This condition is called bacterial wetwood or slime flux. Bacteria enter the woody tissue of the tree through a broken branch or pruning wound and start working to decay the wood.
In response to wounding, pecan trees develop lignin fortified internal walls to block bacteria and fungi from penetrating the entire woody interior. The tree's defense mechanism works well in stopping wood rotting organisms from moving deeper into the wood or expanding radially. However, the tree's natural defenses are weakest above and below a wound.
When you see wetwood, look straight above the weeping bark and you usually find a large pruning wound. Here's what happens. Bacteria enter the tree at the pruning wound and work downward through the wood. Once the bacteria move far enough away from the pruning wound (and source of oxygen), the wood rotting process becomes anaerobic, producing methane and nitrogen gasses. The gas pressure becomes so great it actually bursts the bark open and bacterial ooze streams out.
There is no cure for this disease. You can only work to prevent bacterial wetwood by encouraging tree growth and rapid wound healing. A limited amount of wetwood on a large pecan tree does not seem to limit nut production.