An Application of the Truncated Pluri-gaussian Method for Modeling Geology
Published:January 01, 2006
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A. Galli, G. Le Loc′h, F. Geffroy, R. Eschard, 2006. "An Application of the Truncated Pluri-gaussian Method for Modeling Geology", Stochastic Modeling and Geostatistics: Principles, Methods, and Case Studies, Volume II, T. C. Coburn, J. M. Yarus, R. L. Chambers
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The truncated pluri-Gaussian approach is a powerful method for modeling geology. Its main strength is its flexibility in representing complex lateral and vertical facies transitions with different anisotropies. In addition, it is easy to condition simulations to data points. This makes it an excellent method for modeling reservoirs with a complex architecture, such as carbonate bioconstructions or reservoirs affected by diagenesis.
This chapter presents a method for obtaining a tractable and mathematically consistent model for lithotype semivariograms and cross-semivariograms in complex cases. The method is illustrated using an example involving algal bioconstructions in the outcrops of the Paradox basin (Utah). Complex facies transitions, both vertically and laterally between the mound and intermound facies, together with the complex geometry of the algal mounds, make it virtually impossible to simulate these sorts of deposits using object-based models or classical pixel-based methods. The truncated pluri-Gaussian model is introduced to handle these complex facies transitions using the concept of a lithotype rule.
Such a rule, when expressed diagrammatically, can be a valuable tool for synthesizing geological information, and it can serve as one of the key inputs into the stochastic model. As illustrated here, combining the rule with proportion curves is a very effective way for analyzing and modeling geology in terms of facies sequences, even in complex depositional environments.
Stochastic models are widely used to simulate geology and petrophysical properties in petroleum reservoirs. Several modeling approaches are available, and the selection of an appropriate technique depends on the context in question. No matter which method is used, it should respect available well data, reproduce trends and directionality among facies (e.g., anisotropy), respect the experimental crosscorrelations and the transition probabilities among facies, and guarantee the consistency of models used for the semivariograms and cross-semivariograms.
Two broad categories of modeling techniques exist: those that are object based and those that are pixel based. Both categories accommodate the above four requirements but in different and important ways. Consequently, it is both interesting and instructive to consider these differences, particularly as they relate to the ideas of stochastic simulation applied to facies modeling, which is the primary focus of this chapter.
The remainder of this section briefly contrasts object- and pixel-based methods and provides the rationale for using truncated Gaussian (TG) and truncated pluri-Gaussian (TPG) random functions. The next section discusses some basic properties of indicator variables. Subsequent sections describe the geological setting of a particular motivating example from the Paradox basin, along with the development and application of a modeling approach for this area based on the use of TPG random functions.
Object-based Modeling Techniques
In light of the four requirements noted above, it is useful to first consider object-based techniques because of their popularity. With regard to anisotropy, trends and directionality are treated quite naturally in object-based methods, and inconsistencies between models for semivariograms and cross-semivariograms do not arise. In terms of respecting well data, however, conditioning is difficult when many wells are present or when the objects are large. Constraints from seismic and/or production data make this even more challenging. Furthermore, it seems to be difficult for object-based methods to reproduce semivariograms in this case and also when transitions among facies are involved. Semivariograms clearly provide valuable information in the vertical direction, even when only a few wells are available, and in the horizontal direction, when either many wells or seismic information is available. Consequently, the ability to appropriately treat them is important. One point of contention is that, commonly, the need to reproduce the experimental crosscorrelation is not even addressed, which may in part be because of the users' unfamiliarity with computing simple and cross-semivariograms.
