## Abstract

Reservoir production is highly dependent on reservoir models. 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. Therefore, geological models have to be built in three dimensions (3D). Unfortunately, manual construction of 3D geological models (deterministically) is almost impossible, which explains why geologists often limit their interpretation to two dimensional (2D) correlation panels, fence-diagrams or maps. Consequently, geological conceptual models are rarely included or considerably simplified in reservoir models used for flow simulations and replaced by stochastic or geostatistic approaches. In spite of this admission of failure, sedimentological cross-sections and maps contain most of the knowledge and concepts of sedimentologists. They represent the outcome of sedimentological studies, including available well data, seismic interpretation and especially sedimentological and environmental concepts, incorporating all facies transitions and successions in a high-resolution stratigraphic framework. They allow fine temporal- and spatial-scale sedimentological heterogeneities to be identified. The integration of these fine-scale sedimentological heterogeneities is an essential step in improving the precision and accuracy of static reservoir models and volumetric calculations. This paper demonstrates the quantitative influence of introducing sedimentological information into the reservoir characterization workflow using a simple deterministic workflow. The described incorporation of sedimentological knowledge through facies 3D proportions cubes allows a direct assessment to facies distribution multi-realization scheme and associated uncertainties by applying stochastic simulations.

Stochastic simulations can distribute spatially within reservoir models, facies and other properties observed at wells, using calculated or approximated variograms, spatial observed relationship or imposed object shapes. The choice between which stochastic approaches to use is therefore dependent on the geologically realistic distribution of heterogeneities obtained by conditional simulations. Stochastic modelling approaches are described on the assumption of stationarity of the random function, leading most of the time to unrealistic reservoir models. Due to the low density of hard data in the oil industry (well spacing) and geological spatial variability, stationarity is an hypothesis which should not be tested. Recent developments of stochastic simulation were conducted in between two opposing concerns: the quest of objectivity favouring hard data (Journel & Deutsch 1993; Journel 1996) and the quest of trying to integrate additional information on spatial distribution. So when non-stationarity is suspected at the scale of the domain to be modelled, external drifts can be integrated into the modelling workflow to constrain stochastic simulations. Then lateral geological variability has to be extracted using other quantitative means.

Several approaches have been developed during recent years to quantify this lateral variability, using analogue geological situations (e.g. Ravenne & Beucher 1988; Bryant & Flint 1993; Dreyer *et al.* 1993; Grammer *et al.* 2004), seismic surveys (e.g. Beucher *et al.* 1999; Marion *et al.* 2000; Raghavan *et al.* 2001; Strebelle *et al.* 2003; Andersen *et al.* 2006), hand-drawn sections (e.g. Cox *et al.* 1994) or more elaborated approaches in relation to sedimentological concepts (Massonnat 1999; Massonnat & Pernarcic 2002). Another advantage of stochastic simulation techniques is their flexibility in incorporating soft information coded under the format of local prior probabilities (Rudkiewicz *et al.* 1990; Goovaerts 1997; Deutsch 2002; Mallet 2002). This local prior probability can be represented as proportion cube (or facies probability cube) in reservoir models.

## 3D facies proportion cube definition

A 3D facies proportion cube is the description of reservoir models in terms of vectorial properties (each cell of the model contains the probability of occurrence of each facies represented in the model). This cube represents the distribution of facies and contains a 3D estimation of associated uncertainty.

A deterministic gridded model can be described in terms of proportions by a relatively simple method (Figs 1 and 2). For each horizontal layer of the grid, the occurrence of facies probability can be extracted and transferred as 2D vectorial property (Fig. 1a). When these 2D vectorial properties are stacked vertically, a proportion curve (one dimension) representing the vertical evolution of facies proportions is obtained (i.e. facies evolution through depth; Fig. 1b).

This deterministic gridded model can also be described vertically. Each single column of the model can be defined by the proportion of each facies it contains (Fig. 2a). The map of all vertical proportions is a new vectorial proportion, called ‘proportion map’, representing the horizontal evolution of vertical proportions; it is a representation of facies evolution in a 2D map (Fig. 2b). The combination of the proportion curve and map will give the 3D facies proportion cube.

This description of a simple deterministic model is the basis of our workflow. From the available sedimentological dataset, a vertical proportion curve and a proportion map are built.

