Thank you for the question on the Book of Enoch, “How would Noah’s family have been able to keep the Book of Enoch safe and intact throughout Noah’s flood and the years after?”
The family of Noah did not keep the Book of Enoch, but some of the words of Noah was preserved through oral tradition that was passed on from generation to generation until those quotations were written down.
As one reads the Book of Enoch, Enoch didn’t live to see his great grandson as God took him. He could not have written the Book of Enoch since Noah was not born. See Genesis 5 as Enoch was taken from the earth at the age of 365 while Noah was born 434 later to Lamech. This means that there is a difference of 134 years between Enoch when he was taken to Noah being born, if my math is correct. The implication is that the Book of Enoch was written later while incorporating the oral tradition of Enoch saying. Scholars think that the Book of Enoch was written between 5 BCE-2BCE.
The Book of Enoch contains some words of Enoch, not that the Book of Enoch was written by Enoch. Noah and his children remembered the words that was passed down to them from the grandfather and father.
SUMMARY: Noah and his children remembered words or phrases that was orally passed down to their children. Noah didn’t have the Book of Enoch since it was not written yet. That is just my opinion for one’s consideration.
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