Simulating the heterogeneity in braided channel belt deposits: 2 ...

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WATER RESOURCES RESEARCH, VOL. 46, W04516, doi:10.1029/2009WR008112, 2010

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Simulating the heterogeneity in braided channel belt deposits: 2. Examples of results and comparison to natural deposits Arijit Guin,1 Ramya Ramanathan,1 Robert W. Ritzi Jr.,1 David F. Dominic,1 Ian A. Lunt,2 Timothy D. Scheibe,3 and Vicky L. Freedman3 Received 13 April 2009; revised 29 September 2009; accepted 24 November 2009; published 29 April 2010.

[1] In part 1 of this paper (Ramanathan et al., 2010b) we presented a methodology and a code for modeling the hierarchical sedimentary architecture in braided channel belt deposits. Here in part 2, the code was used to create a digital model of this architecture and the corresponding spatial distribution of permeability. The simulated architecture was compared to the real stratal architecture observed in an abandoned channel belt. The comparisons included assessments of similarity which were both qualitative and quantitative. The qualitative comparisons show that the geometries of unit types within the synthetic deposits are generally consistent with field observations. The unit types in the synthetic deposits would generally be recognized as representing their counterparts in nature, including cross stratasets, lobate and scroll bar deposits, and channel fills. Furthermore, the synthetic deposits have a hierarchical spatial relationship among these units consistent with observations from field exposures and geophysical images. In quantitative comparisons the proportions and the length, width, and height of unit types at different scales, across all levels of the stratal hierarchy, compare well between the synthetic and the natural deposits. A number of important attributes of the synthetic channel belt deposits are shown to be influenced by more than one level within the hierarchy of stratal architecture. First, the high‐permeability open‐framework gravels connected across all levels and thus formed preferential flow pathways; open‐framework gravels are known to form preferential flow pathways in natural channel belt deposits. The nature of a connected cluster changed across different levels of the stratal hierarchy, and as a result of the geologic structure, the connectivity occurs at proportions of open‐framework gravels below the theoretical percolation threshold for random infinite media. Second, when the channel belt model was populated with permeability distributions by lowest‐level unit type, the composite permeability semivariogram contained structures that were identifiable at more than one scale, and each of these structures could be directly linked to unit types of different scales existing at different levels within the hierarchy of strata. These collective results are encouraging with respect to our goal that this model be relevant for testing ideas in future research on flow and transport in aquifers and reservoirs with multiscale heterogeneity. Citation: Guin, A., R. Ramanathan, R. W. Ritzi Jr., D. F. Dominic, I. A. Lunt, T. D. Scheibe, and V. L. Freedman (2010), Simulating the heterogeneity in braided channel belt deposits: 2. Examples of results and comparison to natural deposits, Water Resour. Res., 46, W04516, doi:10.1029/2009WR008112.

1. Introduction [2] In part 1 of this paper [Ramanathan et al., 2010b] we presented and discussed a new computer code for simulating reservoir or aquifer heterogeneity. The code creates digital models for the hierarchical sedimentary architecture in braided channel belt deposits. The code uses a geometric‐ 1 Department of Earth and Environmental Sciences, Wright State University, Dayton, Ohio, USA. 2 StatoilHydro Research, Bergen, Norway. 3 Hydrology Technical Group, Energy and Environment Directorate, Pacific Northwest National Laboratory, Richland, Washington, USA.

Copyright 2010 by the American Geophysical Union. 0043‐1397/10/2009WR008112

based approach to simulate strata observed over multiple scales. Larger‐scale unit types form the bounding regions of associations of smaller‐scale unit types. Accordingly, unit types at all scales are organized as a hierarchy. The input parameters are primarily univariate statistics such as the proportions, and mean and variance in the lengths of sedimentary unit types, at each hierarchical level. A digital model is created as a three‐dimensional cubic lattice, which can be used directly in numerical models for fluid flow. [3] The code was developed as a tool for computational research on subsurface fluid flow and transport. The goal was to develop a means for generating a three‐dimensional digital sedimentary deposit with realistic architecture from the kilometer scale to the centimeter scale. The digital deposit is intended for use as a high‐resolution base case

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GUIN ET AL.: HETEROGENEITY IN CHANNEL BELT DEPOSITS

Figure 1. Portion of a channel belt deposit. Deposits from individual unit bars are delineated and in some cases numbered. Locations of cross sections shown in Figure 2 are indicated. in various areas of research focused on understanding the relationship between multiscale heterogeneity and processes such as mass transport, including the testing of upscaling theories in models for flow and transport in reservoirs and aquifers. The intention is to advance work in the vein of Scheibe and Murray [1998], Scheibe and Yabusaki [1998], Arbogast et al. [2000], Willis and White [2000], Christie and Blunt [2001], Rubin et al. [2003], Zinn and Harvey [2003], Caers and Zhang [2004], Liu et al. [2004], Aarnes et al. [2006], Maji et al. [2006], Ronayne and Gorelick [2006], Lee et al. [2007], and Durlofsky et al. [2007] by providing realistic representations of heterogeneity with which these ideas can be tested. [4] In this paper, part 2, we evaluate the digital models created by the code by comparing the model sedimentary architecture with what has been observed and quantified in the well‐studied deposits of the abandoned Sagavanirktok River channel belt (northern Alaska, United States) by Lunt et al. [2004]. The hierarchy of sedimentary architecture in channel belt deposits was reviewed in part 1 of this paper (see Figures 1, 3, and 4 in part 1). In comparing the metrics between the model and natural deposits, we analyzed each hierarchical level separately. This paper is organized so that comparisons are presented in order from the largest (level III) strata to the smallest (level I) strata. Quantitative comparisons are made by sampling the digital deposit in ways that are consistent with how natural deposits were sampled and computing, from those samples, the same metrics quantified for the natural deposits, and then comparing statistics of those metrics between the digital and natural deposits. In section 2 of this paper, we discuss the metrics which were computed and used in the comparisons. [5] The idea of comparing a digital model for sedimentary deposits to natural deposits seems straightforward, and yet one is hard pressed to find existing studies of natural deposits that are suitable for such comparisons. This issue was discussed by Anderson [1990] in a review of the literature on sedimentary facies models. There are few quanti-

