The characterization of aleatory hydrogeological parameter uncertainty has traditionally been accomplished using probability theory. However, when consideration is given to epistemic as well as aleatory uncertainty, probability theory is not necessarily appropriate. This is especially the case where expert opinion is regarded as a suitable source of information. When experts opine upon the uncertainty of a parameter value, both aleatoric and epistemic uncertainties are introduced and must be modeled appropriately. A novel approach to expert-provided parameter uncertainty characterization can be defined that bridges an historical gap between probability theory and fuzzy set theory. Herein, a random set, a generalization of a random variable is employed to formalize expert knowledge, and fuzzy sets are used to propagate this uncertainty to model estimates of contaminant transport. The resultant random set-based concentration estimates are shown to be more general than the corresponding random variable estimates. In some cases, the random set-based results are shown as upper and lower probabilities that bound the corresponding random variable's cumulative distribution function.