# Sequential Simulation Approach to Modeling of Multi-seam Coal Deposits with an Application to the Assessment of a Louisiana Lignite

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DOI: 10.1007/s11053-012-9185-1

- Cite this article as:
- Olea, R.A. & Luppens, J.A. Nat Resour Res (2012) 21: 443. doi:10.1007/s11053-012-9185-1

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## Abstract

There are multiple ways to characterize uncertainty in the assessment of coal resources, but not all of them are equally satisfactory. Increasingly, the tendency is toward borrowing from the statistical tools developed in the last 50 years for the quantitative assessment of other mineral commodities. Here, we briefly review the most recent of such methods and formulate a procedure for the systematic assessment of multi-seam coal deposits taking into account several geological factors, such as fluctuations in thickness, erosion, oxidation, and bed boundaries. A lignite deposit explored in three stages is used for validating models based on comparing a first set of drill holes against data from infill and development drilling. Results were fully consistent with reality, providing a variety of maps, histograms, and scatterplots characterizing the deposit and associated uncertainty in the assessments. The geostatistical approach was particularly informative in providing a probability distribution modeling deposit wide uncertainty about total resources and a cumulative distribution of coal tonnage as a function of local uncertainty.

### Keywords

Krigingsequential simulationprobability distributioncoal bedassessment## Introduction

Mining resource assessments in general and coal resource assessments, in particular, are characterized by fluctuations in the attributes of interest that are difficult to infer from the incomplete information typically available ahead of mining, a situation that invariably leads to uncertain results (e.g., Caers 2011). Olea et al. (2011) presented a methodology for a single bed that was more mathematically satisfactory and yielded better quantitative results than conventional uncertainty approaches that simply evaluate proximity to the nearest coal measurement, such as those proposed by Wood et al. (1983). Also, the new methodology expresses uncertainty about total resources in terms of in-place tonnage, rather than distance units. This article builds on the 2011 article by assessing a multi-bed deposit to continue exploring the promising potential of geostatistical methods that have been recognized progressively as the standard approach to mineral estimation since the middle of the past century (Mwasinga 2001). In the last two decades, the popularity of geostatistical methods has shifted away from kriging to spatial stochastic simulation (e.g., Srivastava 1994; Journel and Kyriakidis 2004). The following authors have applied stochastic simulation to the particular case of coal assessments:

Hohn and McDowell (2001) (a) modeled three coal beds in properties of different areal extensions at two 7½ min quadrangles in southern West Virginia and (b) reported the resources of each seam separately.

de Souza et al. (2004) (a) applied the modeling to a single layer deposit and (b) assessed a portion of the deposit delimited by the lease rather than by the natural boundary of the coal bed.

Heriawan and Koike (2008) (a) used the spatial modeling method OPSIM for interpolation of discontinuous bodies (Koike and Matsuda 2005) and (b) did not aggregate the resources.

It may be relevant to note that none of these authors checked the results against an exhaustive sampling or some form of ground truthing. In addition, no account was taken for oxidized coal of no commercial value.

- (a)
determine the outer areal boundary amenable for modeling;

- (b)
combine surface elevation and model of depth of oxidation to establish the upper boundary of minable resources in a deposit;

- (c)
make simultaneous use of drilling information in terms of holes penetrating and missing seams for estimating missing coal areas within the model boundary;

- (d)
model coal bed thickness and related uncertainty away from drill holes;

- (e)
prepare a probabilistic assessment of total resources aggregating three beds using an average coal density;

- (f)
use a function of expected cell tonnage as a model for local uncertainty.

The measurements are supposed to be free of institutional uncertainties of non-geologic nature, such as insufficient depth, improper sampling, or inconsistencies in the correlations.

This article is an expansion of the modeling of a single-bed deposit (Olea et al. 2011). The main objective is testing the capability of geostatistics to properly assess uncertainty for the more complex case of multiple coal beds.

