The Munster front row sustained the damage in training on Wednesday and a scan in the afternoon revealed he will be out for up to six weeks.
Sean Cronin will fill the vacancy on the bench for Saturday's Pool C clash with Australia at Eden Park, with Damien Varley being called up to complete the 30-man squad.
Commenting on the injury to Flannery, Ireland team manager Paul McNaughton said: "Obviously it is particularly disappointing for Jerry to have suffered this injury at this point in the tournament.
"With the fact that he is out for four to six weeks, we needed to make the decision to fly Damien Varley down as soon as possible.
"Sean Cronin will step into the replacements seamlessly this weekend which shows the importance of having experience and depth in the squad."
The injury casts doubts over Flannery's career, with the Munster forward having spent much of the last year in the treatment room with calf problems.
He was selected on the bench for all four of last month's warm-up internationals and the 22-10 victory over the United States in last Sunday's World Cup opener.
Flannery was pushing hard for a place in the starting line-up after impressing during his substitute appearances.
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