Pixel-based Modeling Techniques
In contrast to object-based methods, conditioning simulations to many wells is generally not too difficult with pixel-based methods, and it is easier to integrate seismic and/or production data. However, some methods are plagued with an inability to reproduce cross-correlations. Stochastic indicator simulation proposed by Journel and Isaaks (1984) is one such method. This is why simulated annealing is sometimes used to postprocess resulting images (e.g., see Goovaerts, 1996). Most of the first-generation methods, except for those based on Markov random fields, fail to reproduce the transition properties; but even Markov random field methods are difficult to use because their parameters are not directly related to the properties of interest. Unfortunately, it is not uniformly understood that consistency between semivariograms and cross-semivariograms is not guaranteed if they are fitted separately. It is also not well known that specific semivariogram models should be used for fitting indicator semivariograms. These considerations and the desire to honor the observed cross-correlation are the main motivations that led Matheron et al. (1987) to model facies with a TG random function. The TG model is one of the simplest, complete models for indicators. Furthermore, it makes simulations easy to perform (see Rivoirard, 1993, for a discussion on indicator modeling).
Although the TG random function naturally honors a sequential order within lithofacies (Matheron et al., 1987; Ravenne et al., 1991) and can be extended to some other types of transitions, it cannot be used in the general case. One limitation is that the anisotropy has to be the same for all the lithofacies. The TPG random function is a generalization of the TG random function, which has the same advantage of flexibility and mathematical consistency and overcomes the limitations associated with transitions and anisotropies. One of the key steps in the pluri-Gaussian methodology is defining the lithotype rule. This rule is represented by a diagram that synthesizes contacts between facies. Geological knowledge can be used to help define the rule, and geostatistical tools can be applied to quantify the extent of the transitions.
Before describing the TPG methodology in further detail, it is instructive to review the properties of indicator functions and indicator semivariograms. Proportion curves and three-dimensional (3-D) proportion matrices are also discussed. To enhance the presentation of these ideas, reference is made to an outcrop observed in the Paradox basin, which is described in more detail later in a succeeding section.
Basic Properties Of Indicators
Indicator random functions are very special random functions. They take only two values, 0 or 1. They are used, in particular, when working with categorical data such as lithofacies. In such a case, an indicator is defined for each facies Fi by
More generally, given a random set F (that is, a set generated in a probabilistic way), its indicator is the function whose value is 1 on F and 0 elsewhere. Let be the complement of F. Then, the following elementary properties hold:
If pF(x) = E(1F(x)), pF is called the proportion of F and 0 ≤ pF(x) ≤ 1.
Var(lF(x)) = pF(x)( 1 − pF(x)).
σF(x,y) = E(lF(x)lF(y)) = Prob(x ∈ Fnd y ∈ F). σF(x,y) is called the noncentered covariance.
The indicator semivariogram of F, γF(x,y), always lies between 0 and 0.5.
These elementary properties are the ones that make indicator random functions so special. Property 3 has interesting implications in terms of non-stationarity. Nonstationarity in the mean is equivalent to nonstationarity in the variance for indicators. From Property 5, we see that power semivariograms of the form γ(h) = h α are forbidden for indicators (because h α is not bounded).
Using the previous properties, it is easy to prove the following theorems.
If F is a stationary random set and CF is its covariance, is its cross-covariance, and γF is its semivariogram, then
If A, B, and C are three facies whose indicators are second-order stationary, and if A ∪ B ∪ C fills the entire space, then
Up to this point, the discussion has focused on consistency relations when working with different indicators, but there are also strong requirements for a covariance to be the covariance of an indicator (e.g., more than just being a positive definite function; see Matheron, 1989; Armstrong, 1992). In particular, near the origin, indicator semivariograms must have a behavior in hα with α ≤ 1. Consequently, models of semivariograms with a parabolic behavior close to the origin, such as the Gaussian model or the cardinal sine model, are forbidden for indicators, as previously suggested.
Anisotropy is a simple and intuitive concept. An object is said to be anisotropic if it behaves differently in different directions. A simple example in geology is a channel that is more continuous along its principal direction than along the orthogonal direction. As a simple example, consider channels to be parallelepipeds with constant length and width. Figure 1 shows a diagram corresponding to this model where the main direction is east–west.
