Kelly was placed on the 15-day injured list with right calf inflammation Tuesday.
Kelly struggled in his last start Sunday versus the Giants, allowing seven runs over five frames, and he will now be unavailable until after the All-Star break as he nurses his calf. Right-handed relievers Kevin Ginkel and Justin Martinez were called up from Triple-A Reno on Tuesday, though it's yet to be seen who will replace Kelly in Arizona's starting rotation. Top prospect Brandon Pfaadt could be an option to get called up from Reno.