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Measurements made using specially designed, more elaborate apparatus and more astute sampling-handling techniques have yielded radiocarbon ages for anthracite greater than 70,000 radiocarbon years, the sensitivity limit of this equipment. Continuous series of tree-ring dated wood samples have been obtained for roughly the past 10,000 years which give the approximate correct radiocarbon age, demonstrating the general validity of the conventional radiocarbon dating technique.
Thus, all the researcher was able to say about samples with low levels of radiocarbon was that their age was greater than or equal to 20,000 radiocarbon years (or whatever the sensitivity limit of his apparatus was).
If this water is in contact with significant quantities of limestone, it will contain many carbon atoms from dissolved limestone.