The spatial variability of hydraulic conductivity in a shallow unconfined aquifer located at North Bay, Ontario, composed of glacial-lacustrine and glacial-fluvial sands, is examined in exceptional detail and characterized geostatistically. A total of 1878 permeameter measurements were performed at 0.05 m vertical intervals along cores taken from 20 boreholes along two intersecting transect lines. Simultaneous three-dimensional (3-D) fitting of Ln(K) variogram data to an exponential model yielded geostatistical parameters for the estimation of bulk hydraulic conductivity and solute dispersion parameters. The analysis revealed a Ln(K) variance equal to about 2.0 and 3-D anisotropy of the correlation structure of the heterogeneity (λ1, λ2, and λ3 equal to 17.19, 7.39, and 1.0 m, respectively). Effective values of the hydraulic conductivity tensor and the value of the longitudinal macrodispersivity were calculated using the theoretical expressions of Gelhar and Axness (1983). The magnitude of the longitudinal macrodispersivity is reasonably consistent with the observed degree of longitudinal dispersion of the landfill plume along the principal path of migration. Variably saturated 3-D flow modeling using the statistically derived effective hydraulic conductivity tensor allowed a reasonably close prediction of the measured water table and the observed heads at various depths in an array of piezometers. Concomitant transport modeling using the calculated longitudinal macrodispersivity reasonably predicted the extent and migration rates of the observed contaminant plume that was monitored using a network of multilevel samplers over a period of about 5 years. It was further demonstrated that the length of the plume is relatively insensitive to the value of the longitudinal macrodispersivity under the conditions of a steady flow in 3-D and constant source strength. This study demonstrates that the use of statistically derived parameters based on stochastic theories results in reliable large-scale 3-D flow and transport models for complex hydrogeological systems. This is in agreement with the conclusions reached by Sudicky (1986) at the site of an elaborate tracer test conducted in the aquifer at the Canadian Forces Base Borden.
 Although stochastic theories of subsurface flow and transport for applications involving heterogeneous geologic media have existed for several decades, field attempts at validating these theories in a variety of sediments are limited. To enable a quantitative assessment of the validity of the stochastic approach, it is not sufficient to only observe the spatial patterns of a dissolved tracer plume. It is also critical that the controlling hydraulic conductivity variations be mapped in detail sufficient to elucidate the geostatistical structure of the aquifer through which the tracer migrated. Such exercises have to date been limited to relatively simple hydrogeological settings, where the results of controlled tracer tests are available. Much less attention has been paid to demonstrate whether or not the spread of a large-scale plume such as that emanating from a landfill is consistent with the underlying geostatistical structure of the host aquifer.
 One of the first focused attempts at characterizing the spatial variability of hydraulic conductivity in a sandy aquifer where an intensively monitored tracer test had been conducted was performed by Sudicky . His work at the tracer test site at Canadian Forces Base (CFB) Borden, Ontario, has lead to much optimism concerning the workability of stochastic theories such as those developed by Dagan [1982, 1984, 1988], Gelhar and Axness , Neuman et al. , Neuman and Zhang , Zhang and Neuman , Di Federico and Neuman , among others. While the theories are typically based on the assumption of small perturbations of hydraulic conductivity, Rubin  indicates they are fairly robust, even for Ln(K) variances above 2.0. The analyses performed by Sudicky  suggested that an exponential covariance model can suitably describe the log-transformed hydraulic conductivity, Ln(K), variations in the CFB Borden aquifer. Simple least squares fitting of the exponential model to spatial autocorrelation data based on 1279 regularly spaced hydraulic conductivity measurements yielded a horizontal Ln(K) correlation length equal to 2.8 m and a vertical correlation length equal to 0.12 m. The variance of the Ln(K) process was estimated to be 0.29 after the removal of a nugget value equal to 0.09. Bulk hydraulic conductivity values and macrodispersion parameters calculated by Sudicky  on the basis of stochastic theories of groundwater flow and transport proposed by Dagan  and Gelhar and Axness  were shown to be consistent with the large-scale transport parameters deduced by Freyberg  from his analysis of the tracer plume characteristics (through the moment analysis).
 The purpose of this paper is to present the results of an intensive field study conducted in a sandy aquifer at North Bay, Ontario, that elucidates the three-dimensional (3-D) nature of the hydraulic conductivity variations. The data are analyzed in a geostatistical framework in order to quantify the statistical structure of the variations. We then use the geostatistical parameters to obtain the effective hydraulic conductivity tensor and macrodispersivity values using the theoretical expressions developed by Gelhar and Axness  and Dagan . Finally, we utilize the geostatistically derived effective hydraulic conductivity and macrodispersivity in a numerical groundwater flow and transport model to assess the predictability of ambient hydraulic head and concentration distributions of the chloride plume at the North Bay site.
2. Physical Setting and Site Hydrogeology
 The North Bay landfill site is located approximately 7 km northwest of the city of North Bay in the upper northeastern sector of the Chippewa Creek drainage basin (Figure 1). The study area extends from the landfill to Chippewa Creek located 900 m southwest. The landfill is approximately 28 ha in size and has been receiving both domestic and commercial (solid and minor liquid) waste from the city of North Bay and the adjoining North Himsworth Township since 1962. The landfill is unlined and is in direct contact with the water table.
