I'm currently reading the Knox translation of the bible. Ronald Knox's desire was to translate the bible in such a way so that English readers could read it as if it had been written in English. He aims for a sort of timeless English rather than anything faddish. I like it.
Anyway, I am in Genesis again, because my general program is to read a chapter a day, and I get to Hagar and Ishmael getting kicked out of the household, and she's wandering around Beersheba until God shows her the well. (Though Knox relies on the latin mostly for names I think, so he renders it Bersabee).
Then the next story is the treaty at Beersheba, where there's a well of water and Abraham and Abimelech make peace, and Abraham clearly dwells there. Then there's Abraham and Isaac's trek up the mountain for the sacrifice- and at the end of the passage they return to Beersheba and stay there.
Incidentally, Genesis 22:2 ought to give literalists fits because it has God telling Abraham that Isaac is his only son. I guess maybe they can dodge it by saying Isaac is the only son Abraham 'loves.' It makes more sense that as the story was handed down some details were lost. The lie points to the possibility the original story had someone who is not God deceiving Abraham into thinking this sacrifice was required of him. But I digress.
So through chapters 21 and 22 it is clearly established Abraham (and Isaac, to some extent at least) was living in Beersheba and had an ownership claim on a well- likely the one Hagar had been directed to. In chapter 23 Sarah dies at 127 years of age, in Hebron, and Abraham has to go there to mourn for her. It seems to me extremely likely this means he had two households- one with Hagar and Ishmael at Beersheba, and then another with Sarah.
This is probably what I would do in Abraham's situation, so I am not going to claim that this dispassionate biblical interpretation or anything. I don't like the idea of abandoning a child.