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tative models for natural deposits that provide appropriate metrics such as volume proportions or characteristic stratal length statistics in three dimensions. A quantitative, three‐ dimensional facies model has recently emerged for the hierarchical stratal architecture within channel belt deposits. As summarized by Bridge [2006], the three‐dimensional structure of these deposits has been illuminated by (1) the use of ground‐penetrating radar in combination with cores and trenches; (2) the study of channel deposits in frozen rivers allowing access to whole channel belts; and (3) improved methods for studying the history of evolution of bars and channels by using time series of aerial photos and satellite images. The nature of fluvial deposits has been coupled to the processes creating them by studies of water flow, sediment transport, and channel migration in natural rivers and by laboratory studies in flumes [e.g., Lunt and Bridge, 2007]. The depositional model that has emerged would not be known from any one of these data types alone, and was possible only through their synthesis. Because there are quantitative metrics for the stratal architecture in channel belt deposits in three dimensions, channel belt deposits are a good target for testing ideas about geometric‐based simulation. The research link between these newer studies of natural sedimentary deposits and the associated computational research on modeling reservoir and aquifer heterogeneity should be a two‐way street. Advances in field research allow us to test ideas about geometric‐based modeling approaches. In turn, those attempts at modeling reveal the limits of what we currently know from field studies. Attempting to properly represent the volume proportions, typical geometry, and variation in geometry of a particular unit type may reveal that it has been insufficiently characterized in natural deposits, and this revelation may help focus future field studies on quantifying those attributes. [6] After comparing the digital deposits to natural deposits in sections 3–6, we then examine them in section 7 for the existence of preferential‐flow pathways created by connected high‐permeability units. We studied the connectedness of the high‐permeability strata in the digital model within the framework of the percolation theory. The percolation theory originally addressed the connectivity of cells in random infinite lattices, and it has been used as a framework for considering connectivity of units with geologic structure, in finite domains [e.g., Harter, 2005; Hunt, 2005; Guin and Ritzi, 2008]. We used an approach appropriate for the case of hierarchical stratal architecture, and considered the finite domain effect created by the finite size of larger‐scale bounding units at each level of the stratal hierarchy. [7] In section 8 we populate the model with permeabilty values, and study the permeability spatial correlation structure by linking it to quantifiable attributes of the hierarchical stratal architecture. Correlation length, in general, does not directly correspond to the length of a stratatype. Ritzi et al. [2004], Dai et al. [2005], and Ritzi and Allen‐King [2007] have shown that the correlation scales represented in two‐ point bivariate statistics can be related directly to the probability of transitions across strata types at each scale. In turn, these can be directly related to the proportions and the mean and variance in the length of the strata types at each level within the stratal hierarchy. Thus the distributions of lengths among strata types at each hierarchical level are analyzed. The methodology of Ritzi et al. [2004] is used to show how

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Figure 2. Cross sections, as located in Figure 1. A‐A′ illustrates the maximum apparent length (MAL) metric (radar profile adapted from Lunt et al. [2004]). Cross sections B‐B′ and C‐C′ are along the shorter and longer axis of a unit bar deposit, respectively. Note that cross section C‐C′ is drawn at a different scale. each scale of the stratal architecture affects the two‐point spatial bivariate statistics for permeability within the model.

2. Metrics Used for Comparing the Digital and Natural Deposits [8] The stratal architecture in natural deposits has been studied in cores, in outcrop or trench exposures, or in

images from geophysical transects (e.g., radar or seismic data). In trench exposures and radar transects, the apparent dip of a unit will depend on the orientation of the exposure or transect to the true dip. In the same way, the apparent length will also depend upon the orientation of the exposure or transect, as illustrated in Figures 1 and 2. This fact gives rise to a few issues that must be addressed with regard to how lengths are sampled.

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Table 1. Mean and Variances Used for Defining the Distributions of IGL

Compound bar deposits Unit bar deposits Cross stratasets

Lcb Wcb Hcb Lub Wub Hub Wts Hts

Mean (m)

Variance (m2)

1,000 400 2.50 225 45 1.25 2.97 0.21

53,337 7,368 0.42 5,641 225 0.10 0.05 0.01

[9] Lunt et al. [2004] reported ranges for stratal lengths and heights which they measured in trench exposures and on images from radar transects, as shown in Figure 2. Consider the sampling of the length of a unit bar deposit. Lunt et al. [2004] sampled a unit once along the longest line (maximum length) that could be drawn. For example, if the horizontal length of unit bar 1, as shown, is to be measured, it could be taken along any number of horizontal lines drawn across the unit. In an exhaustive set of all such samples, all will all be less than the maximum extent of the unit as projected onto the horizontal line below. This projection represents the maximum horizontal metric one could define from this exposure of unit bar 1. [10] Importantly, such radar transects are usually not oriented exactly along the principal axes of units. Thus, as shown among the cross sections in Figure 2, the maximum length metric for unit bar 1 in the radar profile is an apparent maximum length that is less than the length measured along the long axis (section C‐C′), and greater than the length measured along the short axis (section B‐B′). Accordingly we will refer to the metric reported by Lunt et al. [2004] as the maximum apparent length (MAL) metric. The MAL metric will be used for comparing the model to the natural deposit. [11] The code creates an occurrence of a unit type starting with an archetypal geometry (see part 1, Figure 7a). The archetypal geometry is a polyhedron defined by piecewise planar elements. A parsimonious number of geometric lengths are used to define the size and shape of a polyhedron. A length of this sort is referred to as an input geometric length (IGL). Initially, variability is introduced by randomly drawing IGL from statistical distribution functions defined by user inputs (the mean and variance). The final geometry of each occurrence in the model will generally differ from its starting, archetypal geometry, as sinuosity is added to its axis, and as parts are removed in later stages of the simulation. Thus the distributions of lengths in the output are quite different from the distributions of the IGL. [12] We emphasize that there is no direct relationship between the values used for input variables to the code (e.g., the mean and variance defining the distribution from which an IGL will be drawn) and the values of metrics or variables of interest quantified in the output (e.g., the distribution of unit lengths in the digital deposit). Whereas the code starts by using IGL generated (in most cases) from normal distributions defined by the user, the length distributions in the output are generally Erlangian and should be expected to have a different mean and variance than the distributions of the IGL. We will examine this below. As a practical matter,