 The parent material underlying the unconfined sand aquifer in the immediate vicinity of the study area is typically fine- to medium-grained, gray biotite gneiss that is middle Precambrian in age. Precambrian syenites and granites are common with numerous crosscutting quartz dikes visible in outcrops. The bedrock is part of a characteristic metamorphic suite found throughout the region. A near-continuous thin lodgement veneer of till material, approximately 1 m thick, covers the parent material. The till is typically a grayish, compact, poorly sorted, unstratified mixture of silt- to boulder-sized material containing less than 5% clay [Hewetson, 1981].
3. Borehole Locations and Hydraulic Conductivity Determination
 Continuous overburden cores were taken from 20 boreholes at 2.0 m and 4.0 m horizontal spacings along two intersecting transect lines located in the central region of the study area (Figure 2a). The ground surface elevations at each borehole were carefully surveyed to ensure proper vertical alignment of each core. Line 1 consists of 13 boreholes along a northeast-southwest transect that corresponds to the northeast-southwest transect of Moore . Line 2 contains eight boreholes along a northwest-southeast transect and is oriented at an angle of 60° with respect to line 1 (Figure 2b). All holes were drilled to refusal using the Waterloo cohesionless core barrel [Zapico et al., 1987] in conjunction with hollow stem augering. The core barrel employs an inner aluminum sleeve for sample collection and a piston device to minimize sample loss. The device allows the retrieval of relatively undisturbed samples. Core recovery was estimated at 86%.
 Each core was subdivided into 5 cm long subsamples which were prepared, repacked, and testing using a falling-head permeameter apparatus and procedure described by Sudicky . The 5 cm long subsamples were visibly found to be relatively homogeneous at this scale such that a hydraulic conductivity value for each represents a local point value. Occasional pebbles or stones found in the subsamples were removed before repacking in the permeameters. The total number of measurements performed along lines 1 and 2 are 1178 and 815, respectively. Duplicate tests performed on a variety of subsamples indicated a high degree of reproducibility. All hydraulic conductivity values were temperature-corrected to an aquifer temperature of 8°C.
 Natural log-transformed hydraulic conductivity values, Ln(K), were treated as point values and contoured using linear interpolation between the data points along the cross sections defined by lines 1 and 2 (Figures 3a and 3b). The spatial patterns of Ln(K) distributions are indicative of a complex network of long, relatively thin, irregularly truncated hydrostratigraphic units of similar Ln(K) material. The hydraulic conductivity values along line 1 range between 4.2 × 10−2 cm/s and 5.4 × 10−5 cm/s, while those along line 2 vary between 2.7 × 10−2 cm/s to 1.4 × 10−5 cm/s. Both cross sections visually exhibit horizontal correlation between neighboring cores. Results for line 1 (Figure 3a) suggest that the average bed length is longer than that observed for line 2 (Figure 3b). The thickness of the beds appears to be slightly greater for line 1 than for line 2. Overall, the orientation of the bedding appears to be mainly horizontal with the deeper beds showing some alignment with the bedrock surface. Details regarding the geostatistical analyses presented in sections 4–6 are given by Goltz .
4. Mean and Variance of Hydraulic Conductivity
 In this section, the suitability of a normal or lognormal probability density function (pdf) to describe the hydraulic conductivity variations will be examined and the first two moments (i.e., mean and variance) of the population parameters will be estimated. It will be assumed here that a trend in the hydraulic conductivity data does not exist and stationarity is assumed. The lack of an obvious horizontal trend is evidenced by the nature of the variations shown in Figures 3a and 3b (and further evidenced by vertically averaged data presented in this section).
 Frequency distributions for the K and Ln(K) data for a random subsample of size 200 and for the entire data set of size 1878 are displayed in Figure 4. Although it is recognized that the shape of the histogram distribution can be influenced by the selected class interval sizes, the Ln(K) distribution visually appears to better resemble a Gaussian distribution as compared with that for the untransformed K values. The Ln(K) histograms for the combined data set and a random sample of size 200 display a slight bimodality, while those for the untransformed hydraulic conductivity values are strongly skewed to the right. The Ln(K) histogram shows slightly skewed tails composed of Ln(K) values less than −11.0. Similar to the findings of Woodbury and Sudicky  for the Borden aquifer, these low values appear as outliers in the data. Although not shown, similar distributions were obtained for frequency histograms generated using the line 1 and line 2 data separately.
 For the class intervals shown in Figure 4, chi-square “goodness-of-fit” tests were performed in order to distinguish between the suitability of the normal and lognormal probability density functions. Using the untransformed hydraulic conductivity data and the randomly selected subsample size of 200, a chi-square statistic equal to 141.3 with 29 degrees of freedom was computed. For the Ln(K) histogram data, the chi-square statistic for the randomly selected subsample is equal to 43.2 with 23 degrees of freedom. Thus it would appear that Ln(K) better follows a normal distribution than K, with the chi-square statistic for the plotted histogram being significant at the 0.1% level. Similar chi-square results were obtained for Ln(K) data for line 1 and line 2 when treated separately. These findings were not sensitive to the size of the subsample randomly selected.
