[1] We appreciate the commentary by Neuman [2006] and the opportunity for discussion and further clarification of our article [Ritzi et al., 2004]. In his commentary, Neuman expounds upon his derivation of a model for the permeability semivariogram [Neuman, 2003, equation (53)] and describes aquifer architecture that would be represented by such a model. As requested by the Associate Editor, our reply focuses on illustrating the distinctions between the types of architecture represented by Neuman [2006] and Ritzi et al. [2004]. [2] Ritzi et al. [2004] considered the structure of permeability semivariograms arising in unconsolidated sedimentary deposits, the most common type of aquifer. Sedimentologists conventionally describe such deposits using a hierarchy of stratal unit types. Each of these stratal unit types is designated based on distinguishable characteristics, such as texture and geometry. A unit type defined at one hierarchical level is composed of smaller-scale unit types defined at the next lower level. Our approach makes a link between this hierarchical stratal architecture and the structure of the permeability semivariogram. To more clearly make this point we refer to Figure 1, which illustrates stratal architecture typically found in the sedimentological literature [e.g., Bridge, 2003]. Figure 1 shows a hierarchy of unit types designated within a sedimentary deposit formed as a braid bar within a river. The braid bar is composed of smaller-scale unit types such as large-scale inclined strata and cross-bar channel fills. In this example we refer to these as level II unit types within the hierarchy. Each of these level II unit types is, in turn, composed of smaller-scale level I unit types. For example, one large-scale inclined stratum (level II) is shown to be made up of level I units including small-scale cross strata composed of fine sand, planar strata composed of medium sand, and medium-scale cross strata composed of sandy gravel. [3] An expanded hierarchy could be created by including unit types that exist at still larger or smaller scales, with any number of levels. For example, we could add a higher level in the hierarchy by representing larger-scale unit types such as point bars and major-channel fills, which occur with braid bars in channel belts. We could add a lower level in the hierarchy by representing smaller-scale stratal unit types, such as avalanche beds and interbeds, which occur within a set of cross strata [see Ritzi et al., 2004]. Dai et al. [2005] showed that hierarchies may not be unique and that a unique hierarchal designation is not required for the approach presented by Ritzi et al. [2004]. [5] Ritzi et al. [2004] studied the relative contribution of each term toward defining the composite semivariogram, along the principal directions sampled, with data representing a fluvial point-bar deposit. Dai et al. [2005] did so with data from the Skull Ridge member of the Tesuque Formation, an alluvial deposit in the Rio Grande Rift basin. In both cases, the shape and range of the composite semivariogram were mostly determined by the sum of the cross-transition terms (last three summation groups on the RHS of equation (1)), with negligible contribution from the autotransition terms (first summation group on the RHS of equation (1)). In both examples the semivariogram could be modeled by knowing just the cross-transition probabilities and the univariate statistics for permeability, further underscoring the importance of the cross-transition terms. [7] The issue from our article upon which Neuman [2006] has focused concerns the dropping of the cross-transition terms, as done by Neuman [2003] and as in other approaches in the literature [e.g. Davis et al., 1997]. The context was that when stratal hierarchies are delineated in sedimentary deposits based, in part, on differences in sediment texture (mean grain size and grain size sorting), the stratal unit types can give rise to a hierarchy of multiple permeability modes. If so, Ritzi et al. [2004] and Dai et al. [2005] have shown that in directions in which lags will cross stratal boundaries (which is true for all directions in Figure 1), the permeability semivariogram can be almost entirely defined by the sum of cross-transition terms (last three summation groups on the RHS of equation (1)). Indeed, in both examples presented in these articles, it was the autotransition terms rather than the cross-transition terms that could be assumed negligible when modeling the permeability semivariogram.
Search from 250 M+ research papersSearch from 250 M+ research papers
Open Access
Jan 1, 2010
Dinah M Smith + 5
