The Wild also will receive a conditional 2013 seventh-round draft pick in the trade.
Christensen, a highly skilled yet inconsistent center, has not played for the Rangers since Dec. 17 and has appeared in only 20 games this season. The 28-year-old was sent on a two-week AHL conditioning stint prior to the All-Star break.
Wellman, 6-foot and 186 pounds, will report to the Rangers' AHL affiliate Connecticut Whale. The UMass-Amherst product has recorded 14 goals and 12 assists in 26 games for the AHL Houston Aeros this season. Wellman also has two goals and five assists in 14 games with the Wild this year.
Rangers coach John Tortorella said he didn't know much about Wellman beyond what assistant general manager Jim Schoenfeld told him.
"I know from talking to Shony, he can play all forward positions, I believe," Tortorella said. "A hard-working guy. Has some skill."
By moving Christensen, the Rangers have thinned the logjam at forward.
"He gets another opportunity," Tortorella said of Christensen, who was claimed off waivers by the Rangers in 2009. "He had some opportunities here. Great guy. I wish the best for him."
Both Christensen and winger Wojtek Wolski have been consistent scratches for the Eastern Conference-leading team.
On Thursday, Wolski was sent along with seventh defenseman Jeff Woywitka on a conditioning assignment with the Whale.
The Wild also reassigned forward Chad Rau to Houston.
Katie Strang covers the NHL for ESPNNewYork.com. Information from The Associated Press was used in this report.