Universal scaling of hydraulic conductivities and dispersivities in geologic media

Abstract

Neuman [1990] provided insight into a problem that has troubled groundwater hydrologists for at least the last ten years. He demonstrated that a fractal model appears to explain the scaling phenomenon of heterogeneity in aquifers, as measured by dispersivity values. The purpose of this comment is to put N euman's model into a geologic context. Neuman developed a universal scaling formula to relate dispersivity to the scale of observation but noted, "A major cause for local deviations from universal behavior appears to be the presence in many geologic media of discrete natural scales at which the log hydraulic conductivity field is homo-geneous" (p. 1756). These "discrete natural scales" are what geologists call facies. Specifically, a facies is a "three-dimensional body of rock having an environment origin that can be inferred from a set of characteristics including external geometry, internal geometry, sedimentary structures, lithology, organic content, stratigraphic relations, and associated facies" [Finley and Tyler, 1986, pp. 2-3]. Geologists apply the concept of facies on at least two scales: the local scale of Neuman's discrete natural scales (Figure 1 a) and a regional scale used in conceptual models of facies relationships (Figure lb). Following the dictates of a fractal model of heterogeneity, Neuman questioned the continued use of the representative elementary volume since "homogeneity is at best a local phenomenon limited to random and relatively narrow intervals of scale" (p. 1756). Yet, mapping geologic facies at a local scale may provide a way of defining meaningful representative elementary volumes. Miall [1985] extended the concept of facies by developing lithologic and geometrical descriptions of eight "architectural elements" that are found in alluvial systems. An element consists of a suite of facies that are associated with a specific depositional process. Examples include gravel bars and channel deposits. Davis et al. [1990] used Maill's architectural elements and facies descriptions when mapping an alluvial aquifer; they also measured permeability and calculated variograms for sections of the aquifer within an element. It is likely that such detailed geologic mapping is required to quantify heteroge-neity for accurate modeling of contaminant transport at certain scales. Given enough information on probable facies characteristics and distribution it may be possible to scale up heterogeneity and calculate effective parameters for contaminant transport modeling at other scales as is done in reservoir simulation [Lasseter et al., 1986]. It is important to note, however, that lumped measures of heterogeneity like effective parameters and Neuman's scaling rule cannot represent local peculiarities such as channel-ing of flow caused by the presence of connected units of high hydraulic conductivity. Indeed, Neuman cautioned that his scaling rule "does not necessarily describe conditions at any given locale but accounts for the self-similarity of log hydraulic conductivities in a mean sense over a la...

Flow in multiscale log conductivity fields with truncated power variograms

“…They showed that this can be done at a given locale by calibrating a truncated variogram model to data observed on a given support in one region and by predicting the autocovariance structure of the corresponding field in regions that are either smaller or larger. They suggested that one may also venture to bridge across both window scales and locales by adopting the generalized Hurst coefficient H = 0.25, originally proposed by Neuman [1990], and other generalized variogram parameters derived in their paper.…”

Evaluating uncertainty in predicting spatially variable representative elementary scales in fractured aquifers, with application to Turkey Creek Basin, Colorado

“…Their model predicts smoothly varying, monotonic parameter behavior. In similar concept, a fractal model is proposed by Neuman [1990], which exhibits a linear log-log relation between sample scale and parameter value, rendering the concept of RES meaningless. While fracture structure may be fractal, at least over some scale ranges in some types of geologic media, Wellman and Poeter [2005] found no such scale behavior using hydraulic head estimation of RES (HYRES), but we acknowledge fractal architectures were not considered.…”

Mathematical Formulation Requirements and Specifications for the Process Models

“…is assumed, which may be taken as the asymptotic limiting form of the dispersion tensor (Neuman, 1990). It should be noted that issues related to the scale dependence of dispersion are not considered.…”