If this culvert is near the road it might be maintained by the city. I'd give them a call first before performing unnecessary labor. They probably have one of those culvert-cleaning tools which is easier to use. If the culvert is just stopped up, then blasting water through the culvert with a pressure washer should help. In the meantime, divert water away from that spot if you can until the situation is fixed or the mud will continue to build up.