Linking hierarchical stratal architecture to plume spreading in a Lagrangian-based transport model: 2. Evaluation using new data from the Borden site

Transcription

1 Click Here for Full Article WATER RESOURCES RESEARCH, VOL. 46, W01510, doi: /2009wr007810, 2010 Linking hierarchical stratal architecture to plume spreading in a Lagrangian-based transport model: 2. Evaluation using new data from the Borden site Ramya Ramanathan, 1 Robert W. Ritzi Jr., 1 and Richelle M. Allen-King 2 Received 2 February 2009; revised 11 August 2009; accepted 3 September 2009; published 22 January [1] A new data set collected at the well-documented Borden research site was used in order to evaluate a newly published but as yet untested idea for stochastic modeling (Ramanathan et al., 2008). The new data set reveals the stratal architecture of the Borden aquifer and allowed us to determine how the stratal architecture, at different scales, controlled the macrodispersion observed in the original natural gradient tracer test. The newly published idea for modeling uses a Lagrangian-based model for the particle displacement variance developed from independent, physically based, quantifiable univariate statistics, including the proportions and mean length of the strata. The method for defining model parameters avoids the often equivocal step of fitting sample bivariate permeability statistics. The model parameters were developed from the new data from the Borden site. The new data included geologic data in much greater abundance than permeability data. There are two scales of unit types delineated in a hierarchy of stratal architecture. Their shapes are complex, not simple layers or lenses. The method facilitated the direct use of both geologic and permeability data, which need not be collocated. The dispersion model developed from these data represents the field-measured particle displacement variance that occurred in the natural gradient tracer test well. The contributions to time-dependent macrodispersion by strata at each scale were computed and analyzed independently. This analysis revealed that macrodispersion at the Borden site is primarily controlled by the proportions, and the mean and variance in length of larger-scale strata of medium sand (M) and strata of fine sand and silt (FZ), with secondary contributions by smaller-scale strata types occurring within the larger-scale units. When sampling the pattern in the longitudinal direction, M and FZ couplets repeat at 10 m intervals on average, with a high length variance. To reach a time-constant macrodispersivity, the stratal length variability must be fully sampled by the plume. The macrodispersivity becomes asymptotic beyond an advective distance of about 60 m, corresponding to about 12 longitudinal transitions between M and FZ unit types. Citation: Ramanathan, R., R. W. Ritzi Jr., and R. M. Allen-King (2010), Linking hierarchical stratal architecture to plume spreading in a Lagrangian-based transport model: 2. Evaluation using new data from the Borden site, Water Resour. Res., 46, W01510, doi: /2009wr Introduction [2] Sudicky [1986] presented the results of collecting a large number of permeability measurements taken at the site of the well-documented natural gradient tracer test conducted at the Borden research site [Mackay et al., 1986]. These data, along with a statistical characterization of the tracer plume [Freyberg, 1986], allowed Sudicky to examine the validity of existing stochastic theories for contaminant transport [Dagan, 1982, 1987; Gelhar and Axness, 1983]. [3] As per Rubin [2003], one attractive point of stochastic modeling is that it can represent both the intrinsic spatial 1 Department of Earth and Environmental Sciences, Wright State University, Dayton, Ohio, USA. 2 Department of Geology, State University of New York at Buffalo, Buffalo, New York, USA. Copyright 2010 by the American Geophysical Union /10/2009WR007810$09.00 variability of subsurface attributes controlling flow and transport and the epistemic uncertainty about that variability at any specific site. Opportunities to test the theories with field-measured data are few for at least two reasons. First, the subsurface is challenging to access and thus is difficult to characterize. Sudicky s effort of collecting some spatially distributed permeability data has seldom been repeated in subsequent research at other sites. Second, a research-quality tracer test requires a huge investment in time and resources before yielding a complete and usable data set. The Borden site is one of only a small number of examples where research-quality data for the spatial distribution of permeability and the spatiotemporal distribution of a tracer are both available, and it was the first of such sites. Because opportunities to evaluate stochastic theories against field data are indeed few, articles containing such evaluations can be significant contributions. Indeed, Sudicky [1986] is the fifth most cited article in Water Resources W of12

2 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 Research (F. Schwartz, personal communication, 2009). We will not review all of these citations but will review only the most relevant to this article, as organized into the following three areas. [4] 1. The models of Dagan [1987] and Gelhar and Axness [1983] which Sudicky [1986] evaluated are based on the integral scale, traditionally determined by fitting model curves to sample bivariate statistics computed from large numbers of spatially distributed permeability data. Discussion of the spatial bivariate correlation of log conductivity and modeling it, including estimating its integral scale, can be found in work by Kemblowski [1988], Sudicky [1988], Woodbury and Sudicky [1991], Turcke and Kueper [1996], and Ritzi and Allen-King [2007]. Modeling the spatial bivariate correlation is problematic. As elucidated by Woodbury and Sudicky [1991, p. 533], even with large amounts of permeability data it is unlikely that one can unequivocally distinguish between various competing variogram models. [5] 2. Further discussion centering on the applicability of stochastic theories to explain the observed tracer behavior can be found in a number of works. Relevant examples include Dagan [1989a, 1989b], Naff et al. [1988, 1989], Kemblowski [1988], Sudicky [1988], Barry and Sposito [1990], Rajaram and Gelhar [1991], and Fitts [1996]. [6] 3. As a part of the discussion under points 1 and 2, there have been a number of conjectures about the stratal architecture of the Borden aquifer. Examples include discussions by Dagan [1989a, 1989b], Naff et al. [1988, 1989], Woodbury and Sudicky [1991], and Rajaram and Gelhar [1991]. [7] This article is in the same vein as Sudicky [1986]. We used a new data set collected at the Borden site (R. Polmanteer, M.S. thesis in preparation, State University of New York at Buffalo, 2010) in order to evaluate a newly published but as yet untested idea for stochastic modeling [Ramanathan et al., 2008]. [8] The new data set reveals the stratal architecture of the Borden aquifer and thus allows us to move beyond conjecture. We can, for the first time, directly discuss how the stratal architecture, at different scales, controlled the macrodispersion observed in the original natural gradient tracer test. [9] The newly published idea for modeling uses a Lagrangian-based model for the particle displacement variance developed from independent, physically based, and field-quantified univariate statistics, including the proportions and mean length of the strata. It avoids the curvefitting problems discussed under point 1 above. The new data set creates the opportunity to develop the model parameters specifically for the Borden aquifer and to evaluate the model against the original tracer data. As stated above, opportunities to test the stochastic theory with field-based methods are rare, and such testing is the main contribution of this article. [10] The article is organized as follows. Section 2 discusses the stratal architecture as now known over a 30 m length of the transect that was traversed by the tracers. Univariate statistics for proportions and length of strata types are computed, and permeability distributions corresponding to those strata types are presented. Section 3 presents a few relevant aspects of the work by Freyberg [1986] and subsequent articles that have computed metrics for the plume spreading observed in the Borden tracer test. Section 4 summarizes the model of Ramanathan et al. [2008] and, using the univariate statistics as parameters, compares it to those metrics for the observed plume spreading. [11] Before moving to section 2, we first give more discussion on the motivation for this work. The goal is to develop models that avoid the fitting of sample bivariate statistics for permeability (which is often equivocal) and which clearly quantify the relative contributions of each scale of stratal architecture to the plume spreading. The Lagrangian-based approaches to modeling relate the particle displacement variance of a conservative solute to the spatial covariance of log permeability [Rubin, 2003]. Lagrangianbased models have gone through several phases of evolution. As first introduced, the model was based on a representing log permeability as a spatial random function, Y(x), with a spatial covariance defined by a single, finite integral scale [Dagan, 1982]. This represented a significant advantage over Eulerian models that were based on a dispersion parameter. This advantage lay in the fact that the dispersion parameter of Eulerian models could only be determined through model calibration (i.e., history matching to concentration data and not field-measured data independent of the state variable), whereas in the Lagrangian-based approach the model for plume spreading could be built up from basic measurements of permeability without concentration history matching. The downside was that in this first phase, the approach to defining the integral scale involved fitting covariance or semivariogram model functions to sample bivariate statistics. In essence, one fitting parameter had been replaced by another. Determining sample bivariate statistics for permeability required an abundance of spatially distributed permeability data, and permeability data with the appropriate spatial distribution are difficult, time consuming, and expensive to collect. This may be the primary reason why, though successfully demonstrated, the Lagrangian-based model has been little used by practitioners. Furthermore, the process of choosing and fitting model functions to sample bivariate statistics was highly equivocal even with relatively abundant permeability data [Woodbury and Sudicky, 1991]. Linkage between the integral scale and stratal architecture was vague, and therefore, geologic data were not particularly helpful in dealing with equivocal models. [12] In a second phase of development, the Lagrangianbased model for plume spreading and the model for bivariate statistics it was based on were linked more closely to sedimentary unit types [see Rubin, 1995; Dai et al., 2004a; Rubin et al., 2006]. For i sedimentary units, the permeability Pwas represented by the spatial random function Y(x) = I i (x)y i (x), where I i (x) is a discrete indicator space random i function assuming the value of 1 if x falls within stratal type i and 0 otherwise. This formulation made a stronger link to geology, but to develop the model for a specific site required defining the integral scales for permeability correlation within each unit [e.g., Dai et al., 2004b]. Defining those in-unit integral scales for permeability requires fitting models to sample bivariate statistics for each unit. Doing so is subject to the same unreasonable data requirements and equivocal outcomes as in the first phase. [13] However, use of deterministic geostatistics [Isaaks and Srivastava, 1988] has shown a way around the problem of modeling bivariate statistics. In this approach, rather than deriving expressions for conjectures on how permeability 2of12

3 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 Figure 1. Photos of exposures of the Borden aquifer. (a) Boundary between two level II unit types (FZ and M) indicated. Arrows at the top point to etchings on the exposure illustrating what would be sampled by cores spaced 0.25 m apart. (b) Two level I unit types (FLD and MM) and a channel-like feature cutting into them. varies by using a probabilistic framework and space random functions, sample statistics are developed from the study of data-rich sites sampled deterministically. A number of studies using this approach showed that the relevant underlying spatial structure of permeability was almost completely defined by the proportions of transitions across strata types as a function of lag [Ritzi et al., 2004; Dai et al., 2005; Ritzi and Allen-King, 2007]. From that insight, it was realized that the global permeability covariance could be modeled by modeling the probability of transitioning across unit types, and, importantly, the permeability integral scales within those units were inconsequential and could be