Injection of supercritical CO2 allows us to sequester an important component of the greenhouse gases in the subsurface to reduce emissions of CO2 in the energy sector. In light of global environmental change, it is of utmost importance to reliably predict the fate and containment of injected CO2 at geological time scales. Therefore, modelling of the CO2 plume migration and its containment within geological carbon storage repositories helps to uncover and understand challenges arising during injection and post-injection phase. Discrete fracture and matrix (DFM) simulations have emerged as a powerful technology to analyse the fundamental flow and transport properties in naturally fractured reservoirs and bridge gaps between geosciences and reservoir engineering: they help to validate upscaling workflows, improve the analysis of pressure transients from welltests, and allow to explore how uncertainties in fracture network properties impact hydrocarbon recovery. The key difference between DFM simulations and traditional dualporosity approaches is that the structurally complex fracture geometries are explicitly discretised as 2D surfaces in a 3D reservoir model, hence provides a more geologically realistic representation of the fracture patterns and rock matrix as compared to dualporosity models. Here I extend the DFM simulation workflow to account for capillary trapping of CO2 in fractured porous media. Fluid flow is described by a fully compositional model including an equation of state for CO2-H2O-NaCl fluids. The governing equations are discretised in space using unstructured and mixed-dimensional finite element - finite volume techniques. An Operator splitting approach is used to decouple capillary diffusion from the mass balance equation. For the first time, relative permeability hysteresis within the DFM method has been performed. I demonstrate how my new DFM approach can be applied to simulate CO2 injection and trapping of CO2 in fractured geological formations with geologically realistic fracture networks. I show how matrix diffusion and capillary forces influence the rate at which CO2 is trapped in the rock matrix.