs in unrestricted ensemble of all clusters in a porous medium, however,one can restrict diffusion to the incipient infinite cluster by including the factor |p 2 pc|b, the probabilitythat a site (or bond) belongs to the incipient infinite cluster, into the following results (equations (8) and(10)) [Havlin and Ben-Avraham, 1987; Ben-Avraham and Havlin, 2000]. Note that b is a universal scaling expo-nent equal to 5/36 and 0.41 in two and three dimensions, respectively [Stauffer and Aharony, 1994].

For p< pc, diffusion is dominated by the largest clusters of size of correlation length v � |p 2 pc|2m in whichm is a scaling exponent whose value is 4/3 in two dimensions and 0.88 in three dimensions [Stauffer andAharony, 1994]. At large time scales (e.g., t !1), hr2ðtÞi � v2 � jp2pcj22m and the probability of being inany of those large clusters is proportional to |p 2 pc|b. Thus [Havlin et al., 1983; Havlin and Ben-Avraham,1987; Ben-Avraham and Havlin, 2000],

Dp / t21jp-pcjb22m; p < pc (8)

where t is time. Note that as t !1, the factor t21 tends to zero, and consequently Dp! 0, which guaran-tees no macroscopic diffusion below the percolation threshold.



At the percolation threshold, diffusion is independent of p because p 5 pc, and hr2ðtÞi is scaled as t2=dw

[Gefen et al., 1983; Harris and Aharony, 1987]. Hence [Havlin et al., 1983; Havlin and Ben-Avraham, 1987; Ben-Avraham and Havlin, 2000]

Dp / t 2-dwð Þ=dw ; p5pc (9)

This time dependence of equation (9) was confirmed experimentally by Knackstedt et al. [1995]. They foundthat the scaling behavior of water diffusion in microemulsions near the structural transition point was con-sistent the theory of percolation. Note that in equation (9), valid for both two- and three-dimensional sys-tems at the percolation threshold, diffusion is non-Fickian with dw> 2 [Ben-Avraham and Havlin, 2000],meaning that Dp vanishes in the limit t !1.

Above the percolation threshold, the contribution of the infinite cluster becomes dominant to diffusion,and at long time scales (e.g., t !1) we have [Havlin et al., 1983; Havlin and Ben-Avraham, 1987; Ben-Avra-ham and Havlin, 2000]

Dp / p-pcð Þl; p > pc (10)

where l is a universal scaling exponent whose value is 1.3 and 2 in two-dimensional and three-dimensionalsystems [Stauffer and Aharony, 1994]. In fact, slight uncertainty regarding the precise value of l exists. Butsince results from Gingold and Lobb [1990] and Normand and Herrmann [1996] are consistent with l 5 2within about 1% precision, nowadays l is often assumed to be exactly 2. Note that the exponent l 5 2includes contributions from both tortuosity and connectivity [see Ghanbarian et al., 2013].

Ben-Avraham and Havlin [2000] pointed out that diffusion coefficient crosses over to a linear dependenceon time above a certain scale. In fact, the nonlinear behavior at smaller (length or time) scales has an impacton the diffusion constant. Ben-Avraham and Havlin [2000] state, ‘‘At short times there exist strong correc-tions to scaling (e.g., equation (8) or (10)) and scaling fails.’’ Therefore, equations (8) and (10) are valid atlarge time scales as we mentioned before.

2.2. Continuum PercolationEquations (8–10) presented above are valid for either bond or site percolation. In order to study gas diffu-sion in natural porous media, we apply continuum percolation and replace the variables p and pc with theair-filled porosity e and its critical value for percolation in the porous medium et, respectively [Berkowitz andBalberg, 1993; Hunt, 2001; Hunt and Ewing, 2003, 2009]. Since in practice gas diffusion is only measuredabove the percolation threshold e> et, we focus on equation (10). Applying continuum percolation repre-sentation to equation (10) gives

Dp / e-etð Þl (11)

As a limiting case, the diffusion coefficient in free space D0 occurs when e approaches 1 which leads to ascale factor 1/(1 2 et)

l in equation (11). The same scale factor was also applied to electrical conductivity[Montaron, 2009] and tortuosity [Ghanbarian et al., 2013] modeling using percolation theory.

Although percolation theory provides precise predictions near the threshold, normalizing equation (11)using the value of Dp 5 D0 at e 5 1 (far from the percolation threshold et), except for a numerical prefactorof about 1.35, still accurately fits gas diffusion results within the measured range of e.

Applying the above scale factor to equation (11) results in the universal scaling law for gas diffusion inporous media

Dp

D05

e-et

1-et

� �l

(12)

Recall that l value is 1.3 and 2 in two and three dimensions, respectively [Stauffer and Aharony, 1994]. Thereexist several techniques including series-expansion analyses, Monte Carlo simulations, and real-spacerenormalization group methods to estimate percolation threshold of continuum models [Torquato, 2002].



As a rough estimate, one can set et 5 0.1/ [Hunt, 2004a] in coarse-textured soils. Alternatively, Liu and Rege-nauer-Lieb [2011] presented a morphological technique to estimate the percolation threshold from three-dimensional soil images. Recently, Ghanbarian-Alavijeh and Hunt [2012] proposed a method to estimate et

from wet end of soil water retention curve data. Their approach was successfully applied to predict air per-meability in porous media. In the absence of knowledge to the contrary one should expect that the samethreshold air fraction would apply to both the air permeability and the gas diffusion.

Note that equation (12) reduces to Buckingham’s relation (equation (3)) when et 5 0. In addition, equation(12) has the same form as the empirical model of Troeh et al. [1982] who found a good match with measure-ments. In contrast to their empirical approach, all percolation-based model (equation (12)) parameters, e.g.,l and et are physically meaningful, and l is fixed (l 5 2 6 0.02).