Figure 2 shows the semivariograms of these channels along the east–west and north–south directions. The curves are consistent with intuition about anisotropy. However, because of Theorem 1, the background semivariogram has to be the same as that of the channels; that is, the background has the same anisotropy as the channels. This result is somewhat disturbing, until it is realized that the background is linked with the channels. Being in the background is different from being in a channel. So, the intuitive feeling about the isotropy of the background derives from local analysis based on visual inspection, not from a global view. Basically, this is the meaning of Theorem 1.
Figure 5 shows the simple and cross-semivario-grams of these three facies. Surprisingly, both the background and the small squares have anisotropic semivariograms, as do the channels. This is a part of the meaning of Theorem 2: to a smaller or greater extent, anisotropy is observed in the indicator semivariogram for a facies as well as in each simple semivariogram.
The other meaning of Theorem 2 is understood when looking at the cross-semivariograms. They have some common features with the simple semivariograms. This indicates that all semivariograms (simple and cross) have to be fitted simultaneously to respect the constraints, and furthermore, that anisotropy has to be present on all the simple and cross-semivariograms. Theorem 2 is a more precise way of stating the relationship that should be followed when fitting the model. Consequently, to respect anisotropy during simulation, a model must be used that honors the way the different anisotropies interact.
Transitions among Facies
Transitions describe contacts between facies. More precisely, they are quantified as probabilities of being one facies or another. Transitions were intensively studied about 20 yr ago in the context of using Markov chains to model geology (Allègre, 1964; Krumbein, 1967). However, transitions can vary with the direction (horizontal vs. vertical) and orientation (upward or downward). Commonly, a vertical nonstationarity is present because of the cyclicity of sedimentation, such as low-order cycles being superimposed on higher order cycles that vary rapidly in a small amount of time (i.e., small variation in depth). This makes it more difficult to characterize the probabilities because they change vertically.
A Motivating Example
To better demonstrate the principles of TPG methodology and to explain its potential, the following example is presented.
The Paradox basin, which extends through southwest Colorado and in southern Utah, is an asymmetric basin that was probably created in a strike-slip tectonic context (Baars and Stevenson, 1981). During the Pennsylvanian, the basin paleogeography was characterized by thick evaporites that interfingered southward with thinner, shallow-marine carbonated platform sediments (the Hermosa Group). These carbonate-platform deposits are now exposed in the canyon of the San Juan River (New Mexico), where they are characterized by a tabular organization at the outcrop scale caused by the vertical stacking of several depositional cycles. The fourth-order sequences have been interpreted as the result of high-frequency, relative sea level variations, probably of climatic origin (Goldhammer et al., 1991). Each sequence corresponds to a transgressive-regressive cycle, commonly bounded by exposure surfaces. Significant algal buildups have been identified in some of the sequences. In outcrops, phylloid algal mounds can be observed in the Akhah and lower Ismay formations, whereas in the subsurface, algal mounds of the Desert Creek Formation constitute the main hydrocarbon reservoir of the Aneth field area (Chidsey et al., 1996, Montgomery et al., 1999).
Sedimentology and Stratigraphic Architecture of the Lower Ismay Formation in Outcrops
The present work focuses on the phylloid algal mounds of the lower Ismay Formation, which are beautifully exposed along the San Juan River canyon walls in the Eight Foot Rapid area (Figure 6). The sedimentary facies and the sequential organization of the mounds have been studied in detail by Grammer et al. (1996). Geostatistical modeling of the area has been conducted by Van Buchem et al. (2000). These authors identify and describe 12 sedimentary facies in detail, which are here regrouped into nine facies for simulation purposes. The main sedimentological characteristics of these nine facies are summarized in Table 1.
