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### NARROW

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Abstract 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.

Abstract In this introduction, we would like to highlight what appear to be the important land-marks 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 variogram-based 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 collo-cated 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 numer-ous techniques developed around the conditional simulation concept. Conditional sim-ulation 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

Abstract Nature often behaves in a very complicated way, and geology is no exception. In petro-leum applications, where we are dealing with reservoirs at depths of several kilometers that are recognized by only a few wells and some seismic, we need to simplify the description of these reservoirs by means of models. A model is a simplification of nature and should never be identified with the natural phenomenon it seeks to describe. However, a model has the advantage of reducing our understanding of the reservoir to the estimation of a few parameters. The best approach is to explain the concepts using a 1D example. Fig. 2-1 shows, on the left, a variable that varies around a constant mean. At any location, the behavior of the variable, although complicated, can be qualified as “homo-geneously heterogeneous.” On the average, it behaves the same everywhere, in the sense that we would make the same kind of error at any location if we were to predict the value of the variable from the value of the horizontal line. This will be discussed later as the stationarity hypothesis. Fig. 2-1 shows, on the left, a variable that varies around a constant mean. At any location, the behavior of the variable, although complicated, can be qualified as “homo-geneously heterogeneous.” On the average, it behaves the same everywhere, in the sense that we would make the same kind of error at any location if we were to predict the value of the variable from

#### Interpolation: Kriging, Cokriging, Factorial Kriging, and Splines

Abstract In the previous two chapters, we discussed the meaning of the geostatistical model and of its parameters. We will now discuss how this model can be applied. We will start with deterministic techniques, known under the generic name of kriging. Here, “deterministic” should be understood in the sense of “providing only one solution.” We will see that, although the model is probabilistic, kriging produces only one solution. Kriging covers a wide range of applications. The first one consists of interpolating one single variable in one, two, or three dimensions, and the second one consists of interpolating one variable but using the extra information provided by another variable that is related, of course, to the first one. Recall the probabilistic model we defined in the previous chapter. The variable Z (x) is interpreted as the sum of a polynomial trend, m (x), plus a residual, R (x), of mean zero. Under this model, universal kriging ( Matheron, 1970 ) addresses the problem of interpolating a variable on the basis of a number of scattered data. This can be the interpolation of layer-averaged porosity from well data or the interpolation of seismic times from a 2D seismic campaign. To understand kriging, let us consider the variogram from another perspective (Fig. 3-1 ). Suppose that layer-averaged porosity has been calculated at a well and that we want to estimate porosity 1 km away from that well by using the value at the well. Obviously, we will make an error, which can be directly read from

#### Conditional Simulation for Heterogeneity Modeling and Uncertainty Quantification

Abstract 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

Abstract In the previous chapter, we saw how geostatistics can be used to generate 3D heterogeneity models satisfying a number of input statistical parameters, such as the vari-ogram, or equivalently, the spectral density. Relationships between simulated parameters and seismic-derived information were statistical, usually coded as a linear correlation between the seismic parameter and the simulated variable. This was the “nprimary- versus secondary-variable” approach. We also saw earlier that the geostatistical paradigm, based on the concept of trend and covariance, can be considered to be an approach for coding the a priori model that is often used in Bayesian terminology. In parallel with geostatistical developments, inversion technology made significant progress in the last three decades (Fig. 5-1 ). Thanks to the Bayesian formalism promoted by authors such as Tarantola (1987) and Duijndam (1988) , the standard optimization-based deterministic approach (minimization of an objective function with a regu-larization term) was improved, and it became possible to produce an estimate of uncertainty together with the inverted acoustic-impedance block. Logically, the idea emerged in the early 1990s to apply the conditional simulation approach to acoustic-impedance inversion to produce multiple 3D realizations, all constrained by seismic data. This resulted in geostatistical-inversion methodology. In the following, we will consider that geostatistical and stochastic inversion are exactly the same thing. The method discussed here is that presented in the papers of Bortoli et al. (1992) and Haas and Dubrule (1994) . Geostatistical inversion (GI) consists of generating 3D acoustic-impedance realizations, all