## Methodology

Spatial modeling methods in general and geostatistics, in particular, do not work well far beyond peripheral observations inferring what some authors denote as hypothetical resources. Hence, a first step in any geostatistical modeling is the definition of the area feasible to model using the data available. In the absence of legal or geological constraints, one possibility is to limit the study area employing some criteria based on distance to drill holes. A related but more optimal approach is to use the kriging variance. Kriging can be regarded as a generalization of the least squares method to take into account spatial correlation among the data (e.g., Isaaks and Srivastava 1989). The method infers values one location at a time, generating an estimate and its error variance—also known as the kriging variance. This variance is sensitive to the distance and pattern of all measurements around any estimation location as well as to the style and magnitude of the fluctuations according to the data. Thus, by defining a threshold, one can use the kriging variance to define the extension of the area amenable of modeling. Given a number of drill holes, the kriging variance is consistent with the experience that more continuous depositional systems, such as a lacustrine deposit, are easier to assess and extrapolate than more complex deposits, such as deltaic deposits.

In the early days of geostatistics, modeling was based on a straight application of kriging (e.g., Tewalt et al. 1983). Today such practice has been abandoned primarily for three reasons. First, locally, estimates are conditionally biased, i.e., estimated values toward the low end are systematically higher than the actual values and the converse is true for the highest values. Hence, the probability distribution and spatial autocorrelation of the estimated attribute do not follow the probability distribution and the autocorrelation of the data, let alone that of the true attribute. Second, if there is interest in assigning confidence intervals to the results, kriging does not provide the distribution for the errors. At best, some results can be obtained by assuming that the estimate and the kriging variance are the two parameters of a distribution for the error, the rule being the normal distribution. It is not possible to verify such an assumption, which leaves in limbo the significance of the modeling. Third, only under multivariate normality is the kriging variance a good approximation to the error variance. Kriging, however, has not been completely abandoned; it remains at the core of several of the improved methodologies that have been developed to correct these drawbacks.

We use sequential simulation (e.g., Zanon and Leuangthong 2005) in the modeling of all attributes relevant to the assessment of a multi-layer coal deposit because it is a satisfactory formulation to characterize coal beds and to take into account multiple sources of partial information. Using an average coal density leaves net thickness as the primary spatial attribute in tonnage calculations. However, if sufficient density data are available, they can easily be added to the modeling; ignoring density fluctuations does not take any generality away from the present approach. Sequential simulation numerically models surfaces or volumes by assigning values to a regular grid of locations. By design, the results are not unique. For any given attribute partially known through limited drilling, sequential simulation can generate multiple plausible results—maps in our case—employing the same empirical data. Each one of these maps has the same probability to be the true unknown surface. These maps are jointly used to numerically simulate the uncertainty of the attribute of interest at each equally spaced cell in a grid or for total values over a study area. Sequential Gaussian simulation (e.g., Nowak and Verly 2005) was used for the modeling of continuous attributes, such as coal bed thickness or elevation of the base of the oxidation zone. For categorical variables, such as the presence or the absence of a coal bed at any given site, we selected sequential indicator simulation (e.g., Deutsch 2006). Today, user friendly software, such as the public domain SGeMS used here (Remy et al. 2009), allows application of the geostatistical methods mentioned above and several others.

Once cell maps of resource tonnage are generated for every bed, the final step is the aggregation of all beds, which depends on the degree of stochastic correlation among seams estimated from collocated data (e.g., Schuenemeyer and Gautier 2010). Final results are more easily understood when summarized in the form of maps, probability distributions, and tables. Mean, variance, and length of the percentile interval *P*_{5}–*P*_{95} in the final results often stabilize using 40–80 realizations (e.g., de Souza et al. 2004).

The following procedure summarizes all necessary modeling steps:

### Procedure

- A.
Set a resolution for the modeling by selecting a cell size and define a regular grid containing a total of

*C*cells extending over the area of interest. - B.
Apply sequential Gaussian simulation to generate at least 100 realizations for the elevation of the base of the oxidation zone.

- C.For every bed:
- 1.Transform the
*n*thickness data at every location \( {\mathbf{z}}_{j} \) into a presence/absence indicator \( i({\mathbf{z}}_{j} ),j = 1,2, \ldots ,n \)$$ i({\mathbf{z}}_{j} ) = \left\{ {\begin{array}{ll} {0,} & {\text{if\;the\;bed\;is\;missing}} \\ {1,} & {\text{if\;thickness} > 0} \\ \end{array} } \right.. $$ - 2.
Krige the indicator values, prepare a standard deviation map, and use the map to set a maximum value. The outer bounds of modeling will be defined by all cells below the threshold. Cells above the threshold will be blanked in all realizations.

