The Toffees full-back has undergone surgery on his broken tibia and fibula but hopes to remain strong enough to recover in time to feature at the World Cup for Costa Rica
The 23-year-old picked up the horror injury after a tangle with midfielder Simon Heslop but Monday's surgery went well.
Danny Donachie, Everton's head of medical services, told the club's official website: "Bryan's surgery was a big success and he is in a comfortable and stable condition.
"He fractured his tibia and his fibula and they were displaced, so we weren't able to put them back into place until Monday.
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"It was a clean break and that means that there are no pieces of bone. Sometimes when there are small fragments of bone in the area, it can be difficult to deal with but these are clean breaks, so they will put the bones back into position and it will heal well from there. We will be working hard to get him back as quickly as we can."
The Costa Rican now faces a race against time to be fit to take his place on the plane to World Cup 2014 in the summer.
"Ever since it happened, I've been thinking about the World Cup," said the left-back. "I have to remain calm but, in the hands of God, I will be ready for it.
"If I am back in time for the finals, then great but, if not, then the world won't stop turning. I have already missed a tournament through injury before.
"You learn from experiences like this. You take knocks in life but the key is to be strong and to fight back."
"I have tried to stay strong and positive but I don't want to watch the video of the incident," Oviedo added. "I don't think I will ever want to see it."