ignored. Furthermore, the deterministic studies showed that the crosstransition probabilities could be modeled quite well on the basis of the proportions and the univariate statistics for length of unit types, as computed directly from geologic data (discussed further in section 4, equations (7) and (8)). Thus, permeability correlation could be modeled without fitting sample bivariate statistics for either indicator or continuous space variables. The approach may not work for all sedimentary deposits. Prior experience has shown that the approach does work well at sites where the sediment can be delineated into stratal unit types which have different distributions for permeability and which repeat together within a facies assemblage. [14] Following these developments in a newer, third phase of evolution, Ramanathan et al. [2008] presented a Lagrangian-based model in which the model parameters governing time-dependent behavior are the proportions and mean lengths of strata. Both of these parameters are univariate statistics directly representing physical attributes which can be directly quantified from geological data. [15] In addition to avoiding the need to fit spatial bivariate statistics, there are other advantages which stem from having a model based on geologic data. Strata types can be directly observed, and their lengths can be measured from outcrop exposures, cores, geophysical transects, or stratigraphic models, as in examples by Dai et al. [2005], Lunt et al. [2005], Rubin et al. [2006], and Ritzi and Allen-King [2007]. As a result, geologic data are typically much more abundant than permeability data, and thus, there is strong benefit in being able to directly use these data. Furthermore, because the model is directly linked with quantifiable attributes of the stratal architecture, it provides insight into how different scales of unit types within a hierarchy of stratal architecture each contribute to solute plume spreading. [16] Because of these advantages, we are motivated to test and evaluate the Ramanathan et al. [2008] model against field data. One reviewer questioned the value of testing the idea within the safe confines of the Borden site (i.e., a site in which the overall variance of ln(k) is less than unity). We feel that an evaluation at the Borden site is a significant contribution for two reasons. First, Borden is the only site for which there is both a research-quality natural gradient tracer test documented in the literature and, now, as described below, a quantitative characterization of the stratal architecture (proportions and horizontal lengths of strata types). The Borden site alone now provides the opportunity to test the theory, and tests of theory are important. Second, if the strata types of the Borden aquifer, with mild permeability contrasts, are shown to control dispersion, then strata types with higher permeability contrasts at other sites likely do also, and thus, if the method is shown to work at Borden, it creates optimism that it will work at other sites. In this latter context, the Borden aquifer provides an especially good test site. 2. Stratal Architecture of the Borden Aquifer [17] The stratal architecture of the Borden aquifer has been investigated and described by Bohla [1986], Allen-King et al. [1998], Divine [2002], and Ritzi and Allen-King [2007]. Importantly, the last of these defined a hierarchy of texturalbased strata types, with larger-scale (level II) strata, each comprising assemblages of smaller-scale (level I) strata types. Figure 1 shows photographs of exposures of some of these strata types, described in Tables 1a and 1b which convey the complete hierarchy. [18] The unit types at any hierarchical level fill space and are mutually exclusive [cf. Ritzi et al., 2006]. The volume proportion of level II unit type r is p r, and the proportion (as per total volume) of a level I unit type o that occurs within it is p ro. [19] The method involves using lithologic data to compute the proportions, p r and p ro, and mean lengths, l r and l ro. Permeability data are used to define the mean and variance of log permeability, ^m ro, and ^s 2 ro, respectively, at level I only. [20] The lithologic and permeability data used in this study are derived from 67 cores taken along a 30.5 m long transect parallel to the axis of the well-known Borden 3of12

4 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 Table 1a. Hierarchy of Unit Types, Proportions, and Univariate Statistics for Permeability at Level II Unit Type Description ^p r (x I ) ^p r (x k ) ^m r (x k ) a ^ r 2 (x k ) a M associations of level I medium sand units FZ b associations of level I fine sand and silt units a Statistics for natural log transform of permeability in cm 2. b Combines the fine sand and silt categories of Ritzi and Allen-King [2007]. natural gradient tracer test, as described by Polmanteer (manuscript in preparation, 2009). This transect was located as close to that axis as possible without disturbing ongoing experiments, starting at the origin of the tracer test but with a western offset of 10 m [see Ritzi and Allen-King, 2007, Figure 2]. The cores were extracted from below the water table and within the depth interval shown in Figure 2. Fiftysix of these cores were newly collected in order to augment data from an existing 11 cores previously described by Divine [2002], Allen-King et al. [2006], and Ritzi and Allen-King [2007]. [21] The previously existing 11 cores are those on the 1 m spaced ticks in Figure 2 from 5 m north (5 N) to 15 N. Ritzi and Allen-King [2007] showed that those cores were insufficient for determining the horizontal length statistics for two reasons. First, the 1 m spacing of those cores was insufficient for characterizing length statistics for level I unit types, with specific concern about the variance in length being underrepresented. Second, the 10 m length of that transect was insufficient for characterizing the length of longer, level II unit types, with specific concern about the mean length being underrepresented because of insufficient lateral exposure. The new cores were taken to augment the previously existing data from the old cores, with the goal to overcome each of these two deficiencies. [22] To sample with higher resolution and to better characterize length statistics for level I unit types, new cores were taken in the following manner. Between 5 N and 15 N, the new cores were taken 0.5 m between each pair of old cores, which was the smallest spacing deemed to be possible if disturbed zones around old core holes were to be