 A geometric mean hydraulic conductivity equal to 3.54 × 10−3 cm/s and a sample variance of Ln(K) equal to 1.79 were calculated for the entire data set (n = 1878; see also Table 1). The mean and variance of each core were plotted against horizontal distance along line 1 (Figure 5a) and line 2 (Figure 5b). Along line 1, some scatter of the individual core Ln(K) mean values about the overall sample mean for line 1 is apparent. The Ln(K) variances of each individual core along line 1 also display some scatter about the overall Ln(K) variance calculated for line 1. No apparent trend is, however, evident for either the mean or variance values of Ln(K) with distance for line 1. A similar degree of scatter in the Ln(K) variances for each core along line 2 is noted (Figure 5b), although the variability of the mean Ln(K) values along line 2 is somewhat larger than that for line 1. Nevertheless, it can be concluded that there is no strong visual evidence of a horizontal trend, and it is believed that the stationarity assumption for the North Bay aquifer is reasonable, at least over the distal scales of the measurements.
Table 1. Mean and Variance of Ln(K) for North Bay Aquifer
Combined Data Set
Sample size n
Geometric mean Kg (cm/s)
3.54 × 10−3
3.74 × 10−3
3.09 × 10−3
 Student's t and F statistics were performed to determine if the Ln(K) mean and variance values between lines 1 or 2 and the combined data set differed significantly. An average F-value (variance ratio test) was calculated between the combined data set and each of the transect lines, and between line 1 and line 2, from 300 randomly selected Ln(K) subsamples each of size 41 (degrees of freedom equal to 40). In each case, the differences in the variances between line 1, line 2, and the combined data set were not significant at the 95% confidence interval. This finding was not sensitive to the size of the subsample drawn from the data or to the number of subsamples compared.
 Student's t values (difference of mean test) were calculated using Ln(K) sample variance and mean values obtained for line 1, line 2, and the combined data set. At 1% level of significance, no statistical difference exists between the mean Ln(K) values for either line 1 or line 2 and the combined data set. Student's t values were also calculated for the 300 randomly selected subsamples used to calculate the average F statistic. In each case, the differences between the subsample means between line 1, line 2, and the combined data set, were not significant at the 95% confidence interval, which further supports the assumption of stationarity. A summary of the mean and variance data are given in Table 1 and are compared to Ln(K) mean and variances for the CFB Borden, Cape Cod, and the Columbus sites in Table 2. In comparison, the geometric mean hydraulic conductivity calculated for the North Bay aquifer is almost an order of magnitude smaller than the values calculated for the Cape Cod and CFB Borden aquifers. The Ln(K) sample variance for the North Bay aquifer being about 6–12 times larger than that for the Borden aquifer [Sudicky, 1986; Woodbury and Sudicky, 1991] and for the Cape Cod aquifer [Wolf, 1988; Hess, 1989], while the Columbus site is considered to be most heterogeneous of the four sites.
Table 2. Mean and Variance for Cape Cod, Canadian Forces Base Borden, Columbus, and North Bay Aquifers
 Spatial correlation in the measured hydraulic conductivity field can be characterized by either autocorrelation or variogram analysis using the Ln(K) data. Here it is elected to perform a variogram analysis because it avoids the need to normalize calculated covariance data by an estimated Ln(K) variance. Moreover, Woodbury and Sudicky  showed that the classical variogram estimator, among the various types available, yields a reasonable interpretation of the structure of the Borden aquifer and hence the classical form will be used. Two of the principal directions for the underlying correlation structure are assumed to lie in the horizontal direction and the third in the vertical direction. Line 1, which will be shown to contain the greatest persistence in spatial correlation, is assumed to follow the principal horizontal direction in the 1-D horizontal and vertical variogram analysis along each of the two core lines. This direction approximately parallels the longitudinal bedding direction observed in local overburden exposures and parallels the mean groundwater flow direction (see Figure 8a). Line 1 also closely parallels the longitudinal axis of the measured chloride plume along section A-A′. In the fully 3-D variogram analysis, however, the direction of maximum horizontal correlation is not assumed a priori to follow the line 1 direction because the horizontal angle of rotation required to achieve the principal directions becomes a fitting parameter in the analysis. Evidence for the selection of one of the two horizontal directions as a principal direction of correlation can be observed from the cross-sectional contours of Ln(K) (Figures 3a and 3b), which indicate that the beds are on average horizontal in orientation.
