We did not build a core in the back of the dam or do any digging at all. Instead onsite dirt/clay was just pushed in layers onto the base of the dam and worked up to the top by a dozer. This resulted in increasing the dam width by about 1/2, maybe more at the base. So far so good. If the dam starts to leak again, I'll let you guys know. It is the least expensive fix by far and if it fails after a while the end result will be that I got a much stronger dam that I don't have to worry about breaching. If it begins to leak at some point in the future, coring the back side will be the way I go.
I found a university site online (Kansas or aTm) that recommended using pond lining material in the core trench. I like that idea as it doesn't have to be compacted and the trench can be as narrow as one can dig it. I think I would pour bentonite in it to seal the bottom though.