avoided. Over the virgin interval from 15.5 N to 25.5 N the new cores were taken with a 0.25 m spacing, using a multitube sampling procedure described by Polmanteer (manuscript in preparation, 2009) to minimize cross-coring disturbances. [23] To characterize level II length statistics, the transect was extended to 30.5 m, a distance thought a priori to be sufficient on the basis of study of an analog quarry pit exposure at another location on base Borden. The project resources did not support taking more than 56 new cores in total, so the transect was extended from 25 N to 35.5 N by taking 10 more cores with a 1 m spacing, with those cores targeted specifically for augmenting the level II analyses. [24] Polmanteer (manuscript in preparation, 2009) studied grain size and lamination in the cored sediment samples. On the basis of that study and reexamination of the older cores, Polmanteer developed a number of alternate hierarchical classifications of the sediment by defining different scales of unit types, each based on textural or depositional attributes. As demonstrated by Dai et al. [2005], our methods can be applied using any alternate hierarchy, but classifications are more parsimonious when the delineation of unit types captures distinctions in permeability distributions. [25] We chose to work with the hierarchy described in Tables 1a and 1b and mapped in Figure 2. The seven texturalbased level I unit types have fairly distinct permeability distributions (shown below). Furthermore, the map in Figure 2b corresponds well to visual features easily identifiable in a photo mosaic of the core sediment. This allows us to relate plume spreading to visually identifiable attributes of the sedimentary architecture. The two unit types at level II capture larger-scale associations of level I unit types and have even greater distinction between the permeability distributions. Note that some of the erosional surfaces traceable across a photo mosaic of the core sediments are indicated in Figure 2b, and these surfaces define channel-like features. Generally, the sediment is coarser within the base of these channel-like features (mostly medium sand types) and is generally finer upward (mostly fine sand and silt types). Thus, the level II unit types, though strictly based on texture, correspond closely to the depositional-based facies associations described by Polmanteer (manuscript in preparation, 2009). [26] The database of indicators identifying level II unit types, I r (x), and level I unit types, I ro (x), used for quantifying proportions and lengths was created by sampling all cores with a 0.01 m vertical spacing. This interval was somewhat arbitrarily chosen, being an even multiple of a meter and being relatively exhaustive in the context of defining horizontal length samples. This gave 9764 sample Table 1b. Hierarchy of Unit Types, Proportions, and Univariate Statistics for Permeability at Level I Unit Type a Description b ^p ro (x I ) ^p ro (x k ) ^m ro (x k ) ^ 2 ro (x k ) MLD medium sand, distinct lamination MLF medium sand, faint lamination MM medium sand, massive FLD fine sand, distinct lamination FLF fine sand, faint lamination FM fine sand, massive Z silt a Level I categories correspond to Ritzi and Allen-King [2007] as follows: MLD and MLF split HPXS according to whether lamination is distinct or faint. MM equals MCG. FLD combines LPXS, CPXS, XSS, and DPL. FLF combines FPXS and FPL. FM equals MFG. b Laminated units may have planar or cross stratification. 4of12

5 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 Figure 2. Maps of lithology identified in the cores as per Tables 1a and 1b for (a) level II unit types and (b) level I unit types. Permeability samples were taken across the black horizontal dotted lines. Yellow lines delineate erosional surfaces, showing channel-like features. locations, x I. Polmanteer (manuscript in preparation, 2009) addressed the issue of bias in estimates of proportion which is created by the clustering of sample locations [Isaaks and Srivastava, 1989], with highest density in the middle third of the transect and lowest density in the northern third. To give equal representation from each section and to remove bias from clustering, the estimates of proportion for unit types at each level of the hierarchy were computed using data from each of the cores on the X.5N locations (X =5,6,...35 with 1 m spacing). These are given for the unit types at each hierarchical level in Tables 1a and 1b as ^p r (x I ) and ^p ro (x I ). [27] Permeability was measured using air-based methods as discussed by Polmanteer (manuscript in preparation, 2009). Measuring permeability is more time consuming than identifying lithology, and thus, only a subset of the locations with indicator data was chosen for measurement. These permeability sample locations, x k, are indicated in Figure 2 where each of six dotted lines intersects a core. These sampling lines have relatively equal vertical spacing and were somewhat arbitrarily chosen. Locations were chosen considering locations of data in previously existing cores such that proportions mirrored those of the full indicator database and also considering issues related to developing a collocated database for chemically reactive attributes of the sediment, which are outside the scope of this article. These intersections give 396 locations at which there are log 5of12

6 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 Figure 3. Cumulative frequency of ln(k) for (a) level II unit types and (b) level I unit types. permeability data, Y ro (x k ). Distributions of these data according to strata type, at each level, are given in Figure 3. [28] Polmanteer (manuscript in preparation, 2009) addressed the issue of how bias is created in estimates of ^m ro and ^s 2 ro because of the clustering of permeability sample locations, just as is true with the lithologic data. Thus, just as with computing proportions from lithologic data, these permeability statistics were computed using data only from the cores on X.5N locations. Note in Tables 1a and 1b that the proportions computed using x k are generally similar to those using x I, indicating that the x k sample locations are distributed in a way that generally represents the unit types in proper proportions. Because level I unit type Z occurs in very low proportion, it is challenging to characterize using x k locations. Core was substituted for 21.5 in these calculations in order to better represent unit type Z. The resulting univariate statistics for permeability are given in Tables 1a and 