WORKSHOP “Fractal patterns…” Morahalom, 21-23 May, 2009

1. Fractal patterns in geology, and their application in mathematical modelling of reservoir properties WORKSHOP MODERATORS: Tivadar M. Tóth Tomislav MALVIĆ WORKSHOP “Fractal patterns…” Morahalom, 21-23 May, 2009

2. Fractals appear similar at all levels of magnification. • Fractals are often considered to be infinitely complex. • Natural objects that could be approximated by fractals are clouds, mountain ranges, coastlines, snow flakes, rock fractures, faults etc. • Not all self-similar objects are fractals, e.g. the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics. • For instance, Euclidean line is regular enough to be described in Euclidean terms.

3. HISTORY Iterative functions (as the base of fractal function) in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincare, Felix Klein, Pierre Fatou and Gaston Julia (Julia set). In the 1975 Mendelbrot coined the word “fractal” to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. Mendelbrot set discovered a many new fractal patterns. Computer graphic made available insight in many hidden fractal forms and natural phenomenon characterised by fractal dimensions. Hausdorff-Besicovitch dimension: (also known as the Hausdorff dimension) in mathematics is an extended non-negative real number associated to any metric space. It generalizes the notion of the dimension of a real vector space. The H-B dimension of a single point is 0, of a line is 1, of the plane is 2 etc. But, many irregular sets have non-integer H-B dimension. The concept was introduced in 1918 by the mathematician Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilovitch Besicovitch. Figure 1: Sierpinski triangle – a space with fractal dimension log23 or ln3/ln2, which is approximately 1.585

4. FAMOUS FRACTALS • Koch snowflake (or Koch star) is a mathematical curve and one of the earliest fractal curves to have been described. It appeared in a 1904 paper titled “On a continuous curve without tangents, constructible from elementary geometry” by Swedish mathematician Helga von Koch. • The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure. It resulted in: • four times multiplication of each line segment, • the length of segment remained only one-third of starting length. • Hence the total length increase by one-third and thus the length at step “n” will be “(4/3)n”. • The fractal dimension of Koch snowflake is “log4/log 3=1.26”. • It is greater than the dimension of a line (1), but lower than triangle (2). Figure 2: The first four iterations of the Koch snowflake

5. FAMOUS FRACTALS (2) Sierpinski triangle (or Sierpinski gasket or S. Sieve) is a fractal named after the Polish mathematician Waclaw Sierpinski who described it in 1915. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Figure 3: The Sierpinski triangle - each removed triangle (a trema) is topologically an open set. A set is called open if the distance between any point in the shape and the edge is always greater than zero. The Hausdorff dimension of Sierpinski triangle Koch snowflake is “log3/log 2=1.585”.

6. FRACTALS IN GEOLOGICAL PATTERNS • Some geological patterns can be modelled by fractals. For example it could be: • Fractures distribution; • Faults patterns; • Granulometry; • Porosity distribution. • The geological rules for such “fractal” pattern are: • It need to include some plane or volume with clearly recognizable and interpretable geological pattern. It is often in scale of e.g. 1 m2 or 1 m3; • Then this pattern need to be described as self-similar and modelled by some known fractal set (Julia Newton, Medelbrot etc.); • Finally, this pattern need to be extrapolated in entire analysed bed or depositional sequence (between top and base of the strata).

7. FRACTALS IN GEOLOGICAL PATTERNS (2) Natural fractures distribution : fractures that are randomly chaotically distributed in small area (planes, volumes), but at larger scale it shows preferable direction with several subordinate, perpendicular axis. It is observed in outcrops of Badenian source rocks in the Northern Croatia. Artifical induced fractures : fractures that are results of well stimulation. Figure 5: Artificial induced fractures – maybe as it looks like in the inflitrated zone Figure 4: Natural fractures distribution that simulate Badenian outcrops