The injury is a recurrence of one which kept him out of two matches earlier this season.
Club doctor Mark Whitaker said: "The operation has been successful and the medical team will monitor his progress."
Richards recently signed a new contract which commits him to City until 2013.
Richards has forged an impressive partnership with captain Richard Dunne at the heart of the City defence this season.
He has won 11 England caps and scored one international goal since making the step up from the Under-21s.
"It is really bad news and a big blow," added Eriksson. "Everyone knows how important he is to us."