3. Materials and Methods

Gas diffusion experiments used in this study are digitized data (e.g., Dp/D0 versus e) from graphs presentedin seven different published papers in the literature, e.g., Moldrup et al. [2000a, 2000b, 2003, 2004, 2005a,2005b, 2005c]. The experiments include undisturbed, disturbed, and volcanic ash soils covering a widerange of porous materials (0.41</< 0.87 where / is porosity). The interested reader is referred to the orig-inal articles for more information.

Following Ghanbarian-Alavijeh and Hunt [2012], we apply the least-square optimization method to findout the best et value which leads to a power of 2 (60.02) by fitting equation (12) to the measured gasdiffusion data. One would expect that the numerical prefactor should be 1 when we fit equation (12) toDp/D0 versus (e 2 et)/(1 2 et) data. When we examined the air permeability [Ghanbarian-Alavijeh andHunt, 2012], which is normalized to its value at e 5 /, the regression did indeed yield a value very closeto 1 (in particular, 0.96) for the first database used. However, because we now have to normalize thegas diffusion in percolation clusters (equation (11)) using Dp 5 D0 at e 5 1, which is far away from perco-lation threshold this numerical prefactor might be different than 1. As it turns out, our analysis of experi-ments generates a numerical prefactor with average value 1.35. But the predicted scaling relationship[Kirkpatrick, 1973] far from the threshold (at values of the air-filled porosity never reached in experiment)deviates from equation (12). Accounting for this deviation yields a modification of the value of thenumerical prefactor to 1.36.

4. Results and Discussion

The obtained universal scaling results are promising. We found that 66 out of 71 experiments followed theuniversal scaling (equation (12)) with the exponent l 5 2 (60.02). The results for different databases aresummarized in Table 1. As can be seen, for most samples (93%) the exponent is 2 (60.02) or very close. Wefound that the mean and the standard deviation of the exponent were 1.983 and 0.066, respectively, overall 71 experiments (not reported in Table 1). In the following, we discuss those five experiments, which didnot scale with the universal exponent of percolation (equation (12)).

For sample Miles in the Moldrup et al. [2000b] database, we should point out that there is a great deal ofscatter through 21 measured data points [see Moldrup et al., 2000b, Figure 3g]. In addition, the range of themeasured air-filled porosity values is narrow, restricted to 0.31< e< 0.4 (Table 1). As we demonstratedbefore [Ghanbarian-Alavijeh and Hunt, 2012], the extracted exponent of the scaling (equation (12)) is mostsensitive to the measured experimental values at low air-filled porosities. Missed data points at low air-filledporosities tend to produce an exponent value significantly less than 2, while introducing greater uncertaintyin the inferred value. This is also the case for sample KyuGL 1 from Moldrup et al. [2003] in which six meas-ured data points are in the range of 0.27–0.4, far away from low air-filled porosity values [Moldrup et al.,2003, Figure 4c] near the percolation threshold.

For sample, Alakawa Gray 200 kPa from Moldrup et al. [2005a] although gas diffusion coefficient was meas-ured also at low air-filled porosities, e.g., 0.09–0.31, the exponent l is still less than 2 (1.667). Note that forother two experiments Tsubaka 200 kPa and Ashurst in the Moldrup et al. [2005a] database, we found theexponent l still nearly 2 (1.897 and 1.860) with R2 values greater than 0.97. However, in two of these sam-ples, there are only five data points, meaning that experimental accuracy may still be an issue.



Table 1. Physical Properties and Fitted Parameters of the Universal Scaling Approach (Equation (12)) for Experiments of SevenDatabases Used in This Studya