The black laminated mudstones (facies 1) were deposited during periods of relative sea level rise, which first induced a rapid flooding of the platform, and then the development of an open carbonate-platform setting (facies 2 and 3). During the period of relative sea level highstand, algal mounds started to aggrade, incipient mounds (facies 4) being followed by bioherm construction (facies 5 and 6). Intermound troughs were progressively filled by apron debris as mounds grew and by open-platform carbonates (facies 7). Mound tops were capped by high-energy bioclastic beds deposited close to sea level. Finally, the sand brought into the basin by Eolian winds during periods of relative sea level lowstand was reworked around the mounds by tidal and wave dynamics (facies 8). The sequence ended with the emersion of the platform and the development of soil horizons (facies 9).
Two sedimentary cycles have been identified in the lower Ismay Formation (Van Buchem et al., 2000). The first cycle starts with a marine transgression and the deposition of the black laminated mudstones (Gothic Shales Member). It ends with the platform exondation, which is marked by an important soil horizon on top of an extensive sandstone bed. Algal mounds were well developed in this cycle. The second cycle is marked by another transgressive event and new development of open-marine platform carbonates. Algal mounds have only low-relief morphology in this interval (Horn Point Member). A well-developed karstified surface marks a significant emersion at the end of the cycle.
Phylloid Algal Mound Geometry
In the outcrops, mound height can reach 15 m (50 ft), for a lateral extension ranging between 200 and 700 m (660 and 2300 ft). Mounds also have relatively symmetrical circular shapes, although outcrops do not allow detailed 3-D reconstructions of their internal architecture. The complexity of the algal mound facies architecture from the point of view of shapes and transitions makes this outcrop a challenge for stochastic modeling.
Outcrops in the Eight Foot Rapid area make it possible to reconstruct the lower Ismay stratigraphic architecture. The model is nearly 3-D because of the bends in the San Juan River and the occurrence of side canyons. A continuous photomosaic of the canyon sides has been constructed and then interpreted in terms of sedimentary facies (Figure 7) with the help of 17 detailed vertical sedimentological sections. This 6-km (3.7-mi)-long transect has been digitized, and pseudowells have subsequently been extracted from the transect and compiled into a working database.
DEVELOPMENT AND APPLICATION OF THE TRUNCATED PLURI-GAUSSIAN
As noted above, the TG methodology was first introduced by Galli et al. (1994). The basic idea is to define facies by truncating several Gaussian random functions. Using more than one Gaussian random function provides greater flexibility to handle complex transitions and anisotropy among facies.
The general approach is to start from a partition Di, i = 1,..., n, of the p-dimensional space (i.e., Di#x2229;Dk = ϕ and ∪Di = RP) and a vector of p Gaussian random functions, γ(x) = (γ 1(x), γ2(x),...γp(x)), where x is the location in 3-D space. By definition, facies Fi consists of the points x of the 3-D space for which γ(x) ∈ Di. This truncation rule is called the lithotype rule because it synthesizes transitions between facies. If there are two Gaussian random functions, the lithotype rule can be visualized as a twodimensional (2-D) diagram. Figure 8 shows a simple example where the partition is defined in terms of rectangles (because this simplifies the computations) and where the two Gaussian random functions are uncorrelated. Even with these simplifications, the results are still realistic. In fact, they correspond to erosion that could have been obtained in two steps using the regular TG approach. Note also that the anisotropy exhibited by the Gaussian random functions is intuitively expected on the facies, although as shown previously, the anisotropy indicated on the indicator semivariograms would be far more complicated. To this extent, TPG is more natural: knowing the anisotropy of facies, it is possible to directly choose the same anisotropy on the semivariograms of the Gaussian random function. The TPG is far more general than the previous example suggests. In some cases, it turns out to be instructive to use more than two Gaussian random functions. In addition, rectangles can be replaced by polygons that could give even more complex transitions.
Furthermore, Gaussian random functions that are correlated could be used. In such a case, one way to obtain a consistent model for the semivariograms and cross-semivariograms is to use the linear model for coregionalization (see Wackernagel, 1998). Another possibility is to use Gaussian random functions linked by linear transforms. Armstrong and Galli (1999) present a case of Gaussian random functions linked by partial differential equations. In Roth et al. (1998) a mining example is presented.
