The estimation of the extent and timing of solute migration in a fractured medium is a fundamental task for verifying the level of protection against contaminant releases (e.g., toxic chemicals or radionuclides) offered by the engineered and natural barriers of a waste repository. In this paper we present a novel approach for modeling solute transport in a fractured medium, based on an extension of the Kolmogorov-Dmitriev theory of stochastic branching processes. The model equations for the expected values of the solute concentration take a form similar to that of classical dual-continua models. On the other hand, the stochastic nature of the modeling approach lends itself to a new particle tracking scheme of resolution, which allows accounting for realistic features of the transport process. The proposed stochastic modeling framework and simulation solution approach are illustrated with reference to the experimental results from a case study of literature. Some of the model parameters are optimally identified by means of a genetic algorithm search aimed at best fitting the experimental data.