#### Stochastic Earth Modeling That Integrates All Subsurface Uncertainties

Abstract 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

Abstract We have presented the model used by geostatistics for 2D and 3D petroleum applications. This quantified geological model assumes that the variable of interest is the sum of a deterministic trend plus a residual characterized by its variogram or its covariance. When there are enough data, the variogram model can be fitted to the experimental function. In other cases, assumptions based on geological knowledge, combined with a variogram analysis of seismic data, will be used to define the variogram model. Variograms can be related to fractal models and to a priori information used by geo-physicists in seismic inversion or in Fourier analysis (Fig. 7-1 ). The geostatistical model can address the problem of deterministic interpolation through kriging. A degree of smoothing can be applied to kriging through the error-cokriging approach, which allows the filtering of random noise, whereas the factorial kriging approach allows the filtering of short-range — or high-frequency — terms due (for instance) to seismic-acquisition artifacts. Kriging based on well data can also incorporate extra information coming from seismic data, through the external drift or the collocated cokriging approach. Kriging is closely related to other interpolation techniques, such as splines or radial-basis functions. Specifying a kriging model amounts to specifying the regularization term of energy-based inversion techniques. Kriging and all its family of associated techniques remain a deterministic method. The production of a minimum variance estimate results in an interpolation that is very smooth away from the data points. In practice, however, geology has no reason to

Abstract The goal of this exercise is to show how a 3D model is fitted to an anisotropic experimental variogram. The example used is from Chu et al. (1994) . This paper shows (in its Fig. 7) experimental porosity variograms calculated in the North-South, East-West, and vertical directions. The data are from an Amoco west-Texas carbonate field of Permian age. The experimental variograms were calculated from the well data in their strati-graphic coordinates, within a layer of thickness ranging from 11 to 27 m. A total of 4697 elementary porosity-log data were available in 90 wells. Experimental variograms are displayed in Fig. 2-26 . The model fitted by Chu et al. is shown in Fig. 2-26 . A screen copy of the spreadsheet used is shown in Figs. E1-1 . Thanks to this spreadsheet, it is possible to evaluate the impact of a parameter change on the variogram model, and thus to better understand the meaning of each parameter. A particular point to discuss is the use of a shortrange model to represent the nugget effect of the lateral variograms, without impacting the vertical variogram fit. The goal of this exercise is to evaluate the impact of the variogram choice on the results of an elementary kriging system. We assume that we work in 2D, and that the value of a variable z at a data point x0 is kriged using the values at four data points located in the neighborhood of xo (Figs. E2-1 ).

Abstract This course is divided in three main parts: geological quantification, data integration and uncertainty quantification. This choice emphasizes applications where petroleum geostatistics has been most successful so far. The course does not pretend to be an exhaustive review of petroleum geostatistics. The author selected the topics which, in his view, provided the clearest and strongest illustration of the technique and its applications. Rather than going through a tedious exposure of the geostatistical formalism, we prefer to review the methodology from an application standpoint. This course will start with a brief history of the technique, then finish with perspectives for the future. It will be stressed that, although it is already very successful, petroleum geostatistics is still a very new and changing discipline.

#### A brief history of the development of geostatistical techniques in the oil industry

Abstract The quantification of geology has always been a fascinating topic, and the first pioneering efforts to reach this goal were those of Vistelius (1949) and his many followers using Markov chain analysis to quantify one-dimensional lithological sequences along wells (Fig. 1). Many successes were encountered with this approach, but it appeared difficult to generalize to the second and third dimension. Then, in the mid Sixties, the giant Hassi-Messaoud field in Algeria was the object of pioneering applications of quantitative reservoir description techniques. The distribution of sand lenses and shale breaks was modelled in a vertical cross-section (see Fig. 2), with the goal of understanding their impact on effective permeability. Shales were represented as thin sheets, whilst sands were given a varying thickness and a constant width. These bodies were distributed randomly in the cross-section. Their lateral extent had been derived from a detailed outcrop study of the Ajjers’ Tassili analogue (Algeria). This model was used as a basis for reservoir simulation and it was observed that, because heterogeneities were modelled in a realistic way, a satisfactory history-match could be achieved more easily. An overview of the approach used is given in Delhomme and Giannesini, 1979. This Hassi-Messaoud application seems to have been rather isolated in the petroleum industry. In the mining industry, things were different. G. Matheron, at the Centre de Géostatistique (France) , had pioneered the use of Mining Geostatistics in the early Sixties (Matheron, 1970, Journel and Huijbregts, 1978), and its use was spreading fast.

