IVANA GAS FIELD

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Fractal patterns in geology, and their
application in mathematical modelling of
reservoir properties
WORKSHOP MODERATORS Tivadar M. Tóth Tomislav
MALVIC
WORKSHOP Fractal patterns Morahalom, 21-23
May, 2009
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• 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.

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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-Be
sicovitch 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
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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 31.26.
• 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
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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 21.585.
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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).

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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