# Diavik Waste Rock Project: Geostatistical Analysis of Sulfur, Carbon, and Hydraulic Conductivity Distribution in a Large-Scale Experimental Waste Rock Pile

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## Abstract

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## 1. Introduction

_{S}). This investigation was focused on these three parameters because all are directly related to the geochemical evolution of waste rock and all can be measured using simple well-defined techniques. Saturated hydraulic conductivity was included, even though most waste rock piles are unsaturated, because it can be estimated using particle-size distribution (PSD) and is a parameter that is readily incorporated into models (compared to D

_{10}—the diameter of particle size for which 10% of a given sample is smaller—for example). Limited investigations have been conducted regarding the heterogeneity of these fundamental parameters. Fala et al. [8] investigated the role of heterogeneity in hydrogeological properties (K

_{S}, suction, and volumetric water content) on water flow through a hypothetical waste rock pile and found that calculated flow patterns generally followed the distribution of K

_{S}within a waste rock pile (i.e., higher flow corresponding to higher K

_{S}zones). The work of Fala et al. [8] showed that coupling a heterogeneous K

_{S}distribution with a heterogeneous S and/or C distribution in a mechanistic implementation could define further the role of mineralogical and K

_{S}heterogeneity in the geochemical evolution of mine-waste rock.

_{4}concentrations that showed no spatial dependence within the pile, and a correlation between these geochemical parameters and particle-size distribution was not identified. Khalil et al. [10] collected mine waste samples at the abandoned Ketarra mine in Morocco to assess the extent of metal impacts. They suggested a normal statistical distribution was appropriate for solid samples collected at their study site, where samples were widely distributed and collected from near the waste rock pile surface, and therefore may not be indicative of distribution through the depth of the pile. Marescotti et al. [11] used geostatistics to quantify the distribution of metals in waste rock samples from a sulfidic waste rock pile. Their semi-variogram work indicated spatial relationships for most of the assessed parameters, but, similar to the work of Khalil et al. [10], samples were collected close to the surface of the waste rock pile. Recent work by Blannin et al. [12] used geostatistics to assess sampling strategies for investigation of spatial variability of selected metals in a tailings impoundment. Although this study focused on mill tailings, which fundamentally differ from waste rock, the results indicated that geostatistics can be successfully used in the assessment of anthropogenically deposited materials.

_{0.852}Ni

_{0.004}Co

_{0.001}S] [16], with small contents of chalcopyrite [CuFeS

_{2}], sphalerite [ZnS], and pentlandite [(Fe,Ni)

_{9}S

_{8}], were primarily contained in the biotite schist [15]. Mineralogical analyses indicated C was present only as carbonate [16].

_{4}, Ni, Co, Cu, Zn) and acidic pH (average annual pH ranging from 4.7 to 4.3 between 2008 and 2012) indicative of sulfide mineral weathering. Effluent geochemistry from the Type I experiment was documented by Bailey [19] and was generally similar to the Type III effluent with lower concentrations of solutes indicative of sulfide mineral weathering and circumneutral to slightly acidic pH (average annual pH ranging from 5.9 to 6.7 between 2008 and 2012).

_{S}of the matrix material (i.e., diameter < 5 mm) based on samples collected during the construction of the Type I and Type III test piles and the deconstruction of the Type I test pile. The geostatistical analysis is used to characterize the geochemical and physical heterogeneity of a waste rock pile from the perspective of the influence of these parameters on effluent quality. Additionally, reactive transport simulations are included to demonstrate the influence of matrix material proportion on the geochemical evolution of waste rock.

## 2. Methodology

_{S}were conducted on samples collected for mineralogical analysis and PSD as part of the construction (Type I and Type III) and deconstruction (Type I) of the test-pile experiments. Samples collected during the construction of the Type I and/or Type III test piles are referred to as ‘construction’; samples collected during the deconstruction of the Type I test pile are referred to as ‘deconstruction’. Samples collected during the construction phase of the experiment generally had limited associated spatial information; samples collected during the Type I deconstruction were spatially located using a Real Time Kinetic Global Positioning System with +/−2 cm accuracy [17].

_{S}. Further detail regarding the collection of mineralogical and PSD samples during the construction of the test piles was documented by Smith et al. [13,20].

