An estimated 50,000 people -- 20,000 of whom are children, the U.N. says -- are trapped in the city.
There are reports that hundreds of families are being used by ISIS as human shields in the center of Falluja; that men and boys who refuse to fight for ISIS are being executed; and that civilians have been killed in heavy shelling, according to the U.N.
And many experts believe that ISIS has booby traps, snipers, IEDs and other deadly surprises waiting.
"This will not be without cost," said Lt. Col. Rick Francona , another CNN military analyst. "The outcome is pretty much assured -- [the question is] what is the cost going to be on either side."
Some analysts say it will be a while before the terror group is driven out of the city.
"This is going to be a tough fight," says retired Lt. Gen. Mark Hertling, a military analyst for CNN. "The Iraqi government says they should close it down within two days and clear the forces. I doubt that... I think it's going to take much longer than that."
Hundreds, mostly women and children, fled Falluja on Friday
as Iraqi soldiers attacked to drive ISIS from the city, the Iraqi military said.
Those who escaped say
the city lacks food, clean water and medical supplies.
"With every moment that passes, their need for safe exits becomes more critical," said Nasr Muflahi, the Norwegian Refugee Council country director in Iraq. "Getting there in the first place is near impossible for those in the city center."
The Iraqi-led operation to liberate Falluja is just days old, and the city is already surrounded.
Experts say they're surprised at the professionalism and efficacy of Iraqi forces, which were in tatters nearly two years ago.
"They seem to have gotten their act together -- they did exactly what they were supposed to do," Francona said.
ISIS pushes back
fierce clashes erupted between Iraqi security forces and ISIS militants at the southern edge of the city.
Iraqi security tried to enter the city at dawn, a senior member of its rank told CNN. The official requested anonymity because he was not authorized to speak to the media.
ISIS fighters used suicide car bombs, RPGs and snipers to beat back the Iraqi forces, the official said. The Iraqi side suffered losses, the official added, declining to say how many.
The most intense fighting took place in Nuaimiya, a farming area considered to be one of the last strips of land that separated Iraqi forces from ISIS south of Falluja.
ISIS claimed that its fighters stopped Iraqi security forces and prevented them from advancing toward southern Falluja. The group said it killed at least 25 Iraqi troops and destroyed six army vehicles. CNN cannot confirm that.
The next step in the military's plan was to push into the heart of the city and drive ISIS out, which may be the most dangerous part of the operation.
If that's true, "that means they have absolutely know way out," CNN military analyst Col. Cedric Leighton said. "They are going to be pawns in the struggle ... and it is going to be one of the worst scenes that we can possibly imagine."
Storm the city
The newest phase of the operation to retake Falluja was announced early Monday morning.
"With God's blessing we have launched the third phase of the operation to storm the center of Falluja city -- by our heroes in the counterterrorism forces, units of the Iraqi army and Anbar police," Iraqi military spokesman Yahya Rasoul said on Iraqi state TV.
Falluja is particularly important because of it's proximity to Baghdad. Analysts say that ISIS has used it as a staging ground to plan attacks it carries out in the Iraqi capital.
It and Mosul are the last two Iraqi cities
under ISIS' control.
In a video released by Iraq's Ministry of Defense, Iraqi Gen. Hamid al-Maliki announced Monday the city was encircled by Iraqi troops.
If the city is liberated, Iraqi authorities will face a whole new set of challenges -- among them, addressing the social and political problems that led to the rise of ISIS
, as well as security issues.
"What seems to be lacking from the United States, from, the coalition and from the Iraqi government is a strategy for the day after [ISIS is defeated]," Ali Khedery, a former adviser to U.S. Central Command in Iraq, told CNN "You have tens or hundreds of thousands of young males, who, again, are not in schools, which means they will not have jobs, which means they will not have hope in the future. And ultimately that is a ripe feeding ground for the next al Qaeda, the next bin Laden, the next ISIS."