"We're all feeling for him at this time and know how disappointed he will be.
"Rutherford was with the test squad earlier in the tour and has been playing for Leicestershire in the T20 Blast."
The news continues a run of misfortune for Williamson. He missed the second test against India with a recurring elbow injury, which extended to the entire home summer.
Rutherford now comes into the reckoning for a first test appearance in the format since meeting Sri Lanka in Wellington during January 2015.
Alternatively, Michael Bracewell could be added to the middle order of the XI as the country's 283rd test player.
Devon Conway, Henry Nicholls and Daryl Mitchell could fill numbers three, four and five respectively. Tom Latham will captain the side in Williamson's absence.