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input values have to be adjusted to achieve target values for metrics as sought in the output. There is no reason the IGL have to be well informed at the start, but good starting guesses will save time. [13] The mean and variance for IGL distributions in Table 1 were found to give acceptable results, as explored below. Most of the analyses presented in this paper are performed on digital deposits sampled from a single, large‐scale geometric model created by running the code with those input values (see Figure 10a in part 1 of this paper for a rendering of the full model). In making the comparisons below, the goal is to show that the methodology is capable of creating a digital deposit which represents, on a quantitative basis, what has been observed in nature. [14] In sections 3, 4, and 5, parts of the geometric model are sampled at different scales and unit types at each hierarchical level (see Figure 3 in part 1 of this paper) are imaged and analyzed separately.

3. Level III Unit Types: Compound Bar Deposits and Major Channel Fills [15] Figure 3 shows an aerial photograph of both active and abandoned channel belts. Some of the individual compound bar deposits in the abandoned channel belt were traced to help clarify that they commonly have an oblong shape in plan view, with the longer axis subparallel to the direction of paleoflow. The abandoned bar assemblages are interwoven with former channel segments now filled with lower‐permeability sand (major channel fills). Tables 2a and 2b give ranges for MAL of these two unit types in the horizontal and vertical directions as reported by Lunt et al. [2004]. [16] The full geometric model was first sampled with a subdomain size of 2 km × 1 km × 5 m, and a digital model was created using a grid spacing of 2 m × 2 m × 0.1 m. A slice along an x‐y plane near the top of the simulation is shown in Figure 4. The image shows only the compound bar deposits, major channel fills, and cross bar channel fills. As in Figure 3, the simulated compound bar deposits are oblong in plan view with the longer axes being subparallel to the predominant direction of paleoflow represented by the model. The major channel fills and cross‐bar channel fills are interwoven within these deposits. Figure 4 (bottom right) shows cross sections through the simulation. [17] In reporting the range of MAL summarized in Tables 2a and 2b, Lunt et al. [2004] stated that lengths could not be distinguished from widths among measurements of compound bar deposits taken from radar profiles. Thus a range is given only for undifferentiated horizontal lengths. As the digital model was sampled, we could make finer distinctions. We generally knew if measurements were closer to the long or short axes of compound bar deposits, and thus we could group the MAL as being more representative of length or width. The MAL metrics for the simulated compound‐bar deposits and major channel fills are given in Tables 2a and 2b. The MAL for the length of simulated compound bar deposits are about twice those for width, consistent with the oblong shape of compound bar deposits as seen in the images in Figures 3 and 4. The range of horizontal lengths is close to the range reported by Lunt et

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Figure 3. Sagavanirktok River showing a portion of the active channel belt (on left) and of the abandoned channel‐belt deposits (on right) with highlighting of some compound bar deposits (adapted from Lunt et al. [2004]). al. [2004] for a natural deposit, for both unit types, in both directions, and also in the vertical direction. [18] Lunt et al. [2004] only report ranges of MAL for width and thickness of the major channel fills, not length, and the MAL for width and thickness in the simulated deposits are in general agreement with their ranges. [19] In summary, the simulated compound bar deposits and major channel fills have expressions in aerial view that are qualitatively similar to those in aerial photographs of abandoned channel belt deposits. The compound bar deposits are oblong in plan view and oriented with longer axes subparallel to the direction of paleoflow. The ranges of MAL in the simulation generally correspond to those quantified in the abandoned channel belt.

4. Level II Unit Types: Unit Bar Deposits and Cross‐Bar Fills [20] Figure 1 conveys a history of unit bar accretion as preserved in compound bar deposits. Figure 2 conveys some of the spatial relationships among the resulting unit bar deposits. Unit bar deposits are generally elongate in the direction of the longer axis of the compound bar deposit. Where unit bars accreted on the upstream side of preceding bars, their deposits generally dip upstream. As unit bars accreted on the flanks of a compound bar that was expanding outward with concomitant channel migration (forming scroll bars), their curvature can increase with concomitant increase in channel curvature. Where heads of unit bars are preserved, they are steeper on their lee side. [21] Figure 5 shows unit bar deposits within a complete level III polyhedron (compound bar and adjacent channel fills), which was sampled from the geometric model, and used to create a digital model with a subdomain size of 450 m × 900 m × 4 m, and a grid spacing of 1.0 m ×

1.0 m × 0.1 m. Note that this polyhedron overlaps with others in the geometric model, and only a portion will be used, but the whole is examined here to better illustrate the relationship among unit bar deposits. The internal architecture within each unit bar has not been rendered. A slice along an x‐y plane through the middle of the simulation is shown in Figure 5 (top). The curvature of the unit bar deposits generally increases away from the center of the compound bar deposit. In Figure 5 (bottom), accreting unit bar deposits dip in the correct direction. Thus, among these images, we see the general spatial relationships that should exist for unit bar deposits, as described above. [22] Table 3a gives the ranges for lengths and heights of unit bar deposits from trench exposures and radar transects as quantified by Lunt [2002]. The MAL metrics for the simulated unit bar deposits were computed from three slices along x‐z planes (1–3 in Figure 5, top) and two slices of y‐z planes (4 and 5 in Figure 5, top). Table 3b gives these MAL. The ranges of MAL in all directions generally correspond to those reported by Lunt [2002]. [23] Images showing the cross‐bar fills were already shown in Figure 4. Table 3a gives the ranges for MAL for the width and height of cross‐bar fills in trench exposures and radar transects as reported by Lunt [2002]. Table 3b

Table 2a. MAL Ranges for Compound Bar Deposits and Major Channel Fills Reported by Lunt et al. [2004]

Compound bar deposits (m) Major channel fills (m)

Horizontala

Vertical

200–1100 30–350b

1.0–3.8 1.0–3.5

a Could not distinguish lengths from widths in ground‐penetrating radar profiles [Lunt et al., 2004, p. 409]. b Width only, as per width to maximum thickness ratio.