 The structure of the Ln(K) variations was determined from experimental variogram data using the classical variogram estimator, defined by Journel and Huijbregts  as
where Y = Ln(K), xi and sj (j = x, y, z) are the spatial coordinates and lag distances, respectively, and n(sj) is the number of pairs of observations separated by a lag distance s1. The experimental variogram data generated by (1) were fitted three-dimensionally using a negative exponential model given by
where sj* is the lag distance in the principal directions, λ(j) is the correlation length of Ln(K), σY2 represents the variogram sill value (equivalent to sample variance plus nugget), and σ02 represents the variogram intercept at zero lag distance which is the nugget value. An exponential model was selected from the numerous other possible models used in geostatistical analysis because it closely resembles the shape of the experimental variogram data and because it has been applied successfully in the past to determine Ln(K) structure of sandy aquifers [Sudicky, 1986; Wolf, 1988; Hess, 1989; Woodbury and Sudicky, 1991]. A discussion on the origins of the shape of a variogram is given by Ritzi , and further evidence of an exponential behavior is given in the analyses by Ritzi et al. , Dai et al. , and Ritzi and Allen-King . In fitting the variogram model given by (2) to the experimental variogram data, the fitting parameters are σY2, σ02, λ(j) (j = 1, 2, 3), and the angle of rotation β. We shall refer to this variogram model hereafter as the 3-D exponential variogram model.
 In order to perform the 3-D inversion the horizontal lag coordinates along lines 1 and 2 (s1 and s2, respectively) were projected onto the unknown principal axes of correlation (i.e., s1* and s2*). In this manner, the experimental variogram data generated in both the vertical and horizontal directions along line 1 and line 2 are considered simultaneously in the inversion process.
 For variogram data along line 1, sj* is defined by
where β represents the angle in the horizontal plane between line 1 and the direction of maximum correlation (positive in counter clockwise direction). For line 2 variogram data, sj* is defined by
where θ is the separating angle between line 1 and line 2 and is equal to 60°. The 3-D variogram model was fitted by nonlinear least squares estimation based on the Marquardt steepest decent scheme [van Genuchten, 1987].
 The fully 3-D form of the exponential variogram model (2) was fitted using both horizontal and vertical full-length variogram data from lines 1 and 2 simultaneously. The 3-D inversion yields the principal correlation lengths, λ1 and λ2, equal to 17.19 and 7.39 m, respectively. For the vertical direction, the principal value is λ3 = 1.02 m. The best fit Ln(K) variance and nugget values were found to be equal to 2.15 and 0.16, respectively. The values of these best fit geostatistical parameters are not substantially different from those obtained from the 1-D analysis because angle β between line 1 and the s1* direction is very small (β = 2.3°). Numerical experiments performed using different initial guesses of the geostatistical parameters and the angle β as input to the nonlinear least squares regression algorithm indicate that the fitted parameters are unique. Figure 6 shows the fit obtained between the 3-D variogram model and the experimental variogram data for line 1 and line 2 in the horizontal direction, and the combined data set in the vertical direction. Although there is considerable scatter in the horizontal data for large lag distances, the fitted variogram model nevertheless agrees with the general trends of the experimental data reasonably well. Thus the experimental model with parameters provided above will be adopted to describe the Ln(K) variations in the North Bay aquifer. The nugget value, σ02, equal to 0.16 corresponds to the variance attributed to small-scale variability and measurement error. Removing the nugget from the regressed Ln(K) variance of 2.15 gives a corrected variance for Ln(K) equal to about 2.0.
6. Effective Hydraulic Conductivity and Macrodispersivity Estimates
 Advances in stochastic theory for application to solute transport in heterogeneous media [e.g., Gelhar et al., 1979; Gelhar and Axness, 1983; Dagan, 1982, 1984; Neuman et al., 1987] have led to a methodology for predicting the bulk effective flow and transport parameters from a knowledge of the geostatistical parameters describing the spatial variations of the underlying Ln(K) process. The use of effective flow and transport parameters should, in principle, allow the replacement of the real heterogeneous aquifer with a macroscopic homogeneous equivalent for the purposes of making the large-scale predictions of groundwater flow and contaminant migration.
 The relationship between the asymptotic macrodispersivity tensor (Aii, i = 1, 2, 3) and the geostatistical parameters describing the spatial variability of the hydraulic conductivity field have been derived for a variety of hydrogeological settings by Gelhar and Axness . For the case of statistical anisotropy in the vertical and horizontal directions where λ1 > λ2 > λ3 (case 2 [Gelhar and Axness, 1983]), the longitudinal macrodispersivity A11 is given by
and ϕ is the angle in the horizontal plane between the mean flow direction and the longitudinal axis of the effective hydraulic conductivity tensor (ii). It should be noted that a longitudinal macrodispersion coefficient calculated using (5) must be augmented by the value of the local longitudinal dispersion coefficient. When the mean flow direction coincides with the λ1 direction (ϕ = 0°), the transverse macrodispersivity values are zero, thus indicating that the transverse macrodispersion process is controlled by local transverse dispersion. The effective hydraulic conductivity tensor used above is given by
 Using the best fit estimate values for λ1, λ2, and λ3 equal to 17.2, 7.4, and 1.02 m, respectively, and a corrected variance σY2 = 2.0, as estimated from the 3-D variogram analysis, the principal values of the effective hydraulic conductivity tensor become
with anisotropy ratios 11/22, 11/33, and 22/33 equal to 1.16, 5.18, and 4.44, respectively. The computed value of the longitudinal macrodispersivity using (5) equals 5.31 m on the basis of the geostatistical parameters derived from the 3-D variogram analysis. We note that the theory is based on the assumption of small perturbations (σY2 < 1) and consequently may not be valid for the very heterogeneous conditions of the North Bay site; however, as indicated by Rubin , comparisons between theory and numerical experiments suggest that the theory remains workable even for a Ln(K) variance as large as that estimated for the North Bay aquifer. Rubin  also argues that the flow factor ξ appearing in (5) should be unity because of an inconsistency in the order of the perturbation expansions between the flow and transport analyses presented by Gelhar and Axness . If ξ = 1, then the estimated value of the longitudinal macrodispersivity with ϕ = 0°, σY2 = 2.0, and λ1 = 17.2 m becomes σY2λ1 = 34.4 m.