_{S}from the results of the PSD analysis were evaluated as part of this investigation. The methods of Hazen [34], Schlichter [35], Terzaghi [36], and Chapuis [37] (Equation (1)) were used to calculate K

_{S}and compared to the results of the Neuner et al. [25] measurements.

_{S}is saturated hydraulic conductivity (cm s

^{−1}), D

_{10}is the grain diameter at which 10% of the mass of a given sample is finer (mm), and e is the void ratio (m

^{3}m

^{−3}). The use of the Kozeny–Carman equation [38] was rejected due to the requirement for a soil specific surface value. Neuner et al. [25] measured K

_{S}on 18 matrix material samples from the Diavik experiments using constant-head permeameter tests.

_{S}within the Type I test pile. The approach included calculation of sample statistics, assessment of stationarity, estimation of semi-variograms, and fitting of theoretical semi-variograms to estimated semi-variograms.

_{S})) from the deconstruction of the Type I test pile. Three common variogram estimation methods were applied to assess the capability of the estimators to describe the distribution of the waste rock in the Type I test pile. The classical variogram estimator [39] is:

_{i}is the log-normally transformed parameter at a given location, and y

_{i}

_{+}

_{s}is the log-normally transformed parameter at a second location of lag distance (s) from y

_{i}.

^{2}

_{Y}is the global variance δ(s) and is the Kronecker delta equaling 1 when lag distance s = 0 and 0 when s > 0.

^{2}

_{0}is the nugget and λ (m) is the correlation length.

## 3. Results

#### 3.1. Calculation of Hydraulic Conductivity

_{S}values calculated using the methods of Hazen [34], Schlichter [35], Terzaghi [36], and Chapuis [37] (Table 1), the Chapuis [37] equation was selected as the preferred method for calculation of K

_{S}for this geostatistical analysis due to comparable geometric mean and standard deviation to the laboratory measured values of Neuner et al. [25]. The values calculated using the Schlichter [35] method were also comparable to the measured values but were rejected due to the reliance of the method on empirical parameters and temperature and a significantly lower standard deviation. The void ratio used in the Chapuis [37] equation was based on the mean porosity of 0.25 reported by Neuner et al. [25].

#### 3.2. Statistical Distribution

_{S}profiles and calculation of sample statistics, including mean and standard deviation for each of the construction and deconstruction S, C, and K

_{S}data sets, did not exhibit any significant trends in the distribution of the parameters (Figure 2). It is important to emphasize that the spatial plot of K

_{S}reflects only the properties of the matrix, and not an effective bulk K

_{S}at the scale of the test pile.

_{S}were observed to have skewness to the right, indicating a non-normal distribution (Figure 3). Chi-square goodness-of-fit tests and quantile–quantile plots for each of the log-normally transformed data sets (e.g., Y = Ln(K

_{S})) indicated acceptance of the normal distribution hypothesis in each case (Figure 3). The assumption of log-normal distribution was further analyzed using 10 subsets of 50 randomly selected values from the Type I deconstruction Ln(K

_{S}). Chi-square goodness-of-fit tests conducted on these subsets indicated 90% of the subsets passed the additional chi-square testing, suggesting the assumption of log-normal distribution for the data sets was reasonable. The construction data set for Type I C contained less than 10 values; therefore, the distribution analysis was not conducted for this data set.

_{S}in the waste rock at the Diavik test piles can be approximated using a log-normal distribution with mean and standard deviation calculated from samples collected during construction. Waste rock piles that are segregated according to S content may require individual distributions for S content associated with each section.

_{S}at co-located sampling points was calculated. The S and C data exhibited low correlation with a correlation coefficient of −0.15 based on 120 co-located samples. The S and K

_{S}and C and K

_{S}data sets exhibited low correlation with correlation coefficients of −0.11 and −0.21, respectively, based on 85 co-located samples for each set. The correlation calculations indicated that the S content at a given location is not related to the C content or K

_{S}at that same location and vice versa.