Sample Number of Data Points Porosity emin emax Exponent et R2

Moldrup et al. [2000a]—Undisturbed SamplesKootwijk B 18 NA 0.13 0.5 1.994 0.109 0.85Kootwijk A 8 NA 0.24 0.55 2.003 0.132 0.95Oss C 33 NA 0.14 0.48 1.997 0.102 0.95Eijsden C 39 NA 0.13 0.49 1.996 0.089 0.95Belvedere C 18 NA 0.07 0.45 1.999 0.047 0.93Harderbos A 20 NA 0.11 0.59 2.006 0.031 0.94Moldrup et al. [2000b]—Repacked SamplesHjorring 6 0.42* 0.26 0.36 2.007 0.057 0.96Luundgard 9 0.47* 0.31 0.42 1.993 0.001 0.98Lergjerg 1 12 0.47* 0.26 0.42 2.007 0.015 0.92Lergjerg 3 11 0.47* 0.13 0.38 2.002 0.039 0.98Hjorring sand clay 8 0.44* 0.08 0.35 2.007 0.049 0.99Lergjerg 5 11 0.49* 0.13 0.37 1.997 0.066 0.99Miles 21 0.41 0.31 0.40 1.780 0 0.70Sharpsburg 15 0.45 0.08 0.40 1.999 0.018 0.98West Covina 8 NA 0.20 0.46 2.005 0.037 0.99Yolo 8 NA 0.12 0.42 1.992 0.057 0.99Moldrup et al. [2003]—Volcanic SamplesTsumagol 1 7 0.705 0.05 0.29 1.999 0.025 0.99Miura 1 8 0.687 0.07 0.37 1.999 0.019 0.97Kyushu 1 6 0.697 0.08 0.33 2.001 0.053 0.99Tsumagol 2 5 0.745 0.05 0.26 2.000 0.022 0.94Miura 2 8 0.715 0.10 0.44 1.997 0.058 0.97Kyushu 2 6 0.783 0.05 0.16 2.006 0.013 0.98Tsumagol 3 9 0.770 0.20 0.52 1.998 0.059 0.99Miura 3 8 0.819 0.13 0.38 1.995 0.083 0.99Kyushu 3 6 0.802 0.05 0.21 1.993 0.025 0.95Tsumagol 5 6 0.732 0.11 0.24 2.001 0.043 0.97Miura 4 8 0.750 0.13 0.46 1.990 0.087 0.99Kyushu 4 5 0.758 0.19 0.47 1.993 0.079 0.99Tsumagol 7 8 0.756 0.13 0.25 1.997 0.052 0.94Miura 5 8 0.753 0.13 0.46 2.003 0.088 0.97Kyushu 5 6 0.685 0.10 0.32 2.002 0.028 0.99Kounuso GL 6 0.613 0.20 0.34 2.004 0.038 0.90KyuGL 2 6 0.488 0.07 0.18 2.005 0.014 0.97KyuGL 1 6 0.600 0.27 0.40 1.631 0 0.94KyuGL 3 5 0.866 0.04 0.21 2.007 0.043 0.99Moldrup et al. [2004]—Undisturbed SamplesUDP (a) 8 0.535 0.16 0.38 2.000 0.070 0.99UDP (b) 7 0.428 0.11 0.22 2.000 0.025 0.94UDP (c) 8 0.486 0.10 0.27 1.999 0.072 0.99NP (d) 7 0.459 0.04 0.14 1.995 0.018 0.98NP (e) 8 0.555 0.17 0.41 1.999 0.099 0.99NP (f) 7 0.456 0.13 0.26 2.000 0.080 0.99NP (g) 8 0.471 0.04 0.21 1.999 0.017 0.98Latosol 5 0.647 0.34 0.48 1.997 0.140 0.99Heavy disk 5 0.625 0.29 0.44 1.991 0.164 0.97Disk plow 5 0.649 0.34 0.48 1.995 0.194 0.99No plow 5 0.643 0.30 0.45 1.998 0.085 0.99Heavy disk (I) 5 0.638 0.31 0.45 1.999 0.085 0.99Disk plow (m) 5 0.627 0.32 0.44 1.997 0.129 0.99Moldrup et al. [2005a]—Repacked SamplesTsubaka 0 kPa 5 0.756 0.27 0.47 1.997 0.028 0.99Tsubaka 50 kPa 5 0.755 0.30 0.47 1.998 0.017 0.98Tsubaka 200 kPa 5 0.739 0.23 0.45 1.897 0 0.97Forbes loam 0 kPa 8 0.630 0.22 0.42 1.997 0.002 0.97Alakawa Gray 50 kPa 5 0.619 0.19 0.40 1.996 0.031 0.99Alakawa Gray 200 kPa 5 0.539 0.09 0.31 1.667 0 0.99Ashurst 11 0.420 0.14 0.38 1.860 0 0.98Brookings 5 0.580 0.26 0.38 2.001 0.095 0.99West Covina 7 0.500 0.20 0.39 1.997 0.029 0.99Moldrup et al. [2005b]—Intact SamplesField I 0–20 8 0.778 0.18 0.52 2.002 0.016 0.98Field I 20–40 8 0.706 0.08 0.34 1.996 0.042 0.99Field I 40–60 8 0.717 0.07 0.27 2.005 0.044 0.99Field II 0–16 8 0.754 0.20 0.53 1.996 0.092 0.99Field II 16–35 8 0.700 0.09 0.30 2.003 0.063 0.99



We should note that equations (10–12) are valid when diffusion is effective in all percolation clusters, assuggested by Ben-Avraham and Havlin [2000] to be applicable ‘‘to many practical purposes.’’ However, fol-lowing Sahimi [1993a, 1993b], one may expect that only diffusion in sample-spanning cluster contributessignificantly to Dp. Therefore, one should replace the exponent l 5 2 in equations (10–12) by l 2 b 5 1.59(where b 5 0.41 in three dimension), as we pointed out above.

Alternatively, as Sahimi [1994a, 2011] explains natural porous media are not necessarily random and mayhave some correlation. For example, packing of particles may contain only short-range correlations, whileheterogeneous field-scale porous media, e.g., aquifers might be long-ranged correlated which make themdifferent than random media [Sahimi, 1994a]. Sahimi’s results [Sahimi, 1994a] indicated that as long-rangecorrelation (Hurst exponent) increased from 0.5 to 0.98, the value of the exponent l (51.3 in 2-D systems,[Stauffer and Aharony, 1994]) decreased as expected from 1.307 to 0.427 (deduced from l̂ values presentedin Table 1 of Sahimi [1994a] given that m 5 4/3). We also point out that with long-range correlations in themedium itself, flow paths may be less tortuous, causing a diminution of the effective exponent for theconductivity.

Given that 66 out of 71 samples followed universal scaling, and that of the remaining five there were poten-tial experimental issues identifiable in four experiments (either a lack of measured data, or no data near thethreshold), one could also postulate that universal scaling from percolation theory would always beobserved if the data were accurate enough. Although our universal scaling model for diffusion (equation(12)) should be only valid near pc, near has not been defined well. Nonetheless ‘‘near’’ can be roughly esti-mated by p – pc� 1/Z where Z is the coordination number (the number of bonds connected to the samesite) [Sahimi, 1993a]. However, universal scaling laws from percolation theory are valid over a much broaderregion in many systems [Sahimi, 1993a, 2011]. In particular, Ewing and Hunt [2006] examined the universal-ity of electrical conductivity across the full range of saturation, and confirmed the universal behavior of elec-trical conductivity in porous media. Hunt and Ewing [2009] indicated that solute diffusion coefficientfollowed a universal power law with the exponent 2. However, percolation threshold was set equal to zero.Recent results of Ghanbarian-Alavijeh and Hunt [2012] demonstrated that their proposed universalpercolation-based air permeability model was robust and predictive. Furthermore, Ghanbarian et al. [2013]proposed a tortuosity model for saturated and unsaturated media using concepts from percolation theory.Their model compared with both numerical simulations and experiments showed universal behavior over awide range of porosity and saturation values, far away percolation threshold.