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Table 2b. MAL Ranges for Compound Bar Deposits and Major Channel Fills Measured From Model in Figure 4

Compound bar deposits (m) Major channel fills (m)

Horizontal, Closer to Long Axis, From y‐z Slices

Horizontal, Closer to Long Axis, From an x‐y Slice

Horizontal, Closer to Short Axis (Width) From x‐z Slices

Horizontal, Closer to Short Axis (Width) From an x‐y Slice

Vertical From All Slices

209.09–1045.45 172.73–454.54

495.24–1104.76 142.86–1180.95

98.1–405.98 28.57–211.45

171.43–380.95 19.04–180.95

0.36–3.45 0.9–3.2

gives the ranges for the corresponding MAL in the model as computed from four slices along x‐z planes (1–4 in Figure 4, left); these compare well to those reported by Lunt [2002]. [24] Thus the simulated unit bar deposits and cross‐bar fills have spatial relationships and have ranges of MAL which generally correspond to those characterized and quantified by Lunt [2002] and Lunt et al. [2004] in the natural Sagavanirktok channel belt deposits.

5. Level I Unit Types: Cross Stratasets [25] As illustrated in Figure 2 (middle and bottom), the shape of the cross stratasets exposed in a trench or radar profile will depend on the orientation of the exposure or

transect (see also Figure 4 in part 1). When exposed parallel to the direction of dune migration (Figure 2, bottom), they have planar boundaries, with dip increasing within the bar in the downstream direction. Figure 6 shows an exposure of cross stratasets within unit bar deposits of the Parana River, Argentina. Although it is not from the Sagavanirktok deposits, this exposure illustrates very clearly some important characteristics common to unit bar deposits [Bridge, 2003]. One of these characteristics is the dip angle of cross stratasets, which can be measured in this image and the results compared with similar measurements from the simulated deposits. Lunt et al. [2004] observed that the dip angle of cross stratasets changes with position in a unit bar deposit; angles are very low near the upstream end and

Figure 4. Simulated compound bar deposits shown as lighter areas, and cross‐bar and major channel fills as darker areas. The level I and the level II strata existing within compound bars are not rendered in these images. (left) Image of a portion of the grid layer (x‐y slice) sampled from close to the top of the digital model. Lines indicate the locations of grid slices along x‐z and y‐z which are used to sample MAL for compound bar deposit and major channel fill unit types. (bottom) Cross sections along A‐A′ and B‐B′. Vertical exaggeration is ×30. 6 of 19

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Figure 5. (top) Image of a portion of the grid layer (x‐y slice) from a digital model of one of the compound bar deposits, before merging with others, showing the internal architecture at level II. (Architecture within unit bars, and cross‐bar fills are not rendered in this image.) Unit bar deposits are shown with different, randomly chosen shades of gray. Note that grid slices showing unit bar morphology along only one elevation will appear different from images like Figure 1 which project surface morphology occurring at more than one elevation onto a single horizontal plane. Lines indicate the locations of grid slices along x‐z and y‐z which are used to sample MAL for unit bar deposits In cross sections A‐A′ (middle) and B‐B′ (bottom), the unit bar deposits are shown as lighter areas, major channel fills are shown as medium gray, and unit bar boundaries (arbitrary thickness) are shown as darker gray. Vertical exaggeration is ×30. 7 of 19

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Table 3a. MAL Ranges for Unit Bar Deposits and Cross‐Bar Fills, Measured From Radar Lines and Trenches by Lunt [2002]

Unit bar deposits (m) Cross‐bar fills (m)

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Horizontal, Closer to Long Axis

Horizontal, Closer to Short Axis (Width)

Vertical

20–150 …

5–40 5–40

0.3–1.7 0.1–1.0

geometry is consistent with the shape of sets of trough cross‐stratified strata, and their dip angle increases toward the downstream end of a unit bar deposit. The MAL metrics and the proportions of cross strataset types in samples from the model are consistent with what was quantified in the abandoned Sagavanirktok channel belt deposits by Lunt [2002].

6. Image of All Unit Types Together increase toward the downstream end. However, they did not report measured dip angles. [26] The ranges of MAL of cross stratasets quantified by Lunt [2002] in the Sagavanirktok channel belt deposits are given in Table 4a. As volume proportions of a unit bar deposit, the cross stratasets were 68% sandy gravel (sg), 27% open framework gravel (ofg), and 5% sand (s). [27] In Figure 6, the dip angles of individual cross strata within cross stratasets are all near the angle of repose, indicating that the exposure is close to parallel with the long axis of the unit bar deposit. Therefore the dip angles of cross stratasets measured here will represent the true dip of these stratasets. The ranges for dip angle of cross stratasets measured in Figure 6 are given in Table 4b. The angles are generally steeper in the downstream half of the unit bar deposit as compared with the upstream half, as observed by Lunt et al. [2004]. [28] To analyze cross stratasets, we extracted a subdomain of 15 m × 35 m × 2.1 m from the geometric model with a grid resolution of 0.05 m × 0.05 m × 0.05 m. In doing so, we extracted one relatively complete unit bar deposit from the simulation that has an axis close to the y‐coordinate direction. Figure 7a shows a slice along the x‐z plane revealing the trough‐shaped nature of the strata in the view along the long axis of the bar. Figures 7b–7d show slices along a y‐z plane from the downstream, midsection, and upstream sections of the unit bar deposit. It is apparent that the dip of the cross stratasets is steeper at the downstream end and systematically decreases toward the upstream end. [29] The simulated volumetric proportions of cross stratasets within unit bar deposits are 70% sg, 24% ofg, and 6% s. Thus the code can create cross stratasets with proportions in these three textural categories in general agreement with those reported by Lunt et al. [2004]. (Note that the geometric model was created using input proportions of 50% sg, 45% ofg, and 5% s, as proportions of cross strataset occurrences, not volumetric proportions, within unit bar deposits. Again, some model metrics cannot be directly specified by input values, and a couple iterations of changing the input proportions were required to achieve cross stratasets simulated with acceptable volume proportions.) [30] The MAL and dip angles were quantified from Figures 7b–7d and are presented in Table 4c, compiled separately for the downstream third, middle third, and upstream third of the unit bar deposit. The quantified dip angles are indeed steeper in the downstream end and significantly lower in the upstream end. The range of the MAL combined from among all three sections of the simulated deposit is generally the same as the range of corresponding horizontal MAL (closer to the long axis) given by Lunt [2002]. [31] To summarize, the simulated cross stratasets have the following aspects consistent with real deposits: Their

