Capturing CO2 directly at industrial complexes and safely storing it in the subsurface is one of the proposed mitigation measures of climate change. One of these challenges is identifying suitable reservoirs that ensure the safe and permanent storage of CO2. Therefore, detailed integrity assessments should be performed on the caprock to accurately determine and manage the risks involved in the permanent storage of CO2 (Kaldi et al., 2013). This should include leakage risk quantification of (sub-seismic) fault zones in the primary seal (Zappone et al., 2021). This thesis aims to understand and predict the probability of caprock leakage through a fault-related fracture network. It specifically focuses on the network organisation of fault damage zone (FDZ) fracture networks and the impact of their topology on the connectivity of the fracture network. Topology, in the form of fracture node classification, has been used to determine the connectivity of fracture networks (Sævik and Nixon, 2017). However, this research points out that the connectivity of a fracture network cannot be solely determined by a linear relationship between the fracture node ratio and the connectivity. This methodology does not effectively succeed in capturing connectivity differences of different fracture networks. To research these differences, a measure of connectivity is introduced based on the graph theory concept of the giant component. In this concept, networks are connected by a certain amount depending on their  node type. By using the total length of the giant component we find the largest component ratio (LCR). This measure of connectivity and its relationship with percolation was tested using a DFN simulator. The concept of the giant component allows us to study the connectivity of various fracture networks. This is done by developing an algorithm that is based on the concept of robustness that removes fracture segments from a fracture network, which in turn enables us to find a relationship between the topology and the connectivity of the network and assess its uncertainty. From the DFN simulations and the robustness algorithm, it is found that the relationship between connectivity and topology is unique for different fracture networks. However, it was concluded that in general a fracture network significantly starts to increase in its connectivity when the average node has 1.6 edges connected to it (k). It also showed that the highest uncertainity of fracture network connectivity is present at 1.9<k<2.2.