 The longitudinal macrodispersivity value estimated here is considerably larger than those values estimated for the CFB Borden and Cape Cod aquifers and is slightly larger than that for the Columbus site, as shown in Table 3. These values for North Bay are about 9 times larger than the values obtained by Sudicky  and Freyberg  at the CFB Borden aquifer, and approximately 6 times greater than the value estimated by Garabedian , Garabedian et al. , and Hess  for the Cape Cod aquifer. These differences indicate that the North Bay aquifer is considerably more heterogeneous and exhibits a more spatially persistent correlation structure than either the Borden or Cape Cod aquifers.
Table 3. Corrected Ln(K) Variances (σY2), Correlation Lengths (λi), and Longitudinal Macrodispersivity Values (A11) for Cape Cod, CFB Borden, Columbus, and North Bay Aquifers
 A 3-D, variably saturated flow and transport model of the site was developed using the control-volume finite element model HydroGeoSphere [Therrien et al., 2005]. The purpose of this modeling effort was to simulate groundwater flow using the effective hydraulic conductivity, and a chloride plume from the landfill using the longitudinal macrodispersivity estimated from the stochastic theory of Gelhar and Axness . We will also simulate the plume migration using a longitudinal macrodispersivity value based on Dagan's  result that takes the value for the flow factor ξ to be unity, as well as for a case with a value reduced by 30% and equal to 3.7 m. This latter value is the smallest that could be used without introducing spurious numerical dispersion for the discretization employed. Our results will show that the different values obtained with the competing dispersion theories make little difference in the long-term longitudinal extent of a plume emanating from a long-term, near-continuous source. Because details on the flow transients were not available over the history of the operation of the landfill site, only a steady state variably saturated flow simulation was performed. Moreover, because the contaminant loading history is also unknown, a constant source input concentration is used, with the input concentration being adjusted within a reasonable range in order to best match the overall plume concentrations.
7.1. Model Domain
 The extent of the model domain was defined based on the migration of the leachate plume and the aquifer thickness. The model area was selected to include the entire extent of the plume as mapped in 1985. The aquifer pinches out near the bedrock outcrop that straddles the Airport Access Road (see Figure 2a), and therefore the model domain was modified to exclude those thinner areas (see also Figure 7). The northwestern boundary of the model domain was assigned along a flow line that follows the outer limits of the plume to the north. The southeastern boundary was similarly chosen along a flow line to the northeast of the bedrock outcrop that flanks the Airport Access Road. Southwest of the outcrop, the aquifer pinches out, and thus this very thin portion of the aquifer was eliminated from the model domain. The southern boundary is the natural boundary condition of Chippewa Creek, and the northeastern boundary was defined to capture the majority of the mass coming from the landfill. The resulting model domain is shown in Figure 2a.
 A 3-D, prism-element mesh was generated between the kriged surfaces generated from the ground surface elevations and the bedrock topography data. The ground surface elevations were taken from the level survey performed by Moore . Sand pit excavations have resulted in a topographic surface that changes over time, and hence for the purposes of the model, the surface elevations surveyed in 1985 were selected. This data set was kriged to generate a surface (Figure 7a). The bedrock topography data were derived from the geophysical survey completed west of Airport Access Road by Ben-Miloud  in combination with bedrock topography data for the landfill site from Moore . Ben-Miloud  performed seismic and resistivity surveys and combined this data with borehole logs to generate a bedrock elevation map. The combined data set was kriged to generate a bedrock surface (Figure 7b).
 The ground and bedrock surfaces were used to define the top and bottom, respectively, of a computational mesh of varying aquifer thickness (Figure 7c). In plan view, the triangular grid cells are relatively uniform and average approximately 8 m on a side. The domain was subdivided vertically into 20 layers, with a thickness that varies from 19 m in the thickest portions of the model domain to minimum of 0.05 m in the thinnest, therefore resulting in prism elements of varying thickness with sloping top and bottom surfaces. There are 4859 nodes per sheet of nodes which make up a 3-D mesh of more than 102,000 nodes.
