Since Russia launched its invasion of Ukraine, about 660,000 Ukrainian refugees have fled to neighboring countries, the United Nations Refugee Agency said Tuesday.

"This figure has been rising exponentially, hour after hour, literally, since Thursday," UN High Commissioner for Refugees Filippo Grandi told the UN Security Council on Monday. "I have worked in refugee crises for almost 40 years, and I have rarely seen such an incredibly fast-rising exodus of people — the largest, surely, within Europe, since the Balkan wars.”

"Over 280,000 have fled to Poland. Another 94,000 to Hungary, nearly 40,000 are currently in Moldova; 34,000 in Romania, 30,000 in Slovakia; tens of thousands in other European countries. We are also aware that a sizable number have gone to the Russian Federation," Grandi continued.

UN aid agencies warned last week that fuel, cash and medical supplies were becoming scarce in Ukraine, Reuters reported, which could cause up to 5 million people to flee abroad.

While Poland has absorbed the largest number of Ukrainian refugees by far, the Associated Press reports that many of those fleeing Ukraine were traveling on to countries further west.

Aksieniia Shtimmerman, 41, arrived in Berlin with her four children Monday morning after a three-day journey from Kyiv.

Sitting inside the German capital's main train station, she tried to understand a flyer with instructions and maps on how to get to a shelter.

Attempting to comfort her crying 3-year-old twin boys, Shtimmerman told the AP she had worked in telecommunications at a Kyiv university but was now seeking a place where she and her children could eat, sleep and rest.

"I grabbed my kids on Friday morning at 7 a.m. to run away from the war," she said. "I can't even count anymore how many different trains we took until we arrived here."

In a press briefing at the Palais des Nations in Geneva Tuesday, UNHCR spokesperson Shabia Mantoo said that the agency is "coordinating the refugee response with other UN agencies and NGO partners, in support of national authorities."

"At this rate, the situation looks set to become Europe's largest refugee crisis this century, and UNHCR is mobilizing resources to respond as quickly and effectively as possible," Mantoo said.

Under-Secretary-General for Humanitarian Affairs and Emergency Relief Coordinator Martin Griffiths told the UN Security Council on Monday that "civilians are already paying the price."

"Humanitarian needs are growing at an alarming pace in the hardest-hit areas," Griffiths said. "Civilian children, women and men have been injured and killed. Homes have been damaged and sometimes destroyed."

"Families are separated," he continued. "The elderly and people with disabilities find themselves trapped and unable to flee, not even able to get that small comfort. The picture is grim — and could get worse still."

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