1b. [29] Samples of horizontal length were exhaustively taken from the I ro (x I ) and I r (x I ) data along horizontal lines, spaced 0.01 m in the vertical. In this way each unit is typically sampled many times through its thickness, and thus, both the mean and variance in the horizontal length of each unit occurrence are represented in these data, as well as the mean and variance among all occurrences of the unit type. The more closely spaced cores from 5 N to 25.5 N were used to characterize the horizontal lengths of level I unit types. The X.5N cores that cover the full 30.5 m transect were used to characterize the horizontal lengths of the longer level II unit types. [30] Samples which have fewer than two terminations create bias, causing the mean length to be underrepresented. Fewer than two terminations per unit arise because of the following: (1) the unit extends beyond the lateral extent of the cores, (2) there are gaps in the data because core sample was not recovered, and (3) there is a fringe at the top and bottom (Figure 2) because the cores do not extend to uniform depth. To avoid bias from the fringe, x I are only used from depths between and m. To reduce the remaining bias, a procedure was used on the basis of termination frequencies and Bayesian updating algorithm developed by White and Willis [2000], as modified by Dai et al. [2005]. The results are given in Table 2. The proportions are further refined in this procedure, but little changed, as reported in Table 2. Thicknesses were computed from the same data following the same procedures and are reported in Table 3. [31] Sudicky [1986, Figure 6] presented a contour map of log-transformed permeability data for each of his transects, and the reader may be expecting the same here with Figure 2. There are 9764 lithologic sample points in Figure 2. Those familiar with the research effort required to sample and measure permeability would understand that generating a permeability value corresponding to each of those 9764 locations, in order to generate a corresponding permeability map, is an unreasonable expectation. In what must have been a large investment of time, Sudicky measured 1279 values. A large research effort was required in order to generate the 396 values for our study. Examining the six lines in Figure 2 along which our permeability data fall, it is clear that permeability cannot be contoured in any meaningful way. These observations in fact underscore the following points. (1) Lithologic data are easier to generate than permeability data. (2) Therefore, lithologic data are usually more abundant. (3) We benefit from being able to use the more abundant lithologic data. (4) We benefit if less permeability data are required as a result. Here we have used less than a third of the number permeability data used by Sudicky [1986]. 3. Metrics Characterizing the Plume Spreading Observed in the Borden Tracer Test [32] As described by Mackay et al. [1986], the concentration data collected to characterize the dispersion of the Table 2. Analysis of Horizontal Length a Level II N ^p r l r C v l r b M FZ Level I N ^p ro l ro C v l ro c MLD MLF MM FLD FLF FM Z a Shown are numbers (N) and proportions ( ^p) of horizontal length samples, univariate statistics for length (mean, l, in meters, and coefficient of variation, C v ) and the associated integral scales (l in meters), for level I and level II unit types. b Proportion weighted average for level II is c Proportion weighted average for level I is of12

7 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 Table 3. Analysis of Thickness a Level II N ^p r l r C v l r b M FZ Level I N ^p ro l ro C v l ro c MLD MLF MM FLD FLF FM Z a Shown are numbers (N) and proportions (^p) of horizontal length samples, univariate statistics for length (mean, l, in meters, and coefficient of variation, C v ) and the associated integral scales (l in meters), for level I and level II unit types. b Proportion weighted average for level II is c Proportion weighted average for level I is chloride and bromide tracer plumes in the Borden natural gradient test were generated from samples extracted from within a dense array of multilevel sampler ports. Synoptic sampling was conducted at different times through the duration of the field experiment. At any one of those times, the data have been used to compute sample spatial moments of the concentration distribution. The second moment around the center of mass defines a sample spatial covariance tensor, ^X ij, used as the metric for plume spreading at that time. It has been computed in two dimensions by Freyberg [1986] and in three dimensions by Barry and Sposito [1990] and Rajaram and Gelhar [1988, 1991] for each of the synoptic sampling rounds. We will not repeat the equations but will only make a few relevant points about the different methods used among these authors and the differences in their results. We start by focusing on ^X 11 (t i ), which represents the variance in the longitudinal displacement of particles around the centroid of the plume at time t i of a synoptic round. [33] Because the data are discrete, ^X 11 (t i ) must be computed by numerical integration. These numerical integrations were performed in two steps. The first step involves a vertical integration of the concentrations using trapezoidal quadrature. The second step involves an integration in the horizontal plane, which requires interpolation between sample locations and extrapolation to the zero concentration boundary. This step has been approached differently among Freyberg [1986], Barry and Sposito [1990], and Rajaram and Gelhar [1988, 1991], and the approaches and differences in the outcomes have been a source of debate within their articles. The differences between ^X 11 (t i ) among their results may be due more to their extrapolation schemes than their interpolation schemes [Farrell et al., 1994]. Before discussing that further, we summarize factors which affect the estimates. [34] There are a number of factors that obscure our understanding of the true longitudinal displacement variance X 11 (t). Some of these lead to ^X 11 (t i ) that are reduced from X 11 (t). First, as pointed out by Sudicky [1988], the water samples collected for the analysis of tracer concentrations comprised volumes on the order of 70 cm 3. Given this sample volume and taking into account the porosity, the measured concentration is representative of an average within a sphere surrounding the point of extraction that is of the order of Sudicky s estimated vertical correlation of ln(k) equal to