7.2. Flow Boundary Conditions
 Both the northwestern and southeastern boundaries were designated as no-flow boundaries. At the southwestern end of the domain, nodes in the top layer along the creek and in the region of the marsh were assigned constant head values equal to the elevation of the nodes. The northeastern boundary in the landfill region was defined as a prescribed head boundary based on the water table elevations from August 1980 [Hewetson, 1981]. The prescribed heads along this boundary vary from about 344 m above mean sea level (msl) at the southern end to about 350 m above msl at the northern end, and were assigned over the entire thickness of the aquifer. This is equivalent to assuming that flow is essentially horizontal over this boundary. A uniform rate of infiltration was applied to the top of the model domain, and the infiltration rate was determined using the calibration procedure described below.
 The saturated values of the hydraulic conductivities input to the model were not altered from the effective values indicated in (7b). Instead, only the infiltration rate over the land surface was adjusted by a trial-and-error procedure. The flux entering the northeastern prescribed-head boundary can be back-calculated from the flow solution and is a function of the infiltration rate applied to the top of the model domain and its hydraulic properties. This boundary, located approximately 1 km southeast of the upper limits of the Chippewa Creek drainage basin, divides the basin approximately in half. Assuming that the principal direction of flow upgradient of the study area is consistent with that within the study area (where data are available), and that the rate of infiltration is the same in both halves of the domain, then the back-calculated flux should be equal to the total infiltration applied to the model. The flow model was calibrated by adjusting the infiltration rate until this condition was satisfied and resulted in a value of 150 mm/yr, which is reasonable given an average annual precipitation rate of 300 mm/yr for the city of North Bay (see http://www.weatheroffice.gc.ca/canada_e.html).
7.3. Aquifer and Solute Transport Properties
 The statistically derived bulk flow and dispersion parameters calculated in section 6 are used to conceptualize a macroscopically homogeneous flow regime that acts as a large-scale substitute for the actual heterogeneous aquifer. In essence, the small-scale perturbations in the hydraulic conductivity field are replaced by a macroscopically equivalent homogeneous aquifer that should control the plume behavior as it evolves over distal scales of hundreds of meters. The calculated principal values of the hydraulic conductivity tensor given by (7b), and the angular rotation with respect to the x-y Cartesian coordinate system used in the finite element mesh, were therefore used as input to the HydroGeoSphere model. The model locally rotates the hydraulic conductivity tensor on an element-by-element basis from the principal values in a standard finite element procedure given the principal values and the orientation of the x-y axes relative to the principal directions of anisotropy; the vertical direction is, however, assumed to be a principal direction.
 The flow simulation was performed to steady state by considering both the saturated and unsaturated zones using the mixed form of Richards' equation which is inherent to the HydroGeoSphere model. The van Genuchten  model was utilized to describe the saturation-pressure-head relationship in the unsaturated zone:
 The relative permeability is calculated using
where α, β, and γ are fitting parameters, Sw is water saturation, and ψ is pressure head. Table 4 lists the aquifer properties and the van Genuchten  parameters assigned in the model. The values of these parameters were chosen to be consistent with the textural properties of the surficial sands as determined from grain-size analyses. The residual saturation was chosen to be 0.06, and the air entry pressure was defined to be zero.
Table 4. Aquifer Properties Input Into Numerical Model
Hydraulic conductivity, principal values (cm/s)
9.02 × 10−3
7.73 × 10−3
1.74 × 10−3
van Genuchten parameters
Air entry pressure (Pa)
 The solute transport simulations considered here involve the migration of the conservative tracer chloride. A retardation factor of 1.0 was therefore used. The porosity of the sandy aquifer equals 0.35 (Table 4), which is a typical value for clean sand [Freeze and Cherry, 1979]. The value of 0.35 is at the lower end of Moore's  range of measured porosities for all of the overburden at the North Bay landfill study area. The free solution diffusion coefficient was assigned a value of 3.153 × 10−2 m2/yr (10−9 m2/s) and the tortuosity was chosen to be 0.8, with the effective diffusion coefficient given by the product of the free-solution diffusion coefficient and the tortuosity.
 Three different values of the longitudinal macrodispersivity were used as input to the model in order to assess the sensitivity of the simulated plume to the value of this parameter. The base-case value, equal to 5.31 m, was determined from the 3-D variogram analysis described above, complemented by the stochastic-analytic theory of Gelhar and Axness . The second value was estimated by assuming that the flow factor ξ = 1, which yields a longitudinal macrodispersivity equal to 34.4 m. Finally, the third value was obtained by arbitrarily reducing the base-case value by 30% to equal 3.7 m. This is the smallest value that could be used in the model without the introduction of spurious numerical dispersion for the given mesh discretization. The macrodispersivities input to the model must be augmented by the local dispersivity values. The theoretical values of the transverse macrodispersivities are zero when the principal direction of flow coincides with the principal direction of correlation, which is the case here. Transverse dispersion in the aquifer is therefore assumed to be due mainly to local transverse diffusion and mixing artifacts due to transient changes in the direction of groundwater flow. Because the details on the transients in the groundwater flow direction are unavailable, the input transverse horizontal and vertical transverse dispersivity values were adjusted within a moderate range by manual calibration of the simulated plume to the measured chloride concentrations. The final value for the horizontal transverse dispersivity is 0.4 m, and that for the transverse vertical dispersivity value is 0.01 m.
