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This chapter describes how to calculate experimental variograms for the facies indicators and how to fit models to them. The relationship between the facies indicators and the underlying gaussian values has been given in Chap. 2. In this chapter we will give the variogram equations in the case when we only have constraints on facies but no other categorical variable or continuous function in the data set. The theoretical relation linking the variograms of the underlying gaussians and those of the indicators is described. From this link we use a specific fitting method: rather than invert it, we use an indirect iterative procedure to fit a variogram model. But first, we show how to calculate the experimental variograms for the facies indicators.

Simulation methods fall into two broad classes: pixel methods and object-based methods. Before presenting any practical examples, it is important to decide which method would be the most appropriate for the problem under study. So we first review the pros and cons of two classes of models, sequence-based pixel methods such as the truncated gaussian and plurigaussian simulations, and object-based methods, in particular boolean simulations. As this book focuses on plurigaussian simulations, we do not intend to present boolean simulations here. Interested readers can consult Matheron (1968, 1975), Lantuejoul (1997a, 1997b and 2002) Chiles and Delfiner (1999), and Molchanov (1997). Sequence-based pixel methods and boolean simulations can also be combined to get the best of both approaches. These are called nested simulations.

In truncated gaussian simulations, the lithofacies are not simulated directly: a stationary gaussian random function is simulated first, and is then transformed into the lithofacies variable by truncation. For example, if we want to simulate two lithofacies, F1 and F2, a very intuitive way to transform the simulated gaussian variable into facies values is to say “if the numerical value of the simulated gaussian is lower than the number t1, we obtain the first facies F1; otherwise, we obtain the second facies F2”. The value t1 is called a threshold.

The aim of this section is to review so me of the basic properties of indicators that will be needed later. We let F be a random set (i.e. all the points belonging to one particular facies). Let \( \overline {\hbox{F}} \) be its complement (i.e. the points that do not belong to F). The indicator function for the set F takes the value 1 at all the points inside F; it takes the value 0 elsewhere. This indicator function is denoted by \( {1_{\rm{F}}}({\hbox{x}}) \). In the Fig. 3.1, the set F has been shaded. Its complement \( \overline {\hbox{F}} \) includes all of the non-shaded area. The set F could be any shape or form, and need not be a single piece. It could be split into several parts.

Our objective is to simulate the facies at a set T of N target points given a data set D. The target is usually a large 2D or 3D grid, often with several hundred thousand nodes. In petroleum, the data points come from wells interpreted in lithofacies typically each foot whereas in mining studies data are core samples from drill-holes. In both cases there are usually a large number of data points too.

This chapter is designed to give an overview of the practical aspects of kriging: negative kriging weights, the impact of the choice of the variogram model on the kriging weights, crossvalidation, the screen effect and last but not least, some criteria for testing the quality of a kriging.

This chapter deals with the effect of the support of a regionalized variable (i.e. its physical shape and volume) on its histogram and its variogram. In an introductory exercise, the histograms and the basic statistics are calculated for two support sizes: 1m x 1m blocks and 2m x 2m blocks. Although the means of the two distributions are identical, the variance of the larger support is much smaller and its histogram isalmost bell shaped whereas the other one is skewed.

Once a suitable prospect has been found, an exploratory drilling campaign is carried out to establish the limits of the mineralization (if they are not already known), and to determine the total ore tonnage and the average grade. As well as knowing the total reserves, it is very important to know how accurate the estimates are. Provided there are not too many samples, kriging can be used to estimate the reserves and the kriging variance will give a measure of its accuracy. However when there are too many points to invert the kriging system a different approach is required.

This chapter presents the theory of kriging. Kriging is an estimation method that gives the best unbiased linear estimates of point values or of block averages. Here “best” means minimum variance. Three types of kriging estimators are discussed: ordinary kriging (OK) used when the mean is unknown, kriging the unknown mean value and simple kriging (SK) used when the mean is known.