The 22-year-old South Australia player, who has appeared in one Test and five one-day internationals, will travel from Brisbane to join up with the squad in India after being told of the news while training for the new domestic season.
Cricket Australia said that under tournament rules, International Cricket Council approval was still needed to confirm Cullen's nomination to replace Clark as a member of the 14-man squad.
"It's really disappointing for Stuart to miss out with an injury and I'm sure that everyone wishes him the best for a speedy recovery," Cullen said in a statement.
"However, it's now up to me to take this opportunity and put my best foot forward for the team and I'm really looking forward to going to India."
Clark, 31, pulled out of Australia's squad on Wednesday after suffering a thigh injury. Australia, who have yet to win the Champions Trophy, open their campaign against a qualifier in Mumbai on Oct. 18.
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