Rivers hurt his right knee during the Cats' 21-point win over the Western Bulldogs at Etihad Stadium on Saturday night.
"The medical staff thought it was a slight medial (ligament strain)," Geelong coach Chris Scott said on Tuesday morning.
"But it was more painful than they would expect with that kind of injury.
"His medial ligament is fine. There's just some cartilage damage that potentially will need to be repaired, but we haven't made a decision on exactly what process we'll go through there.
"He's going to miss at least a few weeks, maybe more, which is unfortunate.
"He's had a bad run."
Rivers missed two games after suffering a posterior cruciate ligament strain in the same knee against Hawthorn in round one, but the club says the injuries are not related.
"Both are completely isolated," Scott said. "This one was very much a bad luck thing in a contest."
Rivers' injury is a blow to the Cats' plan to play Harry Taylor up forward more often this season.
Taylor booted five goals against the Bulldogs last weekend, but he is likely to be needed in the backline when Geelong takes on Richmond at the MCG on Saturday night.
"The Rivers injury is a little bit of an impediment, but we've got others who can come in and play at both ends," Scott said.
"We spent a bit of time in the pre-season training Podsiadly as a defender and Mitch Brown forward and back as well."
On the positive side, Tom Hawkins and Paul Chapman, who were rested last weekend, will return for the clash with the Tigers.
Both Hawkins and Chapman trained strongly on Tuesday morning.
Josh Caddy is recovering from the high bump he copped from Bulldogs defender Brett Goodes on Saturday night.
The former Gold Coast midfielder was concussed in the incident, which cost Goodes a two-match suspension.
Caddy trained strongly on Tuesday, but Scott said, "We're still taking our time to assess him."
Rookie-listed midfielder Jackson Sheringham will be sidelined for at least a couple of weeks after dislocating his shoulder in the VFL last weekend.
