Published online 29 June 2007
Published in Soil Sci Soc Am J 71:1418-1419 (2007)
DOI: 10.2136/sssaj2007.0152l
© 2007 Soil Science Society of America
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COMMENTS & LETTERS TO THE EDITOR
Comments on "Fractal Fragmentation, Soil Porosity, and Soil Water Properties: I. Theory"
A. G. Hunt*
Dep. of Physics and Dep. of Earth and Environmental Sciences, Wright State Univ., Dayton, OH 45435
* Corresponding author (allen.hunt{at}wright.edu).
A recent comment (Yu, 2007) and response (Sposito, 2007) evaluate the relative merits of the fractal model of porous media proposed by Rieu and Sposito (1991), as opposed to that of Katz and Thompson (1985), or related work by Nigmatullin et al. (1992). The chief contrast alleged lies in the two results for the porosity,
 | [1] |
and
 | [2] |
In these two formulas 
is the porosity, D is the fractal dimensionality, and r0 and rm are minimum and maximum radii, respectively [note the difference from the notation of Rieu and Sposito (1991)]. Whether pore or solid space is meant depends on the author and the context. On the face of it this difference appears troublesome. The response to the comment (Sposito, 2007) notes that the Rieu and Sposito (1991) model was developed to describe soil aggregates and the tendency for the porosity of aggregates to increase with increasing aggregate size. But this is unnecessarily restrictive, and the model can be applied to the textural pore space, and to explain associated water retention curves (Hunt and Gee, 2002). In this case it is possible to use Eq. [1] and [2] to describe the same object.
Hunt (2005a) showed that if the Katz and Thompson (1985) and Thompson et al. (1987) interpretation involves solid structures (describing the original images of structures in the solid phase) provided that the Rieu and Sposito (1991) interpretation involves pore space, the results in Eq. [1] and [2] are precisely compatible. In order for this assertion to be true, in Eq. [1] D must refer to the solid space, whereas in Eq. [2] D must refer to the pore space. If particle and pore radii are proportional, as is often assumed (Arya and Paris, 1981; Gvirtzman and Roberts, 1991) for example, then r0 and rm can refer in either formula to either particle or pore radii, since the ratio r0/rm would be the same in either case. If not, then in Eq. [1] they must refer to particle radii and in Eq. [2] to pore radii.
Katz and Thompson (1985), however, make the comment that D is the same for both the solid portion of the medium and the pore space, a comment which is not justifiable. Note that, from Eq. [1] and [2] D of the pore space will usually be larger than D for the solid medium, provided that r0/rm is the same for both particles and pores and that the porosity is less than half, which is typically true though with many exceptions.
That the Rieu and Sposito (1991) result is valid for the pore space and that Eq. [1] cannot be valid if D refers to the pore space can be relatively easily seen by summing a finite geometric series (Hunt, 2006), as long as the final term in the series represents the solid medium. The successive terms involve two ratios: (1) a ratio, p < 1 of the size of pores in the n+1th fractal iteration to those in the nth iteration, and (2) the ratio, q > 1 of the number of pores in successive iterations. The fractal dimensionality, D, is determined by the method of Mandelbrot (1983) in terms of q and p. Rieu and Sposito (1991) showed that the probability that a given pore has radius r and is proportional to r–D. However, the equivalence of Eq. [1] and [2] is much more easily established with a continuum model (Hunt and Gee, 2002; Hunt, 2005a), in which the probability density function (pdf) that a given pore has radius between r and r + dr is proportional to r–D-1. Then the porosity is expressed as an integral over the product of r3 and r–D-1, for which the fractal dimensionality clearly refers to the pores. Evaluation of this definite integral yields the difference of rm3-D and r03-D, corresponding to the upper and lower limits of the integration, respectively. Thus the porosity must correspond to the difference of two terms, and Eq. [2] results immediately when the normalization factor is chosen proportional to 1/rm3-D. It is impossible to derive Eq. [1] directly from consideration of the pore space, but it can be obtained if one uses an analogous process for the solid part of the medium and a fractal dimensionality appropriate for the solid portion. Then one obtains 1– as the difference of two terms, and thus a formula for that relates the porosity to a single term. With suitable normalization, Eq. [1] is generated, but with D referring to the solid portion of the medium. More details can be supplied, but they may not make the results simpler and the interested reader is referred to Hunt (2005a, 2006).
An additional check on these results is made possible by analyzing the hydraulic conductivity. In this context it is important that the geometric series for the porosity satisfies exactly the criteria for the definition of a fractal dimensionality. Thus, in the Rieu and Sposito (1991) model, one can hold D constant while taking the limit 
1 by allowing r0/rm0. In such a case the pore-size distribution follows a power law to arbitrarily small pore sizes, and percolation theory (Stauffer and Aharony, 1994) yields a hydraulic conductivity and electrical conductivity with non-universal exponents (Balberg, 1987). The result for the hydraulic conductivity as a function of moisture content, 
, and D for the pore space is (Hunt, 2005a, 2005b),
 | [3] |
where t is a threshold moisture content for percolation. Note that this percolation theoretical result in terms of a related power, 
, was quoted by Berkowitz and Balberg (1993) in their review. Hunt and Gee (2002) and Hunt (2005a, 2005b) showed that when applied to the Rieu and Sposito (1991) model, critical path analysis leads to the following result for the hydraulic conductivity,
 | [4] |
Note that the original derivation of Eq. [4], which was based on the scaling of the "bottleneck" or critical resistance (Hunt and Gee, 2002) found the exponent 3/(3-D), whereas using the harmonic mean conductance (average resistance) along the quasi-one-dimensional percolation path, as in Balberg (1987) yields D/(3-D). In the case that 1, i.e., r00, Eq. [4] yields Eq. [3] (Hunt, 2005b). Clearly use of Eq. [1] with the assumption that D refers to the pore space would not allow Eq. [4] to be consistent with Eq. [3]. The importance of this argument lies in the question of how the porosity relates to the fractal dimensionality.
If D refers to the pore space, then the limit D3, for any finite smallest pore size, does indeed yield = 0, but if D refers to the solid portion of the medium, then the same limit yields = 1. The distinction between D for the pore and the particle spaces is not particularly important at typical soil porosities of about 0.4, but at porosities of 0.05, common in rocks, it is. Making the assumption that the value of D for the pore space is that value obtained from visual observations of the solid medium will lead to a value of the porosity of 0.95, clearly unacceptable. What this limit D3 really means, however, is that if is to remain finite, the ratio r0/rm must be zero. And indeed, what is typically found (Hunt and Gee, 2002) is that D values near 3 are associated primarily with very wide ranges of pore sizes, though to a lesser extent, also with a small value of . Thus, it is easier to relate the limit r0/rm0 to a fractal process than the limit D3, which can be associated either with very small porosity or a wide range of pore sizes or both.
Finally, at least three independent studies (Filgueira et al., 1999; Bird et al., 2000; Hunt and Gee, 2002) have shown that fractal analysis of particle-size data makes application of the Rieu and Sposito (1991) model to water retention curves predictive. It must be concluded that the criticism of the Rieu and Sposito (1991) model is not warranted. But it is also true that, if the assumption made by Katz and Thompson (1985), that the fractal dimensionality of the solid portion of the medium is equal to that of the pore space, is dropped, both models can be used equivalently. And in fact Hunt and Gee (2002) implicitly did this in calculating a specific surface area from the particle-size data and comparing with experimental values (Moldrup et al., 2001) of a threshold moisture content expressed in terms of the specific surface area.
Received for publication April 19, 2007.
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