All-rounder Oram is suffering from a bruised heel and Kyle Mills has a groin strain. Both will also miss the opening game of the tour next Thursday when the two sides play a Twenty20 match in Auckland.
Coach John Bracewell said Mills would return to club cricket this week and would be considered for selection for the remainder of the one-day series but it was unknown how long Oram would be unavailable.
Other changes to the team that beat Sri Lanka 4-1 in their one day series early last month include the recall of pace bowler Michael Mason.
Mason has not played an international since March 2004 but Bracewell said he had remained one of New Zealand's most consistent line and length bowlers.
"With the gap left by injuries to Kyle Mills, Andre Adams and Jacob Oram, he has a real opportunity to prove himself at international level," he said.
Veteran all-rounder Chris Cairns has been included for the Twenty20 match, which will be his final game before he retires from international cricket.
The first One-day international is on February 18 in Wellington.
Squad: Stephen Fleming (captain), Jamie How, Lou Vincent, Peter Fulton, Hamish Marshall, Nathan Astle, Scott Styris, Brendon McCullum, Chris Cairns, Daniel Vettori, James Franklin, Jeetan Patel, Shane Bond, Michael Mason.