In Figure 1, the graphical results are presented for each database. We show the measured relative gas diffu-sion coefficient Dp/D0 versus measured relative air-filled porosity (e 2 et)/(1 2 et) in log-log scale. The expo-nent of 2.006 [Moldrup et al., 2000a], 1.908 [Moldrup et al., 2000b], 1.983 [Moldrup et al., 2003], 2.124[Moldrup et al., 2004], 1.825 [Moldrup et al., 2005a], 1.938 [Moldrup et al., 2005b], and 1.979 [Moldrup et al.,2005c] with high correlation coefficients R2> 0.90 indicate the successful prediction of universal scaling(equation (12)) in the modeling of gas diffusion coefficient in natural porous media.

Figure 2 also presents the measured Dp/D0 versus (e 2 et)/(1 2 et) values, but for all 71 samples from theseven databases used in this study. The exponent 2.014 and the numerical prefactor 1.35 close to theoreti-cal values of 2 and 1, respectively, with correlation coefficient value R2 5 0.95 suggest the practical applica-tion of percolation theory in the modeling of gas diffusion in natural porous media.

Table 1. (continued)

Sample Number of Data Points Porosity emin emax Exponent et R2

Moldrup et al. [2005c]—Intact SamplesDune sand 10 0.44 0.09 0.44 1.998 0.03 0.99Alakawa Gray 5 0.62 0.20 0.40 1.996 0.031 0.99Forbes loam 8 0.63 0.22 0.45 2.000 0.006 0.97Yolo loam 15 0.49 0.12 0.42 2.002 0.051 0.99Hjorring sandy clay loam 8 0.43 0.08 0.35 1.999 0.053 0.98Lerbjer 5 11 0.49 0.13 0.37 2.001 0.063 0.99Tsukuba andisol 0–20 7 NA 0.12 0.36 2.001 0.087 0.99Tsukuba andisol 37–39 7 NA 0.08 0.21 1.999 0.008 0.98Kootwijk C 10 NA 0.17 0.38 2.012 0.151 0.98

aPorosity was calculated from bulk density and particle density data.



Figure 1. Relative gas diffusion coefficient Dp/D0 versus relative air-filled porosity (e 2 et)/(1 2 et) for (a) Moldrup et al. [2000a], (b) Moldrup et al. [2000b], (c) Moldrup et al. [2003], (d)Moldrup et al. [2004], (e) Moldrup et al. [2005a], (f) Moldrup et al. [2005b], and (g) Moldrup et al. [2005c] databases used in this study.



As discussed in section 2, the normaliza-tion process gave an initial estimate of thenumerical prefactor of 1 for the power-lawfunction presented in Figures 1a–1g and2. However, we found a range of 0.9–1.66indicating a discrepancy between ourtheory and use of the normalization inequation (11) to the gas diffusion coeffi-cient in free space (Dp 5 D0 at e 5 1). Wewish to point out that there is no necessityto choose such a normalization constantthat cannot be observed in the context ofthe limit of a sequence of attainablemedia. In particular, measurements withair-filled porosity e near 1 are almostimpossible in natural media. Although wecould thus suggest using a normalization

constant obtainable from direct measurements on the same medium (say, evaluated at the largest air-filledporosity measured), alternatively we can use theoretical arguments to understand the discrepancy. Weshow that obtaining a numerical prefactor larger than 1 is due to the implied assumption of the validity ofpercolation theory in a range of media, for which it is not suited. In particular, at very large distances fromthe percolation threshold (e � 1 >> et), effective medium theories are known to perform better. Conse-quently, the normalization constant at e 5 1 can only be incorporated into our theoretical description ifeffective medium approximations are employed at such large air-filled porosity values, but percolationtheory closer to the threshold. In fact, this problem was already addressed 40 years ago [Kirkpatrick, 1973].

The lowest order effective medium approximations of Kirkpatrick [1973] and Keffer et al. [1996], derived fornetwork representations of a medium, indicate that at large p values (e.g., p> 0.75 deduced from Kirkpatrick[1973, Figure 6]) the effective conductance g scales as

g51-1-p

1-2=Z(13)

instead of (p 2 pc)l in which l is 1.3 and 2 in two and three dimensions, respectively. Z in equation (13) isthe coordination number. Below, we adapt this expression to continuum percolation by substituting e for p.In 2-D bond percolation, it is known that 2/Z is a good approximation to the percolation threshold, pc [Vys-sotsky et al., 1961; Shante and Kirkpatrick, 1971], which would then be et.

Following Kirkpatrick [1973] and Keffer et al. [1996], at large p values the normalized gas diffusion coefficientwould scale linearly as [Sahimi, 1994b]

Dp

D05

p-2=Z1-2=Z

� p-pc

1-pc5

e-et

1-et(14)

which means at p 5 1 (or equivalently e 5 1), the ratio Dp/D0 would be 1 as well. Because of the known pro-portionality between pc and Z21 in all dimensions [Hunt and Ewing, 2009], we use the same substitution ofet for 2/Z in 3-D as well, even though the actual value of pc is typically somewhat smaller, 2/Z � 4/3 pc

[Shante and Kirkpatrick, 1971].

Direct evaluation of this hypothesis (equation (14)) for gas diffusion coefficients in natural porous mediaused in this study is not possible, since no measurements are available in the range 0.75< e< 1 (emax< 0.59presented in Table 1). However, one can renormalize equation (11) using the concept of this linear scaling.For this purpose, we assume that the universal scaling from percolation theory (equation (11)) and the linearscaling from effective medium theory are valid in the ranges et� e< ex and ex� e� 1, respectively, where ex

is the crossover point in which gas diffusion behavior changes. The schematic interpretation is presented inFigure 3 in which we set et 5 0.05 and ex 5 0.75 [Kirkpatrick, 1973].

Figure 2. Relative gas diffusion coefficient Dp/D0 versus relative air-filledporosity (e 2 et)/(1 2 et) for all seven databases (including 632 data points)used in this study.



