Improving permeability semivariograms with transition probability models of hierarchical sedimentary architecture derived from outcrop analog studies

Abstract

[1] As analogs for aquifers, outcrops of sedimentary deposits allow sedimentary units to be mapped, permeability to be measured with high resolution, and sedimentary architecture to be related to the univariate and spatial bivariate statistics of permeability. Sedimentary deposits typically can be organized into hierarchies of unit types and associated permeability modes. The types of units and the number of hierarchical levels defined on an outcrop might vary depending upon the focus of the study. Regardless of how the outcrop sediments are subdivided, a composite bivariate statistic like the permeability semivariogram is a linear summation of the autosemivariograms and cross semivariograms for the unit types defined, weighted by the proportions and transition probabilities associated with the unit types. The composite sample semivariogram will not be representative unless data locations adequately define these transition probabilities. Data reflecting the stratal architecture can often be much more numerous than permeability measurements. These lithologic data can be used to better define transition probabilities and thus improve the estimates of the composite permeability semivariogram. In doing so, bias created from the incomplete exposure of units can be reduced by a Bayesian approach for estimating unit proportions and mean lengths. We illustrate this methodology with field data from an outcrop in the Española Basin, New Mexico.Citation: Dai, Z., R. W. Ritzi Jr., and D. F. Dominic (2005), Improving permeability semivariograms with transition probability models of hierarchical sedimentary architecture derived from outcrop analog studies, Water Resour. Res., 41, W07032,

A Markov chain model for characterizing medium heterogeneity and sediment layering structure

“…Conventional geostatistical methods (e.g., kriging) use the two-point variogram (or covariance) to describe spatial structures; recently the three-point variogram has been advocated [Strebelle, 2002;Krishnan and Journel, 2003;Journel and Zhang, 2006;Feyen and Caers, 2006]. An alternate geostatistical method, one that does not use a variogram, uses transition probability and Markov chains (TP/MC) to describe spatial structures of categorical data [e.g., Carle andFogg, 1996, 1997;Fogg et al, 1998;Carle, 1999;Weissmann et al, 1999;Li et al, 1999;Ritzi, 2000;Elfeki and Dekking, 2001;Lu and Zhang, 2002;Park et al, 2004;Dai et al, 2005;Sivakumar et al, 2005;Maji et al, 2006;Zhang et al, 2006;Li, 2007aLi, , 2007bLee et al, 2007;Dai et al, 2007;Sun et al, 2008]. An advantage of the TP/MC method relative to the conventional geostatistical methods is that it can incorporate geological information such as mean lengths and directional orientation of lithofacies [Carle andFogg, 1996, 1997].…”

“…For example, in a study of tritium transport at a site in Rulison, Colorado, because the mean horizontal length of shale could not be identified from the borehole data, the mean horizontal length estimates for an outcrop were used [Cooper et al, 2007]. Outcrop data were also used by Dai et al [2005] for a better definition of transition probability. Recent studies [Maji et al, 2006;Lee et al, 2007] demonstrate that the TP/MC method outperforms conventional geostatistical methods by reconstructing the linkage between lithology and contaminant plumes.…”

Modeling 3-D permeability distribution in alluvial fans using facies architecture and geophysical acquisitions

“…The optimal parameters of the latter are estimated through a generalized output least-squares criterion. Then, the composite semivariograms are computed using a hierarchical sedimentary architecture Dai et al, 2005) to obtain the K variance in each zone. Finally, the configuration of log 10 (K) is simulated through a multiple-zone sequential Gaussian algorithm with estimated statistic parameters reflecting the K spatial structures in the alluvial fan.…”

“…Equation (8) delineates the composite semivarigoram with respect to the individual facies semivariogram and transition probability. The general shape function and range of the composite semivarigoram can be obtained from individual facies mean length and volumetric proportion with the methods described in Dai et al (2005). The transition probability t ki (h ϕ ) has an analytical solution as derived by Dai et al (2007):…”

“…Hydraulic conductivity distributions in alluvial fans can be assigned according to the various hydrofacies simulated by conditional indicator geostatistical methods (Eggleston and Rojstaczer, 1998;Fogg et al, 1998;Weissmann et al, 2002a, b;Ritzi et al, 2004Ritzi et al, , 2006Proce et al, 2004;Dai et al, 2005;Harp et al, 2008;Hinnell et al, 2010;Maghrebi et al, 2015;Soltanian et al, 2015;Zhu et al, 2016a). However, the geostatistical methods require the stationary assumption; i.e., the distribution of the volumetric proportions and correlation lengths of hydrofacies converge to their mean values in the simulation do-main.…”