Comment on “Spatial correlation of permeability in cross-stratified sediment with hierarchical architecture” by Robert W. Ritzi, Zhenxue Dai, David F. Dominic, and Yoram N. Rubin


[1] Following their analysis of permeability variations in cross-stratified sediments with hierarchical architecture, Ritzi et al. [2004, paragraph 63] conclude that “Davis et al. [1997] and Di Federico and Neuman [1997] give fractal scaling models for the semivariogram, which do not represent cross-semivariograms between units i and j with ij. Neuman [2003, equation 53] also assumes that cross-semivariograms can be ignored. The application in this paper shows the importance of the cross-transition terms in certain geologic settings.”

[2] Ritzi et al. [2004] are correct that the fractal scaling model of Di Federico and Neuman [1997] does not include cross-semivariograms between units. They are, however, incorrect in stating that this, and equation (53) of Neuman [2003], represent an assumption (that cross-semivariograms can be ignored). Quite the contrary: Neuman's [2003] development starts from his equation (47) which explicitly includes both cross transitions between units and cross correlations between corresponding log permeabilities; none are ignored. Next, Neuman considers a special case represented by his (50), which in turn leads directly and rigorously to his equation (53). The latter ultimately leads to the fractal scaling model of Di Federico and Neuman [1997]. Neuman's [2003] analysis thus shows in a rigorous manner that, rather than disregarding cross-semivariograms between units as implied erroneously by Ritzi et al. [2004], the fractal model of Di Federico and Neuman [1997] and Neuman's (53) do not include them simply because they drop out formally in the special case represented by this model. The relevant question therefore is: under what circumstances does the fractal model of Di Federico and Neuman [1997] correspond to the real world?

[3] As explained by Neuman [2003], his (50) represents a stretching of the categories (geologic facies, soil or rock types) in a way that causes them to be found in fixed proportions within a representative sampling volume centered about any mathematical point throughout space. In this sense, the categories overlap; the idea is analogous to the well-known dual-continuum concept in which two categories, most commonly fractures and porous blocks, are considered to overlap. Neuman clearly states (in full agreement with Ritzi et al. [2004]) that in reality the categories do not overlap, a situation clearly represented by his (47). His special case, embodied in (50) and (53), becomes relevant when an attribute (such as permeability) is sampled jointly over all overlapping categories, as in the case of dual continua. It is important to recognize that Neuman's [2003] concept of overlapping categories (like that of dual continua) is not any less physical than that of a single porous continuum, which represents a useful (and widely accepted) abstraction of nonoverlapping solid and void spaces.

[4] The presence of each category within each representative sampling volume throughout space is a necessary (but insufficient) condition for any attribute to exhibit fractal behavior: fractal objects fill space so as to be found everywhere on each scale. That some hydrogeologic attributes (most notably log permeability) do in fact tend to exhibit fractal behavior, especially when sampled jointly in varied media over a wide range of scales, has been amply documented in the literature (see recent reviews by Neuman and Di Federico [2003] and Molz et al. [2004]). Neuman's [2003] theory provides a hydrogeologic rationale for such behavior.

[5] The finding by Ritzi et al. [2004] that cross-transition terms are important in the synthetic geologic setting they consider does not contradict in any way either Di Federico and Neuman [1997] or Neuman [2003]. All of the above authors agree that if an attribute is sampled over volumes which are small compared to any individual unit (or facies), as are the “points” in the variograms of Ritzi et al., then cross transitions between units and cross correlations between the corresponding attributes must be considered, as in equation (47) of Neuman. Neuman however makes clear that when an attribute is sampled on scales that are larger, over volumes containing several categories (fractures and porous blocks, various units or facies), the attribute may (but need not) exhibit fractal behavior (more precisely, behave as fractional Brownian motion, or FBM); if it does, its behavior can be validly represented by the theories of Di Federico and Neuman [1997] and Neuman [2003].

[6] The difference between Ritzi et al. [2004] and Neuman [2003] thus lies not in the types of subsurface architecture they consider but in the way they sample this architecture: up to and including his equation (47), Neuman considers a sampling volume that is much smaller than the characteristic length of any category, as do Ritzi et al. throughout their paper. This means that both Ritzi et al. and Neuman sample each category individually. However, starting with his equation (50) Neuman, in contrast to Ritzi et al., introduces (through the artifact of stretching) a mathematical equivalent to the sampling of a much larger volume, within which all categories are found in fixed proportions as illustrated schematically in his Figure 1. This is equivalent to sampling the categories not individually but as if they overlapped, rendering the cross-transition terms in Neuman's (47) as well as in Ritzi et al. irrelevant. The latter authors have overlooked this crucial distinction between the two sampling schemes.