Setting equations (11) and (14) (within thecontinuum percolation representation) at ex

equal to each other results in the followinggas diffusion model (Figure 3)

Dp

D05

1-et

ex -et

e-et

1-et

� �2

; et � e<ex (15a)

Dp

D05

e-et

1-et; ex � e � 1 (15b)

The numerical prefactor (1 2 et)/(ex 2 et) inequation (15a) justifies the experimental val-ues that we found by fitting our universalscaling to the measured gas diffusion data(reported in Figures 1 and 2). For example, ifone sets ex 5 0.75 [Kirkpatrick, 1973] and theaverage value of et 5 0.053 found over all 71experiments, then one would have (1 2 et)/(ex 2 et) 5 1.36 which is not different than1.35 presented in Figure 2.

We should point out that Zhou et al. [1997]also combined the effective medium approach with percolation theory in their study of electrical conductiv-ity in reservoir rocks, but their power law is not universal, meaning that they allow the exponent l tochange from sample to sample. The potential relevance of nonuniversal exponents to permeability or con-duction properties has actually been discussed for decades [e.g., Kogut and Straley, 1979; Berkowitz and Bal-berg, 1992, 1993]. But the kinds of percolation models cited in order to justify nonuniversal exponents (theoverlapping spherical pore or particle models of Bunde et al. [1986], Feng et al. [1987], and Wagner and Bal-berg [1987], also reviewed in Berkowitz and Balberg [1993]), were developed for single-phase flow, and donot apply to nonwetting phase flow under unsaturated conditions. For single-phase flow, in either saturatedor completely dry media, all pores (from the smallest to the largest) contribute to fluid flow. In the case ofpartial saturation one must consider the wetting and nonwetting fluids from different perspectives. At inter-mediate saturations, the wetting phase (e.g., water) flow is restricted to the smallest pores (or equivalentlyconductances), while the air phase is found in the largest pores. Thus, the limiting behavior of the conduct-ance distribution at zero pore size, which is known to be the origin of nonuniversality [Hunt, 2004b], is irrele-vant to properties of the nonwetting phase, although it could be relevant to the wetting phase. Since in thiswork our focus is the nonwetting phase, we do not discuss the conditions that determine the relevance ofnonuniversal exponents of percolation theory to the wetting phase in detail.

The approach of Zhou et al. [1997] could be justified by the following argument. Since the universal conduc-tion exponent of percolation theory holds only in a given range of air-filled porosity e values near thethreshold et, but the behavior crosses over to a linear dependence at values of e approaching one, onecould choose a single (nonuniversal) exponent between 1 and 2 to describe conduction through the entirerange of e values. We propose that defining a crossover ex, consistent with theory, is also superior choice forpractical reasons. In particular, it has been shown that et values obtained by comparing with universal scal-ing of the air permeability could be reasonably predicted from examination of the soil water retention curve[Ghanbarian-Alavijeh and Hunt, 2012]. If the crossover to linear behavior (at larger e values) is allowed toinfluence analysis by permitting the choice of an arbitrary exponent, this would allow effects from the diffu-sion process at large e values to influence any extracted value of the threshold (including 0), but the thresh-old is a measure of the minimum air content necessary to generate a connected air-filled path.

We should point out that varying the air-filled porosity threshold rather than the exponent leads to morereasonable values of the numerical prefactor. Consider the implications of fixing et at zero and allowing l tovary vis-�a-vis fixing l at 2 and allowing et to vary. In order to study the effect of the value of threshold ongas diffusion, we compare two air-filled porosity threshold values in the fitting process for the undisturbedsample Oss C in the Moldrup et al. [2000a] database. In Figure 4a, where we simply set et 5 0, the power-law

Figure 3. A schematic interpretation of gas diffusion, including both theuniversal quadratic and linear scaling results from percolation theory andthe effective medium approach, respectively. Note that, we set et 5 0.05,the average experimental value, and stipulated that ex 5 0.75 [Kirkpatrick,1973]. Although numerical simulations confirmed the crossover to linearscaling, of course they did not show an actual kink, but rather, a gradualchange from a quadratic to a linear dependence on p 2 pc in the vicinityof ex 5 0.75.



exponent (3.364) and the numerical prefactor (3.782) were found greater than 2 and 1, respectively, withR2 5 0.96. In contrast, increasing the threshold value to et 5 0.102, reduces the exponent and the numericalprefactor values, to 1.997 and 1.53, respectively, with goodness of fit R2 5 0.95 (Figure 4b). Note that theobtained numerical prefactor 1.53 has a physical interpretation in our theoretical framework, consistent

Figure 4. Comparison of two different values of air-filled porosity threshold (a) et 5 0, and (b) et 5 0.102 in the fitting process for the undis-turbed sample Oss C in the Moldrup et al. [2000a] database.

Figure 5. Predicted gas diffusion versus measured one using (a) universal scaling model (equation (15a)) with et 5 0.1/ [Hunt, 2004a] andex 5 0.75 [Kirkpatrick, 1973], (b) Buckingham’s model (equation (3)) with m 5 2, and (c) Millington and Quirk’s model (equation (5)) for all71 experiments used in this study. Note that the largest measured air-filled porosity was applied where porosity value was unknown.



with a crossover air-filled porosity value of ex 5 0.69 (using et 5 0.102) which is not greatly different from0.75, as suggested above.

