Extrapolation of fractal dimensions of natural fracture networks from one to two dimensions in dolomites of Slovenia

Abstract

If fractal properties of fractures are available only through 1-D measurements (boreholes or scanlines), it is essential to extrapolate the fractal dimensions from 1-D to 2-D values correctly. This paper focuses on such an extrapolation, obtained from twenty-two natural fracture networks in Triassic dolomites in Slovenia. Fractures are analyzed by 1-D (in x- and y-directions) and 2-D box-counting methods. Networks are analyzed by several box-counting methods, by box-flex and box-rotate methods to determine the fractal dimensions in 2-D and consequently by the ‘full’ method (using all data points in the log-log plot) and the more appropriate ‘cut-off’ method (using data greater than the cut-off points), which are used for comparison of 1-D and 2-D data. According to theoretical presumptions, extrapolation of fractal dimension from 1-D to 2-D should be straight-forward: D2-D=D1-D + 1 (D2-D being fractal dimension measured in 2-D environment and the D1-D being fractal dimension measured in 1-D environment). Results show that the values of fractal dimensions obtained in a 1-D environment are very similar and lie in a very narrow data range. This can be attributed to the similar fracturing style of dolomites or isotropy of fractures. Results obtained by the ‘cut-off’ method give higher values of D than the ‘full’ method, as only appropriate data values were considered in calculations. Values of one-dimensional values of D can be reliably extrapolated to a two-dimensional environment by equation D*2-D=D*1-D + 1.03 for the ‘cut-off’ method and D2-D=D1-D + 1.06 for the ‘full’ method. Both differences between D1-D and D2-D values are very close to a theoretical value of 1.00, so the fracture networks in dolomites can be described as nearly ideal non-mathematical and isotropic fractal objects.