He was replaced by Peter Fulton for the Twenty20, but the injury means he will miss the first three ODIs, starting with the match at Chester-le-Street on Sunday. His possible return date would be the fourth match at The Oval on June 25.
Oram is crucial to New Zealand's one-day form, his hard-hitting batting providing power to the middle order, while his bowling can strangle the scoring. He is an injury-prone cricketer, and missed the final Test against England in Napier in March, although managed to play throughout the recent series.
They have a few options in the squad to cover for Oram. Tim Southee has been out of the side since the first Test, while Jeetan Patel provides a second spin possibility, but no one can give the all-round balance that Oram provides.
New Zealand have been hit by a few injuries on their tour. Daniel Vettori cut his spinning finger before the Test series, Daniel Flynn lost two teeth after being hit by a bouncer in the Test at Old Trafford, and Brendon McCullum was unable to keep at Trent Bridge due to a bad back.