Oddly it does. Their is little to no vegetation on the surface and associated organic matter to absorb the water. This is what always happens in the desert when it has short but great rainfall.
In our west in Arizona and Utah we have great canyons carved out in the desert due to this. It does not rain frequently but when it does it becomes a raging river and carves out those canyons. Note: Not Grand Canyon as this is a river basin that always has water flowing and carving out that magnificent canyon.
When camping in the desert, put your campsite on high ground.