To model the facies, the Gaussian random functions are first simulated, and then the lithotype rule is applied. This means that, at each observed location, the values of the Gaussian random functions have to belong to the set D corresponding to the lithofacies. In other words, for facies Fi at point x0 in a well, the values at x0, have to be in the set Di. To honor this information, Gibbs sampling is used to reconstitute consistent values at the well locations (Le Loc′h and Galli, 1997; Armstrong et al., 2003). Once the values at observed locations are obtained, classical Gaussian conditional simulation is performed (see Lantuejoul, 1994), where the conditioning part has to be accomplished by cokriging involving all the Gaussian random functions considered.
As is the case for the TG approach, there is a well-known correspondence between simple and cross-semivariograms of indicators and simple and cross-semivariograms of a Gaussian random function. The relationship is even simpler when written in terms of noncentered indicator covariances, σFiFk between facies Fi and fades Fk. As previously noted, this noncentered indicator covariance (property 4) is a bi-variate probability. Because of the definition of the indicators in the TPG approach, this becomes
The right-hand side of equation 1 represents a multi-Gaussian probability, so it can be written as an integral of the Gaussian density over Di × Dk (that is, an integral in 2p dimensions). Because this density depends only on the covariance matrix between γ(x) and γ(γ), the required relationship between the indicator (cross-)covariances and the Gaussian random function covariances and cross-covariances is obtained. In practice, it is used in the reverse way: the experimental semivariograms and cross-semivariograms of the indicators are computed, then the semivariograms and cross-semivariograms of the Gaussian random functions are selected, and the resulting fit is obtained using equation 1. Note that the previous formula is also valid in the nonstationary case. For more details on these relationships and their use in fitting indicator semivariograms, see Galli et al. (1994), Le Loc′h and Galli (1997), and Armstrong et al. (2003).
It is now necessary to choose a partition of the space RP (R2 in the present case). The simplest way is to use rectangles. Information on lithotype transitions is used to determine where to place these rectangles in the space. In the present example, the unit has been split into upper and lower units, with one lithotype rule for each of them. For the lower unit, which is simpler (Figure 9b), only the first Gaussian random function is used for defining the facies. This corresponds to the regular TG approach. The upper unit is more complex. For application of the lithotype rule illustrated in Figure 9a, the three green facies and the yellow one are defined by the union of two rectangles. This is merely for convenience. It avoids having to split this unit into two subunits or to define different facies that would need to be regrouped after simulation. The lithotype rule shows which facies are in contact. For example, the red facies can touch (or have transitions with) the dark blue, the yellow, and the medium green ones. The red facies separates the dark blue one and the green one below, and it also separates the two yellow facies and green one. This pattern corresponds to what is seen on the outcrop. Looking carefully at the lithotype rule, it is apparent that the facies starting from the red to the light blue have properties determined only by the first Gaussian random function; but below the red facies, the yellow one and the three green ones have properties determined by both Gaussian random functions. This means that the spatial characteristics of the green and the yellow facies will be different according to the rectangle from which they originate.
When the lithotype rule has been decided, the precise boundaries of all the rectangles have to be computed. For a fixed partition, these boundaries depend on the proportions of each lithofacies. By definition, pFi(x), the proportion of facies Fi. around location x, is given by
So knowing the experimental proportion of each facies (and the lithotype rule), finding the location of the boundaries of the Di is what remains.
As an aside, it is worthwhile to emphasize that these formulas are valid even if the proportions are nonstationary (as previously noted, this also means that the indicators are nonstationary). This non-stationarity can be complex, involving variations in the proportions in the whole space, or can be simple, in which case they depend only on the elevation (after flattening according to a reference level). The latter situation is always present simply because of the cyclic character of sedimentation.