[32] All levels are shown together in a voxel rendering in Figure 8a. The digital model was created by sampling the geometric model with a subdomain of 650 m × 1200 m × 10 m and a voxel size of 2 m × 2 m × 0.05 m. Thus there are 39 million voxels. The cross stratasets are difficult to make out when imaged at this scale using this resolution, and Figures 7a–7d give a better impression of the structure of cross stratasets. [33] In sections 7 and 8 we first study the connectivity of the open‐framework gravel cross stratasets within the model. We then populate the model with permeability distributions. Finally, we examine the relationship between the permeability correlation structure and the sedimentary architecture.

7. Connectivity (Percolation) of Open‐Framework Gravels [34] Open‐framework gravels are known to be connected and to form preferential flow pathways in natural channel belt deposits [Lunt et al., 2004]. The open‐framework gravels in the digital model are highly connected and form pathways which span (percolate) opposing boundaries of the domain in all three coordinate directions. Here we examine the nature of that connectivity. In future work we are interested in using the digital channel belt model within finite difference models for fluid flow, and in that context connectivity exists between two voxels if they have adjacent faces (adjacent corners do not form connections). In the percolation theory, this is the same problem as for percolation on a simple cubic lattice [Stauffer and Aharony, 1994]. We used a code [Guin, 2009] that searches the digital model, finds clusters of connected of open‐framework gravel voxels, and quantifies them as discussed below. [35] It is known from the percolation theory that if open‐ framework gravel voxels were placed randomly within an infinite cubic lattice they would percolate at proportions above 0.3116 [Stauffer and Aharony, 1994]. In the finite digital model, with the structure imposed by the stratal architecture, they percolate at a lower proportion. Percolation in structured media within finite domains has been previously studied, for example, by Harter [2005] and Guin and Ritzi [2008]. Here the problem needs to be approached

Table 3b. MAL Ranges for Unit Bar Deposits and Cross‐Bar Fills, Measured From the Model in Figure 5

Unit bar deposits (m) Cross‐bar fills (m)

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Horizontal, Closer to Long Axis, From y‐z Slices

Horizontal, Closer to Short Axis (Width) From x‐z Slices

Vertical

41.7–161.7 …

7.5–52.5 17.1–38.46

0.13–1.33 0.11–0.69

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Figure 6. An exposure of the stratal architecture within a compound bar deposit excavated near the Parana River, Argentina. Flow was from right to left. The arrows are on a trench face parallel to the long axis of the unit bar deposit. The white arrows indicate the upper and lower boundaries of one unit bar deposit. The black arrows indicate the boundary between two cross stratasets within the bar.

with a methodology that considers the hierarchy of unit types. The level N units are irregularly shaped, finite boundaries of level N‐1 unit types. We define a spanning cluster to exist at level N, in a coordinate direction, if the cluster connects opposing level N boundaries with respect to that coordinate direction. [36] To understand the nature of the percolating clusters, we first examined the clusters at each level individually. The IGL used to create the internal architecture in all these realizations were drawn from normal distributions defined in Table 1 and the different realizations were created by changing the seed numbers for the IGL random generators in the code. [37] Ten realizations of an assemblage of cross stratasets were generated inside of a unit bar deposit polyhedron with a length, width, and height of 150.0 m, 40.0 m, and 1.7 m, respectively. The average of the results from the 10 realizations is given in Table 5. In all 10 realizations, the open‐ framework gravel percolated across the unit bar deposit in all three coordinate directions, in clusters distributed throughout the polyhedra. More clusters percolate in the z‐coordinate direction than in the x or y directions. The percolating clusters are numerous, and most contain a minor fraction of open‐framework gravel voxels. [38] To study what happens at the next higher level, 10 realizations of an assemblage of unit bar deposits (with internal architecture) and cross‐bar fills were then generated inside of a level III polyhedron containing a compound bar deposit with length, width, and height of 800.0 m, 400.0 m, and 3.0 m, respectively. In all 10 realizations, open‐framework gravel cells percolated across the compound bar deposit in all three coordinate directions. However, as shown in Table 5, the cells are in one large cluster

Table 4a. MAL Ranges for Cross Stratasets, Measured From Radar Lines and Trenches by Lunt [2002]

Cross Stratasets (m)

Horizontal, Closer to Long Axis

Horizontal, Closer to Short Axis (Width)

Vertical

2–7

1–5

0.1–0.4

that percolates in all three coordinate directions. The average volume fraction of all percolating open‐framework gravel cells almost equals the volume fraction of the largest percolating cluster. Thus most of the numerous but separate clusters percolating each unit bar deposit in the z direction become connected to other such clusters across unit bar boundaries and combine as one larger cluster as they are merged into a compound bar deposit. [39] As shown in Table 5, in the full‐scale realization where the compound bar deposits are merged, 70% of the open framework gravel cells formed a single large cluster which percolates the domain in all three coordinate directions. Guin [2009] found that this connectivity exists when the volume fraction of open‐framework gravels exceeds approximately 0.2, even when the IGL distributions were changed to the lower or higher end of the range of lengths given by Lunt et al. [2004]. Isosurface renderings of percolating clusters at different levels are given by Guin [2009]. [40] To summarize, when simulated in volume proportions above 20%, the open‐framework gravels percolate a unit bar deposit in numerous separate clusters, mainly across boundaries opposed in the z coordinate direction. Most of these clusters become connected across the boundaries of adjacent unit bar deposits and, in turn, compound bar deposits. As a result, there is essentially one large cluster that percolates the finite portion of a channel belt deposit in all three coordinate directions.

