Stochastic Earth Modeling That Integrates All Subsurface Uncertainties
In the previous pages, we have focused on the use of geostatistical conditional simulation for 3D heterogeneity modeling. We saw that GCS provided a satisfactory solution to the problem of generating realistic 3D representations of the subsurface. We also saw that, thanks to GCS, we were able to generate not one, but a large number of realizations, all of which were compatible with the well data, the a priori geostatistical constraints (histogram and variogram), and, in many cases, the seismic data. The variability from one realization to another was a representation of the remaining uncertainty left after constraining our models by all this input information. We will now discuss how this quantification of uncertainties can be applied to all parameters of the earth model to lead to uncertainties attached to gross-rock volume, oil-in-place, reserves, or production profiles (Fig. 6-1). But why should we be interested in quantifying uncertainties?
An uncertainty calculation is a useless exercise if no decision making is attached to it (Fig. 6-2). But which kinds of decisions shall we be able to support with an uncertainty calculation? Fig. 6-3 lists some of the most important decisions geoscientists are led to support with their uncertainty studies (see examples in Tyler et al., 1996 and Charles et al., 2001). Usually, these decisions are related to a significant financial investment. Instead of one production profile, a typical uncertainty study will produce a family of production profiles or the field reserves
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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.