"It's a silent arrival of a lot of people who are crossing the border and staying illegally on the Colombian side," said Martin Gottwald, the United Nations Refugee Agency's representative in Colombia.

No exact figures are available, but the number of Venezuelans fleeing to Colombia is already "quite large," and Colombia should prepare itself for more, Gottwald told AFP in an interview.

"The avalanche is probably going to increase, with or without the reopening of the border," he said.

Venezuelan President Nicolas Maduro closed the countries' border in August 2015 after an attack on an army patrol. He blamed right-wing paramilitaries from Colombia.

The leftist leader briefly reopened it last weekend to allow Venezuelans to stock up on food, medicine and other basic supplies amid severe shortages in Venezuela.

Gottwald said a sizeable number of Venezuelans who entered Colombia probably never returned.

"If you consider that 100,000 people crossed the border to stock up on supplies at the weekend, if a minimal percentage, let's say 10 percent, stayed (in Colombia), we're already talking about quite large numbers," he said.

Venezuelans are also sneaking across the border even when it is closed, driven abroad by the economic crisis, violent crime and a health care system teetering on the brink of collapse.