8. Analysis of Exhaustively Sampled Length Distributions and Consequent Permeability Correlation Structure [41] The digital model was populated using one scenario for the distribution of natural log permeability as defined for each level I unit type (and only for level I unit types), as

Table 4b. Apparent Dip for Cross Stratasets, Measured From Parana River Exposure in Figure 6

Dip Angle (deg)

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Downstream

Upstream

10–26

2–9

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Figure 7. Cross sections sampled from the geometric model showing cross stratasets. Orange lines delineate boundaries between units, and are not themselves strata. The thickness of boundary lines is arbitrary except that thicker lines are used to mark boundaries of the unit bar or compound bar. (a) Section perpendicular to the long axis of a unit bar deposit. Figures 7b–7d show sections parallel to the axis of the unit bar deposit and taken from (b) the downstream section, (c) middle section, and (d) upstream section.

given in Table 6. The result is shown in Figures 8b and 9. The purpose was to study the impact that each level in the hierarchy had on the composite spatial correlation structure of permeability manifest in the model. This is a scenario with fairly high permeability contrast between level I unit types and low variance within them, intended to emphasize the impact of different scales of geologic structure on the spatial correlation structure of permeability. [42] The exhaustively sampled length (ESL) metrics become important in this context [Ritzi, 2000; Ritzi et al., 2004], and statistics for them from exhaustively sampling the digital model are compiled in Table 7. The IGL are normally distributed (e.g., Figure 10), but the ESL have Erlangian‐like distributions (e.g., Figure 11). Hereinafter in this section, “length” will refer to the ESL metric. [43] The model permeability field in Figure 8b was exhaustively sampled and the sample permeability semivariogram was computed in the +y coordinate direction as shown in Figure 12a. The sample permeability semivariogram appears to have a smooth and monotonically increasing, exponential‐like spatial correlation structure that

levels off to a sill, though the model was populated without assuming any function for the spatial bivariate structure of permeability. Any spatial correlation structure that exists is imparted by the hierarchical geometric model for stratal architecture shown in Figure 8a. Thus the exponential‐like structure observed in Figure 12a can be explained and analytically linked to the stratal architecture through the method of Ritzi et al. [2004]. In this method, the shape of the permeability correlation structure is linked to the variance in the length of strata at each level, the range of correlation is linked to the mean length of strata at each level, and the sill is linked to the univariate statistics for permeability at the lowest level. The method has been used in a Table 4c. MAL Ranges and Apparent Dip for Cross Stratasets, Measured From Model in Figures 7b–7d Downstream Midsection Upstream Horizontal, closer to long axis (m) Vertical (m) Dip angle (deg)

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0.14–1.09 0.03–0.3 12–22

0.21–1.53 0.03–0.33 7.5–10.0

2.0–13.0 0.03–0.16 1–5

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Figure 8. Images representing the same 650 m × 1200 m × 10 m region sampled from the geometric model, rendered with a voxel size of 2 m × 2 m × 0.05 m. (a) Rendering of unit types. Layers of voxels are marked yellow along level III unit boundaries, and are marked orange along the boundaries of unit bar deposits. (The thickness of marker layers is arbitrary, but made thick enough so boundaries are easily discerned. The marker layers create orange patches where they intersect the top grid layer.) Unit types at level I are hard to make out at this scale (see Figures 7 and 9). (b) Rendering of natural‐log permeability assigned as per distributions defined in Table 6 for each level I unit type.

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Table 5. Percolation Metrics Level I Units Within a Level II Polygona

Level I and II Units Within a Level III Polygona

All Levels Within a Channel Beltb

0.25 5.4 4.8 61.2 0.94 0.23

0.26 1.0 1.2 1.0 0.82 0.81

0.24 1.0 1.0 2.0 0.7 0.7

Fraction of all voxels that are ofg Number of percolating clusters in x Number of percolating clusters in y Number of percolating clusters in z Fraction of all ofg voxels in percolating clusters Fraction of all ofg in the largest cluster a

Average values of metrics computed from 10 different realizations at levels I and II. Metrics computed from a large scale realization with all levels.

b

number of cases to explain permeability correlation in natural sedimentary deposits [Dai et al., 2005; Ritzi and Allen‐ King, 2007]. Here it can be applied without regard to measurement error or sample bias. [44] Consider two points x and x′ separated by a lag vector h within the three level stratal hierarchy. Let x occur in level III unit type s, level II unit type r, and level I unit type o (hereinafter referred to as region type sro). Let x′ occur in level III unit type k, level II unit type j, and level I unit type i (hereinafter referred to as region type kji). The composite sample mean, variance, and semivariogram for the natural log transform of permeability, Y, are exactly equal to mY ¼

2Y ¼

XXX s

þ

r

o

XXX s

r

o

psro msro ;

2sro psro

2 1XXXXXX psro pkji msro mkji ; 2 s k r j o i

ðhÞ ¼

XXXXXX s

k

ð1Þ

r

j

o

sro;kji ðhÞpsro ðhÞtsro;kji ðhÞ;

ð2Þ

ð3Þ

i

where msro and s2sro are the mean and variance of log permeability for region type sro, psro(h) is the proportion of region type sro sampled by the tails of lag vector h, and tsro,kji(h) are the sample transition probabilities which represent the fraction of lag vectors that end in region type kji given that they start in region type sro. Equation (3) is a linear sum of 25 terms which can be organized into four groups based on stratal lag vector type. The possible unit types in the heads and tails of lag vectors are listed in Table 8 and organized into four groups: an aaa group (auto transitions at all three levels) that includes five of the terms summed in equation (3), an aac group (cross transition at level I) with six of the summed terms, an acc group with six of the summed terms, and a ccc group with eight of the summed terms. [45] Figure 12a also shows the relative contribution of each group of terms. We analyzed the contribution from each group, and decided which were important to explaining and modeling the semivariogram structure and which could be ignored on the basis of making negligible contributions.

