Topology of Fracture Networks

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.