#### 3.3. Stationarity

_{S}) was considered. Stationarity is defined in this context as the absence of significant trends in mean and variance of the analyzed parameters with depth in the test-pile experiment. The stationarity of the Ln(S), Ln(C), and Ln(K

_{S}) was tested by calculating the mean and variance at each bench for the Type I deconstruction data and compared to the overall mean and variance of the deconstruction data set (Figure 4). There was some fluctuation in the mean and variance about the average in each case.; however, it appears that stationarity in the Ln(S), Ln(C), and Ln(K

_{S}) was a reasonable assumption for the Type I test pile. The spatial information for the Type I and Type III construction data sets was limited; however, analysis of the mean and variance of samples by instrumentation face compared to the overall mean and variance indicated that the assumption of stationarity appears reasonable for Ln(S) and Ln(K

_{S}) of the construction samples. The stationarity of Ln(C) was not investigated using construction samples due to the low number of samples analyzed for this parameter during construction of the test piles.

#### 3.4. Experimental Semi-Variogram Estimation

_{S}) using each method (Figure 5). Horizontal semi-variograms were estimated from parameter pairs that were separated by benches. The maximum vertical separation of horizontal Ln(S) and Ln(C) pairs was 2.6 m; the maximum vertical separation of horizontal Ln(K

_{S}) pairs was 3.7 m. Vertical semi-variograms were estimated from parameter pairs that were horizontally separated by a maximum of 3 m. Lag intervals were selected based on number of pairs available to provide a reasonable estimate. The horizontal lag interval selected for Ln(S) and Ln(C) was 4 m; the vertical lag interval selected for Ln(S) and Ln(C) was 1.5 m. The lag intervals selected for Ln(K

_{S}) were 3 m and 1 m for horizontal and vertical lags, respectively. Visual inspection of the experimental semi-variograms indicated that the estimates calculated by the classical method provided the most stable trends in each case, and as such the classical method was selected as the estimator for fitting of the theoretical variograms.

#### 3.5. Theoretical Semi-Variogram Fitting

_{S}). To test this observation, three common theoretical semi-variogram models were applied to the estimates. Models were evaluated for fit by calculating the standard error of the regression (sum of squares). The pure nugget model (Equation (5)) was selected for analysis because visual inspection of the semi-variograms for Ln(S) and Ln(C) indicated no spatial dependence at the selected lag intervals. The exponential model (Equation (6)) was selected based on the success of other researchers in fitting the model to semi-variograms of Ln(K

_{S}) for fine-grained sands (e.g., [44,45,46]). The Gaussian model (Equation (7)) was selected based on visual inspection of the experimental Ln(K

_{S}) semi-variograms.

^{2}

_{Y}) was 0.28 for Ln(S) (horizontal and vertical) and 0.05 for Ln(C) (horizontal and vertical). The horizontal and vertical semi-variograms for Ln(K

_{S}) indicated limited spatial dependence with the Gaussian model providing the best fit with σ

^{2}

_{Y}of 0.043, λ of 10 m, and σ

^{2}

_{0}of 0.025 (Figure 6).

#### 3.6. Reactive Transport Simulations

_{4}in the higher matrix proportion scenarios. For some parameters (e.g., Cu, Al) the matrix proportion significantly influenced the effluent concentrations, whereas the impact of an increase or decrease in the matrix proportion had little influence on the pH, which was constrained by the dissolution of carbonate minerals and aluminum-bearing phases. The proportion of matrix material within a pile is an important factor in assessing the geochemical evolution of a waste rock pile.

## 4. Discussion

_{S}was used to investigate the spatial distribution of these parameters that are fundamental to the assessment of the geochemical evolution of waste rock. The statistical distributions of S, C, and K

_{S}were determined to be log-normal, which is common in natural systems [46,47,48].

#### 4.1. Statistical Comparison of Construction Samples

_{S}) for the Type I and Type III construction sample results indicated that the assumption of log-normally distributed data and stationarity in the first two moments was reasonable at each test pile. The variances of Ln(S) and Ln(C) were consistent between the Type I and Type III test piles. The means of Ln(S) and Ln(C) were not consistent between the test piles, reflecting the slightly differing S and C contents of the two piles. The mean and variance of Ln(K

_{S}) were consistent between the two test piles. The similarity in parameter distribution, mean variance, and stationarity suggested that the K

_{S}of the two test piles are statistically similar, and that PSD of the matrix material collected at the Type I pile could be used to describe the heterogeneity in K

_{S}of the Type III pile despite difference in S content between the two piles. This consistency is expected given the similar blast design and transport methods used for the source material, the similar construction methods used for the test piles, and the similar mineralogy of host material in the two piles. This result could likely be applied to other waste rock piles where host rock mineralogy is relatively homogeneous.

