NOGALES, Ariz. — On an embankment that runs along a towering steel fence separating this border town from its Mexican sister city, a patch of new concrete with a date carved into it stands out, marking the exit of a tunnel Border Patrol agents sealed in May.

Dozens more like it snake around town, part of a vast underground network that Mexican drug cartels have used for years to funnel hundreds of pounds of illicit drugs into the United States. When Border Patrol agents find the tunnels, they dump concrete to seal them and stamp them with the date that they are shut down.

But they struggle to stay ahead of the digging. Last Friday, the Border Patrol, in a joint operation with Mexican authorities, discovered an unfinished tunnel that started in a Mexican cemetery.

“The clock is ticking as soon as they complete a tunnel,” said Kevin Hecht, a Border Patrol tunneling expert. “They know that we will eventually find them. But if even one load gets through before we find it, they consider it a success.”