 In 1980, it was estimated that waste had been disposed over an area of 28 ha (2.8 × 105 m2) and the landfill has not expanded further since that time. Only about one quarter of the landfill site area is confined within the model domain, and therefore it is not possible to define the nature of the contaminant input zone over the entire 28 ha of the landfill region. The extent and location of the source was defined based on chloride plume concentration maps from 1980 and 1985 [Hewetson, 1981; Moore, 1986]. The source zone covers an area of 4.5 ha and was assigned a constant chloride concentration of 500 mg/L in the upper third of the model in the source zone, from slightly ground surface to a depth of 10 m. The nonsource finite elements in the top layer of the model, where solute-free infiltration occurs, were assigned as third-type boundaries with an input concentration equal to zero.
8. Model Results
 Using the statistically derived flow parameters, limited calibration was necessary to simulate the measured hydraulic head distribution at steady state. The simulated chloride plumes at a time equal to 23 years are compared with the measured 1985 chloride plume to evaluate the predictive effectiveness of the bulk flow and macrodispersion parameters.
8.1. Groundwater Flow Results
 The simulated steady state water table closely resembles the observed water table shown in Figure 8. Both the observed and simulated water table maps (Figures 8a and 8b) show relatively smooth equipotentials across the site. The sinuosity in the 345 m hydraulic head contour is evident at about the same location in both the simulated and measured heads, and is due to a slight mounding of the water table in this area where the aquifer thickness is relatively small. The hydraulic gradient steepens near the creek/marsh area, which can also be seen in both Figures 8a and 8b. While the hydraulic head patterns shown in Figure 8b depict a relatively smooth flow regime, the vertical flow velocity at depth in the aquifer becomes much more complex due to the variable thickness of the aquifer and the irregular bedrock topography. The complexity of the deeper portions of the flow regime has a major impact on the solute plume migration.
Figure 8c compares the observed and simulated hydraulic heads and was constructed from data presented by Moore . These data consist of head measurements obtained at various depths in the aquifer as well as water table measurements, and demonstrate a very good match between observed and simulated values.
8.2. Solute Transport Results
 Solute transport was simulated using the standard 3-D form of the advection-dispersion equation for the conservative tracer chloride. One snapshot of the measured chloride plume was used for comparison of the model to the field data (August 1985 data assembled by Moore ). As mentioned earlier, the calibration exercise mainly involved the adjustment of the transverse dispersivity values. The observed chloride plume extends from the landfill site to the marsh near Chippewa Creek following the principal direction of flow (Figure 9a). The background chloride concentration was measured to be less than 1 mg/L [Moore, 1986]. Moore  found that the highest chloride concentrations were only found at the southwest corner of the landfill, as shown in Figure 9a. Although the simulated plume for August 1985 (Figure 9b) discharges into the creek further upgradient than the observed plume, the toe of the plume is approximately the correct width. Seasonal changes in the flow regime as well as local fluctuations in hydraulic conductivity, unaccounted for in this macroscopically homogeneous model, could cause such discrepancies in the location of the plume centerline near its toe.
Figure 10 compares the measured and simulated chloride concentrations for 1985 along cross section A-A′. For the base case with a longitudinal macrodispersivity equal to 5.31 m (Figure 10b), this cross section shows that the simulated plume spreads longitudinally toward its toe, having a length that is in agreement with the observed plume. Also, the 50 and 100 mg/L contours extend upward as the plume migrates over the bedrock high located beneath the escarpment, as do the 50, 100, and 200 mg/L contours for the observed data. Intricacies exhibited by the observed plume, such as the fingering of the 50 and 100 mg/L contours seen near boreholes AP1 and MM29, or the extent and fingering of the 200 mg/L contour are not reflected in the simulated plume. This is not surprising given that the complex hydraulic conductivity field as shown in Figure 3 has been replaced by a homogeneous equivalent in the model. Figures 10c and 10d show the simulated plumes for the cases with the reduced (3.7 m) and increased (34.4 m) values of the longitudinal macrodispersivity. These changes produce almost no apparent difference in the simulated longitudinal extent of the 1985 plume. This is because the plume is approaching steady state conditions with the underlying assumptions of a steady flow field and constant source strength. Under such conditions, the length of a plume is dominated by the strength of the transverse dispersion process and becomes relatively insensitive to the value of the longitudinal dispersivity.