[46] In plotting Figure 12a, the aaa group is also further subdivided, so that the contribution from autotransitions of cross stratasets is shown separately from those of channel fills. The aaa group contributes significantly at all lags, and most of that contribution arises from the autotransitions of cross stratasets. The aaa contribution from channel fills is negligible and is ignored hereinafter. At all lags, the aac group of terms makes the greatest contribution. The acc group of terms contributes negligibly and is ignored hereinafter. The ccc group of terms contributes significantly. Thus the shape and the range of the structure of the composite semivariogram can be explained by the sum of contributions from the aaa group of autotransitions among cross stratasets and the aac and ccc groups of cross‐ transition terms. [47] An understanding of the shape and correlation range (range hereinafter) of the curves for each of these three groups can be obtained by examining the psro(h)tsro,kji(h) terms in equation (3). These are plotted for each group in Figure 12b. At any lag, the psro(h)tsro,kji(h) gives the fraction of lags that start in region sro and end in region kji. The sum of all psro(h)tsro,kji(h) terms equals 1, and each curve indicates the fraction of transition types falling into that group as a function of lag distance. At any lag distance, the aaa group of terms makes up a significant fraction (sill at ∼0.5). The aac group rises to make up a significant fraction of transition types (sill at ∼0.3). Note that the aac group contributes more to the semivariogram than does the aaa group (Figure 12a) at all nonzero lags because the variability between the heads and tails of cross transitions reflects the difference in mean permeability between open‐ framework gravel and sandy gravel or sand, whereas the variability in autotransitions does not. Similarly, though the fraction of ccc cross transitions (sill at ∼0.15) is less than that of the aaa transitions, it contributes more to the semivariogram at lags beyond 150 m as compared with aaa Table 6. Mean and Variance for Distributions of Natural Log of Permeabilitya

s ofg sg xbar mcf a

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Units are darcies.

Mean

Variance

4.58 11.00 5.98 4.58 4.58

0.50 0.59 1.87 0.50 0.50

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Figure 9. Enlargement of the structure at the scale of cross stratasets rendered at a sample size and resolution similar to that in Figure 7. Layers of cells marking the boundaries of (top) unit bar deposits, and (top and bottom) cross stratsets are of arbitrary thickness varying with the orientation of the unit. In lower plot, marker cells have a dark blue color helpful for distinguishing individual cross stratasets, and do not indicate permeability.

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Table 7. Mean and Variances of ESL by Unit Type, as Sampled in Coordinate Directionsa

Compound bar deposits Major channel fills Unit bar deposits Cross‐bar channel fills Sand cross stratasets Open‐framework gravel cross stratasets Sandy gravel cross stratasets

Direction

Mean

Variance

y x z y x z y x z y x z y x z y x z y x z

291 121 1.07 74 23 0.90 41 20 0.24 31 27 0.85 3.40 2.11 0.12 4.23 2.49 0.156 19 11 0.59

70,740 8,134 0.82 4,432 319 0.37 1,376 246 0.022 196 358 0.31 5.50 0.22 0.0030 10.94 1.21 0.01 600 149 0.28

a

Contiguous units of the same type are part of one sample. Lengths are given in meters.

(Figure 12a) because of the larger difference in mean permeability in the heads and tails of its cross transitions. The fraction of all cross transitions (aac and ccc groups) is close to, and just a bit more than, the aaa group beyond about 100 m lag. However, because the cross transitions reflect differences in mean permeability between open‐ framework gravel and sandy gravel or sand, whereas the autotransitions do not, the combined cross transitions, as shown in Figure 12a, explains most of the composite permeability semivariogram. [48] The ranges of the psro(h)tsro,kji(h) curves for the aac and the aaa groups in Figure 12b are smaller than that of the ccc group. The range of the aac curve is small because it is mostly defined by cross transitions across different strataset types within a unit bar deposit. Likewise, the range of the aaa curve is small because it is mostly defined by autotransitions among strataset types within a unit bar deposit. However, the range of the ccc group is considerably longer because it is mostly defined by cross transitions across the larger‐scale compound bar deposits and major channel fills. On the basis of this understanding, the hierarchical levels are decoupled to explain and model the psro(h)tsro,kji(h). [49] Following from Ritzi [2000], Ritzi et al. [2004], Dai et al. [2005], and Ritzi and Allen‐King [2007], we know that the exponential‐like structures exist because the coefficient of variation (Cv, defined as the ratio of the standard deviation to the mean) among the ESL of unit types is close to unity. Therefore at level I the transition probabilities can be expressed as 3h to;i ðhÞ ¼ pi þ o;i pi e ao;i ;

ð4Þ

Figure 10. Histograms of IGL for archetypal polygons of unit bar deposits generated by a normal random number generator (see Figure 7a in part 1 of this paper): (a) length, (b) width, and (c) height.

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but we have shown that the acc group can be ignored. At level III 3h ts;k ðhÞ ¼ ps 1 eas;k

ð6Þ

as;k ¼ 3ls ð1 ps Þ;

ð7Þ

and

where l s is the mean length of compound bar deposits or major channel fills. Equation (3) is rewritten as ðhÞ ¼

X X X s

r

o

i h 3h 2sro psro psro þ ea ð1 psro Þ ;

i h i X X X X 1h 3h þ 2sro þ 2sri þ ðmsro msri Þ2 psro psri 1 ea ; 2 s r o i6¼o XXXXXX1h 2 i þ 2sro þ 2kji þ msro mkji psro pkji 2 s k6¼s r j6¼r o i6¼o h i 3h ð8Þ 1 ea ;

Figure 11. Histograms of ESL for unit bar deposits in the digital model (see Figure 7d in part 1 of this paper): (a) length, (b) width, and (c) height. where d o,i is the Kronecker delta and ao;i ¼ ao;o ¼ 3l o ð1 po Þ;