Since the value of the threshold air-filled porosity is a key input into universal scaling predictions, it would,in analogy to our study of air permeability [Ghanbarian-Alavijeh and Hunt, 2012], be important to find anindependent source of this parameter. In that study, we were able to estimate et from the soil water reten-tion curve. However, accurate soil water retention data are not available for the present databases. Thus, weused a previous estimate of the air-filled porosity threshold as et 5 0.1/ [Hunt, 2004a], and set ex 5 0.75[Kirkpatrick, 1973]. Then, for comparison we predict gas diffusion using two other methods: the Buckingham(equation (3)), and Millington and Quirk [1961] (equation (5)). In all these three methods, gas diffusion is pre-dicted solely from the soil porosity value and the relative air-filled porosity. Results are presented in Figure5. As can be seen, the universal scaling approach predicts gas diffusion slightly better than the Buckinghammodel and considerably more accurately than the Millington and Quirk method. Although the root-mean-square error (RMSE) value calculated for the universal scaling method (Figure 5a) is not much smaller thanthat of the Buckingham model (Figure 5b), the analysis reveals greater systematic error in both Buckinghamand Millington and Quirk models. In particular, in contrast to the universal scaling predictions, it can beseen that the data in Figures 5b and 5c do not follow the 1:1 line as well as for the universal scaling, particu-larly at higher air-filled porosities. That the Buckingham model (with m 5 2) may nevertheless seem to be alegitimate alternative to universal scaling is not really surprising since it is actually a special case of the per-colation prediction.

Further study is warranted to determine the value of the threshold of the air-filled porosity from, say,detailed 3-D image analysis. This would remove the option to treat it as a fitting parameter, and allow aclearer determination of the potential relevance of any power different from 2. It thus also makes clearwhether diffusion is restricted to the sample-spanning cluster as Sahimi [1993b] assumes or occurs in allclusters without restriction as Ben-Avraham and Havlin [2000] suggest.

5. Conclusion

We developed a model for gas diffusion in porous media based on cluster topology concepts from percola-tion theory. Our model (equation (12)), derived from fundamental theory of Havlin et al. [1983] is a powerlaw in the air-filled porosity (less a threshold value) with an exponent of 2.0 whose parameters are physicallymeaningful. Comparison with a wide range of experiments indicated the developed approach is robust. Inparticular, 66 out of 71 samples followed universal scaling with the exponent 2 from percolation theory. Inthe remaining five samples, the deviation from universal scaling may have been due to insufficient: (1) num-ber of measurements or (2) data points near the threshold. Our results, in conjunction with those of Ghan-barian-Alavijeh and Hunt [2012], confirm the universal scaling of gas transport (e.g., diffusion andpermeability) in porous media. In this study, we also applied the linear scaling of the diffusion constantfrom effective medium theory [Kirkpatrick, 1973] and Keffer et al. [1996], appropriate at large air-filled poros-ities (specifically e> 0.75 [Kirkpatrick, 1973]), which allowed more accurate calculation of the numerical pre-factor, and thus a more accurate prediction of experimental measurements.

ReferencesBen-Avraham, D., and S. Havlin (2000), Diffusion and Reactions in Fractals and Disordered Systems, Cambridge Univ. Press, New York.Berkowitz, B., and I. Balberg (1992), Percolation approach to the problem of hydraulic conductivity in porous media, Transp. Porous Media,

9, 275–286.Berkowitz, B., and I. Balberg (1993), Percolation theory and its application to groundwater hydrology, Water Resour. Res., 29, 775–794.Buckingham, E. (1904), Contributions to our knowledge of the aeration of soils, Bur. Soils Bull., vol. 25, U.S. Gov. Print. Off., Washington, D. C.Bunde, A., H. Harder, and S. Havlin (1986), Nonuniversality of diffusion exponents in percolation systems, Phys. Rev. B, 34, 3540–3542.Burdine, N. T. (1953), Relative permeability calculations from pore-size distribution data, Trans. Am. Inst. Min. Metall. Pet. Eng., 198, 71–77.Campbell, G. S. (1974), A simple method for determining unsaturated conductivity from moisture retention data, Soil Sci., 117, 311–314,

doi:10.1097/00010694-197406000-00001.Chamindu Deepagoda, T. K. K., P. Moldrup, P. Schj�nning, L. W. de Jonge, K. Kawamoto, and T. Komatsu (2011a), Density-corrected models

for gas diffusivity and air permeability in unsaturated soil, Vadose Zone J., 10, 226–238, doi:10.2136/vzj2009.0137.Chamindu Deepagoda, T. K. K., P. Moldrup, P. Schj�nning, K. Kawamoto, T. Komatsu, and L. W. de Jonge (2011b), Generalized density-

corrected model for gas diffusivity in variably saturated soils, Soil Sci. Soc. Am. J., 75, 1315–1329.Chamindu Deepagoda, T. K. K., P. Moldrup, P. Schj�nning, K. Kawamoto, T. Komatsu, and L. W. de Jonge (2012), Variable pore connectivity

model linking gas diffusivity and air-phase tortuosity to soil matric potential, Vadose Zone J., 11(1), doi:10.2136/vzj2011.0096.

AcknowledgementB.G. thanks Daniel Ben-Avraham,Department of Physics ClarksonUniversity, for his fruitful comments.



Currie, J. A. (1960), Gaseous diffusion in porous media: 2. Dry granular materials, Br. J. Appl. Phys., 11, 314–317, doi:10.1088/0508-3443/11/8/302.

Currie, J. A. (1961), Gaseous diffusion in porous media: 3. Wet granular materials, Br. J. Appl. Phys., 12, 275–281, doi:10.1088/0508-3443/12/6/303.

Currie, J. A., and D. A. Rose (1985), Gas diffusion in structured materials: The effect of tri-modal pore size distribution, J. Soil Sci., 36, 487–493.