Abstract A key problem faced in the development of a hydrocarbon reservoir is that of constructing a reservoir model that can generate reliable production forecasts under various development scenarios. After a few appraisal wells have been drilled, or after a few years of production, the reservoir geologist will provide a model of the inter-well geological architecture. This model (see example in Fig. 10) may be a conceptual representation of the architecture of genetic bodies (e.g. fluvial channels, floodplain shales) within which petrophysical variations can later be distributed. Such representations can have a very important impact on economical decisions. Examples are the location of an in-fill well, or a reservoir simulation exercise, the results of which influence the choice of a development strategy. Correlation panels such as those shown in Fig. 10 are a representation of the subsurface that incorporates the well data and the interpretation of the depositional environment, facies associations and geometries. It is well known that if geological models are to be used for reservoir simulation, and hence as a basis for development decisions, they must be generated in three dimensions. Two-dimensional models do not provide a realistic representation of reservoir connectivity. Unfortunately, manual construction of 3-D geological models is close to impossible, which explains why geologists often limit their interpretations to 2-D correlation panels, fence-diagrams or maps. As a result of this, there is often a gap between conceptual geological representations and models used in reservoir simulations because the gridded 3-D model used by reservoir engineers does not incorporate the geological knowledge embedded in detailed 2-D correlation panels.

Abstract Actually, some of the earlier applications of geostatistical simulations were twodimensional. In their simpler form, these applications used as input the layer-averaged values of petrophysical properties at wells. For instance, Fig. 80 shows four realisations of a continuous parameter map generated by conditional simulation. The values at the three control points are the same in each realisation. These four realisations share the following properties: an identical (spherical) variogram, a similar histogram, and they all honour the wells. The realisations are often called “equiprobable” in the sense that they all share the statistical properties believed to be those of the actual (unknown) map. The use of such straightforward 2D conditional simulations has decreased. They can prove useful for mapping permeability in the case of high net/gross layers (as encountered for instance with braided-streams deposits ) where lateral variations are more important than vertical ones. For training purposes, Fig. 81 shows four other realisations of the parameter of Fig. 80, obtained this time with a gaussian variogram, the practical range of which was chosen to be equal to that of the spherical model used in Fig. 80. The realisations are smoother with the gaussian than with the spherical model. This is because a gaussian variogram implies that the modelled variable is very smooth (indefinitely differentiable, in mathematical terms). In spite of this somewhat unrealistic assumption, the gaussian model often proves useful in applications, because the models produced are smooth and easy to interpret. More important is the use of 2D conditional simulation for the evaluation of structural uncertainties.

Abstract So far, we have presented approaches for generating conditional simulations of the distribution of geological or petrophysical parameters in one, two and three dimensions. The goal of these approaches is to generate representations of the variable of interest that mimic the unknown spatial distribution of this variable: object-based models honour width-thickness distributions, variogram-based models honour the input variogram model, etc… We have seen that, for each set of wells and input statistical relationships, a large number of representations could be generated, all matching the wells and the statistical parameters. This is simply a consequence of the non-uniqueness of the problem. The wells and the statistical parameters are not sufficient to uniquely constrain the model between the wells. Variability between different realisations is simply a measure of the uncertainty affecting the knowledge of the parameter at any given location. The standard deviation of all values of the realisations at a fixed location is a quantification of this uncertainty. In many instances, geostatistical realisations offer an interesting solution to the problem of generating realistic images of the variability in the inter-well volume, and of quantifying the associated uncertainty. In some situations, the large number of realisations may overwhelm the user of geostatistical techniques, who may just need a simple map of the variable of interest in the inter-well volume. Consider the average of several geostatistical realisations at the same location. A 1D example (Fig. 55) shows that the average of all conditional simulations is a smooth function, far smoother than any of the realisations.