ð5Þ

where l o is the mean length of s, ofg, or sg cross stratasets [Ritzi, 2000]. Level II unit types would have been appropriate to modeling the acc cross‐transition group,

where aaaa and aaac are equal to ao,i, and accc is equal to as,k. These correlation ranges, computed using equations (5) and (7), are given in Tables 9a and 9b. [50] Figure 13 shows the semivariogram modeled using equation (8) and the sample semivariogram. The two are quite close. Most of the differences between the model and the sample semivariogram arise from differences occurring with the aac group between 10 and 20 m lag. We emphasize that the model curves have not been fitted to the sample bivariate statistics, but were independently developed from univariate statistics on stratal length. [51] In summary, most of the semivariogram structure arises from the structure of cross transitions between different level I cross stratasets, and between the larger scale level III compound bar deposits and major channel fills. The cross stratasets impart a short‐range (101 m) structure defined by their proportions, and mean and variance in length. The ordinate scaling of their contribution arises specifically from the variance of permeabilty within stratasets and the difference in mean permeability between different strataset types. [52] The level III unit types impart a longer range (102 m), exponential‐like structure. The ordinate scaling arises from the variance of permeability within stratasets occurring within a compound bar, the variance of permeability within major channel fills, and the difference in mean permeability between different strataset types and between them and major channel fills. [53] The sum of the shorter‐ and longer‐range cross‐ transition structures (green curve) explains the shape and range of the composite semivariogram in Figure 13 quite well, except for a relatively constant offset of about one unit on the ordinate scale. This relatively constant offset is the contribution given by autotransitions within stratasets. The ordinate scaling of their contribution arises only from the variance in permeability within stratasets, and is not influenced by the difference in mean permeability between stratasets. [54] The analysis shows that more than one level in the hierarchy of stratal architecture is relevant in the spatial bivariate structure of the permeability field. More details are given by Ramanathan [2009]. Ramanathan et al. [2008,

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Figure 12. Analysis of sample semivariogram and underlying structure (computed in the +y direction with 2‐m lag spacing; every fifth point plotted). (a) Sample semivariogram with the relative contributions of each group of terms. (b) The psro(h)tsro,kji(h) for each group of terms.

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Table 8. Unit Types in the Heads and Tails of Lag Vectors, as Grouped in the Analyses Transition Group

Heads

Tails

aaa

ofg sg s xbar mcf ofg ofg sg sg s s ofg sg s xbar xbar xbar ofg sg s xbar mcf mcf mcf mcf

ofg sg s xbar mcf sg s s ofg sg ofg xbar xbar xbar ofg sg s mcf mcf mcf mcf ofg sg s xbar

aac

acc

ccc

2010a] have linked multilevel heterogeneity in this vein directly to contaminant plume spreading through the Lagrangian‐based stochastic theory. The channel belt model is being used in ongoing work to further test the theory within that context.

9. Summary and Conclusions [55] A digital model for the hierarchical sedimentary architecture in braided channel belt deposits was created using the code presented in part 1 of this paper. The simulated architecture was compared with the real stratal architecture observed by Lunt [2002] and Lunt et al. [2004] in the preserved Sagavanirktok channel belt deposits. The comparisons included assessments of similarity which were both qualitative and quantitative. The qualitative assessments included comparing cross sections sampled from the digital deposit to field photographs and profiles from radar transects and trenches. The quantitative assessments were based on comparing proportions and three‐dimensional length metrics from the model to those reported by Lunt et al. [2004] for natural deposits, for all scales of unit types, across the hierarchical levels.

Table 9a. Correlation Ranges Computed From Equation (7), Level III Units

cb mcf Proportion weighted average

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Number of Occurrences

Proportion of Occurrences

Mean Length (m)

Cv

43,692 20,281

0.682 0.317

290.47 74.4

0.91 0.81

as,k (m) 276.68 152.44 236.99

Table 9b. Correlation Ranges Computed From Equation (5), Level I Cross Stratesets

s ofg sg Proportion weighted average

Number of Occurrences

Proportion of Occurrences Within Unit Bar Deposits

Mean Length (m)

Cv

102,218 705,448 768,944

0.065 0.447 0.488

3.42 4.89 13.51

0.70 0.84 1.41

ao,o and ao,i (m) 9.60 8.11 20.76 14.38

[56] From the qualitative comparisons we conclude that a synthetic deposit created by the code has unit types, at each level, with a geometry that is generally consistent with the geometry of unit types observed in the field. The digital unit types would generally be recognized as representing their counterparts in nature, including lobate‐ shaped unit bar deposits, trough‐shaped cross strata, and concave‐up channel fills. Furthermore, the synthetic deposit has a hierarchical spatial relationship among these units which represents how the unit types are observed in field exposures and geophysical images. From the quantitative comparisons, we conclude that length metrics for all unit types within the stratal hierarchy are generally within ranges quantified in the natural deposits by Lunt et al. [2004]. [57] A number of important attributes of the channel belt model were shown to be influenced by more than one level within the hierarchy of stratal architecture. First, the high‐permeability open‐framework gravels percolate at all levels and thus form preferential flow pathways. Open‐ framework gravels are indeed known to form preferential flow pathways in natural channel belt deposits [Lunt et al., 2004]. The formation of percolating clusters develops differently across different levels of the hierarchy of stratal architecture. As a result of the geologic structure, percolation occurs at proportions of open‐framework gravels below the theoretical threshold for random infinite media. When the channel belt model was populated with permeability distributions defined only at level I and without a permeability correlation model, the composite permeability semivariogram had a clear, multiscale, exponential‐like correlation structure, controlled by more than one level (scale) in the stratal hierarchy. These collective results are encouraging with respect to our goal that this model be relevant as a base case in future studies testing ideas for addressing the upscaling problem. [58] On the basis of collective qualitative and quantitative comparisons of the digital model with natural deposits, we conclude that the geometric‐based simulation methodology is quite appropriate for generating multiscale models which represent the hierarchical stratal architecture in channel belt deposits. The model presented here is indicated to be appropriate for future work testing ideas about addressing the upscaling problem. The success in this first phase encourages us to continue developing the methodology and to expand the code to represent more aspects of the stratal architecture.

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Figure 13. Model semivariogram and its components as compared with the sample semivariogram. The model was created without curve fitting. The model parameters were independently obtained from univariate statistics for quantifiable physical attributes including the mean lengths and the proportions of unit types at each relevant scale of the stratal architecture. [59] Acknowledgments. This research was supported by the National Science Foundation under grants EAR‐0510819 and EAR‐0810151. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect those of the National Science Foundation. A portion of the research was performed using EMSL, a national scientific user facility sponsored by the Department of Energy’s Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. The paper was improved based on comments and helpful suggestions from two anonymous reviewers.

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