Currie, J. A. (1984), Gas diffusion through soil crumbs: The effects of compaction and wetting, J. Soil Science, 35, 1–10.Einstein, A. (1905), On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat,

Ann. Phys., 17, 549–560.Ewing, R., and A. Hunt (2006), Dependence of the electrical conductivity on saturation in real porous media, Vadose Zone J., 5(2), 731–741,

doi:10.2136/vzj2005.0107.Ewing, R. P., and R. Horton (2003), Scaling in diffusive transport, in Scaling Methods in Soil Physics, edited by Y. Pachepsky et al., pp. 49–61,

CRC Press, Boca Raton, Fla.Feder, J. (1988), Fractals, Plenum, New York.Feng, S., B. I. Halperin, and P. N. Sen (1987), Transport properties of continuum systems near the percolation threshold, Phys. Rev. B, 35,

197–214.Gefen, Y., and A. Aharony (1983), Anomalous diffusion on percolating clusters, Phys. Rev. Lett., 50, 77–80.Ghanbarian, B., A. G. Hunt, M. Sahimi, R. P. Ewing, and T. E. Skinner (2013), Percolation theory generates a physically based description of

tortuosity in saturated and unsaturated porous media, Soil Sci. Soc. Am. J., 77, 1920–1929.Ghanbarian-Alavijeh, B., and A. G. Hunt (2012), Comparison of the predictions of universal scaling of the saturation dependence of the air

permeability with experiment, Water Resour. Res., 48, W08513, doi:10.1029/2011WR011758.Gingold, D. B., and C. J. Lobb (1990), Percolative conduction in three dimensions, Phys. Rev. B, 42, 8220–8224.Hamamoto, S., P. Moldrup, K. Kawamoto, L. W. de Jonge, P. Schonning, and T. Komatsu (2011), Two-region extended Archie’s law model

for soil air permeability and gas diffusivity, Soil Sci. Soc. Am. J., 75, 795–806.Harris, A. B., and A. Aharony (1987), Anomalous diffusion, superlocaliztion and hopping conductivity on fractal media, Europhys. Lett., 4,

1355–1360.Havlin, S., and D. Ben-Avraham (1987), Diffusion in disordered media, Adv. Phys., 36, 695–798.Havlin, S., D. Ben-Avraham, and H. Sompolinsky (1983), Scaling behavior of diffusion on percolation clusters, Phys. Rev. A Gen. Phys., 27,

1730–1733.Hunt, A. G. (2001), Applications of percolation theory to porous media with distributed local conductances, Adv. Water Resour., 24(3–4),

279–307.Hunt, A. G. (2004a), Continuum percolation theory for water retention and hydraulic conductivity of fractal soils: Estimation of the critical

volume fraction for percolation, Adv. Water Resour., 27, 175–183.Hunt, A. G. (2004b), Percolative transport and fractal porous media, Chaos Solitons Fractals, 19, 309–325.Hunt, A. G. (2005), Percolation Theory for Flow in Porous Media, Lect. Notes Phys., vol. 674, Springer, Berlin.Hunt, A. G., and R. P. Ewing (2003), On the vanishing of solute diffusion in porous media at a threshold moisture content, Soil Sci. Soc. Am.

J., 67, 1701–1702.Hunt, A. G., and R. P. Ewing (2009), Percolation Theory for Flow in Porous Media, Lect. Notes Phys., 2nd ed., vol. 771, Springer, Berlin.Iiyama, I., and S. Hasegawa (2005), Gas diffusion coefficient of undisturbed peat soils, Soil Sci. Plant Nutr., 51, 431–435.Jin, Y., and W. A. Jury (1996), Characterizing the dependence of gas diffusion coefficient on soil properties, Soil Sci. Soc. Am. J., 60, 66–71,

doi:10.2136/sssaj1996.03615995006000010012x.Kawamoto, K., P. Moldrup, P. Schj�nning, B. V. Iversen, D. E. Rolston, and T. Komatsu (2006), Gas transport parameters in the vadose zone:

Gas diffusivity in field and lysimeter soil profiles, Vadose Zone J., 5, 1194–1204, doi:10.2136/vzj2006.0014.Kirkpatrick, S. (1973), Percolation and conduction, Rev. Mod. Phys., 45, 574–588, doi:10.1103/RevModPhys.45.574.Keffer, D., A. V. McCormick, and H. T. Davis (1996), Diffusion and percolation on zeolite sorption lattices, J. Phys. Chem., 100, 967–973.Knackstedt, M. A., B. W. Ninham, and M. Monduzzi (1995), Diffusion in model disordered media, Phys. Rev. Lett., 75, 653–656.Kogut, P. M., and J. Straley (1979), Distribution-induced non-universality of the percolation conductivity exponents, J. Phys. C Solid State

Phys., 12, 2151–2159.Kristensen, A. H., A. Thorbj�rn, M. P. Jensen, M. Pedersen, and P. Moldrup (2010), Gas-phase diffusivity and tortuosity of structured soils, J.

Contam. Hydrol., 115, 26–33.Liu, J., and K. Regenauer-Lieb (2011), Application of percolation theory to microtomography of structured media: Percolation threshold,

critical exponents, and upscaling, Phys. Rev. E, 83, 016106, doi:10.1103/PhysRevE.83.016106.Millington, R. J., and J. P. Quirk (1961), Permeability of porous solids, Trans. Faraday Soc., 57, 1200–1207.Moldrup, P., T. Olesen, D. E. Rolston, and T. Yamaguchi (1997), Modeling diffusion and reaction in soils: VII. Predicting gas and ion diffusivity

in undisturbed and sieved soils, Soil Sci., 162, 632–640.Moldrup, P., T. Olesen, T. Yamaguchi, P. Schj�nning, and D. E. Rolston (1999), Modeling diffusion and reaction in soils: IX. The Buckingham-

