It was announced earlier this week that the 25-year-old would miss this weekend's game against England, but now he has been ruled out for up to eight weeks.
Scotland team doctor James Robson said: "Sean was examined by consultant orthopaedic surgeon Graham Lawson. He found that Sean has sustained some damage to the structures of his knee and ankle but, happily, does not require surgery.
"The expected recovery time for soft tissue damage of this sort is up to eight weeks."
Scotland's final fixture of the tournament is against Wales on Saturday, March 15.