The first step modeling an oil bearing geological structure is to define the topology. The topology defines the the coordinate system, grid dimensions, the orientation of the axis, and the cell dimensions.
Standard practice is to use the Cartesian coordinate system. The Cartesian coordinate system defines a location in space by an x, y, and z coordinate. In terms of notation, geostatistics uses u to denote location.The grid dimensions are the maximum and minimum coordinates that the grid must cover to post all of the data. A good first approach is to plot the data points on a location map. In order to do this the minimum and maximum data locations are required so that the extents of the plot can be set. One way to determine these parameters is to plot a histogram and calculate the summary statistics for each of the coordinate axis. The advantage to this approach is that the minimum, maximum, and mean location for each axis is posted allowing you to determine the required parameters for the location map and get a feel for the sampling scheme. The distribution of locations can reveal a biased sampling scheme.
Consider the following 2-D example:
Figure 3.3, A histogram of the X and Y data.
Notice that the x axis data seems well distributed while the y axis data seems a little skewed. This implies that the sampling scheme was a little biased toward the northerly end of the map. The corresponding location map is:
Figure 3.5, A contour map using the sample data set. The accuracy of the map is not critical. Its purpose is to simply illustrate trends.
The contour map illustrates that any areas of high potential (red areas) are heavily sampled; a biased sampling procedure. The contour map also illustrates that we may want to extend the map in the east direction.
It is common practice to use the Cartesian coordinate system and corner-point grids for geological modeling. The corner-point grid system is illustrated in Figure 3.6.
Fig. 3.6 The standard grid system used for geological modeling
Notice that the Z dimension b in Figure 3.6 is not the same as the dimension a in the areal grid, but the XY dimension for both the areal and vertical grids are the same. For the sake of computational efficiency the stacked areal grids are aligned with Z axis, but for flexibility the Z axis need not be of the same dimensions as the areal grid. This technique proves valuable for:
1. modeling the hydrocarbon bearing formation as a stack of stratigraphic layers: It is intuitively obvious that a model should be built layer by layer with each layer derived from a homogenous depositional environment. Although each depositional environment occurred over a large span of time in our context the depositional environment actually occurred for only a brief period of geological time and for our purposes can be classified as a homogenous depositional environment.
2. volume calculations: The model must conform to the stratigraphic thickness as closely as possible. Modeling the formation as a "sugar cube" model leads to poor estimates.
3. flow calculations: Flow nets must have equipotential across facies. A "sugar cube" model would
This permits modeling the geology in stratigraphic layers. The stratigraphic layers are modeled as 2-D surface maps with a thickness and are then stacked for the final model. Thus having a non regular grid in the Z direction allows for conformity to thickness permitting accurate volume calculations, also allows for flow nets (must be equipotential across any face).
Geological events are rarely oriented with longitude and latitude. There is usually some azimuth, dip, or plunge to the formation. If the angle between the formation and the coordinate axis is large there will be error a' in the cell dimensions as indicated by Figure 3.7. Also, it is confusing to have to deal with the angles associated with azimuth, dip, and plunge, so we remove them and model in some more easily understood coordinate system.
Figure 3.7, Notice that with large deviations in dip that there will be some cell dimension error.
It is common practice to rotate the coordinate axis so that it aligns with the direction of maximal continuity. The direction of maximal continuity can be derived from the contour map. A note about continuity; it is assumed that the direction of maximal continuity is that direction which the formation has the greatest continuity, and the direction of minimal continuity is perpendicular to the direction of maximal continuity. The rotations are performed in two steps. The first step removes the azimuth the second removes the dip. In the event that there is plunge to the formation the procedure for removing dip is repeated. The procedure is illustrated in Figure 3.8. The transform for removing the azimuth is indicated in Matrix 3.1, where x1, y1 are the original coordinates, x1 0, y1 0 are the translated coordinates for the origin, α is the azimuth angle, x2, y2 are the new coordinates.
(3.1)
The dip is removed using the transform indicated in Matrix 3.2, where where x , z are the coordinates
Figure 3.8 Illustrates the process of rotating the coordinate axis to be aligned with the major axis of the reservoir. First the axis are rotated about the z axis to accommodate the the azimuth of there reservoir, second axis are rotated about the y axis to accommodate dip in the reservoir.
The rotations can be removed using the the transforms indicated in Matrix 3.3 and 3.4.
(3.3)
(3.4)
Sometimes fluvial channels can be difficult to model because they deviate significantly. In this case it is possible to straighten the channel using the transform in Equation 3.5. Figure 3.8 diagrammatically illustrates the straightening transform.
(3.5)
Figure 3.8, Transforming a twisty channel into a straight channel.
Stratigraphic Coordinates
Reservoirs often consist of stratigraphic layers separated by a surfaces that correspond to some sequence of geologic time events, much like growth rings in a tree. The bounding surfaces that differentiate the strata are the result of periods of deposition or periods of deposition followed by erosion. The surfaces are named according to these geologic events:
• Proportional: The strata conform to the existing top and base. The strata may vary in thickness due to differential compaction, lateral earth pressures, different sedimentation rates, but there is no significant onlap or erosion (Deutsch, 1999).
• Truncation: The strata conform to an existing base but have been eroded on top. The stratigraphic elevation in this case is the distance up from the base of the layer.
• Onlap: The strata conform the existing top (no erosion) but have "filled the existing topography so that a base correlation grid is required.
• Combination: The strata neither conform to either the existing top or bottom surfaces. Two additional grids are required.
Figure 3.9 Illustrates proportional, truncation, onlap, and combination type correlation surfaces.
The stratigraphic layers must me be moved so that they conform to a regular grid. This is done by transforming the z coordinate to a relative elevation using:
(3.5)
Figure 3.10 shows how the strata is moved to a regular grid. Note that features remain intact, just the elevation has been altered to a relative elevation.
Figure 3.10 Illustrates the result of transferring the z coordinate to a regular grid using formula 3.9.
It is important to note that these are coordinate transforms; the data is transformed to a modeling space and transformed back to reality. There are no property or distance changes here, just the movement from reality to some virtual space then back to reality.