## Introducing sedimentological concepts in cross-sections

Based on sequence stratigraphy concepts and knowledge of the studied area, the sedimentologist describes the lateral and vertical evolution of facies tracts from one well location to another. Facies not encountered at well locations, but conceptually identified to be part of the studied facies tract (Fig. 3), can be introduced. This example shows three alternative sedimentological interpretations of facies extensions between two wells. The lateral facies extent is based on the geologist’s knowledge of the studied system tract, therefore this knowledge can be integrated into the designed sedimentological cross-section. In the Figure 3a example, the facies 7 (dark blue) is recognized to have narrow extensions, whereas in the Figure 3b interpretation, its extension is known to be wide. These differences will induce contrasting probabilities in the final 3D proportion cube. In scenario A, facies 7 associated proportions will be constrained near well B location, whereas in scenario B, these proportions will be extended away from well B. Subsequent stochastic simulations, driven by the created proportions, will follow the defined sedimentological trends. In Figure 3c, the sedimentologist integrates a facies (Facies 4, pale yellow) to be part of the studied facies tract. The introduction of this facies within the cross-section will imply a modification of the derived proportions and therefore of stochastic facies distributions in the final model. The introduction of sedimentological facies extensions does not change the global certainty zones usually defined as high certainty at well location and low certainty away from well bores, but introduces the sedimentological concept as an external drift for subsequent stochastic simulations.

The sedimentological cross-sections represent the spatial distribution of facies and the only uncertainty carried by sedimentological cross-sections is located at facies transition zones. As imaged in the following simple example (Fig. 4), the cross-section shows facies where they are known to occur (interpreted with high degree of certainty). Conversely, intermediate areas between facies A and B are symbolised by interfingering A and B facies, representing lower degree of certainty. Furthermore, the drawing design of the facies transition zone contains the related uncertainty (Fig. 4). This allows, in cases where facies interfingering does not physically occur (e.g. erosional limit of channel in floodplain shale), to introduce an uncertainty on the facies limit position.

## Extracting proportion curves and maps from sedimentological cross-sections

As the sedimentological cross-sections contain all the information necessary for the construction of a 3D proportion cube and if we assume that they are representative of the entire 3D modelled volume, then from each sedimentological cross-section a vertical and a horizontal proportion curve can be extracted (Fig. 5). This extraction procedure is divided into two phases. The first allows the construction of a vertical proportion curve, by calculating the facies proportions along horizontal layers defined on each cross-section (Fig. 5a). The second creates a horizontal proportion curve, by calculating the facies proportions along vertical columns defined on each cross-section (Fig. 5b).

In the described workflow, vertical proportion curves are considered first. All vertical proportion curves can be merged (with a different weighting factor if necessary, depending on sedimentological certainty on correlations) to build a single proportion curve for the stratigraphic interval modelled.

Once vertical proportion curves are constructed, horizontal proportion curves positioned along the selected cross-sections (one curve per sedimentological cross-section) are used to build a proportion map through a phase of interpolation. The interpolation between previously defined horizontal proportion curves need not be linear between cross-sections, and is typically guided by ‘trend lines’ defined by any available source of information, typically seismic interpretation (e.g. amplitude map) or conceptual sedimentary model (e.g. expected geometry, curvature of channels or palaeogeography maps), (Fig. 6).

The final step is to implement the 3D Discrete Smoothed Interpolation (3D DSI) technique after Mallet (1989). 3D DSI interpolates facies between wells using vertical proportion curves and proportion maps as key constraints (Fig. 7). The final result is a distribution of probability values for each facies where the sum of facies probabilities at any given node of the grid is equal to one.

The 3D DSI process allows the integration of additional constraints acting as external drift for the interpolation. These external constraints can be introduced within the modelling workflow as 3D *a priori* facies probabilities derived from seismic attributes (Ruijtenberg *et al.* 1990; Haas & Dubrule 1994; Fontaine *et al.* 1998; Beucher *et al.* 1999; Grijalba-Cuenca *et al.* 2000; Marion *et al.* 2000; Strebelle *et al.* 2003; Andersen *et al.* 2006; Escobar *et al.* 2006), or stratigraphic simulation results derived from tools e.g. DIONISOS (Granjeon 1997; Granjeon & Joseph 1999; Burgess *et al.* 2006), SEDSIM (Griffiths *et al.* 2001) or FLUVSIM (Duan *et al.* 1998). These different key constraints can have various relative weightings according to their importance or to the certainty associated with each of them (Fig. 8).