Following from Kemblowski [1988, equation (7)] or Dagan [1989a, equation (4.7.1)], the reduction of the variance of a vertically graded (averaged) process would consequently be a factor of 2e 1 or Sudicky [1988] uses this logic in arguing support for the use of a leading constant b = 0.74 in his model for plume spreading, as was suggested to him by Dagan. [35] Second, the sample locations did not capture the full plume in a number of the synoptic rounds in both the vertical and the horizontal directions. Among the extrapolation schemes, the Rajaram and Gelhar case 4 had the maximum extrapolation distance, which led to a significant improvement in the mass accounted for in the computation, and a 75% increase in ^X 11 for the time at which it was implemented. We take this as a strong indication that the estimates computed with other methods have underrepresented X 11 (t). [36] In addition to extrapolation between ports, the interpolation between ports creates some smoothing, more so with the Freyberg [1986] method than with the Rajaram and Gelhar [1991] method [Farrell et al., 1994]. Another smoothing effect would be created by the use of a constant porosity in the computations. [37] Barry and Sposito [1990] have also pointed out that the true plume spreading may have been reduced from what theories [Dagan, 1987, 1988] would suggest because of boundary effects created by the close proximity of the water table above and clay layer below (see elucidation of boundary effects by Rubin and Dagan [1988, 1989]). Furthermore, in contrast to the model theory, the tracer experiment represents only one realization within the ensemble result that the model random variables conceptually represent. [38] Some of the results for ^X 11 (t i ) from among Freyberg [1986], Barry and Sposito [1990], and Rajaram and Gelhar [1991] are given in Figure 4. The collective results create an envelope of interpretations for X 11 (t), to which we compare model results below. [39] Models for the particle displacement variance, ^ X 11 (t), for both the 2-D isotropic and 3-D anisotroptic forms by Dagan [1987, 1988], have been compared to these results [e.g., Sudicky, 1986; Naff et al., 1988; Barry and Sposito, 1990; Woodbury and Sudicky, 1991]. The largetime asymptotic limit of the effective macrodispersivity ( D ^ 11 (t) = dx ^ 11 (t)/2dt) is the same for the 2-D and 3-D models. Furthermore, the D ^ 11 (t) of the 2-D and 3-D models are close, before that limit is reached, with the 2-D form being lower. Neither model accounts for all of the factors listed above which bias ^X 11 (t i ) to be lower. As stated above, Sudicky [1986] presented the 2-D form as reduced by a leading constant b = 0.74 in order to account for the reduction due to sample averaging effects. This seems a reasonable correction to us and would also apply to a 3-D model. Barry and Sposito [1990] presented the 3-D form but reduced it by a leading constant of Part of this was attributed to 2 correcting s lnk for temperature; however, the variance of ln(k) is not affected by correcting K by a constant multiplier. Another part was attributed to the nugget in Sudicky s [1986] covariance model. However, from Ritzi and Allen-King [2007] it is clear that any apparent nugget is mostly due to 7of12

8 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 spatial covariance tensor are so small that they cannot be distinguished from noise and chose not to present them. Though Barry and Sposito [1990] and Rajaram and Gelhar [1991] provide noisy ^X 33 (t i ) estimates, comparing the 3-D model to them would be virtually meaningless. [42] As elucidated by Naff et al. [1989], the estimates for the transverse horizontal displacement variance ^X 22 (t i ) are much larger than can be explained by the Dagan [1989a] models and probably reflect changes in the direction of the gradient. Naff et al. [1989] presented a model which accounts for temporal variations in the gradient. However, as with the Dagan [1989a] models, the model we examine below does not. Therefore, we are not able to evaluate the transverse moments computed from the model, and we will only examine X ^ 11 (t) from the model. Figure 4. Borden plume moments ^X 11 (t i ) from various investigators and 3-D model X ^ 11 (t). Model is shown (a) with b reduction and (b) without b reduction. Shown in both are the separate contributions of the ac and cc model terms to particle displacement variance. For ^X 11 (t i ) in both Figure 4a and Figure 4b, circles denote bromide, and crosses denote chloride from Freyberg [1986]. Pluses denote bromide with normal extrapolation, asterisks denote chloride, and squares denote bromide with case 4 extrapolation from Rajaram and Gelhar [1991]. Solid triangles denote upper jackknife bound, and open triangles denote the lower jackknife bound for bromide from Barry and Sposito [1990]. not representing the unsampled, finite-length transitions less than the 1 m sampling distance, which can affect dispersion, and is not due to measurement error as implied by those authors (see also Sudicky s comments about highly reproducible K measurements). Thus, neither of the two Barry and Sposito [1990] corrections seem to be supported to us, though they do make the model fall closer to ^X 11 (t i ). [40] Our conclusion from reviewing this prior literature is that it is logical that a model reduced by b should indeed be closer to those ^X 11 (t i ) which are most affected (reduced) by the various effects summarized above and thus which fall at the bottom of the envelope of interpretations plotted in Figure 4. A model without the empirical b adjustment [e.g., Naff et al., 1988] should indeed be closer to and perhaps above the ^X 11 (t i ) which are least affected by these effects and fall at the top of the envelope of interpretations. This is consistent with the opinion of Rajaram and Gelhar [1991, p. 1247]. We will examine the Ramanathan et al. [2008] model within this context, with and without b. It will allow study of the relationship between the stratal architecture and plume spreading, which will be shown to be the same in either case. [41] Now we comment on the transverse moments. Freyberg [1986] pointed out that the general temporal variation in estimates of the vertical components of the 4. Summary of the Transport Model and Method [43] The model of Ramanathan et al. [2008] for the particle displacement variance can be written as adapted to any stratal architecture and is here written in a form specifically for the Borden stratal hierarchy as in Tables 1a and 1b. Furthermore, the model