We meet with this name, Isaiah 15:8. But it is more than probable, that it is the name of well so sweetly spoken of Numbers 21:16-18. I beg the reader to consult the Scripture, and let him judge for himself, whether it be not so. Beerelim means, the well of the princes. And the princes are said to have digged it. But when the reader hath satisfied his mind on this point, there is another object, and that of an higher nature, that I would request the reader to attend to. In those wells, I humbly conceive, we discover gospel lessons beautifully represented. Hence, the prophet sings, "Because God (saith he) is my salvation, therefore, with joy shall ye draw water out of those wells of salvation." (Isaiah 12:2-3) And hence, if, with an eye to the Lord Jesus Christ, who is himself, in the souls of all his redeemed, a well of water springing up unto everlasting life, (John 4:14) we accept those Beer-elim in the word, we then join the Lord's song, in the Lord's own words, as he directed Moses. This is the well whereof the Lord spake unto Moses, "Gather the people together, and I will give them water." Then Israel sang this song. "Spring up, O well! sing ye unto it. The princes digged the well; the nobles of the people digged it, by the direction of the lawgiver with their staves." (Numbers 21:16-18)

Bibliography Information
Hawker, Robert D.D. Entry for 'Beer-Elim'. Hawker's Poor Man's Concordance and Dictionary. https://www.studylight.org/dictionaries/eng/pmd/b/beer-elim.html. London. 1828.

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Hawker's Poor Man's Concordance And Dictionary

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Beer-Elim