## The influence of the stratigraphic framework

An understanding of the depositional model and thus of facies occurrences is necessary to compute facies proportions correctly. Most of the time, the easiest way to understand the depositional model is to link the interpretation to sequence stratigraphy analysis. Furthermore, the reliability of our modelling workflow needs to be connected to stratigraphic framework. Most of the time, stratigraphic cycles display coherent facies tract trends along wells and cross-sections, which are necessary to ensure their 3D representation. This can be illustrated with a simple geological model with two distinct sedimentological facies distribution trends (Fig. 9). If this model is considered as a single cycle in the workflow, the resulting 3D proportion cube is incoherent (Fig. 9a). As this model has two distinct trends, treating them as a single stratigraphic entity provides a single proportion map which is fairly homogeneous; the two sedimentary trends cancel each other, resulting in a poor rendering of the sedimentary concepts and architecture. If the initial model is divided into two distinct cycles and our workflow is applied separately on each sequence, then the result is coherent and is a good facies distribution outcome (Fig. 9b).

## Workflow results

### Basis for multi-realizations of facies distribution/uncertainty assessment

This deterministic 3D proportion cube is a background trend for subsequent geostatistical facies simulations, such as Truncated Gaussian Simulation (TGS) and Sequential Indicator Simulation (SIS). According to the stochastic approach chosen, facies distributions will vary. By using SIS, facies are modelled separately and independently; in this way, possible constraints of their relative position are not taken into account (Journel & Alabert 1990; Journel & Deutsch 1993). TGS is based on direct algorithms, without any iterative process. Ravenne & Beucher (1988) and Rudkiewicz *et al.* (1990) proposed this direct approach to take into account the facies spatial relationship. This approach allows the simulation of concomitant facies, such as those in shoreface deposits and those in most carbonate depositional settings. All the facies have the same variogram reflecting the same spatial continuity and implying the same anisotropy and correlation length. Figure 10 shows an example of Truncated Gaussian Simulation applied on a proportion cube for an oolitic ramp from the Middle East Gulf region.

The result of our modelling workflow is several ‘equiprobable’ 3D facies distributions of realistic appearance. They contain an external drift generated by the geologist and have an element of uncertainty reflected by the multi-realization scheme, all this with classical geostatistical constraints to well data. The stratigraphic grid populated with facies or major facies associations can then be used as an input to petrophysical modelling or seismic inversion.

### Basis for object-based models

The 3D proportion cube can also be used as a soft-conditioning constraint for object simulations, which are widely used methods for facies simulations (Dubrule 1989; Haldorsen & Damsleth 1990; Caers 2005). The object distribution is guided by the 3D proportion cube. Figure 11 shows an example of mouth bar distribution achieved using the object-based methodology in Tunu Field (Mahakam, Indonesia). Channel distributaries are introduced deterministically in the model, whereas mouth bar architectural elements are simulated according to their width, thickness, length or width-to-thickness ratio, using a 3D proportion cube derived from sedimentological cross-sections and maps. As previously discussed, the final model provides several ‘equiprobable’ 3D object distributions and can then be used as an input to petrophysical modelling or seismic inversion.

## Conclusions

The workflow is based on simple modelling techniques; it allows sedimentologists to deterministically integrate their interpretations and concepts into the reservoir characterization workflow, with all available attributes to guide them. This workflow also integrates sedimentological uncertainty on heterogeneity distribution, leading to the construction of a 3D proportion cube used in uncertainty studies. Also the resulting 3D proportion cube is a unique output (vectorial property), which acts as an input for various facies distribution methods (e.g. TGS, SIS or Object-based) without any distortion of initial inputs. These results are then populated with petrophysical properties using classical geostatistical methods.

Deterministic modelling coupled with stochastic or geostatistic models provide interesting solutions to the main challenges of reservoir modelling, the construction of 3D geologically realistic representation of heterogeneity and the quantification of uncertainty through the generation of, not one, but a variety of possible models or ‘realizations’.

## Acknowledgments

The authors would like to thanks Yannick Boisseau, Dominique Marion, Bruno Michel, Olivier Robbe and Philippe Samson for their involvement in the workflow construction, and John K. Williams and Adam Robinson for their constructive comments and suggestions on the manuscript.

- © The Geological Society of London 2008