written for this hierarchy is a linear sum of terms, which can be organized (combined) in alternate ways in order to study contributions from among the possible transitions across different strata types. Here the terms are organized to study contributions from transitions across level I unit types collectively, as separate from transitions across level II unit types collectively, consistent with the focus of Ritzi and Allen-King [2007] and with how they are presented by Ramanathan et al. [2008]. The model can be written in 2-D or 3-D forms. [44] The model for X ^ 11 (t) has two terms as written here. The first term represents transitions across smaller-scale level I unit types within the same level II unit type (ac, autotransition at level II, cross-transition at level I). The second term represents transitions across units that are different with respect to both level II and level I unit types (cc, cross transitions at both level II and level I): where B ¼ X r ^ X 11 t ðþ¼al 2 ac xt ac i6¼o þ Bl 2 cc xt cc ; ð1þ A ¼ X X X 1 h i 2 ^s2 ro þ ^s2 ri þ ð^m ro ^m ri Þ 2 ^p ro ^p ri r o X X X 1 h 2 ^s2 ro þ i 2 ^s2 ji þ ^m ro ^m ji ^p ro ^p ji : j6¼r o i6¼o In the 2-D model 3 xt ac ¼ 2tac 3ln t ac þ 2 3E þ 3Ei t ac 2 þ 3 e ta þ e ta t ac 1 = t ac ð4þ 3 ¼ 2tcc 3ln t cc þ 2 3E þ 3Ei t cc 2: þ 3 e tcc þ e tcc t cc 1 = t cc ð5þ xt cc ð2þ ð3þ 8of12

9 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 Equations (4) and (5) have dimensionless time parameters t ac ¼ U 1t l ac ; t cc ¼ U 1t l cc ; and each level has an integral scale defined by ð6þ l ac ¼ l ro ð1 ^p ro Þ ð7þ l cc ¼ l r ð1 ^p r Þ; ð8þ and U 1 is the mean groundwater velocity. [45] The model can be developed by computing l ac and l cc directly from geologic data (equations (7) and (8)) without the need for collocated permeability data. Permeability enters in only through univariate statistics (A and B) that essentially scale the ordinate. The time-dependent terms in the model do not involve permeability. [46] Equations (7) and (8) apply when the coefficient of variation in the length of unit types is close to unity, which has consistently been found true in investigations of different sedimentary deposits having enough data for such evaluation [Ritzi, 2000; Ritzi et al., 2004; Dai et al., 2005; Ritzi and Allen-King, 2007]. These studies have shown that when this is true, the log permeability covariance is well approximated by C Y ðhþ ¼ Aelac h þ Be h lcc ; as assumed in deriving equation (1). Furthermore, l ac and l cc represent the integral scales of exponential crosstransition probabilities, which represent the underlying geologic structure [see Ritzi and Allen-King, 2007]. Here the transition probabilities (^t) are decoupled for the unit types at each hierarchical level: ^t ro;ri ðhþ ¼ ^p ri 1 elac h ð10þ ^t r;j ðhþ ¼ ^p j 1 e h lcc ð9þ ; ð11þ where i and j represent the level II and level I unit types, respectively, and r 6¼ j and i 6¼ o. Thus, the spatial structure of equation (9) is defined entirely by the geologic data and not by permeability data. In all of the studies cited in this paragraph, equation (9) or its semivariogram equivalent has been directly and rigorously evaluated and found to indeed be a good representation of the underlying permeability structure. These studies have used data representing a variety of sedimentary deposits representing point bar, alluvial, glaciofluvial, and beach or nearshore deposition. The success in these studies is used as the rationale for indeed skipping direct evaluation of the permeability covariance here and assuming that equations (7) and (8) can be computed directly and used in equation (1). We only check to make sure the coefficient of variation in length is indeed close to unity. Thus, the method does not require directly computing or fitting the sample permeability covariance. [47] The 3-D model assumes the ratios of the vertical l ac and l cc to their horizontal counterparts are the same value, e. This implies that the vertical to horizontal aspect ratio of the geometry of level I and level II unit types is the same, and e is indeed shown to be about the same below. The model is written as in equation (1) above except ¼ 2tac þ 2ðe tac 1Þþ8e xt ac xt cc Z 1 0 ½J 0 ðfþ 1Š 1 v 2 er er Z 1 dr 2e J v 2 u1=2 2vu 3=2 0 ðfþ J 1ðfÞ f " # e 3 R 3 ðe 2 R 2 þ 5 þ 5R 2 Þ ðe 2 R 2 1 R 2 Þ 3 þ 1 þ R2 5e 2 R 2 u 3=2 v 3 dr ð12þ ¼ 2tcc þ 2ðe tcc 1Þþ8e Z 1 0 ½J 0 ðfþ 1Š 1 v 2 er er Z 1 dr 2e J v 2 u1=2 2vu 3=2 0 ðfþ J 1ðfÞ f " # e 3 R 3 ðe 2 R 2 þ 5 þ 5R 2 Þ ðe 2 R 2 1 R 2 Þ 3 þ 1 þ R2 5e 2 R 2 u 3=2 v 3 dr; ð13þ where f = Rt zc, u =1+R 2, v =1+R 2 e 2 R 2, z =[a, c], J 0 and J 1 are the zero and first order Bessel functions, and R is the variable of integration. [48] Ramanathan et al. [2008] also derived the corresponding equation for the large-time limit or asymptote of the macrodispersivity, D ^ 11,1 /U 1,whichis ^ D 11;1 U 1 ¼ Al ac þ Bl cc : ð14þ It is the same for 2-D and 3-D forms. [49] As discussed in detail by Ramanathan et al. [2008], the equations are built by simply adapting the Dagan framework to an empirically determined covariance expression and thus will work best under the same assumptions as those of Dagan [1982]: (1) the average gradient is assumed to be aligned with the principal axis, (2) flow is at steady state, (3) the velocity field is uniform, (4) the domain is unbounded, (5) the log conductivity is weakly stationary, and (6) the global variance of log conductivity is less than unity. 5. Computing X 11 (t) and Comparing to Observed Tracer Moments [50] A and B were computed as and 0.258, respectively, by applying equations (2) and (3) to the univariate statistics in Tables 1a and 1b. As per Table 2, both the level II unit types and the more abundant and thus the better characterized of the level I unit types (MLD, FLD, and FLF; see Table 1) all have a coefficient of variation in length close to or exceeding unity. As per Ritzi [2000], this gives rise to exponential transition probabilities (equations (10) and (11)) and justifies using equations (7) and (8). [51] The horizontal l ac and l cc were computed from equations (7) and (8). The results using each unit type in turn are given in Table 2. If the unit types occurred in equal numbers so that proportion was only determined by mean 9of12