#### 4.2. Theoretical Semi-Variogram Fitting

_{S}) horizontal and vertical semi-variograms, with the spatial dependence being slightly more prominent in the horizontal direction. The poorer fit at larger lag distances in the vertical direction is likely the result of relatively low numbers of pairs (i.e., <10 for lag >7.5 m). Both directions were fit using a relatively large nugget value (59% of the underlying variance) and, coupled with the relatively small range of K

_{S}values (within one order of magnitude), suggests limited spatial dependence (i.e., the K

_{S}of the matrix material was not depth dependent).

_{S}lenses associated with traffic surfaces created during pile construction [49,50]. Internal traffic surfaces were not created as part of the construction of the test piles due to the method of construction; however, it is likely that the presence of surface and internal traffic surfaces could significantly alter the hydrogeologic regime within a pile.

_{S}) indicated that K

_{S}did not vary significantly with depth within the core of the Type I test pile experiment. The lack of depth dependence in K

_{S}likely results from using only the PSD of the matrix material within the pile. Using the Chapuis equation (Equation (1)) and the PSD for particle sizes < 75 mm from the Type I test pile deconstruction, Barsi et al. [33] found that the bulk K

_{S}increased with depth in the Type I test pile. The results of this study indicate that the matrix material K

_{S}was not depth dependent.

#### 4.3. Reactive Transport Simulations

_{S}in the test pile experiments is significant because matrix material is a controlling component in the test pile hydrology; most of the water flow is conveyed through the fine-grained matrix of the test pile. Furthermore, due to the large surface area of the fine fraction, the geochemical evolution of the pore water is dominantly controlled by the matrix material (e.g., [51]). The uniformity of the matrix material within the pile suggests all portions of the test pile that include matrix material could contribute to the solute loading, with the exception of stagnant zones that are isolated from flow by obstructions (e.g., matrix material located below large boulders). This observation is consistent with previous research (e.g., [52]) that indicated relatively uniform solute loading (approximately one order of magnitude) from different parts of a single pile. The results of the reactive transport simulations indicate that most parameters are at least somewhat influenced by the matrix material content within the pile and are consistent with the observation of the importance of matrix material proportion within a waste rock pile.

## 5. Conclusions

_{S}field within the matrix material of the Type I test pile at a scale of 3 m horizontal and 1 m vertical. The K

_{S}of the matrix material was not depth dependent. It was determined that no spatial dependence in the structures of the S and C distribution was present in the Type I test pile at a scale of 4 m horizontal and 1.5 m vertical. This result was expected given the methods used to construct the test-pile experiments. Sampling density during the deconstruction of the Type I test pile likely contributed to the results of the geostatistical characterization; however, given the extremely large size of typical waste rock piles, a denser characterization of the spatial distribution of S, C, and K

_{S}is not likely to contribute significantly to an overall assessment of the geochemical evolution of waste rock.

_{S}in the test piles, which can be characterized as randomly heterogeneous, is likely due to construction methods; the distributions of S and C within the test pile could also be influenced by their distribution within the original lithology. The heterogeneity of S, C, and K

_{S}in the waste rock at the Diavik test piles can be approximated using a log-normal distribution with mean and standard deviation calculated from samples collected during construction. This is an important conclusion, suggesting that heterogeneity representative of physical and mineralogical conditions of a pile can be implemented in a spatially random manner according to measured log-normal distribution statistics. Waste rock piles that are portioned according to S content may require individual distributions for S content associated with each segregated component. The lack of spatial dependence in matrix K

_{S}is significant because of the importance of the matrix material in controlling water flow through the test-pile experiments. Due to the large surface area of the matrix material, the geochemical evolution of pore water is dominantly controlled by this finer fraction. Reactive transport simulations conducted using differing proportions of matrix material indicate that most parameters are at least somewhat influenced by the matrix material content within the pile.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Overview of test piles area during initial stages of the Type I test pile deconstruction (

**left**) and the Type I test pile at excavation of bench three with two sampling trenches (

**right**). The white line on the right photo shows the approximate location of the 15 m profile. Photographs courtesy of Sean Sinclair.

