Relating the Spatial Bivariate Statistics of Permeability to Hierarchical Aquifer Architecture
A suite of computer programs and user’s guide (as a 2.8 MB zip file), which facilitate analyses with this methodology, can be downloaded for free from the following link:
We have developed a geostatistical method which can be used to study how the permeability semivariogram, and its parameters (shape, range, sill) are directly related to hierarchical stratal architecture. The method can be used to determine the semivariogram directly from quantifiable physical attributes including the proportions and lengths of unit types within the stratal hierarchy. The methodology has been developed and applied to different types of deposits by Ritzi et al. (2004), Popken (2004), Dai et al. (2005), and Ritzi and Allen-King (2007). We encourage the research community to continue such studies at many other sites. Here we are distributing a set of software for conducting such analyses, along with a user's manual and sample data set. The example data set is from the well-known Borden research site and allows repeating all analyses published in Ritzi and Allen-King (2007). Our goal is to foster more well-studied examples of real deposits through making the necessary tools readily available and easily understood by the research community.
This methodology can be used to:
- Relate the length statistics of stratal unit types, at different hierarchical levels, to length scales of correlation embodied within the spatial correlation structure
- Utilize geologic information at locations beyond where permeability is measured in order to better define correlation structures
- Understand and remove bias introduced by the distribution of sample locations
- Model correlation structures based on quantifiable attributes of the unit types within the stratal hierarchy (as an alternative to curve fitting).
Sedimentary deposits can be conceptualized as an aggregation of stratal unit types. These units can be defined at different spatial scales within a hierarchical framework, i.e., larger scale unit types are made up of smaller-scale unit types which, in turn, are made up of still smaller unit types, and so on. The spatial correlation of permeability can be strongly related to this hierarchical stratal architecture.
Over some range of larger scales, stratal architecture can be represented directly within numerical models for flow and transport (e.g. Weissmann and Fogg, 1999; Proce et al., 2004). However, there often are practical limitations to the number and the size of the grid cells that can be included within a numerical model, and this limits the direct representation of stratal architecture below a certain scale. There may in fact be stratal unit types that exist over a range of scales smaller than a model grid cell, which significantly influence flow and/or transport, in the real deposit. This smaller-scale architecture must be indirectly represented in the model through upscaling methods, i.e. methods which represent the influence of stratal architecture and the associated spatial variation of permeability with one or more lumped measures or parameters. Whatever the lumped measures/parameters may be (e.g. bivariate spatial statistics like the semivariogram, dual-domain mass transfer rates, or memory functions), we would prefer that they be based on quantifiable physical attributes of the sedimentary deposit, rather than being just fitting parameters.
The method and its application to a number of specific sedimentary deposits are given in the following publications from this work:
- Ritzi, R.W., and R.M. Allen-King, 2007, “Why did Sudicky (1986) find an exponential-like spatial correlation structure for hydraulic conductivity at the Borden research site?” Water Resources Research, 43, W01406doi: 10.1029/2006WR004935 (pdf for personal use only)
- Dai, Z., R.W. Ritzi, and D.F. Dominic, 2005, “Improving permeability semivariograms with transition probability models of hierarchical sedimentary architecture derived from outcrop-analog studies,” Water Resources Research, 41, W07032, doi:10.1029/2004WR003515. (pdf for personal use only)
- Dai, Z., R.W. Ritzi, C. Huang, D.F. Dominic, and Y.N. Rubin, 2004, Transport in heterogeneous sediments with multimodal conductivity and hierarchical organization across scales, Journal of Hydrology, 294 (1-3), p 68-86. (pdf for personal use only)
- Popken, E., 2005, “Spatial Correlation of Permeability at the Oyster Site, Virginia,” M.S. Thesis, Wright State University
- Ritzi, R. W., Z. Dai, D.F. Dominic, and Y.N Rubin, Y. N., 2004, Spatial correlation of permeability in cross-stratified sediment with hierarchical architecture, Water Resour. Res.,40(3), W03513doi: 10.1029/2003WR002420 (pdf for personal use only)
- Ritzi, R.W., 2000, "Behavior of indicator variograms and transition probabilities in relation to the variance in lengths of hydrofacies," Water Resources Research, 36(11), p. 3375-3381. (pdf for personal use only)
- Ritzi, R. W., Z. Dai*, D. F. Dominic, and Y. Rubin, 2006, Reply to comment by Shlomo P. Neuman on “Spatial correlation of permeability in cross-stratified sediment with hierarchical architecture,” Water Resources Research, 42, W05602doi: 10.1029/2005WR004402 (pdf for personal use only)
The ideas presented here were developed with support from the National Science Foundation under grants NSF-EAR 00-01125, EAR-0510819, and EAR-0538037. Any opinions, findings and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect those of the National Science Foundation.
Robert Ritzi, firstname.lastname@example.org, (937) 775-2460