It is clear from equation 2 that the location of the boundaries depends on the correlation between the two Gaussian random functions, as well as on the proportions. This is true because equation 2 involves the p-dimensional integral of the density of the vector Y(x). In the present example, it is only a 2-D integral involving the bivariate density of the two Gaussian random functions at the same point. This density involves the covariance matrix of γ 1(x) and γ 2(x), which has diagonal terms of 1 and off-diagonals terms that are just the correlation between γ 1(x) and γ 2(x).
Vertical Proportion Curves and 3-D Matrices of Proportions
In most cases, a reservoir comprises different genetic units. Each of them corresponds to a parasequence of high order. Because of the relative change of position compared to the source of material, the amount of each facies varies in time for a given areal position. In a chronostratigraphical system (with changing coordinates), time is roughly equivalent to depth or thickness, so a marked evolution in the proportion of each facies is expected with depth. Mathematically speaking, this implies that the proportion pFi of a given facies Fi is a function of z.
The proportion is calculated, then, level by level, by computing the number of wells showing a particular facies as a proportion of the number of wells that penetrate that level. It is important to have an overview of this vertical variation. This is why vertical proportion curves have been designed (Matheron et al., 1987; Volpi et al., 1997).
Having chosen an order for the lithotypes, a vertical proportion curve is the representation of the accumulated proportions of the previous facies vs. depth. At a given depth, the space between curve i and i + 1 is just the proportion of facies i +1 at that depth. This is illustrated in Figure 9 for the two units from the Paradox basin. In the vertical proportion curve of the upper unit (Figure 9a), the dark green facies and the yellow one are present before and after the red one.
Note that an interaction exists between the vertical proportion curve and the lithotype rule. The vertical proportion curve also controls the transitions. If the proportion of a facies is small at a given level, the transition probability between this facies and another is generally also small, so that the lithotype rule is adjusted for each level. Figure 8 illustrates the situation based on the general rules, along with the possible transitions, but for each level that will be modulated at each level according the vertical proportion curve.
In some cases where the situation is more complex or where more information is available from seismic data, it might be useful to consider more than one proportion curve for the genetic unit. In addition, proportion curves can be computed for different areas, or alternatively, vertical proportion curves can be computed on the nodes of a coarse grid regrouping several simulation nodes. This gives a 3-D matrix of proportions (Beucher et al., 1993).
Modeling Reservoir Architecture With Pluri-Gaussian Random Functions: The Paradox Basin Example
As noted previously, geological complexity can be difficult to reproduce in simulations using conventional object- or pixel-based algorithms. The facies architecture of the algal mounds in the example from the Paradox basin is a clear example. In such situations, the facies have a complex sequential organization that varies both vertically and laterally. Spatial relationships among facies depend on the stratigraphic level. In the Paradox basin example, the lower half of the first cycle has a tabular geometry with a strict vertical ordering among the facies. This layer-cake geometry contrasts with the mound and trough architecture of the bioherms, in which intermound facies laterally interfinger with the mounds. In addition, in this example, early diagenetic processes may affect some of the facies differentially, making the ordering more complex. Furthermore, the mounds have a complex geometry. Their shape is irregular, which influences the facies distribution within and around each mound. The system is characterized by concentric facies belts in the mounds and draping geometries for the intermound facies and capping beds.
The main mound complexes are located preferentially in the easternmost part of the regional transect. Van Buchem et al. (2000) have simulated the algal mounds at a regional scale with a pixel-based model, and in doing so, they have consequently insisted on a nonstationary configuration of the mound distribution. However, this observation is less crucial at the scale of the present study, in which mound distribution can be considered stationary along the 6-km (3.7-mi)-long transect.