10 W01510 RAMANATHAN ET AL.: LINKING HIERARCHICAL STRATAL ARCHITECTURE, 2 W01510 lengths, the result would be the same when computed among unit types at a given level. However, as can be seen in Table 2, the number of occurrences is not the same, and there is variation among the computed l ac and l cc. Following the procedure shown to be successful in a number of prior studies [Ritzi et al., 2004; Dai et al., 2005; Ritzi and Allen-King, 2007], the proportion-weighted average is used at each level. This gives a horizontal l ac equal to 1.19 m and horizontal l cc equal to 2.11 m. [52] The vertical l ac and l cc were computed following the same procedure, and the results are given in Table 3. This gives l ac equal to 0.09 m and l cc equal to m. The e ac, and e cc are close at 0.07 and 0.05, respectively. Accordingly, we use an average equal to 0.06 for e in the model. [53] U 1 is given as 0.09 m/d by Sudicky [1986]. Thus, during the first 330 days of tracer advection, the center of the plume moved approximately 30 m and thus covered the distance represented by the transect (Figure 2) from which the parameters for the model were developed. Following ^ Woodbury and Sudicky [1991], X 11 (t) is augmented by X 11 (t = 0) using the field data from Freyberg [1986]. [54] Figure 4a shows the 3-D model premultiplied by Sudicky s b of 0.74 following from the discussion in section 3. The 2-D model plots essentially the same. The 3-D model without b is plotted in Figure 4b. In each plot, the model can be compared to published estimates of ^X 11 (t i ) which were reviewed in section 3. It is logical that the model reduced by b should indeed be closer to those ^X 11 (t i ) which are most affected (reduced) by the various effects summarized in section 3 and which thus fall at the bottom of the envelope of interpretations plotted in Figure 4. Likewise, the model without the b adjustment should indeed be closer to, and perhaps above, the ^X 11 (t i ) which are least affected by these effects and fall at the top of the envelope of interpretations. This result is consistent with prior published model results discussed above. [55] A more important point of discussion is with regard to the insight we gain from the model about the dispersion process, with specific focus here on how it relates to the hierarchy of stratal architecture. The insight is the same from either the 2-D or the 3-D model and is independent of the b adjustments. In Figures 4a and 4b the first and second terms in equation (1) are plotted independently. It is clear that the second term (cc), representing the effect of the level II architecture, explains almost all of the particle displacement variance. The first term (ac) explains only a small percentage of the variance. [56] The A in the model represents the variability in permeability in transitions across level I unit types. Breaking it down further, the P P P 1 r o i6¼o 2 [s ro 2 + s 2 ri ]p ro p ri term represents the variability at the head or tail of a level I cross transition, and P P P 1 r o i6¼o 2 [(m ro m ri ) 2 ]p ro p ri represents the difference in mean permeability between the heads and the tails. These are equal to and 0.005, respectively. For transitions across level II unit types, represented by B, the P P P P 1 r j6¼r o i6¼o 2 [s ro 2 + s 2 ji ]p ro p ji term is 0.094, and the P P P P 1 r j6¼r o i6¼o 2 [(m ro m ji ) 2 ]p ro p ji term is Thus, in transitions across level I unit types, the variability at the head or tail of the transition, within those respective unit types, contributes more to the model than does the mean contrast in permeability across those units. The combined contribution is smaller than that from transitions across level II unit types. In transitions across level II unit types, the contrast in mean permeability between the head and tail unit types contributes more to the model than differences at the head or tail of the cross transition. [57] From the collective results, it is clear that the proportions and length statistics for the level II unit types in Figure 2a are the most important attributes of the stratal architecture and subsurface spatial structure in explaining the plume spreading. Note that these attributes (proportion and length) for both the M and the FZ strata define the probability of cross transitions as a function of lag distance in the model (through equations (9) (11) from which the model was developed [Ramanathan et al., 2008]). Thus, it is not the geometry of just one of the unit types alone that controls the plume spreading but the occurrence of both of them and the consequent cross-transition probability structure. The sum of the mean lengths for M and FZ are of the order of 10 m. The variance in length around the mean is high, and length samples along horizontal lines range from many tens of meters down to a meter, as apparent from Figure 2a. Thus, the plume must sample many transitions across M and FZ units before fully sampling the average heterogeneity. These are the attributes of stratal architecture that combine to most affect the plume spreading. [58] These attributes are different than those that were inferred to be important by Sudicky [1986]. Sudicky [1986] attributed his 2.8 m integral scale to what he saw as the appearance of discontinuous lenses of higher and lower ln(k) in the contour map of his data. He described the lenses as typically of the order of 1 or 2 m in length. Our experience suggests that those features were probably artifacts arising from interpolating permeability measurements among the smaller level I unit types across the 1 m lateral spacing as between 26.5 N to 35.5 N in Figure 2b. In examining between 15 N and 25 N in Figure 2b, the level I strata types are generally not lenticular. More importantly, the strata actually controlling the plume spreading are the larger-scale level II units in Figure 2a. Furthermore, as a more fundamental point, equations (7) and (8) make clear that the relevant integral scales are related to and generally only a fraction of the mean length among unit types, and thus, one should not try to match the magnitude of the integral scale directly to the typically size of any one unit type. Indeed, Ramanathan et al. [2008] showed that the global integral scale is given by l ¼ 1 s 2 Al ac þ Bl cc : ð15þ Y [59] In light of this, and the results presented here, it is clear that Sudicky s global integral scale is a mathematical abstraction of stratal architecture, embodying the proportions, mean length, and variance in lengths of more than one unit type, at more than one hierarchical level, and thus should not be expected to correspond directly to the typical length of a lens. [60] Equation (14) gives the large-time limit for macrodispersivity. The asymptotic macrodispersivity is 0.62 m and is almost entirely defined by Bl cc. A time of 667 days 10 of 12