**Figure 2.**Location and distribution of S and C content and K

_{S}estimates from the deconstruction dataset at the east–west oriented 15 m profile of the Type I test pile (see Figure 1 for approximate location). Data were contoured using irregular data point triangulation in Tecplot 360 EX 2015. The K

_{S}estimates shown are estimated from the PSD of the matrix material. Vertical axis is elevation (m above sea level); horizontal axis starting point is arbitrary. Black dots represent sample locations; the outline represents the profile of the deconstructed test pile.

**Figure 3.**Frequency histograms for Type I deconstruction S (wt.%), C (wt.%), K

_{S}(m s

^{−1}), Ln(S), Ln(C), and Ln(K

_{S}) and quantile–quantile plots for Ln(S), Ln(C), and Ln(K

_{S}). The dashed line on the histograms represents the normal distribution calculated from the population mean and standard deviation; the solid line on the quantile–quantile plots represents the log-normal distribution.

**Figure 4.**Mean (μ) and variance (σ

^{2}) of Ln(S), Ln(C), and Ln(K

_{S}) from the deconstruction samples calculated by deconstruction bench. Overall mean and variance shown with dashed line. PSD analysis was conducted for samples from benches 1 to 5 only. Bench 1 was at the top of the test pile; bench 6 was at the bottom.

**Figure 5.**Horizontal and vertical experimental semi-variograms for Ln(S), Ln(C), and Ln(K

_{S}) using classical, Cressie–Hawkins, and SMAD estimators.

**Figure 6.**Theoretical semi-variogram fits to classical horizontal and vertical semi-variogram estimates for Ln(S), Ln(C), and Ln(K

_{S}). The pure nugget model is fit to Ln(S) and Ln(C) experimental semi-variograms; the Gaussian model is fit to Ln(K

_{S}) experimental semi-variograms.

**Figure 7.**Concentrations of mineral weathering products SO

_{4}, Fe, Ni, Co, Cu, Zn, Al, Ca, and K (mg L

^{−1}), pH (SU) versus time (year) measured at the Type III test pile basal collection lysimeters compared to aqueous concentration exiting the simulation domain. Simulated concentrations are for 0.2 proportion (solid line), 0.2 ± 0.05 matrix proportion (dashed line), and 0.2 ± 0.1 matrix proportion (dotted line) matrix proportion.

Parameter | Measured | Hazen | Schlichter | Terzaghi | Chapuis |
---|---|---|---|---|---|

geometric mean | 9 × 10^{−6} | 5 × 10^{−5} | 5 × 10^{−6} | 3 × 10^{−4} | 4 × 10^{−5} |

standard deviation | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−6} | 8 × 10^{−5} | 8 × 10^{−6} |

minimum | 2 × 10^{−6} | 2 × 10^{−5} | 3 × 10^{−6} | 1 × 10^{−4} | 2 × 10^{−5} |

maximum | 3 × 10^{−5} | 9 × 10^{−5} | 1 × 10^{−5} | 6 × 10^{−4} | 7 × 10^{−5} |

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## Share and Cite

**MDPI and ACS Style**

Wilson, D.; Smith, L.; Atherton, C.; Smith, L.J.D.; Amos, R.T.; Barsi, D.R.; Sego, D.C.; Blowes, D.W. Diavik Waste Rock Project: Geostatistical Analysis of Sulfur, Carbon, and Hydraulic Conductivity Distribution in a Large-Scale Experimental Waste Rock Pile. *Minerals* **2022**, *12*, 577.
https://doi.org/10.3390/min12050577

**AMA Style**

Wilson D, Smith L, Atherton C, Smith LJD, Amos RT, Barsi DR, Sego DC, Blowes DW. Diavik Waste Rock Project: Geostatistical Analysis of Sulfur, Carbon, and Hydraulic Conductivity Distribution in a Large-Scale Experimental Waste Rock Pile. *Minerals*. 2022; 12(5):577.
https://doi.org/10.3390/min12050577

**Chicago/Turabian Style**

Wilson, David, Leslie Smith, Colleen Atherton, Lianna J. D. Smith, Richard T. Amos, David R. Barsi, David C. Sego, and David W. Blowes. 2022. "Diavik Waste Rock Project: Geostatistical Analysis of Sulfur, Carbon, and Hydraulic Conductivity Distribution in a Large-Scale Experimental Waste Rock Pile" *Minerals* 12, no. 5: 577.
https://doi.org/10.3390/min12050577