Lithotype rules are a very convenient tool for reproducing the complex facies organization of the platform deposits and of the bioherms. For simulation purposes, the series has been divided into two units. This layering is based on the facies architecture and does not necessarily correspond to the sedimentary cycles identified by Van Buchem et al. (2000). The lower unit comprises the layer-cake platform facies, from the organic-rich mudstone at the base to the tabular phylloid layer. Its lithotype rule shows adjoining facies bands (facies 1–5, Figure 9b), completed by a marked vertical organization in layers identified by the vertical proportion curve (Figure 10b). This will prevent interfingering between facies (e.g., facies 1 and 5), which is in agreement with the facies sequence. The upper unit comprises the bioherms, the intermound facies, the capping and the sandstone beds, and the upper part of the series where mounds have low relief. Its lithotype rule (Figure 9a) is more complex. Intermound facies 7 can be in contact with the mound facies 5 and 6 and also with the draping sandstone beds (facies 8). The more tabular organization of the upper part of the series is respected by the different facies bands in the upper part of the lithotype rule. The corresponding vertical proportion curve (Figure 10a) also reflects this more complex geometry.
As previously noted, outcrop observations suggest the existence of large-scale lateral nonstationarity, but this is difficult to check or to quantify with the available information. Because the aim of the present work is to concentrate on the complex geometry of the mounds, the possible nonstationarity is ignored. The resulting layout of the 17 vertical sections (Figure 7) makes it difficult to calculate meaningful horizontal semivariograms for a stationary model. For these reasons, the semivariogram models are chosen to honor the spatial continuity of the facies that can be seen in the outcrop. After a few trials, a model consisting of two Gaussian random functions, each based on a Gaussian semivariogram, can be shown to be adequate. The first semivariogram has an isotropic horizontal range of 400 m (1312 ft) and a vertical range of 10 m (33 ft), whereas the second one has an isotropic horizontal range of 2000 m (6561 ft) and a vertical range of 7 m (23 ft). Furthermore, a correlation coefficient of —0.5 between the two Gaussian random functions has been chosen to accentuate the draping of the red facies over the mounds. With these models and the lithotype rule, the yellow and green lithofacies below the red facies in the lithotype rule are expected to be less horizontally continuous than the same colors above it. For the lower unit only, a Gaussian random function based on a Gaussian semivariogram is used. It has an isotropic horizontal range of 2000 m (6561 ft) and a vertical range of 7 m (23 ft).
Key features of these carbonate-platform deposits can be reproduced very realistically, as illustrated in Figures 11–13. The cross section (Figure 11) shows the distribution of mound and intermound facies, with the overlying facies draped over them. Below and above the mounds, the layer-cake geometry of the carbonate-platform deposits is apparent. In plan view (Figure 12), the mounds show an irregular, rounded shape with concentric facies belts, which is precisely what is expected from a sedimentological point of view. Figure 13 shows these features in an excavated block diagram.
Although stochastic models are widely used for simulating geology in the petroleum industry, few methods can honor the transitions between facies together with nonstationarity because of the cyclic aspect of sedimentation. This chapter illustrates how the TPG approach can be used to successfully accomplish this result. One feature of the method is its simple and intuitive way of representing transitions. Furthermore, it is easy to incorporate anisotropy. The directions are chosen intuitively, for example, following the direction of channels. This contrasts with the complexity of modeling indicator semivariograms directly, where part of the anisotropy has to be incorporated in each of the semivariograms. One of the strong points of the TPG approach is that it provides a very simple way to consistently model all
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Since publication of the first volume of Stochastic Modeling and Geostatistics in 1994, there has been an explosion of interest and activity in geostatistical methods and spatial stochastic modeling techniques. Many of the computational algorithms and methodological approaches that were available then have greatly matured, and new, even better ones have come to the forefront. Advances in computing and increased focus on software commercialization have resulted in improved access to, and usability of, the available tools and techniques. Against this backdrop, Stochastic Modeling and Geostatistics Volume II provides a much-needed update on this important technology. As in the case of the first volume, it largely focuses on applications and case studies from the petroleum and related fields, but it also contains an appropriate mix of the theory and methods developed throughout the past decade. Geologists, petroleum engineers, and other individuals working in the earth and environmental sciences will find Stochastic Modeling and Geostatistics Volume II to be an important addition to their technical information resources.