- 3.
Use the indicator data to model no coal areas inside the model boundary by applying sequential indicator simulation to generate

*N*realizations, with*N*being to at least equal 100. - 4.
Apply sequential Gaussian simulation for generating

*N*realizations of thickness. - 5.
Randomly pair each thickness realization with one indicator realization. Then, cell by cell, eliminate from the modeling all cells for which the indicator is 0. The resulting maps are realizations of the locations where the bed is present today after non-deposition and erosion.

- 6.
Based on the bed elevation data, generate

*N*sequential Gaussian realizations for the bed top elevation. - 7.Associate every one of the realizations in the previous Step to one of the oxidation maps in Step B. For every pair of realizations, make a cell-by-cell comparison to generate an oxidation indicator \( o(c),c = 1,2, \ldots ,C \)$$ o(c) = \left\{ {\begin{array}{ll} {0,} & {\text{if\;bed\;roof\;elevation} > {\text{elevation\;for\;the\;oxidized\;zone\;base}}} \\ {1,} & {\text{otherwise}} \\ \end{array} } \right.. $$
- 8.
Further eliminate from the

*N*realizations in Step C.5 all cells for which*o*(*c*) is 0 to discard cells of no commercial value. The resulting maps are the*N*realizations of thickness after considering non-deposition, erosion, and oxidation. - 9.
Convert cell thickness to cell tonnage using a constant conversion factor.

- 10.
If this is not the last bed to model, go back to Step C.3.

- 1.
- D.
Using collocated thickness data, calculate the correlation coefficient between every possible pair of beds.

- E.
Use the results in the previous Step to aggregate the resources cell by cell.

- F.
Use the

*N*realizations resulting from the previous Steps to summarize the results. - G.
End.

SGeMS (Remy et al. 2009) was supplemented with utility programs to handle procedures such as determination of the weathered zone and aggregation of resources. The next two sections illustrate the application of the procedure to a site in the Gulf Coast basin of Louisiana. Although the reader will not be able to reproduce our results without adequate geostatistical background or software, everyone should be able to follow and understand the results. This article is not a tutorial in geostatistics; the reader is encouraged to consult the references for a more in-depth understanding of geostatistics (e.g., Olea 2009).

## Geology of a Test Site

## Geological Modeling of the Case Study

Modeling bed by bed and aggregating the partial results at the end yield more geological details and more precise results than aggregating the data up front. With three coal beds and an equal number of drilling stages, modeling of all the stages and layers in the deposit required the preparation of nine sets of partial results. Modeling using only the reconnaissance data is the hardest to do correctly because of the scant amount of information available. Modeling based on the reconnaissance drilling of deepest coal bed B5 has been selected for illustrating the methodology in detail. Data from the infill and development stages were used solely to check the results of the reconnaissance modeling.

In a situation like ours with multiple datasets with different spacings, the densest is the one to analyze for deciding a systematic grid cell spacing (Step A). A square grid cell size of 200 ft was selected based on experience gained modeling over 50 lignite deposits in the Gulf Coast and heuristic rules often cited in the literature: (a) the spacing must be smaller than the average distance among drill holes, (b) at least 95% of the drill holes must land in different cells, and (c) the resolution of the grid should be sufficient, without being excessive, to model all details relevant to the assessment (e.g., Jones et al. 1986; Deutsch 2002; Hengl 2007).

## Stochastic Resource Assessment of the Case Study

## Conclusions

Stochastic simulation allows both a detailed geological characterization of coal deposits and a quantification of associated uncertainties inherent to the modeling. Geological factors considered in the modeling can be bed boundaries, erosion, oxidation, bed thickness, and coal density.

The modeling provides detailed as well as synoptic results for uncertainties at both the deposit and the cell level, as well as either for total resources or for resources per bed. For total resources, a probability distribution provides likely total resources and associated probabilities. At the cell level, it is possible to map cell tonnage, generate probability distributions cell by cell, and summarize the expected cell tonnage in the form of a cumulative distribution as a function of cell uncertainty. These criteria are more informative and theoretically sounder than those put forth in U.S. Geological Survey Circular 891 (Wood et al. 1983).

## Acknowledgments

We are grateful to Lawrence Drew (U.S. Geological Survey) and Gerald Weisenfluh (Kentucky Geological Survey) for reviewing a preliminary version of the manuscript and making suggestions for its improvement. Two anonymous reviewers contributed additional suggestions as part of the evaluation of the article by the journal.