Figure 11 compares the measured and simulated 1985 plumes along the transverse cross section B-B′. The maximum concentration occurs at the same location for both the simulated and observed plumes. The observed plume is, however, narrower than the simulated plume. This is because the 3-D bedrock topography input into the model, based on kriging of all the available depth-to-bedrock data (see Figure 7b), differs significantly from that depicted in Figure 11a. The large bedrock high in the simulated bedrock surface explains the steep concentration gradients that exist toward the northwest side of the plume in the plan view diagram (Figure 8). Figures 11c and 11d again illustrate that changing the longitudinal macrodispersivity value has a minor effect on the plume, except for the case with large longitudinal macrodispersivity equal to 34.4 m. The slightly larger lateral spreading observed in Figure 11d compared with the base case of Figure 11b is because the large longitudinal macrodispersivity results in enhanced values of the dispersive cross terms appearing in the advection-dispersion equation given that the local velocity field is nonuniform on account of the irregular surface and bedrock topography.
9. Summary and Conclusions
 A detailed analysis of the geostatistics of hydraulic conductivity obtained from an extensive series of permeameter measurements on repacked 0.05 m long subsamples of core material collected from a sand aquifer at North Bay, Ontario, was conducted. The cores were obtained from 20 boreholes drilled along two transects intersecting at a 60° angle. Frequency distributions of the data indicate that the sample population is lognormally distributed with a geometric mean of 3.54 × 10−3 cm/s and sample variance of Ln(K) equal to 1.79. Distinct outliers comprising fine-grained material appear in the Ln(K) frequency distributions. Hydraulic conductivity values were observed to range over about 3 orders of magnitude and indicate the strong heterogeneity of the aquifer materials.
 Contoured plots of the Ln(K) data for each of the transects reveal that the aquifer consists of long irregularly truncated hydrostratigraphic units of similar Ln(K) material. The lengths of the beds are considerably longer and slightly thicker along line 1 than line 2, thus indicating statistical anisotropy in the horizontal directions. Overall, the beds appear to be flat-lying with some minor alignment of the deep beds with the lower bedrock/till surface. A detailed geostatistical analysis of the spatial variability of hydraulic conductivity revealed the 3-D statistical anisotropy for Ln(K) aquifer. A 3-D variogram analysis indicated that an exponential variogram model provides a reasonable description of the correlation structure of the sandy aquifer. The values of the geostatistical parameters obtained by fitting the exponential model to the experimental variogram are a Ln(K) variance of 2.0, principal horizontal correlation lengths given by λ1 = 17.2 m, λ2 = 7.4 m, and a vertical correlation length (λ3) equal to about 1.0 m. These correlation lengths appear to be in good agreement with observations of the vertical profiles of hydraulic conductivity between neighboring cores, and with the contoured profiles of Ln(K) presented for cross sections along the two transects. A small angle (β < 3°) in the horizontal plane between the maximum correlation and the orientation of line 1 was estimated from the 3-D variogram fit, indicating that line 1 essentially coincides with the principal axis of Ln(K) correlation. This conclusion is reinforced by close alignment of line 1 with local sediment bedding orientation, and with the longitudinal axis of a leachate plume emanating from a landfill located in the study area.
 The effective hydraulic conductivity and macrodispersivity tensors were calculated based on the 3-D stochastic theory presented by Gelhar and Axness . An effective hydraulic conductivity in the mean flow direction, K11, equal to 9.0 × 10−3 cm/s and a longitudinal macrodispersivity value equal to 5.31 m are calculated for the aquifer. Using K11 = 9.0 × 10−3 cm/s, an average horizontal gradient equal to 0.022 [Moore, 1986], and a porosity of 0.35, an average linear groundwater velocity of about 180 m/yr is calculated. This value is in good agreement with a value calculated on the basis of the rate of advance of the chloride plume. A 3-D variably saturated flow and transport model of the site was developed using the numerical model HydroGeoSphere [Therrien et al., 2005]. The purpose of this modeling effort was to simulate groundwater flow using the effective hydraulic conductivity tensor and the transport of the chloride plume from the landfill using macrodispersion parameters estimated from the stochastic theory. The statistically derived effective hydraulic conductivity values based on data at the North Bay site allowed a reasonably close calibration of the flow system to the measured water table. The flow regime is highly complex due to variable aquifer thickness and the irregular bedrock surface. Seasonal variations in the groundwater flow regime appear to affect the plume but were not explicitly accounted for in the model. Rather, the effects of the seasonal flow variations were addressed by an augmentation of the values of the transverse macrodispersivities used as input for the transport simulations. On the basis of a longitudinal macrodispersivity equal to 5.31 m, the simulated solute plume was similar in length to the observed plume. It was also shown the length of plume was relatively insensitive to the value of the longitudinal macrodispersivity.
 This study demonstrated that the use of statistically derived parameters, based on stochastic theories, results in a reliable 3-D model for complex hydrogeological systems. This is in agreement with the conclusions reached by Sudicky  at the site of an elaborate tracer test conducted in the aquifer at the Canadian Forces Base Borden and represents one of the few attempts [e.g., Sudicky, 1986; Zhang and Neuman, 1990; Hyun et al., 2002] at validating stochastic theories of groundwater flow and/or solute transport in three dimensions at a site where extensive field data have been collected.
 Support for this research was provided by grants from the Natural Sciences and Research Council (NSERC) of Canada awarded to E. Sudicky and W. Illman, and a Canada Research Chair in Quantitative Hydrogeology held by E. Sudicky.