Burdine-Campbell equation for gas diffusivity in undisturbed soil, Soil Sci., 164, 542–551, doi:10.1097/00010694-199908000-00002.Moldrup, P., T. Olesen, P. Schj�nning, T. Yamaguchi, and D. E. Rolston (2000a), Predicting the gas diffusion coefficient in undisturbed soil

from soil water characteristics, Soil Sci. Soc. Am. J., 64, 94–100.Moldrup, P., T. Olesen, J. Gamst, P. Schj�nning, T. Yamaguchi, and D. E. Rolston (2000b), Predicting the gas diffusion coefficient in repacked

soil: Water-induced linear reduction model, Soil Sci. Soc. Am. J., 64, 1588–1594.Moldrup, P., T. Olesen, T. Komatsu, P. Schj�nning, and D. E. Rolston (2001), Tortuosity, diffusivity, and permeability in the soil liquid and gas-

eous phases, Soil Sci. Soc. Am. J., 65, 613–623.Moldrup, P., S. Yoshikawa, T. Olesen, T. Komatsu, and D. E. Rolston (2003), Gas diffusivity in undisturbed volcanic ash soils: Test of soil-

water-characteristic-based prediction models, Soil Sci. Soc. Am. J., 67, 41–51.Moldrup, P., T. Olesen, S. Yoshikawa, T. Komatsu, and D. E. Rolston (2004), Three-porosity model for predicting the gas diffusion coefficient

in undisturbed soil, Soil Sci. Soc. Am. J., 68, 750–759.Moldrup, P., T. Olesen, S. Yoshikawa, T. Komatsu, and D. E. Rolston (2005a), Predictive-descriptive models for gas and solute diffusion coeffi-

cients in variably saturated porous media coupled to pore-size distribution: I. Gas diffusivity in repacked soil, Soil Sci., 170, 843–853.Moldrup, P., T. Olesen, S. Yoshikawa, T. Komatsu, and D. E. Rolston (2005b), Predictive-descriptive models for gas and solute diffusion coef-

ficients in variably saturated porous media coupled to pore-size distribution: II. Gas diffusivity in undisturbed soil, Soil Sci., 170, 854–866.



Moldrup, P., T. Olesen, S. Yoshikawa, T. Komatsu, A. M. McDonald, and D. E. Rolston (2005c), Predictive-descriptive models for gas and sol-ute diffusion coefficients in variably saturated porous media coupled to pore-size distribution: III. Inactive pore space interpretations ofgas diffusivity, Soil Sci., 170, 867–880.

Moldrup, P., T. Olesen, H. Blendstrup, T. Komatsu, L. W. de Jonge, and D. E. Rolston (2007), Predictive-descriptive models for gas and solutediffusion coefficients in variably saturated porous media coupled to pore-size distribution: IV. Solute diffusivity and the liquid phaseimpedance factor, Soil Sci., 172, 741–750.

Montaron, B. (2009), Connectivity theory—A new approach to modeling non-Archie rocks, Petrophysics, 50, 102–115.Naveed, M., et al. (2013), Correlating gas transport parameters and X-Ray computed tomography measurements in porous media, Soil Sci.,

178, 60–68.Normand, J. M., and H. J. Herrmann (1996), Precise determination of the conductivity exponent of 3D percolation using ‘‘PERCOLA’’, Int. J.

Modern Phys. C, 6, 813–817.Penman, H. L. (1940), Gas and vapor movements in soil: The diffusion of vapors through porous solids, J. Agric. Sci., 30, 437–462, doi:

10.1017/S0021859600048164.Resurreccion, A. C., K. Kawamoto, T. Komatsu, P. Moldrup, N. Ozaki, and D. E. Rolston (2007), Gas transport parameters along field transects

of a volcanic ash soil, Soil Sci., 172, 3–16.Sahimi, M. (1993a), Fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media, Transp. Porous

Media, 13, 3–40.Sahimi, M. (1993b), Flow phenomena in rocks—From continuum models to fractals, percolation, cellular automata, and simulated anneal-

ing, Rev. Mod. Phys., 65(4), 1393–1534.Sahimi, M. (1994a), Long-range correlated percolation and flow and transport in heterogeneous porous media, J. Phys. I, 4, 1263–1268.Sahimi, M. (1994b), Applications of Percolation Theory, 258 pp., Taylor and Francis, London.Sahimi, M. (2011), Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches, 709 pp., Wiley-

VCH, Weinheim, Germany, doi:10.1002/9783527636693.Sahimi, M. (2012), Dispersion in porous media, continuous-time random walks, and percolation, Phys. Rev. E, 85, 016316.Sallam, A., W. A. Jury, and J. Letey (1984), Measurement of the gas diffusion coefficient under relatively low air-filled porosity, Soil Sci. Soc.

Am. J., 48, 3–6.Shante, V. K. S., and S. Kirckpatrick (1971), An introduction to percolation theory, Adv. Phys., 20, 325–357.Stauffer, D., and A. Aharony (1994), Introduction to Percolation Theory, 2nd ed., Taylor and Francis, London.Torquato, S. (2002), Random Heterogeneous Materials: Microstructure and Macroscopic Properties, 701 pp., Springer, Berlin.Troeh, F. R., J. D. Jabro, and D. Kirkham (1982), Gaseous diffusion equations for porous materials, Geoderma, 27, 239–253.Tuli, A., and J. W. Hopmans (2004), Effect of degree of fluid saturation on transport coefficients in disturbed soils, Eur. J. Soil Sci., 55(1), 147–

164, doi:10.1046/j.1365-2389.2003.00551.x.van Bavel, C. H. M. (1952), Gaseous diffusion and porosity in porous media, Soil Sci., 73, 91–104.Vyssotsky, V. A., S. B. Gordon, H. L. Frisch, and J. M. Hammersley (1961), Critical percolation probabilities (bond problem), Phys. Rev., 123,

1566–1567.Wagner, N., and I. Balberg (1987), Anomalous diffusion and continuum percolation, J. Stat. Phys., 49, 369–382.Werner, D., P. Grathwohl, and P. Hohener (2004), Review of field methods for the determination of the tortuosity and effective gas-phase

diffusivity in the vadose zone, Vadose Zone J., 3, 1240–1248.Zhou, D., S. Arbabi, and E. H. Stenby (1997), A percolation study of wettability effect on the electrical properties of reservoir rocks, Transp.

Porous Media, 29, 85–98.



universal scaling of gas diffusion in porous media

Documents