Product Tour

Statistical Distribution Analysis

All four data types are characterized by statistical distributions through cumulative probability graphs and frequency histograms. The cumulative density function (CDF) displays the data type versus percentile. This graph can be easily altered to show a CCDF graph. CDF and CCDF plots can be graphed on the following scales: linear, log - linear, log - log, linear - log. To fit a curve to the CDF or CCDF function the user must decide if the data is most accurately fit using a normal, log normal, power law, or exponential curve. Statistics are calculated for the user to access the fit.

Histograms represent data type versus percent of total. Interval range and increment spacing can be changed as specified by the user. To fit a curve to the histogram, the user decides on the appropriate fit using normal, log normal, power law, and exponential functions. Calculated statistics guide the user in accessing the best fit.

Box Fractal Dimension Analysis

GeoFractal analyzes 2-D irregularly spaced data and fracture trace data to find the box fractal dimension. The graph displays the box size versus a count of the fractures in doubly logarithmic axes. The Box fractal model may be a useful model for the spatial pattern of the traces if the points lie on a straight line.

There are three methods used to count the fractures:

• fracture centers
• whole fractures
• random points on fractures

To calculate the Box dimension, the User must specify the geometry of the calculation grids. GeoFractal calculates the minimum or starting grid size. The largest grid size incorporates all imported data. The user needs to specify the number of grid cell sizes in between these two extreme values. This analysis can be carried out on the entire region or on a subregion of the data set.

Mass Fractal Dimension Analysis

Mass fractal analysis compares the distance from a specified origin on the studied region to the number of fracture centers found within the distance from the origin. This analysis can be carried out on 2-D irregularly spaced data nad fracture trace data. One of the original uses of this dimension was to examine the spatial scaling of mass in the universe, in which planets, suns, etc. are essentially "points" with a scalar value of mass associated with them. The calculation started with Earth at the center, and looked at how much mass was contained in spheres of ever-increasing radius.

The Mass Fractal dimension is computed by determining how much of some parameter is in a region of a particular size. The user has the option to modify the origin for this analysis and the interval range for the analysis of various radii. The calculation method depends on the method chosen to count fractures. Four options exist:

• fracture centers
• whole fractures
• fracture intersection
• fracture density (P21)

The results are plotted as the Log(mass) vs. Log(radius). If this plot is a straight line, then a mass fractal model may be a useful model for the data. GeoFractal displays the statistics to allow the user to make the best fit.

An alternative to using a single circle center is the multiple centers option. The user is free to select the starting points for this calculation with a mouse, or to allow the program to select a user-specified number of random starting points. The calculation is carried out for the circles surrounding each point, and the results for all of these starting points is displayed.

The Mass Fractal dimension probably is most useful for assessing the scaling properties of fracture intensity, and for fluid flow. If the intersection points option is selected, then the result is akin to a 2D-lattice percolation problem, for which it is known that critical percolation occurs when the mass fractal dimension is approximately equal to 1.89.

Spectral Analysis

Spectral analysis makes it possible to evaluate the self-affine fractal dimension of gridded data. This is done by computing the empirical variogram and displaying it on doubly-logarithmic axes. If the empirical variogram plots along a straight line, then a self-affine fractal model is probably a useful representation of the spatial correlation in the data. The slope, q, of this line, is related to the self-affine fractal dimension by:

D= 2- theta/2

where D is the self-affine fractal dimension. In the special case where the line is approximately horizontal, there is no spatial correlation in the data. Spectral analysis often benefits from detrending and filtering the data.

Baecher Analysis

Baecher analysis focuses on whether the spatial pattern in 2D conforms to a spatial Poisson process. The analysis includes both the cumulative and probability density functions and can be carried out on fracture tracer data.

Baecher analysis can be carried out on fracture tracer data in the The CDF for Baecher Analysis and the PDF for Baecher Analysis. The Baecher analysis focuses on whether the spatial pattern in 2D conforms to a spatial Poisson process. The hallmark of such a process is that there is no spatial correlation in the data (This can also be assessed through the semivariogram, which should be a horizontal line if the data has no spatial correlation, a horizontal power spectrum, a box dimension equal to 2.0, and a mass dimension equal to 2.0).

GeoFractal grids the fracture trace data, and estimates the mean data density as a function of grid cell size. Next, it creates a PDF and CDF for a Poisson process that has the same mean density for the same grid cell size, and compares it to the actual results. For the PDF, a Chi-Square test is performed to check the goodness-of-fit. For the CDF, a Kolmogorov-Smirnof test is carried out. Test statistics are reported for both tests.

Geostistical Analysis

Geostistical analysis describes the spatial correlation between observations in terms of a second order moment defined as:

where

• x(z) is the value of some parameter at a location z
• x(z+h) is the value of the parameter at some location a vectorial distance h from z
• E[ ] is the mathematical expectation of the quantity in braces, and
• g(h) is the semivariance
• h is referred to as the lag or the lag distance

The plot showing the semivariance function vs. the lag is termed the variogram or semivariogram. The lag is the distance or separation between data points and the semivariogram is a measure of the spatial correlation. The user must specify a model type to fit a curve to the data. The model types available are: Spherical, Exponential, Gaussian, Power, or de Wijs.

The standard deviation or the sum of squares error is displayed to guide the user as to how well the model with the parameter values displayed fits the empirical semivariogram. The parameters for each model can be computed using a minimum likelihood estimator (MLE) or can be directly specified by the user.