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Topology of Fracture Networks

Hope, Sigmund Mongstad
Doctoral thesis
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URI
http://hdl.handle.net/11250/2360237
Date
2015
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Abstract
This thesis is submitted as partial fulfilment of the requirements for the degree

of philosophiae doctor.

Methods from network science are applied to investigate the topology of fracture

networks. The main focus is on fracture outcrops and artificial fracture networks

generated by discrete fracture network models. For two-dimensional networks the

analysis has also been performed for fracture networks in sea ice, while the lack

of real data for three-dimensional fracture networks has led to the investigation

of the three-dimensional structure of a desert rose.

In a work unrelated to the main topic, a method for automatic detection of signal

correlations is introduced.

Paper I We propose a mapping from fracture systems consisting of intersecting

fracture sheets in three dimensions to an abstract network consisting of nodes and

links. This makes it possible to analyse fracture systems with the methods developed

within modern network theory. We test the mapping for two-dimensional

geological fracture outcrops and find that the equivalent networks are small-world

and disassortative. By analysing the Discrete Fracture Network model, which is

used to generate artificial fracture outcrop networks, we also find small-world

networks. However, the networks turn out to be assortative.

Paper II Fracturing and refreezing of sea ice in the Kara Sea is investigated using

complex network analysis. By going to the dual network, where the fractures are

nodes and their intersections links, we gain access to topological features which are

easy to measure and hence compare with modelled networks. Resulting networks

reveal statistical properties of the fracturing process. The dual networks have

a broad degree distribution, with a scale-free tail, high clustering and efficiency.

The degree-degree correlation profile shows disassortative behaviour, indicating

preferential growth. This implies that long, dominating fractures appear earlier

than shorter fractures, and that the short fractures which are created later tend

to connect to the long fractures. The knowledge of the fracturing process is used

to construct a growing fracture network (GFN) model which provides insight

into the generation of fracture networks. The GFN model is primarily based on

the observation that fractures in sea ice are likely to end when hitting existing

fractures. Based on an investigation of which fractures survive over time, a simple

model for refreezing is also added to the GFN model, and the model is analysed

and compared to the real networks.

Paper III The topology of two discrete fracture network models is compared to

investigate the impact of constrained fracture growth. In the Poissonian discrete

fracture network model the fractures are assigned length, position and orientation

independent of all other fractures, while in the mechanical discrete fracture

network model the fractures grow and when fractures intersect the growth of the

smallest fracture stops. While the Poissonian model results in assortative networks

the mechanical model results in disassortative networks. This may explain

why the mechanical model is better at producing flow channelling.

Paper IV The topology of three-dimensional fracture networks is studied by mapping

them onto equivalent graphs. The fracture networks considered are generated

by two variants of the Discrete Fracture Network model. In both variants,

fracture characteristics are similar (random centers and orientations, power-law

distribution for fracture sizes). The two models differ, however, in the general

organization of fractures. In the Poissonian model, fracture size is randomly

assigned from a power-law distribution, whereas in the mechanical model, the

fracture size is obtained from a growth process that mimics geological fracturing.

The details of the fracture growth rules are found to significantly impact the

topology of the ensuing graphs. Not only between the Poissonian and mechanical

models, but also between different implementations of what happens to the

growth of the cracks after they touch. In particular, a strong connection is found

between the degree mixing of the graphs and whether the fracture network model

is mechanical or Poissonian.

Paper V Desert roses are gypsum crystals that consist of intersecting discs. We

determine their geometrical structure using computer assisted tomography. By

mapping the geometrical structure onto a graph, the topology of the desert rose

is analysed and compared to a model based on diffusion limited aggregation. By

comparing the topology, we find that the model gets a number of the features of

the real desert rose right, whereas others do not fit so well.

Paper VI Automatic detection of correlated patches between two signals has a

wide range of applications. Dynamic Time Warping is a much-used method for

detecting such correlations. A problem with this method is that it can only detect

one correlated patch at a time. To detect others, realignment of the two signals is

necessary. Here a new method is presented, Global Correlation Analysis, which

is able to detect and place in an importance hierarchy all the correlated patches

without realignment.
Publisher
NTNU
Series
Doctoral thesis at NTNU;2015:200

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