Understanding the flow and contaminant transport process in rocks has received increased attention in the past two decades as these geological formations serve as potential sites to contain different types of wastes in geological repositories. In this paper, a discrete fracture network model is developed to understand the process of contaminant transport in a fractured rock mass. This model integrates a stochastic fracture generating algorithm for sedimentary rocks and a two-dimensional numerical model that simulates the flow and transport of a contaminant through the heterogeneous system of fractures. Further, the uncertainty in geological and transport properties of the contaminant are considered as random variables. The probability of concentration exceeding the permissible value at the endpoint is quantified by carrying out reliability analysis using the subset simulation method. The probabilistic framework developed in this study helps assess the behaviour of contaminant migrating through a complex heterogeneous system and provides a basis for gaining a level of confidence in the model results.