Conditional Simulation for Heterogeneity Modeling and Uncertainty Quantification
In the previous chapter, kriging proved to be an interpolation technique that was flexible enough to filter correlated or uncorrelated noise from the data or to combine seismic and well information. Thanks to the flexibility in the choice of the trend and the covariance model, kriging is closely related with splines, multiquadrics, and trend-surface analysis. Kriging, when it is based on the trend and covariance models fitted to the data, also provides an estimate of the uncertainty at every location of the map. However, kriging remains a deterministic approach that provides a very smooth image of geological variables that are, in most cases, very erratic.
As an example, Fig. 4-1 shows in red the kriged surface already used in Fig. 3-24 (Hohn data set). This surface is very smooth. A variogram calculated on the points of this surface would be extremely different from the spherical variogram fitted on the data and used as input to kriging. Is there not a contradiction here? Should not the vari-ogram of the kriged surface be the same as the spherical model used as input to kriging? The answer is definitely no, because the goal of kriging is not to generate a surface that mimics the actual variations of the interpolated variable, but to provide, at each location, an estimate that is as close as possible — on average — to the unknown value.
A simpler way to understand this is to use the example of a random variable, Z, taking
Figures & Tables
In this introduction, we would like to highlight what appear to be the important landmarks in the history of geostatistical applications in the petroleum industry. What do we mean by "geostatistics?" In this course, this term will cover the petroleum applications resulting from the pioneering work of Prof. Georges Matheron and his Research Group at the Centre de Géostatistique de l'Ecole des Mines de Paris. As far as this course is concerned, the main pillars of this work are the developments of variogrambased modeling applications.
Variogram-based modeling applications can be classified in two broad categories, the first of which can be called deterministic geostatistics and is essentially all the development around kriging. We will see later that this covers a very wide number of techniques, including external drift kriging, error cokriging, factorial kriging, and collocated cokriging. Although kriging is a technique based on a stochastic model, it generates one single model as a result, and it is deterministic in that sense.
The second category can be called stochastic geostatistics, and it covers the numerous techniques developed around the conditional simulation concept. Conditional simulation is stochastic in the sense that, as with the Monte-Carlo simulation, it generates a family of "realizations" of 1D, 2D, or 3D models, all compatible with the a priori model and the existing data. With regard to kriging, conditional simulation includes several techniques, such as indicator simulation, collocated cosimulation, or geostatistical inversion. This explains why this one-day course is subdivided in two half-days, the first half-day presenting the basic concepts and the deterministic family of applications, the second half-day covering the stochastic applications (Fig. 1-1). The most complete synthesis of Matheron's work can be found in Chilès and Delfiner (1999). Isaaks and Srivastava (1989), Hohn (1988), and Deutsch (2002) are also other excellent presentations of geostatistics.
Following the work of Matheron, petroleum applications went through different episodes (Fig. 1-2). The first one could be qualified as deterministic mapping. This was the first development of kriging for mapping applications; see, for instance, the papers of Haas and Viallix (1974) or Haas and Jousselin (1976). This period saw the development of commercial mapping applications, such as Bluepack (Renard, 1990). Another important step in the development of 2D mapping applications was Doyen's (1988) paper showing the potential of cokriging for mapping porosity using seismic-derived information and well data.
The mid-1980s to mid-1990s saw the explosion of 3D stochastic (simulationbased) reservoir modeling.