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Simulating contaminant transport in fractured geologic media is challenging. Aside from the difficulties encountered in properly modeling the heterogeneities in the hydraulic properties of the fractures and the matrix, it is difficult to quantify and model the disorder in the fracture connectivity. Correct prediction of the spread of contaminants in fractured geologic media is not possible without considering this inherent morphological disorder. Here, we develop a network model of fractures, and use the model to study transport of contaminants. We investigate the influence of morphology on the transport process by introducing disorder in the fracture connectivity through a novel percolation scheme. The network close to the percolation threshold is very complex and allows the contaminant particles to follow many slow paths. This closely captures the physical situation. We show, how the disorder in the network changes the residence time distributions and its various temporal moments. We also show how the residence time distribution and the temporal moments are influenced by the interaction of the disorder with the various transport mechanisms, such as convection, dispersion, adsorption, and first-order decay.