Other radiometric dating methods such as potassium-argon or rubidium-strontium are used for such purposes by those who believe that the earth is billions of years old.
For this reason special precautions need to be exercised when sampling materials which contain only small amounts of radiocarbon.
Reports of young radiocarbon ages for coal probably all stem from a misunderstanding of one or both of these two factors.
In the early days of radiocarbon analysis this limit was often around 20,000 radiocarbon years.
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.