4. Spatial: Spatial Location Analysis

4.1 Algorithm
4.2 Spatial User Interface
4.3 Spatial Verification

4. SPATIAL: SPATIAL LOCATION ANALYSIS

Spatial analysis of fracture patterns is an essential aspect of the discrete feature network modeling processes. Current spatial analysis methods include geostatistical (Isaaks and Srivastava, 1989), spatial statistics (Ripley, 1988), and fractal (La Pointe and Barton, 1995). Rule-based methods developed within the scope of this project provide an important additional tool for use in defining structural patterns.

The key to spatial fracture analysis of discrete features is the recognition that fracture formation is not a purely stochastic process. Rather, there is a physical, mechanical basis for every fracture's location and geometry. While it is frequently not productive to attempt to understand the entire stress-strain and material property history related to each fracture, the basic rules of fracture formation can be used to derive more realistic DFN model spatial location processes.

Spatial pattern analysis derives correlation structures which can be used to explain the location, size, and orientation of discrete features. A rule-based approach relies on geologically based correlations rather than on purely statistical or stochastic/fractal process based correlations. The geological correlations behind fracture spatial patterns can be expressed quantitatively as (Figure 4-1):

• precedence: the relative order in the sequence of fracture creation
• intensity trends: the variation of fracture intensity with distance from specific types of pre-existing features, or in a specific direction
• orientation trends: the variation of orientation with distance from specific types of pre-existing features, or in a specific direction
• size trends: the variation of orientation with distance from specific types of pre-existing features, or in a specific direction

These correlation structures provide the underlying underpinings for the Hierarchical Fracture Model (Ivanova, 1998). The rule-based spatial analysis approach quantifies spatial trends and correlations to provide input parameters for hierarchical fracture models.

4.1 Algorithm

The rule-based algorithm developed for analysis of spatial fracture patterns is illustrated in Figure 4-2. The algorithm starts with digitized fracture spatial data from wells or outcrops (Figure 4-3). The analysis proceeds as follows:

1. Set Definition: The first step in the algorithm is to preprocess fractures into sets using a method which allows definition of fracture sets in terms of precedence and geological characteristics as well as orientation. Examples of these algorithms include the Neural network (see Section 3), and stochastic/probabilistic methods (Figure 4-4; Dershowitz et al, 1996).
2. Data Gridding: Data is gridded by defining a grid over the data and marking cells (a) (0,1) fractured/unfractured; (b) number of fractures; or (c) intensity P₂₁ (tracemeters/meters squared), as illustrated in Figure 4-5.
3. Same-Set Trends: The second step in the algorithm is to identify possible spatial trends in intensity, size, and orientation for each set. This is done by calculating the statistics for the set on a grid (Figure 4-6).
4. Prior Set Correlations: Once a list of possible spatial trends have been established, the algorithm looks for prior-set features which could explain the trends. Examples of prior set correlation include, for example, decreases in intensity away from identified "fracture zones", and increased intensity within identified "fracture zones" (Figure 4-7).
5. Between Set Correlations: The fourth step is to determine whether there are correlations between the trends observed for different sets. For example, the spatial variation in intensity or orientation should correlate strongly for conjugate shears (Figure 4-8).
6. Statistical Probabilities: For each of the possible correlations, and trends identified in (2), (3), and (4) above, the algorithm calculates the probability that the correlation will apply for a particular fracture generation. The rules and statistical probabilities are then reported.

4.2 Spatial User Interface

As a result of the increasing interest in Web-based software, the project team decided to implement spatial data analysis as a Java 1.1 application, which can be implemented as a web server application. As a server application, users would not need to download and install Spatial in order to evaluate the software. Spatial 1.0 was coded using Java 1.1, and is currently compiled as a Windows 95 application.

Spatial 1.0 spatial data analysis software is designed to facilitate spatial analysis of lineament map files, using the algorithms described in Section 4.1 above. The user interface for Spatial 1.0 is illustrated in Figures 4-9 through 4-12. Spatial 1.0 provides the following four features for spatial analysis of lineament data:

• data gridding (Figure 4-9a)
• contour plotting of intensity, size, and termination mode to facilitate visual evaluation of spatial trends (Figure 4-9b)
• trend analysis of intensity, size, termination mode, and orientation to facilitate analysis of non-stationarity (Figure 4-10)
• dependency analysis for evaluation of trends with respect to location of primary sets (Figure 4-11)
• correlation analysis for comparison of spatial trends for different sets (Figure 4-12)

To execute Spatial, select the Spatial icon, and double click with the mouse. The general procedure for analysis is summarized in Table 4-1. The operation of the individual Spatial menu items are described in the next section. Navigation through Spatial is done using Microsoft Windows mouse conventions. In general, the left hand mouse button is used for making selections.

Table 4-1 Spatial Lineament Analysis Sequence

4.3 Spatial Verification

Spatial analyses are based on values of intensity P₂₁ (m/m²), fracture length (m), and fracture orientation calculated on a cell-by-cell basis. The center of the cell is used as a spatial reference of each cell value. The cell values are presented in either colored grids or figures. Spatial is therefore verified primarily by checking that the cell values are calculated correctly.

Each of the cells in the Spatial verification case (Figure 4-13) have been assigned a cell number (1 to 16). Figure 4-13a presents a simple fracture pattern in which it is possible to calculate intensity P₂₁ (m/m²), fracture length (m), and fracture orientation manually for comparison to spatial results. The verification of Spatial's correlation features depends on the assignment of fractures to different sets. To evaluate this feature, additional "Set 2" fractures were added to the original "Set 1" fractures, as illustrated in Figure 4-13b.

Figure 4-14 presents the comparison between expected and calculated "Set 1" cell values for the cases illustrated in Figure 4-13a and 4-13b. Spatial results are identical to verification case calculations for both these cases.

The fracture pattern in Figure 4-13a was used to verify Spatial's trend analysis feature, which calculates projected distance along a user specified projection angle Cell No. 10 was selected to verify the projected distance calculation. Figure 4-15 presents comparisons of projected distances to cell No. 10 at angles of 30°, 90° and 135°. For angles of 30° and 90°, the distance is calculated for them low-left corner of the grid. For angles greater than 90°, such as 135°, the projected distance is calculated from the upper-left corner of the grid. The projected distances obtained from Spatial match the expected values for the verification case from hand calculations.