Generally I tell my patients who want to resume running that if they are painfree and compliant with their back exercises (McKenzie, stabilization, etc) and lower extremity flexibility that they can resume running/jogging but really limit the time/distance that they run initially. In other words, stop before you start having pain. And limit running to once or twice weekly for the first month gradually building up from there. We might start with a time of 5 minutes and slowly increase as long as they stay painfree and don't neglect the corrective exercises.
Disc injuries are compression injuries, and the compressive forces you body experiences are much higher during running than walking. You want to be sure that things are well healed before introducing that compression to the spine again.
But then again, if you are running in mud, you probably aren't going that fast ... and the compression would be less
(We played a charity game of Ooozeball - volleyball in the mud - when I was in college ... it was a challenge to run in that stuff!!)